OE- REESE; LIBRA UNIVERSITY OF CALIFORNIA. /fo.3 *.She/f + -I- A MANUAL OF RULES, TABLES, AND DATA FOR MECHANICAL ENGINEERS. A MANUAL OF RULES, TABLES, AND DATA FOR MECHANICAL ENGINEERS BASED ON THE MOST RECENT INVESTIGATIONS: OF CONSTANT USE IN CALCULATIONS AND ESTIMATES RELATING TO STRENGTH OF MATERIALS AND OF ELEMENTARY CONSTRUCTIONS; LABOUR; HEAT AND ITS APPLICATIONS, STEAM AND ITS PROPERTIES, COMBUSTION AND FUELS, STEAM BOILERS, STEAM ENGINES, HOT-AIR ENGINES, GAS-ENGINES ; FLOW OF AIR AND OF WATER; AIR MACHINES; HYDRAULIC MACHINES; MILL-GEARING; FRICTION AND THE RESISTANCE OF MACHINERY, &c. ; WEIGHTS, MEASURES, AND MONIES, BRITISH AND FOREIGN, WITH THE RECIPROCAL EQUIVALENTS FOR THE CONVERSION OF BRITISH AND FRENCH COMPOUND UNITS OF WEIGHT, PRESSURE, TIME, SPACE, AND MONEY; SPECIFIC GRAVITY AND THE WEIGHT OF BODIES ; WEIGHT OF METALS, &c. WITH TABLES OF LOGARITHMS, CIRCLES, SQUARES, CUBES, SQUARE-ROOTS, AND CUBE-ROOTS; AND MANY OTHER USEFUL MATHEMATICAL TABLES. BY DANIEL KINNEAR CLARK, MEMBER OF THE INSTITUTION OF CIVIL ENGINEERS ; AUTHOR OF "RAILWAY MACHINERY," "EXHIBITED MACHINERY OF 1862," ETC. FOURTH EDITION. LONDON: BLACKIE & SON: OLD BAILEY, E.G. GLASGOW AND EDINBURGH. 1889. All Rights Reserved. PREFACE. THIS Work is designed as a book of general reference for Engineers : to give within a moderate compass the leading rules and data, with numerous tables, of constant use in calculations and estimates relating to Practical Mechanics. The Author has endeavoured to concentrate the results of the latest investigations of others as well as his own, and to present the best information, with perspicuity, conciseness, ana scientific accuracy. Amongst the new and original features of this Work, the follow- ing may be named : In the section on Weights and Measures, the weight, volume, and relations of water and air as standards of measure, are concisely set forth. The various English measures, abstract and technical, are given in full detail, with tables of various wire-gauges in use: and equivalent values of compound units of weight, power, and measure as, for example, miles per hour and feet per second. The French Metric Standards are defined, according to the latest determinations, with tables of metric weights and measures, equi- valents of British and French weights and measures, and a number of convenient approximate equivalents. There is, in addi- tion, a full table of equivalents of French and English compound units of weight, pressure, time, space, and money as, for example, pounds per yard and kilogrammes per metre; which will be found of great utility for the reciprocal conversion of English and French units. The tables of the Weight of bars, tubes, pipes, cylinders, plates, sheets, wires, &c., of iron and other metals, have been calculated expressly for this Work, and they contain several new features designed to add to their usefulness. They are accompanied by a summary of the various units of weight of wrought iron, cast iron, and steel, with plain rules for the weight. In the section on Heat and its Applications, the received mechan- ical theory is defined and illustrated by examples. The relations of the pressure, volume, and temperature of air and other gases, VI PREFACE. with their specific heat, are investigated in detail. The transmission of heat through plates and pipes, between water and water, steam and air, &c., for purposes of heating or cooling, is verified by many experimental data, which are reduced to units of performance. The physical properties of steam are deduced from the results of Regnault's experiments, with the aid of the mechanical theory of heat. A very full table of the Properties of Saturated Steam is given. The table is, for the most part, reproduced from the article " Steam," contributed by the Author to the Encyclopedia Britannica, 8th edition, and it was the first published table of the same extent, in the English language, based on Regnault's data. An original table of the properties of saturated mixtures of air and aqueous vapour is added. In the section on Combustion, new and simple formulas and data are given for the quantity of air consumed in combustion, and of the gaseous products of combustion, the heat evolved by combus- tion, the heating power of combustibles, and the temperature of combustion ; with several tables. On Coal as a Fuel, both English and Foreign, its composition, with the results of many series of experiments on its combustion, are collected and arranged. The quantity of air consumed in its com- bustion, and of the gaseous products, with the total heat generated, are calculated in detail. Coke, lignite, asphalte, wood, charcoal, peat, and peat-charcoal, are similarly treated; whilst the combus- tible properties of tan, straw, liquid-fuels, and coal-gas, are shortly treated. The section on Strength of Materials is wholly new. The great accumulation of experimental data has been explored, and the most important results have been abstracted and tabulated. The results of the experiments of Mr. David Kirkaldy occupy the greater por- tion of the space, since he has contributed more, probably, than any other experimentalist to our knowledge of the Strength of Materials. The Author has investigated afresh the theory of the transverse strength and deflection of solid beams, and has deduced a new and simple series of formulas from these investigations, the truth of which has been established with remarkable force by the evidence of experi- ment. These investigations, based on the action of diagonal stress, throw light upon the element called by Mr. W. H. Barlow, "the resistance of flexure:" revealing, in a simple manner, the nature of that hitherto occult entity; and showing that flexure is not the cause, but the effect of the resistance. In addition to formulas PREFACE. vii for beams of the ordinary form, special formulas have been deduced for the transverse strength and deflection of railway rails, double-headed or flanged, of iron or steel; in the estab- lishment of which he has availed himself of the important experimental data published by Mr. R. Price Williams, and by Mr. B. Baker. To our knowledge of the strength of timber, Mr. Thomas Laslett has recently made important additions, and the results of his experiments have been somewhat fully abstracted and analyzed. But woods, by their extremely variable nature, are not amenable, like wrought-iron and steel, to the unconditional applica- tion of formulas for transverse strength. The Author has, never- theless, deduced from the evidence, certain formulas for the trans- verse strength and deflection of woods, with tables of constants, which, if applied with intelligence and a knowledge of the uncer- tainties, cannot fail to prove of utility. The Torsional Strength of Solid Bodies has also been investigated afresh, and reduced to new formulas. In dealing with the Strength of Elementary Constructions, the Author has brought together many important experimental results. In treating of rivet-joints and their employment in steam-boilers, he has, he believes, clearly developed the elements of their strength and their weakness. By a close comparison of the results of tests of cast-iron flanged beams, it is plainly shown that the ultimate strength of a cast-iron beam is scarcely affected by the proportionate size of the upper flange, and that the lower flange and the web are, practically, the only elements which regulate the strength. The tests of solid-rolled and rivetted wrought-iron joists are also ana- lyzed ; and for the strength and deflection of these, as for those of cast-iron flanged beams, new and simple rules and formulas are given. A new investigation, with appropriate formulas, is given for the bursting strength of hollow cylinders, of whatever thickness. It is shown that the variation of stress throughout the thickness, follows a diminishing hyperbolic ratio from the inner surface to- wards the outer surface. The resistance of tubes and cylindrical flues to collapsing pressure is also investigated, and formulas based on the results of experience are given. On the subject of Mill-gearing, a new and compact table of the pitch, number of teeth, and diameter of toothed wheels is given, with new formulas and tables for the strength and horse-power of the teeth of wheels, and for the weight of toothed wheels. New formulas and tables are given for the driving power of leather Vlll PREFACE. belts, and the weight of cast-iron pulleys. For the strength of Shafting, cast-iron, wrought-iron, and steel, a new and complete series of formulas has been constructed, comprising its resistance to transverse deflection and to torsion, with very full tables of the weight, strength, power, and span of shafting. The Evaporative Performance of Steam-boilers is exhaustively investigated with respect to the proportions of fuel, water, grate- area, and heating-surface, and the relations of these quantities are reduced to simple formulas for different types of boilers. The actual evaporative performances of boilers are abstracted in tabular form. The Performance of Steam worked expansively, in single and in compound cylinders, is exhaustively analysed by the aid of diagrams; the similarity and the dissimilarity of its action in the Woolf-engine and the Receiver-engine, are investigated; and the principles of calculation to be applied respectively to these, the leading classes of compound engines, are explained. The best working ratios of expansion are deduced from the results of numerous experiments and observations on the performance of steam-engines. The principles of Air-compressing Machines, and Compressed-air Engines are investigated, and convenient formulas and tables for use are deduced. The whole of the materials for the preparation of this work have been drawn from the best available sources, foreign as well as English. Vast stores of the results of experience are accumulated in the Proceedings of the Institution of Civil Engineers, the Proceedings of the Institution of Mechanical Engineers, and other journals. From these and other sources, the Author has drawn much of his material. D. K. CLARK. 8 Buckingham Street, Adelphi, LONDON, zoth March, 1877. NOTE ON THE FOURTH EDITION. I have thoroughly revised this book, and, besides correcting it up to date, I have introduced much new matter, which will render this edition even more valuable than the last. D. K. CLARK. January r , 1889. CONTENTS. GEOMETRICAL PROBLEMS. PAGE Straight Lines Straight Lines and Circles Circles and Rectilineal Figures The Ellipse The Parabola The Hyperbola The Cycloid and Epicycloid The Catenary Circles Plane Trigonometry Mensuration of Surfaces Solids Heights and Distances, ......... I MATHEMATICAL TABLES. Explanation of the following Tables : 32 Logarithms of Numbers from i to 10,000, ........ 38 Hyperbolic Logarithms of Numbers from 1. 01 to 30, 60 Numbers or Diameters of Circles, Circumferences, Areas, Squares, Cubes, Square Roots, and Cube Roots, ........ 66 Circles : Diameter, Circumference, Area, and Side of Equal Square, . . 87 Lengths of Circular Arcs from i to 180, 95, 97 Areas of Circular Segments, .......... 100 Sines, Cosines, Tangents, Cotangents, Secants, and Cosecants of Angles, . . 103 Logarithmic Sines, Cosines, Tangents, and Cotangents of Angles, . . .no Rhumbs, or Points of the Compass, 117 Reciprocals of Numbers from I to I ooo, . . . . . . . .118 WEIGHTS AND MEASURES. WATER as a Standard Weight and Volume of pure Water The Gallon and other Measures of Water Pressure of Water Sea- water Ice and Snow French and English Measures of Water, . . . . . . . . .124 AlR as a Standard Pressure of the Atmosphere Measures of Atmospheric Pres- sure Weight of Air Volume Specific Heat, ...... 127 GREAT BRITAIN AND IRELAND Imperial Weights and Measures, . . . 128 Measures of Length : Lineal Land Nautical Cloth, ..... 129 Wire-gauges, 130 Inches and their Equivalent Decimal Values in parts of a Foot Fractional Parts of an Inch, and their Decimal Equivalents, ...... 135 Measures of Surface : Superficial Builders' Measurement Land, . . . 136 Measures of Volume : Solid or Cubic Builders' Measurement, . . -137 Table of Decimal Parts of a Square Foot in Square Inches, .... 138 Measures of Capacity : Liquid Dry Definition of the Standard Bushel Coal Old Wine and Spirit Old Ale and Beer Apothecaries' Fluid, . 138 Measures of Weight: Avoirdupois Troy Diamond Apothecaries' Old Apothecaries' Weights of Current Coins Coal Wool Hay and Straw Corn and Flour, ........... 140 Miscellaneous Tables : Drawing Papers Commercial Numbers Stationery Measures relating to Building Commercial Measures Measures for Ships, 143 Comparison of English Compound Units : Measures of Velocity Of Volume and Time Of Pressure and Weight Of Weight and Volume Of Power, 144 X CONTENTS. PAGE FRANCE The Metric Standards of Weights and Measures Metre Kilogramme, . 146 Countries where the system is legalized, . . . . . . . .146 Measures of Length, ........... 147 Wire-gauges, 148 Measures of Surface, . . . . . . . . . . .149 Measures of Volume : Cubic Wood, 149 Measures of Capacity : Liquid Dry, ........ 149 Measures of Weight, . . . . . . . . . .150 EQUIVALENTS of British Imperial and French Metric Weights and Measures, . 150 Measures of Length Tables of Equivalent Values of Millimetres and Inches Square Measures or Measures of Surface Cubic Measures Wood Mea- sure Measures of Capacity Measures of Weight, 150 Approximate Equivalents of English and French Measures, 156 Equivalents of French and English Compound Units of Measurement : Weight, Pressure, and Measure Volume, Area, and Length Work Heat Speed Money, ............. 157 GERMAN EMPIRE : Weights and Measures : Length Surface Capacity Weight, 160 Values of the German Fuss or Foot in the various States, . . . . . 161 Old Weights and Measures in Prussia (Kingdom of) Bavaria (Kingdom of) Wurtemberg (Kingdom of) Saxony (Kingdom of) Baden (Grand-duchy of) Hanse Towns: Hamburg Bremen Lubec Old German Customs Union Oldenburg Hanover, &c., ........ 162 Austrian Empire, . 170 Russia, 171 Holland Belgium Norway and Denmark Sweden . . . . . .173 Switzerland Spain Portugal Italy, . . . . . . . . 175 Turkey Greece and Ionian Islands Malta, . . . . . . . .178 Egypt Morocco Tunis Arabia Cape of Good Hope, . . . . 1 79 Indian Empire Bengal Madras Bombay Ceylon, . . . . . .180 Burmah China Cochin-China Persia Japan Java, . . . . .183 United States of North America, . 186 British North America, . . . . . . . . . . .187 Mexico Central America and West Indies West Indies (British) Cuba Guate- mala and Honduras British Honduras Costa Rica St. Domingo, . . 187 South America Colombia Venezuela Ecuador Guiana Brazil Peru Chili Bolivia Argentine Confederation Uruguay Paraguay, . . . .188 Australasia: New South Wales Queensland Victoria New Zealand, &c., . 189 MONEY BRITISH AND FOREIGN. Great Britain and Ireland : Value, Material, and Weight of Coins Mint Price of Standard Gold, &c., .......... 190 France: Material and Weight of French Coins, and Value in English Money, . 190 German Empire : Names and Equivalent Values of Coins, . . . . .191 North and South Germany (Old Currency of), ...... 191 Hanse Towns (Old Monetary System of): Hamburg, Bremen, Lubec, . . . 191 Austria Russia Holland Belgium Denmark Sweden Norway, . . .192 Switzerland Spain Portugal Italy Turkey Greece and Ionian Islands Malta, 1 93 Egypt Morocco Tunis Arabia Cape of Good Hope, . . . . 194 Indian Empire China Cochin-China Persia Japan Java, . . . 195 United States of North America, .......... 195 Canada British North America, . . . . . . . . . .196 Mexico West Indies (British) Cuba Guatemala Honduras Costa Rica St. Domingo 196 CONTENTS. XI PAGE South America Colombia Venezuela Ecuador Guiana Brazil Peru Chili Bolivia Argentine Confederation Uruguay Paraguay, . . .196 Australasia, 197 WEIGHT AND SPECIFIC GRAVITY. Standard Bodies and Temperatures for Comparative Weight Rules for Specific Gravity, 198 General Comparison of the Weights of Bodies, . . ...'.. . . .199 Tables of the Volume, Weight, and Specific Gravity of Metallic Alloys Metals Stones, ............. 200 Coal Peat Woods Wood-Charcoal, . . 206 Animal Substances Vegetable Substances, 212 Weight and Volume of various Substances, by Tredgold, ..... 213 Weight and Volume of Goods carried over the Bombay, Baroda, and Central Indian Railway, . 213 Weight and Specific Gravity of Liquids, . . . . . . . .215 Weight and Specific Gravity of Gases and Vapours, . . . . . .216 WEIGHT OF IRON AND OTHER METALS. Data for Wrought Iron for Steel for Cast Iron, . . . . . .217 Tables of Weights: Weights of given Volumes of Metals Volumes of given Weights of Metals Weight of One Square Foot of Metals Weight of Metals of a given Sectional Area, . . . . . . . . . . .218 Special Tables for the Weight of Wrought Iron: Rules for the Weight of Wrought Iron Cast Iron and Steel, . . . 223 Rule for the Length of I cwt. of Wire of different Metals, of a given thickness, 224 Weight of French Galvanized Iron Wire, ....... 225 Special Tables of the Weight of Wrought-Iron Bars, Plates, &c. ; Multipliers for other Metals : Flat Bar Iron Square Iron Round Iron Angle Iron and Tee Iron Wrought-Iron Plates Sheet Iron Black and Galvanized- Iron Sheets Hoop Iron Warrington Iron Wire Wrought-Iron Tubes, by Internal Diameter Wrought-Iron Tubes, by External Diameter, . 226 Weight of Cast Iron, Steel, Copper, Brass, Tin, Lead, and Zinc Special Tables : Cast-Iron Cylinders, by Internal Diameter Cast-Iron Cylinders, by External Diameter Volumes and Weight of Cast-Iron Balls, for given Diameters; Multipliers for other Metals Diameter of Cast-Iron Balls for given Weights, 253 Weight of Flat-Bar Steel Square and Round Steel Chisel Steel, . . . 259 Weight of One Square Foot of Sheet Copper Copper Pipes and Cylinders, by Internal Diameter Brass Tubes, by External Diameter One Square Foot of Sheet Brass, 261 Size and Weight of Tin Plates Weight of Tin Pipes and Lead Pipes Dimen- sions and Weight of Sheet Zinc, 268 FUNDAMENTAL MECHANICAL PRINCIPLES. FORCES IN EQUILIBRIUM : Solid Bodies Fluid Bodies, 271 MOTION : Uniform Motion Velocity Accelerated and Retarded Motion, . . 277 GRAVITY : Relations of Height, Velocity, and Time of Fall Rules and Tables, . 277 ACCELERATED AND RETARDED MOTION IN GENERAL: General Rules Descent on Inclined Planes, 282 MASS, 287 MECHANICAL CENTRES: Centre of Gravity Centre of Gyration Radius of Gyration Moment of Inertia Centre of Oscillation The Pendulum Length of Seconds Pendulum Centre of Percussion 287 CENTRAL FORCES : Centripetal Force Centrifugal Force, 294 Xll CONTENTS. PAGE MECHANICAL ELEMENTS: The Lever The Pulley The Wheel and Axle The Inclined Plane Identity of the Inclined Plane and the Lever The Wedge The Screw, 296 WORK: English and French Units of Work Work done by the Mechanical Ele- ments By Gravity Work accumulated in Moving Bodies Work done by Percussive Force, 312 HEAT. THERMOMETERS: Table of Equivalent Degrees by Centigrade and Fahrenheit Pyrometers, 317, 967 MOVEMENTS OF HEAT: Radiation Conduction Convection, .... 329 THE MECHANICAL THEORY OF HEAT: Mechanical Equivalent of Heat Joule's Equivalent in English and French Units Illustrations, .... 332 EXPANSION BY HEAT: Linear and Cubical Expansion, 335 Table of Linear Expansion of Solids, ........ 336 Expansion of Liquids, ........... 338 Expansion of Gases The Absolute Zero-point Table of the Compression of Gases by Pressure under a Constant Temperature, ..... 342 Relations of the Pressure, Volume, and Temperature of Air and other Gases General Rules Special Rules for One Pound weight of a Gas, with Table of Coefficients Table of the Volume, Density, and Pressure of Air at various Temperatures, .......... 346 SPECIFIC HEAT: Specific Heat of Water, with Table Specific Heat of Air- Specific Heat of Solids Specific Heat of Liquids Specific Heat of Gases, . 352 FUSIBILITY OR MELTING POINTS OF SOLIDS: Table, 363 Latent Heat of Fusion of Solid Bodies, with Rule and Table, .... 367 BOILING POINTS OF LIQUIDS, 368 Latent Heat and Total Heat of Evaporation of Liquids, ..... 370 Boiling Points of Saturated Vapours under various Pressures, . . . '371 Latent Heat and Total Heat of Evaporation of Liquids under One Atmosphere, 372 LIQUEFACTION AND SOLIDIFICATION OF GASES, ...... 372 SOURCES OF COLD: Siebe's Ice-making Machine Carre's Cooling Apparatus Frigorific Mixtures, . . . . . . . . . . 373 STEAM. Physical Properties of Steam, .......... 378 Gaseous Steam Its Expansion Its Total Heat, ...... 383 Specific Heat of Steam Specific Density of Steam Density of Gaseous Steam, 384 Properties of Saturated Steam from 32 to 212 F., . . . . . . 386 Properties of Saturated Steam for Pressures of from I Ib. to 400 Ibs., . . 387 Comparative Density and Volume of Air and Saturated Steam, . . . 391 MIXTURE OF GASES AND VAPOURS. Respective Pressures of Gas and Vapours in Mixture, 392 Hygrometry, ............. 392 Properties of Saturated Mixtures of Air and Aqueous Vapour, with Table, . 394 COMBUSTION. Combustible Elements of Fuel Process of Combustion, ..... 398 AIR CONSUMED IN THE COMBUSTION OF FUELS : Quantity of the Gaseous Pro- ducts of the Complete Combustion of One Pound of Fuel Surplus Air, . 400 HEAT EVOLVED BY THE COMBUSTION OF FUEL : Heat of Combustion of Simple and Compound Bodies Heating Powers of Combustibles, .... 402 Temperature of Combustion, .......... 407 CONTENTS. Xlll FUELS. PAGE Fuels or Combustibles generally used, 409 COAL: Its Varieties Small Coal: Its Utilization Washing Small Coal Deterioration of Coal by Exposure to Atmosphere, 409 British Coals Composition of Bituminous Coals Dr. Richardson's Analysis, . 412 Weight and Composition of British and Foreign Coals, by Delabeche and Playfair, 413 Weight and Bulk of British Coals, 416 Hygroscopic Water in British Coals, 416 Torbanehill or Boghead Coal, with Table of its Composition, . . . .417 American and Foreign Coals : Composition, Weight and Bulk, .... 418 French Coals: Utilization of the Small Coal Composition of French Coals Mean Density, Composition, and Heating Power, ..... 420 Indian Coals : Australian and Indian Coals Composition, 423 COMBUSTION OF COAL : Process of Combustion Gaseous Products of the Com- bustion of Coal Surplus Air Total Heat of Combustion of British Coals, . 426 COKE : Proportion of Coke from Coals Anthracitic Coke Weight and Bulk of Coke Composition of Coke Moisture in Coke Heating Power of Coke, . 430 LIGNITE AND ASPHALTE : Density, Composition, and Heating Power of Lignites and Asphaltes, 436 WOOD: Moisture in Wood Composition Weight and Bulk of Wood, with Table Firewood Quantity of Air Chemically Consumed in the Complete Combustion of Wood Gaseous Products Total Heat of Combustion Temperature of Combustion, . 439 WOOD-CHARCOAL : Yield of Charcoal Composition, with Table of Composition at various Temperatures Carbonization of Wood in Stacks, and Yield of Charcoal Manufacture of Brown Charcoal Distillation of Wood Charbon de Paris (artificial fuel) Weight and Bulk of Wood-Charcoal Absolute Density of Charcoal Moisture in Charcoal Air Consumed in the Combus- tion of Charcoal Gaseous Products Heat of Combustion, . . . 444 PEAT: Nature and Composition Condensed Peat Average Composition Pro- ducts of Distillation Heating Power of Irish Peat, ..... 452 PEAT-CHARCOAL : Composition and Heating Power, 455 TAN : Composition and Heating Power, ........ 455 STRAW: Composition, . 456 LIQUID FUELS : Petroleum, Petroleum-Oils, Schist Oil, and Pine-wood Oil ; their Composition and Heating Power, . . . . . . . .456 COAL-GAS : Composition and Heating Power, . . . . . . -457 APPLICATIONS OF HEAT. TRANSMISSION OF HEAT THROUGH SOLID BODIES FROM WATER TO WATER THROUGH SOLID PLATES AND BEDS: M. Peclet's Experiments Mr. James R. Napier's Experiments Circumstances which affect the Ratio of Trans- mission Mr. Craddock's Experiments, ....... 459 HEATING AND EVAPORATION OF LIQUIDS BY STEAM THROUGH METALLIC SURFACES: Experiments by Mr. John Graham, by M. Clement, by M. Peclet, by MM. Laurens and Thomas, by M. Havrez, by Mr. William Anderson, by Mr. F. J. Bramwell Table of Performance of Coiled Pipes and Boilers in Heating and Evaporating Water by Steam, with Deductions, 461 COOLING OF HOT WATER IN PIPES: Observations of M. Darcy Experiments by Tredgold Deductions, 469 COOLING OF HOT WORT ON METAL PLATES IN AIR: Results of Experiments at Trueman's Brewery, 47 COOLING OF HOT WORT BY COLD WATER IN METALLIC REFRIGERATORS: Table of Results of Performance, and Deductions, . . . . 47 1 XIV CONTENTS. PAGE CONDENSATION OF STEAM IN PIPES EXPOSED TO AIR: Experiments by Tred- gold, and by M. Burnat, on Pipes with various Coverings, with Table Experiments by Mr. B. G. Nichol, by M. Clement, by M. Grouvelle Condensation of Steam in a Boiler Exposed in Open Air, .... 472 CONDENSATION OF VAPOURS IN PIPES OR TUBES BY WATER: M. Audenet's Experiments on Steam Mr. B. G. Nichol's Experiments Condensation of other Vapours, ............ 475 WARMING AND VENTILATION: Allowance of Air for Ventilation, . . . 477 VENTILATION OF MINES BY HEATED COLUMNS OF AIR. Furnace- Ventilation Mr. Mackworth's Data, .......... 479 COOLING ACTION OF WINDOW-GLASS: Mr. Hood's Data, .... 480 HEATING ROOMS BY HOT WATER: Mr. Hood's Estimates Total Quantity of Air to be Warmed per Minute Table of the Length of 4-inch Pipe required to Warm any Building Boiler-power French Practice Perkins' System, . 481 HEATING ROOMS BY STEAM: Length of 4-inch Pipe required French Practice, 486 HEATING BY ORDINARY OPEN FIRES AND CHIMNEYS: M. Claudel's Data, . 488 HEATING BY HOT AIR AND STOVES: Sylvester's Cockle-Stove French Prac- tice House- Stoves placed in the Rooms to be Warmed House- Stoves placed outside the Rooms to be Warmed, . 488 HEATING OF WATER BY STEAM IN DIRECT CONTACT: Mr. D. K. Clark's Experiments, ............ 490 EVAPORATION (SPONTANEOUS) IN OPEN AIR: Dalton's Experiments, and Deduc- tions Rule for Spontaneous Evaporation Dr. Pole's Formula, . . . 491 DESICCATION BY DRY WARM AIR: Design of a Drying Chamber Results of Experiments Drying-house for Calico Drying Linen and Various Stuffs Drying Stuffs by Contact with Heated Metallic Surfaces Drying Grain Drying Wood, ............ 493 HEATING OF SOLIDS: Cupola Furnace Plaster Ovens Metallurgical Furnaces Blast Furnaces, 497 STRENGTH OF MATERIALS. DEFINITIONS, 500 WORK OF RESISTANCE OF MATERIAL, 501 COEFFICIENT OF ELASTICITY, 503 TRANSVERSE STRENGTH OF HOMOGENEOUS BEAMS, 503 SYMMETRICAL SOLID BEAMS: Investigation and Generalized Formula, . . 503 Formula for the Transverse Strength of Solid Beams of Symmetrical Section, without Overhang, and Flanged or Hollow For Unsymmetrical Flanged Beams Neutral Axis Elastic Strength, ...... 509 FORMS OF BEAMS OF UNIFORM STRENGTH: Semi-Beams Loaded at One End Uniformly Loaded, . . . . . . . . . . 517 Forms of Beams of Uniform Strength, Supported at Both Ends Under a Con- centrated Rolling Load, 521 SHEARING STRESS IN BEAMS AND PLATE-GIRDERS 525 DEFLECTION OF BEAMS AND GIRDERS : Investigation Rectangular Beams Double-flanged Uniform Beams Supported at Three or more Points, . . 527 TORSIONAL STRENGTH OF SHAFTS: Round Hollow Square Deflection, . 534 STRENGTH OF TIMBER: Results of Experiments, 537 Transverse Strength of Timber of Large Scantling, ...... 542 Elastic Strength and Deflection of Timber: Experiments by MM. Chevandier and Wertheim, by Mr. Laslett, by Mr. Kirkaldy, by Mr. Barlow, . . 545 Rules for the Strength and Deflection of Timber, 548 STRENGTH OF CAST IRON: Tensile Strength and Compressive Strength Results of Experiments, 553 Shearing Strength, 561 CONTENTS. XV Transverse Strength: Results of Experiments Test Bars Transverse Deflection and Elastic Strength, 561 Torsional Strength, 565 STRENGTH OP WROUGHT IRON: Tensile Strength, &c. Mr. Kirkaldy's Experi- ments, ............. 567 Experiments of the Steel Committee of Civil Engineers, ..... 579 Hammered Iron Bars (Swedish) Krupp and Yorkshire Plates Prussian Plates, 581 Iron Wire, 586 Shearing and Punching Strength, ......... 587 Transverse Strength Deflection and Elastic Strength, 588 Torsional Strength, 590 STRENGTH OF STEEL: Mr. Kirkaldy's Early Experiments Hematite Steel Krupp Steel, 593 Experiments of the Steel Committee, 596 Experiments at H.M. Gun Factory, Woolwich Fagersta Steel, Mr. Kirkaldy's Experiments, in seven series, 604 Siemens- Steel Plates and Tyres Mr. Kirkaldy's Experiments, . . . .612 Whitworth's Fluid-compressed Steel, 614 Sir Joseph Whitworth's Mode of Expressing the Value of Steel, . . .615 Chernoff s Experiments on Steel, . .616 Steel Wire, 617 Shearing Strength of Steel, . . . . . . . . . .617 Transverse Strength and Deflection, . . . . . .. .617 Torsional Strength, . . . .619 Strength Relatively to the Proportion of Constituent Carbon, . . . .621 Resistance of Steel and Iron to Explosive Force, ...... 622 RECAPITULATION OF DATA ON THE DIRECT STRENGTH OF IRON AND STEEL: Tensile and Compressive Strength of Cast Iron, Wrought Iron, and Steel Diagram of the Relative Elongation of Bars of Cast Iron, Wrought Iron, and Steel, 623 WORKING STRENGTH OF MATERIALS FACTORS OF SAFETY: Factors of Safety for Cast Iron, Wrought Iron, Steel, and Timber Load on Foundations, Mason-work Ropes Dead Load Live Load, 625 TENSILE STRENGTH OF COPPER AND OTHER METALS: Tables of the Strength of Copper and its Alloys: Tin, Lead, Zinc, Solder, ..... 626 TENSILE STRENGTH OF WIRE OF VARIOUS METALS: Tenacity of Metallic Wires at Various Temperatures Wires of Various Metals, . . . 628 STRENGTH OF STONE, BRICKS, &c. : Table of the Tensile Strength of Sandstones and Grits, Marbles, Glass, Mortar, Plaster of Paris, Portland Cement, Roman Cement, Granites, Whinstone, Limestone, Slates, Bricks, Brickwork in Cement Adhesion of Bricks, . . . . . . . . . 629 STRENGTH OF ELEMENTARY CONSTRUCTIONS. RIVET-JOINTS : In Iron Plates, 633 In Steel Plates, 640 PILLARS OR COLUMNS : Compressive Strength, 648 CAST-IRON FLANGED BEAMS: Transverse Strength, 647 Deflection and Elastic Strength, 652 WROUGHT-IRON FLANGED BEAMS OR JOISTS: Solid Wrought-iron Joists Transverse Strength and Deflection, -653 Rivetted Wrought-iron Joists, 657 BUCKLED IRON PLATES, 660 XVI CONTENTS, PAGE RAILWAY RAILS: Transverse Strength of Rails of Symmetrical Section, . .661 Rails of Unsymmetrical Section, . 665 Deflection of Rails, 668 STEEL SPRINGS: Laminated and Helical, 671 ROPES: Hemp and Wire, 673 CHAINS, 677 LEATHER BELTING, 679 BOLTS AND NUTS, 680 SCREWED STAY-BOLTS AND FLAT SURFACES, 685 HOLLOW CYLINDERS TUBES, PIPES, BOILERS, &c. .-Resistance to Internal or Bursting Pressure Transverse Resistance, ....... 687 Longitudinal Resistance to Bursting Pressure, ....... 692 Wrought-iron Tubes, 693 Cast-iron Pipe, . 693 Resistance to External or Collapsing Pressure Solid- drawn Tubes Large Flue Tubes Lead Pipes, 694 FRAMED WORK CRANES, GIRDERS, ROOFS, c.: The Triangle the Funda- mental Feature, ............ 697 Warren- Girder Loaded at the Middle, and at an Intermediate Point Uniformly Loaded Rolling Load, , = .,...... 699 Parallel Lattice- Girder, 708 Parallel Strut-Girder, 708 Roofs, 713 WORK, OR LABOUR. UNITS OF WORK OR LABOUR: Horse-power Mechanical Equivalent of Heat Labour of Men, ............ 718 Labour of Horses Work of Animals Carrying Loads, 720 FRICTION OF SOLID BODIES. LAWS OF FRICTION: Friction of Journals Friction of Flat Surfaces, . . 722 FRICTION ON RAILS: M. Poiree's Experiments, 724 WORK AND HORSE-POWER ABSORBED BY FRICTION: Formulas, . . . 725 MILL-GEARING. TOOTHED GEAR: Pitch of the Teeth of Wheels Spur Fly-wheels Toothed Wheels for Mill work Rules, . . 727 Form.of the Teeth of Wheels, . . .731 Proportions of the Teeth of Wheels, 734 Transverse Strength of the Teeth of Wheels Working Strength, . . . 735 Breadth of the Teeth of Wheels, 737 Horse-power Transmitted by Toothed Wheels, 737 Weight of Toothed Wheels, 739 FRICTIONAL WHEEL-GEARING, 741 BELT-PULLEYS AND BELTS: Tensile Strength, 742 Horse-power Transmitted by Belts, 743 Adhesion and Power of Belts Examples of very wide Belts, .... 744 India-rubber Belting, 750 Weight of Belt- Pulleys, 750 ROPE GEARING: Transmission of Power by Ropes to Great Distances, . . 753 Cotton Ropes, 755 CONTENTS. Xvii SHAFTING: Transverse Deflection of Shafts, 756 Ultimate Torsional Strength of Round Shafts, 758 Torsional Deflection of Round Shafts, 759 Power Transmitted by Shafting, 760 Weight of Shafting, . . 761 Strength and Horse-power of Round Wrought-iron Shafting, .... 762 Frictional Resistance of Shafting, 763 Ordinary Data for the Resistance of Shafting, ... ... 763 Journals of Shafts, 766 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. NORMAL STANDARDS, 768 HEATING POWER OF FUELS: Table of Heating Power, 769 EVAPORATIVE PERFORMANCE OF STATIONARY AND MARINE STEAM-BOILERS, WITH COAL: Surplus Air Admitted to the Furnace, .... 770 Experiments on the Evaporative Power of British Coals, by Delabeche and Playfair, 770 Evaporative Performance of Lancashire Stationary Boilers at Wigan With Economizer and Without Economizer Water-tubes Temperature of the Products of Combustion, and of the Feed-water Trials of D. K. Clark's Steam-Induction Apparatus Of Vicars' Self- feeding Fire-grate, . 771 Evaporative Performance of South Lancashire and Cheshire Coals in a Marine Boiler, at Wigan, 781 Trials of Newcastle and Welsh Coals in the Wigan Marine Boiler, . . . 784 Evaporative Performance of Newcastle Coals in a Marine Boiler, at Newcastle- on-Tyne, 785 Trials of Newcastle and Welsh Coals in the Marine Boiler at Newcastle, for the Board of Admiralty, .......... 787 Trials of Welsh and Newcastle Coals in a Marine Boiler at Keyham Factory, . 790 Evaporative Performance of American Coals in a Stationary Boiler, . . -791 Evaporative Performance of an Experimental Marine Boiler, Navy Yard, New York, 795 Evaporative Performance of Stationary Boilers in France, .... 796 Evaporative Performance of Locomotive Boilers, ...... 798 Evaporative Performance of Portable-Engine Boilers, . . . . 801 RELATIONS OF GRATE-AREA AND HEATING SURFACE TO EVAPORATIVE PER- FORMANCE: Mr. Graham's Experiments Experiments by Messrs. Woods and Dewrance Experimental Deductions of M. Paul Havrez, . . . 802 FORMULAS FOR THE RELATIONS OF GRATE- AREA, HEATING SURFACE, WATER, AND FUEL: General Equations, . 804 Formulas for the Experimental Boilers, . . . . . . ... 807 General Formulas for Practical Use, . . . . . . . .819 Table of the Equivalent Weights of Best Coal and Inferior Fuels, . . . 820 STEAM-ENGINE. ACTION OF STEAM IN A SINGLE CYLINDER: The Work of Steam by Expan- sion Clearance Formulas for the Work of Steam Initial Pressure in the Cylinder Average Total Pressure in the Cylinder Average Effective Pres- sure Period of Admission and the Actual Ratio of Expansion Relative Performance of Equal Weights of Steam Worked Expansively Proportional Work Done by Admission and by Expansion Influence of Clearance in Reducing the Performance of Steam, ........ 822 Table of Ratios of Expansion of Steam, with Relative Periods of Admission, /YY, Pressures, and Total Performance, . 835 b XV111 CONTENTS. Total Work Done by One Pound of Steam Expanded in a Cylinder, . . 838 Consumption of Steam Worked Expansively per Horse-power of Net Work per Hour, ............ 840 Table of the Work Done by One Pound of Steam of loo-lbs. Pressure per Square Inch, ............ 841 Net Cylinder-Capacity Relative to the Steam Expanded and Work Done in One Stroke, 843 Table of ditto, . 844 COMPOUND STEAM-ENGINE: Woolf Engine Receiver-Engine Ideal Diagrams, without Clearance Work of Steam as Affected by Intermediate Expansion Intermediate Expansion Work, with Clearance Comparative Work of Steam in the Woolf Engine and the Receiver- Engine, ..... 849 Formulas and Rules for Calculating the Expansion and the Work of Steam, . 869 COMPRESSION OF STEAM IN THE CYLINDER, 878 PRACTICE OF THE EXPANSIVE WORKING OF STEAM: Actual Performance- Data Deductions Conclusions, ........ 879 FLOW OF AIR AND OTHER GASES. Discharge of Air through Orifices Anemometer, . . . . . . .891 Outflow of Steam through an Orifice, ......... 893 Flow of Air through Pipes and Other Conduits, ....... 894 Resistance of Air to the Motion of Flat- Surfaces, ....... 897 Ascension of Air by Difference of Temperature, ....... 897 WORK OF DRY AIR OR OTHER GAS, COMPRESSED OR EXPANDED. WORK AT CONSTANT TEMPERATURES: Isothermal Compression or Expansion, 899 WORK IN A NON-CONDUCTING CYLINDER, ADIABATICALLY, . . . .901 EFFICIENCY OF COMPRESSED-AIR ENGINES, 909 Compression and Expansion of Moist Air, . . . . . . .912 AIR MACHINERY. MACHINERY FOR COMPRESSING AIR AND FOR WORKING BY COMPRESSED AIR: Compression of Air by Water at Mont Cenis Tunnel Works By Direct-action Steam-pumps Compressed-air Machinery at Powell Duffryn Collieries, ............. 915 HOT- AIR ENGINES: Rider's Belou's, 917 GAS-ENGINES: Lenoir's Otto & Langen's Otto's Clerk's, .... 918 GASEOUS FUEL : Wilson Gas Producer Dowson Generator Gas, . . . 922 FANS OR VENTILATORS : Common Centrifugal Fan Mine- Ventilators, . . 924 BLOWING ENGINES, ....... v .... 926 ROOT'S ROTARY PRESSURE-BLOWERS, 927 FLOW OF WATER. FLOW OF WATER THROUGH ORIFICES: Formulas Mr. Bateman's Experi- ments, ............. 929 Mr. Brownlee's Experiments with a Submerged Nozzle, 931 FLOW OF WATER OVER WASTE-BOARDS, WEIRS, &c., 932 FLOW OF WATER IN CHANNELS, PIPES, AND RIVERS, ..... 932 CAST-IRON WATER PIPES, 934 CAST-IRON GAS PIPES, 936 CONTENTS. XIX WATER-WHEELS. PAGE WHEELS ON A HORIZONTAL Axis: Undershot- Wheels Paddle- Wheels Breast- Wheels Overshot- Wheels, 937 WHEELS ON A VERTICAL Axis: Tub Whitelaw's Water-mill Turbines Tangential Wheels, 939 MACHINES FOR RAISING WATER. PUMPS: Reciprocating Pumps Centrifugal Pumps Chain Pump Noria, . 944, 968 Water-works Pumping Engines, ......... 948 Hydraulic Rams, 949 HYDRAULIC MOTORS. HYDRAULIC PRESS, " . " 95 ARMSTRONG'S HYDRAULIC MACHINES, 950 FRICTIONAL RESISTANCES. STEAM ENGINES, 951 TOOLS: Shearing Machines Plate-bending Machines Circular Saws, . -951 Work of Ordinary Cutting Tools, in Metal, ....... 952 Screw-cutting Machines Wood-cutting Machines Grindstones, . . . 954 COLLIERY WINDING ENGINES, 956 WAGGONS IN COAL PITS, ........... 956 MACHINERY OF FLAX MILLS: M. Cornut's Experiments, 957 Horse-power Required, 959 MACHINERY OF WOOLLEN MILLS: Dr. Hartig's Experiments, .... 959 MACHINERY FOR THE CONVEYANCE OF GRAIN, ...... 960 TRACTION ON COMMON ROADS: M. Dupuit's Experiments M. Debauve's De- ductions M. Tresca's Experiments, ........ 961 CARTS AND WAGGONS ON ROADS AND ON FIELDS, 962 RESISTANCE ON RAILWAYS, 965 RESISTANCE ON STREET TRAMWAYS, 966 APPENDIX. DR. SIEMENS' WATER PYROMETER, 967 ATMOSPHERIC HAMMERS, 967 BERNAYS' CENTRIFUGAL PUMPS, . . 968 STEAM-VACUUM PUMP, 969 INDEX, 971 AUTHORITIES CONSULTED OR QUOTED. American, United, Railway Master Car- Builders' Association, Standard Sizes of Bolts and Nuts by, 683. American Society of Civil Engineers, Journal of: Mr. J. F. Flagg, on Steam-vacuum Pumps, 969. Anderson, Dr., on the Strength of Cast Iron, 555- Anderson, William, on Heating Water by Steam, 465, 466, 468 ; Translation of Cher- noff 's Paper on Steel, 616. Annales des Mines: M. Krest, on the Slip of Belts, 742. Annales des Fonts et Chaussdes: M. Hirn's Rope Transmitter of Power, 754. Annales du Ge"nie Civile: M. Paul Havrez, on Heating Surface of Locomotives, 803. Annals of Philosophy : Mr. Dunlop, on Tor- sional Strength of Cast Iron, 565. Annua ire de FA ssociation des Inge"n ieurs sortis de Vcole de Lie"ge: Rivetted Joints, 641. Armengaud, French Standard Bolts and Nuts, by, 683. Armstrong, Sir Wm., on Evaporative Power of Coals, 785; his Hydraulic Machinery, 95- Arson, Anemometer by, 892. Ashby & Co., Work of Steam in Portable Engine by, 883. Audenet, on Surface-Condensers, 475. B Baker, B., on the Strength of Beams, 512; of Oak, 544, 549; of Columns, 645, 646; of Rails, 662, 666. Barlow, Peter, on Strength of Timber, 547 ; of Cast Iron, 561 ; of Wrought Iron, 567, 588, 590 ; of Iron Wire, 586. Barlow, W. H., on the " Resistance of Flex- ure," 507. Barnaby, Mr., on Strength of Punched Steel Plate, 642. Barrow Hematite Steel Company, Strength of Steel made by, 594, 618, 619, 620, 621. Bateman, J. F., on Flow of Water through Submerged Openings, 930; his Cast-Iron Pipes, 934. Baudrimont, on Strength of Metallic Wires, 628. Beardmore, on the Work of Horses, 720 ; on Limits of Velocity at the Bottom of a Channel, 934. Beaufoy, Colonel, on Resistance of Air, 897. Bell, J. Lothian, on the Heat in Blast Fur- naces, 498. Berkley, George, on the Strength of Cast- iron Beams, 647-650. Berkley, J., Specific Gravity of Indian Woods, by, 209. Bernays, Joseph, on Centrifugal Pumps, 968. Bertram, W., on Rivetted Joints, 634-637. Borsig, Herr, Strength of Wrought-Iron Plates, 586. Box, Thomas, on the Load on Journals, 766 ; Thickness of Gas Pipes, by, 936. Boyden, Outflow Turbine by, 940. Bradford, W. A., on Otto and Langen's Gas- Engine, 924. Bramwell, F. J., on Heating Water by Steam, 467, 468 ; on the Strength of Cast Iron, 556 ; on Portable Steam Engines, 801, 883, 886 ; on the Expansive Working of Steam, 889. Brereton, R. P., on Strength of Timber Piles, 646. Briggs, Blowing Engine by, 927. British Associatiom, Transactions of: F. W. Shields, on Strength of Cast-Iron Columns, 645. Brown & May, Work of Steam in Portable Engine by, 882. Brownlee, J., on Saturated Steam, 382; on the Outflow of Steam, 893 ; Flow of Water through a Submerged Nozzle, 931. Bruce, G. B., on the Work of a Labourer, 719. Brunei, on the Strength of Rivetted Joints, 638 ; and of Bolts and Nuts, 680. Buchanan, W. M., on Saturated Steam, 379. Buckle, W., on Fans, 924. Buel, R. H., on the Slip of Belts, 742. Bulletin de la Socidtd Industriette de Mul- house: M. Leloutre on Steam Engines, 886. Burnat, on Condensation of Steam in Pipes, 472, 474. Bury, Wm., on Strength of Flat Stayed Sur- faces, 686. b 2 XX11 AUTHORITIES CONSULTED OR QUOTED. Cameron, Dr. , Analysis of Peat by, 454. Charie-Marsaines, on Flemish Horses, 964. Chenot Aine", Atmospheric Hammer by, 967. Chernoft, on Steel, 616. Chevandier, on Composition of Wood, 440; on its Weight and Bulk, 442, 443. Chevandier & Wertheim, on Strength of Tim- ber, 538, 545, 546, 549. Clark, D. K., on Properties of Saturated Steam, 387; on Locomotive Boilers, 798; on the Work of Steam, 879, 880, 884 ; on Resistance on Railways, 965 ; Tramways, 966. Clark, Edwin, on the Strength of Beams, 512; of Red Pine, 543, 544, 549; of Cast Iron, 562 ; of Bar Iron, 570, 588, 590, 623. Clark, Latimer, on Wire Gauges, 130. Claudel, on Fuels and Woods, by, 207, 211, 212 ; tints of Heated Iron, 328 ; on Heating Factories, 486 ; on Heating Rooms, 488, 489; on Belts, 743, 746; on Blowing En- gines, 927; on Pumps, 944. Clement, on Transmission of Heat, 462, 468 ; on Condensation of Steam in Pipes, 474; on Drying Stuffs, 496 ; on the Heat to Melt Iron, 497. Cochrane, J., on Strength of Perforated Bar Iron, 633. Cockerill, John, Blowing Engines by, 927. Colliery Guardian : Mr. Mackworth on Ven- tilation of Mines, 480. Conservatoire des Arts et Matters, Annales du: Hot-Air Engines by Laubereau, and by Belou, 917-919 ; Gas- Engines by Lenoir, 920; by Hugon, 921; by Otto & Langen, 923- Cooper, J. H., on Very Wide Belts, 747, 749. Cornet, on the Work of a Labourer in France, 720. Cornut, E., on Mill-Shafting, 766 ; on Machin- ery of Flax-Mills, 957 ; on Flow of Air in Pipes, 896. Cotterill, J. H., on Work of Compression of Air, 903. Cowper, E. A., Compound Engine by, 889. Craddock, Thomas, on Cooling through Plates, 461. Crighton & Co., on Drying Grain, 496. Crookewitt, on Specific Gravities of Alloys, 200. Crossley, F. W., on Otto & Langen's Gas- Engines, 923. Cubitt, Mr., on Strength of Cast-Iron Beams, 649. D Daglish, G. H., on Resistance of Colliery Winding Engines, 956. Dalton, Dr., on " Spontaneous " Evaporation of Water, 491. Daniel, W., on Ventilation of Mines, 925. Danvers, F. C., on Coal Economy, 410. Darcy, on Cooling Hot Water in Pipes, 469. D'Aubuisson, on Flow of Compressed Air, 896 ; on Hydraulic Rams, 949. Davey, Paxman, & Co., Work of Steam in Portable Engine by, 883. Davies, Thomas, on Strength of Rivetted Joists, 658. Davison, R., on Resistance of Shafting, 766; Duty of Pumps by, 944; on Resist- ance of Grain Machinery, 961. Day, Summers, & Co., Work of Steam in Marine Engines by, 882. Debauve, on Resistance on Common Roads, 961. Delabeche & Playfair, on British and Foreign Coals, 206, 413, 416, 770. Despretz, on Conducting Powers of Bodies, 33*' Deville, Sainte-Claire, on Composition of Petroleum and other Oils, 456, 457. Dewrance, John, on the Heating Surface of a Locomotive, 803. Donkin, Bryan, & Co., Work of Steam in Stationary Engines by, 882. Downing, on Flow of Water in Pipes, 933, 934- Dunlop, on Strength of Cast Iron, 565. Dupuit, on Resistance on Common Roads, 961. Durie, James, on Rope-Gearing, 753. Duvoir, Rene, Drying House by, 495. Eastons & Anderson, on Portable Steam Engines, 801 ; on Rider's Hot-Air Engine, 917 ; on Resistance of Waggons, 962. Elder, John, & Co., on the Strength of Boilers, 638, 693 ; Work of Steam in Marine Engine by, 882. Emery, on American Marine Engines, 884. Engineer, The: Crighton & Co. on Drying Grain, 496 ; Mr. W. S. Hall on the Strength of Rivetted Joints, 641 ; Messrs. Woods & Dewrance on Locomotive Boilers, 803 ; Mr. C. L. Hett on Hydraulic Rams, 949. Engineering: on Heating Water by Steam, 464 ; on Cooling Wort, 470, 471 ; Mr. B. G. Nichol on Surface Condensation, 476 ; Mr. G. Graham Smith on Strength of Timber, 544; Factor of Safety for Wrought Iron, by Roebling, 625 ; Mr. W. S. Hall on the Strength of Rivetted Joints, 641 ; Mr. John Mason on Strength of Untanned Leather Belts, 680; Mr. Phillips on Strength of Flat Plates, 686; Mr. Bury on the Strength of Flat Stayed Surfaces, 686 ; Messrs. John Elder & Co. on the Strength of Boilers, 638, 693 ; Mr. J. Durie on Rope Gearing, AUTHORITIES CONSULTED OR QUOTED. XXlll 753; Dr. Hartig on Resistance of Tools, j 951 ; Resistance of Waggons, by Messrs. Eastons & Anderson, 962. English Mechanic: Mr. W. A. Bradford on Otto & Langen's Gas-Engine, 924. Evrard, A., on the Work of Animals, 720. Fagersta Steel Works, Strength of Steel made at, 604, 618, 619, 620, 621. Fairbairn, Sir William, on Hot-Blast Iron, 556 ; on the Strength of Cast Iron, 557 ; on the Strength of Wrought Iron, 567-569; of Rivetted Joints, 633 ; of Screwed Stay- Bolts and Flat Stayed Plates, 685 ; on the Proportions of Spur Wheels, 729, 734, 737 ; on the Load on Journals, 766, 767; on Water Wheels, 938. Fairbairn & Tate, on the Expansion of Steam, 383- Fairweather, James C. , on Resistance of Air, 897. Faraday, Dr., on the Liquefaction of Gases, 372. Favre & Silbermann, on the Heating Powers of Combustibles, 404. Field, Joshua, on the Work of Labourers, 719. Fincham, on Strength of Timber, 542, 543, 549- Flagg, J. F., on Steam- vacuum Pumps, 969. Fletcher, L. E., on the Strength of a Boiler, 638, 693 ; his Reports, 696 ; his Report on Boiler and Smoke Prevention Trials, 771- 784. Fowke, Captain, on Colonial Woods, 209. Fowler, G., on Resistance of Waggons in Coal Pits, 956. Fowler, John, Strength of Steel Rails de- signed by, 666, 670. Fowler, J., & Co., Compressed-air Machinery by, 916. Fox, Head & Co., on Condensation of Steam in a Boiler, 475. Francis, J. B., on a Swain Turbine, 943. Franklin Institute, Journal of: the Shear- ing Resistance of Bar Iron, by Chief Engineer W. H. Shock, 588 ; Mr. R. H. Buel on Belts, 742 ; Mr. H. R. Towne on Belts, 742, 745; Mr. J. H. Cooper on Belts, 747 ; Mr. S. Webber on Mill Shaft- ing, 763, 764; Mr. Emery on American Marine Engines, 884; Mr. Briggs on Blowing Engines, 927; Mr. J. B. Francis on a Swain Turbine, 943 ; Mr. E. D. Leavitt's Pumping Engines, 948. Gammelbo & Co., Hammered Bars made by, Strength of, 581. Gaudillot, on Heating Apparatus, 486. Gay-Lussac, on Cold by Evaporation, 376. Glynn, Mr., on Strength of Ropes, 673 ; on the Work of a Labourer, 718. Gooch, Sir Daniel, on Consumption of Water by the "Great Britain" Locomotive, 884. Gordon, L. D. B., on Strength of Columns, 645- Graham, John, on Heating Water, 461 ; on Heating Surface, 802. Grant, on Strength of Cements, &c., 630. Greaves, on Pumping Engines, 948. Grouvelle, on Condensation of Steam in Pipes, 474; on Heating Factories, 486, 487. H Hackney, W., on Anthracitic Coke, 432. Haines, R., on Indian Coals, 423. Hall, W. S., on the Strength of Rivetted Joints, 641. Harcourt, Vernon, on Analysis of Coal-Gas, 458. Harmegnies, Dumont, & Co. , on French Wire Ropes, 677. Hartig, Dr., on Driving Belts, 743; on Re- sistance of Tools, 951 ; on Resistance of Machinery of Woollen Mills, 959. Havrez, P., on Heating Water by Steam, 464, 468 ; on Heating Surface of Loco- motives, 803. Hawksley, Thomas, on Flow of Air through Pipes, 894 ; on Velocity of Air in Up-cast Shaft, 897 ; on Flow of Water in Pipes, 933 ; on Thickness of Water Pipes, 935. Hett, C. L., on Hydraulic Rams, 949. Hick, John, M.P., on Friction of Leather Collars, 950. Hirn, on Work of Expanded Steam in Sta- tionary Engines, 886. Hodgkinson, on the Strength of Cast Iron, 553-555- 558, 559- 563. 564: of Columns, 643, 646 ; of Cast-Iron Flanged Beams, 647-650. Holtzapffel, his Wire-Gauges, 131, 132, 134. Hood, on Warming and Ventilation, 477-485. Hopkinson, on the Performance of a Corliss Engine, 88 1. Hunt, R., on Combustion of Coal, 770. Hutton, Dr., Law of Resistance of Air by, i. Institute of Naval Architects, Transactions of the: Strength of Rivet Joints of Steel Plates, 642. Institution of Civil Engineers, Proceedings of: Mr. Wm. Anderson on Heating Water by Steam, 465; M. Burnat on Condensation of Steam in Pipes, 472; Dr. Pole on Spon- taneous Evaporation, 493 ; Regenerative Hot-Blast Stoves, 556 ; Mr. Bramwell on XXIV AUTHORITIES CONSULTED OR QUOTED. Strength of Cast Iron, 556; Mr. Grant on | the Strength of Cements, &c., 630; Mr. J. Cochrane on the Strength of Punched Bar Iron, 633 ; Mr. R. Price Williams on Strength of Rails, 662 ; Mr. J. T. Smith on the Strength of Bessemer Steel Rails, 664; Mr. R. Davison on Resistance of Shafting, 766 ; Evaporative Performance of Steam Boilers in France, 796 ; Composition of Coals and Lignites, 797; M. Paul Havrez on Heating Surface of Locomotives, 803 ; Mr. Emery on American Marine Engines, 884; Mr. Hawksley on Flow of Air through Pipes, 894 ; and on Velocity of Air in Up- cast Shaft, 897; M. Piccard on the Work of Compressed Air, 911; Mr. J. B. Francis' trial of a Swain Turbine, 943 ; Mr. R. Davison on Duty of Pumps, 944 ; Hon. R. C. Parsons on Centrifugal Pumps, 947 ; Mr. Henry Robinson on Armstrong's Hydraulic Machines, 950. Institution of Engineers and Ship-Builders in Scotland, Transactions of the: on Strength of Helical Springs, 672 ; Report on Safety Valves, 893; Mr. J. Brownlee's Experi- ments on Flow of Water, 931. Institution of Mechanical Engineers, Pro- ceedings of: Mr. C. Little on the Shearing and Punching Strength of Wrought Iron, 587 ; Mr. Vickers on the Strength of Steel, 621; Mr. W. R. Browne's paper on Rivetted Joints, 637; Mr. Robertson on Grooved Frictional Gearing, 741 ; Mr. H. M. Mor- rison on Hirn's Rope Transmitter, 755; Mr. Ramsbottom on Cotton-Rope Trans- mitter, 755; Mr. Westmacott and Mr. B. Walker on Resistance of Shafting, 766 ; Mr. D. K. Clark on the Expansive Working of Steam in Locomotives, 879, 880; Data of the Practical Performance of Steam, 880; Mr. F. J. Bramwell on Economy of Fuel in Steam Navigation, 889 ; Compressed-Air Machinery by Messrs. John Fowler & Co., 916 ; Wenham's Hot-Air Engine, 919 ; Mr. F. W. Crossley on Otto and Langen's Gas- Engine, 923 ; Mr. Buckle on Fans, 924 ; Mr.J.S.E. Swindell on Ventilation of Mines, 925; Mr. W. Daniel on Ventilation of Mines, 925 ; Mr. A. C. Hill on Blowing Engines, 927 ; Mr. J. F. Bateman's Experiments on Flow of Water, 930 ; Mr. David Thomson on Pumping Engines, 948 ; Mr. G. H. Daglish on Winding Engines, 956 ; Mr. G. Fowler on Resistance of Waggons in Coal Pits, 956 ; Mr. Westmacott on Corn- Ware- housing Machinery, 961. Iron and Steel Institute, Journal of the: Mr. J. Lothian Bell on the Cleveland Blast Furnaces, 498. Isherwood, Trials of Evaporative Performance of a Marine Boiler, 795. J James, Captain, on the Strength of Cast Iron, 555- Jardine, Mr., on the Strength of Lead Pipes, 696. Johnson, Professor W. R., on American Coals, 418, 770, 79!-79S- Joule, Dr., Mechanical Equivalent of Heat, K Kane, Sir Robert, on Peat, 453. Kennedy, Colonel J. P., on Weight and Volume of Goods carried on Railways, 213. Kirkaldy, David, on Compressive Strength of Timber, 546, 547, 647 ; on the Tensile Strength of Wrought Iron and Steel, 571- 578 ; of Swedish Hammered Bars, 581, 590; of Krupp and of Yorkshire Iron Plates, 583-586 ; of Borsig's Iron Plates, 586 ; Ten- sile Strength of Bar Steel, 593, 594 ; of He- matite Steel, 594 ; of Krupp Steel, 595 ; of Steel Bars, for the Steel Committee, 597- 600; of Fagersta Steel, 604-611 ; of Siemens- Steel Plates and Tyres, 612-614; on Shear- ing Strength of Steel, 617 ; on Strength of Phosphor-Bronze, 628, 629; of Wires, 629; of Rolled Wrought-iron Joists, 654; of Rails, 662, 663, 666-668; of Ropes, 674; of Belt- ing, 680; of Plates of a Marine Boiler, 694. Krest, on the Slip of Belts, 742. Krupp, Herr, Strength of Wrought-iron Plates made by, 583 ; of his Cast Steel, 595, 618- 621. L Landore Siemens-Steel Company, Strength of Steel Plates and Tyres made by, 612-614. Laslett, Thomas, on the Strength of Timber, 538-542, 546, 548, 550, 647. Leavitt, E. D., Pumping Engines by, 948. Legrand, on Boiling Points, 370. Leigh, Evan, on Belting, 746. Leloutre, on M. Hirn's Experiments on Work of Steam, 886. Leplay, on Moisture in Wood, 439; on Drying Wood, 496. Literary and Philosophical Society of Man- chester, Memoirs of : Dr. Dalton on "Spontaneous" Evaporation, 491 1 Mr. John Graham on Heating Surface, 802. Little, C., on the Shearing and Punching Strength of Wrought Iron, 587. Lloyd, Thomas, on the Slrength of Bar Iron, 569. 570. London Association of Foremen Engineers, Proceedings of: Mr. David Thomson on Expansive Work of Steam, 822. Longridge, J. A., on Combustion and Evap- orative Power of Coals, 770, 785. Longsdon, Mr., on Strength of Krupp Steel, 595- AUTHORITIES CONSULTED OR QUOTED. XXV M MacColl, on the Strength of Rivetted Joints, 641. Mackintosh, Charles, Weight of Belt-Pulleys by, 752. Mackworth, H., on Ventilation of Mines, 479- Maclure, H. H., on Strength of Timber, 542, 543- 549- Macneil, Sir John, on Resistance on Common Roads, 964. Mahan, Lieutenant F. A., on Outward-Flow Turbines, 941. Mallard, on Compressed- Air Machines, 902 ; on Compressed Air, 907, 912. Mallet, R., Strength of Buckled Iron Plates by, 660. Marshall, Sons, & Co., Work of Steam in Portable Engine by, 883. Mason, John, Strength of Untanned Leather Belts by, 680. M'Donnell, A., on Composition of Peat, 454. Menelaus, on Portable Steam Engines, 801. Miller, T. W., Trials of Coals by, 790. Miller & Taplin, Trials of Coals by, 787. Montgolfier, on Drying by Forced Currents, 494- Monthly Reports to the Manchester Steam- Users Association:^. L. E. Fletcher's Data, 696. Morin, on Transverse Strength of Timber, 537; on the Friction of Journals, 722 ; and of Solid Bodies, 723 ; on Leather Belts, 743-745 ; on Breast Wheels, 938 ; on a Fourneyron Turbine, 940; on Centrifugal Pumps, 946. Morrison, H. M., on M. Hirn's Rope Trans- mitter, 755. Morton, Francis, & Co., Weight of Iron Sheets by, 245 ; Strength of Cable Fencing Stands by, 676. Moser, Strength of Beams tested for, 654. Muspratt, Dr., Analyses of Coke by, 433. N Napier, James R., on Transmission of Heat, 460 ; on Drying Stuffs, 496. Nau, on Moisture in Charcoal, 451. Newall, R. S., & Co., Strength of Hemp and Wire Ropes by, 674. Nichol, B. G., on Condensation of Steam in Pipes and Tubes, 474, 476. Nicoll & Lynn, Trials of Coals by, 784. Norris & Co., Strength of Leather Belts by, 680. North British Rubber Company, Driving Belts by, 750. North of England Mining Institute, Transac- tions of: Rivetted Joints, 588. Oldham, Dr., on Indian Coals, 424. Ott, Karl Von, on Strength of Ropes, 674, 679. P Parsons, on Strength of Oak Trenails, 551. Parsons, Hon. R. C., on Centrifugal Pumps, 947- Payen, on Explosive Mixture of Gas and Air, 921. Pearce, W. A., on Rope Gearing, 754. Peclet, on Radiation of Heat, 329 ; on French Coals, 420 ; on Coke, 431 ; on Moisture in Tan, 455 ; on Transmission of Heat, 459, 462, 463, 468; on Condensing Power of Air and Water, 475 ; on Ventilation, 477 ; on Heating Apparatus, 488, 489 ; on Drying by Air Currents, 494 ; on a Drying House, 495 ; on Cupola Furnaces, 497. Penot, on Drying Houses, 496. Penrose & Richards, their Anthracitic Coke, 432. Perkins, Heating Apparatus by, 486. Perkins, Jacob, Invention of the Ice-Making Machine by, 373. Person, on the Latent Heat of Fusion, 367. Phillips, on Strength of Flat Plates, 686. Piccard, on Work of Compressed Air, 911. Poiree, on Friction on Rails by, 724. Pole, Dr., on Spontaneous Evaporation, 493; on the Strength of Steel Wire, 617. Poncelet, on Water Wheels, 938. Portefeuille de John Cockerill : Blowing Engines, 927. Porter, C. T., on Expansion of Steam, 886. Pouillet, on Luminosity at High Temper- atures, 328. R Radford, R. Heber, Weight of Belt-Pulleys by, 7S J . 752. Ramsbottom, J., on Cotton- Rope Transmitter, 755- Rankine, Dr., on Expansion of Water, 340; on the Melting Point of Ice, 364; on Transmission of Heat, 461 ; on Shearing Strength of Oak Trenails, 551; and of Cast Iron, 561 ; Factors of Safety, 625, 626 ; on Stresses in Roofs, 715, 717 ; on Load on Working Surfaces, 767. Reading Engine Works Co., Work of Steam in Portable Engine by, 883. Reclus, Specific Gravity of Sea Water by, 126. Regnault, Air Thermometer by, 325 ; on the Expansion of Air, 344 ; on Specific Heat of Metals, 353; and Gases, 359; Boiling Points of Vapours, 371 ; on Steam, 378, 379- 3^3. 384 ; on the Mixture of Gases and XXVI AUTHORITIES CONSULTED OR QUOTED. Vapours, 392 ; on French Coals, 420, 421 ; on Lignite and Asphalte, 436. Reilly, Calcott, on the Varieties of Stress, 500. Rennie, on the Work of Horses, 720. Revue Industrielle: Atmospheric Hammer by M. Chenot Aine", 967. Reynolds, Dr., on Peat, 454. Richardson, Dr., on Coals, 412 ; on Coke, 433 ; Report on Evaporative Power of Coals, 785- Robertson, James, on Grooved Frictional Gearing, 741. Robinson, Henry, on Armstrong's Hydraulic Machines, 950. Roebling, on the Strength of Iron Wire, 587 ; and of Steel Wire, 617 ; Factor of Safety for Iron, 625; on the Strength of Wire Rope and Hemp Rope, 676. Ross, Owen C. D., on Coal Gas, 457. Rouget de Lisle, on Drying Stuffs, 496. Royal Society of Edinburgh, Proceedings of: Mr. Fairweather on Resistance of Air, 897. Royer, on Drying Houses, 496; on Drying Stuffs, 496. Russell & Sons, J., on the Strength of Wrought-Iron Tubes, 692, 693. Ryland Brothers, Warrington Wire Gauge by, 133. 247- Sauvage, on Charcoal, 447, 449, 452. Scheurer - Kestner & Meunier - Dollfus, on French and other Coals, and Lignites, 422, 797- Sharp, Henry, on Rivetted Joints of Steel Plates, 642. Shields, F. W., on Cast-Iron Columns, 645. Shock, Chief Engineer W. H., on Shearing Strength of Bar Iron, 587. Siemens, Dr. C. W., on Isolated Steam, 383; on the Consumption of Fuel in Metallurgical Furnaces, 497; on the Strength of Hot- Blast Iron, 556; on Hot-Air Engines, 920; his Water Pyrometer, 967. Simms, F. W., on the Work of Horses, 720. Smeaton, on the Power of Labourers, 718. Smith, C. Graham, on Strength of Timber, 543- 544. 549- Smith, J. T., on Punching Resistance of Steel, 617 ; on the Strength of Rails, 664. Snelus, G. J., Analysis of Welsh Coal by, 413. Socidte" Industrielle de Mulhouse: on Steam Boilers, 796. Socidtd Industrielle Minerale, Bulletin de la: M. Cornut on Compressed- Air Machi- nery, 896 ; M. Mallard on Compressed- Air Machines, 902. Sod did des Ingdnieurs Civils, Comptes Rendus de la: Anemometer by M. Arson, 892. Socidtd Vaudoise des Ingdnieurs et des Archi- tectes, Bulletin de la: M. Piccard on Compressed Air, 911. Society of Arts, Committee of, on Resistance on Common Roads, 963. Society of Arts, Journal of: on Resistance on Common Roads, 963. Spill, Strength of Belting by, 680. Steel Committee of Civil Engineers, on the Strength of Wrought Iron, 579, 580 ; and of Steel, 596-603, Stephenson, Robert, on the Strength of Cast Iron, 555, 561. Stoney, on Stress in a Curved Flange, 525 ; on Sectional Area of a Continuous Web, 526; on Shearing Strength of Cast Iron, 561 ; his Factors of Safety, 625 ; on the Re- sistance of Columns, 643, 645, 646 ; on Stresses in Roofs, 715. Sullivan, Dr., on Peat, 207. Sutcliffe, on Condensation of Steam in the Cylinder, 880. Swindell, J. S. E., on Ventilation of Mines, 925- Sylvester, Cockle Stove by, 488. Tangye, J., on the Compressive Resistance of Wrought Iron, 582. Tasker, Work of Steam in Portable Engine by, 883. Telford, Thomas, on the Strength of Wrought Iron, 567 ; and of Iron Wire, 586. Thomas & Laurens, on Brown Charcoal, 449; on Heating by Steam, 463, 468. Thomson, David, on Expansive Action of Steam, 822, 882; on Centrifugal Pumps, 946 ; Duty of Pumping Engines, 948. Thomson, Professor James, Vortex Wheel by, 943- Thurston, on the Strength of Iron Wire, 587. Thwaites & Carbutt, on Root's Blower, 928. Towne, H. R., on Leather Belts, 679, 742, 745, 748-750- Tredgold, Weight and Volume of Various Substances by, 213 ; on Cooling Hot Water, 469; on Cooling of Steam in Pipes, 472, 474 ; on the Work of a Horse, 720. Tresca, on Laubereau's Hot-Air Engine, 917 ; on Gas-Engines, 920, 921, 923 ; on Pumps, 945, 946 ; on Resistance of Tram- way Omnibus, 961. Turner, Work of Steam in Portable Engine by, 883. Tweddell, R. H., on Shafting, 763. U Umber, on M. Hirn's Wire Ropes, 754. Unwin, on Strength of Columns, 645. Ure, Specific Gravity of Alloys by, 200. AUTHORITIES CONSULTED OR QUOTED. XXVll V Vickers, T. E., on the Strength of Steel, 621, 622. Violette, on Wood, 439, 441, 442, 445; on Charcoal, 446-448, 450, 451 w Wade, Major, on the Strergth of Cast Iron, 557- Walker, B., on Resistance of Shafting, 766. Walker, John, on the Work of Labourers, 718. Webb, F. W., on the Strength of Steel, 614, 621. Webber, S., on Mill Shafting, 763, 764, 766. Westmacott, Percy, on Shafting, 766 ; on Corn - Warehousing Machinery, 961 ; on Armstrong's Hydraulic Machines, 950. Whitelaw, James, Water Mill by, 939. Whitworth, Sir Joseph, Standard Wire-Gauge by, 133, 134; Strength of his Fluid-Com- pressed Steel, and of Iron, 614, 615 ; on Resistance of Steel and Iron to Explosive Force, 622; his System of Standard Sizes of Bolts and Nuts, 68 r ; Standard Pitches of Screwed-Iron Piping, 683. Wiesbacb, Coefficients for Flow of Water, 892. Williams, R. Price, on the Transverse Strength of Rails, 662, 664. Williams, Foster, & Co., Weight of Sheet Copper by, 261. Wilson, A., on the Work of Bullocks, 720. Wilson, R., on Strength of Perforated Iron Plates, 633. Wilson, Robert (Patricroft), on Teeth of Wheels, 732. Wood, J. & E., Work of Steam in Stationary Engine by, 882. Woods, E., andj. Dewrance, on the Efficiency of Heating Surface of a Locomotive, 803. Wright, J. G., on Rivetted Joints, 637. A MAN UAL OF RULES, TABLES, AND DATA* FOR MECHANICAL ENGINEERS. GEOMETRICAL PROBLEMS. PROBLEMS ON STRAIGHT LINES. PROBLEM I. To bisect a straight line, or an arc of a circle, Fig. i. From the ends A, B, as centres, de- scribe arcs intersecting at c and D, and draw c D, which bisects the line, or the arc, at the point E or F. PROBLEM II. 72? draw a perpen- dicular to a straight line, or a radial line to a circular arc, Fig. i. Operate Fig. i. Probs. I. and II. as in the foregoing problem. The line CD is perpendicular to A B : the line c D is also radial to the arc A B. PROBLEM III. To draw a perpen- dicular to a straight line, from a given point in that line, Fig. 2. With any radius, from the given point A, in the line B c, cut the line at B and c ; with a longer radius describe arcs from B A ~Tc Fig. 2. Prob. III. and c, cutting each other at D, and draw the perpendicular D A. 2d Method, Fig. 3. From any cen- tre F, above BC, describe a circle passing through the given point A, Fig. 3. Prob. III. ad method. and cutting the given line at D ; draw D F, and produce it to cut the circle at E; and draw the perpendicular A E. GEOMETRICAL PROBLEMS $d Method, Fig. 4. From A de- scribe an arc E c, and from E, with the same radius, the arc A c, cutting E A Fig. 4. Prob. III. 3d method. the other at c ; through c draw a line E c D, and set oft" c D equal to c E ; and through D draw the perpendicu- lar A D. 4//z Method, Fig. 5. From the given point A set off a distance A E E 1 3 A Fig. 5. Prob. III. 4th method. equal to three parts, by any scale; and on the centres A and E, with radii of four and five parts respec- tively, describe arcs intersecting at c. Draw the perpendicular A c. Note. This method is most useful on very large scales, where straight edges are inapplicable. Any multi- ples of the numbers 3, 4, 5 may be taken with the same effect, as 6, 8, 10, or 9, 12, 15. PROBLEM IV. To draw a perpen- dicular to a straight line from any point without it, Fig. 6. From the point A, with a sufficient radius, cut the given line at Fanda; and from these points describe arcs cutting at E. Draw the perpendicular A E. Note. If there be no room below the line, the intersection may be taken above the line; that is to say, be- tween the line and the given point. 2d Method, Fig. 7. From any two points B, c, at some distance apart, Fig. 7 . Prob. IV. ad method. in the given line, and with the radii B A, c A, respectively, describe arcs cutting at A D. Draw the perpendi- cular A D. PROBLEM V. To draw a straight line parallel to a given line, at a given distance apart, Fig. 8. From the cen- Fig. 8.-Prob. V. tres A, B, in the given line, with the given distance as radius, describe arcs c, D; and draw the parallel line CD touching the arcs. PROBLEM VI. To draw a parallel through a given point, Fig. 9. With a radius equal to the distance of the ON STRAIGHT LINES. given point c from the given line A B, describe the arc D from B, taken Fig. 9. Prob. VI. considerably distant from c. Draw the parallel through c to touch the arc D. 2d Method, Fig. 10. From A, the A/; E F Fig. 10. Prob. VI. ad method. given point, describe the arc F D, cut- ting the given line at F; from F, with the same radius, describe the arc E A, and set off F D equal to E A. Draw the parallel through the points A, D. Note, Fig. n. When a series of parallels are required perpendicular to a base line A B, they may be drawn, as in Fig. i, through points in the base line, set off at the required dis- Fig. n. Prob. VI. tances apart. This method is con- venient also where a succession of parallels are required to a given line, CD; for the perpendicular A B may be drawn to it, and any number of par- allels may be drawn upon the per- pendicular. PROBLEM VII. To divide a straight line into a number of equal parts, Fig. 12. To divide the line AB into, say, five parts. From A and B draw par- allels A c, B D, on opposite sides. Set off any convenient distance four times A-"' JK" \ A"" \ \ \ Fig. 12. Prob. VII. (one less than the given number) from A on AC, and from B on B D ; join the first on AC to the fourth on B D, and so on. The lines so drawn divide A B as required. 2d Method, Fig. 13. Draw the line A c at an angle from A, set off, say, 2 3 4 Fig. 13. Prob. VII. sd method. five equal parts; draw B 5, and draw parallels to it from the other points of division in AC. These parallels divide A B as required. Note. By a similar process a line may be divided into a number of unequal parts; setting off divisions on A c, proportional by a scale to the required divisions, and drawing par- allels cutting A B. PROBLEM VIII. Upon a straight GEOMETRICAL PROBLEMS line to draw an angle equal to a given angle, Fig. 14. Let A be the given angle, and FG the line. With any radius, from the points A and F, de- scribe arcs D E, i H, cutting the sides of the angle A, and the line F G. Set Fig. 14. Prob. VIII. off the arc i H equal to D E, and draw F H. The angle F is equal to A, as required. To draw angles of 60 and 30, Fig. 15. From F, with any radius F i, de- scribe an arc i H ; and from i, with the same radius, cut the arc at H, and F x T Fig. 15. Prob. VIII. draw F H to form the required angle i F H. Draw the perpendicular H K to the base line, to form the angle of 30 F H K. To draw an angle of 45, Fig. 16. Set off the distance F i, draw the F I Fig. 16. Prob. VIII. perpendicular i H equal to i F, and join H F, to form the angle at F as re- quired. The angle at H is also 45. PROBLEM IX. To bisect an angle, Fig. 17. Let ACB be the angle; on the centre c cut the sides at A, B. On A and B, as centres, describe arcs cutting at D. Draw c D, dividing the angle into two equal parts. PROBLEM X. To bisect the inclina- tion of two lines, of which the intersec- tion is inaccessible, Fig. 18. Upon the Fig. 18. Prob. X. given lines c B, CH, at any points, draw perpendiculars E F, G H, of equal lengths, and through F and G draw parallels to the respective lines, cut- ting at s; bisect the angle FSG, so formed, by the line s D, which divides equally the inclination of the given lines. ON STRAIGHT LINES AND CIRCLES. PROBLEMS ON STRAIGHT LINES AND CIRCLES. PROBLEM XI. Through two given points to describe an arc of a circle with a given radius, Fig. 19. On the points Fig. 19. Prob. XI. A and B as centres, with the given radius, describe arcs cutting at c ; and from c, with the same radius, describe an arc A B as required. PROBLEM XII. To find the centre of a circle, or of an arc of a circle. ist, for a circle, Fig. 20. Draw the PROBLEM XIII. To describe a cir- cle passing through three given points, Fig. 21. Let A, B, c be the given points, and proceed as in last pro- Fig. 21. Prob. XII. XIII. blem to find the centre o, from which the circle may be described. Note. This problem is variously useful: in striking out the circular arches of bridges upon centerings, when the span and rise are given; describing shallow pans, or dished Fig. 20. Prob. XII. chord A B, bisect it by the perpendi- cular c D, bounded both ways by the circle ; and bisect c D for the centre G. 2d, for a circle or an arc, Fig. 21. Select three points, A, B, c, in the circumference, well apart; with the same radius, describe arcs from these three points, cutting each other; and draw the two lines, D E, F G, through their intersections, according to Fig. i. The point o, where they cut, is the centre of the circle or arc. Fig. 22. Prob. XIV. ist method. covers of vessels ; or finding the dia- meter of a fly-wheel or any other object of large diameter, when only a part of the circumference is ac- cessible. PROBLEM XIV. To describe a circle passing through three given points when the centre is not available. \st Method, Fig. 22. From the extreme points A, B, as centres, de- scribe arcs AH, EG. Through the third point c, draw A E, B F, cutting GEOMETRICAL PROBLEMS the arcs. Divide A F and B E into any number of equal parts, and set off a series of equal parts of the same length on the upper portions of the arcs beyond the points E, F. Draw straight lines, B L, B M, &c., to the divi- sions in A F; and A i, A K, &c., to the divisions in EG; the successive inter- sections N, o, &c., of these lines, are points in the circle required, between the given points A and c, which may be filled in accordingly: similarly the remaining part of the curve B c may be described. zd Method, Fig. 23. Let A, D,B be the given points. Draw A B, A D, D B, Fig. 23. Prob. XIV. 2d method. and ef parallel to A B. Divide D A into a number of equal parts at i, 2, 3, &c., and from D describe arcs through these points to meet ef. Divide the arc A e into the same number of equal parts, and draw straight lines from D to the points of division. The inter- sections of these lines successively with the arcs i, 2, 3, &c., are points in the circle which may be filled in as before. Note. The second method is not perfectly exact, but is sufficiently near to exactness for arcs less than one- fourth of a circle. When the middle point is equally distant from the ex- tremes, the vertical c D is the rise of the arc; and this problem is service- able for setting circular arcs of large radius, as for bridges of very great Fig. 24. Prob. XV. span, when the centre is unavailable; and for the outlines of bridge-beams, and of beams and connecting-rods of steam-engines, and the like. PROBLEM XV. To draw a tangent to a circle from a given point in the circumference, Fig. 24. Through the given point A, draw the radial line Fig. 25. Prob. XV. 2d method. A c, and the perpendicular F G is the tangent. 2d Method, when the centre is not available, Fig. 25. From A, set off equal segments A B, A D; join B D, and draw A E parallel to it for the tangent. PROBLEM XVI. To draw tangents to a circle from a point without it. Fig. 26. Prob. XVI. ist method. ist Method, Fig. 26. Draw AC from the given point A to the centre ON STRAIGHT LINES AND CIRCLES. c; bisect it at D, and from the centre D, describe an arc through c, cutting the circle at E, F. Then A E, A F, are tangents. 2d Method, Fig. 27. From A, with the radius A c, describe an arc BCD, and from c, with a radius equal to the Fig. 27. Prob. XVI. 2d method. diameter of the circle, cut the arc at B, D; join EC, CD, cutting the circle at E, F, and draw A E, A F, the tan- gents. Note. When a tangent is already drawn, the exact point of contact may be found by drawing a perpen- dicular to it from the centre. PROBLEM XVII. Between two in- clined lines to draw a series of circles touching these lines and touching each other, Fig. 28. Bisect the inclination Fig. 28. Prob. XVII. of the given lines A B, c D by the line NO. From a point P in this line, draw the perpendicular P B to the line A B, and on P describe the circle B D touching the lines and cutting the centre line at E. From E draw E F perpendicular to the centre line, cut- ting A B at F, and from F describe an arc E G, cutting A B at G. Draw G H parallel to B p, giving H, the centre of the next circle, to be described with the radius H E, and so on for the next circle i N. Inversely, the largest circle may be described first, and the smaller ones in succession. Note. This problem is of frequent use in scroll work. PROBLEM XVIII. Between two inclined lines to draw a circular seg- ment to fill the angle, and touching the lines, Fig. 29. Bisect the inclination F A Fig. 29. Prob. XVIII. of the lines A B, D E by the line F c, and draw the perpendicular A F D to define the limit within which the cir- cle is to be drawn. Bisect the angles A and D by lines cutting at c, and from c, with radius c F, draw the arc H F G as required. PROBLEM XIX. To describe a cir- cular arc joining two circles, and to touch one of them at a given point, Fig. 30. To join the circles A B, F G, by an arc touching one of them at F, draw the radius E F, and produce it both ways; set off FH equal to the radius AC of the other circle, join CH 8 GEOMETRICAL PROBLEMS and bisect it with the perpendicular L i, cutting E F at i. On the centre i, Fig. 30. Prob. XIX. with radius i F, describe the arc F A as required. PROBLEMS ON CIRCLES AND RECTILINEAL FIGURES. PROBLEM XX. To construct a tri- angle on a given base, the sides being given. i st. An equilateral triangle, Fig. 31. Fig. 31. -Prob. XX. On the ends of the given base, A, B, with A B as radius, describe arcs cut- ting at c, and draw AC, c B. 2d. A triangle of unequal sides, Fig. 32. On either end of the base A D, with the side B as radius, describe an arc ; and with the side c as radius, on the other end of the base as a centre, cut the arc at E. Join A E, D E. Note. This construction may be used for finding the position of a point, c or E, at given distances from the ends of a base, not necessarily to form a triangle. Fig. 32. Prob. XX. PROBLEM XXI. To construct a square or a rectangle on a given straight line. ist. A square, Fig. 33. On the Fig. 33. Prob. XXI. ends A, B, as centres, with the line A B as radius, describe arcs cutting at c; on c, describe arcs cutting the others at D E ; and on D and E, cut these at F G. Draw A F, EG, and join the in- tersections H, i. 2d. A rectangle, Fig. 34. On the base E F, draw the perpendiculars E H, Fig. 34. Prob. XXI. F G, equal to the height of the rect- angle, and join G H. When the centre lines, A B, CD, Fig- 35, of a square or a rectangle are given, cutting at E. Set off E F, EG, ON CIRCLES AND RECTILINEAL FIGURES. the half lengths of the figure, and E H, E T, the half heights. On the centres H, T, with a radius of half the length, M Fig. 35. Prob. XXI. describe arcs; and, on the centres F, G, with a radius of half the height, cut these arcs at K, L, M, N. Join these intersections. PROBLEM XXII. To construct a parallelogram, of which the sides and me of the angles are given, Fig. 36. Fig. 36. Prob. XXII. Draw the side D E equal to the given length A, and set off the other side D F equal to the other length B, form- ing the given angle c. From E, with D F as radius, describe an arc, and from F, with the radius D E, cut the arc at G. Draw F G, EG. Or, the remaining sides may be drawn as parallels to D E, D F. The formation of the angle D is readily done as indicated, by taking the straight length of the arc H i and c i as radius, and finding the inter- section L. PROBLEM XXIII. To describe a circle about a triangle, Fig. 37. Bisect two sides A B, A c of the triangle at E, F, and from these points draw per- pendiculars cutting at K. On the centre K, with the radius K A, draw the circle ABC. Fig. 37 . Prob. XXIII. PROBLEM XXIV. To inscribe a circle in a triangle, Fig. 38. Bisect two of the angles A, c, of the triangle by lines cutting at D; from D draw a perpendicular D E to any side, and with D E as radius describe a circle. When the triangle is equilateral, the centre of the circle may be found by bisecting two of the sides, and Fig. 38. Prob. XXIV. drawing perpendiculars as in the pre- vious problem. Or, draw a perpen- dicular from one of the angles to the opposite side, and from the side set off one-third of the perpendicular. Fig. 39. Prob. XXV. PROBLEM XXV. To describe a circle about a square, and to inscribe a square in a circle, Fig. 39. 10 GEOMETRICAL PROBLEMS i st. To describe the circle. Draw the diagonals A B, CD of the square, cutting at E; on the centre E, with the radius E A, describe the circle. 2d. To inscribe the square. Draw the two diameters A B, CD at right angles, and join the points A, B, c, D to form the square. Note. In the same way a circle may be described about a rectangle. PROBLEM XXVI. To inscribe a circle in a square, and to describe a square about a circle. Fig. 40. i st. To inscribe the circle. Draw Fig. 40. Prob. XXVI. the diagonals A B, c D of the square, cutting at E; draw the perpendicular E F to one side, and with the radius E F describe the circle. 2(1. To describe the square. Draw two diameters A B, c D at right angles, and produce them; bisect the angle D E B at the centre by the diameter F G, and through F and G draw per- Fig. 41. Prob. XXVII. pendiculars AC, ED, and join the points A D and B c, where they cut the diagonals, to complete the square. PROBLEM XXVII. To inscribe a pentagon in a circle, Fig. 41. Draw two diameters A c, B D at right angles, cutting at o; bisect A o at E, and from E, with radius E B, cut A c at F ; from B, with radius B F, cut the circumference at G, H, and with the same radius step round the circle to i and K; join the points so found to form the pentagon. PROBLEM XXVIII. To construct a hexagon upon a given straight line, Fig. 42. From A and B, the ends of the given line, describe arcs cutting at g\ from g, with the radius g A, de- F Fig. 42. Prob. XXVIII. scribe a circle; with the same radius set off the arcs A G, G F, and B D, D E. Join the points so found to form the hexagon. PROBLEM XXIX. To inscribe a hexagon in a circle, Fig. 43. Draw a diameter ACB; from A and B as centres, with the radius of the circle A c, cut the circumference at D, E, F, G; and draw AD, D E, &c. to form the hexagon. Fig. 43. Prob. XXIX. The points D, E, &c., may also be found by stepping the radius six times round the circle. ON CIRCLES AND RECTILINEAL FIGURES. II PROBLEM XXX. To describe a hex- agon about a circle, Fig. 44. Draw a Fig. 44. Prob. XXX. diameter ADB, and with the radius A D, on the centre A, cut the circum- ference at c; join AC, and bisect it with the radius D E: through E draw the parallel F G cutting the diameter at F, and with the radius D F describe the circle F H. Within this circle de- scribe a hexagon by the preceding problem; it touches the given circle. PROBLEM XXXI. To describe an octagon on a given straight line, Fig. 45. H k A. B 7 Fig. 45. Prob. XXXI. Produce the given line AB both ways, and draw perpendiculars A E, B F; bisect the external angles A and B, by the lines A H, B c, which make equal to A B. Draw c D and H G par- allel to A E, and equal to A B ; from the centres G, D, with the radius A B, cut the perpendiculars at E. F, and draw E F to complete the octagon. PROBLEM XXXII. To convert a square into an octagon, Fig. 46. Draw the diagonals of the square cutting at e; from the corners A, B, c, D, with A e as radius, describe arcs cutting the Fig. 46. Prob. XXXII. sides at g, h, &c.; and join the points so found to form the octagon. PROBLEM XXXIII. To inscribe an octagon in a circle, Fig. 47. Draw Fig. 47 . Prob. XXXIII. two diameters AC, B D at right angles ; bisect the arcs AB, BC, &c., at e,f, &c., and join A e, , &c., at L, M, N, &c. These are points in the circumference of the ellipse, and the curve may be traced through them. Points in the other half of the ellipse are formed by ex- tending the intersecting lines as indi- cated in the figure. PROBLEM XLI. To describe an ellipse approximately by means of cir- cular arcs. First, with arcs of two radii, Fig. 60. Find the difference i6 GEOMETRICAL PROBLEMS of the two axes, and set it off from the centre o to a and c, on o A and o c ; Fig. 60. Prob. XLI. draw a c, and set off half a c to d; draw d i parallel to a c, set ofT o e equal to od, join ei, and draw the parallels em, d m. From m, with radius m c, describe an arc through c ;and from i describe an arc through D ; from d and e describe arcs through A and B. The four arcs form the ellipse approximately. Note. This method does not ap- ply satisfactorily when the conjugate axis is less than two-thirds of the transverse axis. o M equal to c L, and on D describe an arc with radius DM; on A, with radius OL, cut this arc at a. Thus the five centres D, a, b, H, H' are found, from which the arcs are described to form the ellipse. Note. This process works well for nearly all proportions of ellipses. It is employed in striking out vaults and stone bridges. PROBLEM XLII. To draw a tan- Fig. 61. Prob. XLI. ad method. Second, with arcs of three radii, Fig. 6 1. On the transverse axis AB draw the rectangle B G, on the height o c ; to the diagonal A c draw the per- pendicular G H D ; set off OK equal to o c, and describe a semicircle on A K, and produce oc to L; set off Fig. 62. Prob. XLII. gent to an ellipse through a given point in the curve, Fig. 62. From the given point T draw straight lines to the foci F, F'J produce F T beyond the curve to c, and bisect the exterior angle c T F, by the line T d, which is the tangent. PROBLEM XLIII. To draw a tangent to an ellipse from a given point without the curve, Fig. 63. From the given point T, with a radius to the nearest focus F, de- scribe an arc on the other focus F', with a radius equal to the trans- verse axis, cut the arc at K L, and Fig. 63. Prob. XLIII. draw K F', L F', cutting the curve at M, N. The lines T M, T N are tangents. ON THE PARABOLA. PROBLEMS ON THE PARABOLA. A parabola, DAC, Fig. 64, is a curve such that every point in the curve is equally distant from the di- rectrix K L and the focus F. The focus lies in the axis A B drawn from the vertex or head of the curve A, so as to divide the figure into two equal parts. The vertex A is equidistant from the directrix and the focus, or A e= A F. Any line parallel to the axis is a diameter. A straight line, as E G or D c, drawn across the figure at right angles to the axis is a double ordinate, and either half of it is an ordinate. The ordinate to the axis E F G, drawn through the focus, is called the parameter of the axis. A segment of the axis, reckoned from the vertex, is an absciss of the axis; and it is an absciss of the ordinate drawn from the base of the absciss. Thus, A B is an absciss of the ordinate B c. Abscisses of a parabola are as the squares of their ordinates. PROBLEM XLIV. To describe a parabola when an absciss and its ordi- nate are given; that is to say, when the height and breadth are given, Fig. 64. Bisect the given ordinate 7 Fig. 64. Prob. XLIV. B c at a ; draw A a, and then a b per- pendicular to it, meeting the axis at b. Set off A e, A F, each equal to B b; and draw K e L perpendicular to the axis. Then K L is the directrix and F is the focus. Through F and any number of points, _2? a, b c d J. Fig. 66. Prob. XLIV. sd method. the given axis, and c D a double ordi- nate or base; to describe a parabola 2 18 GEOMETRICAL PROBLEMS of which the vertex is at A. Through A draw E F parallel to c D, and through c and D draw c E and D F parallel to the axis. Divide B c and B D into any number of equal parts, say five, at #, b, &c., and divide c E and D F into the same number of parts. Through the points a, b, c, d in the base c D, on each side of the axis, draw perpendiculars, and through a, b, c, d, in c E and D F, draw lines to the vertex A, cutting the perpendicu- lars at e,f, -, h. These are points in the parabola, and the curve c A D may be traced as shown, passing through them. PROBLEMS ON THE HYPERBOLA. The vertices A, B, Fig. 67, of oppo- site hyperbolas, are the heads of the curves, and are points in their centre or axial lines. The transverse axis A B is the distance between the ver- tices, of which the centre c is the centre. The conjugate axis G H is a straight line drawn through the centre at right angles to the transverse axis. An ordinate F K is a straight line drawn from any point of the curve perpendicular to the axis. The seg- ments of the transverse axis A F, B F, between an ordinate F K and the ver- tices of the curves, are abscisses. The parameter is the double ordinate drawn through the focus. The asymptotes are two straight lines, s s, R R, drawn from the centre through the ends of a tangent ED at the vertex, equal and parallel to the conjugate axis, and bisected by the transverse axis. The nature of the hyperbola is such that the difference of the distances of any point in the curve from the foci is always the same, and is equal to the transverse axis. In a hyperbola the squares of any two ordinates to the transverse axes are to each other as the rectangles of their abscisses. PROBLEM XLV. To describe a hyperbola, the transverse and conjugate axes being given. Fig. 67. Draw AB Fig. 67. Prob. XLV. equal to the transverse axis, and D E perpendicular to it and equal to the conjugate G H. On c, with the radius c E, describe a circle cutting A B pro- duced, at F/; these points are the foci. In A B produced take any number of points o, o, &c., with the radii KO, BO, and on centres F,/ describe arcs cut- ting each other at n, n, &c. These are points in the curve, through which it may be traced. 2d Method, Fig. 67. The curve may be drawn thus: Let the ends of two threads/p Q, F p Q, be fastened at the points /, F, and be made to pass through a small bead or pin p, and knotted together at Q. Take hold of Q, and draw the threads tight; move the bead along the threads, and the point P will describe the curve. If the end of the long thread be fixed at F, and the short thread at f, the opposite curve may be described in the same manner. Or, the line /Q may be replaced by a straight-edge turning on a pin at/ and the cord F Q joined to it at Q. The curve may then be described by means of a point or pencil in the same manner as for the parabola, Fig. 65. $d Method ; when the breadth c D, ON THE HYPERBOLA, CYCLOID, EPICYCLOID. A B, and transverse axis A A.' of the curve are given. Fig. 68. Divide Os O v & -B Ct> ~fa O ct, J) Fig. 68. Prob. XLV. 3 d method. the base or double ordinate c D into a number of equal parts on each side of the axis at a, b, &c. ; and divide the parallels c E, D F, into the same number of equal parts at a, b, &c. From the points a, b, &c., in the base, draw lines to A', and from the points a, b, &c., in the verticals, draw lines to A, cutting the respective lines from the base. Trace the curve through the intersections thus obtained. THE CYCLOID AND EPICYCLOID. PROBLEM XLVI. To describe a cycloid, Fig. 69. When a wheel or a circle D G c rolls along a straight line A B, Fig. 69, beginning at A and end- ing at B, where it has just completed one revolution, it measures off a straight line A B exactly equal to the circumference of the circle D G c, which is called the generating circle, and a point or pencil fixed at the point D in the circumference traces out a curvilinear path A D B, called a cycloid. A B is the base and c D is the axis of the cycloid. Place the generating circle in the middle of the cycloid, as in the figure, draw a line E H parallel to the base, cutting the circle at G; and the tan- gent H i to the curve at the point H. Then the following are some of the properties of the cycloid : The horizontal line H G = arc of the circle G D. The half-base A c = the half-circum- ference CGD. The arc of the cycloid D H = twice the chord D G. The half-arc of the cycloid D A = twice the diameter of the circle D c. Or, the whole arc of the cycloid A D B = four times the axis c D. The area of the cycloid A D B A = three times the area of the generating circle D c. The tangent H i is parallel to the chord G D. PROBLEM XLVII. To describe an Fig. 7 o.-Prob. XLVII. exterior epicycloid, Fig. 70. The epicy- cloid differs from the cycloid in this, 20 GEOMETRICAL PROBLEMS that it is generated by a point D in one circle D c rolling upon the cir- cumference of another circle A c B, instead of on a flat surface or line; the former being the generating circle, and the latter the fundamental circle. The generating circle is shown in four positions, in which the generating point is successively marked D, D'. D", D'". A D'" B is the epicycloid. PROBLEM XLVIIL To describe Fig. 71. Prob. XLVIIL an interior epicycloid, Fig. 71. If the generating circle be rolled on the in- side of the fundamental circle, as in Fig. 71, it forms an interior epicycloid, or hypocycloid, A D'" B, which becomes in this case nearly a straight line. The other points of reference in the figure correspond to those in Fig. 70. When the diameter of the generating circle is equal to half that of the fun- damental circle, the epicycloid be- comes a straight line, being in fact a diameter of the larger circle. THE CATENARY. When a perfectly flexible string, or a chain consisting of short links, is suspended from two points M, N, Fig. 72, it is stretched by its own weight, and it forms a curve line known as the catenary, M c N. The point c, where the catenary is horizontal, is the vertex. PROBLEM XLIX. To describe a catenary, Fig. 72. Draw the vertical c G equal to -the length of the arc of the chain, M c, on one side of the vertex, and divide it into a great number of equal parts,at ( i ), ( 2), (3),&c. Draw the horizontal line c H equal to the length of so much of the rope or chain as measures by its weight the horizontal tension of the chain. From the point c as the vertex, set off c ( i) on the horizontal line equal to c i" on the vertical; and (i) (2) from the point (i), parallel to H i and equal to c (i); and again (2) (3) from the point (2) parallel to H 2 and equal to c (i); and so on till the last segment (6) M is drawn parallel to H G. The poly- gon c (i) (2) (3) . . . M, thus formed, is approximately the catenary curve, which may be traced through the middle points of the sides of the polygon. A similar process being performed for the other side of the curve, the catenary is completed. Fig. 72. Prob. XLIX. 2(t Method. Suspend a finely linked chain against a vertical wall. The curve may be traced from it, on the wall, answering the conditions of given length and height, or of given width or length of arc. A cord having numerous equal weights suspended from it at short and equal distances may be used. CIRCLES, PLANE TRIGONOMETRY. 21 CIRCLES. The circumference of a circle is commonly signified in mathematical discussions by the symbol T, which indicates the length of the circumfer- ence when the diameter = i. The area of a circle is as the square of the diameter, or the square of the circumference. The ratio of the diameter to the circumference is as- 1 to 3*141593 commonly abbreviated, as i to 3*1416 approximately, as i to 3^ or as 7 to 22 When the diameter = i, the area is equal to 785398 + or, commonly abbreviated, 7854 approximately, -fths. When the circumference = i, the area is equal to ' 795 7 7 + or, abbreviated, '0796 approximately, Aths, or -08. In these ratios, the diameter and the circumference are taken lineally, and the area superficially. So that if the diameter = i foot, the circum- ference is equal to '3*1416 feet, and the area is equal to "7854 square foot. Note. If the first three odd figures, i, 3, 5, be each put down twice, the first three of these will be to the last three, that is 113 is to 355, as the diameter to the circumference. PLANE TRIGONOMETRY. The circumference of a circle is supposed to be divided into 360 degrees or divisions, and as the total angularity about the centre is equal to four right angles, each right angle contains 90 degrees, or 90, and half a right angle Fig- 73- Definitions in Plane Trigonometry. contains 45. Each degree is divided into 60 minutes, or 60'; and, for the sake of still further minuteness of measurement, each minute is divided into 60 seconds, or 60". In a whole circle there are, therefore, 360 x 60 x 60 = 22 GEOMETRICAL PROBLEMS. 1,296,000 seconds. The annexed diagram, Fig. 73, exemplifies the rela- tive positions of the sine, cosine, versed sine, tangent, co-tangent, secant, and co-secant of an angle. It may be stated, generally, that the correlated quantities, namely, the cosine, co-tangent, and co-secant of an angle, are the sine, tangent, and secant, respectively, of the complement of the given angle, the complement being the difference between the given angle and a right angle. The supplement of an angle is the amount by which it is less than two right angles. When the sines and cosines of angles have been calculated (by means of formulas which it is not necessary here to particularize), the tangents, co-tan- gents, secants, and co-secants are deduced from them according to the following relations : rad. x sin. rad. 2 rad.'-' rad. 2 tan. = - ; cotan. = - j sec. = ; cosec. = . cos. tan. cos. sin. For these the values will be amplified in tabular form. A triangle consists of three sides and three angles. When any three of these are given, including a side, the other three may be found by cal- culation : CASE i. When a side and its opposite angle are two of the given parts. RULE i. To find a side, work the following proportion: as the sine of the angle opposite the given side is to the sine of the angle opposite the required side, so is the given side to the required side. RULE 2. To find an angle: as the side opposite to the given angle is to the side opposite to the required angle, so is the sine of the given angle to the sine of the required angle. RULE 3. In a right-angled triangle, when the angles and one side next the right angle are given, to find the other side: as radius is to the tangent of the angle adjacent to the given side, so is this side to the other side. CASE 2. When two sides and the included angle are given. RULE 4. To find the other side: as the sum of the two given sides is to their difference, so is the tangent of half the sum of their opposite angles to the tangent of half their difference add this half difference to the half sum, to find the greater angle; and subtract the half difference from the half sum, to find the less angle. The other side may then be found by Rule i. RULE 5. When the sides of a right-angled triangle are given, to find the angles: MENSURATION OF SURFACES. 23 as one side is to the other side, so is the radius to the tangent of the angle adjacent to the first side. CASE 3. When the three sides are given. RULE 6. To find an angle. Subtract the sum of the logarithms of the sides which contain the required angle, from 20; to the remainder add the logarithm of half the sum of the three sides, and that of the difference between this half sum and the side opposite to the required angle. Half the sum of these three logarithms will be the logarithmic cosine of half the required angle. The other angles may be found by Rule i. RULE 7. Subtract the sum of the logarithms of the two sides which con- tain the required angle, from 20, and to the remainder add the logarithms of the differences between these two sides and half the sum of the three sides. Half the result will be the logarithmic sine of half the required angle. Note. In all ordinary cases either of these rules gives sufficiently accur- ate results. It is recommended that Rule 6 should be used when the required angle exceeds 90; and Rule 7 when it is less than 90. MENSURATION OF SURFACES. To find the area of a parallelogram. Multiply the length by the height, or perpendicular breadth. Or, multiply the product of two contiguous sides by the natural sine of the included angle. To find the area of a triangle. Multiply the base by the perpendicular height, and take half the product. Or, multiply half the product of two contiguous sides by the natural sine of the included angle. To find the area of a trapezoid. Multiply half the sum of the parallel sides by the perpendicular distance between them. To find the area of a quadrilateral inscribed in a circle. From half the sum of the four sides subtract each side severally; multiply the four re- mainders together; the square root of the product is the area. To find the area of any quadrilateral figure. Divide the quadrilateral into two triangles; the sum of the areas of the triangles is the area. Or, multiply half the product of the two diagonals by the natural sine of the angle at their intersection. Note. As the diagonals of a square and a rhombus intersect at right angles (the natural sine of which is i), half the product of their diagonals is the area. To find the area of any polygon. Divide the polygon into triangles and trapezoids by drawing diagonals ; find the areas of these as above shown, for the area. To find the area of a regular polygon. Multiply half the perimeter of the polygon by the perpendicular drawn from the centre to one of the sides. Note. To find the perpendicular when the side is given 2 4 GEOMETRICAL PROBLEMS. as radius to tangent of half-angle at perimeter (see table No. i), so is half length of side to perpendicular. Or, multiply the square of a side of any regular polygon by the corres- ponding area in the following table: TABLE No. i. ANGLES AND AREAS OF REGULAR POLYGONS. NAME. Number of Sides. One half Angle at the Perimeter. Area. (Side=i) Perpendi- cular. (Side=i) Equilateral triangle ? O'A.'Z'ZO 0-2887 Square, . 3 4' o w 4 c * ^TOO W I 'OOOO O' '(OOO Pentagon, $ 54 I'72O^ 0*6882 Hexagon, 6 60 2x081 o'S66o Heptagon, 7 64? * Jy^ JJ 2*6770 i '0787 Octagon, 8 67A 4-8284 w o"o I '2O71 Nonagon Q 70 6'i8i8 I *7 1 "2 1 Decagon, y IO 72 7'6oA2 1 61 61 T'^88 Undecagon 1 1 7V 3 7 vjy^^s Q'76^6 1 JO 00 I '7028 Dodecagon, I 2 / O 1 1 7 c y o^D^ 1 1 '1962 I '8660 / J To find the circumference of a circle. Multiply the diameter by 3 '141 6. Or, multiply the area by 12 '5664; the square root of the product is the circumference. To find the diameter of a circle. Divide the circumference by 3 '141 6. Or, multiply the circumference by '3183. Or, divide the area by 7854; the square root of the quotient is the diameter. To find the area of a circle. Multiply the square of the diameter by 7854. Or, multiply the circumference by one-fourth of the diameter. Or, multiply the square of the circumference by '07958. To find the length of an arc of a circle. Multiply the number of degrees in the arc by the radius, and by '01745. Or, the length may be found nearly, by subtracting the chord of the whole arc from eight times the chord of half the arc, and taking one-third of the remainder. To find the area of a sector of a circle. Multiply half the length of the arc of the sector by the radius. Or, multiply the number of degrees in the arc by the square of the radius, and by '008727. To find the area of a segment of a circle. Find the area of the sector which has the same arc as the segment; also the area of the triangle formed by the radial sides of the sector and the chord of the arc; the difference or the sum of these areas will be the area of the segment, ac- cording as it is less or greater than a semicircle. To find the area of a ring included between the circumferences of two con- MENSURATION OF SURFACES. centric circles. Multiply the sum of the diameters by their difference, and by 7854- To find the area of a cycloid. Multiply the area of the generating circle by 3- To find the length of an arc of a parabola, cut off by a double ordinate to the axis. To the square of the ordinate add four-fifths of the square of the absciss ; twice the square root of the sum is the length nearly. Note. This rule is an approximation which applies to those cases only in which the absciss does not exceed half the ordinate. To find the area of^ a parabola. Multiply the base by the height; two- thirds of the product is the area. To find the circumference of an ellipse. Multiply the square root of half the sum of the squares of the two axes by 3*1416. To find the area of an ellipse. Multiply the product of the two axes by 7854- Note. The area of an ellipse is equal to the area of a circle of which the diameter is a mean proportional between the two axes. To find the area of an elliptic segment, the base of which is parallel to either axis of the ellipse. Divide the height of the segment by the axis of which it is a part, and find the area of a circular segment, by table No. VII. , of which the height is equal to this quotient; multiply the area thus found by the two axes of the ellipse successively; the product is the area. To find the length of an arc of a hyperbola, beginning at the vertex. To 19 times the transverse axis add 21 times the parameter to this axis, and multiply the sum by the quotient of the absciss divided by the transverse. 2d. To 9 times the transverse add 2 1 times the parameter, and multiply the sum by the quotient of the absciss divided by the transverse. 3d. To each of these products add 1 5 times the parameter, and then as the latter sum is to the former sum, so is the ordinate to the length of the arc, nearly. To find the area of a hyperbola. To the product of the transverse and absciss add five-sevenths of the square of the absciss, and multiply the square root of the sum by 21; to this product add 4 times the square root of the product of the transverse and absciss; multiply the sum by 4 times the product of the conjugate and absciss, and divide by 75 times the transverse. The quotient is the area nearly. To find the area of any curvilineal figure, bounded at the ends by parallel straight lines, Fig. 74. Divide the length of the figure a b into any even number of equal parts, and draw ordinates c, d, e, &c., through the points of division, to touch the boundary lines. Add together the first and last ordinates (c and k), and call the sum A; add together the even ordinates (that is, Fi s- 7 4--For Area of Curvilinear Figure. d,f<> h)j\ and call the sum B; add together the odd ordinates, except the first and last (e,g, i}, and call the sum c. Let D be the common distance of the ordinates, then 26 GEOMETRICAL PROBLEMS. (A + 4 B + 2 c) x D = area of figure. This is known as Simpson's Rule. zd Method, Fig. 74. Having divided the figure into an even or an odd number of equal parts, add together the first and last ordinates, making the sum A; and add together all the intermediate ordinates, making the sum B. Let L = the length of the figure, and n = the number of divisions, then A + 2 B 2 n x L = area of figure. That is to say, twice the sum of the intermediate ordinates, plus the first and last ordinates, divided by twice the number of divisions, and multi- plied by the length, is equal to the area of the figure. This method is that commonly used; it is sufficiently near to exactness for most purposes. $d Method, Fig. 74. Having divided the figure as above, measure by a scale the mean depth of each division, at the middle of the division; add together the depths of all the divisions, and divide the sum by the number of divisions, for the average depth; multiply the average depth by the length, which gives the area. For the sake of obtaining a more nearly exact result, the figure may be divided into two half-parts, c, k, Fig. 75, one at each end, and a number of whole equal parts, d,e,f,g,h,i,j, intermediately. Then the ordinates separating these parts, excluding the extreme ordinates, may be measured rT x*-* . 7 c * . f ^ *. ,; 7 t Fig. 75- For Area Fig. 76. direct, and the sum of the measurements divided by the number of them, and multiplied by the length, for the area. Note. In dealing with figures of excessively irregular outline, as in Fig. 76, representing an indicator-diagram from a steam-engine, mean lines, ab, cd, may be substituted for the actual lines, being so traced as to intersect the undulations, so that the total area of the spaces cut off may be com- pensated by that of the extra spaces inclosed. Note 2. The figures have been supposed to be bounded at the ends by parallel planes. But they may be terminated by curves or angles, as in Fig. 76, at b, when the extreme ordinates become nothing. MENSURATION OF SOLIDS. 2/ MENSURATION OF SOLIDS. To find the surface of a prism or a cylinder. The perimeter of the end multiplied by the height gives the upright surface; add twice the area of an end. To find the cubic contents of a prism or a cylinder. Multiply the area of the base by the height. To find the surface of a pyramid or a cone. Multiply the perimeter of the base by half the slant height, and add the area of the base. To find the cubic contents of a pyramid or a cone. Multiply the area of the base by one-third of the perpendicular height. To find the surface of a frustum of a pyramid or a cone. Multiply the sum of the perimeters of the ends by half the slant height, and add the areas of the ends. To find the cubic contents of a frustum of a pyramid, or a cone. Add together the areas of the two ends, and the mean proportional between them (that is, the square root of their product), and multiply the sum by one-third of the perpendicular height. Or, when the ends are circles, add together the square of each diameter, and the product of the diameters, and multiply the sum by 7854, and by one-third of the height. To find the cubic contents of a wedge. To twice the length of the base add the length of the edge; multiply the sum by the breadth of the base, and by one-sixth of the height. To find the cubic contents of a prismoid (a solid of which the two ends are un- egual but parallel plane figures of the same number of sides}. To the sum of the areas of the two ends, add four times the area of a section parallel to and equally distant from both ends; and multiply the sum by one-sixth of the length. Note. This rule gives the true content of all frustums, and of all solids of which the parallel sections are similar figures; and is a good approxima- tion for other kinds of areas and solidities. To find the surface of a sphere. Multiply the square of the diameter by 3-1416. Note. The surface of a sphere is equal to 4 times the area of one of its great circles. 2. The surface of a sphere is equal to the convex surface of its circum- scribing cylinder. 3. The surfaces of spheres are to one another as the squares of their diameters. To find the curve surf ace of any segment or zone of a sphere. Multiply the diameter of the sphere by the height of the zone or segment, and by 3 '141 6. Note. The curve surfaces of segments or zones of the same sphere are to one another as their heights. To find the cubic contents of a sphere. Multiply the cube of the diameter by -5236. Or, multiply the surface by one-sixth of the diameter. 28 GEOMETRICAL PROBLEMS. Note. The contents of a sphere are two-thirds of the contents of its circumscribing cylinder. 2. The contents of spheres are 'to one another as the cubes of their diameters. To find the cubic contents of a segment of a sphere. From 3 times the diameter of the sphere subtract twice the height of the segment; multiply the difference by the square of the height, and by '5236. Or, to 3 times the square of the radius of the base of the segment, add the square of its height; and multiply the sum by the height, and by '5236. To find the cubic contents of a frustum or zone of a sphere. To the sum of the squares of the radii of the ends add ^3 of the square of the height ; multiply the sum by the height, and by 1-5708. To find the cubic contents of a spheroid. Multiply the square of the re- volving axis by the fixed axis and by '5236. Note. The contents of a spheroid are two-thirds of the contents of its circumscribing cylinder. 2. If the fixed and revolving axes of an oblate spheroid be equal to the revolving and fixed axes of an oblong spheroid respectively, the contents of the oblate are to those of the oblong spheroid as the greater to the less axis. To find the cubic contents of a segment of a spheroid. ist. When the base is parallel to the revolving axis. Multiply the difference between thrice the fixed axis and double the height of the segment, by the square of the height, and the product by "5236. Then, as the square of the fixed axis is to the square of the revolving axis, so is the last product to the content of the segment. 2d. When the base is perpendicular to the revolving axis. Multiply the difference between thrice the revolving axis and double the height of the segment, by the square of the height, and the product by '5236. Then, as the revolving axis is to the fixed axis, so is the last product to the content of the segment. To find the solidity of the middle frustum of a spheroid. ist. When the ends are circular, or parallel to the revolving axis. To twice the square of the middle diameter, add the square of the diameter of one end; multiply the sum by the length of the frustum, and the product by "2618 for the content. 2d. When the ends are elliptical, or perpendicular to the revolving axis. To twice the product of the transverse and conjugate diameters of the middle section, add the product of the transverse and conjugate diameters of one end; multiply the sum by the length of the frustum, and by '2618 for the content. To find the cubic contents of a parabolic conoid. Multiply the area of the base by half the height. Or, multiply the square of the diameter of the base by the height, and by -3927. To find the cubic contents of a frustum of a parabolic conoid. Multiply half the sum of the areas of the two ends by the height of the frustum. MENSURATION OF SOLIDS. 2 9 Or, multiply the sum of the squares of the diameters of the two ends by the height, and by '3927. To find the cubic contents of a parabolic spindle, Multiply the square of the middle diameter by the length, and by '41888. To find the cubic contents of the middle frustum of a parabolic spindle. Add together 8 times the square of the largest diameter, 3 times the square of the diameter at the ends, and 4 times the product of the diameters; multiply the sum by the length of the frustum, and by '05236. To find the surface and the cubic contents of any of the five regular solids, Figs. Fig. 77. Fig. 78 Fig. 80. Fig. 81. 77) 7 8 > 79? 8o > 8 1. For the surface, multiply the tabular area below, by the square of the edge of the solid. For the contents, multiply the tabular contents below, by the cube of the given edge. Note. A regular solid is bounded by similar and regular plane figures. There are five regular solids, shown by Figs. 77 to 81, namely: The tetrahedron, bounded by four equilateral triangles. The hexahedron, or cube, bounded by six squares. The octahedron, bounded by eight equilateral triangles. The dodecaAedronJxyvs&ed. by twelve pentagons. The icosahedron, bounded by twenty equilateral triangles. Regular solids may be circumscribed by spheres; and spheres may be inscribed in regular solids. SURFACES AND CUBIC CONTENTS OF REGULAR SOLIDS. Number of sides. Name. AREA. Edge = i. CONTENTS. Edge = i. 4 6 8 Tetrahedron Hexahedron Octahedron 17320 6'oooo 3*464.1 0*1178 I *OOOO 0*4.714 12 Dodecahedron 20*64=18 7*6631 2O Icosahedron. . 8*6603 2*1817 To find the cubic contents of an irregular solid. Suppose it divided into parts, resembling prisms or other bodies measurable by preceding rules; find the content of each part; the sum of the contents is the cubic contents of the solid. Note. The content of a small part is found nearly by multiplying half the sum of the areas of each end by the perpendicular distance between them. 30 GEOMETRICAL PROBLEMS. Or, the contents of small irregular solids may sometimes be found by im- mersing them under water in a prismatic or cylindrical vessel, and observing the amount by which the level of the water descends when the solid is withdrawn. The sectional area of the vessel being multiplied by the descent of the level, gives the cubic contents. Or, when the solid is very large, and a great degree of accuracy is not requisite, measure its length, breadth, and depth in several different places, and take the mean of the measurement for each dimension, and multiply the three means together. Or, when the surface of the solid is very extensive, it is better to divide it into triangles, to find the area of each triangle, and to multiply it by the mean depth of the triangle for the contents of each triangular portion ; the contents of the triangular sections are to be added together. The mean depth of a triangular section is obtained by measuring the depth at each angle, adding together the three measurements, and taking one-third of the sum. MENSURATION OF HEIGHTS AND DISTANCES. To find the height of an accessible object. Measure the distance from the base of the object to any convenient station on the same horizontal plane; and at this station take the angle of altitude. Then as radius to tangent of the angle of altitude, so is the horizontal distance to the height of the object above the horizontal plane passing through the eye of the observer. Add the height of the eye, and the sum is the height of the object. Note. The station should be chosen so that the angle of altitude should be as near to 45 as practicable; because the nearer to 45, the less is the error in altitude arising from error of observation. When the angle of elevation is 45, the height above the plane of the eye is equal to the distance. When it is 26 34', the height is half the dis- tance. To find approximately the height of an accessible object. There are four methods based on the principle of similar triangles, i st By a geometrical square, Fig. 82. This is a square, a b, with two sights on one of its sides, a n, a plumb-line hung from one extremity, n, of that side, and each of the two sides opposite to that extremity, mb,ma, divided into 100 equal parts; the division beginning at the remote ends, so that the looth divisions meet at the corner m. Let re be the object, and the sights be directed to the summit e, at the known distance ad. When the '~He e hT ati n f a plummet cuts the side b m at, say, c, then by similar triangles, n b :b c :\ a d : d e. Or, if the plumb-line cuts the side a m, then the part of a m cut off is to a n : : a d : de. Adding to de the height of the eye rd, the sum is the height of the object, re. MENSURATION OF HEIGHTS AND DISTANCES. 2d. By shadows. Fig. 83. When the sun shines, fix a pole be in the ground, vertically, and measure its shadow ab. Measure also the shadow de Fig. 83. Mensuration of a Height. Fig. 84. of the object e m; then, by similar triangles, a b : b c : : de : e m, the height of the object. 3d. By reflection, Fig. 84. Place a basin of water, or any horizontal reflecting surface, at a, level with the base of the object de, and retire from it till the eye at c sees the top of the object e, in the centre of the basin at a. Then, by similar triangles, ab\ be \ \ad\de. 4th. By two poles, Fig. 85. Fix two poles a m, en, of unequal lengths, parallel to the object er, so that the eye of the observer at a, the top of the shorter pole, may see c, the top of the longer pole, in a line with e, the summit of the object re. By similar triangles, a b : b c : : a d : de; and adding r d, the height of the eye, to de, the sum re is the height of the object. To find the distance of the visible horizon. To half the logarithm of the height of the eye, add 3-8105; the sum is the logarithm of the distance in feet, nearly. To find the distance of an object by the motion of sound. Multiply the number of seconds that elapse between the flash or other sign of the gene- ration of the sound and the arrival of the sound to the ear, by 1120. The product is the distance in feet. Note. When a sound generated near the ear returns as an echo, half the interval of time is to be taken, to find the distance of the reflecting surface. Fig. 85. Mensuration of a Height. 32 MATHEMATICAL TABLES. MATHEMATICAL TABLES. TABLE No. I. OF LOGARITHMS OF NUMBERS FROM i TO 10,000. Logarithms consist of integers and decimals; but, for the sake of com- pactness, the integers have been omitted in the table, except in the short preliminary section containing the complete logarithms of numbers from i to 100. The table No. I. contains the decimal parts, to six places, of the loga- rithms of numbers from i to 10,000. The integer, or index, or character- istic of a logarithm, standing on the left-hand side of the decimal point, is a number less by i than the number of figures or places in the integer of the number. If a number contains both integers and decimals, the index is regulated according to the integers. If it contain only decimals, the index is equal to the number of cyphers next the decimal point, plus i; moreover, the index is negative, and is so distinguished by the sign minus, - , written over it. For example, to illustrate the adjustment of the integer of the logarithm to the composition of the number : Number. Logarithm. 4743 3-676053 474-3 2.676053 47-43 1-676053 4-743 0.676053 -4743 1-676053 04743 ^.676053 004743 3-676053 Still more for the sake of compactness, the first two figures of the loga- rithms are given only at the beginning of each line of logarithms, to save repetition, only the remaining four decimal places being given for each logarithm. In seeking for a logarithm, the eye readily takes in the prefixed two digits at the commencement of each line. Rules. To find the logarithm of a number containing one or two digits, look for the number in the preliminary tablet in one of the columns marked No., and find the logarithm next it. Or, look in the body of the table for the given number in the columns marked N, with one or two cyphers following it; the decimal part of the logarithm is in the column next to it. For example, the decimal part of the logarithm of 3 is found, in the column next to the number 300, to be .477121, and as there is but one digit, the logarithm is completed with a cypher, thus, 0.477121. The same logarithm stands for 30, except that, when completed, it becomes 1.477121. Again, take the number 37; look for 370 in column N, and the decimal part of the logarithm is found, in the column next it, to be .568202, which, being completed, becomes 1.568202. If the number be .37, the logarithm becomes 1.568202. To find the logarithm of a number consisting of three digits, look for the EXPLANATION AND USES OF THE TABLES. 33 number in column N, and find the logarithm in the column next it, as already exemplified, for which the index is to be settled and prefixed as before. If the number consist of four digits, look for the first three in column N, and the fourth in the horizontal line at the head or at the foot of the table. The decimal part of the logarithm is found opposite the three first digits and under or over the fourth. Take the number 5432; opposite 543 in column N, and in the column headed 2, is the logarithm .734960, to which 3 is to be" prefixed, making 3.734960. If the number be 5.432, the complete logarithm is 0.734960. If the number consist of five or more digits, find the logarithm for the first four as above; multiply the difference, in column D, by the remaining digits, and divide by 10 if there be only one digit more, by 100 if there be two more, and so on; add the quotient to the logarithm for the first four. The sum is the decimal part of the required logarithm, to which the index is to be prefixed. For example, take 3.1416. The logarithm of 3141 is .497068, decimal part; and the difference, 138 x 6 -=- 10 = 83, is to be added, thus 0.497068 making the complete logarithm, .................. 0.497151 To find the number corresponding to a given logarithm, look 'fce W logarithm without the index. If it be found exactly or within two or &re$V/x< units of the right-hand digit, then the first three figures of the indicated number will be found in the number column, in a line with the logarithm, and the fourth figure at the top or the foot of the column containing the logarithm. Annex the fourth figure to the first three, and place the decimal point in its proper position, on the principles already explained. If the given logarithm differs by more than two or three units from the nearest in the table, find the number for the next less tabulated logarithm, which will give the four first digits of the required number. To find the fifth and sixth digits, subtract the tabulated logarithm from the given loga- rithm, add two cyphers, and divide by the difference found in column D opposite the logarithm. Annex the quotient to the four digits already found, and place the decimal point. For example, to find the number represented by the logarithm 2.564732: 2-564732 given logarithm. Log ................... 367.0= ......... 2.564666 nearest less. D 118)6600 (56 nearly. 590 700 708 Showing that the required number is 367.056. To multiply together two or more numbers, add together the logarithms 3 34 MATHEMATICAL TABLES. of the numbers, and the sum is the logarithm of the product. Thus, to multiply 365 by 3.146: Log 365 = 2.562293 Log 3.146 =0.497759 3.060052 Log 1148 3- 59942 29 D 380)11000 (29 nearly. 760 1148.29 3400 3420 Showing that the product is 1148.29. To divide one number by another, subtract the logarithm of the divisor from that of the dividend, and the remainder is the logarithm of the quotient. To find any power of a given number, multiply the logarithm of the num- ber by the exponent of the power. The product is the logarithm of the power. To find any root of a given number, divide the logarithm of the number by the index of the root. The quotient is the logarithm of the root. To find the reciprocal of a number, subtract the decimal part of the logarithm of the number from o.oooooo; add i to the index of the loga- rithm, and change the sign of the index. This completes the logarithm of the reciprocal. For example, to find the reciprocal of 230: o.oooooo Log 230 = 2.361728 3.638272 = log 0.004348 (reciprocal). Inversely, to find the reciprocal of the decimal .004348: o.oooooo Log .004348 = 3^638272 2.361728 = ^ 230 (reciprocal). Note. It will be found in practice, for the most part, unnecessary to note the indices of logarithms, as the decimal parts are in most cases suffi- ciently indicative of the numbers without the indices. The exact calcula- tion of differences may also in most cases be dispensed with rough mental approximations being sufficiently near for the purpose particularly when the numbers contain decimals. The indices are, however, indispensable in the calculation of the roots of numbers. EXPLANATION AND USES OF THE TABLES. 35 TABLE No. II. OF HYPERBOLIC LOGARITHMS OF NUMBERS. In this table, the numbers range from i.oi to 30, advancing by .01, up to the whole number 10; and thence by larger intervals up to 30. The hyperbolic logarithms of numbers, or Neperian logarithms, as they are sometimes called, are calculated by multiplying the common logarithms of the given numbers, in table No. I., by the constant multiplier, 2.302585. The hyperbolic logarithms of numbers intermediate between those which are given in the table, may be readily obtained by interpolating proportional differences. TABLE No. III. OF CIRCUMFERENCES, CIRCULAR AREAS, SQUARES AND CUBES; AND OF SQUARE ROOTS AND CUBE ROOTS. It has been shown how to calculate the powers and roots of numbers by means of logarithms. The table No. III. will be useful for reference. It contains the powers and roots of numbers consecutively from i to 1000. The circumferences and areas of circles, due to the numbers contained in the first columns, considered as diameters, are also given. By a suitable adjustment of decimal points the circumferences, areas, squares and cubes, may be determined from the contents of the table for diameters ten or a hundred times as much as, or less than, the values given in the first column. For example, if the number 378 in the first column, page 73, be taken as 37.8, the corresponding circumference, area, square and cube are as follows: Original. Decimalized, Number 378 37.8 Circumference 1,187.52 118.752 Circular area 112,221.09 1122.2109 Square 142,884 1,428.84 Cube 54,010,152 54,010.152 TABLE No. IV. OF CIRCUMFERENCES AND AREAS OF CIRCLES WITH SIDES OF EQUAL SQUARES. The table No. IV. gives the circumferences and areas of circles from Jg- inch to 120 inches in diameter, advancing by sixteenths of an inch up to 6 inches diameter; thence by eighths of an inch to 50 inches diameter; thence by quarters of an inch to 100 inches diameter; and thence by half- inches to 1 20 inches diameter. Whilst the diameters are here expressed as inches, they may be taken as feet, or as measures of any other denomination. The column of sides of equal squares, contains the sides of squares having the same area as the circles in the same lines of the table respectively. TABLES Nos. V. AND VI. OF LENGTHS OF CIRCULAR ARCS. The lengths of circular arcs are given proportionally to that of the radius, and to that of the chord, in the tables Nos. V. and VI. In the first of these tables, the radius is taken - i, and the number of degrees in the arc are given in the first column. The length of the arc as compared with the radius is given decimally in the second column. 36 MATHEMATICAL TABLES. In the second table, the chord is taken = i, and the rise or height of the arc, expressed decimally as compared with the chord, is given in the first column. The length of the arc relatively to the chord is given in the second column. To use the first table, No. V., find the proportional length of the arc corresponding to the degrees in the arc, and multiply it by the actual length of the radius; the product is the actual length of the arc. To use the second table, No. VI., divide the height of the arc by the chord for the proportional height of the arc, which find in the first column of the table; the proportional length of the arc corresponding to it being multi- plied by the actual length of the chord, gives the actual length of the arc. Note. The length of an arc of a circle may be found nearly thus: Subtract the chord of the whole arc from 8 times the chord of half the arc. A third of the remainder is the length nearly. TABLE No. VII. OF AREAS OF CIRCULAR SEGMENTS. The areas of circular segments are given in Table No. VII., in proportional superficial measure, the diameter of the circle of which the segment forms a portion being = i. The height of the segment, expressed decimally in proportion to the diameter, is given in the first column, and the relative area in the second column. To use the table, divide the height by the diameter, find the quotient in the table, and multiply the corresponding area by the square of the actual length of the diameter; the product will be the actual area. TABLE No. VIII. SINES, COSINES, TANGENTS, COTANGENTS, SECANTS, AND COSECANTS OF ANGLES FROM o TO 90. This table, No. VIII., is constructed for angles of from o to 90, advancing by 10', or one-sixth of a degree. The length of the radius is equal to i, and forms the basis for the relative lengths given in the table, and which are given to six places of decimals. Each entry in the table has a duplicate significance, being the sine, tangent, or secant of one angle, and at the same time the cosine, cotangent, or cosecant of its complement. For this reason, and for the sake of compactness, the headings of the columns are reversed at the foot ; so that the upper headings are correct for the angles named in the left hand margin of the table, and the lower headings for those named in the right hand margin. To find the sine, or other element \ to odd minutes ; divide the difference between the sines, &c., of the two angles greater and less than the given angle, in the same proportion that the given angle divides the difference of the two angles, and add one of the parts to the sine next it. By an inverse process the angle may be found for any given sine, &c., not found in the table. TABLE No. IX. OF LOGARITHMIC SINES, COSINES, TANGENTS, AND CO- TANGENTS OF ANGLES FROM o TO 90. This table, No. IX., is constructed similarly to the table of natural sines,. &c., preceding. To avoid the use of logarithms with negative indices, the radius is assumed, instead of being equal to i, to be equal to io 10 , or EXPLANATION AND USES OF THE TABLES. 37 10,000,000,000; consequently the logarithm of the radius = 10 log 10= 10. Whence, if, to log sine of any angle, when calculated for a radius = i, there be added 10, the sum will be the log sine of that angle for a radius = io 10 . For example, to find the logarithmic sine of the angle 15 50'. Nat. sine 15 50'= "272840; its log = 1*435908 add = io Logarithmic sine of 15 50'= 9*435908 When the logarithmic sines and cosines have been found in this manner, the logarithmic tangents, cotangents, secants, and cosecants are found from those by addition or subtraction, according to the correlations of the trigonometrical elements already given, and here repeated in logarithmic form: Log tan = io + log sin. log. cosin. Log cotan = 20 - log tan. Log sec = 20 - log cosin. Log cosec = 20 - log sin. To find the logarithmic sine, tangent, &*c., of any angle. When the number of degrees is less than 45, find the degrees and minutes in the left hand column headed angle, and under the heading sine, or tangent, &c., as required, the logarithm is found in a line with the angle. When the number of degrees is above 45, and less than 90, find the degrees and minutes in the right hand column headed angle, and in the same line, above the title at the foot of the page, sine or tangent, &c., find the logarithm in a line with the angle. When the number of degrees is between 90 and 180, take their supple- ment to 1 80; when between 180 and 270, diminish them by 180; and when between 270 and 360, take their complement to 360, and find the logarithm of the remainder as before. If the exact number of minutes is not found in the table, the logarithm of the nearest tabular angle is to be taken and increased or diminished as the case may be, by the due proportion of the difference of the logarithms of the angles greater and less than the given angle. TABLE No. X. RHUMBS, OR POINTS OF THE COMPASS. The Mariner's Compass is a circular card suspended horizontally, having a thin bar of steel magnetized, the needle, for one of its diameters; the circumference of the card being divided into 32 equal parts, or points, and each point subdivided into quarters. A point of the compass is, therefore, equal to (360 ---32 = ) 11 15'. TABLE No. XI. OF RECIPROCALS OF NUMBERS. The table No. XI. contains the reciprocals of numbers from i to 1000. It has already been shown how to find the reciprocal of a number by means of logarithms. MATHEMATICAL TABLES. TABLE No. I. LOGARITHMS OF NUMBERS FROM I TO IO,OOO. No. Log. No. Log. No. Log. No. Log. 1 2 3 4 5 0.000000 0.301030 0.477121 0.602060 0.698970 26 27 28 29 30 414973 43^64 .447158 .462398 .477121 51 S 2 53 54 55 1.707570 1.716003 .724276 .732394 740363 76 77 78 79 80 .880814 .886491 .892095 .897627 .903090 6 9 10 0.778151 0.845098 0.903090 0.954243 I.OOOOOO 31 32 33 34 35 .491362 505150 .518514 531479 .544068 56 57 58 59 60 .748188 : 763428 .770852 .778151 81 82 83 84 85 .908485 .913814 .919078 .924279 .929419 11 12 13 H 15 .041393 .079181 113943 .146128 .176091 36 ? 39 40 556303 . 568202 579784 .591065 .602060 61 62 63 64 65 .785330 .792392 .799341 .806180 .812913 86 87 88 89 90 934498 939519 944483 949390 954243 16 17 18 19 20 .204120 .230449 .255273 .278754 .301030 41 42 43 44 45 .612784 .623249 .633468 .643453 .653213 66 67 68 69 70 .819544 .826075 .832509 .838849 .845098 91 92 93 94 95 .959041 .963788 968483 .973128 .977724 21 22 23 24 25 .322219 342423 .361728 .380211 1.397940 46 47 48 49 5 .662758 .672098 1.681241 1.690196 1.698970 71 72 73 74 75 .851258 857332 .863323 .869232 .875061 96 97 98 99 100 .982271 .986772 .991226 1.995635 2.OOOOOO N 01234 56 7 89 D 100 101 102 102 I0 3 104 I0 4 oo- oooo 0434 0868 1301 1734 oo- 4321 4751 5181 5609 6038 oo- 8600 9026 9451 9876 01- 0300 oi- 2837 3259 3680 4100 4521 OI ~ 733 745 r 7868 8284 8700 02- 2l66 2598 3029 3461 3891 6466 6894 7321 7748 8174 0724 1147 1570 1993 2415 4940 5360 5779 6197 6616 9116 9532 9947 0361 0775 432 428 425 424 420 417 416 105 106 107 107 108 109 109 O2- 1189 1603 2Ol6 2428 2841 02- 5306 5715 6125 6533 6942 02- 9384 9789 03- 0195 600 1004 03- 3424 3826 4227 4628 5029 03- 7426 7825 8223 8620 9017 04 3252 3664 4075 4486 4896 7350 7757 8164 8571 8978 1408 1812 2216 2619 3021 5430 5830 6230 6629 7028 94H 9811 0207 0602 0998 412 408 405 404 400 398 3Q7 N 01234 56789 D LOGARITHMS OF NUMBERS. 39 N o i 2 34 56789 D 110 in 1 12 04- 1393 1787 2182 2576 2969 04- 5323 57H 6105 6495 6885 O4 Q2l8 0606 QQQ3 3362 3755 4148 4540 4932 7275 7664 8053 8442 8830 393 389 388 112 "3 114 114. 05- 0380 0766 05- 3 78 3463 384 6 4230 4613 05- 6905 7286 7666 8046 8426 06 1153 1538 1924 2309 2694 4996 5378 576o 6142 6524 8805 9185 9563 9942 0320 386 383 383 379 115 116 117 06- 0698 1075 1452 1829 2206 06- 4458 4832 5206 5580 5953 06- 8186 8557 8927 9298 9668 2582 2958 3333 3709 4083 6326 6699 7071 7443 7815 376 373 380 117 118 119 07- 07- 1882 2250 2617 2985 3352 7- 5547 59 12 6276 6640 7004 0038 0407 0776 1145 1514 3718 4085 4451 4816 5182 7368 7731 8094 8457 8819 370 366 363 120 O7 Ql8l Q^43 QQO4 362 1 20 08 O266 0626 0087 1347 1707 2067 2426 O"-" 360 121 122 08- 2785 3144 3503 3861 4219 08- 6360 6716 7071 7426 7781 08 QQCX 4576 4934 5291 5647 6004 8136 8490 8845 9198 9552 357 355 7CC I2 3 124 09- 0258 0611 0963 1315 09- 3422 3772 4122 4471 4820 1667 2018 2370 2721 3071 5169 5518 5866 6215 6562 353 349 125 127 128 128 09- 6910 7257 7604 7951 8298 IO- io- 0371 0715 1059 1403 1747 io- 3804 4146 4487 4828 5169 io- 7210 7549 7888 8227 8565 8644 8990 9335 9681 0026 2091 2434 2777 3119 3462 5510 5851 6191 6531 6871 8903 9241 9579 9916 0253 348 346 343 34i 338 337 129 ii- 0590 0926 1263 1599 1934 2270 2605 2940 3275 3609 335 130 ii- 3943 4277 4611 4944 5278 ii- 7271 7603 7934 8265 8595 12- 5611 5943 6276 6608 6940 8926 9256 9586 9915 0245 OCOCO o Co Co HHCO 132 133 134 12- 0574 0903 1231 1560 1888 12- 3852 4178 4504 4830 5156 12- 7105 7429 7753 8076 8399 2216 2544 2871 3198 3525 5481 5806 6131 6456 6781 8722 9045 9368 9690 OOI2 328 325 323 323 o\r)\D\r)\j-> 775 776 776 777 778 779 88- 9302 9358 9414 9470 9526 88- 9862 9918 9974 89- 0030 0086 89- 0421 0477 0533 0589 0645 89- 0980 1035 1091 1147 1203 89- 1537 1593 1649 1705 I76o 9582 9638 9694 9750 9806 0141 0197 0253 0309 0365 0700 0756 0812 0868 0924 1259 1314 1370 1426 1482 1816 1872 1928 1983 2039 56 56 5 ? 56 56 56 780 781 782 783 784 89- 2095 2150 2206 2262 2317 89- 2651 2707 2762 2818 2873 89- 3207 3262 3318 3373 3429 89- 3762 3817 3873 3928 3984 89- 4316 4371 4427 4482 4538 2373 2429 2484 2540 2595 2929 2985 3040 3096 3151 3484 3540 3595 3651 3706 4039 4094 4150 4205 4261 4593 4648 4704 4759 4814 56 56 56 55 55 785 786 787 788 789 89- 4870 4925 4980 5036 5091 89- 5423 5478 5533 5588 5644 89- 5975 6 3 6085 6140 6195 89- 6526 6581 6636 6692 6747 89- 7077 7132 7187 7242 7297 5146 5201 5257 5312 5367 5699 5754 5809 5864 5920 6251 6306 6361 6416 6471 6802 6857 6912 6967 7022 7352 7407 7462 7517 7572 55 55 55 55 55 790 791 792 793 794 89- 7627 7682 7737 7792 7847 89- 8176 8231 8286 8341 8396 89- 8725 8780 8835 8890 8944 89- 9273 9328 9383 9437 9492 8g- 9821 9871; Q91O 098; 7902 7957 8012 8067 8122 8451 8506 8561 8615 8670 8999 9054 9109 9164 9218 9547 9602 9656 9711 9766 55 55 55 55 ec 7Q4- QO- OO3Q OOQ4. OI4.Q O2O3 O2^8 O7I2 rc jj 795 796 797 798 799 go- 0367 0422 0476 0531 0586 go- 0913 0968 1022 1077 1131 go- 1458 1513 1567 1622 1676 go- 2OO3 2O57 21 12 2l66 2221 go- 2547 2601 2655 2710 2764 0640 0695 0749 0804 0859 1186 1240 1295 1349 1404 1731 1785 1840 1894 1948 2275 2329 2384 2438 2492 2818 2873 2927 2981 3036 55 55 54 54 54 800 80 1 802 803 804 go- 3090 3144 3199 3253 3307 90- 3633 3687 374i 3795 3849 go- 4174 4229 4283 4337 4391 go- 4716 4770 4824 4878 4932 go- 5256 5310 5364 5418 5472 3361 3416 3470 3524 3578 3904 3958 4012 4066 4120 4445 4499 4553 4607 4661 4986 5040 5094 5148 5202 5526 5580 5634 5688 5742 54 54 54 54 54 805 90- 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 54 N 01234 56789 D LOGARITHMS OF NUMBERS. 55 N 01234 56789 D 806 80*9 90- 6335 6389 6443 6497 6551 go- 6874 6927 6981 7035 7089 go- 7411 7465 7519 7573 7626 go- 7949 8002 8056 8 no 8163 6604 6658 6712 6766 6820 7143 7196 7250 7304 7358 7680 7734 7787 7841 7895 8217 8270 8324 8378 8431 54 54 54 54 810 811 812 812 go- 8485 8539 8592 8646 8699 go- 9021 9074 9128 9181 9235 9~ 9556 9610 9663 9716 9770 QI- 8753 8807 8860 8914 8967 9289 9342 9396 9449 9503 9823 9877 9930 9984 0037 54 54 54 e-j 813 814 gi- 0091 0144 0197 0251 0304 gi- 0624 0678 0731 0784 0838 0358 0411 0464 0518 0571 0891 0944 0998 1051 1104 JO 53 53 815 816 817 818 819 gi- 1158 i2ii 1264 1317 1371 gi- 1690 1743 1797 1850 1903 gi- 2222 2275 2328 2381 2435 91- 2753 2806 2859 2913 2966 91- 3284 3337 3390 3443 3496 1424 1477 1530 1584 1637 1956 2009 2063 2116 2169 2488 2541 2594 2647 2700 3019 3072 3125 3178 3231 3549 3602 3655 3708 3761 53 53 53 53 53 820 821 822 823 824 gi- 3814 3867 3920 3973 4026 91- 4343 4396 4449 4502 4555 gi- 4872 4925 4977 5030 5083 gi- 5400 5453 5505 5558 5611 9i- 5927 598o 6033 6085 6138 4079 4132 4184 4237 4290 4608 4660 4713 4766 4819 5136 5189 5241 5294 5347 5664 5716 5769 5822 5875 6191 6243 6296 6349 6401 53 53> 53 53 53 825 826 827 828 829 gi- 6454 6507 6559 6612 6664 gi- 6980 7033 7085 7138 7190 gi- 7506 7558 7611 7663 7716 gi- 8030 8083 8135 8188 8240 gi- 8555 8607 8659 8712 8764 6717 6770 6822 6875 6927 7243 7295 7348 7400 7453 7768 7820 7873 7925 7978 8293 8345 8397 8450 8502 8816 8869 8921 8973 9026 53 53 52 52 52 830 831 3 ' 832 ? 33 834 gi- 9078 9130 9183 9235 9287 gi- 9601 9653 9706 9758 9810 g2- ga- 0123 0176 0228 0280 0332 g2- 0645 0697 0749 0801 0853 g2- 1166 1218 1270 1322 1374 9340 9392 9444 9496 9549 9862 9914 9967 0019 0071 0384 0436 0489 0541 0593 0906 0958 1010 1062 1114 1426 1478 1530 1582 1634 52 52 52 52 52 52 835 836 837 838 839 ga-: 1686 1738 1790 1842 1894 g2- 2206 2258 2310 2362 2414 g2- 2725 2777 2829 2881 2933 ga- 3244 3296 3348 3399 3451 92- 3762 3814 3865 3917 3969 1946 1998 2050 2102 2154 2466 2518 257O 2622 2674 2985 337 3089 3HQ 3 J 92 353 3555 3607 3658 37 4021 4072 4124 4176 4228 52 52 52 52 52 840 841 842 843 844 92- 4279 4331 4383 4434 4486 g2- 4796 4848 4899 4951 5003 92- 5312 5364 5415 5467 5518 92- 5828 5879 5931 5982 6034 g2- 6342 6394 6445 6 497 6548 4538 4589 4641 4693 4744 5054 5106 5157 5209 5261 5570 5621 5673 5725 5776 6085 6137 6188 6240 6291 6600 6651 6702 6754 6805 52 52 52 5i 5i 845 846 847 848 849 g2- 6857 6908 6959 7011 7062 92- 7370 7422 7473 7524 7576 92- 7883 7935 7986 8037 8088 g2- 8396 8447 8498 8549 8601 g2- 8908 8959 9010 9061 9112 7114 7165 7216 7268 7319 7627 7678 7730 7781 7832 8140 8191 8242 8293 8345 8652 8703 8754 8805 8857 9163 9215 9266 9317 9368 5i 5i 5i 5i 51 850 851 g2- 9419 9470 9521 9572 9623 92- 993O 9981 9674 9725 9776 9827 9879 5i Cl N 01234 56789 D MATHEMATICAL TABLES. N 01234 56789 D 851 852 853 854 93- 0032 0083 0134 93- 0440 0491 0542 0592 0643 93- 0949 1000 1051 i 102 1153 93- 1458 1509 1560 1610 1661 0185 0236 0287 0338 0389 0694 0745 0796 0847 0898 1203 1254 1305 1356 1407 1712 1763 1814 1865 1915 51 5i 5i 5i 855 856 857 858 859 93- 1966 2017 2068 2118 2169 93- 2474 2524 2575 2626 2677 93- 2981 303*1 3082 3133 3183 93- 3487 3538 3589 3639 3690 93- 3993 4044 4094 4H5 4195 2220 2271 2322 2372 2423 2727 2778 2829 2879 2930 3234 3285 3335 3386 3437 3740 3791 3841 3892 3943 4246 4296 4347 4397 4448 5i 51 5i 51 5i 860 861 862 863 864 93- 4498 4549 4599 4650 4700 93- 53 554 5 I0 4 5154 5205 93- 557 5558 5608 5658 5709 93- 6011 6061 6ni 6162 6212 93- 6514 6564 6614 6665 6715 4751 4801 4852 4902 4953 5255 5306 5356 5406 5457 5759 5809 5860 5910 5960 6262 6313 6363 6413 6463 6765 6815 6865 6916 6966 5o 50 50 50 5o 865 866 867 868 869 93- 7016 7066 7117 7167 7217 93- 75i8 7568 7618 7668 7718 93- 8019 8069 8119 8169 8219 93- 8520 8570 8620 8670 8720 93- 9020 9070 9120 9170 9220 7267 7317 7367 7418 7468 7769 7819 7869 7919 7969 8269 8319 8370 8420 8470 8770 8820 8870 8920 8970 9270 9320 9369 9419 9469 50 50 50 50 50 870 871 872 873 874 93- 95*9 95 6 9 96i9 9669 9719 94- 0018 0068 01 18 0168 0218 94- 0516 0566 0616 0666 0716 94- 1014 1064 1114 1163 1213 94- 1511 1561 1611 1660 1710 9769 9819 9869 9918 9968 0267 0317 0367 0417 0467 0765 0815 0865 0915 0964 1263 1313 1362 1412 1462 1760 1809 1859 1909 1958 50 50 5 5o 50 875 876 877 878 879 94- 2008 2058 2107 2157 2207 94- 2504 2554 2603 2653 2702 94- 3000 3049 3099 3148 3198 94- 3495 3544 3593 3643 3692 94- 3989 4038 4088 4137 4186 2256 2306 2355 2405 2455 2752 2801 2851 2901 2950 3247 3297 3346 3396 3445 3742 3791 3841 3890 3939 4236 4285 4335 4384 4433 5o 50 49 49 49 880 881 882 883 884 94- 4483 4532 4581 4631 4680 94-. 4976 5025 5074 5124 5173 94- 5469 55i8 5567 5616 5665 94- 5961 6010 6059 6108 6157 94- 6452 6501 6551 6600 6649 4729 4779 4828 4877 4927 5222 5272 5321 5370 5419 5715 5764 5813 5862 5912 6207 6256 6305 6354 6403 6698 6747 6796 6845 6894 49 49 49 49 49 885 886 887 888 889 94- 6943 6992 7041 7090 7140 94- 7434 7483 7532 7581 7630 94- 7924 7973 8022 8070 8119 94- 8413 8462 8511 8560 8609 94- 8902 8951 8999 9048 9097 7189 7238 7287 7336 7385 7679 7728 7777 7826 7875 8168 8217 8266 8315 8364 8657 8706 8755 8804 8853 9146 9195 9244 9292 9341 49 49 49 49 49 890 891 8qi 94- 9390 9439 9488 9536 9585 94- 9878 9926 9975 Q^" OO2J. OO7^ 9634 9683 9731 9780 9829 OI2I OI7O O2I9 0267 0316 49 49 4Q 892 893 894 95- 0365 0414 0462 0511 0560 95- 0851 0900 0949 0997 1046 95- 1338 1386 1435 ^83 1532 0608 0657 0706 0754 0803 1095 1143 1192 I24O 1289 1580 1629 1677 1726 1775 49 49 49 895 896 897 95- 1823 1872 1920 1969 2017 95- 2308 2356 2405 2453 2502 95- 2792 2841 2889 2938 2986 2066 2114 2163 221 I 2260 255 2599 2647 2696 2744 3034 3083 3131 3180 3228 48 48 48 N 01234 56789 D LOGARITHMS OF NUMBERS. 57 N o i 2 3. 4 56789 D 898 899 95- 3276 3325 3373 342i 3470 95- 376o 3808 3856 3905 3953 3518 3566 3615 3663 37H 4001 4049 4098 4146 4194 48 48 900 901 902 903 904 95- 4243 4291 4339 4387 4435 95- 4725 4773 4821 4869 4918 95- 5207 5255 5303 5351 5399 95- 5688 5736 5784 5832 588o 95- 6168 6216 6265 6313 6361 4484 4532 4580 4628 4677 4966 5014 5062 5110 5158 5447 5495 5543 5592 5640 5928 5976 6024 6072 6120 6409 6457 6505 6553 6601 4 8 48 4 8 4 8 48 905 906 907 908 909 95- 6649 6697 6745 6793 6840 95- 7128 7176 7224 7272 7320 95- 7607 7655 7703 775 i 7799 95- 8086 8134 8181 8229 8277 95- 8564 8612 8659 8707 8755 6888 6936 6984 7032 7080 7368 7416 7464 7512 7559 7847 7894 7942 7990 8038 8325 8373 8421 8468 8516 8803 8850 8898 8946 8994 4 8 48 48 4 8 48 910 911 912 912 9i3 914 95- 9041 9089 9137 9185 9232 95- 9518 9566 9614 9661 9709 95-9995 96- 0042 0090 0138 0185 96- 0471 0518 0566 0613 0661 96- 0946 0994 1041 1089 1136 9280 9328 9375 9423 9471 9757 9804 9852 9900 9947 0233 0280 0328 0376 0423 0709 0756 0804 0851 0899 1184 1231 1279 1326 1374 4 8 48 48 4 8 4 8 47 915 916 917 918 919 96- 1421 1469 1516 1563 1611 96- 1895 1943 1990 2038 2085 96- 2369 2417 2464 2511 2559 96- 2843 2890 2937 '2985 3032 96- 3316 3363 3410 3457 3504 1658 1706 1753 1801 1848 2132 2180 .2227 2275 2322 2606 2653 2701 2748 2795 3079 3126 3174 3221 3268 3552 3599 3646 3693 374i 47 47 47 47 47 920 921 922 923 924 96- 3788 3835 3882 3929 3977 96- 4260 4307 4354 4401 4448 96- 4731 , 4778 4825 4872 4919 96- 5202 5249 5296 5343 5390 96- 5672 5719 5766 5813 5860 4024 4071 4118 4165 4212 4495 4542 4590 4637 4684 4966 5013 5061 5108 5155 5437 5484 5531 5578 5625 5907 5954 6001 6048 6095 47 47 47 47 47 925 926 927 928 929 96- 6142 6189 6236 6283 6329 96- 6611 6658 6705 6752 6799 96- 7080 7127 7173 7220 7267 96- 7548 7595 7642 7688 7735 96- 8016 8062 8109 8156 8203 6376 6423 6470 6517 6564 6845 6892 6939 6986 7033 7314 7361 7408 7454 7501 7782 7829 7875 7922 7969 8249 8296 8343 8390 8436 47 47 47 47 47 930 93 1 932 0-7 -3 96- 8483 8530 8576 8623 8670 96- 8950 8996 9043 9090 9136 96- 9416 9463 9509 9556 9602 q6- 0882 QQ28 007^ 8716 8763 8810 8856 8903 9183 9229 9276 9323 9369 9649 9695 9742 9789 9835 47 47 47 4.7 933 934 97- OO2I OO68 97- 0347 0393 0440 0486 0533 0114 0161 0207 0254 0300 0579 0626 0672 07^9 0765 47 46 935 936 937 938 939 97- 0812 0858 0904 0951 0997 97- 1276 1322 1369 1415 1461 97- 1740 1786 1832 1879 1925 97- 2203 2249 2295 2342 2388 97- 2666 2712 2758 2804 2851 1044 1090 1137 1183 1229 1508 1554 1601 1647 1693 1971 2018 2064 21 10 2157 2434 2481 2527 2573 2619 2897 2943 2989 3035 3082 46 46 46 46 46 940 941 942 943 97- 3128 3174 3220 3266 3313 97- 3590 3636 3682 3728 3774 97- 4051 4097 4143 4189 4235 97- 4512 4558 4604 4650 4696 3359 3405 345 J 3497 3543 3820 3866 3913 3959 4005 4281 4327 4374 4420 4466 4742 4788 4834 4880 4926 46 46 46 46 N 01234 56789 D MATHEMATICAL TABLES. N 01234 56789 D 944 97- 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 46 945 946 948 949 97- 5432 5478 5524 557 5616 97- 5891 5937 5983 6029 6075 97- 6350 6396 6442 6488 6533 97- 6808 6854 6900 6946 6992 97- 7266 7312 7358 7403 7449 5662 5707 5753 5799 5845 OI2I 6167 6212 6258 6304 6579 6625 6671 6717 6763 7037 7083 7129 7175 7220 7495 754i 7586 7632 7678 46 46 46 46 46 950 95i 952 953 954 97- 7724 7769 7815 7861 7906 97- 8181 8226 8272 8317 8363 97- 8637 8683 8728 8774 8819 97- 9093 9138 9184 9230 9275 97- 9548 9594 9639 9685 973 7952 7998 8043 8089 8135 8409 8454 8500 8546 8591 8865 8911 8956 9002 9047 9321 9366 9412 9457 9503 9776 9821 9867 9912 9958 46 46 46 46 46 955 956 957 958 959 98- 0003 0049 0094 0140 0185 98- 0458 0503 0549 0594 0640 98- 0912 0957 1003 1048 1093 98- 1366 1411 1456 1501 1547 98- 1819 1864 1909 1954 2000 0231 0276 0322 0367 0412 0685 0730 0776 0821 0867 1139 1184 1229 1275 I 3 2 o 1592 1637 1683 1728 1773 2045 2090 2135 2181 2226 45 45 45 45 45 960 961 962 963 964 98- 2271 2316 2362 2407 2452 98- 2723 2769 2814 2859 2904 98- 3175 3220 3265 3310 3356 98- 3626 3671 3716 3762 '3807 98- 4077 4122 4167 4212 4257 2497 2543 2588 2633 2678 2949 2994 3040 3085 3130 3401 3446 3491 3536 3581 3852 3897 3942 3987 4032 4302 4347 4392 4437 4482 45 45 45 45 45 965 966 967 968 969 98- 4527 4572 4617 4662 4707 98- 4977 522 5 6 7 5 "2 5 J 57 98- 5426 5471 5516 5561 5606 98- 5875 5920 5965 6010 6055 98- 6324 6369 6413 6458 6503 4752 4797 4842 4887 4932 5202 5247 5292 5337 5382 5651 5696 5741 5786 5830 6100 6144 6189 6234 6279 6548 6593 6637 6682 6727 45 45 45 45 45 970 971 972 973 974 98- 6772 6817 6861 6906 6951 98- 7219 7264 7309 7353 7398 98- 7666 7711 7756 7800 7845 98- 8113 8157 8202 8247 8291 98- 8559 8604 8648 8693 8737 6996 7040 7085 7130 7175 7443 7488 7532 7577 7622 7890 7934 7979 8024 8068 8336 8381 8425 8470 8514 8782 8826 8871 8916 8960 45 45 45 45 45 975 976 077 98- 9005 9049 9094 9138 9183 98- 945 9494 9539 9583 9628 08- 9891; QQ3Q QQ8-? 9227 9272 9316 9361 9405 9672 9717 9761 9806 9850 45 44 A/\ 977 978 979 99- OO28 0072 99- 0339 0383 0428 0472 0516 99- 0783 0827 0871 0916 0960 0117 01 6 i 0206 0250 0294 0561 0605 0650 0694 0738 1004 1049 1093 1137 1182 44 44 44 980 981 982 983 984 99- 1226 1270 1315 1359 1403 99- 1669 1713 1758 1802 1846 99- 21 I I 2156 2200 2244 2288 99- 2554 2598 2642 2686 2730 99- 2995 3039 3083 3127 3172 1448 1492 1536 1580 1625 1890 1935 1979 2023 2067 2333 2377 2421 2465 2509 2774 2819 2863 2907 2951 3216 3260 3304 3348 3392 44 44 44 44 44 985 986 $ 989 99- 3436 348o 3524 3568 3613 99- 3877 3921 3965 4009 4053 99- 4317 4361 4405 4449 4493 99- 4757 4801 4845 4889 4933 99- 5196 5240 5284 5328 5372 3657 3701 3745 3789 3833 4097 4141 4185 4229 4273 4537 4581 4625 4669 4713 4977 5021 5065 5108 5152 5416 5460 5504 5547 5591 44 44 44 44 44 N 01234 56789 D i LOGARITHMS OF NUMBERS. 59 N 01234 56789 D 990 991 992 993 994 99- 5635 5679 5723 5767 58" 99- 6074 6117 6161 6205 6249 99- 6512 6555 6599 6643 6687 99- 6949 6993 7037 7080 7124 99- 7386 743 7474 75 i 7 756 i 5854 5898 5942 5986 6030 6293 6337 6380 6424 6468 6731 6774 6818 6862 6906 7168 7212 7255 7299 7343 7605 7648 7692 7736 7779 44 44 44 44 44 995 996 997 998 999 99- 7823 7867 7910 7954 7998 99- 8259 8303 8347 8390 8434 99- 8695 8739 8782 8826 8869 99- 9 I 3 I 9 r 74 9218 9261 9305 99- 95 6 5 9609 9652 9696 9739 8041 8085 8129 8172 8216 8477 8521 8564 8608 8652 8913 8956 9000 9043 9087 9348 9392 9435 9479 9522 9783 9826 9870 9913 9957 44 44 44 44 43 N 01234 56789 D 6o MATHEMATICAL TABLES. TABLE No. II. HYPERBOLIC LOGARITHMS OF NUMBERS FROM I.OI TO 3O. Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. I.OI .0099 I. 3 6 3075 I.7I 5365 2.06 .7227 1.02 .0198 i-37 .3148 1.72 5423 2.07 7275 1.03 .0296 1.38 .3221 i-73 .5481 2.08 7324 I.O4 .0392 i-39 3293 1.74 5539 2.09 7372 1.05 ..0488 1.40 3365 i-75 5596 2.10 .7419 1. 06 .0583 1.41 3436 1.76 5 6 53 2. II .7467 1.07 .0677 1.42 3507 1.77 5710 2.12 75 J 4 i. 08 .0770 1-43 3577 1.78 .5766 2.13 75 61 1.09 .0862 1.44 .3646 1.79 5822 2.14 .7608 1. 10 953 i-45 .3716 i. 80 .5878 2.15 7655 i. ii .1044 1.46 .3784 1.81 5933 2.16 .7701 1. 12 ii33 1.47 3853 1.82 .5988 2.17 7747 I-I3 .1222 1.48 .3920 1.83 .6043 2.18 7793 I.I4 .1310 1.49 .3988 1.84 .6098 2.19 7839 l -*5 .1398 1.50 4055 1.85 .6152 2.20 .7885 1.16 .1484 I-5I .4121 1.86 .6206 2.21 7930 1.17 1570 1.52 .4187 1.87 .6259 2.22 7975 1.18 1655 i-53 4253 1.88 6313 2.23 .8020 1.19 .1740 1-54 .4318 1.89 .6366 2.24 .8065 1.20 .1323 i-55 4383 1.90 .6419 2.25 .8109 1. 21 .1906 1.56 4447 1.91 .6471 2.26 8154 1.22 .1988 i-57 45 11 1.92 6523 2.27 .8198 1.23 .2O70 1.58 4574 i-93 6575 2.28 .8242 1.24 .2151 i-59 4637 1.94 .6627 2.2 9 .8286 1.25 .2231 i. 60 .4700 I -95 .6678 2.30 8329 1.26 .2311 1.61 .4762 1.96 .6729 2.31 8372 1.27 .2390 1.62 .4824 1.97 .6780 2.32 .8416 1.28 .2469 1.63 .4886 1.98 .6831 2-33 .8458 1.2 9 .2546 1.64 4947 1.99 .6881 2-34 .8502 1.30 .2624 1.65 .5008 2.0O .6931 2-35 .8544 LSI .2700 1.66 .5068 2.OI .6981 2.36 .8587 1.32 .2776 1.67 .5128 2. 02 .7031 2-37 .8629 1-33 .2852 1.68 .5188 2.03 .7080 2.38 .8671 i-34 .2927 1.69 5 2 47 2.04 .7129 2-39 8713 1-35 .3001 1.70 53o6 2.05 7i78 2.40 i 8755 HYPERBOLIC LOGARITHMS OF NUMBERS 61 Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. 2.41 .8796 2.8l 1.0332 3.21 1.1663 3 .6l 1.28 3 7 2.42 .8838 2.82 1.0367 3.22 1.1694 3-62 1.2865 2-43 .8879 2.83 1.0403 3-23 1725 3-63 1.2892 2.44 .8920 2.84 1.0438 3-24 I75 6 3-64 1.2920 2-45 .8961 2-8 5 1-0473 3-25 .1787 3-65 1.2947 2.46 .9002 2.86 1.0508 3.26 .1817 3.66 1-2975 2.47 .9042 2.87 1-0543 3-27 .1848 3-67 1.3002 2.48 .9083 2.88 1.0578 3.28 .1878 3 .68 1.3029 2.49 .9123 2.89 1.0613 3-29 .1909 3.69 1-3056 2.50 .9163 2.90 1.0647 3-30 I - I 939 3.70 1.3083 2-51 .9203 2.91 1.0682 3-31 1.1969 3-71 I.3IIO 2.52 .9243 2.92 1.0716 3-32 1.1999 3-72 I-3I37 2 -53 .9282 2-93 1.0750 3-33 1.2030 3-73 1.3164 2-54 .9322 2-94 1.0784 3-34 1.2060 3-74 I.3I9I 2-55 .9361 2-95 I. O8l8 3-35 1.2090 3-75 I.32I8 2.56 .9400 2.96 1.0852 3-36 1.2119 3.76 L3244 2.57 9439 2-97 1. 0886 3-37 1.2149 3-77 1.3271 2.58 .9478 2.98 1.0919 3-38 1.2179 3-78 1.3297 2-59 95 1 ? 2-99 1-0953 3-39 1.2208 3-79 L3324 2.60 9555 3.00 1.0986 3-4 1.2238 3.80 I -335 2.61 9594 3.01 I.IOI9 3-4i 1.2267 3-8i I-3376 2.62 .9632 3.02 I-I053 3-42 1.2296 3.82 1-3403 2.63 .9670 3-3 I.I086 3-43 1.2326 3-83 1-3429 2.64 .9708 3-4 I.III9 3-44 r -2355 3-84 I -3455 2.65 .9746 3-5 I.II5I 3-45 1.2384 3-85 1.3481 2.66 9783 3.06 1.1184 3 : 46 1.2413 3-86 1.3507 2.67 .9821 3-7 I.I2I7 3-47 1.2442 3-87 L3533 2.68 .9858 3.08 I.I249 3-48 1.2470 3-88 -1.3558 2.69 9895 3-9 I.I282 3-49 1.2499 3-89 1-3584 2.70 9933 3.10 I.I3I4 3-5o 1.2528 3-9 1.3610 2.71 .9969 3- 11 1.1346 3-5i i-255 6 3-9 1 1-3635 2.72 i. 0006 3.12 I.I378 3-52 1-2585 3-9 2 1.3661 2-73 1.0043 3-i3 I.I4IO 3-53 1.2613 3-93 1.3686 2.74 1.0080 3-!4 I.I442 3-54 1.2641 3-94 1.3712 2-75 1.0116 3-15 I.I474 3-55 1.2669 3-95 1-3737 . 2.76 1.0152 3.16 1.1506 3.56 1.2698 3-96 1.3762 2.77 1.0188 3-i7 I-I537 3-57 1.2726 3-97 1.3788 2.78 1.0225 3.18 1.1569 3.58 1.2754 3-98" 1-3813 2-79 1.0260 3- x 9 I. l6oO 3-59 1.2782 3-99 1-3838 2.80 1.0296 3.20 1.1632 3.60 1.2809 4.00 1-3863 MATHEMATICAL TABLES. Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. 4.01 4.02 4-03 4-04 4-05 1.3888 L3938 1.3962 1.3987 4.41 4.42 4-43 4-44 4-45 1.4839 1.4861 1.4884 1.4907 1.4929 4.8l 4.82 4.83 4.84 4-85 I-5707 1.5728 L5748 L5769 1-5790 5-22 5.23 5-24 5-25 1.6506 1.6525 1.6514 1-6563 1.6582 4.06 4.07 4.08 4.09 4-10 I.40I2 1.4036 1.4061 1.4085 I.4IIO 4.46 4-47 4.48 4-49 4-5 I-495 1 1.4974 1.4996 1.5019 1.5041 4.86 4-87 4.88 4.89 4.90 1.5810 L583I L585I 1.5872 1.5892 5-26 5-27 5-28 5-30 1. 66oi 1.6620 1.6639 1.6658 1.6677 4.II 4.12 4-13 4.14 4.15 L4I34 L4I59 1.4183 1.4207 1.4231 4-5 1 4.52 4.53 4-54 4-55 1-5063 1.5085 1.5107 1.5129 4.91 4.92 4-93 4.94 4.95 I-59I3 1-5933 1-5953 1-5974 T -5994 5.31 5.32 5-33 5-34 5-35 1.6696 1.6715 1.6734 1.6752 1.6771 4.16 4.17 4.l8 4.19 4.2O 1.4255 1.4279 1.4303 1.4327 I-435 1 4-56 4.57 4.58 4-59 4.60 I -5 I 95 1.5217 1-5239 1.5261 4-96 4-97 4.98 4.99 5-o 1.6014 1.6034 1.6054 1.6074 1.6094 5.36 5-37 5-38 5-39 5-40 1.6790 1. 6808 1.6827 1.6845 1.6864 4.21 4.22 4.23 4.24 4.25 1-4375 1.4398 1.4422 1.4446 1.4469 4.61 4.62 4-63 4.64 4-65 1.5282 1-5304 1-5326 I -5347 J -53 6 9 5.01 5-02 5-3 5-4 5-05 1.6114 1.6134 1.6154 1.6174 1.6194 5-4i 5-42 5-43 5-44 5-45 1.6882 1.6901 1.6919 1.6938 1.6956 4.26 4.27 4.28 4.29 4-30 1-4493 1.4516 1.4540 1.4563 1.4586 4.66 4-67 4.68 4-69 4.70 1.5412 L5433 1-5454 L5476 5.06 5-07 5.08 5-09 5.10 1.6214 1.6233 1-6253 1.6273 1.6292 5-46 5-47 5-48 5-49 5-5 1.6974 1.6993 I.7OII 1.7029 1.7047 4.31 4.32 4-33 4.34 4-35 1.4609 1.4633 1.4656 1.4679 1.4702 4.71 4-72 4-73 4-74 4-75 L5497 1.5539 5- 11 5- 12 1.6312 1.6332 1-6351 1.6371 1.6390 5-5 1 S-S 2 5-53 5-54 5-55 1.7066 1.7084 I.7I02 I.7I20 I.7I38 4-36 4-37 4-38 4-39 4.40 1.4725 1.4748 1.4770 1-4793 1.4816 4-76 4-77 4.78 4.79 4.80 1.5602 1.5623 1.5644 1.5665 1.5686 vo t-00 O\ O M H M M M to to to to to 1.6409 1.6429 1.6448 1.6467 1.6487 5.56 5-57 5.58 5-59 5.60 I.7I56 I.7I74 I.7I92 I.72IO 1.7228 HYPERBOLIC LOGARITHMS OF NUMBERS. Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. 5.61 5-62 5.63 5-64 5.65 .7246 .7263 .728l .7299 .7317 6.01 6. 02 6.03 6.04 6.05 1-7934 L795 1 1.7967 1.7984 1.8001 6. 4 I 6.42 6-43 6.44 6-45 1.8579 1.8594 1. 86lO 1.8625 1.8641 6.81 6.82 6.83 6.84 6.85 1.9184 1.9199 1.9213 1.9228 1.9242 5.66 5.67 5.68 5-69 5-70 7334 735 2 7370 .7387 I-7405 6.06 6.07 6.08 6.09 6.10 1.8017 1.8034 1.8050 1.8066 1.8083 6.46 6.47 6.48 6.49 6.50 1.8656 1.8672 1.8687 1.8703 1.8718 6.86 6.87 6.88 6.89 6.90 I -9 2 57 1.9272 1.9286 .9301 -93^5 5-7i 5-72 5-73 5-74 5-75 1.7422 1.7440 1-7457 1-7475 1.7492 6.ii 6.12 6.13 6.14 6.15 1.8099 1.8116 1.8132 1.8148 1.8165 6.51 6.52 6-53 6-54 6-55 1.8733 1.8749 1.8764 1.8779 1-8795 6.91 6.92 6-93 6.94 6-95 933 9344 9359 9373 9387 5.76 5-77 5.78 5-79 5.80 i-759 i.75 2 7 1-7544 1.7561 1-7579 6.16 6.17 6.18 6.19 6. 20 1.8181 1.8197 1.8213 1.8229 1.8245 6.56 6-57 6.58 6-59 6.60 1. 88lO 1.8825 1.8840 1.8856 1.8871 6.96 6.97 6.98 6.99 7.00 .9402 .9416 .9430 9445 -9459 5-81 5-82 5-83 5-84 5-85 1.7596 1.7613 1.7630 1.7647 1.7664 6.21 6.22 6.23 6.2 4 6.25 1.8262 1.8278 1.8294 1.8310 1.8326 6.61 6.62 6.63 6.64 6.65 1.8886 1.8901 1.8916 1.8931 1.8946 7.01 7.02 7-03 7.04 7.05 9473 .9488 -952 .9516 9530 5-86 5.87 5.88 5-89 5-90 1.7681 1.7699 1.7716 1-7733 1-775 6.26 6.27 6.28 6.29 6.30 1.8342 1-8358 1.8374 1.8390 1.8405 6.66 6.67 6.68 6.69 6.70 1.8961 1.8976 1.8991 1.9006 1.9021 7.06 7.07 7.08 7.09 7.10 1-9544 1-9559 J-9573 1-9587 1.9601 5.9i 5-92 5-93 5-94 5-95 1.7766 1-7783 1.7800 1.7817 1-7834 6. 3 I 6.32 6-33 6-34 6-35 1.8421 1.8437 1.8453 1.8469 1.8485 6.71 6.72 6-73 6-74 6.75 1.9036 1.9051 1.9066 1.9081 I.9095 7.11 7.12 7.13 7.14 7.15 1.9615 1.9629 1.9643 1.9657 1.9671 5-96 5-97 5.98 5-99 6.00 I 1.7851 1.7867 1.7884 1.7901 1.7918 6.36 6-37 6.38 6-39 6.40 1.8500 1.8516 1-8532 1.8547 1.8563 6.76 6-77 6.78 6-79 6.80 1.9110 1.9125 1.9140 i.9i55 1.9169 7.16 7.17 7.18 7.19 7.20 1.9685 1.9699 i.97i3 1.9727 1.9741 6 4 MATHEMATICAL TABLES. Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. 7.21 7.22 7.23 7.24 7.25 1-9755 1.9769 1.9782 1.9796 1.9810 7 .6l 7.62 7-63 7.64 7.65 2.0295 2.0308 2.0321 2.0334 2.0347 8.01 8.02 8.03 8.04 8.05 2.0807 2.0819 2.0832 2.0844 2.0857 8.41 8. 4 2 8-43 8.44 8-45 2.1294 2.1306 2.1318 2.1330 2.1342 7.26 7.27 7.28 7.29 7-30 1.9824 1.9838 1.9851 1.9865 1.9879 7.66 7.67 7.68 7.69 7.70 2.0360 2.0373 2.0386 2.0399 2.0412 8.06 8.07 8.08 8.09 8.10 2.0869 2.0882 2.0894 2.0906 2.0919 8.46 8-47 8.48 8.49 8.50 2.1353 2.1365 2.1377 2.1389 2.1401 7-31 7.32 7-33 7-34 7-35 1.9892 1.9906 1.9920 1-9933 1.9947 7.71 7.72 7-73 7-74 7-75 2.0425 2.0438 2.0451 2.0464 2.0477 8.ii 8.12 8.13 8.14 8.15 2.0931 2.0943 2.0956 2.0968 2.0980 8.51 8.52 8.53 8.54 8-55 2.1412 2.1424 2.1436 2.1448 2-1459 7.36 7-37 7.33 7-39 7.40 1.9961 1.9974 1.9988 2.0001 2.0015 7.76 7-77 7.78 7-79 7.80 2.0490 2.0503 2.0516 2.0528 2.0541 8.16 8.17 8.18 8.19 8.20 2.0992 2.1005 2.IOI7 2.1029 2.I04I 8.56 8.57 8.58 8-59 8.60 2.1471 2.1483 2.1494 2.1506 2.1518 7.41 7.42 7-43 7-44 7-45 2.0028 2.0042 2.0055 2.0069 2.0082 7.81 7.82 7-83 7.84 7.85 2.0554 2.0567 2.0580 2.0592 2.0605 8.21 8.22 8.2 3 8.2 4 8.2 5 2.1054 2.1066 2.1078 2.1090 2.II02 8.61 8.62 8.63 8.64 8.65 2.1529 2.1541 2.I55 2 2.1564 2.1576 7.46 7-47 7.48 7-49 7-5 2.0096 2.0IO9 2.0122 2.0136 2.0149 7.86 7.87 7.88 7.89 7.90 2.o6l8 2.0631 2.0643 2.0656 2.0669 8.26 8.2 7 8.28 8.29 8.30 2.III4 2.II26 2.1138 2.II50 2.1163 8.66 8.67 8.68 8.69 8.70 2.1587 2.1599 2.1610 2.1622 2.1633 7-5i 7-52 7-53 7-54 7-55 2.0l62 2.0176 2.0189 2.0202 2.0215 7.91 7.92 7-93 7-94 7-95 2.0681 2.0694 2.0707 2.0719 2.0732 8. 3 I 8.32 8-33 8.34 8.35 2.II75 2.II87 2.1199 2.I2II 2.1223 8.71 8.72 8-73 8.74 8-75 2.1645 2.1656 2.1668 2.1679 2.1691 7.56 7-57 7.58 7-59 7.60 2.0229 2.0242 2.0255 2.0268 2.028l 7.96 7-97 7.98 7-99 8.00 2.0744 2.0757 2.0769 2.0782 2.0794 8.36 8-37 8.38 8-39 8.40 2.1235 2.1247 2.1258 2.I27O 2.1282 8.76 8.77 8.78 8.79 8.80 2.1702 2.1713 2.1725 2.1736 2.1748 HYPERBOLIC LOGARITHMS OF NUMBERS. Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. 8.81 2.1759 9.II 2.2094 9.41 2.2418 9.71 2.2732 8.82 2.1770 9.12 2.2105 9.42 2.2428 9.72 2.2742 8.83 2.1782 9.13 2.2Il6 9-43 2.2439 9-73 2.2752 8.84 2.1793 9.14 2.2127 9.44 2.2450 9-74 2.2762 8.85 2.1804 9.15 2.2138 9-45 2.2460 9-75 2.2773 8.86 2.l8l5 9.16 2.2148 9.46 2.2471 9.76 2.2783 8.87 2.1827 9.17 2.2159 9.47 2.2481 9-77 2.2793 8.88 2.1838 9.18 2.2170 9.48 2.2492 9.78 2.2803 8.89 2.1849 9.19 2.2l8l 9-49 2.2502 9-79 2.2814 8.90 2.1861 9.20 2.2192 9-5 2.2513 9.80 2.2824 8.91 2.1872 9 .2I 2.2203 9-51 2.2523 9.81 2.2834 8.92 2.1883 9.22 2.2214 9-52 2.2534 9.82 2.2844 8-93 2.1894 9. 2 3 2.2225 9-53 2.2544 9-83 2.2854 8.94 2.1905 9.24 2.2235 9-54 2-2555 9.84 2.2865 8-95 2.1917 9.25 2.2246 9-55 2-2565 9-85 2.2875 8.96 2.1928 9.26 2.2257 9.56 2.2576 9.86 2.2885 8.97 2.1939 9.27 2.2268 9.57 2.2586 9.87 2.2895 8.98 2.1950 9.28 2.2279 9.58 2.2597 9.88 2.2905 8.99 2.1961 9.29 2.2289 9-59 2.2607 9.89 2.2915 9.00 2.1972 9.30 2.2300 9.60 2.26l8 9.90 2.2925 9.01 2.1983 9.31 2.23II 9.61 2.2628 9.91 2-2935 9.02 2.1994 9.32 2.2322 9.62 2.2638 9.92 2.2946 9-03 2.2006 9-33 2.2332 9.63 2.2649 9-93 2.2956 9.04 2.2017 9-34 2.2343 9.64 2.2659 9-94 2.2966 9-05 2.2028 9-35 2.2354 9.65 2.2670 9-95 2.2976 9 t o6 2.2039 9.36 2.2364 9.66 2.2680 9.96 2.2986 9.07 2.2050 9-37 2.2375 9.67 2.2690 9-97 2.2996 9.08 2.2061 9.38 2.2386 9.68 2.2701 9.98 2.3006 9.09 2.2072 9-39 2.2396 9.69 2.27II 9.99 2.3016 9.10 2.2083 9.40 2.2407 9.70 2.2721 IO.OO 2.3026 10.25 2.3279 12.75 2-5455 I 5-S Q 2.7408 21.0 3-0445 10.50 2 -35 I 3 13.00 2.5649 16.0 2.7726 22.0 3.09II iQ-75 2-3749 13-25 2.5840 16.5 2.8034 23.0 3-1355 II. OO 2.3979 13.5 2.6027 17.0 2.8332 24.0 3.I78I 11.25 2.4201 13.75 2.6211 J 7-5 2.8621 25.0 3.2l8 9 11.50 2.4430 14.00 2.6391 18.0 2.8904 26.0 3.2581 11.75 2.4636 14-25 2.6567 18.5 2.9173 27.0 3-2958 12.00 2.4849 14.50 2.6740 19.0 2-9444 28.0 3-3322 12.25 2.5052 14-75 2.6913 19-5 2.9703 29.0 3.3673 12.50 2.5262 15.00 2.7081 2O.O 2-9957 30.0 3.4012 66 MATHEMATICAL TABLES. TABLE No. III. NUMBERS, OR DIAMETERS OF CIRCLES, CIR- CUMFERENCES, AREAS, SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. I 3.I4I6 0.7854 I I 1. 000 1. 000 2 6.28 3.14 4 8 I.4I4 1.2^0 3 9.42 o- *- 1 7.07 T- 9 27 T" T^ 1.732 D ? 1.442 , . 12X7 12X7 16 64 2. OOO I ^87 5 *" j 1 "' o 1 19.63 2 5 I2 5 2.236 1.709 6 ... 18.85 28.27 16 . ... 216 2.44Q I 8l7 / 1 1 S ****/ 7 21.99 38.48 49 343 2.645 I.9I2 8 ... 25.13 50.27 ...... 64 512 2.828 2. OOO 9 28.27 63.62 81 729 3.000 2.080 10 .. 31.42 78.154 100 1,000 3.162 2 I ^4 ii , 34.56 / O ^ 95.03 121 O 3-3 l6 2.223 12 ... 37.70 113.10 144 1^728 3.464 2.289 13 40.84 132.73 169 2,197 3-605 2-35 1 14 . . 43.Q8 1^3 04 . 106 2,744 3.741 2.4IO * *T 15 to y - 47.12 176.71 7 225 *j / *r*r 3,375 O' 1 ^ 3.872 Wi.i{.i W 2.466 16 ... 50.26 201.06 4,096 4.000 2.519 17 53-41 226.98 289 4,913 4.123 2.571 18 .. c6. s z ..2^4.47 324 e 832 4.242 2.62O J J J OT^ *T / o T- D,^O^ 19 59.69 283.53 3 6l 6,859 4.358 2.668 20 6^ 83 3 IzL 1 6 4OO 8 ooo A 472 2 7 T A. t t* i tl * * / x ^f 21 65.97 346.36 441 9,261 4.582 2.758 22 ... 69.11 380.13 484 10,648 4.690 2.802 23 72.26 415.48 529 12,167 4-795 2.843 24 ... 75.40 452.39 576 13,824 4.898 2.884 2 5 78.54 490.87 62 5 15.625 5.0OO 2.924 26 ... 81.68 530.93 676 17,576 5-99 2.962 27 84.82 572.56 729 19,683 5.196 3.000 28 ..87.96 615.75 784 21,952 5.291 3.036 29 91.11 660.52 841 24,389 5.385 3.072 30 ... 94.25 706.86 900 27,000 5-477 3.107 3 1 97-39 754-77 961 29,791 5-567 3.141 32 100.53 804.25 ... 1,024 32,768 5-656 3,174 33 103.67 855.30 ,08 9 35,937 5-744 3.207 34 106.81 907.92 ... ,I 5 6 39,304 5-830 3-239 35 109.96 962.11 ,225 42,875 5.916 3.271 36 113.10 ... 1017.88 ... ,296 46,656 6. ooo 3-3 01 37 116.24 1075.21 ,369 50,653 6.082 3-332 38 119.38 ... 1134.11 ... ,444 54,872 6.164 3-361 39 122.52 1194.59 ,521 59,3i9 6.244 3-39 1 40 125.66 ... 1256.64 ,600 64,000 6.324 3-4I9 41 128.80 1320.25 ,681 68,921 6.403 3-448 42 w-n ... 1385.44 ... ,764 74,o88 6.480 3-476 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 6 7 Number, or Diameter, j Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 43 1-35-09 I452.2O 1,849 79,507 6-557 3.503 44 138.23 ... I5 2 0.53 ... 1,936 85,184 6.633 3-530 45 MI.37 I590-43 2,025 91,125 6.708 3.556 46. I44-5 1 ... 1661.90 ... 2,116 97,336 6.782 3.583 47 147-65 1734.94 2,209 103,823 6.855 3.608 48 150.80 ... 1809.56 ... 2,304 H0,592 6.928 3-634 49 I53.94 1885.74 2,401 117,649 7-000 3.659 5 157.08 ... 1963.50 ... 2,500 125,000 7.071 3.684 5 1 160.22 2O42.82 2,601 132,651 7.I4I 3.708 5 2 163.36 ... 2123.72 ... 2,704 I4O,6o8 7.2II 3.73 2 53 166.50 2206.18 2,809 148,877 7.280 3.756 54 169.65 ... 2290.22 ... 2,916 157,464 7.348 3-779 55 172.79 2375^3 3,025 166,375 7.416 3.802 56 175-93 ... 2463.01 ... 3.136 I75,6l6 7.483 3.825 57 179.07 255I-76 3 ? 249 185,193 7-549 3.848 58 182.21 ... 2642.08 .- 3>3 6 4 i95, 112 7-6i5 3.870 59 185.35 2733.97 3,481 205,379 7.681 3.892 60 188.50 ... 2827.43 ... 3,600 216,000 7-745 3-9 J 4 61 191.64 2922.47 3,721 226,981 7.810 3.936 62 194.78 ... 3019.07 ... 3,844 238,328 7-874 3-957 63 197.92 3II7-25 3,969 250,047 7-937 3-979 64 201.06 ... 3216.99 ... 4,096 ... 262,144 8.000 4.000 65 204.20 33 l8 -3 I 4,225 274,625 8.062 4.020 66 207.34 ... 3421.19 ... 4,356 287,496 8.124 4.041 67 210.49 3525- 6 5 4,489 300,763 8.185 4.061 68 213.63 ... 3631.68 ... 4,624 3 J 4,43 2 8.246 4.081 69 216.77 3739-28 4,761 328,509 8.306 4.101 70 219.91 . 3848.45 ... 4,900 343,00 8.366 4.121 7i 223.05 3959.19 5,04! 357,9n 8.426 4.140 72 226.19 ... 4071-5 ... 5,184 373, 2 48 8.485 4.160 73 229.34 4185-39 5,329 389,017 8.544 4.179 74 232.48 ... 4300.84 ... 5,476 405,224 8.602 4.198 75 235.62 4417.86 5,625 421,875 8.660 4.217 76 238.76 ... 4536.46 5,776 438,976 8.717 4-235 77 241.90 4656.63 5,929 456,533 8-744 4-254 78 245.04 ... 4/78.36 ... 6,084 474,552 8.831 4.272 79 248.19 4901.67 6,241 493,039 8.888 4.290 So 2 5 r -33 ... 5026.55 ... 6,400 512,000 8.044 4.308 81 254.47 5i53-o o 6,561 53M4i X I ^ 9.000 \J 4.326 82 257.61 ... 5281.02 ... 6,724 551,368 9-055 4-344 83 260.75 5410.61 6,889 571,787 9.110 4.362 84 263.89 5541-77 ... 7,056 592,704 9.165 4-379 85 267.03 5674-5 7,225 614,125 9.219 4.396 86 270.18 ... 5808.80 ... 7,396 636,056 9-273 4.414 87 273.32 5944-68 7,569 658,503 9-327 4-43 1 88 276.46 ... 6082.12 ... 7,744 681,472 9.380 4-447 89 279.60 6221.14 7,921 704,969 9-433 4.461 go 282.74 ... 6361.73 ... 8,100 729,000 9.486 4.481 68 MATHEMATICAL TABLES. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 9 1 285.88 6503.88 8 5 28l 753,571 9-539 4-497 92 289.03 ... 6647.61 ... 8,464 778,688 9-59i 4.514 93 292.17 6792.91 8,649 804,357 9-643 4-530 94 295-3 1 ... 6939.78 ... 8,836 830,584 9-695 4.546 95 298.45 7088.22 9,025 857,375 9.746 4-562 96 3oi-59 ... 7238.23 ... 9,216 884,736 9-797 4.578 97 304.73 7389.81 9,409 912,673 9.848 4-594 08 ^07.88 . 7=542.06 . 0,604 .. 04.1,102 0.800 4.610 ? 99 O / 311.02 / O -/ 7697.69 ... y y WVf 9,801 yf-*-, * y** 970,299 y* ss 9-949 T.. **:*** 4.626 IOO 314.16 ... 7853.98 ...10,000 ,000,000 10.000 4.641 IOI 3*7-3 8011.85 10,201 .030,301 10.049 4-657 102 320.41 ... 8171.28 ...10,404 ,061,208 10.099 4.672 103 323-58 8332.29 10,609 ,092,727 10.148 4.687 IO4 326.73 ... 8494.87 .. .10,816 ... ,124,864 10.198 4.702 105 329.87 8659.01 11,025 ^57,625 10.246 4.7I7 106 333- 01 8824.73 ...11,236 ,191,016 10.295 4-732 107 336.15 8992.02 n,449 ,225,043 10.344 4-747 108 339.29 ... 9160.88 ...11,664 ... ,259,712 10.392 4.762 109 342-43 933L32 11,881 .295,029 10.440 4.776 no 345-57 9503-32 ...12,100 ... ,331,000 10.488 4.791 in 348.72 9676.89 12,321 I >367, 6 3 1 10.535 4.805 112 351.86 ... 9852.03 ...12,544 ... 1,404,928 10.583 4.820 "3 355-oo 10028.75 12,769 1,442,897 10.630 4-834 114 358.14 ...10207.03 ...12,996 ... 1,481,544 10.677 4.848 JI 5 361.28 10386.89 13.225 ^520,875 10.723 4.862 116 364-42 ...10568.32 ...13.456 ... 1,560,896 10.770 4.876 117 367-57 10751.32 13.689 1,601,613 10.816 4.890 118 370.71 ...10935.88 ...13,924 ... 1,643,032 10.862 4-904 119 373.85 11122.02 14,161 1,685,159 10.908 4.918 I2O 376.99 ...11309.73 ...14,400 ... 1,728,000 10.954 4.932 121 380.13 11499.01 14,641 I ,77 I ,5 6 i II.OOO 4.946 122 383-27 ...11689.87 ...14,884 ... 1,815,848 11.045 4-959 I2 3 386.42 11882.29 i5> 12 9 1,860,867 11.090 4-973 124 389.56 ...12076.28 ...15.376 ... 1,906,624 11-135 4.986 125 392.70 12271.85 15.625 I ,953, I2 5 11.180 5.00 126 395.84 ...12468.98 ...15,876 ... 2,000,376 11.224 5-013 127 398.98 12667.69 16,129 2,048,383 11.269 5.026 128 402.12 ...12867.96 ...16,384 ... 2,097,152 11-313 5-039 I2 9 405.26 13069.81 16,641 2,146,689 n-357 S-oS 2 I 3 408.41 13273-23 ...16,900 ... 2,197,000 11.401 5-065 I 3 I 411-55 13478.22 17,161 2,248,091 n.445 5-078 132 414.69 ..13684.78 ...17,424 ... 2,299,968 11.489 5-09I 133 417-83 13892.91 17,689 2,352,637 n-53 2 5.104 !34 420.97 ...I4I02.6l ...17,956 ... 2,406,104 n-575 $"7 J 35 424.11 14313.88 18,225 2,460,375 11.618 5.129 136 427.26 ...14526.72 ...18,496 2,515,456 11.661 5-U2 137 430.40 14741.14 18,769 2,57i,353 11.704 5-J55 138 433-54 ...14957.12 ...19,044 ... 2,620,872 11.747 5- T 67 NUMBERS, OR DIAMETERS OF CIRCLES, &c. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 139 436.68 15174.68 19.321 2,685,619 11.789 5.180 140 439.82 ...15393.80 ...19,600 ... 2,744,000 11.832 5.1 9 2 141 442.96 15614.50 I9,88l 2,803,221 11.874 5.204 142 446. 1 1 ...15836.77 ...20,l64 ... 2,863,288 II.9I6 5.217 143 449.25 l6o6o.6l 20,449 2,924,207 11.958 5.22 9 144 45 2 -39 ...16286.02 ...20,736 2,985,984 I2.OOO 5.241 145 455-53 16513.00 21,025 3,048,625 12.041 5-253 146 458.67 ...16741.55 ...21,316 ... 3,112,136 12.083 5-265 147 461.81 16971.67 21,609 3,176,523 12.124 5-277 148 464.96 ...17203.36 ...21,904 ... 3,241,792 12.165 5.289 149 468.10 17436.62 22,201 3,307,949 I2.2O6 5.301 150 471.24 ...17671.46 ...22,500 3,375,ooo 12.247 5.313 151 474.38 17907.86 22,801 3,442,95! 12.288 5.325 J 5 2 477-52 ...18145.84 ...23,104 ... 3,5^,808 12.328 5-336 J 53 480.66 18385.39 23,409 3,58i,577 12.369 5-348 J 54 483.80 ...18626.50 ...23,716 ... 3,652,264 12.409 5.360 J55 486.95 18869.19 24,025 3,723,875 12.449 5-371 156 490.09 ...19113.45 ---24,336 ... 3,796,416 12.489 5.383 i57 493-23 19359.28 24,649 3,869,893 12.529 5-394 158 496.37 ...19606.68 ...24,964 3>944,3 12 12.569 5.406 !59 499-5 1 19855.65 25,28l 4,019,679 12.609 5-4I7 160 502.65 ...2OI06.I9 ...25,600 ... 4,096,000 12.649 5.428 161 505.80 20358.34 25,921 4,173,281 12.688 5-440 162 508.94 ...20611.99 ...26,244 4,251,528 12.727 5-451 163 512.08 20867.24 26,569 4,330,747 12.767 5.462 164 5 J 5.22 ...21124.07 ...26,896 ... 4,410,944 12.806 5-473 165 518.36 21382.46 27,225 4,492,125 12.845 5-484 1 66 521.5 ...21642.43 27,556 ... 4,574,296- 12.884 5-495 167 524.65 21903.97 27,889 4,657,463 12.922 5-506 168 527.79 ...22167.08 ...28,224 ... 4,741,632 12.961 5.5I7 169 530.93 22431.76 28,561 4,826,809 13.000 5-528 170 534.07 ...22698.01 ...28,900 ... 4,913,000 13-038 5-539 171 537-21 22965.83 29,241 5,000,21 1 13.076 5-550 172 540.35 ...23235.22 ...29,584 ... 5,088,448 13.114 5-561 173 543.50 23506.18 29,929 5,177,717 I 3^S 2 5-572 174 546.64 ...23778.71 ...30,276 ... 5,268,024 13.190 5-582 175 549.78 24052.82 30,025 5,359,375 13.228 5-593 176 552-92 ...24328.49 ...30,976 5,45^776 13.266 5-604 177 556.o6 24605.79 31,329 5,545,233 13-304 5.614 178 559-20 ...24884.56 ...31,684 5,639,752 i3.34i 5-625 179 562.34 25164.94 32,041 5,735,339 13-379 5.635 180 565.49 ...25446.90 ...32,400 ... 5,832,000 13.416 5.646 181 568.63 25730.43 32,76l 5,929,741 13-453 5.656 182 571-77 ...26015.53 33, I 24 ... 6,028,568 13.490 5.66 7 183 574-91 26302.20 33,489 6,128,487 13-527 5.677 184 578.05 ...26590.44 - -.33,856 ... 6,229,504 13-564 5.687 185 581.19 26880.25 34,225 6,331*625 13.601 5.698 186 584-34 ...27171.63 34,596 ... 6,434,856 13-638 5.7o8 MATHEMATICAL TABLES. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. I8 7 587.48 27464.59 34,969 6,539,203 13.674 5.718 188 590.62 ...27759.11 35,344 ... 6,644,672 13.711 5.728 189 593-76 28055.21 35.721 6,751,269 13.747 5.738 190 596.90 ...28352.87 ...36,100 ... 6,859,000 13.784 5.748 191 60O.O4 28652.11 36,481 6,967,871 13.820 5.758 192 603.19 ...28952.92 ...36,864 ... 7,077,888 13.856 5-768 !93 606.33 29255-30 37,249 7,189,057 13.892 5.778 194 609.47 ...29559.26 ...37,636 ... 7,301,384 13.928 5-788 195 612.61 29864.77 38,025 7 5 4M,875 13.964 5-798 196 6i5.75 ...30171.86 ...38,416 7,529,536 I4.OOO 5.808 197 618.89 30480.52 38,809 7,645,373 14.035 5-8l8 198 622.03 ...30790.75 ...39,204 ... 7,762,392 14.071 5.828 199 625.18 3H02.55 39,601 7,880,599 14.106 5.838 200 628.32 3 I 4i5-93 ...40,000 ... 8,OOO,OOO 14.142 5.848 201 631.46 31730.87 40,401 8,120,601 14.177 5.857 202 634.60 ...32047.39 ...40,804 ... 8,242,408 14.212 5.867 203 637-74 32365-47 41,209 8,365,427 14.247 5.877 204 640.88 ...32685.13 ...41,616 ... 8,489,664 14.282 5.886 205 644.03 33006.36 42,025 8,615,125 I4.3I7 5.896 206 647.17 ...33329.16 ...42,436 ... 8,741,816 14.352 5.905 207 650-31 33653.53 42,849 8,869,743 14.387 5.915 208 653-45 33979.47 ...43,264 ... 8,998,912 14.422 5-924 209 656-59 34306.98 43,681 9,123,329 14.456 5-934 210 659.73 ...34636.06 ...44,100 ... 9,26l,000 14.491 5-943 211 662.88 34966.71 44,52i 9,393,931 I4-525 5-953 212 666.02 ..35298.94 ...44,944 ... 9,528,128 14.560 5-962 2I 3 669.16 35632.73 45.369 9,663,597 14-594 5-972 214 672.30 ..35968.09 ...45,796 ... 9,800,344 14.628 5.981 215 675.44 36305-03 46,225 9,938,375 14.662 5-990 216 678.58 ..36643.61 ...46,656 ...10,077,696 14.696 6.000 217 681.73 36983.61 47,089 10,218,313 14.730 6.009 218 684.87 37325-26 ...47.524 ...10,360,232 14.764 6.018 219 688.01 37668.48 47,961 !0,503,459 14.798 6.027 220 691.15 ..38013.27 ...48,400 ...10,648,000 14.832 6.036 221 694.29 38359-63 48,841 10,793,861 14.866 6.045 222 697.43 ..38707.56 ...49,284 ...10,941,048 14.899 6.055 223 700.57 39057.07 49,729 11,089,567 J 4-933 6.064 224 703.72 ..39408.14 ...50,176 ...11,239,424 14.966 6.073 225 706.86 39760.78 50,625 11,390,625 15.000 6.082 226 710.00 ..40115.00 ...51,076 n, 543,176 !5.033 6.091 227 7I3-I4 40470.78 5^529 11,697,083 15.066 6.100 228 716.28 ..40828.14 ...51,984 ...11,852,352 15.099 6.109 229 719.42 41187.07 52,44i 12,008,989 I 5- I 32 6.118 230 722.57 41547.56 ...52,900 ...12,167,000 15-165 6.126 2 3 I 725-71 41909.63 53.361 12,326,391 15.198 6.135 232 728.85 -.42273-27 53,824 ...12,487,168 J 5-23i 6.144 233 73!-99 42638.48 54,289 12,649,337 15.264 6.153 234 735- I 3 ..43005.26 54,756 ...12,812,904 15-297 6.162 NUMBERS, OR DIAMETERS OF CIRCLES, &c. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 235 738.27 43373-61 55.225 12,977,875 !5-329 6.171 236 741.42 43743-54 ...55,696 ...13,144,256 15.362 6.179 237 744.56 44II5-03 56,169 i3>3 12 .53 J 5-394 6.188 238 747.70 ...44488.09 .56,644 ...13,481,272 15-427 6.197 239 750.84 44862.73 57. 121 13,651,919 I 5-459 6.205 240 753.98 45238.93 ...57,600 ...13,824,000 I5-49 1 6.214 241 757.12 45616.71 58,081 i3,997,52i i5-5 2 4 6.223 242 760.26 ...45996.06 .-58,564 ...14,172,488 6.231 243 763.4I 46376.98 59.049 14,348,907 15-588 6.240 244 766.55 ...46759.47 59.536 ...14,526,784 15.620 6.248 245 769.69 47143.52 60,025 14,706,125 15.652 6.257 246 772.83 ...47529.16 ...60,516 ...14,886,936 15.684 6.265 247 775-97 47916.36 61,009 15,069,223 15.716 6.274 248 779.11 ...48305.13 ...61,504 ...15,252,992 I5-748 6.282 249 782.26 48695.47 62,001 15,438,249 !5-779 6.291 250 785.40 ...49087.39 ...62,500 ...15,625,000 15.811 6.299 251 788.54 49480.87 63,001 15,813,251 15.842 6.307 252 791.68 ...49875.92 -63,504 ...16,003,008 15.874 6.316 253 794.82 50272.55 64,009 16,194,277 6.324 254 797.96 ...50670.75 ...64,516 ...16,387,064 15-937 6-333 255 801.11 51070.52 65,025 16,581,375 15.968 6.341 256 804.25 ...51471.86 65,536 ...16,777,216 16.000 6-349 257 807.39 51874.76 66,049 i6,974,593 16.031 6.357 258 810.53 ...52279.24 ...66,564 17,173,512 16.062 6.366 259 813.67 52685.29 67,081 17,373,979 16.093 6-374 260 816.81 ...53092.96 ...67,600 17,576,000 16.124 6.382 26l 819.96 53502.11 68,121 I 7,779,58 r 16.155 6.390 262 823.10 ...53912.87 ...68,644 ...17,984,728 16^186 6.398 263 82.6.24 54325.21 69,169 18,191,447 16.217 6.406 264 829.38 54739-I 1 ...69,696 18,399,744 16.248 6.415 265 832.52 55 J 54-59 70,225 18,609,625 16.278 6.423 266 835.66 -55571-63 ...70,756 ...18,821,096 16.309 6.431 267 838.80 55990.25 71,289 19,034,163 16.340 6-439 268 841.95 ...56410.44 ...71,824 ...19,248,832 16.370 6.447 269 845.09 56832.20 72,361 19,465,109 16.401 6-455 270 848.23 57255-53 ...72,900 ...19,683,000 16.431 6.463 271 85L37 57680.43 73,441 19,902,511 16.462 6.471 272 854-51 ...58106.90 73.984 ...20,123,648 16.492 6-479 273 857.65 58534-94 74,529 20,346,417 16.522 6.487 274 860.80 58964.55 ...75,076 ...20,570,824 16.552 6-495 275 863.94 59395-74 75,625 20,796,875 16.583 6.502 2 7 6 867.08 ...59828.49 ...76,176 ...21,024,576 16.613 6.510 277 870.22 60262.82 76,729 21,253,933 16.643 6.518 2 7 8 873.36 ...60698.72 ...77,284 ...21,484,952 16.673 6.526 279 876.50 61136.18 77,84i 21,717,639 16.703 6-534 280 879.65 ...61575.22 ...78,400 ...21,952,000 16.733 6-542 28l 882.79 62015.82 78,961 22,188,041 16.763 6-549 282 885-93 ...62458.00 ...79,524 ...22,425,768 16.792 6-557 MATHEMATICAL TABLES. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 283 889.07 62901.75 80,089 22,665,187 16.822 6.565 284 892.21 ...63347.07 ...80,656 ...22,906,304 16.852 6-573 285 895-35 63793-97 81,225 23,149,125 16.881 6.580 286 898.49 ...64242.43 ...81,796 23,393>656 16.911 6.588 287 901.64 64692.46 82,369 23,639,903 16.941 6.596 288 904.78 ...65144.07 ...82,944 ...23,887,872 16.970 6.603 289 907.92 65597-24 83,521 24,137,569 17.000 6.611 2QO 911.06 ...66051.99 ...84,100 ...24,389,000 17.029 6.619 291 914.20 66508.30 84,681 24,642,171 17-059 6.627 292 9*7-34 ...66966.19 ...85,264 ...24,897,088 17.088 6.634 293 920.49 67425.65 85,849 25^53,757 17.117 6.642 294 923.63 ...67886.68 ...86,436 ...25,412,184 17.146 6.649 295 926.77 68349.28 87,025 25>672,375 17.176 6.657 296 929.91 ...68813.45 ...87,616 25*934,336 17.205 6.664 297 933-05 69279.19 88,209 26,198,073 17-234 6.672 298 936.19 ...69746.50 ...88,804 26,463,592 17.263 6.679 2 99 939-34 70215.38 89,401 26,730,899 17.292 6.687 300 942.48 ...70685.83 ...90,000 . . .27,OOO,OOO 17.320 6.694 3 OI 945.62 71157.86 90,601 27,270,901 *7.349 6.702 302 948.76 ...71631.45 ...91,204 --27,543,608 17-378 6.709 33 951.90 72106.62 91,809 27,818,127 17.407 6.717 34 955-4 --72583-36 ...92,416 ...28,094,464 I7-436 6.724 305 958.19 73061.66 93> 02 5 28,372,625 17.464 6.731 306 9 6l -33 73541-54 ...93,636 ...28,652,616 !7-493 6.739 307 964.47 74022.99 94,249 28,934,443 17.521 6.746 308 967.61 ...74506.01 ...94,864 ...29,2l8,II2 17.549 6-753 309 970-75 74990.60 95.481 29,503,629 17.578 6.761 310 973-89 ...75476.76 ...96,100 ...29,791,000 17.607 6.768 3H 977.03 75964-50 96,721 30,080,231 17-635 6.775 312 980.18 ...76453.80 97,344 ...30,371,328 17.663 6.782 3i3 9 8 3-32 76944-67 97,969 30,664,297 17.692 6.789 3U 986.46 ...77437.12 ...98,596 ...30,959,144 17.720 6.797 3i5 989.60 7793I-I3 99,225 3I,255, 8 75 17.748 6.804 316 992-74 ...78426.72 ...99,856 31,554,496 17.776 6.8n 3i7 995.88 78923.88 100,489 3 r , 8 55,oi3 17.804 6.8i8 3i8 999.03 ...79422.60 101,124 ...32,157,432 17.832 6.826 319 1002.17 79922.90 101,761 32,461,759 17.860 6.833 320 1005.31 ...80424.77 102,400 ...32,768,000 17.888 6.839 321 1008.45 80928.21 103,041 33,076,161 17.916 6.847 322 1011.59 ...81433.22 103,684 ...33,386,248 17.944 6.854 323 1014.73 81939.80 104,329 33,698,267 17.972 6.861 324 1017.88 ...82447.96 104,976 ...34,012,224 18.000 6.868 325 IO2I.O2 82957.68 105,625 34,328,125 18.028 6.8 75 326 I024.I6 ...83468.98 106,276 ...34,645,976 18.055 6.882 327 1027.30 83981.84 106,929 34,965,783 18.083 6.889 328 1030.44 ...84496.28 107,584 35>287,552 18.111 6.896 329 1033.58 85012.28 108,241 35,611,289 18.138 6.903 330 1036.73 ...85529.86 108,900 35,937,000 18.166 6.910 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 73 Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 33 1 1039.87 86049.01 109,561 36,264,691 18.193 6.917 332 1043.01 ...86569.73 IIO,224 ...36,594,368 18.221 6.924 333 1046.15 87092.02 110,889 36,926,037 18.248 6.931 334 1049.29 ...87615.88 in,556 37,259,704 18.276 6.938 335 1052.43 88141.31 112,225 37,595^375 18.303 6-945 336 1055.57 ...88668.31 112,896 37,933,056 18.330 6.952 337 1058.72 89196.88 113.569 38,272,753 18-357 6-959 338 1061.86 ...89727.03 114,244 ...38,614,472 18.385 6.966 339 IO65.0O 90258.74 114,921 38,958,219 18.412 6-973 340 IO68.I4 ...90792.03 115,600 ...39,304,000 18.439 6.979 34i 1071.28 91326.88 116,281 39,651,821 18.466 6.986 342 1074.42 ...91863.31 116,964 ...40,001,688 18.493 6-993 343 1077.57 92401.31 117,649 40,353,607 18.520 7.000 344 1080.71 ...92940.88 118,336 40,707,584 18.547 7.007 345 1083.85 93482.02 119,025 41,063,625 18.574 7.014 346 1086.99 ...94024.73 119,716 ...41,421,736 18.601 7.020 347 1090.13 94569.01 120,409 41,781,923 18.628 7.027 348 1093.27 ...95114.86 121,104 ...42,144,192 18.655 7.034 349 1096.42 95662.28 121,801 42,508,549 18.681 7.040 350 1099.56 ...96211.28 122,500 ...42,875,000 18.708 7.047 35i 1102.70 96761.84 123,201 43,243,5s 1 18.735 7.054 352 1105.84 ...97314.76 123,904 ...43,614,208 18.762 7.061 353 1108.98 97867.68 124,609 43,986,977 18.788 7.067 354 1 1 12. 12 ...98422.96 i25,3 l6 ...44,361,864 18.815 7.074 355 1115.26 98979.80 126,025 44,738,875 18.842 7.081 356 III8.4I ...99538.22 126,736 ...45,118,016 18.868 7.087 357 II2 i.55 100098.21 127,449 45,499,293 18.8-94 7.094 358 1124.69 100659.27 128,164 ...45,882,712 18.921 7.101 359 1127.83 IOI222.9O 128,881 46,268,279 18.947 7.107 360 1130.97 101787.60 129,600 ...46,656,000 18.974 7.114 361 1134.11 102353.87 130,321 47,045,881 19.000 7.120 362 1137.26 102921.72 131,044 47,437,928 19.026 7.127 363 1140.40 103491.13 131,769 47,832,147 19.052 7.133 364 1143.54 IO4062.I2 132,496 ...48,228,544 19.079 7.140 365 1146.68 104634.67 133,225 48,627,125 19.105 7.146 366 1149.82 105208.80 133,956 ...49,027,896 19.131 7.153 367 1152.96 105784.49 134,689 49,430,863 I 9- I 57 7-159 368 1156.11 106361.76 i35>424 49,836,032 19.183 7.166 369 1159.25 106940.60 136,161 50,243,409 19.209 7.172 370 1162.39 107521.01 136,900 ...50,653,000 19-235 7.179 37i 1165.53 108102.99 137,641 51,064,811 19.261 7.185 372 1168.67 108686.54 138,384 ...51,478,848 19.287 7.192 373 1171.81 109271.66 139,129 51,895,117 19-313 7.198 374 1174.96 109858.35 139,876 ...52,313,624 19-339 7.205 375 1178.10 110446.62 140,625 52,734,375 19-365 7.211 376 1181.24 111036.45 !4i,376 .53^57,376 I9-39 1 7.218 377 1184.38 111627.86 142,129 53,582,633 19.416 7.224 378 1187.52 II222O.83 142,884 ...54,010,152 19.442 7.230 74 MATHEMATICAL TABLES: Number, or | Diameter. Circum- Circular ference. Area. Square. Cube. Square Root. Cube Root. 379 1190.66 112815.38 143,641 54,439.939 19.468 7-237 380 II93.80 113411.49 144,400 ...54,872,000 *9-493 7-243 38i 1196.95 114009.18 I45,l6l 55,306,341 19-519 7.249 382 1200.09 114608.44 145.924 ...55,742,968 !9-545 7.256 383 1203.23 115209.27 146,689 56,181,887 J 9-57o 7.262 384 1206.37 115811.67 147,456 ...56,623,104 19.596 7.268 385 1209.51 116415.64 148,225 57,066,625 19.621 7-275 386 1212.65 117021.18 148,996 ...57,512,456 19.647 7.28l 387 1215.80 117628.30 149,769 57,960,603 19.672 7.287 388 1218.94 118236.98 I 5>544 ...58,411,072 19.698 7.294 389 1222.08 118847.24 15^321 58,863,869 19.723 7.299 390 1225.22 119459.06 152,100 59,3 I 9,oo 19.748 7.306 39i 1228.36 120072.46 152,881 59,776,471 19.774 7-312 39 2 1231.50 120687.42 J 53,664 ...60,236,288 19.799 7-319 393 1234.65 121303.96 1 54,449 60,698,457 19.824 7.325 394 1237.79 121922.07 I 55> 2 36 ...61,162,984 19.849 7.331 395 1240.93 122541.75 156,025 61,629,875 19.875 7-337 396 1244.07 123163.00 156,816 ...62,099,136 19.899 7-343 397 1247.21 123785.82 157,609 62,570,773 19.925 7-349 398 1250.35 124410.21 158,404 63,044,792 19.949 7.356 399 1253-49 125036.17 159,201 63,5 2I , I 99 19-975 7.362 400 1256.64 125663.71 160,000 . ..64,000,000 20.000 7-368 401 1259.78 126292.81 160,801 64,481,201 20.025 7-374 402 1262.92 126923.48 161,604 ...64,964,808 20.049 7.380 403 1266.06 127553-73 162,409 65,450,827 20.075 7.386 404 1269.20 128189.55 163,216 65,939,264 20.099 7-392 405 1272.34 128824.93 164,025 66,430,125 20.125 7-399 406 1275.49 129461.89 164,836 ...66,923,416 2O.I49 7-405 407 1278.63 130100.42 165,649 67,4i9> 1 43 20.174 7.411 408 I28I.77 130740.52 166,464 ...67,911,312 20.199 7.417 409 1284.91 131382.19 167,281 68,417,929 2O.224 7.422 410 1288.05 132025.43 168,100 ...68,921,000 20.248 7.429 411 1291.19 132670.24 168,921 69,426,531 20.273 7-434 412 1294.34 1333*6.63 169,744 69,934,528 20.298 7.441 4i3 1297.48 133964-58 170,569 70,444,997 20.322 7-447 414 I30O.62 134614.10 171,396 70,957,944 20.347 7-453 4i5 1303.76 135265.20 172,225 71,473,375 20.371 7-459 416 1306.90 135917.86 ^3,056 ...71,991,296 20.396 7-465 417 1310.04 136572.10 173,889 72,511,713 20.421 7.471 418 I3I3-I9 137227.91 174,724 ...73,034,632 20.445 7-477 419 I3I6.33 137885.29 I 755 61 73,560,059 20.469 7-483 420 i3 J 9-47 138544.24 176,400 ...74,088,000 20.494 7.489 421 1322.61 139204.70 177,241 74,618,461 20.518 7-495 422 I325-75 139866.85 178,084 75, I 5 I ,448 20.543 7-5 01 423 1328.89 140530.51 178,929 75,686,967 20.567 7-507 424 1332-03 I4II95.74 179,776 ...76,225,024 20.591 7-5*3 425 1335-iS 141862.54 . 180,625 76,765,625 20.6l5 7-518 426 1338.32 142530.92 181,476 ...77,308,776 20.639 7-524 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 75 ' Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 427 1341.46 143200.86 182,329 77,854,483 20.664 7-530 428 1344.60 143872.38 183,184 ...78,402,752 20.688 7.536 429 1347-74 144545.46 184,041 78,953,589 20.712 7-542 430 1550.88 145220.12 184,900 ...79,507,000 20.736 7.548 43? I354-03 145896.35 185,761 80,062,991 20.760 7-554 43 2 1357.17 I46574.I5 186,624 ...80,621,568 20.785 7-559 433 1360.31 147253-52 187,489 81,182,737 20.809 7.565 434 I363-45 147934.46 188,356 ...81,746,504 20.833 7-571 435 1366.59 148616.97 189,225 82,312,875 20.857 7-577 436 I3 6 9-73 149301.05 190,096 ...82,881,856 20.881 7-583 437 1372.88 149986.70 190,969 83,453,453 20.904 7-588 438 1376.02 I5 673.93 i 19^844 ...84,027,672 20.928 7-594 439 I379.I6 151362.72 I92J2I 84,604,519 20.952 7.600 440 1382.30 152053.08 ! 193,600 ...85,184,000 20.976 7.606 441 I385-44 152745.02 ! 194,481 85,766,121 21.000 7.612 442 1388.58 153438-53 ;; 195,364 ...86,350,388 21.024 7.617 443 i39 I -73 154133.60 196,249 86,938,307 21.047 7.623 444 1394.87 154830.25 197,136 ...87,528,384 21.071 7.629 445 1398.01 155528.47 198,025 88,121,125 21.095 7.635 446 1401.15 156228.26 198,916 ...88,716,536 21.119 7.640 447 1404.29 156929.62 199,809 89,314,623 21.142 7.646 448 1407.43 157632.55 2OO,7O4 - 8 9>9 I 5,392 21.166 7.652 449 1410.57 158337.06 201,601 90,518,849 21.189 7-657 450 1413.72 i5943- I 3 202,5OO ...91,125,000 21.213 7-663 451 1416.86 159750.77 203,401 9i,733,85i 21.237 7.669 452 1420.00 160459.99 204,304 ...92,345,408 21.260 7-674 453 1423.14 161170.77 205,209 92,959,677 21.284 7.680 454 1426.28 161883.13 2O6,IO6 ...93,576,664 21.307 7.686 455 1429.42 162597.06 207,025 94,196,375 21. 33 1 7.691 456 1432.57 163312.55 207,936 ...94,818,816 21-354 7.697 457 i435-7i 164029.62 208,849 95,443,993 21.377 7-703 458 1438-85 164748.26 209,764 ...96,071,912 21.401 7.708 459 1441.99 165468.47 2IO,68l 96,702,579 21.424 7.714 460 i445- I 3 166190.25 211,600 ...97,336,000 21.447 7.719 461 1448.27 166913.60 212,521 97,972,181 21.471 7.725 462 1451.42 167638.53 2I3>444 ...98,611,128 21.494 - 7-731 463 1454.56 168365.02 214,369 99,252,847 2i.5 J 7 ! 7.736 464 1457.70 169093.08 215,296 99,897,345 21.541 7.742 465 1460.84 169822.72 216,225 100,544,625 21.564 7-747 466 1463.98 170553-92 217,156 101,194,696 21.587 7-753 467 1467.12 171286.70 218,089 101,847,563 21.610 ; 7.758 468 1470.26 172021.05 219,024 102,503,232 21.633 7-764 469 i473'4i 172756.97 219,961 103,161,709 21.656 7.769 470 1476.55 1 73494-45 1 220,900 103,823,000 21.679 7-775 471 1479.69 I74233-5 1 l! 221,841 104,487,111 21.702 7.780 472 1482.83 174974.14 222,784 105,154,048 21.725 7.786 473 1485.97 175716.35 223,729 105,823,817 21.749 7.791 474 1489.11 176460.12 224,676 106,496,424 21.771 7-797 MATHEMATICAL TABLES. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 475 1492.26 177205.46 225,625 107,171,875 21.794 7.802 476 1495.40 177952.37 226,576 107,850,176 2I.8I7 7.808 477 1498.54 178700.86 227,529 108,531,333 21.840 7.813 478 1501.68 179450.91 228,484 109.215,352 21.863 7.819 479 1504.82 180202.54 229,441 109,902,239 21.886 7.824 480 1507.96 180955.74 230,400 110,592,000 21.909 7.830 481 I5II.II 181710.50 231,361 111,284,641 21.932 7.835 482 i5 J 4-25 182466.84 232,324 III,98o,l68 21-954 7.8 4 483 1517-39 183224.75 233.289 112,678,587 21.977 7.8 4 6 484 I 5 20 -53 183984.23 234,256 113.379.904 22.000 7.851 485 i J 5 2 3.67 184745.28 235^25 114,084,125 22.023 7.857 486 j 1526.81 185507.90 236,196 114,791,256 22.045 7.862 487 | 1529.96 186272.10 237,169 ii5.5 OI >33 22.069 7.868 488 1533-10 187037.86 238,144 116,214,272 22.091 7.873 489 ! i53 6 - 2 4 187805.19 239,121 116,936,169 22.113 7.878 490 I539.38 188574.10 240,100 117,649,000 22.136 7.884 491 i54 2 -52 i89344.57 24I,o8l 1*8,370,771 22.158 7.88 9 492 1545.66 190116.62 242,064 119*095,488 22.l8l 7.894 493 1548.80 190890.24 243,049 119.823,157 22.2O4 7.899 494 I 55 I -9S 191665.43 244,036 120,553,784 22.226 7.905 495 : !555-o9 192442.19 245 5 025 121,287,375 22.248 7.910 496 ! !55 8 .23 193220.51 246,016 122,023,936 22.271 7.9I5 497 1561.37 194000.42 247,009 122,763,473 22.293 7.921 498 1564-51 194781.89 248,004 I2 3>55.99 2 22.3l6 7.926 499 1567-65 !95564-93 249,001 124,251,499 22.338 7.932 500 1570.80 i9 6 349.54 250,000 125,000,000 22.361 7-937 5 01 I573.94 197135-72 251,001 125,751,501 22.383 7-942 502 1577.08 197923.48 252,004 126,506,008 22.405 7-947 53 1580.22 198712.80 253.009 127,263,527 22.428 7-953 54 1583-36 199503-70 254,016 128,024,864 22.449 7.958 55 1586.50 200296.17 255.025 128,787,625 22.472 7-963 506 i5 8 9.65 201090.20 256,036 129.554,216 22.494 7-969 57 1592.79 201885.81 257.049 130,323.843 22.517 7-974 508 I595.93 202682.99 258,064 131,096,512 22-539 7-979 59 1599.07 203481.74 259,081 131,872,229 22.561 7-984 5 10 1602.21 204282.06 260,100 132,651,000 22.583 7.989 5 11 1605.35 205083.95 261,121 133.432,831 22.605 7-995 5 12 1608.49 205887.42 262,144 134,217,728 22.627 8.000 5 J 3 1611.64 206692.45 263,169 i35.oo5.697 22.649 8.005 5 J 4 1614.78 207499.05 264,196 135.796,744 22.671 8.010 5 I 5 1617.92 208307.23 265,225 136,590,875 22.694 8.016 5i6 1621.06 209116.97 266,256 137,388,096 22.716 8.021 5 J 7 1624.20 209928.29 267,289 138,188,413 22.738 8.026 5i8 1627.34 210741.18 - 268,324 138,991,832 22-759 8.031 5 J 9 1630.49 211555-63 269,361 139.798,359 22.782 8.036 520 1633-63 212371.66 270,400 140,608,000 22.803 8.041 5 21 1636.77 213189.26 271,441 141,420,761 22.825 8.047 522 1639.91 214008.43 272,484 142,236,648 22.847 8.052 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 77 Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 5 2 3 1643.05 214829.17 273,529 143,055,667 22.869 8.057 5 2 4 1646.19 215651.49 274,576 143,877,824 22.891 8.062 5 2 5 1649.34 2I6475-37 275,625 144,703,125 22.913 8.067 526 1652.48 217300.82 276,676 145,531,576 22-935 8.072 S 2 ? 1655.62 218127.85 277,729 146,363,183 22.956 8.077 528 1658.76 218956.44 278,784 147,197,952 22.978 8.082 529 1661.90 219786.61 279,841 148,035,889 23.000 8.087 530 1665.04 220618.32 280,900 148,877,000 23.022 8-093 531 1668.19 221451.65 281,961 149,721,291 23-043 8.098 532 1671.33 222286.53 283,024 150,568,768 23.065 8.103 533 1674.47 223122.98 284,089 i5 I ,4i9>437 23.087 8.108 534 1677.61 223961.00 285,156 152,273,304 23.108 8.II3 535 1680.75 224800.59 286,225 !53,i3o,375 23.130 8.II8 536 1683.89 225641.75 287,296 153,990,656 23.152 8.123 537 1687.04 226484.48 288,369 154,854,153 2 3 I 73 8.128 538 1690.18 227328.77 289,444 155,720,872 2 3 I 95 8.133 539 1693.32 228174.66 290,521 156,590,819 23.216 8.138 540 1696.46 229022.IO 291,600 157,464,000 23.238 8.143 54i 1699.60 229871.12 292,681 158,340,421 23-259 8.148 542 1702.74 230721.71 293,764 159,220,088 23.281 8.153 543 1705.88 231573.86 294,849 160,103,007 23.302 8.158 544 1709.03 232427.59 295,936 160,989,184 23-324 8.163 545 1712.17 233282.89 297,025 161,878,625 23-345 8.168 546 I7I5-3 1 234139.76 298,116 162,771,336 23.367 8.173 547 1718.45 234998.20 299,209 163,667,323 23-388 8.178 548 1721.59 235858.21 300,304 164,566,592 23.409 8-183 549 1724.73 236719.79 301,401 165,469,149 23-43 1 8.188 550 1727.88 237582.94 302,500 166,375,000 23-452 8.193 55i 1731.02 238447.67 303,601 167,284,151 23-473 8.198 552 1734.16 239313.96 304,704 168,196,608 23-495 8.203 553 I737-30 240181.83 305,809 169,112,377 23-516 8.208 554 1740.44 241051.26 306,916 170,031,464 23-537 8.213 555 I743-58 241922.27 308,025 i7o,953,875 23-558 8.218 556 1746.73 242794.85 309,136 171,879,616 23-579 8.223 557 1749.87 243668.99 310,249 172,808,693 23.601 8.228 558 1753.00 244544.61 3H,364 173,741,112 23.622 8.233 559 1756-15 245422.00 3I2, 4 8l 174,676,879 23-643 8.238 560 1759.29 246300.86 313,600 175,616,000 23.664 8.242 56i 1762.43 247181.30 3!4,72i 176,558,481 23-685 8.247 562 1765-57 248062.30 3i5>844 177,504,328 23.706 8.252 563 1768.72 248946.87 316,969 178,453,547 23.728 8.257 564 1771.86 249832.01 318,096 179,406,144 23.749 8.262 565 1775.00 25 7l8.73 319,225 180,362,125 23.769 8.267 566 1778.14 251607.01 320,356 181,321,496 23.791 8.272 567 1781.28 252496.87 321,489 182,284,263 23.812 8.277 568 1784.42 253388.30 322,624 183,250,432 23.833 8.282 569 1787.57 254281.30 323,761 184,220,009 23-854 8.286 570 1790.71 255 I 75-86 324,900 185,193,000 23-875 8.291 MATHEMATICAL TABLES. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 571 I793-85 256072.00 326,041 186,169,411 23.896 8.296 572 1796.99 256969.71 327,184 187,149,248 23.916 8.301 573 I80O.I3 257868.99 328,329 188,132,51-7 23-937 8.306 574 1803.27 258769.85 329,476 189,119,224 23-958 8. 3 II 575 1806.42 259672.27 33 ? 625 190,109,375 23-979 8.315 576 1809.56 260576.26 33^776 191,102,976 24.000 8.320 577 l8l2.70 261481.83 332,929 192,100,033 24.021 8.325 573 1815.84 262388.96 334,084 !93,ioo,552 24.042 8-330 579 1818.98 263297.67 335,241 194,104,539 24.062 8-335 5 8o I822.I2 264207.94 336,400 195,112,000 24.083 8-339 58i 1825.26 265119.79 337,56i 196,122,941 24.104 8-344 582 1828.41 266033.21 338,724 197,137,368 24.125 8-349 583 I83L55 266948.20 339,889 198,155,287 24.145 8-354 584 1834.69 267864.76 341,056 199,176,704 24.166 8-359 585 1837.83 268782.80 342,225 200,201,625 24.187 8-363 586 1840.97 269702.59 343,396 201,230,056 24.207 8.368 587 1844.11 270623.86 344,569 202,262,003 24.228 8-373 588 1847.26 271546.70 345,744 203,297,472 24.249 8.378 589 1850.40 272471.12 346,921 204,336,469 24.269 8-382 590 I853.54 273397. 10 348,ioo 20 5, 379> 000 24.289 8.387 59i 1856.68 274324.66 349,281 206,425,071 24.310 8.392 592 1859.82 275253-78 350 5 464 207,474,688 24.331 8-397 593 1862.96 276184.48 35^649 208,527,857 24-35 1 8.401 594 l866. 1 1 277116.75 352,836 209,584,584 24.372 8.406 595 1869.25 278050.59 354,025 210,644,875 24-393 8.411 596 l872. 39 278985.99 355,216 211,708,736 24-413 8.415 597 I875-53 279922.97 356,409 212,776,173 24-433 8.420 598 1878.67 280861.53 357,604 213,847,192 24.454 8.425 599 1881.81 281801.65 358,8oi 214,921,799 24.474 8.429 600 1884.96 282743.34 360,000 216,000,000 24-495 8-434 601 1888.10 283686.60 361,201 217,081,801 24-5*5 8-439 602 1891.24 284631.44 362,404 218,167,208 24-536 8.444 603 1894.38 285577.84 363,609 219,256,227 24-556 8.448 604 -1897.52 286525.82 364,816 220,348,864 24.576 8.453 605 1900.66 287475.36 366,025 221,445,125 24-597 8.458 606 1903.80 288426.48 367,236 222,545,016 24.617 8.462 607 1906.95 289379.17 368,449 223,648,543 24.637 8.467 608 1910.09 29 333.43 369,664 224,755,712 24.658 8.472 609 1913.23 291289.26 370,881 225,866,529 24.678 8.476 610 1916.37 292246.66 3.72,100 226,981,000 24.698 8.481 6n 1919.51 293205.63 373,321 228,099,131 24.718 8-485 612 1922.65 294166.17 374,544 229,220,928 24-739 8.490 6i 3 1925.80 295128.28 375.769 230,346,397 24.758 8.495 614 1928.94 296091.97 376,996 23^475,544 24.779 8.499 615 1932.08 297057.22 378,225 232,608,375 24.799 8.504 616 1935.22 298024.05 379,456 233,744,896 24.819 8.509 617 1938.36 298992.44 380,689 234,885,113 24.839 8-513 618 1941.50 299962.41 381,924 236,029,032 24.859 8.518 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 79 Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 619 1944.65 300933.95 383^61 237,1^6,659 24.879 8.522 620 1947.79 301907.05 384,400 238,628,000 24.899 8.527 621 1 9S-93 302881.73 385,641 239,483,061 24.919 8-532 622 1954.07 303857.98 386,884 240,641,848 24-939 8-536 623 1957.21 304835.80 388,I2 9 241,804,367 24-959 8.541 624 1960.35 305815.20 389 ? 376 242,970,624 24.980 8-545 625 1963.50 306796.16 390,625 244,140,625 25.000 8-549 626 1966.64 307778.69 391,876 245,3 J 4>376 25.019 8-554 627 1969.78 308762.79 393,129 246,491,883 25.040 8.559 628 1972.92 309748.47 394,384 247,673,152 25-059 8.563 629 1976.06 3I0735-7I 395,641 248,858,189 25.079 8.568 630 1979.20 3 IJ 724-53 396,900 250,047,000 25.099 8-573 6 3 I 1982.34 312714.92 398,l6l 25^239,591 25.119 8-577 6 3 2 1985.49 313706.88 399.424 252,435,968 2 5- I 39 8.582 633 1988.63 314700.40 400,689 253,636,137 2 5-!59 8.586 634 1991.77 315695-5 401,956 254,840,104 2 5-!79 8.591 635 1994.91 316692.17 403,225 256,047,875 2 5-i99 8-595 6 3 6 1998.05 317690.42 404,496 257,259,456 25-219 8-599 637 2001.19 318690.23 405,769 258,474,853 25-239 8.604 6 3 8 2004.34 319691.61 407,044 259,694,072 25-259 8.609 639 2007.48 320694.56 408,321 260,917,119 25-278 8.613 640 2010.62 321699.09 409,600 262,144,000 25-298 8.618 641 2013.76 322705.18 4IO,88l 263,374,721 25-318 8.622 642 2016.90 323712.85 412,164 264,609,288 25-338 8.627 643 2020.04 324722.09 413,449 265,847,707 25-357 8.631 644 2023.19 325732.89 414,736 267,089,984 25-377 8.636 645 2026.33 326745.27 416,025 268,836,125 25-397 8.640 6 4 6 2029.47 327759.22 417,316 269,586,136 25.416 8.644 647 2032.61 328774.74 418,609 270,840,023 25-436 8.649 648 2035-75 329791.83 419,904 272,097,792 25-456 8.653 649 2038.89 330810.49 421,201 273,359,449 25-475 8.658 6 5 2042.04 331830.72 422,500 274,625,000 2 5-495 8.662 651 2045.18 332852-53 423,801 275,894,451 2 5-5!5 8.667 652 2048.32 333875-90 425,104 277,167,808 25-534 8.671 653 2051.46 334900.85 426,409 278,445,077 25-554 8.676 654 2054.60 3359 2 7-3 6 427,716 279,726,264 25-573 8.680 655 2057.74 336955-45 429,025 281,011,375 25-593 8.684 656 2060.88 337985-!o 430,336 282,800,416 25.612 8.689 657 2064.03 339016.33 431,649 2 83,593,393 25-632 8.693 658 2067.17 340049.13 432,964 284,890,312 25-651 8.698 659 2070.31 341083.50 434,28l 286,191,179 25.671 8.702 660 2073-45 342119.44 435,600 287,496,000 25.690 8.706 661 2076.59 343156.95 436,921 288,804,781 25.710 8.711 662 2079.73 344196.03 438,244 290,117,528 25.720 8.715 663 2082.88 345236-69 439,569 291,434,247 25-749 8.719 664 2086.02 346278.91 440,896 292,754,944 25-768 8.724 665 2089.16 347322.70 442,225 294,079,625 25-787 8.728 666 2092.30 348368.07 443,556 295,408,296 25-807 8-733 8o MATHEMATICAL TABLES. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 667 2095.44 349415.00 444,889 296,740,963 25.826 8-737 668 2098.58 35 463.5I 446,224 298,077,632 25.846 8.742 669 2101-73 35I5I3-59 447,561 299,418,309 25.865 8.746 670 2104.87 35 2 5 6 5- 2 4 448,900 300,763,000 25.884 8.750 671 2108. oi 353 6l8 -43 450,241 302,111,711 25.904 8.753 672 2III.I5 354673-24 451,584 303,464,448 2 5-923 8.759 673 2114.29 355729.60 452,929 304,821,217 25.942 8.763 674 2117.43 356787.54 454,276 306,182,024 25.961 8.768 675 2120.58 357847.04 455,625 307,546,875 25.981 8.772 676 2123.72 358908.11 456,976 308,915,776 26.OOO 8.776 677 2126.86 359970.75 458,329 310,288,733 26.019 8.781 678 2130.00 361034.97 459,684 311,665,752 26.038 8.785 679 2I33-I4 362100.75 461,041 313,046,839 26.058 8.789 680 2136.28 363168.11 462,400 314,432,000 26.077 8.794 681 2139.42 364237.04 463,761 315,821,241 26.096 8.798 682 2142.57 3653 7.54 465,124 317,214,568 26.115 8.802 683 2145.71 366379.60 466,489 318,611,987 26.134 8.807 684 2148.85 367453-24 467,856 320,013,504 26.153 8.811 685 2151.99 368528.45 469,225 321,419,125 26.172 8.815 686 2I55-I3 369605.23 470,596 322,828,856 26.192 8.819 687 2158.27 370683.59 471,969 324,242,703 26.211 8.824 688 2l6l.42 37I763-5 1 473,344 325,660,672 26.229 8.828 689 2164.56 372845.00 474,721 327,082,769 26.249 8.832 690 2167.70 373928.07 476,100 328,509,000 26.268 8.836 691 2170.84 375012.70 477,481 329,939,371 26.287 8.841 692 2173.98 376098.91 478,864 331,373,888 26.306 8.845 693 2177.12 377186.68 480,249 332,812,557 26.325 8.849 694 2l8o.27 378276.03 481,636 334,255,384 26.344 8-853 6 95 2183.41 379366.95 483,025 335,702,375 26.363 8.858 696 2186.55 380459.44 484,416 337,153,536 26.382 8.862 697 2189.69 38i553.5 485,809 338,608,873 26.401 8.866 698 2192.83 382649.43 487,204 340,068,392 26.419 8.870 699 2195.97 383746.33 488,601 341,532,099 26.439 8-875 700 2199.12 384845.10 490,000 343,000,000 26.457 8.879 701 2202.26 385945-44 491,401 344,472,101 26.476 8.883 702 2205.40 387047-36 492,804 345,948,088 26.495 8.887 703 2208.54 388150.84 494,209 347,428,927 26.514 8.892 704 2211.68 389255-90 495,616 348,913,664 26.533 8.896 705 2214.82 390362.52 497,025 350,402,625 26.552 8.900 706 2217.96 391470.32 498,436 351,895,816 26.571 8.904 707 2221. II 392580.49 499,849 353,393,243 26.589 8.908 708 2224.25 393691.83 501,264 354,894,9 12 26.608 8.913 709 2227.39 394804.74 502,681 356,400,829 26.627 8.917 710 2230.53 3959 I 9- 21 504,100 357,911,000 26.644 8.921 711 2233.67 397035-27 505,521 359,425,431 26.664 8.925 712 2236.81 398152.89 5o6,944 360,944,128 26.683 8.929 7i3 2239.96 399272.08 508,369 362,467,097 26.702 8-934 7i4 2243.10 400392.84 509,796 363,994,344 26.721 8.938 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 8l Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 715 ^246.24 401515.18 511,225 365.525.875 26.739 8.942 7 l6 2249.38 402639.08 512,656 367,061,696 26.758 8.946 717 2252.52 403764.56 514,089 368,601,813 26.777 8.950 7 l8 2255.66 404891.60 515.524 370,146,232 26.795 8-954 719 2258.81 4O6O2O.22 516,961 371,694,959 26.814 8-959 720 2261.95 407150.41 5l8,400 373,248,000 26.833 8.963 721 2265.09 408282.17 519,841 374,805,361 26.851 8.967 722 2268.23 409415.50 521,284 376,367,048 26.870 8.971 7 2 3 2271.37 410550.40 522,729 377,933,067 26.889 8-975 724 2274.51 411686.87 524,176 379.503,424 26.907 8.979 725 2277.66 412824.91 525.625 381,078,125 26.926 8.983 726 2280.80 413964.52 527,076 382,657,176 26.944 8.988 727 2283.94 4i5 I0 5-7* 528,529 384,240,583 26.963 8.992 728 2287.08 416248.46 529,984 385,828,352 26.991 8.996 729 2290.22 417392.79 531.441 387,420,489 27.000 9.000 730 2293.36 418538.68 532,900 389,017,000 27.018 9.004 731 2296.50 419686.15 534,361 390,617,891 27.037 9.008 732 2299.65 420835.19 535.824 392,223,168 27.055 9.012 733 2302.79 421985.79 537.289 393. 8 32,837 27.074 9.016 734 2305-93 423137-97 538,756 395,446,904 27.092 9.020 735 2309.07 424291.72 540,225 397,065,375 27.111 9.023 736 2312.21 425447.04 541,696 398,688,256 27.129 9.029 737 2315-35 426603.93 543.169 400,315,553 27.148 9-033 738 2318.50 427762.40 544,644 401,947,272 27.166 9-037 739 2321.64 428922.43 546,121 403,5 8 3.4i9 27.184 9.041 740 2324.78 430084.03 547.600 405,224,000 27.203 9-045 741 2327.92 431247.21 549,081 406,869,021 27.221 9.049 742 2331.06 432411.95 550.5 6 4 408,518,488 27.239 9-053 743 2334.20 433578.27 552,049 410,172,407 27.258 9-057 744 2337.35 434746.16 553.536 411,830,784 27.276 9.061 745 2340.49 4359 I 5- 62 555.025 4i3.493.625 27.295 9.065 746 2343-63 437086.64 556,5 16 415,160,936 27-3I3 9.069 747 2346.77 438259.24 558,009 416,832,723 27-33 1 9.073 748 2349.91 439433-41 559.504 418,508,992 27-349 9.077 749 2353.05 440609.16 561,001 420,189,749 27.368 9.081 750 2356.20 441786.47 562,500 421,875,000 27.386 9.086 75i 2359-34 442965-35 564,001 423,564,75! 27.404 9.089 752 2362.48 444145.80 565.504 424,525,900 27.423 9.094 753 2365.62 445327-83 567,009 426,957,777 27.441 9.098 754 2368.76 446511.42 568,516 428,661,064 27.459 9.102 755 2371.90 447696.59 570,025 430,368,875 27.477 9.106 756 2375.04 448883.32 571,536 432,081,216 27-495 9.109 757 2378.19 450071.63 573.049 433798,o93 27-5 I 4 9.114 758 23 8 i-33 451261.51 574,564 435.5 I 9>5 12 27-532 9.118 759 2384.47 452452.96 576,o8i 437,245,479 27.549 9.122 760 2387.61 453645.98 577,600 438,976,000 27.568 9.126 761 2 39o.75 454840.57 579. T 2i 440,711,081 27.586 9.129 762 2393-89 456036.73 580,644 442,450,728 27.604 9.134 82 MATHEMATICAL TABLES. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 763 2397.04 457234.46 582,169 444, 1 94,947 27.622 9.138 764 240O.I8 45 8 433.77 583,696 445,943,744 27.640 9.142 765 2403.32 459 6 34-64 585,225 447,697,125 27.659 9.146 766 2406.46 460837.08 586,756 449,455,096 27.677 9.149 7 6 7 2409.60 462041.10 588,289 .451,217,663 27.695 9.154 768 2412.74 463246.69 589,824 452,984,832 27.713 9.158 769 2415.89 464453.84 59^361 454,756,609 27.731 9.162 770 2419.03 465662.57 592,900 456,533,ooo 27.749 9.166 771 2422.17 466872.87 594,441 45 8 ,3 I 4,on 27.767 9.169 772 2425.31 468084.74 595,984 460,099,648 27.785 9-173 773 2428.45 469298.18 597,529 461,889,917 27.803 9.177 774 243L59 475 I 3- I 9 599,076 463,684,824 27.821 9.l8l 775 2434-73 471729.77 600,625 465,484,375 27.839 9.185 776 2437.88 472947.92 602,176 467,288,576 27-857 9.189 777 2441.02 474167.65 603,729 469,097,433 27.875 9.193 778 2444.16 475388.94 605,284 470,910,952 27.893 9.197 779 2447.30 476611.81 606,841 472,729,139 27.910 9.2OI 780 2450.44 477836.24 608,400 474,552,ooo 27.928 9.205 78i 2453-58 479062.25 609,961 476,379,541 27.946 9.209 782 245 6 -73 480289.83 611,524 478,211,768 27.964 9.213 783 2459.87 481518.97 613,089 480,048,687 27.982 9.217 784 2463.01 482749.69 614,656 481,890,304 28.000 9.221 785 2466.15 483981.98 616,225 483,736,025 28.017 9.225 786 2469.29 485215.84 617,796 485,587,656 28.036 9.229 787 2472.43 486451.28 619,369 487,443,403 28.053 9-233 788 2475-5 8 487688.28 620,944 489,303,872 28.071 9-237 789 2478.72 488926.85 622,521 491,169,069 28.089 9.240 790 2481.86 490166.99 624,100 493,039,ooo 28.107 9-244 791 2485.00 491408.71 625,681 494,913,671 28.125 9.248 792 2488.14 492651.99 627,264 496,793,o88 28.142 9.252 793 2491.28 493896.85 628,849 498,677,257 28.160 9.256 794 2494.43 495143.28 630,436 500,566,184 28.178 9.260 795 2497-57 496391.27 632,025 502,459,875 28.196 9.264 796 2500.71 497640.84 633,6l6 504,358,336 28.213 9.268 797 2503-85 498891.98 635,209 506,261,573 28.231 9.271 798 2506.99 500144.69 636,804 508,169,592 28.249 9-275 799 2510.13 501398.97 638,401 510,082,399 28.266 9.279 800 2513.27 502654.82 640,000 512,000,000 28.284 9.283 801 2516.42 5Q39 I2 -25 641,601 513,922,401 28.302 9.287 802 2519.56 505171.24 643,204 515,849,608 28.319 9.291 803 2522.70 506431.80 644,809 517,781,627 28.337 9.295 804 2525-84 507693.94 646,416 519,718,464 28.355 9.299 805 2528.98 508957.65 648,025 521,660,125 28.3 7 2 9.302 806 2532.12 510222.92 649,636 523,606,616 28.390 9.306 807 2535-27 511489.77 651,249 525,557,943 28.408 9.310 808 2538.41 512758.19 652,864 527,514,112 28.425 9-3 J 4 809 254L55 514028.19 654,481 529,474,129 28.443 9.318 810 2544.09 515299.74 656,100 531,441,000 28.460 9.321 NUMBERS, OR DIAMETERS OF CIRCLES, &c. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 811 2547.83 516572.86 657,721 533,4H,73I 28.478 9.325 812 2550.97 5I7847-57 659,344 535,387,328 28.496 813 2554.12 519123.84 660,969 537,366,797 28.513 9-333 814 2557.26 520401.68 662,596 539,353,144 28.531 9-337 815 2560.40 52l68l.IO 664,225 541,343,375 28.548 9.341 816 2563.54 522962.08 665,856 543,338,496 28.566 9-345 817 2566.68 524244.63 667,489 545,338,513 28.583 9.348 818 2569.82 525528.76 669,124 547,343,432 28.601 9.352 819 2572.96 526814.46 670,761 549,353,259 28.6l8 9-356 820 2576.11 528101.73 672,400 551,368,000 28.636 9.360 821 2579.25 529390.56 674,041 553,387,661 28.653 9-364 822 2582.39 530680.97 675,684 555,412,248 28.670 9.367 823 2585.53 531972.95 677,329 557,441,767 28.688 9.371 824 2588.67 533266.50 678,976 559,476,224 28.705 9-375 825 2591.81 534561.63 680,625 561,515,625 28.723 9-379 826 2594.96 535858.32 682,276 563,559,976 28.740 9-383 827 2598.10 537156.58 683,929 565,609,283 28.758 9.386 828 26OI.24 538456.41 685,584 567,663,552 28.775 9-390 829 2604.38 539757-82 687,241 569,722,789 28.792 9-394 830 2607.52 541060.79 688,900 571,787,000 28.810 831 2610.66 542365.34 690,561 573,856,191 28.827 9.401 832 2613.81 543671.46 692,224 575,930,368 28.844 9-405 833 2616.95 544979.15 693,889 578,009,537 28.862 9.409 834 2620.O9 546288.40 695,556 580,093,704 28.879 9-4I3 835 2623.23 547599-23 697,225 582,182,875 28.896 9.417 836 2626.37 548911.63 698,896 584,277,056 28.914 9.420 837 2629.51 550225.61 700,569 586,376,253 28.931 9.424 838 2632.64 55I54LI5 702,244 588,480,472 28.948 9.428 839 2635.80 552858.26 703,921 590,589,719 28.965 9-432 840 2638.94 554176.94 705,600 592,704,000 28.983 9-435 841 2642.08 555497-20 707,281 594,823,321 29.000 9-439 842 2645.22 556819.02 708,964 596,947,688 29.017 9-443 843 2648.36 558142.42 710,649 599,077,107 29.034 9-447 844 2651.50 559467.39 712,336 601,211,584 29.052 9-450 845 2654.65 560793.92 714,025 603,351,125 29.069 9-454 846 2657.79 562122.03 605,495,736 29.086 9.458 847 2660.93 563451.71 717,409 607,645,423 2 9 .103 9.461 848 2664.07 564782.96 719,104 609,800,192 29.120 9-465 849 2667.21 566115.78 720,801 611,960,049 2 9 .I 3 8 850 2670.35 56745. ][ 7 722,500 6l4,I25,OOO 9-473 851 2673.50 568786.14 724,201 616,295,051 29.172 9-476 852 2676.64 570123.67 725,904 618,470,208 29.189 9.480 853 2679.78 571462.77 727,609 620,650,477 29.206 9.483 854 2682.92 572803.45 729,316 622,835,864 2 9 .223 9.487 855 2686.06 574145.69 73 I > 02 5 625,026,375 29.240 9.491 856 2689.20 575489.5 1 732,736 627,222,Ol6 29.257 9-495 857 2692.35 576834.90 734,449 629,422,793 29.274 9-499 858 2695.49 578181.85 736,164 631,628,712 29.292 9.502 8 4 MATHEMATICAL TABLES. Number, or Diameter. Circum- ' ference. Circular Area. Square. Cube. Square Root. Cube Root. 859 2698.63 579530-38 737,881 633,839,779 29.309 9.506 860 2701.77 580880.48 739,600 636,056,000 29.326 9-509 861 2704.91 582232.15 74I532I 638,277,381 29-343 9.513 862 2708.05 583585-39 743,044 640,503,928 29.360 9.517 863 2711.19 584940.21 7445769 642,735,647 29-377 9.520 864 2714.34 586296.59 746,496 644,972,544 29.394 9.524 865 2717.45 587654.54 748,225 647,214,625 29.411 9.528 866 2720.62 589014.07 749,956 649,461,896 29.428 9.532 867 2723.76 59375- 16 751,689 651,714,363 29-445 9-535 868 2726.90 59I737-83 753,424 653,972,032 29.462 9-539 869 2730.04 593102.06 755,!6i 656,234,909 29.479 9-543 870 2733-I9 594467.87 756,900 658,503,000 29.496 9-546 871 2/36.33 595835-25 758,641 660,776,311 29-5I3 9-550 872 2739-47 597204.20 760,384 663,054,848 29.529 9-554 873 2742.61 598574.72 762,129 665,338,617 29.546 9-557 874 2745-75 599946.81 763,876 667,627,624 29-563 9.561 875 2748.89 601320.47 765,625 669,921,875 29.580 9-565 876 2752.04 602695.70 767,376 672,221,376 29-597 9.568 877 2755- 18 604072.50 769,129 674,526,133 29.614 9-572 878 2758.32 605450.88 770,884 676,836,152 29.631 9-575 879 2761.46 606830.82 772,641 679, I 5 I ,439 29.648 9-579 880 2764.60 608212.34 774,400 681,472,000 29.665 9-583 881 2767.74 609595-42 776,161 683,797,841 29.682 9-586 882 2770.89 610980.08 777,924 686,128,968 29.698 9-590 883 2774.03 612366.31 779,689 688,465,387 29-715 9-594 884 2777.17 613754.11 781,456 690,807,104 29.732 9-597 885 2780.31 615143.48 783,225 693,154,125 29-749 9.601 886 2783-45 616534.42 784,996 695,506,456 29.766 9.604 887 2786.59 617926.93 786,769 697,864,103 29.782 9.608 888 2789.73 619321.01 788,544 700,227,072 29.799 9.612 889 2792.88 620716.66 790,321 702,595,369 29.816 9.615 890 2796.02 622113.89 792,100 704,969,000 29-833 9.619 891 2799.16 623512.68 793,881 707,347,971 29.850 9.623 892 2802.30 624913.04 795,664 709,732,288 29.866 9.626 893 2805.44 626314.98 797,449 712,121,957 29-883 9-630 894 2808.58 627718.49 799, 2 3 6 714,516,984 29.900 9-633 895 2811.73 629123.56 801,025 716,917,375 29.916 9-637 896 2814.87 630530.21 802,816 7i9>3 2 3, 1 3 6 29-933 9.640 897 2818.01 631938.43 804,609 721,734,273 29.950 9-644 898 2821.15 633348.22 806,404 724,150,792 29.967 9.648 899 2824.29 634759-58 808,201 726,572,699 29.983 9.651 goo 2827.43 636172.51 810,000 729,000,000 30.000 9.655 901 2830.58 637587-oi 811,804 73 1 >43 2 ,7oi 30.017 9.658 902 2833.72 639003.09 813,604 733,870,808 30.033 9.662 903 2836.86 640420.73 815,409 736,3*4,327 30.050 9.666 904 2840.00 641839.95 817,216 738,763,264 30.066 9.669 905 2843.14 643260.73 819,025 741,217,625 30.083 9-673 906 2846.28 644683.09 820,836 743,677,4i6 30.100 9.676 NUMBERS, OR DIAMETERS OF CIRCLES, &c. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 907 2849.43 646107.01 822,649 746,142,643 30.1l6 9.680 908 2852.57 647532.5 1 824,464 748,613,312 30.133 9.683 909 2855-7I 648959.58 826,281 751,089,429 30.150 9.687 gio 2858.85 650388.21 828,100 753,571,000 30.163 9.690 911 2861.99 651818.43 829,921 756,058,031 30.183 9.694 912 2865.13 653250.21 831,744 758,550,528 30.199 9.698 9i3 2868.27 654683.56 833,569 761,048,497 30.216 9.701 914 2871.42 656118.48 835,396 763,551,944 30.232 9-705 9i5 2874.56 657554.98 837,225 766,060,875 30.249 9.708 916 2877.70 658993. 4 839,056 768,575,296 30.265 9.712 917 2880.84 660432.68 840,889 771,095,213 30.282 9-7I5 918 2883.98 661873.88 842,724 773,620,632 30.298 9 .7l8 919 2887.12 663316.66 844,561 776,151,559 30-3I5 9.722 920 2890.27 664761.01 846,400 778,688,000 30-33 1 9.726 921 2893.41 666206.92 848,241 781,229,961 30-348 9.729 922 2896.55 667654.41 850,084 783,777,448 3 -364 9-733 923 2899.69 669103.47 851,929 786,330,467 30-381 9-736 924 2902.83 670554.10 853,776 788,889,024 30.397 9.740 925 2905.97 672006.30 855,625 791,453,125 30.414 9-743 926 2909.12 673460.08 857,476 794,022,776 30-430 9-747 927 2912.26 674915.42 859,329 796,597,983 30-447 9-75 928 2915.40 6763/2.33 861,184 799,^8,752 30-463 9-754 929 2918.54 677830.82 863,041 801,765,089 30-479 9.757 930 2921.68 679290.87 864,900 804,357,000 30.496 9.761 93i 2924.82 680752.50 866,761 806,954,491 30.512 9.764 932 2927.96 682215.69 868,624 809,557,568 30-529 9.768 933 2931.11 683680.46 870,489 812,166,237 30-545 9.771 934 2934-25 685146.80 872,356 814,780,504 30.561 9-775 935 2937-39 686614.71 874,225 817,400,375 30-578 9.778 936 2940-53 688084.19 876,096 820,025,856 30-594 9-783 937 2943.67 689555.24 877,969 822,656,953 30.610 9-785 938 2946.81 691027.86 879,844 825,293,672 30.627 9.789 939 2949.96 692502.05 88l,72I 827,936,019 30-643 9.792 940 2953.10 693977.82 883,600 830,584,000 30-659 9.796 941 2956.24 695455^5 885,481 833'237,62I 30.676 9-799 942 2959-38 696934.06 887,364 835,896,888 30.692 9.803 943 2962.52 698414.53 889,249 838,561,807 30.708 9.806 944 2965.66 699896.58 891,136 841,232,384 30.724 9.810 945 2968.81 701380.28 893,025 843,908,625 30-74I 9-813 946 2971.95 702865.38 894,916 846,590,536 30-757 9.817 947 2975.09 704352.14 896,809 849,278,123 30.773 9.820 948 2978.23 705840.47 898,704 851,971,392 30.790 9.823 949 2981.37 707330.37 900,601 854,670,349 30.806 9.827 950 2984.51 708821.84 902,500 857,375,000 30.822 9.830 95 1 2987.66 710314.88 904,401 860,085,351 30-838 9.834 952 2990.80! 711809.58 906,304 862,801,408 30.854 9.837 953 2 993-94 7I3305-68 908,209 865,523,177 30.871 9.841 954 2997.08 714803.48 9IO,Il6 868,250,664 30.887 9.844 86 MATHEMATICAL TABLES. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root. 955 3000.22 716302.76 912,025 870,983,875 30.903 9.848 95 6 3003.36 717803.66 9 J 3,936 87 3 ,722,8l6 30.919 9.851 957 3006.50 719306.12 9 J 5, 8 49 876,467,493 30.935 9.854 958 3009.65 72o8l0.l6 917,764 879,217,912 30-95 1 9.858 959 3012.79 722315.77 919,681 881,974,079 30.968 9.861 960 3 l5-93 723822.95 921,600 884,736,000 30.984 9.865 961 3019.07 72533L70 9 2 3,5 21 887,503,681 31.000 9.868 962 3022.21 726842.02 925 5 444 890,277,128 3I.OI6 9.872 963 3025.35 728353.91 927,369 893,056,347 31.032 9-875 964 3028.50 729867.37 929,296 895,841,344 31.048 9.878 965 3031.64 731382.40 931,225 898,632,125 31.064 9.881 966 3034.78 732899.01 933^56 901,428,696 3I.O80 9.885 967 3037.92 734417.18 935>o89 904,231,063 31.097 9.889 968 3041.06 735936.93 937,024 907,039,232 SJ-iiS 9.892 969 3044.20 737458.25 938,961 909,853, 2 09 31.129 9-895 970 3047-35 738981.13 940,900 912,673,000 31.145 9.899 971 3050.49 740505.59 942,841 915,498,611 31.161 9.902 972 3053.63 742031.62 944,784 918,330,048 31.177 9.906 973 3056.77 743559-22 946,729 921,167,317 3LI93 9.909 974 3059-9 1 745088.39 948,676 924,010,424 31.209 9.912 975 3063.05 746619.13 950,625 926,859,375 3L225 9.916 976 3066.19 748151.44 952,576 929,714,176 31.241 9.919 977 3069.34 749685.32 954,529 932,574,833 3 I - 2 57 9-923 978 3072.48 751220.78 956,484 935,441,352 3 I - 2 73 9.926 979 3075.62 752757.8o 958,441 938,313,739 31.289 9.929 980 3078.76 754296.40 960,400 941,192,000 31-305 9-933 981 3081.90 755836.56 962,361 944,076,141 3l.3 2 i 9-936 982 3085.04 757378.30 964,324 946,966,168 3*-337 9.940 983 3088.19 758921.61 966,289 949,862,087 3 T -353 9-943 984 309L33 760466.48 968,256 952,763,904 3 I -369 9.946 985 3094.47 762012.93 970,225 955,671,625 3L385 9-950 986 3097.61 763560.95 972,196 958,585,256 31.401 9-953 987 3100.75 765110.54 974,169 961,504,803 31.416 9-956 988 3103.89 766661.71 976,144 964,430,272 3M3 2 9.960 989 3107.04 768214.44 978,121 967,361,669 31.448 9-963 990 3IIO.I8 769768.74 980,100 970,299,000 31.464 9.966 991 3II3.32 771324.61 982,081 973,242,271 31.480 9-970 992 3116.46 772882.06 984,064 976,191,488 31.496 9-973 993 3II9.60 774441.07 986,049 979,146,657 3i-5 12 9-977 994 3122.74 776001.66 988,036 982,107,784 3L528 9.980 995 3I25.8 9 777563.82 990,025 985,074,875 31-544 9-983 996 3129.03 779127.54 992,016 988,047,936 31-559 9.987 997 3132.17 780692.84 994,009 991,026,973 31-575 9-990 998 3135.31 782259.71 996,004 994,011,992 3I-59 1 9-993 999 3138.45 783828.15 998,001 997,002,999 31.607 9-997 1000 3141.60 785398.16 1,000,000 1,000,000,000 31.623 IO.OOO CIRCLES: DIAMETER, CIRCUMFERENCE, &C. TABLE No. IV. CIRCLES: DIAMETER, CIRCUMFERENCE, AREA, AND SIDE OF EQUAL SQUARE. Side of Side of Diameter. Circum- ference. Area. Equal Square (Square Root Diameter. Circum- ference. Area. Equal Square (Square Root of Area). of Area). 3 9.4248 7.0686 2.6586 '/i6 .1963 .00307 0553 3 x /i6 9.62II 7.3662 2.7140 % .3927 .01227 .1107 3 1 A 9.8175 7.6699 2.7694 3/i6 .5890 .02761 .l66l 33/i6 10.014 7.9798 2.8248 X .7854 .04909 .2215 3X IO.2IO 8.2957 2.8801 5/i6 .9817 .07670 .2770 3 5 /i6 10.406 8.6 1 80 2-9355 % .1781 .1104 3323 3 3 /s 10.602 8.9462 2.9909 7/i6 3744 .1503 .3877 37/i6 10.799 9.2807 3.0463 % .5708 .1963 4431 3/2 10.995 9.6211 3.IOI7 9/i6 .7771 .2485 .4984 39/i6 II.I9I 9.9680 3.!57i # 9635 .3068 5539 3% 11.388 10.320 3.2124 "/i6 2.1598 .3712 .6092 3 IJ /i6 11.584 10.679 3.2678 * 2.3562 .4418 .6646 3X II.78I 11.044 3-3232 Sfo 2.5525 .5185 .7200 3'3/i6 11.977 11.416 3.3786 # 2.7489 .6013 7754 3 7 /s 12.173 n.793 3-4340 '5/16 2.9452 .6903 .8308 3 I5 /i6 12.369 12.177 3.4894 i 3.1416 .7854 .8862 4 12.566 12.566 3-5448 I z /i6 3-3379 .8866 .9416 4 J /i6 12.762 12.962 3.6002 I# 3-5343 .9940 .9969 4 l /s 12.959 13-364 3-6555 I 3/ l6 3.7306 1.1075 1.0524 43/i6 I3.I55 13.772 3.7109 iX 3.9270 1.2271 1.1017 4X !3-35i 14.186 3.7663 I S/i6 4-1233 1-3530 1.1631 4 5 /i6 13-547 14.606 3.8217 I# 4.3197 1.4848 1.2185 4 3 A 13.744 15-033 3.8771 I 7/i6 4.5160 1.6229 1.2739 4 7 /i6 13.940 15.465 3-9325 I# 4.7124 1.7671 1.3293 4/2 14.137 15.904 3.9880 I 9/i6 4.9087 1.9175 1-3847 49/i6 14-333 16.349 4.0434 I# 5.1051 2.0739 1.4401 4K 14.529 16.800 4-0987 I"/i6 5-3014 2.2365 1-4955 4 TI /i6 14.725 17.257 4.1541 If* 5.4978 2.4052 1.5508 4X 14.922 17.720 4.2095 1*3/16 5.6941 2.5800 1.6062 4 x 3/i6 15.119 18.190 4.2648 I# 5.8905 2.7611 1.6616 4 7 /8 T 5-3i5 18.665 4.3202 1 15/16 6.0868 2.9483 1.7170 4 J 5/i6 15.511 19.147 4.3756 2 6.2832 3.1416 1.7724 5 15.708 19-635 4.4310 2 '/i6 6.4795 3.338o 1.8278 5 Vx6 15.904 20. 1 29 4.4864 2^ 6.6759 3-5465 1.8831 S/B 16.100 20.629 4.5417 2 3/ l6 6.8722 37584 1.9385 5 3 /i6 16.296 21.135 4-5971 2X 7.0686 3.9760 1-9939 5X 16.493 21.647 4.6525 2 S/ l6 7.2649 4.2000 2.0493 5 5/i6 16.689 22.166 4.7079 2^ 7.4613 44302 2.1047 5 3 /s 16.886 22.690 4.7633 2 7/ l6 7.6576 4.7066 2.1601 5 7 /i6 17.082 23.221 4.8187 2^ 7.8540 4.9087 2.2155 S/2 17.278 23-758 4.8741 2 9/ l6 8.0503 5- J 573 2.2709 5 9 /i6 17.474 24.301 4.9295 2^ 8.2467 5.4119 2.3262 S 5 /s 17.671 24.850 4.9848 2"/i6 8.4430 5.6723 2.3816 5 x ?/6 17.867 25.406 5.0402 2X 8.6394 5-9395 2.4370 5X 18.064 25.967 5.0956 2 I 3/ l6 8.8357 6.2126 2.4924 5 13 /i6 18.261 26.535 5.1510 2^ 9.0321 6.4918 2.5478 5 7 /s 18.457 27.108 5.2064 2*S/,6 9.2284 6.7772 2.6032 5 l S/i 6 18.653 27.688 5.2618 88 MATHEMATICAL TABLES. Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). 6 6/8 w 6/8 6/2 6/8 6% 6/8 18.849 19.242 I9-635 20.027 20.420 20.813 21.205 21.598 28.274 29.464 30.679 31.919 33.183 34-471 35784 37.122 5.3I72 5.4280 5.5388 5.6495 5.7603 5.87II 5.9819 6.0927 12 12/8 12% 12/8 12& 37.699 38.091 38.484 38.877 39.270 39.662 40.055 40.448 113.097 115.466 117.859 120.276 I22.7l8 125.184 127.676 130.192 10.634 10.745 10.856 10.966 11.077 II.I88 11.299 11.409 NM NW No, N NUJ N- Nw OoN -P\ 0N N\ 0N -PS OON 21.991 22.383 22.776 23.169 23.562 23-954 24-347 24.740 38.484 39.871 41.282 42.718 44.178 45.663 47.173 48.707 6.2034 6.3142 6.4350 6.5358 6.6465 6-7573 6.8681 6.9789 13 l i\x l 3/ i3>6 40.840 41.626 42.018 42.411 42.804 43-197 43.589 132.732 135.297 137.886 140.500 I43.I39 145.802 148.489 I5I.20I 11.520 11.631 11.742 11.853 11.963 12.074 I2.I85 12.296 NOO \-4-NOONN NOO X.4.NOO X -^ X i-N X ^ X oooooooooocooooo 25.132 25.515 25.918 26.310 26.703 27.096 27.489 27.881 50.265 51.848 53.456 55.088 56.745 58.426 60. 1 32 61.862 7.0897 7.2005 7.3II2 7.4220 7.5328 7.6436 77544 7.8651 H 14% 43.982 44-375 44.767 45.160 45-553 45-945 46.338 46.731 153.938 156.699 159.485 162.295 165.130 167.989 170.873 173782 12.406 12.517 12.628 12.739 12.850 12.960 13.071 I3.I82 NOO Vj-NOO\N NpO\-4.NOO X^tX^xX^tX ONONONONONONONON 28.274 28.667 29.059 29.452 29.845 30.237 30.630 31.023 63.617 65.396 67.200 69.029 70.882 72.759 74.662 76.588 7.9760 8.0866 8.1974 8.3081 8.4190 8.5297 8.6405 8.7513 15 I5# 47.124 47.516 47.909 48.302 48.694 49.087 49.480 49.872 176.715 179.672 182.654 185.661 188.692 191.748 194.828 197-933 I3-293 I3-403 I3-5I4 13.625 13736 13.847 13-957 14.068 10 10/8 io)4 io/ 2 10% 3I.4I6 31.808 32.201 32.594 32.986 33-379 33-772 34.164 78.540 8o. 5I5 82.516 84.540 86.590 88.664 90.762 92.885 8.8620 8.9728 9.0836 9-1943 9.3051 9.4159 9.5267 9-6375 16 16/8 \6/8 50.265 50.658 51.051 5M43 51.836 52.229 52.621 53.014 201.062 204.216 207.394 210.597 213.825 217.077 220.353 223.654 14.179 14.290 14.400 14.511 14.622 14732 14.843 14.954 II H/8 11/8 H/8 34.558 34-950 35-343 35-735 36.128 36.521 36.913 37.306 95.033 97-205 99.402 101.623 103.869 106.139 108.434 110.753 97482 9.8590 9.9698 10.080 10.191 10.302 10.413 10.523 17 I'jYz 17^ 53407 53-799 54.192 54.585 54.978 55.370 55763 56.156 226.980 230.330 233.705 237.104 240.528 243.977 247.450 250.947 15.065 15.176 15.286 15-397 15.508 I5.6I9 I5-730 15.840 CIRCLES: DIAMETER, CIRCUMFERENCE, &c. 8 9 Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). IS i8> i$X 8^ 8K 8^ 8^ 8^ 56.548 56.941 57-334 57.726 58.119 58.512 58.905 59.297 254.469 258.016 261.587 265.182 268.803 272.447 276.117 279.811 I5.95I 16.062 16.173 16.283 16.394 16.505 16.616 16.727 24 24 l /& 24X 24^ 24X 24^ 24^ 24 7 /& 75-398 75-791 76.183 76.576 76.969 77.361 77-754 78.147 452.390 457-115 461.864 466.638 471.436 476.259 481.106 485.978 21.268 21-379 21490 21.601 21.712 21.822 21-933 22.044 19 '9# 19* i9# i9# 19% 19* 197/8 59.690 60.083 60.475 60.868 61.261 61.653 62.046 62.439 283.529 287.272 291.039 294.831 298.648 302.489 306.355 310.245 16.837 16.948 17.060 17.170 17.280 I7-39I 17.502 17.613 25 25^ 25X 25 3 A 25 X 25^ 25^ 25^ 78.540 78.932 79.325 79718 8o.no 80.503 80.896 81.288 490.875 495796 500.741 505.711 510.706 5I5-725 520.769 525.837 22.155 22.265 22.376 22.487 22.598 22.709 22.819 22.930 20 20^ 20X 20^ 20^ 20^ 20^ 207/& 62.832 63.224 63.617 64.010 64.402 64.795 65.188 65.580 314.160 318.099 322.063 326.05 1 330.064 334-101 338.163 342.250 17.724 17-834 17-945 18.056 18.167 18.277 18.388 18.499 26 26^ 26^ 26^ 26^ 26^ 26^ 26^ 81.681 82.074 82.467 82.859 83-252 83.645 84.037 84.430 530.930 536.047 541.189 546.356 551.547 556.762 562.002 567.267 23.041 23.152 23.062 23.373 23.484 23-595 23.708 23.816 21 2I l /i 21* 21^ 21% 21# 2t# 2IJ/8 6 5-973 66.366 66.759 67.151 67.544 67.937 68.329 68.722 346-361 350.497 354.657 358.841 363.051 367.284 371-543 375.826 18.610 18.721 18.831 18.942 19-053 19.164 19.274 19-385 27 27^ 27X 27^ 27% 27tt 27% 27 7 A 84.823 85.215 85.608 86.001 86.394 86.786 87.179 87.572 572.556 577.870 583.208 588.571 593-958 599-370 604.807 610.268 23-927 24.038 24.149 24.259 24.370 24.481 24.592 24.703 22 22> 22% 22% 22% 22% 22% 227/ & 69.115 69.507 69.900 70.293 70.686 71.078 71.471 71.864 380.133 384.465 388.822 393-203 397.608 402.038 406.493 410.972 19.496 19.607 19.718 19.828 19-939 20.050 20.161 20.271 28 28^ 28X 28H 28^ 28^ 28% 2Sj/ 8 87.964 88.357 88.750 89.142 89-535 89.928 90.321 90.713 6I5-753 621.263 626.798 632.357 637.941 643-594 649.182 654.839 24.813 24.924 25.035 25.146 25.256 25.367 25-478 25.589 23 23^ 23X 23^ 23K 23^ 23^ 23^ 72.256 72.649 73.042 73434 73.827 74.220 74.613 75.005 415476 420.004 424.557 429.135 433-731 438-363 443.014 447.699 20.382 20.493 20.604 20.715 20.825 20.936 21.047 21.158 29 29^ 29X 29^ 2 9 % 29% 29% 29^ 91.106 91.499 91.891 92.284 92.677 93.069 93462 93.855 660.521 666.227 671.958 677.714 683.494 689.298 695.128 700.981 25.699 25.810 25.921 26.032 26.143 26.253 26.364 26.478 MATHEMATICAL TABLES. Diameter Circum- ference. Area. Side of Equal Squar (Square Roo of Area). Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). 30 3o l /8 3oX 30 3 /8 30/2 30% 30% 3o 7 /s 94.248 94.640 95-033 95426 95.818 96.21 1 96.604 96.996 706.860 712.762 718.690 724.641 730.6l8 736.619 742.644 748.694 26.586 26.696 26.807 26.918 27.029 27.139 27.250 27.361 36 36/8 36X 36M 36^ 36^ 36X 36? 113.097 II3490 113.883 114.275 114.668 II5.06I H5-453 1 1 5.846 1017.88 1024.95 1032.06 1039.19 1046.35 1053.52 1060.73 1067.95 3L903 32.014 32.124 32.235 32.349 32.457 32.567 32.678 3i 3* l /8 3iX 3'# 3iX 3i^ 3iX 3i# 97.389 97782 98.175 98.567 98.968 99-353 99-745 100.138 754.769 760.868 766.992 773.140 779-3I3 785.510 791.732 797.978 27.472 27.583 27.693 27.804 27.915 28.026 28.136 28.247 37/8 37X 37 3 A 37^ 37*3 37X 37^ 116.239 116.631 117.024 II74I7 II7.8IO II8.202 118.595 118.988 1075.21 1082.48 1089.79 1097.11 1104.46 IIII.84 1119.24 1126.66 32.789 32.900 33-011 33-021 33-232 33-343 33-454 33-564 32 32^ 32X 32^ 32^ 32^ 32X 32^ 100.531 100.924 101.316 101.709 102.102 102.494 102.887 103.280 804.249 810.545 816.865 823.209 829.578 835.972 842.390 848.833 28.358 28.469 28.580 28.691 28.801 28.912 29.023 29.133 38 38^ 38X 38^ 38K 38^ 38^ 38^ 119.380 II 9 .773 I20.I66 120.558 120.951 121.344 121.737 122.129 II34.II 1141.59 II49.08 1156.61 1164.15 II7I.73 1179.32 1186.94 33.675 33.786 33.897 34.008 34.118 34.229 34.340 3445 J 33 33> 33X 33 3 /8 33% 33% 33% 33 7 A 103.672 104.055 104.458 104.850 105.243 105.636 106.029 I06.42I 855-30 861.79 868.30 874.84 881.41 888.00 894.61 901.25 29.244 29-355 29.466 29-577 29.687 29.798 29.909 3O.020 39 39^ 39X 39^ 39X 39^ 39X 39^ 122.522 122.915 123.307 123700 124.093 124.485 124.878 125.271 1194.59 1202.26 1209.95 1217.67 122542 1233.18 1240.98 1248.79 34.56i 34.672 34.783 34.894 35-005 35-115 35.226 35-337 34 34^ 34X 34^ 34X 34% 34X 34-J/s. Io6.8l4 107.207 107.599 107.992 108.385 108.777 109.170 109.563 907.92 914.61 921.32 928.06 934.82 941.60 948.41 955-25 30.131 30.241 30.352 30'463 30.574 30.684 30.795 30.906 40 40^ 4oX 40^ 4oX 40^ 4oX 40^ 125.664 126.056 126.449 126.842 127.234 127.627 128.020 128.412 1256.64 1264.50 1272.39 1280.31 1288.25 1296.21 1304.20 1312.21 35-448 35-558 35-669 35.78o 35-891 36.002 36.112 36.223 35 35^ 35X 35 3 A 35/2 35% 35% 35 7 /s 109.956 110.348 II0.74I III.I34 III.526 III.9I9 II2.3I2 112.704 962.11 968.99 975.90 982.84 989.80 996.78 1003.78 1010.82 31.017 31.128 3L238 3L349 31.460 3L57I 3I.68I 31.792 4i 4iM 4iX 4iK 4i /2 4*H 41 % 4*# 128.805 129.198 129.591 129.983 130.376 130.769 I3I.l6l I3L554 1320.25 1328.32 1336.40 1344.51 1352.65 1360.81 1 369.00 1377.21 36.334 36.445 3 !I! 36.666 36.777 36.888 36.999 37.109 CIRCLES: DIAMETER, CIRCUMFERENCE, &C. Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). 42 42/8 42^ 42^ 42/8 42X 42J/8 I3I-947 132.339 132.732 133.125 I33-5I8 133.910 134.303 134.696 1385.44 I393-70 1401.98 1410.29 1418.62 1426.98 I435.36 1443-77 37.220 37.331 37-442 37.663 37-774 37.885 37.996 4%/8 48X 48^ 48^ 48/8 48X 48^ 150.796 151.189 151.582 151.974 152.367 152.760 153.153 153.545 1809.56 1818.99 1828.46 1837.93 1847.45 1856.99 1866.55 1876.13 42-537 42.648 42.759 42.870 42.980 43.091 43.202 43.3I3 43 / 43/s 43X 43^ 43^ 43^ 43X 135.088 135.481 I35.874 136.266 136.659 137.052 137-445 137.837 1452.20 1460.65 1469.13 1477.63 1486.17 1494.72 1503.30 1511.90 38.106 38.217 38.328 38.439 38.549 38.660 38.771 38.882 49 49^ 49X 49^ 49 1 A 49/8 49% 49% 153.938 154.331 154723 I55.II6 155.509 155.901 156.294 I 56.687 1885.74 I895.37 1905.03 I9H.70 1924.42 1934.15 1943.91 1953.69 43423 43-534 43-645 43-756 43-867 43.977 44.088 44.199 44 44/8 44X 44^ 44/8 44% 138.230 138.623 139.015 139.408 139.801 140.193 140.586 140.979 1520.53 1529.18 1537.86 1546.55 1555.28 1564.03 I572.8I I58l.6l 38-993 39-103 39.214 39-325 39-436 39-546 39-657 39-768 $0% I 57.080 157.865 158.650 159436 1963.50 1983.18 2002.96 2O22.84 44.310 44-531 44-753 44.974 5 J X S l % I60.22I l6l.007 161.792 162.577 2042.82 2062.90 2083.07 2103.35 45.196 45-4I7 45-639 45-861 45 45^ 45X 45^ 45K 45 /8 45% 4S 7 /8 141.372 141.764 142.157 142.550 142.942 H3-335 143728 144.120 1590.43 1599.28 1608.15 1617.04 1625.97 1634.92 1643.89 1652.88 39-879 39.989 40.110 40.211 40.322 40.432 40.543 40.654 52 52X 52^ 52X 163.363 164.148 164.934 165.719 212372 2144.19 2164.75 2185.42 46.082 46.304 46.525 46.747 53X 166.504 167.290 168.075 I68.86I 2206.18 2227.05 2248.01 2269.06 46.968 47.190 47.411 47.633 ON ON ON ON ON ON ON ON Ssq SU) So! SM SW SM S^H oo\ -P\ os N\ oos -P\ o^s 144.513 144.906 145.299 145.691 146.084 146.477 146.869 147.262 l66l.90 1670.95 1680.01 1689.10 1698.23 1707.37 1716.54 1725.73 40.765 40.876 40.986 41.097 41.208 4L3I9 41.429 41.540 54 54X 54^ 54% 169.646 170.431 I7I.2I7 172.002 2290.22 2311.48 2332.83 2354.28 47.854 48.076 48.298 48.519 i 55% 172.788 173.573 I74.358 175.144 2375.83 2397.48 2419.22 2441.07 48.741 48.962 49.184 49.405 47 47 l /s 47X 47^ 47/2 47/8 47% 47 7 A 147.655 148.047 148.440 148.833 149.226 149.618 I5O.OI I 1 50.404 1734-94 1744.18 1753-45 1762.73 1772.05 1781.39 1790.76 1800.14 41.651 41.762 41.873 41.983 42.094 42.205 42.316 42.427 56 56X 56^ 56X 175.929 176.715 177.500 178.285 2463.01 2485.05 2507.19 2529.42 49.627 49.848 50.070 50.291 92 MATHEMATICAL TABLES. Diameter. Circum- ference. Area. Side of Squal Square Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). 57 57* 57^ 179.071 179.856 180.642 181.427 2551.76 2574.19 2596.72 2619.35 50.513 50.735 50.956 51.178 68 68# 68^ 213.628 214.414 215.199 215.985 3631.68 3658.44 3685.29 3712.24 6o.26l 60.483 60.704 60.926 58 58X 182.212 182.998 183783 184.569 2642.08 2664.91 2687.83 2710.85 51-399 51.621 51.842 52.064 69 69^ 216.770 217-555 2I8.34I 219.126 3739-28 3766.43 3793-67 3821.02 61.147 61.369 61.591 61.812 59 1 59X 185.354 186.139 186.925 187.710 2733-97 2757.19 2780.51 2803.92 52.285 52.507 52.729 52.950 70 70% 219.912 220.097 221.482 222.268 3848.45 3875.99 3903-63 393I-36 62.034 62.255 62.477 62.698 60 6o# 188.496 189.281 190.066 190.852 2827.43 2851.05 2874.76 2898.56 53-I72 53-393 53-836 7i 71* 71/2 223.053 223.839 224.624 225.409 3959-19 3987.13 40I5.I6 4043.28 62.920 63.141 63-363 63.545 ON ON ON ON XXX ~ 191.637 192.423 193.208 193-993 2922.47 2946.47 2970.57 2994-77 54.048 54.279 54.501 54723 72 72# 226.195 226.980 227.766 228.551 4071.50 4099.83 4128.25 4156.77 63.806 64.028 64.249 64.471 ON ON ON ON to to to to XXX 194.779 195.564 196.350 197.135 3019.07 304347 3067.96 3092.56 54-944 55.166 55.387 55-609 73^ 73^ 229.336 230.122 230.907 231.693 4185.39 4214.11 4242.92 4271.83 64.692 64.914 65-I35 65.357 ON ON ON ON Oo Oo Oo Oo XXX 197.920 198.706 I9949I 200.277 3117.25 3142.04 3166.92 3191.91 55-830 56.052 56.273 56.495 74 74* 232.478 233.263 234.049 234.834 4300.84 4329-95 4359.16 4388.47 65.578 65.800 66.022 66.243 64 1 64^ 64^ 2OI.o62 201.847 202.633 203.418 3216.99 3242.17 326746 3292.83 56.716 56.938 57.159 75 i 75# 235.620 236.405 237.190 237.976 4417.86 4447-37 4476.97 4506.67 66.465 66.686 66.908 67.129 ON ON ON ON J-l Ol Ol Ol XXX 204.204 204.989 205.774 206.560 3318.31 3343.88 3369.56 3395.33 57.603 57.824 58.046 58.267 76 , 76^ 76^ 238.761 239-547 240.332 24I.II7 4536.46 4566.36 4596.35 4626.44 67-351 67.572 67794 68.016 66 66% 66^ 66^ 207.345 208.131 208.916 209.701 3421.19 3447.16 3473-23 3499-39 58.489 58.710 58.932 59-154 77 7 77/2 77ti 241.903 242.688 243-474 244.259 4656.63 4686.92 4717.30 4747-79 68.237 68.459 68.680 68.902 67 , 67 1 /2 67^ 210.487 211.272 212.058 212.843 3525.66 3552.01 3578.47 3605.03 59-375 59-597 59.818 60.040 78 , 78K 78^ 245.044 245.830 246.615 247401 4778.36 4809.05 4839-83 4870.70 69.123 69-345 69.566 69.788 CIRCLES: DIAMETER, CIRCUMFERENCE, &C. 93 Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). 79 79% 79% 248.186 248^71 249-757 250.542 4901.68 493275 4963.92 4995.1 9 70.009 70.231 70-453 70.674 9 , 90% 282.744 283.52 9 284.314 2 85.o 99 6361.73 6399.12 6432.62 6468.16 79-758 79.980 80.201 80.423 80 So% 251.328 252.113 252.8 9 8 253.683 5026.55 5058.00 5089.58 5121.22 70.896 7I.II8 71-339 7I.56I 91 i 91 /^2 9 I^ 285.885 286.670 287.456 288.242 6503.88 653^68 6573.56 6611.52 80.644 80.866 81.087 81.308 81 254.469 255.254 256.040 256.825 5153.00 5184.84 5216.82 5248.84 71.782 72.004 72.225 72.447 92 9* 1 A 9* l /2 92^ 28 9 .027 28 9 .8l2 2 9 o.5 9 8 291.383 6647.61 6683.80 6720.07 6756.40 81.530 81.752 81.973 82.195 82 82X S2/ 2 82^ 257.611 258.3 9 6 25 9 .l82 259-967 5281.02 53I3-28 5345-62 5378.04 72.668 72.890 73.111 73-333 93 93 1 / 91 1 A 93% 2 9 2.l68 292.953 293-739 294.524 67 9 2. 9 i 682 9 .48 6866.16 6882.92 82.416 82.638 82.859 83.081 00 00 00 OO Oo 00 Oo Oo K^x 260.752 261.537 262.323 263.108 54I0.6l 5443-24 5476.00 5508.84 73-554 73.776 73-997 74-219 94 94 1 / 94/2 94% 295.310 2 9 6.o 9 5 2 9 6.88i 2 9 7.666 6939.78 6976.72 7013.81 7050.92 83.302 83.524 83746 83.968 84 1 841! 263.8 9 4 264.67 9 265.465 266.250 5541-77 5574.80 5607.95 5641.16 74.440 74.662 74.884 75.106 95 9S 1 / 9S/2 9S% 2 9 8.452 299.237 300.022 300.807 7088.22 7125.56 7163.04 7200.56 84.i8 9 84.411 84.632 84.854 x^x un u-i un u-i OO CO OO OO 267.035 267.821 268.606 269.392 5674.51 5707.92 5741.47 5775-09 75-549 75.770 75.992 96 9W 9W 9 6% 301.593 302.378 302.164 303.948 7238.23 7275.96 73 I 3- 8 4 7351.72 85.077 85.299 85.520 85.742 86 86X 86^ 86^ 270.177 270.962 271.748 272.533 5808.80 5842.60 5876.55 5910.52 76.213 76.435 76.656 76.878 97 1 97 /2 97% 304.734 305.520 306.306 307-090 7389.81 7427.96 7474.20 7504.52 85.963 86.185 86.407 86.628 87 1 87^ 87^ 273-319 274.104 274.890 275.675 5944-68 5978.88 6013.21 6047.60 77-099 77.321 77.542 77764 9 8' 98^ 307.876 308.662 309-446 310.232 7542.96 7581.48 7620.12 7658.80 86.850 87.072 87.293 87.515 88 88X 88^ 88X 276.460 277.245 278.031 278.816 6082.12 6116.72 6151.44 6186.20 78.207 78.428 78.650 99 99 1 A 99/2 99% 311.018 311.802 312.588 313.374 7697.69 7736.60 7775.64 7814.76 87.736 88.180 88.401 100 3H.I59 315.730 7853.98 7932.72 88.623 8 9 .o66 89 8 9 X 89^ 279.602 280.387 281.173 281.958 6221.14 6256.12 6291.25 6326.44 78.871 79-093 79.3I5 79-537 101 101% 317.301 318.872 8011.85 8091.36 8 9> 5o 9 8 9 . 9 52 94 MATHEMATICAL TABLES. Diameter. Circum- ference. Area. Side of Equal Square Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). 102 102^ 320.442 322.014 8171.28 8251.60 90-395 90.838 112 112& 351.858 353-430 9852.03 9940.20 99.258 99.701 103 I03/ 2 323.584 325.154 8332.29 8413.40 91.282 91.725 H3 H3/2 355-ooo 356.570 10028.75 10117.68 IOO.I44 100.587 104 104^ 326.726 328.296 8494.87 8576.76 92.168 92.6ll 114 114/2 358.142 359.712 10207.03 10296.76 IOI.03I 101.474 I0 5 105^ 329.867 33I-438 8659.01 8741.68 93-054 93-497 H5 "5# 361.283 362.854 10386.89 10477.40 IOI.9I7 102.360 106 io6K 333-009 334.580 882473 8908.20 93-940 94.383 116 116% 364-425 365.996 10568.32 10659.64 102.803 103.247 107 10 7 / 2 336.150 337722 8992.02 9076.24 94.826 95.269 117 "7# 367.566 369.138 10751.32 10843.40 103.690 104.133 108 io8X 339.292 340.862 9160.88 9245.92 9 l' 7I l 96.156 118 u8# 370.708 372.278 10935.88 11028.76 104.576 105.019 109 109 > 342.434 344.004 933I-32 9417.12 96.599 97.042 119 H 9 /2 373-849 375.420 1 1 122.02 11215.68 105.463 105.906 IIO 110% 345-575 347.146 9503.32 9589.92 97.485 97.928 120 376.991 II30973 106.350 in HI/2 348.717 350.288 9676.89 9764.28 98.371 98.815 LENGTHS OF CIRCULAR ARCS. 95 TABLE No. V. LENGTHS OF CIRCULAR ARCS FROM i TO 180. GIVEN, THE DEGREES. (RADIUS = i.) Degrees. Length. Degrees. Length. Degrees. Length. Degrees. Length. I -0175 40 .6981 79 1.3788 117 2.O42O 2 0349 41 .7156 118 2 -595 3 .0524 42 7330 80 /-> I-3963 119 2.0769 4 .0698 43 .7505 81 I -4 I 37 6 .0873 .1047 44 45 .7679 .7854 82 83 1.4312 1.4486 120 121 2.0944 2.III8 7 .1222 46 .8028 84 1.4661 122 2.1293 8 .1396 47 .8203 85 1.4835 I2 3 2.1468 9 48 8377 86 1.5010 I2 4 2.1642 40 8552 '87 1.5184 125 2.I8I7 10 -1745 Tx A E N 24 w s 24 E s 24 w i ii 15 o N by E N by w s by E s by w 1*4 i4 3 45 N by E ^4 E N by w Y(. w s by E % E s by w ^ w i^ 16 52 30 N by E YZ E N by w YZ w s by E YZ E s by w YZ w I3 A 19 4i 15 N by E 24 E N by w 24 w s by E 24 E s by w 24 w 2 22 30 NNE NNW SSE ssw 2/4 25 18 45 NNE 14 E NNW 14 W SSE 14 E ssw 14 w 2 /2 28 7 30 NNE % E NNW YZ w SSE YZ E ssw % w 23/ 4 30 56 15 NNE 24 E NNW 24 W SSE 24 E ssw 24 w 3 33 45 o NE by N NW by N SE by s sw by s 3% 36 33 45 NE^N NW 24 N SE 24 s sw24s $% 39 22 30 NE YZ N NW^N SE % S sw Y^ s &A 42 ii 15 NE ^4 N NW % N SE Y*. S sw Yt s 4 45 o o NE NW SE sw 4^4 47 48 45 NE 14 E NW 14 W SE ^4 E sw 14 w 45* 5 37 30 NE YZ E NW % w SE YZ E sw YZ w 4^4 53 26 15 NE 24 E NW 24 w SE24E sw24w 5 56 15 o NE by E NW by w SE by E sw by w 5^4 59 3 45 ENE 24 N WNW 24 N ESE 24 S wsw 24 s 5/^ 61 52 30 ENE YZ N WNW % N ESE YZ S wsw YZ s 5^4 64 41 15 ENE 14 N WNW ^4 N ESE ^4 s wsw ^4 s 6 67 30 o ENE WNW ESE wsw 6^4 70 18 45 ENE ^4 E WNW % W ESE ^4 E wsw 14 w 6^j 73 7 30 ENE YZ E WNW YZ w ESE YZ E wsw YZ w 624 75 56 15 ENE 24 E WNW 24 w ESE 24 E wsw 24 w 7 78 45 o E by N w by N E by s w by s 7^4 8 1 33 45 E24N w 24 N E 24 s w 24 s 7^2 84 22 30 W YZ N E YZ s w YZ s 7^C 87 II 15 E 14 N W I^ N E 14 S w^ s 8 90 o o EAST. WEST. EAST. WEST. u8 MATHEMATICAL TABLES. TABLE No. XL RECIPROCALS OF NUMBERS FROM I TO IOOO. No. Reciprocal. No. Reciprocal. No. Reciprocal. No. Reciprocal. I I.OOOOOO 40 .O2500O 79 .012658 118 .008475 2 .500000 41 .024390 80 119 .008403 3 4 6 7 8 9 10 333333 .250000 .200000 .166667 .142857 .125000 .iimi .100000 42 43 44 45 46 47 48 49 .023810 .023256 .022727 .022222 .021739 .021277 .020833 .O2O408 \J\J 81 82 83 84 85 86 87 88 .012346 .012195 .012048 .011905 .011765 .011628 .011494 .011364 120 121 122 123 124 I2 5 126 127 008333 .008264 .008197 .008130 .008065 .OO8000 .007937 .007874 ii .090909 50 .O2OOOO 89 .011236 128 .007813 12 083333 5 1 .019608 go .OIIIII 129 .007752 13 14 15 16 .076923 .071429 .066667 .062500 .058824 52 53 54 55 56 .019231 .018868 .018519 .Ol8l82 .017857 9 1 92 93 94 95 .010989 .010870 .010753 .010638 .010526 130 132 J 34 .007692 007634 .007576 .007519 .007463 10 .052632 57 58 .017544 .017241 96 97 .010417 .010309 136 .007407 007353 20 .050000 59 .016949 98 .OIO2O4 137 .007299 21 .047619 60 .016667 99 .010101 138 .007246 22 045455 61 .016393 100 .010000 139 .007194 23 .043478 62 .Ol6l29 101 .009901 140 .007143 24 .041667 63 OI 5 8 73 102 .009804 141 .007092 25 .040000 64 .015625 103 .009709 142 .007042 26 .038462 65 015385 I0 4 .009615 143 .006993 27 037037 66 .015152 I0 5 .009524 144 .006944 28 035714 67 014925 106 .009434 J 45 .006897 29 034483 68 .014706 107 .009346 146 .006849 30 3 1 32 033333 .032258 .031250 69 70 .014493 .014286 .014085 108 109 no .009259 .009174 .009091 147 148 149 .006803 .006757 .006711 33 .030303 72 .013889 in .009009 150 .006667 34 .029412 73 .013699 112 .008929 151 .006623 35 .028571 74 OI 35 I 4 113 .008850 152 .006579 36 .027778 75 OI 3333 114 .008772 153 .006536 37 .027027 76 .013158 115 .008696 154 .006494 38 .026316 77 .012987 116 .008021 155 .006452 39 .025641 78 .012821 117 008547 156 .006410 RECIPROCALS OF NUMBERS. No. Reciprocal. No. Reciprocal. No. Reciprocal. No. Reciprocal. 157 .006369 2O2 .004950 247 .004049 2 9 2 003425 158 .006329 20 3 .004926 248 .004032 293 .003413 159 .006289 2O4 .004902 249 .OO40l6 294 .003401 1 60 161 162 i6 3 164 .006250 .OO62II .006173 .006135 .006098 205 2O6 207 208 209 .004878 .004854 .004831 .004808 .004785 2 5 251 252 253 254 .OO4OOO .003984 .003968 003953 .003937 295 296 297 298 299 .003390 .003378 .003367 003356 003344 165 .006061 2IO .004762 255 .003922 300 03333 1 66 .006024 211 .004739 256 .003906 3 OI .003322 167 .005988 212 .004717 257 .003891 3 02 .003311 168 .005952 2I 3 .004695 258 .003876 303 003301 169 .005917 214 .004673 2 59 .003861 34 .003289 170 171 172 173 174 .005882 .005848 .005814 .005780 .005747 215 216 217 218 219 .004651 .004630 .004608 .004587 .004566 26O 26l 262 263 264 .003846 .003831 .003817 .003802 .003788 305 306 307 308 309 .003279 .003268 .003257 003247 .003236 J 75 .005714 220 004545 265 .003774 310 .003226 176 .005682 221 .004525 266 .003759 311 .003215 177 .005650 222 .004505 267 003745 3 I2 .003205 178 .005618 223 .004484 268 .003731 3*3 .003195 179 .005587 224 .004464 269 .003717 3 J 4 003185 180 181 182 183 184 005556 .005225 005495 .005464 005435 225 226 227 228 229 .004444 .004425 .004405 .004386 .004367 27O 271 272 273 274 .003704 .003690 .003676 .003663 .003650 3i5 316 3i7 3i8 3i9 .003175 .003165 003155 .003145 003135 185 1 86 187 188 189 .005405 005376 .005348 .005319 .005291 230 231 232 233 234 .004348 .004329 .004310 .004292 .004274 275 276 277 2 7 8 279 .003636 .003623 .003610 .003597 .003584 320 321 322 323 324 .003125 .003115 .003106 .003096 .003086 igo .005263 235 .004255 280 003571 325 .003077 191 .005236 236 .004237 28l 003559 326 .003067 /-> 192 .005208 237 .004219 282 .003546 327 .003058 *93 .005181 2 3 8 .OO42O2 283 003534 328 .003049 194 .005155 239 .004184 284 .003522 329 .003040 *95 196 197 198 199 .005128 .005102 .005076 .005051 .005025 24O 241 242 243 244 .004167 .004149 .004132 .004115 .004098 285 286 287 288 289 .003509 .003497 .003484 .003472 .003460 330 33 1 332 333 334 .003030 .O03O2I .003012 .003003 .002994 200 .OO50OO 245 .004082 2QO .003448 335 .002985 201 .004975 246 .004065 2 9 I .003436 336 .002976 I2O MATHEMATICAL TABLES. No. Reciprocal. No. Reciprocal. No. Reciprocal. No. Reciprocal. 337 .002967 382 .0026l8 427 .002342 472 .002119 338 .002959 383 .002611 428 .002336 473 .OO2II4 339 .002950 384 .002604 429 .002331 474 .002IIO 340 .002941 385 3 86 .002597 .002591 430 .002326 475 476 .002105 .002IOI 34i 342 343 .002933 .002924 .002915 387 3 88 389 .002584 .002577 .002571 431 432 433 .O0232O .002315 .002309 477 478 479 .002096 .002092 .002088 344 .002907 434 .002304 345 .002899 390 .002564 435 .002299 480 .002083 346 .002890 391 .002558 436 .O02294 481 .OO2079 347 .002882 39 2 .002551 437 .002288 482 .OO2O75 348 .002874 393 .002545 438 .002283 483 .002070 349 .002865 394 .002538 439 .OO2278 484 .002066 350 35' 352 353 354 .002857 .002849 .002841 .002833 .002825 395 396 397 398 399 .002532 .002525 .002519 .002513 .002506 440 441 442 443 444 .OO2273 .OO2268 .OO2262 .OO2257 .O02252 485 486 487 488 489 .002062 .002058 .002053 .OO2049 .OO2045 355 356 .002817 .002809 400 401 .OO25OO .002494 445 446 .OO2247 .OO2242 490 49 1 .002041 .002037 357 .OO28OI 402 .002488 ! 447 .002237 49 2 .002033 358 .002793 403 .002481 44 .OO2232 493 .OO2O28 359 .002786 404 .002475 449 .002227 494 .OO2O24 360 361 .002778 .002770 405 406 .002469 .002463 450 45 1 .OO2222 .OO22I7 495 496 .00202O .002016 362 363 .002762 .002755 407 408 .002457 .002451 452 453 .OO22I2 .OO2208 497 498 .002012 .OO2008 3 6 4 .002747 409 .002445 454 .O022O3 499 .002OO4 365 366 367 368 3 6 9 .002740 .002732 .002725 .002717 .002710 410 411 412 4i3 414 .002439 .002433 .002427 .002421 .002415 455 456 457 458 459 .002198 .002193 .002188 .002183 .002179 500 5 01 502 503 54 .00200O .001996 .001992 .001988 .001984 370 .002703 415 .002410 460 .OO2I74 55 .001980 37i .002695 416 .002407 461 .002169 506 .001976 372 .002688 4i7 .002398 462 .002165 507 .001972 373 .O0268l 418 .002392 463 .OO2l6o 508 .001969 374 .002674 419 .002387 464 .OO2I55 509 .001965 375 376 .002667 .00266O 420 .002381 465 466 .002151 .002146 5 J o .001961 377 .002653 421 .002375 467 .002141 5n .001957 378 379 .002646 .002639 422 423 424 .002370 .002364 .002358 468 469 .002137 .OO2I32 5" 513 5*4 .001953 .001949 .001946 380 .002632 425 002353 470 .002128 5i5 .OOI942 38i .002625 426 .002347 471 .002123 5i6 .001938 RECIPROCALS OF NUMBERS. 121 No. Reciprocal. No. Reciprocal. No. Reciprocal. No. Reciprocal. 5*7 .001934 562 .001779 607 .001647 652 .001534 518 .001931 563 .001776 608 .001645 653 .001531 519 .001927 564 .001773 609 .001642 654 .001529 520 521 522 523 524 .001923 .001919 .001916 .OOI9I2 .001908 565 566 567 5 68 569 .001770 .001767 .001764 .001761 .001757 610 6n 612 613 614 .001639 .001637 .001634 .001631 .001629 655 656 657 6 5 8 659 .001527 .001524 .OOI522 .OOI52O .001517 525 .001905 570 .001754 615 .OOl626 660 .001515 526 .OOI90I 571 .001751 616 .001623 661 .001513 527 .001898 572 .001748 617 .OOl62I 662 .OOI5II 528 .001894 573 .001745 618 .00l6l8 663 .001508 529 .001890 574 .OOI742 619 .OOl6l6 664 .001506 530 .001883 575 576 .001739 .001736 620 621 .OOl6l3 .OOl6lO 665 666 .001504 .001502 53 2 \J .OOl88o 577 .001733 622 .OOl6o8 667 .001499 533 534 .001876 .001873 578 579 .001730 .001727 623 624 .001605 .001603 668 669 .OOI497 .001495 535 536 .001869 .001866 580 .001724 .001721 625 626 .OOI6OO .001597 670 671 .001493 .001490 537 r> .001862 582 .OOI7l8 627 .001595 672 .001488 538 .001859 f\ 583 .001715 628 .001592 673 .001486 539 .001855 584 .001712 629 .001590 674 .001484 540 542 543 544 .001852 .001848 .001845 .001842 .001838 585 586 587 588 589 .001709 .001706 .001704 .OOI7OI .001698 630 631 632 633 634 .001587 .001585 .001582 .001580 .001577 675 676 677 678 679 .001481 .001479 .001477 .001475 .001473 545 546 547 548 549 .001835 .001832 .001828 .001825 .001821 590 592 593 594 .001695 .001692 .001689 .OOI686 .001684 635 636 637 638 639 001575 .OOI572 .OOI57O .001567 .001565 680 681 682 683 684 .OOI47I .001468 .001466 .001464 .001462 55 .00l8l8 595 .OOl68l 640 .00156^ 685 .001460 55i .00l8l5 596 .001678 641 > >5 39 *} 773-27 ?> j> jj , 62, 772.4 62, 819.4 62. 52-3, 82 The volume of one pound of air at 32 F., and under one atmosphere of pressure, is 12.387 cubic feet. The volume at 62 F., is 13.141 cubic feet. The specific heat of air at constant pressure is .2377, and at constant volume .1688, that of water being * i. GREAT BRITAIN AND IRELAND. IMPERIAL WEIGHTS AND MEASURES. The origin of English measures is the grain of corn. Thirty-two grains of wheat, dried and gathered from the middle of the ear, weighed what was called one pennyweight; 20 pennyweights were called one ounce, and 20 ounces one pound. Subsequently, the pennyweight was divided into 24 grains. Troy weight was afterwards introduced by William the Conqueror, from Troyes, in France ; but it gave dissatisfaction, as the troy pound did not weigh so much as the pound then in use; consequently, a mean weight was established, making 16 ounces equal to one pound, and called avoir- dupois (avoir du poids). Three grains of barleycorn, well-dried, placed end to end, made an inch the basis of length. The length of the arm of King Henry I. was made the length of the ulna, or ell, which answers to the modern yard. The imperial standard yard is a solid square bar of gun-metal, kept in the office of the Exchequer at Westminster, 38 inches in length, i inch square, at the temperature 62 F., composed of copper 16 ounces, tin 2^ ounces, and zinc i ounce. Two cylindrical holes are drilled half through the bar, one near each end, and the centres of these holes are 36 inches, or 3 feet, apart the length of the imperial standard yard. Compared with a pendu- lum vibrating seconds of mean time, at the level of the sea, in the latitude of London, in a vacuum, the yard is as 36 inches in length to 39.1393 inches, the length of the pendulum. Measures of capacity were based on troy weight; it was enacted that 8 pounds troy of wheat, from the middle of the ear, well dried, should make i gallon of wine measure, and that 8 such gallons should make i bushel. The imperial gallon is now the only standard measure of capacity, and it contains 277.274 cubic inches. It is said to be the volume of 10 pounds avoirdupois of distilled water, weighed in air, at 62 F. Note. The exact volume of 10 pounds of distilled water at 62 F. is 277.123 cubic inches. GREAT BRITAIN AND IRELAND. LENGTH. 1 29 Tables of weights and measures are conveniently classified thus i. Length; 2. Surface; 3. Volume; 4. Capacity; 5. Weight. The following are some of the principal units of measurement : The acre, for land measure. The mile, for itinerary measure. ' The yard, for measure of drapery, &c. The coomb, for capacity of corn, &c. The gallon, for capacity of liquids. The grain, for chemical analysis. Impound, for grocers' ware, &c. The stone of 8 pounds, for butchers' meat. The stone of 14 pounds, for flour, oatmeal, &c. I. MEASURES OF LENGTH. Tables No. 12. Lineal Measure. 3 barleycorns, or 12 lines, or f 72 points, or > Ilnch - 1000 mils 3 inches 4 inches 9 inches 12 inches 1 8 inches, 3 feet... 'Yz feet ... 5 feet ... 2 yards.. yards palm. hand. span. foot. cubit. yard. military pace. geometrical pace. fathom. rod, pole, or perch. 40 poles, or ) r i 220 yards / '"' l furlon & 8 furlongs, or } 1 760 yards, or > i mile. 5280 feet j 3 miles i league. 2240 yards, or ) ... 1.272 miles } ' Insh mlle ' The inch is also divided into halves, quarters, eighths, and sixteenths; sometimes into tenths. The hand is used as a measure of the height of horses. The military pace is the length of the ordinary step of a man. The geometrical pace is the length of two steps. A thousand of such paces were reckoned to a mile. The fathom is used in soundings to ascertain depths, and for measuring cordage and chains. 9 130 WEIGHTS AND MEASURES. Land Measure. 7.92 inches i link. 100 links, or \ 66 feet, or ( , . 22 yards, or f I cham ' 4 poles 10 chains '. i furlong. 80 chains, or ) ., 8 furlongs } ' mile - They*?*, or woodland pole or perch, is 18 feet. The for tst pole is 21 feet. Nautical Measure. 6086.44 f ee t? or j 1000 fathoms, or ( f i nautical mile, 10 cables, or [ " 1 or knot. 1.1528 statute miles ) 3 nautical miles i league. 60 nautical miles, or \ 69.168 statute miles or > i degree. 20 leagues j ( Circumference 360 degrees < of the earth at ( the equator. The above value of the nautical mile is that which is commonly taken, and is the length of a minute of longitude at the equator. The mean length of a minute of latitude at the mean level of the sea is nearly 6076 feet, or 1.1508 statute miles. The nautical fathom is the thousandth part of a nautical mile, and is, on an average, about ^-th longer than the common fathom. Cloth Measure. 2% inches i nail. 2 nails i finger-length. 4 nails, or 9 inches i quarter. 4 quarters i yard. 5 quarters i ell. WIRE-GAUGES. The " Birmingham Wire-Gauge " is a scale of notches in the edge of a plate, of successively increasing or decreasing widths, to designate a set of arbitrary sizes or diameters of wire, ranging from about half an inch down to the smallest size easily drawn, say, four-thousands of an inch. The practical utility of such a gauge is obvious, when it is considered how far beyond the means supplied by the graduations of an ordinary scale of feet and inches is the measurement of the gradations of the wire-gauge. But the "Birming- ham Wire-Gauge" is a variable measure. The principle, if there was any, on which it was originally constructed, is not known. Mr. Latimer Clark states that, when plotted, the widths of the gauge range in a curve approxi- GREAT BRITAIN AND IRELAND. WIRE-GAUGES. mating to a logarithmic curve, such as would be found by the successive addition of 10 or 12 per cent, to the width of the notches of the gauge. However that may be, there are many varieties of the wire-gauge in existence. The oldest and best-known gauge is that of which the numbers were care- fully measured by Mr. Holtzapffel, and published by him in 1847. It has been, and still is, widely followed in the manufacture of wire; and also of tubes in respect of their thickness. It gives 40 measurements ranging from .454 inch to .004 inch, and is contained in Table No. 13. Although there are only 40 marks in the table, there are 60 different sizes of wire made, for which intermediate sizes have been added to the gauge. This table has also been used in rolling sheet iron, sheet steel, and other materials, and for joiners' screws; but it appears to be falling into disuse for these purposes. BIRMINGHAM WIRE-GAUGE (HoltzapffeFs). Table No. 13. For Wire and Tubes chiefly; and for Sheet Iron and Steel formerly. Mark. Size. Mark. Size. Mark. Size. Mark. Size. No. Inch. No. Inch. No. Inch. No. Inch. OOOO 454 7 .180 17 .058 27 .Ol6 OOO 425 8 .165 18 .049 28 .014 00 .380 9 .148 J 9 .042 2 9 .013 O 340 10 134 20 035 30 .012 I .300 ii .120 21 .032 31 .OIO 2 .284 12 .ICQ 22 .028 32 .009 3 259 13 095 23 .025 33 .008 4 .238 14 .083 24 .022 34 .007 5 .220 15 .072 25 .020 35 .005 6 .203 16 .065 26 .018 36 .004 BIRMINGHAM METAL-GAUGE, or PLATE-GAUGE (HottzapffeTs). Table No. 14. For Sheet Metals, Brass, Gold, Silver, &c. Mark. Size. Mark. Size. Mark. Size. Mark. Size. No. Inch. No. Inch. No. .Inch. No. Inch. I .004 IO .024 19 .064 28 .120 2 .005 II .029 20 .067 29 .124 3 .008 12 034 21 .072 30 .126 4 .OIO J 3 .036 22 .074 3 1 133 5 .OI2 14 .041 23 .077 32 143 6 .013 15 .047 24 .082 33 145 7 .015 16 .051 25 95 34 .148 8 .Ol6 17 057 26 .103 35 .158 9 .019 18 .O6l 27 113 36 .167 Another of Holtzapffel's tables, No. 14, the Plate-Gauge, has been, and may now, to some extent, be, employed for most of the sheet metals, except- 132 WEIGHTS AND MEASURES. LANCASHIRE GAUGE (Holtzapffel' s). Table No. 15. For Round Steel Wire, and for Pinion Wire. Mark. Size. Mark. Size. Mark. Size. Mark. Size. Mark. Size. No. Inch. No. Inch. No. Inch. No. Inch. No. Inch. 80 .013 57 .042 34 .109 II .189 M 295 79 .014 56 .044 33 .III IO .I9O N .302 78 .015 55 .050 32 H5 9 .191 .316 77 .Ol6 54 055 3i .118 8 .192 P 3 2 3 76 .018 53 .058 30 .125 7 195 Q 332 75 .019 52 .060 29 134 6 .198 R 339 74 .022 5 1 .064 28 138 5 .2OI S .348 73 .023 5o .067 27 .141 4 .204 T .358 72 .024 49 .070 26 143 3 .209 U .368 7i .026 48 073 25 .146 2 .2I 9 V 377 70 .027 47 .076 24 .148 I .227 w .386 69 .029 46 .078 2 3 .150 A 234 X 397 68 .030 45 .080 22 152 B .238 Y .404 67 .031 44 .084 21 157 C .242 Z 413 66 .032 43 .086 20 .160 D .246 Ai .420 65 033 42 .091 19 .164 E .250 Bi 431 64 034 4i 095 18 .167 F 257 Ci 443 63 035 40 .096 17 .169 G .261 Di 452 62 .036 39 .098 16 .174 H .266 E! .462 61 .038 38 .IOO 15 175 I .272 Fi 475 60 39 37 .102 14 .177 J .277 Gi .484 59 .040 36 .105 13 .180 K .281 Hi 494 58 .041 35 .107 12 .185 L .290 ing iron and steel : as copper, brass, gilding-metal, gold, silver, and platinum. The intervals are closer or smaller than those of the wire-gauge, and the maximum size, for No. 36, is J /6 inch. When thicker sheets are wanted, their measures are sought in the Birmingham wire-gauge. The last table, No. 15, by HoltzaprTel, the Lancashire Gauge, is employed exclusively for the bright steel wire prepared in Lancashire, and the steel pinion-wire for watch and clock makers. The larger sizes are marked by capital letters, to distinguish them from the others. This, the second part of the table, is known as the Letter-Gauge. Needle- Gauge, for needle wire. The sizes correspond with some of those of the Holtzapffel wire-gauge. The following are the relative marks for equal sizes on the two gauges : Needle wire -gauge Nos. i, 2, 2^, 3, 4, 5, thence to 21, corresponding to B. W.-G. 18^, 19, 19^2, 20, 21, 22, thence to 38. Music Wire-gauge, for the strings of pianofortes. The marks used are Nos. 6 to 20. The following are the relative marks for equal sizes with the Holtzapffel wire-gauge : Music wire-gauge Nos. 6, 7, 8, 9, 10, n, 12, 14, 16, 18, 20, corresponding to B. W.-G. 26, 25%, 25, 24^, 24, 23^, 23, 22, 21, 20, 19. No. 6, the thinnest wire now used, measures about one fifty- fifth of an inch in diameter, and No. 20 about one twenty-fifth of an inch. GREAT BRITAIN AND IRELAND. WIRE-GAUGES. 133 The preceding Tables of Gauges have been extracted from Holtzapffel's estimable work on Turning and Mechanical Manipulation, 1847. Messrs. Rylands Brothers, of Warrington, manufacture iron wire accord- ing to the gauge in Table No. 16. WARRINGTON WIRE-GAUGE (Rylands Brothers). Table No. 16. Mark. Size. Mark. Size. Mark. Size. Mark. Size. No. Inch. No. Inch. No. Inch. No. Inch. 7/0 1/2 .326 8 159 15 .069 6/0 15/32 I .300 9 .146 16 .0625, or J /i6 5/o 7/16 2 .274 TO 133 i7 053 4/0 13/32 3 25, or y io# .125, or ^6 18 .047 3/o 3/8 4 .229 II .117 *9 .041 2/O 1 1/32 5 .209 12 .10, or Vio 20 .036 6 .191 13 .090 21 .0315, or V 32 7 .174 14 .079 22 .028 For sheets, the wire-gauge that seems to be adhered to by the iron-sheet rollers of South Staffordshire, is a scale comprising 32 measurements, ranging from .3125 inch to .0125 inch, contained in Table No. 17. BIRMINGHAM WIRE-GAUGE. Table No. 17. For Iron Sheets chiefly. No. Size. No. Size. No. Size. No. Size. Inch. Inch. Inch. Inch. I .3125 (s/ l6 ) 9 .15625(5/32) 17 .05625 25 02344 2 .28125 10 .140625 18 05 ('/) 26 .021875 3 25 (X) ii I2 5 '(#) 19 04375 27 .020312 4 234375 12 .1125 20 0375 28 .01875 5 .21875 13 .10 ( r /io) 21 034375 29 .01719 6 .203125 14 .0875 22 031250/32) 30 .015625 7 .1875 (3/ x 6) 15 075 23 .028125 3 1 .01406 8 .171875 16 0625 0/ l6 ) 24 025 (V 4 o) 32 OI2 5 ( x /8o) Sir Joseph Whitworth, in 1857, introduced his Standard Wire-Gauge, ranging from a half inch to a thousandth of an inch, and comprising 62 measurements, as given in Table No. 18. It commences with the smallest size, and increases by thousandths of an inch up to half an inch. The smallest size, Vioooth of an inch, is No. i ; No. 2 is 2 /ioooths of an inch, and so on, increasing up to No. 20 by intervals of Vioooth of an inch; from No. 20 to No. 40 by 2 /ioooths; f rO m No. 40 to No. 100 by s/ I000 ths of an inch. The sizes are designated or marked by their respective values in thousandths of an inch. The Standard Imperial Wire Gauge came into force on the ist March, 1884. It supersedes other gauges, which are rendered illegal. 134 WEIGHTS AND MEASURES. SIR JOSEPH WHITWORTH & Co.'s STANDARD WIRE-GAUGE. Table No. 18. Mark. Size. Mark. Size. Mark, Size. Mark. Size. No. Inch. No. Inch. No. Inch. No. Inch. I .001 17 .017 55 55 200 .200 2 .OO2 18 .018 60 .060 220 .220 3 .003 !9 .019 65 .065 240 .240 4 .004 20 .O2O 70 .070 260 .260 5 .005 22 .022 75 075 280 .280 6 .006 24 .024 80 .080 300 .300 7 .007 26 .026 85 .085 325 .325 8 .008 28 .028 90 .090 350 350 9 .OO9 3 .030 95 095 375 375 10 .OIO 3 2 .032 IOO .IOO 400 .400 ii .Oil 34 034 no .no 425 425 12 .OI2 36 .036 I2O .120 45 45 13 .013 38 .038 135 135 475 475 14 .014 40 .040 150 .150 500 .500 15 .015 45 045 165 .165 16 .Ol6 5 .050 1 80 .180 STANDARD IMPERIAL WIRE-GAUGE. Table No. 19. Mark. Size. Mark. Size. Mark. Size. 7/0 .500 16 .064 38 .0060 /o .464 i7 .056 39 .0052 5/o 432 18 .048 40 .0048 4/0 .400 i9 .040 4i .0044 '/o 372 20 ,036 42 .0040 / 348 21 .032 43 .0036 o .324 22 .028 44 .0032 I .300 2 3 .024 45 .OO28 2 .276 24 .022 46 .0024 3 .252 25 .O2O 47 .OO2O 4 .232 26 .018 48 .O0l6 5 .212 27 .0164 49 .OOI2 6 .192 28 .0148 50 .OOIO 7 .176 2 9 .0136 8 .l6o 3 .OI24 9 .144 3i .OIl6 10 .128 32 .OI08 ii .116 33 .OIOO 12 .IO4 34 .0092 13 .092 35 .0084 14 .080 36 .0076 15 .072 37 0068 GREAT BRITAIN AND IRELAND. FRACTIONS OF INCH. 135 INCHES AND THEIR EQUIVALENT DECIMAL VALUES IN PARTS OF A FOOT. Table No. 20. Inches. Fraction of foot. Foot. I / -083? 2 '/6 .. X A . .1667 25 4 e... g 3333 .4167 J" 6 7 ? 5 .5873 8 9 10 1 1 2 ^ 3 /4 M 6 ."/_, .6667 75 .8333 .0167 12 I I.O FRACTIONAL PARTS OF AN INCH, AND THEIR DECIMAL EQUIVALENTS. Tables No. 21. Eighths. Eighths. Fractions. Inch. I I/O ., .12$ 2 3 4 5" 1 % 5/8 2 5 375 .621; 6 7... S "0 75 . .875 8 I I.O Twelfths. Twelfths. Fractions. Inch. I .- X / M . . .OS'?'?'? '# 2 3 4 6 "/ / A i/ .125 16667 25 33333 .41667 c 8 9 10 ii 12 /2 y ::!:: *& i .... .58333 66666 75 83333 .91667 i.o 136 Sixteenths WEIGHTS AND MEASURES. and Thirty-seconds. Tables No. 21 (continued). Thirty- Seconds. Sixteenths. Fractions. Inch. I i/, 2 O3.I 2 1 2 -2 I /32 Vi6 3/_ a W O X ^ J .0625 OQ'?7i; 4 $ 2 / 32 V. 5/,_ . ^VO / .125 I ^62^ 6 7 8 3 4 '3 2 V.6 Vs. v, ' /4 *3**^j 1875 .21875 25 28125 10 1 1 5 / 3 2 V,6 II / 3125 3/1 -?7 t 12 17 6 / 3 2 '/. '3/, 2 JT-O / J 375 4062^ 14 1C 7 /3 2 7 A6 15 A 2 .. 4375 4.6871; 16 17 8 A ' 7 A 2 - 5 C-2I2C 18 10 9 /3 2 9 /,6 9/ M jo^-^o 5625 CQ^7C 20 21 10 / 3 2 5 /8 2I A 2 . OVo / D 625 6^62^ 22 2^ ii /3 2 ii/ , /i6 23 A 2 :, .v^^^-j .6875 7l87S 2 4 25 12 /3 2 3 /4 2 s/, 2 .. / ^^/O 75 .7812^ 26 27 13 732 ...... 13 /i6 27 A 2 .8125 8/1771; 28 20 14 / 3 2 V. 2 9/, 2 . "T-O / D .875 0062 1; 30 31 15 732 I5 A6 3I A 2 .. y w '-'^o 9375 06871; 32 16 / 3 2 I y- M ~' / o 1.0 II. MEASURES OF SURFACE. Tables No. 22. Superficial Measure. 144 square inches, or 7 r 183.35 circular inches ) ........................... i square foot. 9 square feet ...................................... i square yard. 100 square feet ...................................... i square. 272^ square feet, or ) . 30^ square yards } ' The square is used in measuring flooring and roofing. The rod is used in measuring brick-work. GREAT BRITAIN AND IRELAND. SURFACE, VOLUME. 137 Builders' Measurement. i superficial part ........................ i square inch. 12 parts ..................................... "i inch" (12 square inches). 12 "inches" ............................... i square foot. This table is employed in the superficial or flat measure of boards, glass, stone, artificers' work, &c. Land Measure. 9 square feet ................. . .................. i square yard. 1 6 square poles ............................. . . . . i square chain. 40 square poles, or ) , 1 2 10 square yards J 4 roods, or 10 square chains, or 1 60 square poles, or i acre.* 4,840 square yards, or 43,560 square feet 640 acres, or ) ., 3,097,600 square yards \ i square mile. 30 acres i yard of land. 100 acres i hide of land. 40 hides i barony. * The side of a square having an area of one acre is equal to 69.57 lineal yards. III. MEASURES OF VOLUME. Tables No. 24. Solid or Cubic Measure. 1728 cubic inches ^ 2200.15 cylindrical inches ( i cubic foot. 3300.23 spherical inches f 6600.45 conical inches ) 2 7 cubic feet i cubic yard, or load. 35.3156 cubic feet or 1 ; 1.308 cubic yards J Note. The numbers of cylindrical, spherical, and conical inches in a cubic foot, are as I, 1.5, 3. Builders' Measurement. i solid part 12 cubic inches. 1 2 solid parts i "inch" (144 cubic inches). 12 "inches" i cubic foot. This table is used in measuring square-sided timber, stone, &c. 138 WEIGHTS AND MEASURES. Note. The cubic contents of a piece, 6 inches square and 4 feet long is i cubic foot. 12 17 24 DECIMAL PARTS OF A SQUARE FOOT, IN SQUARE INCHES. Table No. 23. Hundredth Parts. Square Inches. Hundredth Parts. Square Inches. Hundredth Parts. Square Inches. Hundredth Parts. Square Inches. I 1.44 26 37-4 51 73-4 7 6 109.4 2 2.88 27 38.9 52 74-9 77 II0.9 3 4-32 28 40-3 53 76.3 78 II2.3 4 5.76 29 41.8 54 77.8 79 II3.8 5 7.20 30 43-2 55 79.2 80 II5.2 6 8.64 3 1 44.6 56 80.6 81 116.6 7 10. 1 32 46.1 57 82.1 82 118.1 8 "5 33 47-5 58 83-5 83 II9-5 9 13.0 34 49.0 59 85.0 84 121. 10 14.4 35 5-4 60 86.4 85 122.4 ii 15-8 36 51-8 61 87.8 86 123.8 12 17-3 37 53-3 62 89-3 87 125.3 13 18.7 38 54-7 63 90.7 88 126.7 14 20. 2 39 56-2 64 92.2 89 128.2 15 21.6 40 57-6 65 93-6 90 129.6 16 23.0 4i 58.0 66 95- 9i I3I.O i7 24.5 42 60.5 67 96-5 9 2 132.5 18 25-9 43 61.9 68 97-9 93 133.9 i9 27.4 44 63-4 69 99.4 94 135.4 20 28.8 45 64.8 70 100.8 95 136.8 21 30.2 46 66.2 7i IO2.2 96 138.2 22 31.7 47 67.7 72 103.7 97 139.7 23 33.1 48 69.1 73 IO5.I 98 I4I.I 24 34-6 49 70.6 74 106.6 99 142.6 25 36.0 50 72.0 75 108.0 IOO 144.0 IV. MEASURES OF CAPACITY. Tables No. 25. Liquid Measure. 8.665 cubic inches i gill or quartern. 4 gills (34.659 cubic inches) i pint. 2 pints i quart. 2 quarts i pottle. 4 quarts, or 8 pints (277.274 cubic inches) i gallon. 6.2355 gallons i cubic foot. The barn-gallon, for milk, is equal to 2 imperial gallons. GREAT BRITAIN AND IRELAND. CAPACITY. 139 Dry Measure. 2 pints ................................................ i quart. 4 quarts ................................................ i gallon. 2 gallons .............................................. i peck. 8 gallons 01 } ( T ' 28 3 66 cubic feet) ................ i bushel. 2 bushels .............................................. i strike. 4 bushels ....................................... . ..... i coomb. 5 bushels .............................................. i sack. 8 bushels ....... ....................................... i quarter. 4 quarters (41.077 cubic feet) ..................... i chaldron. 5 quarters ............................................. i wey or load. 2 loads ................................................ i last. In the Weights and Measures Act of 1878, it is only declared that the Imperial Standard Gallon contains 10 pounds of water at 62 F., and that 8 gallons shall be a bushel. Assuming that i cubic inch of water weighs 252.458 grains, the Imperial Standard Gallon has a capacity of 277.27384 cubic inches, or, say, 277.274 cubic inches, as before announced, page 125. The Imperial Standard bushel, which is equal to 8 gallons, has a capacity of 2218.19072 cubic inches. The internal diameter of the standard bushel, 1 7.8 inches, is double its internal depth, 8.9 inches. Heaped measure, which was used for such goods as could not be stricken as coals, potatoes, fruit, is now legally abandoned. Coals are sold by weight; and for other round goods, the measure is filled level with the brim as nearly as is practicable. The Market Garden bushel is made large enough to hold as much fruit as the heaped bushel held, filled level, so as to pack one on another. Coal and Coke Measure. 3 bushels (heaped) ........................... i sack. 9 bushels .............................. . ........ i vat. 36 bushels, or 12 sacks (58.66 cubic feet) i chaldron. 5^ chaldrons .................................... i room. 2 1 chaldrons .................................... i score. Old Wine and Spirit Measure. 4 gills or quarterns ....................... ... i pint. canons. 2 pints ......................................... i quart. 4 quarts (231 cubic inches) ......... . ..... i gallon = -8333 10 gallons ...................................... i anker = 8.333 1 8 gallons ...................................... i runlet = 15. 3 1 ^ gallons . . . ................................... i barrel = 26.250 42 gallons ...................................... i tierce = 35. .............................. , puncheon = 7 o. 126 gallons, or } 2 hogsheads, or V ........................... i pipeorbutt= 105. i^4 puncheons j 2 pipes or ) ^ Itun =2IO< 3 puncheons J 140 WEIGHTS AND MEASURES. By this measure wines, spirits, cider, perry, mead, vinegar, oil, &c., are measured; but the contents of every cask are reckoned in imperial gallons when sold. The imperial gallon is one-fifth larger than the old wine gallon. Old Ale and Beer Measure. * pnts ............ ..... .................. i quart. 4 quarts (282 cubic inches) ............ i gallon = 1.017 9 gallons .................................. i firkin = 9. 1 53 2 firkins, or 1 8 gallons .................. i kilderkin = 18.306 - 54.9*8 3 barrels, or , 108 gallons } ........................... ' butt '09.836 The imperial gallon is one-sixtieth smaller than the old beer gallon. Apothecaries' Fluid Measure. 60 minims (rn,) ........................... i fluid drachm (/ 5). 8 drachms (water, 1.732 cubic ) n -j inches, 437^ grains) }' ""'d ounce (/|). 20 ounces ..................... . ........... i pint ( ). 8 pints (water, 70,000 grains) ........ i gallon (gall.). 1 drop 60 drops 4 drachms 2 ounces (water, 875 grains) 3 ounces gram. drachm. tablespoonful. wineglassful. teacupful. V. MEASURES OF WEIGHT. Tables No. 26. Avoirdupois Weight. ms, or { 437^2 grains 1 6 drachms, or I , . > i ounce (oz.). 16 ounces, or ) . ,. . 1V } ................................. ' P ound (Pil) 7000 grans 8 pounds .................................. i stone (London meat market). 14 pounds ......................................... i stone. 28 pounds, or ) , stones | ............................... ' Barter 4 quarters, or \ 8 stones, or > ............................... i hundredweight (cwt). 112 pounds ' 20 hundredweights ............................... i ton. The grain above noted, of which there are 7000 to the pound avoirdupois, is the same as the troy grain, of which there are 5760 to the troy pound. Hence the troy pound is to the avoirdupois pound as i to 1.215, or as 14 to 17. GREAT BRITAIN AND IRELAND. WEIGHTS. 141 The troy ounce is to the avoirdupois ounce as 480 grains, the weight of the former, to 4S7/4 grains, the weight of the latter; or, as i to .9115. In Wales, the iron ton is 20 cwt. of 120 Ibs. each. Troy Weight. 24 grains ........................................ i pennyweight (dwt.}. 25 pounds ...... . ................................ i quarter. 4 quarters, or i oo pounds .................. i hundredweight. By troy weight are weighed gold, silver, jewels, and such liquors as are sold by weight. Diamond Weight. i diamond grain ............................. 0.8 troy grain. i carat ......................................... 4 diamond grains. iS/4 carats ..................................... i troy ounce. Apothecaries' Weight. The revised table of weights of the British Pharmacopeia is as follows: it is according to the avoirdupois scale : 437 YZ grains .................................. . . i ounce. 1 6 ounces ................................... i pound. In the old table of Apothecaries' Weight, superseded by the preceding table, the troy scale was followed, thus : Old Apothecaries' Weight. 20 grains .................................... i scruple ( 9). ,8 56}.' ..drachm ( 3 ). 576 1 2 ounces, or ) j 60 grains ' P Ound Weights of Current Coins. i farthing, . 8 inch diameter, ............. x / 10 ounce. i halfpenny, i.o ............. Vs i penny, 1.2 ............. / 3 i threepenny piece ............................ I / 20 i fourpenny piece ............................ .. */ I5 i sixpence ....................................... '/IQ i shilling .......................................... Vs i florin ........................................... 2 /s i half-crown ..................................... T / 2 5 shillings or 10 sixpences ............ % ...... i i sovereign ...................................... x /4 ounce, fully. For the exact weight in grains of these coins, see Table of British Money. 142 WEIGHTS AND MEASURES. Coal Weight. 14 pounds i stone. 28 pounds i quarter hundredweight. 56 pounds i half hundredweight. 88 pounds i bushel.* i sack, of n 2 pounds i hundredweight. i double sack, of 224 pounds... 2 hundredweights. 20 hundredweights, or I 10 double sacks J " 26^/2 hundredweights i chaldron (London). 53 hundredweights i chaldron (Newcastle). 7 tons T room. 21 tons 4 cwt i barge or keel. * Sundry Btishel Measures. i Cornish bushel of coal is 90 or 94 pounds; heaped, 101 pounds. i Welsh bushel, average weight 93 pounds. I Newcastle bushel is 80 or 84 pounds. Bradley Main, 92^ pounds. I London bushel, 80 or 84 pounds. t In Wales the miners' coal-ton is 21 cwt. of 120 Ibs. each. Wool Weight. 7 pounds 2 cloves, or 14 pounds 2 stones y 2 tods 2 weys 12 sacks, or 39 hundredweight. 12 score, or 240 pounds clove. stone. tod. wey. sack. last. pack. Hay and Straw Weight. truss of straw 36 pounds. load of straw 1 1 hundredweights, 64 pounds. truss of old hay 56 pounds. load of old hay 18 hundredweight. cubic yard of old hay 15 stone. truss of new hay 60 pounds. load of new hay 19 hundredweights, 32 pounds. cubic yard of new hay 6 stone. Corn and Flour Weight. 1 peck, or stone of flour 14 pounds. i o pecks i boll =140 2 bolls i sack =280 14 pecks i barrel =196 i bushel of wheat 60 i bushel of barley 47 i bushel of oats 40 Six bushels of wheat should yield one sack of flour; i last of corn is 80 bushels. GREAT BRITAIN AND IRELAND. MISCELLANEOUS. 143 MISCELLANEOUS TABLES. No. 27. Whatmaris Drawing Papers. Sizes of Sheets. Antiquarian 53 inches long, 3 1 inches wide. Double-elephant 40 27 Atlas 34 26 Colombier 34 23 Imperial 30 22 Elephant 28 23 Super-royal 27 19 Royal 23 19 Medium 22 17 Demy 20 15 Commercial Numbers and Stationery. 1 2 articles i dozen. 13 articles i long dozen. 1 2 dozen i gross. 20 articles i score. 5 score i common hundred. 6 score i great hundred. 30 deals 4 quarters 24 sheets of paper 20 quires 2 1 ^ quires 5 dozen skins of parchment. quarter. hundred. quire. ream. printers' ream. roll. Measures relating to Building. Load of timber, unhewn or rough 40 cubic feet. Load, hewn or squared ( 5 cubic feet, reckoned ( to weigh 20 cwt. Stack of wood 108 cubic feet. Cord of wood 128 (In dockyards, 40 cubic feet of hewn timber are reckoned to weigh 20 cwt. ; 50 cubic feet is a load.) i oo superficial feet i square. Hundred of deals 120 deals. Load of i-inch plank 600 square feet. (Load of plank more than i-inch thick = 600 -f- thickness in inches. Planks, section 1 1 by 3 inches. Deals, section 9 by 3 Battens, section 7 by 2^ A reduced deal is i % inches thick, 1 1 inches wide, and 1 2 feet long. Bundle of 4 feet oak-heart laths 120 laths. Load of 37^ bundles. Bundle of 5 feet oak-heart laths 100 laths. Load of 30 bundles. 144 WEIGHTS AND MEASURES. Measures relating to Building (continued.} Load of statute bricks 500. Load of plain tiles 1000. Load of lime 32 bushels. Load of sand 36 Hundred of lime 35 Hundred of nails, or tacks 1 20. Thousand of nails, or tacks 1 200. Fodder of lead I 9/ / 2 cwt. Sheet lead 6 to 10 pounds per sq. ft. Hundred of lead 112 pounds. Table of glass 5 feet. Case of glass 45 tables. Case of glass { (Newcastle and Normandy ( glass, 25 tables). Stone of glass 5 pounds. Seam of glass 24 stone. Sundry Commercial Measures. Dicker of hides 10 skins. Last of hides 20 dickers. Weigh of cheese 256 pounds. Barrel of herrings 26 2 / 3 gallons. Cran of herrings 37^ Pocket of hops i y 2 to 2 cwt. Bag of hops 3^ cwt, nearly. Last of potash, cod-fish, white her- ) rings, meal, pitch, tar. } ' 2 barrels ' Barrel of tar 26^ gallons. Barrel of anchovies 30 pounds. Barrel of butter 224 Barrel of candles 120 Barrel of turpentine 2 to 2 y 2 cwt. Barrel of gunpowder 100 pounds. Last of gunpowder 24 barrels. Measures for Ships. i ton, displacement of a ship, 35 cubic feet. i ton, registered internal capacity of do., 100 do. i ton, shipbuilders' old measurement, 94 do. COMPARISON OF COMPOUND UNITS. Tables No. 28. Measures of Velocity. j 1.467 feet per second, unite per hour | 88.0 feet per minute. i knot per hour 1.688 feet per second. i foot per second .682 mile per hour. i foot per minute .01136 mile per hour. GREAT BRITAIN AND IRELAND. COMPOUND UNITS. 145 Measures of Volume and Time. i cubic foot per second.. I 2 ' 222 cub ! c y ards P er minute - I * 3 3- 333 CUD1C yards per hour. i cubic foot per minute 2.222 cubic yards per hour. i cubic yard per hour .45 cubic foot per minute. i cubic inch per second . . j 2>o8 3 cubic foot P er hour - ( 12.984 gallons per hour. i gallon per second 569. 1 24 cubic feet per hour. i gallon per min ute 9.485 cubic feet per hour. Measures of Pressure and Weight. (See also page 127.) (. 144 Ibs. per square foot, i Ib. per square inch i METRE. 1000 millimetres ) 10 metres i decametre. i o decametres i hectometre. 10 hectometres, or 1000 metres i KILOMETRE (kilo.) 10 kilometres i myriametre. i toise (old measure) =1.949 metres. 1000 toises i mille = 1.949 kilometres. 2000 toises i itinerary league =3.898 2280.329 toises i terrestrial league = 4. 444 2850.411 toises i nautical league =5-555 i noeud (British nautical mile) = 1.855 148 WEIGHTS AND MEASURES. FRENCH WIRE-GAUGES (Jauges de Fits de JFer). The French wire-gauge, like the English, has been subject to variation. Table No. 30 contains the values of the "points," or numbers, of the Limoges gauge; table No. 31 gives the values of a wire-gauge used in the manufacture of galvanized iron; and table No. 32 the values of a gauge which comprises wire and bars up to a decimetre in diameter. FRENCH WIRE-GAUGE (Jauge de Limoges]. Table No. 30. Number. Diameter. Number. Diameter. Number. Diameter. Millimetre. Inch. Millimetre. Inch. Millimetre. Inch. 39 .0154 9 '35 053 2 18 3-40 134 I 45 .0177 10 1.46 575 i9 3-95 .156 2 .56 .0221 ii 1.68 .O66l 20 4-50 .177 3 .67 .0264 12 i. 80 .0706 21 5.10 .201 4 79 .O3II 13 1.91 .0752 22 5-65 .222 5 .90 0354 14 2.02 0795 23 6.20 .244 6 1. 01 .0398 15 2.14 .0843 24 6.80 .268 7 1. 12 .0441 16 2.25 .0886 8 1.24 .0488 17 2.84 .112 FRENCH WIRE-GAUGE FOR GALVANIZED IRON WIRE. Table No. 31, Number. Diameter. Number. Diameter. Number. Diameter. M'metre. Inch. M'metre. Inch. M'metre. Inch. I .6 .0236 9 1.4 Q55 1 17 3-o .118 2 7 .0276 10 i-5 .0591 18 3-4 134 3 .8 03^ ii 1.6 .0630 *9 3-9 154 4 9 354 12 1.8 .0709 20 4.4 173 5 I.O 0394 J 3 2.0 .0787 21 4.9 193 6 i.i 0433 14 2.2 .0866 22 5-4 .213 7 1.2 0473 15 2.4 0945 2 3 5-9 .232 8 '3 .0512 16 2.7 .I06 FRENCH WIRE- AND BAR -GAUGE. Table No. 32. Mark. Size. Mark. Size. Mark. Size. Mark. Size. Millimetre. Millimetre. Millimetre. Millimetre. P 5 8 13 16 27 24 6 4 I 6 9 14 i7 30 25 70 2 7 10 15 18 34 26 7 6 3 8 II 16 I 9 39 27 82 4 9 12 18 20 44 28 88 5 10 13 20 21 49 29 94 6 ii !4 22 22 54 30 IOO 7 12 15 24 23 59 FRANCE. THE METRIC STANDARDS. 149 II. FRENCH MEASURES OF SURFACE. Table No. 33. i oo square millimetres ............. i square centimetre. i oo square centimetres ............. i square decimetre. 'square metre, or ioo square metres, or centiares... i square decametre, or ARE. 100 square decametres, or ares... i square hectometre, or HECTARE. ioo square hectometres, or hectares i square myriametre. Land is measured in terms of the centiare, the are, and the hectare or arpent metrique (metric acre). There is also the decare, of 10 ares. III. FRENCH MEASURES OF VOLUME. Tables No. 34. Cubic Measure. 1000 cubic millimetres ................ i cubic centimetre. i ooo cubic centimetres ............... i cubic decimetre. i ooo cubic decimetres ................ i cubic metre. Wood Measure. 10 decisteres ........................... i stere* (i cubic metre). i voie (Paris) ......................... 2 steres. i voie de charbon (charcoal) ...... 0.2 stere ( x / s cubic metre). i corde ................................. 4 steres. * The stere measures 1. 14 metres x 0.88 metre x i metre, the billets of wood being x. 14 metres in length. IV FRENCH MEASURES OF CAPACITY. Tables No. 35. Liquid Measure. i o cubic centimetres .................... i centilitre. 10 centilitres .............................. i decilitre. i o decilitres .............................. i LITRE. 10 litres .................................... i decalitre. Dry Measure. 10 litres. ....................................... i decalitre. 10 decalitres, or ioo litres 1000 \ i hectolitre. 10 hectolitres, or ) . .. .. , , . . oo litres / i kilolitre (i cubic metre). The use of measures equal to a double-litre, a half -litre, a double-decilitre, a half-decilitre, is sanctioned by law. ISO WEIGHTS AND MEASURES. V. FRENCH MEASURES OF WEIGHT. Table No. 36. i o milligrammes .................. i centigramme. 10 centigrammes .................. i decigramme. 10 decigrammes .................. i GRAMME. 10 grammes ........................ i decagramme. 10 decagrammes .................. i hectogramme. 10 hectogrammes, or I ......... % KILOGRAMME 1000 grammes j i o kilogrammes ................... i myriagramme. M } !es ' or i quintal metrique. ioo kilogrammes j 10 quintaux, or ) f i millier, tonneau de mer, or tonne j " " ( (weight of i cubic metre of water at 39. i). 1000 kilogrammes EQUIVALENTS OF BRITISH IMPERIAL AND FRENCH METRIC WEIGHTS AND MEASURES. I. MEASURES OF LENGTH. Tables No. 37. A DECIMETRE DIVIDED INTO CENTIMETRES AND MILLIMETRES. INCHES AND TENTHS. METRIC DENOMINATIONS AND VALUES. EQUIVALENTS IN IMPERIAL DENOMINATIONS. Metres. Inches. Feet. Yards. Miles. r i millimetre i/ / IOOO ; 0.03937 i centimetre I/ / IOO - 0.39370 i decimetre 'Ao * 3-93704 I METRE .... I -39-37043 - 3.28087 1.09362 I'dekametre IO = 32.80869 10.93623 i hectometre IOO 109.36231 I KILOMETRE 1,000 = 3280.87 7f 1,093.6231 -0.62138 i mynametre 10,000 = 10,936.231 -6.21377 IMPERIAL AND METRIC EQUIVALENTS. Tables No. 37 (continued). IMPERIAL DENOMINATIONS. \ - EQUIVALENTS IN METRIC DENOMINATIONS, ^x Centimetres. Metres. Kilometres. inch (25 4 millimetres) . ... = 2.53995 0.30480 = 0.91439 1.82878 5.02915 = 2O.II662 = 2OI.l662 = 1,609.3296 = 0.20117 - 1.60933 foot or 1 2 inches yard, or 3 feet, or 36 inches.... fathom, or 2 yards, or 6 feet.... pole or 5 /^ yards chain, or 4 poles, or 22 yards... furlong, 40 poles, or 220 yards i mile, 8 furlongs, or 1760 yards EQUIVALENT VALUES OF MILLIMETRES AND INCHES. Tables No. 38. MILLIMETRES = INCHES. Millimetres. Inches. Millimetres. Inches. Millimetres. Inches. Millimetres. Inches. I 0394 27 1.0630 53 2.0866 79 3-II03 2 .0787 28 I.I024 54 2.I26O 80 3.1496 3 .Il8l 2 9 I.I4I7 55 2.1654 81 3.1890 4 1575 30 1.1811 56 2.2047 82 3.2284 5 .1968 31 1.2205 57 2.2441 83 3.2677 6 .2362 3 2 1.2598 58 2.2835 84 3.3071 7 .2756 33 1.2992 59 2.3228 85 3-3465 8 3I5 34 1.3386 60 2.3622 86 3-3859 9 3543 35 1.3780 61 2.4016 87 3-4252 10 3937 36 1-4173 62 2.4410 88 3.4646 ii 433 1 37 1.4567 63 2.4803 89 3-5040 12 .4724 38 1.4961 64 2.5197 90 3-5433 13 .5118 39 I -5354 65 2.5591 9i 3.5827 14 55 12 40 1.5748 66 2.5984 92 3.6221 15 .5906 4i 1.6142 67 2.6378 93 3.6614 16 .6299 42 1-6536 68 2.6772 94 3.7008 17 .6693 43 1.6929 69 2.7166 95 3.7402 18 .7087 44 1.7323 70 2.7559 96 3.7796 *$ .7480 45 1.7717 7i 2-7953 97 3.8189 20 .7874 46 1.8110 72 2.8347 98 3-8583 21 .8268 47 1.8504 73 2.8740 99 3.8977 22 .8661 48 1.8898 74 2.9134 100 3-9370 2 3 9055 49 1.9291 75 2.9528 = i decimetre. 24 .9449 50 1.9685 76 2.9922 2 5 .9843 5 1 2.0079 77 3-3 I 5 26 1.0236 52 2.0473 78 3.0709 152 WEIGHTS AND MEASURES. Tables No. 38 (continued}. INCHES DECIMALLY = MILLIMETRES. Inches. Millimetres. Inches. Millimetres. Inches. Millimetres. Inches. Millimetres. .OI 25 .26 6.60 .60 15.2 .94 23.9 .02 5 1 .28 7.II .62 15-7 .96 24.4 03 .76 30 7.62 .6 4 I6. 3 . 9 8 24.9 .04 i. 02 32 8.13 .66 16.8 I.OO 25-4 5 1.27 34 8.64 .68 17-3 2.0O 50.8 .06 1-52 36 9.14 .70 17.8 3.00 7 6.2 .07 1.78 .38 9- 6 5 .72 18.3 4.OO 101.6 .08 2.03 .40 10.2 74 18.8 5.OO 127.0 .09 2.29 .42 10.7 .76 19-3 6.00 152.4 .IO 2.54 .44 II. 2 .78 19.8 7.00 177.8 .12 3-05 .46 II.7 .80 20.3 8.00 203.2 .14 3.56 .48 12.2 .82 20.8 9.00 228.6 .16 4.06 5 12.7 .84 21.3 10.00 254.0 .18 4-57 S 2 13.2 .86 21.8 11.00 279.4 .20 5.08 54 13-7 .88 22.4 12.00 304.8 .22 5-59 .56 14.2 .90 22.9 = i foot. .24 6.10 58 14.7 .92 23-4 INCHES IN FRACTIONS = MILLIMETRES. Eighths. Sixteenths. Thirty-seconds. Millimetres. Eighths. Sixteenths. Thirty-seconds. Millimetres. I 79 17 13-5 I 2 1-59 9 18 14-3 3 2.38 I 9 I 2 4 3-17 5 IO 20 15-9 5 3-97 21 16.7 3 6 4.76 ii 22 17-5 7 5.56 23 18.3 2 4 8 6-35 6 12 24 19.0 9 7.14 25 19.8 5 IO 7-94 13 26 2O.6 ii 8.73 27 21.4 3 6 12 9-52 7 14 28 22.2 X 3 10.32 29 23.0 7 14 ii. ii 15 30 2 3 .8 15 11.91 31 24.6 4 8 16 12.7 8 16 32 25-4 By means of the preceding tables of equivalent values of inches and millimetres, the equivalent values of inches in centimetres and decimetres, and even in metres, may be found by simply altering the position of the decimal point. This method naturally follows from the decimal subdivisions of French measure. Take, for example, the tabular value of i millimetre, and shift the IMPERIAL AND METRIC EQUIVALENTS. 153 decimal point successively, by one digit, towards the right-hand side; the values of a centimetre, a decimetre, and a metre are thereby expressed in inches, as follows : i millimetre 394 inches. i centimetre Q-394 i decimetre 3.94 i metre 39.4 At the same time, it appears that, by selecting the tabular value of 10 millimetres, the value of its multiples are given more accurately, thus, 10 millimetres, or i centimetre -3937 inches. i decimetre 3.937 i metre 39.37 Again : 100 millimetres, or i decimetre = 3-937 inches. i metre =39-37 Similarly, for example : .32 inch = 8.13 millimetres. 3.2 = 81.3 = ( 813.0 or " \ .813 metre. 32.0 II." SQUARE MEASURES, OR MEASURES OF SURFACE. Tables No. 39. METRIC i square millimetre i square centimetre i square decimetre i square metre, or centiare i ARE, or square dekametre, or 100 square metres i hectare, or metrical acre, or 100 ares, or 10,000 square metres j IMPERIAL SQUARE MEASURES. .00155 square inch. .155 square inch. 15.5003 square inches. 10.7641 square feet, or 1.1960 square yards. 1076.41 square feet, or 119.60 square yards. 11,960.11 square yards, or 2.4711 acres, or 2 acres and 2280.1240 square yards. IMPERIAL METRIC SQUARE MEASURES. Imperial Measures. Square Centimetres. Square Metres. Ares. Hectares. i square inch 6.4SIA8 i square ft., or 144 sq. inches i square yard, or 9 square ) feet, or 1296 sq. inches ( i perch or rod, or 30^ square yards = 0.092901 = 0.836112 = 25.292 i rood, or 40 perches, or' 1210 square yards i acre, or 4 roods, or 4840 SQua.re yards = 1011.696 = 4046.782 = 10.11696 = 40.4678 = 0.40468 i square mile, or 640 acres = 258.98944 154 WEIGHTS AND MEASURES. III. CUBIC MEASURES. Tables No. 40. METRIC IMPERIAL CUBIC MEASURES. i cubic centimetre = 0.061025 cubic inch. , . , . f 61.021524 cubic inches, or i cubic decimetre - { 0^3*56 cubic foot / 35-3*56 cubic feet, or i cubic metre = | 1.308 cubic yards. IMPERIAL = METRIC CUBIC MEASURES. i cubic inch = 16.387 cubic centimetres. , f ( 28.3153 cubic decimetres, or icubicfoot = 1 0.028315 cubic metre. i cubic yard = 0.764513 cubic metre. WOOD MEASURE. i stere, or cubic metre i decistere 3.53 16 cubic feet. . , . / ,v ,v T ( 70.6312 cubic feet, or i voie de bois (wood), or 2 steres, Pans < \.- ^^ cubic ards i voie de charbon (charcoal) = i sack f 5 ^ bushels, or = '/j stere t 7.063 cubic feet. i corde of wood = 4 cubic metres 141.26 cubic feet. IV. MEASURES OF CAPACITY. Tables No. 41. METR AN?V N AL M u ! ES ATIONS EQUIVALENTS IN IMPERIAL DENOMINATIONS. Litres. Gills. Pints. Quarts. Gallons. Bushels. Quarters. Centilitre X / JOO 0.0704 0.0176 Decilitre J /io 0-7043 0.1761 Dekalitre 10 2.2009 - 2 75 I Hectolitre 100 22.009 2.7511 0.344 Kilolitre 1000 220.09 2 7-5 IZ 3-439 EQUIVALENTS IN METRIC DENOMINATIONS. IMPERIAL DENOMINATIONS. Litres. Dekalitres. Hectolitres. gill = 0.1420 pint, or 4 gills = 0.5679 quart, or 2 pints = 1-1359 gallon, or 4 quarts = 4-5435 peck, or 2 gallons = 9.0869 = 0.9087 bushel, or 8 gallons = 36.3477 = 3.6348 quarter, or 8 bushels =290.7816 =29.0782 =2.9078 IMPERIAL AND METRIC EQUIVALENTS. 155 V. MEASURES OF WEIGHT. Tables No. 42. METRIC WEIGHTS = IMPERIAL AVOIRDUPOIS WEIGHTS. i kilogramme 2 Ibs. 3 oz. 4 drachms, 10.473 7 4 grains. METRIC WEIGHTS. EQUIVALENTS IN IMPERIAL DENOMINATIONS. Grammes. Grains. Ounces. Pounds. Hundred- weights. Tons. Milligramme Centigramme x /iooo Vioo '/xo I 10 100 1,000 10,000 100,000 1,000,000 0.0154 0.1543 1-5432 I5-4323 154.3235 I543-2349 15432.3487 0.3527 3.5274 35.2739 2.2046 22.0462 220.462 1 2204.6212 1.9684 19.6841 0.9842 Decigramme GRAMME Dekagramme Hectogramme . . KILOGRAMME Myriagramme Quintal, or 100 kilog. Millier, or metric ton IMPERIAL AVOIRDUPOIS METRIC WEIGHTS. IMPERIAL AVOIRDUPOIS WEIGHTS. Grammes. Decigrammes. Kilogrammes. Millier, or Metric Ton. i drachm = 1.77184 i ounce, or 1 6 drams = 28.34954 = 2.83495 i pound, or 1 6 ounces = 453.59265 -45.35926 0-45359 i hundredweight, ) or 112 pounds J = 50.80237 i ton, or 20 hun- ) dredweights J = 1016.04754 = I.OI6O4 METRIC WEIGHTS = IMPERIAL TROY WEIGHTS. i kilogramme = 2 troy Ibs. 8 oz. 3 dwts., .34874 grain. METRIC WEIGHTS. Grains. Pennyweights. Ounces. Troy Pound. Milligramme . . . Centigramme ... Decigramme . . . GRAMME 0.01543 O.I543 2 I-54323 jc A727A Dekagramme... Hectogramme.. KILOGRAMME... A j'^rj^o^r 154.32349 - 1543.23487 = 15,432.34874 -0.64301 = 6.43014 = 0.32I5I = 3-2I507 -32.15073 = 2.67922 i S 6 WEIGHTS AND MEASURES. IMPERIAL TROY = METRIC WEIGHTS. IMPERIAL TROY WEIGHTS. EQUIVALENTS IN METRIC DENOMINATIONS. Milligramme. Gramme. Dekagramme. Hecto- gramme. Kilo- gramme. i troy grain 64.79895 0.06480 J-SSS 1 ? 3 I - I 349 373- 2 4i95 3-II035 37-324I9 3-73 2 42 0-37324 i dwt, or 24 gr. i oz., or 480 i Ib.,or5,y6o APPROXIMATE EQUIVALENTS OF ENGLISH AND FRENCH MEASURES. The following are approximately equal English and French measures of length : i pole, or peflch (5^ yards)... 5 metres (exactly 5.029 metres). i chain (22 yards) 20 metres (exactly 20.1166 metres). i furlong (220 yards) 200 metres (exactly 201.166 metres). 5 furlongs i kilometre (exactly 1.0058 kilometres). f 3 decimetres (exactly 3. 048 decimetres), or " jj 30 centimetres. One metre = 3. 28 feet = 3 feet 3 inches and 3 eighths all but l / 5I2 inch; = 40 inches nearly ( T /6 4 th or 1.6 per cent. less). .100 metre (i decimetre) = 4 inches nearly (exactly 3 I5 /i6 inches). .010 metre (i centimetre) = .4 inch, or */ 10 ths inch, nearly. .001 metre (i millimetre) = .04 inch, or Viooths inch, or two-thirds of I /\ 6 inch, or x / 25 inch, nearly. One inch is about 2^ centimetres (exactly 2.54). One inch is about 25 millimetres (exactly 25.4). One yard is "/i 2 ths of a metre, n metres are equal to 12 yards. Approximate rule for converting metres, or parts of metres, into yards : Add Vnth (i per cent. less). For converting metres into inches: Multiply by 40; and to convert inches into metres, or parts of metres, divide by 40. One kilometre is about ^6 mile (it is 0.6 per cent. less). One mile is about 1.6 or i s/ s kilometres (it is 0.6 per cent. Iess) = i6io metres, about. With respect to superficial measures: One square centimetre is about ye. 5 part of a square inch. One square inch is equal to about 6.5 square centimetres. One square metre contains fully 10^ square feet, or nearly i I / 5 square yards. One square yard is nearly 6 / 7 ths of a square metre. One acre is over 4000 square metres (about 1.2 per cent. more). One square mile is nearly 260 hectares (about 0.4 per cent. less). FRENCH AND ENGLISH COMPOUND UNITS. 157 With respect to cubic measures, and to capacity : One cubic yard is about ^ cubic metre (it is 2 per cent. more). One cubic metre is nearly ij^ cubic yard (it is i^i per cent. less). One cubic metre is nearly 35 x / 3 cubic feet (it is .05 per cent. less). One litre is over i^ pints (it is 0.57 per cent. more). One gallon contains above 4^ litres (it holds about i per cent. more). One kilolitre (a cubic metre) holds nearly i ton of water at 62 F. (i^ per cent, less), or 220^ gallons. One cubic foot contains 28.3 litres. With respect to weights: The ton and the gramme stand at nearly equal distances above and below the kilogramme, thus : i ton is 1,016,047.5 grammes, i kilogramme is 1,000.0 grammes, i gramme i.o gramme, in the ratio of about 1,000,000 : 1,000 : i. One gramme is nearly 15^ grains (about ^ per cent. less). One kilogramme is about 2 I / 5 pounds avoirdupois (about J / 4 per cent, more). A thousand kilogrammes, or a metric ton, is nearly one English ton (about i ]/?, per cent. less). One hundredweight is nearly 5 1 kilogrammes ( 2 / 5 per cent. less). EQUIVALENTS OF FRENCH AND ENGLISH COMPOUND UNITS OF MEASUREMENT. Weight, Pressure, and Measure. i kilogramme per metre j ' 6 ^ P oun 3 2 9 tonnes per cubic metre. i cubic metre per kilogramme 16.019 cubic feet per pound. i cubic foot per pound .0624 cubic metre per kilogramme. j" 1.329 cubic yards per ton. i cubic metre per tonne < J 7 94 cubic feet per cwt. (35.882 cubic feet per ton. i cubic yard per ton 752 cubic metre per tonne. i cubic foot per cwt 557 cubic metre per tonne. i cubic foot per ton .0279 cubic metre per tonne. Volume, Area, and Length. i cubic metre per lineal metre 1.196 cubic yards per lineal yard. i cubic yard per lineal yard 836 cubic metre per lineal metre. i cubic metre per square metre 3.281 cubic feet per square foot. i cubic foot per square foot 3.048 cubic metres per square metre. i litre per square metre .0204 gallon per square foot. i gallon per square foot 48.905 litres per square metre. ( .405 cubic metre per acre, i cubic metre per hectare \ .529 cubic yard per acre. (89.065 gallons per acre. i cubic metre per acre 2.47 1 cubic metres per hectare. i cubic yard per acre 1.902 cubic metres per hectare. i ooo gallons per acre 11.226 cubic metres per hectare. Work. i kilogrammetre (k x m) 7. 233 foot-pounds. i foot-pound 1382 kilogrammetre. i cheval-vapeur or cheval (75 k x m ) g horse-power, per second) / horse-power 1.0139 chevaux. kilogramme per cheval 2.235 pounds per horse-power. pound per horse-power 447 kilogramme per cheval. square metre per cheval 10.913 square feet per horse-power. square foot per horse-power .0916 square metre per cheval. cubic metre per cheval 3 5. 80 6 cubic feet per horse-power. i cubic foot per horse-power 0279 cubic metre per cheval. FRENCH AND ENGLISH COMPOUND UNITS. 159 Heat. i calorie, or French unit 3.968 English heat-units. i English heat-unit .252 calorie. French mechanical equivalent (425 ) 3074 foot-pounds = 774.70 foot- kilogrammetres) J pounds per English unit. English mechanical equivalent (772 ) , , ., foot-pounds) ... | Ia6 ? kilogrammetres. i calorie per square metre .369 heat-unit per square foot. i heat-unit per square foot 2 - 7 * 3 calories per square metre. i calorie per kilogramme i . 800 or 9/ 5 heat-units per pound. i heat-unit per pound -5555 r 5 /9 calorie per kilo- gramme. Speed, &c. ( 3.281 feet per second, i metre per second < 196.860 feet per minute. ( 2.236 miles per hour, i kilometre per hour .621 mile per hour. i foot per second, or per minute I '35 metre . P er second . OT P ( minute. i mile per hour J -447 metre per second. ( 1.609 kilometres per hour. i cubic metre per second.. .. ] 35-3i6 cu bjc feet per second. ( 2119 cubic feet per minute. i cubic foot per second, or per minute i ' 028 3 cubic metre P er second > I or per minute. i cubic metre per minute 1.308 cubic yards per minute. i cubic yard per minute .765 cubic metre per minute. Money. {4.320 pence per pound. .360 shilling per pound. 40.320 shillings per cwt., or 40. 3 2 per ton. i penny per pound .23 1 franc per kilogramme. i shilling per pound 2.772 franc per kilogramme. i shilling per cwt, or i per ton ... I 24.802 francs per tonne | 2.48 francs per quintal, i franc per quintal .403 shilling per cwt. i franc per tonne ( 4 8 4 penny per cwt. ( .806 shilling per ton. i franc per metre ... . . / -7 26 shillin S P er ^ d ( 8.709 pence per yard. i shilling per yard 1.378 francs per metre. i franc per kilometre ... . . { ' 6 3 86 P er mil .^ ( 15.326 pence per mile. ^"i per mile 15.660 francs per kilometre. i penny per mile .0652 franc per kilometre. i franc per square metre / 7 '^ 3 /?^ Per *^ K ^ A \ .6636 shilling per square yard. l6o WEIGHTS AND MEASURES. i shilling per square yard 1.510 francs per square metre. \ per square yard 30. 1 94 francs per square metre. {.270 penny per cubic foot. 7.281 pence per cubic yard. .607 shilling per cub.c yard. 33 P er cubic yard. i penny per cubic foot 3. 708 francs per cubic metre. i penny per cubic yard 137 franc per cubic metre. i shilling per cubic yard i .648 francs per cubic metre. i per cubic yard 32.962 francs per cubic metre. i franc ner litre I 43 ' 27 pence per gallon< "\ 3.606 shillings per gallon. i franc per hectolitre I -%93 shillings per hogshead (wine). i shilling per hogshead 528 franc per hectolitre. GERMAN EMPIRE. WEIGHTS AND MEASURES. Tables No. 43. From the ist January, 1872, the French metric system of weights and measures became compulsory throughout the German Empire, as follows : I. GERMAN MEASURES OF LENGTH. French Measure. Strich i millimetre. i o Strichs ................ New-Zoll = i centimetre. i oo New-Zolls ............ Stab i metre. 10 Stabs .................. Kette i dekametre. i oo Kettes ................ Kilometre = i kilometre. 7 Kilometres ........... , Mile II. GERMAN MEASURES OF SURFACE. mile , i Quadrat-Stab = i square metre. i oo Quadrat-Stabs ........ i Ar = i oo square metres. ioo Ars ..................... i Hectar = | IO ' 00 sc * uare metres ' or ( 2.47 acres. III. GERMAN MEASURES OF CAPACITY. i Schoppen J4 litre. (Beer Measure.) 2 Schoppens ......... i Kanne = i litre. 50 Kannes. . . . . Scheffel (bushel) = { * g^^tf bushels . -,-. / , v ( i hectolitre, or 2 Scheffels ........... ' Fass (*> = 1 M .oi gallons. The kanne is further divided into measures of % kanne, j^ kanne, and / kanne. GERMAN EMPIRE. WEIGHTS, THE FUSS. l6l IV. GERMAN MEASURES OF WEIGHT. i Milligramm = i milligramme. 10 Milligramms ...... i Centigramm = i centigramme. 10 Centigramms ..... i Dezigramm = i decigramme. ioo Dezigramms ..... x New-Loth = { j 500 grammes, or 50 New-Loths ........ i Pfund j^ kilogramme, or ( 1.1023 pounds avoirdupois. ioo Pfunds ............ i Centner L { 5 kilogrammes, or ( 110.23 pounds avoirdupois. = I00 kilogrammes, or 2000 Pfunds. / ' i 2204.6 pounds avoirdupois. 20 Centners, or ) Tonne = \ I00 kilogrammes, or i OLD WEIGHTS AND MEASURES OF THE GERMAN STATES. These vary for every state. The chief measures of length are the Fuss, and the Elle, of which the second is in general twice the first. The following are the values of the Fuss, which is the German foot, in the principal states. VALUES OF THE GERMAN Fuss IN THE STATES AND FREE TOWNS OF THE GERMAN EMPIRE. Table No. 44. Prussia 12.356 inches. Bavaria 11.491 Wiirtemberg 11.279 Saxony 11.149 Baden 11.811 Mecklenburg-Schwerin 1 1.457 Hesse-Darmstadt 9.843 Hesse-Cassel < 11-328 Oldenburg 11.649 Brunswick IJ - 2 35 Hanover 11.500 Mecklenburg-Strelitz n-457 Anhalt 12.356 Saxe-Coburg-Gotha 11-324 Saxe-Altenburg 11.122 Waldeck 11.512 Lippe -. ii. 398 Schwarzburg-Rudolstadt 15.047 Schwarzburg-Sondershausen : (1) High Sovereignty and Arnstadt ... 11.149 (2) Low Sovereignty and SonderShausen 1 1.33 1 Reuss 11.280 Schaumburg-Lippe 11.421 Hamburg 11.283 Liibeck 11.324 Bremen 11.392 11 162 WEIGHTS AND MEASURES. KINGDOM OF PRUSSIA. OLD WEIGHTS AND MEASURES.- Tables No. 45. I. PRUSSIAN MEASURES OF LENGTH. English Measure. i Linie = .0858 inch. 12 Linien i Zoll 1.0297 inches. 1.0297 feet. 2 Fuss i Elle = 2.0596 feet. 4.1192 yards. 82, Used by Miners. i Lachterlinie = .0927 inch. 10 Lachterlinien i Lachterzoll = .9268 inch. i o Lachterzoll i Achtel = .7723 foot. 6 -p^ g > i Lachter =2.0596 yards. 9 Fuss i Spanne =6.1788 yards. Surveyors' Measure. Scrupel = .0148 inch. 10 Scrupel Linie = .1483 inch. i o Linien Zoll = 1.4828 inches. 10 Zoll Land-Fuss = 1.2356 feet. i o Land- Fuss Ruthe = 4. 1 1 92 yards. 2000 Ruthen Meile =4.6809 miles. II. PRUSSIAN MEASURES OF SURFACE. i Square Linie = .00736 square inch. 144 Square Linien i Square Zoll = 1.0603 square inches. 144 Square Zoll i Square Fuss = 1.0603 square feet. 144 Square Fuss i Square Ruthe =16.967 square yards. 180 Square Ruthen... i Morgen = .63103 acre. 30 Morgan i Hufe = 18.931 acres. III. PRUSSIAN MEASURES OF VOLUME. Cubic Measure. 1728 Cubic Linien.... 1728 Cubic Zoll 1728 Cubic Fuss ...... i Cubic Linie = .000632 cubic inch. i Cubic Zoll = 1.092 cubic inches. i Cubic Fuss = 1.092 cubic feet. i Cubic Ruthe =69.893 cubic yards. For measuring stone and brickwork, earth, peat, fascines, and firewood, the following are used : PRUSSIA. CAPACITY, WEIGHTS. 163 i Cubic Klafter, or ) , . 108 Cubic Fuss }="7.93 cubic feet 4^ Klafters i Haufe =530.70 i Schachruthe (in architecture) 144 Cubic Fuss = 157.25 IV. PRUSSIAN MEASURES OF CAPACITY. Dry Measure. i Maasche = .7560 quart. 4 Masschen, or ) AT , 7 v* > i Metze = 3.0242 quarts. 4 Metzen i Viertel = 3.0242 gallons. 4 Viertel, or ) Scheff 1 - / I -5 121 DUSne l s J or 48 Quarts f (1.941 cubic feet. 4 Scheffeln i Tonne = 6. 0484 bushels. > i Maker 2.26815 quarters. i Last = 11.3407 quarters. The Tonne in the table is the measure for salt, lime, and charcoal. A Tonne of flax-seed is 2.354 Scheffeln. Liquid Measure (for Wine and Spirits). 32 Cubic Zoll i Ossel = 1.0079 pints. 2 Ossel i Quart = 1.0079 quarts. 30 Quarts, or ) . , ,, 60 Ossel } r Anker 7-559 gallons. 2 Ankers i Eimer = 15.118 2 Eimers i Ohm = 30.237 -? Eimers, or ) ~ , * i^Ohm \ I0xhoft ' 45-355 4 Oxhoft, or ) , 6 Ohm I lFuder =''-4 V. PRUSSIAN MEASURES OF WEIGHT. Corn 4.115 grains. 10 Corns Cent .09406 dram. i o Cents Quentche = .9406 dram. i o Quentchen Loth . 5 88 ounce. 30 Loth Zollpfund = 1.1023 pounds. 100 Zollpfund Centner = 1 10. 23 pounds. 20 Zollpfund Stein 2 2.046 pounds. 3 Centners i Schiffspfund = < 40 Centners i Schiffslast = < The Tonne of coals is 2270 pounds avoirdupois, or 1.013 tons. 1 64 WEIGHTS AND MEASURES. KINGDOM OF BAVARIA. OLD WEIGHTS AND MEASURES. Tables No. 46. I. BAVARIAN MEASURES OF LENGTH. 12 Linien 12 Zoll 6 Fuss 10 Fuss .. Linie = .0798 inch. Zoll = .95756 inch. Fuss = .95756 foot. Klafter = 5.74536 feet. Ruthe =9.5756 feet. In surveying, the Fuss is divided into 10 Zoll, and i Zoll into 10 Linien. The Elle contains 2 Fuss 10% Zoll, = 2.733 II. BAVARIAN MEASURES OF SURFACE. i Square Zoll .91692 square inch. 144 Square Zoll .... i Square Fuss .91692 square foot. 100 Square Fuss ... i Square Ruthe 10. 188 square yards. 400 Square Ruthen { < ^J&f^ } T { ^.g aST III. BAVARIAN MEASURES OF VOLUME. i Cubic Zoll = .878 cubic inch. 1728 Cubic Zoll ..................... i Cubic Fuss - .878 cubic foot. 126 Cubic Fuss (6x6x3^ Fuss) i Klafter = I II0 ' 628 cu ^ c feet ' or ( 4.097 cubic yards. IV. BAVARIAN MEASURES OF CAPACITY. Dry Measure. Dreisiger= .12745 peck. 4 Dreisigers 4 Maassls . . . 2 Viertel ... 6 Metzen... 4 Schaffel . . Maassl = .12745 bushel. Viertel = .5098 bushel. Metze = 1.0196 bushels. Schaffel = 6. 1 1 76 bushels. Muth = 3.0588 quarters. Liquid Measure. i Maaskanne ~ .23529 gallon. 64 Maaskannen i Eimer 15.05856 gallons. 25 Eimer i Fass =376.464 gallons. The Schenk-Eimer, ordinarily used in the Wine trade, contains only 60 Maaskannen, equal to 14.1174 imperial gallons. V. BAVARIAN MEASURES OF WEIGHT. i Quentchen^ I 5433 ounce. 4 Quentchen i Loth .6173 ounce. 32 Loth i Pfund 1.23457 pounds. zoo Pfund i Centner = { I2 3-7 pounds or ( 1. 102 hundredweights. WURTEMBERG. LENGTH, SURFACE, ETC. 165 KINGDOM OF WURTEMBERG. OLD WEIGHTS AND MEASURES. Tables No. 47. I. WURTEMBERG MEASURES OF LENGTH. Punkte .01128 inch. ip Punkte Linie .1128 inch. 10 Linien Zoll = 1.128 inches. 10 Zoll Fuss = .93995 foot. 10 Fuss Ruthe 9-3995 f eet - 2.144 Fuss EHe 2.015 feet. 6 Fuss Klafter = 5.6397 feet. 26,000 Fuss i Meile ( 8146.25 yards or ) 4.6285 miles. II. WURTEMBERG MEASURES OF SURFACE. i Square Zoll = 1.272 square inches. 100 Square Zoll i Square Fuss -8835 square foot. 100 Square Fuss i Square Ruthe = 88.3506 square feet. 384 Square Ruthen. . . i Morgen = { 3769-626 square yards, or ( .779 acre. ..j III. WURTEMBERG MEASURES OF VOLUME. i Cubic Linie = .001434 cubic inch. i ooo Cubic Linien i Cubic Zoll = 1.434 cubic inches. i ooo Cubic Zoll i Cubic Fuss = .83045 cubic foot. 144 Cubic Fuss i Cubic Klafter = 119.583 cubic feet. IV. WURTEMBERG MEASURES OF CAPACITY. Dry Measure. i Viertlein = .305 pint. 4 Viertlein i Ecklein = 1.219 pints. 8 Ecklein i Vierling =1.219 gallons. 4 Vierling i Simri = 4.876 gallons. 8 Simri i Scheffel = 4.876 bushels. Liquid Measure. Quart or Schoppen = .4043 quart. 4 Quarts Helleich Maass = 1.6173 quarts. 10 Helleich Maass Imi = 4.0433 gallons. 16 Imi Eimer = 64.6928 gallons. 6 Eimer Fuder = 388. 1568 gallons. V. WURTEMBERG MEASURES OF WEIGHT. i Quentchen .1289 ounce. 4 Quentchen iJLoth 5 I 5^> ounce. 32 Loth i Light Pfund = 1.03115 pounds. 100 Heavy Pfund, or ) ^ , , 104 Light Pfund...... } ' Centner ' I0 7- 2 39 6 Pds. ioo Light Pfund = 103.115 pounds. i66 WEIGHTS AND MEASURES. KINGDOM OF SAXONY. OLD WEIGHTS AND MEASURES. Tables No. 48. I. SAXON MEASURES OF LENGTH. Linie = .07742 inch. 2 Linien Zoll = .9291 inch. 12 Zoll Fuss = .9291 foot. 2 Fuss Elle = 1.8582 feet. 2 Ellen Stab = 3.7165 feet. 1 5 Fuss, 2 Zoll i Ruthe (Land Measure) = 4. 69 7 2 yards. 16 Fuss i Ruthe (Road Measure) = 4.9553 yards. i Lachter (Mining) = 2.1873 yards. 1324.987 Ellen i Meile Post = 4.6604 miles. II. SAXON MEASURES OF SURFACE. i Square Zoll .8632 square inch. 144 Square Zoll i Square Fuss = .863 2 square foot. 300 Square Ruthen i Acker = 1.4865 acres. III. SAXON MEASURES OF VOLUME. i Cubic Zoll = .8021 cubic inch. 1728 Cubic Zoll i Cubic Fuss = .8021 cubic foot. 108 Cubic Fuss i Klafter = 86.624 cubic feet. 3 Klafter i Schragen = 259.873 cubic feet. The Klafter is 6 Fuss by 6 Fuss by 3 Fuss. The Schragen is used in the measurement of firewood. IV. SAXON MEASURES OF CAPACITY. Dry Measure. i Maasche 4 Maaschen i Metze 4 Metzen i Viertel 4 Viertel i Scheffel 12 Scheffel i Malter 2 Malter i Wispel Liquid Measure. i Quartier 4 Quartier i Nossel 2 Nossel i Kanne 36 Kannen i Anker 2 Anker i Eimer 3 Eimer i Oxhoft 6 Eimer . . . i Fass or Barrel = 1.4463 quarts. = 1.4463 gallons. = 5-7 8 5 2 gallons. = 2.8926 bushels. = 34.7124 bushels. = 69.4249 bushels. .2059 pint. .8237 pint. 1.6474 pints. 7.4237 gallons. 14.8262 gallons. 44.4687 gallons. 88.9374 gallons. V. SAXON MEASURES OF WEIGHT. The old Saxon measures of weight are the same as those of Prussia. BADEN. LENGTH, SURFACE, ETC. I6 7 GRAND DUCHY OF BADEN. OLD WEIGHTS AND MEASURES. Tables No. 49. I. BADEN MEASURES OF LENGTH. 10 Punkte. 10 Linien. loZoll .... 2 Fuss.... 10 Fuss... i Punkte Linie Zoll Fuss Elle Ruthe 6 Fuss 14814.815 Fuss 2 Stunden.. .0118 inch. .118 inch. 1.181 inches. .9842 foot. 1.9685 feet. 9.8427 feet. Klafter = 5.9055 feet. Stunde =4860.59 yards. Meile = 5.5234 miles. II. BADEN MEASURES OF SURFACE. 1.3951 square inches. .9688 square foot. 10.7643 square yards. = 1076.43 square yards. I 4305.72 square yards, or " ( .8896 acre. i Square Zoll 100 Square Zoll i Square Fuss 100 Square Fuss i Square Ruthe 100 Square Ruth en... i Viertel 4 Viertel i Morgen III. BADEN MEASURES OF VOLUME. i Cubic Fuss = -95335 cubic foot. 144 Cubic Fuss i Klafter = 137.28 cubic feet. IV. BADEN MEASURES OF CAPACITY. Liquid Measure. i Glass - 1.0563 gills. i o Glass i Maass = 1.3204 quarts. 10 Maass i Stutze = 3.3014 gallons. 10 Stutzen i Ohm = 33.014 gallons. 10 Ohm i Fuder = 330.14 gallons. Dry Measure. i Becher = .2643 P mt - 10 Becher i Maasslein = .1652 peck. 10 Maasslein i S ester .4127 bushel. 10 Sester i Malter = 4.1268 bushels. 10 Maker i Zuber = 41.2679 bushels. V. BADEN MEASURES OF WEIGHT. i As = .7716 grain. 10 As i Pfennig = 7.716 grains. 10 Pfennig i Centas = .1764 ounce. 10 Centas i Zehnling = 1.7637 ounces. 10 Zehnling i Pfund = 1.1023 pounds. 100 Pfund i Centner = 110.230 pounds. i68 WEIGHTS AND MEASURES. THE HANSE TOWNS. OLD WEIGHTS AND MEASURES. Tables No. 50. HAMBURG. WEIGHTS AND MEASURES. I. HAMBURG MEASURES OF LENGTH. i Achtel 8 Achtel i Zoll 1 2 Zoll i Fuss 2 Fuss i Elle 6 Fuss i Klafter, or Faden = 14 Fuss i Marsch-Ruthe = 1 6 Fuss... . i Geest-Ruthe inch. .9402 inch. .9402 foot. 1.8804 feet. 5.6413 feet. 13.1629 feet. 15.0434 feet. The Hamburg Elle above is used for silk, linen, and cotton goods. The Brabant Elle is equal to i J / 5 Hamburg Elle; and 4 of them are reckoned equal to 3 yards. The Prussian Ruthe is also used. The Prussian Fuss is used in surveying. II. HAMBURG MEASURES 144 Square Zoll... 196 Square Fuss.. 256 Square Fuss.. 200 Square Geest- Ruthen 600 Sq. Marsch- Ruthen . . i Square Zoll i Square Fuss i Square Marsch-Ruthe i Square Geest-Ruthe i Scheffel Geest-Land i Morgen OF SURFACE. .8840 square inch. .8840 square foot. 173.26 square feet. 226.30 square feet 5028.98 square yards, or 1.039 acres. 11550.93 square yards, or 2.386 acres. III. HAMBURG MEASURES OF VOLUME. i Cubic Zoll = .8311 cubic inch. 1728 Cubic Zoll i Cubic Fuss = .8311 cubic foot. 88.9 Cubic Fuss.... i (Cubic) Klafter - 73.88 cubic feet. 120 Cubic Fuss i Tehr =99-73 cubic feet. IV. HAMBURG MEASURES OF CAPACITY. Liquid Measure. 2 Ossel Ossel Quartier = .09965 gallon. . iQQT. gallon. 2 Quartier 2 Kannen i Stubchen 4 Viertel ... . Kanne Stubchen Viertel Eimer 39 8 7 gallon. .7974 gallon. 1.5947 gallons. 6.3788 gallons. 5 Viertel 6 Eimer 4 Anker Anker Tonne Ohm = 7-9735 gallons. 38.2728 gallons. 3.1.804.0 gallons. 6 Anker 6 Ohm... Oxhoft Fuder. or Tonneau = ^ A . *_* **JL^\J C^Cll.LVyi.lhJ. 47.8410 gallons. TO 1.76/10 gallons. The above are measures for Wines and Spirits. For Beer, there are three sizes of Tonne, containing respectively 48, 40, and 32 Stubchen. HAMBURG. WEIGHTS. 169 2 Small Maass. 4 Large Maass 4 Spint 2 Himten 2 Fass... Dry Measure. Small Maass = .0236 bushel. Large Maass = .0473 bushel. Spint = .1890 bushel. Himten = .7560 bushel. Fass = 1.5121 bushels. Scheffel = 3.0242 bushels. 10 Scheffeln i Wispel =30.2416 bushels. 3 Wispel i Last = 90.7248 bushels. For barley and oats, the Scheffel contains 3 Fass. V. HAMBURG MEASURES OF WEIGHT. i Half Gramme = .0011 pound =.5 gramme. 10 Half Grammen i Quint .01102 pound = 5 grammes. 10 Quinten i (New) Unze .11023 pound = 50 10 (New) Unzen.. i (New) Pfund 1. 10232 pounds = 500 100 (New) Pfund i Centner = 110.232 pounds = 50 kilog. 60 Centners. i (Commercial) Last - { ^^o^tons 8 ' } =3 oookil g- This, it is apparent, is a metric system of weights, which was comparatively recently introduced and adopted at Hamburg. It is now, of course, over- ruled by the French metric system enforced for the German Empire. BREMEN. OLD WEIGHTS AND MEASURES. The Fuss is equal to 11.392 inches, and the Klafter is equal to 5.696 feet. The Morgen = .6368 acre. The principal measures for wines and spirits are the Viertel= 1.56 gallons; the Anker = 5 Viertels = 7.80 gallons; the Oxhoft = 46.8o gallons. The Scheffel, for dry goods = 2.0388 bushels. The old weights are the same as those of Hamburg. LUBEC. OLD WEIGHTS AND MEASURES. The Fuss is equal to 11.324 inches. The Viertel=i.6o gallons; the Anker = 8 gallons; the Oxhoft = 48.04 gallons. The Scheffel, for dry goods, = .9545 bushel. The old Pfund =1.0725 pounds, and the Centner = 1.0725 cwts. GERMAN CUSTOMS UNION. OLD WEIGHTS AND MEASURES. Table No. 51. Centner 110.23 pounds (50 kilogrammes). Ship-Last of timber about 80 cubic feet. Scheffel 1.512 bushels. Klafter 6 feet. In Oldenburg, Hanover, Brunswick, Saxe-Altenbourg, Birkenfeld, Anhalt, Waldeck, Reuss, and Schaumburg-Lippe, the old system of weights is the same as that of Prussia. I/O WEIGHTS AND MEASURES. AUSTRIAN EMPIRE. WEIGHTS AND MEASURES. Tables No. 52. I. AUSTRIAN MEASURES OF LENGTH. i Punkte .0072 inch. 12 Punkte ............... i Linie .0864 inch. 12 Linien ............... i Zoll 1.0371 inches. 12 Zoll .................. i Fuss 1.0371 feet. 2 Fuss .................. i Elle 2.0742 feet. 6 Fuss .................. i Klafter 6.2226 feet ' 4=o= Klafter .............. . Meile(post) = II. AUSTRIAN MEASURES OF SURFACE. i Square Zoll 1.0756 square inches. 1 44 Square Zoll ........... i Square Fuss = 1.0756 square feet. 36 S^are Fuss ........ : Square Klafter - { ^jSg ": Sf- or } ' Square Ruthe 35 - 854 square yards - 64 Square Ruthen i Metze = 2294.7 square yards. 3 Metzen, or 1 i T h I ^^ 4 sc l uare y ar s, or 1600 Square Klafter ...... J 1 1.4223 acres. III. AUSTRIAN MEASURES OF VOLUME. Cubic Measure. i Cubic Zoll = i* i 1 55 cubic inches. 1728 Cubic Zoll ..... i Cubic Fuss = 1-1155 cubic feet. .,6 Cubic Fuss.... x Cubic Klafter - { IV. AUSTRIAN MEASURES OF CAPACITY. Dry Measure. i Probmetzen = \ 8 Probmetzen i Becher .8460 pint. 4 Becher i Futtermassel = 1.6920 quarts. 2 Futtermassel .. . i Muhlmassel - ( 3 'fj qu ,f ts ' or [ .8460 gallon. 2 Muhlmassel i Achtel = 1.6920 gallons. IT- i ( 3.3840 gallons, or 2 Achtel i Viertel = < 2 o bushel 4 Viertel i Metze = 1.6918 bushels. f So.7S^6 bushels, or 3 Metzen i Muth = | 5^75^ ^^ AUSTRIAN EMPIRE. CAPACITY, WEIGHTS. Liquid Measure. i Pfiff 2 Pfiff '... . i Seidel 2 Seidel Kanne 2 Kannen Mass 10 Mass Viertel 4 Viertel Eimer 32 Eimer Fuder 1.246 gills, or 10.781 cubic inches. 2.491 cubic inches, or .6229 pint. 1.2457 pints. 1.2457 quarts. 3.1143 gallons. 12.4572 gallons. 398.6304 gallons. V. AUSTRIAN MEASURES OF WEIGHT. 4 Pfenning 4 Quentchen.. 2 Loth 4 Unzen i Vierdinge 2 Vierdinges . . . i Mark i Pfenning i Quentchen i Loth i Unze ( 270.1 grains, or ( .6173 dram. i 2.4694 drams. 9.8776 drams, or .6173 ounce. 1.2347 ounces. 4.9388 ounces. 9.8776 ounces, or .6173 pound avoirdupois. 1.2347 pounds avoirdupois. {123.47 pounds avoirdupois, or 1.1024 hundredweights. In 1853, a pfund of 500 grammes, with decimal subdivisions, was adopted for customs and fiscal purposes. 2 Marks, or ) 16 Unzen j " i Pfund 100 Pfund... . i Centner RUSSIA. WEIGHTS AND MEASURES. Tables No. 53. I. RUSSIAN MEASURES OF LENGTH. English Equivalent. i Vershok = 1.75 inches. 16 Vershpks i Arschine = 28 3 Arschines i Sajene = 7 feet. C 3500 feet, or 500 Sajenes i Verst = < 1166^3 yards, or ( 0.6629 mile. The Fuss, or Russian foot, is 13.75 inches; but, since 1831, the English foot of 1 2 inches has been used as the ordinary standard of length, each inch being divided into 12 parts. i Lithuanian Meile 5-55 74 English miles. i Rhein Fuss, used in calculating ) ^ r , f export duties on timber / *-3 En & llsh feet ' 172 WEIGHTS AND MEASURES. II. RUSSIAN MEASURES OF SURFACE. i Square Arschine = ( ^ st l uare inch f ' or 5.444 square feet. 9 Square Arschines.. i Square Sajene = I ^ sc * uare feet ' or ^ 5.444 square yards. 2400 Square Sajenes ..... i Desatine I ( : 3' 66 *<&*** y ards > or [ 2.70 acres. For earthworks, masonry, &c., the Sajene is divided into tenths (dessiatka), hundredth* (sotka), and thousandths (tisiatchka), which are used as a basis for lineal, superficial, and cubic measurements, similarly to the French metre with its sub-multiples. III. RUSSIAN MEASURES OF CAPACITY. Liquid Measure. iTscharkey .8656 gill or ( .2104 pint. 10 Tscharkeys ...... i Kruschka 1.0820 quarts. 100 Tscharkeys ...... i Vedro 2.7049 gallons. 3 Vedros ............ i Anker 8.1147 ^3 V 6 Ankers j ' * Sarokowa J a Boshka = 108.196 Dry Measure (Grain). Garnietz 2.885 quarts. 2 Garnietz ......... Tschetwerka = 1.4424 gallons. 4 Tschetwerkas . . . Tschetwerik = -7 2I 3 bushel. 2 Tschetweriks. . . . Pajak i .4426 bushels. 2 Pajaks ............ Osmin 2.8852 2 Osmins .......... Tschetwert* = 5.7704 16 Tschetwerts... . i Last = II '^. q uar *f, or } 1.154 imperial lasts. * A Tschetwert is usually reckoned as 5^ bushels, and 100 Tschetwerts as 72 quarters, though they are more exactly 72.1308 quarters. loo quarters are equal to 138.637 Tschetwerts. IV. RUSSIAN MEASURES OF WEIGHT. i Dolis -68576 grain. rZolotnick 3 Zolotnicks... i Lotti .4514 8 Zolotnicks... i Lana 1-2037 ounces. 12 Lanas, or *\ f .90285 pound avoirdupois, or 32 Lottis, or > i Funt, or pound = 14.446 ounces, or 96 Zolotnicks ) (6320 grains. 40 pounds i Pood 36.114 pounds avoirdupois. T, T f ^61.14 pounds avoirdupois, or IoPoods - iBerkovitz 3 3.224 hundredweights. 3 Berkovitz i Packen 9.672 hundredweights. HOLLAND, BELGIUM, NORWAY, ETC. 1/3 62.0257 Poods i English ton. 2481.0268 Russian pounds i The Pood is commonly estimated at 36 pounds avoirdupois. The Nuremberg pound, used for apothecaries' weight, weighs 5527 grains, or about .96 pound troy. The Ship-Last is equal to 2 tons nearly. The Carat, for weighing pearls and precious stones, is about 3 x /e grains. HOLLAND. The metric system was adopted in Holland in 1819; the denominations corresponding to the French are as follows : Length. Millimetre, Streep; centimetre, Duim; decimetre, Palm; metre, El; decametre, Roede; kilometre, Mijle. Surface. Square millimetre, Vierkante Streep; square centimetre, Vier- kante Duim; and so on. Hectare, Vierkante Bunder. Cubic Measure. Millistere, Kubicke Streep, and so on. Capacity. Centilitre, Vingerhoed; decilitre, Maatje; liquid litre, Kan; dry litre, Kop; decalitre, Schepel; liquid hectolitre, Vat or Ton; dry hectolitre, Mud or Zak; 30 hectolitres = i Last= 10.323 quarters. Weight. Decigramme, Korrel; gramme, Wigteje; decagramme, Lood; hectogramme, Onze; kilogramme, Pond. BELGIUM. The French metric system is used in Belgium. The name Livre is substituted for kilogramme, Litron for litre, and Aune for metre. DENMARK. : WEIGHTS AND MEASURES. Tables No. 54. I. DANISH MEASURES OF LENGTH. i Linie .0858 inch. 12 Linier ........ i Tomme 1.0297 inches. 12 Tommer ..... i Fod 1.0297 feet. 2 Fod ........... i Alen 2.0594 or ' ^ 2,000 Roder, or ) x Mn f 8237.77 yards, or 24,000 Fod J ( 4.68055 miles. 23,642 Fod ........... i nautical mile= 4.61072 English miles. II. DANISH MEASURES OF SURFACE. 144 Square Linie ...... i Square Tomme = 1.0603 square inches. 144 Square Tomme... i Square Fod = 1.0603 square feet. 144 Square Fod ........ i Square Rode = 16.966 square yards. WEIGHTS AND MEASURES. III. DANISH MEASURES OF VOLUME. 1728 Cubic Linier i Cubic Tomme = 1.0918 cubic inches. 1728 Cubic Tomme.... i Cubic Fod = 1.0918 cubic feet. The Favn of firewood measures 6x6x2 Fod = 72 cubic Fod = 78.60 cubic feet. In forest measure it is 6^x6^x2 Fod = 84^ cubic Fod = 92.26 cubic feet. IV DANISH MEASURES OF CAPACITY. Liquid Measure. Paegle = .4248 pint. 4 Paegle Pot = 1.6991 pints. 2 Potter Kande 3-39^3 38 Potter Anker = 8.0709 gallons. 136 Potter Tonde = 28.885 6 Ankerne Oxehoved == 48.4256 4 Oxehoveder Fad =193.7027 Dry Measure. Pot = 1.6991 pints. 18 Potter Skeppe = 3.8232 gallons. 2 Skepper Fjerdingkar .9558 bushel. 4 Fj erdingkar Tonde = 3.8231 bushels. 12 Tender Laest =45.8769 V. DANISH MEASURES OF WEIGHT. i Ort 7-7 J 63 grains. 10 Ort Kvint 77- l6 3 100 Kvinten Pund 1.1023 pounds. 100 Pund Centner =110.23 40 Centner Last 1.9684 tons. 52 Centner Skip-Last 2 -559 >> 1 6 Pund i Lispund 17.637 pounds. 320 Pund i Skippund = 3.149 cwts. SWEDEN. WEIGHTS AND MEASURES. Tables No. 55. I. SWEDISH MEASURES OF LENGTH. i o Linier 10 Turner 10 Fot 10 Stanger 2 Fot 6 Fot i Linie = i Turn i Fot i Stang i Ref i Men. i Aln i Faden = .1169 inch. 1.1689 inches. 11.6892 9.7411 feet. 32.4703 yards. 1.942 feet. 5-845 SWEDEN, SWITZERLAND. 175 II. SWEDISH MEASURES OF SURFACE. ioo Square Linier... i Square Turn = 1.3666 square inches, ioo Square Turner., i Square Fot = .9489 square foot, ioo Square Fot i Square Stang = 3.5146 square yards. ioo Square Stanger i Square Ref = 4 Square Fot i Square Aln = 3.7956 square feet. ,6 Square Ref.... , Tunn,and = { *f m y+S*> w III. SWEDISH MEASURES OF VOLUME. Cubic Measure. i Cubic Turn =1.5972 cubic inches. 1000 Cubic Turner i Cubic Fot = .9263 cubic foot. 8 Cubic Fot i Cubic Aln = 7.4104 cubic feet. Liquid and Dry Measure. 1000 Cubic Linier i Cubic Turn = . 1 843 gill. ioo Cubic Turner i Kanna = 2.3096 quarts. 10 Kanna i Cubic Fot = 5.774 gallons. 8 Cubic Fot i Cubic Aln =46.192 IV. SWEDISH MEASURES OF WEIGHT. i Korn .6564 grain. ioo Korn i Ort = 2.4005 drams. ioo Ort i Skalpund = .9377 pound. ioo Skalpund : Centner . { 93 ; 779 P-nds, or ioo Centner i Ny-Last = 4.1892 tons. The metric system will become obligatory in 1889. NORWAY. The French metric system is in force in Norway. SWITZERLAND. WEIGHTS AND MEASURES. Tables No. 56. I. Swiss MEASURES OF LENGTH. i Striche .01181 inch. i o Striche i Linie = .11811 10 Linien i Zoll = 1.18112 inches. 10 Zoll i Fuss (3 decimetres).... 11.81124 2 Fuss i Elle = 1.9685 feet. 6 Fuss i Klafter = 5-956 ,, i o Fuss i Ruthe = 9.8427 ,, 1600 Ruth en .... i Schweizer-stunde, or Lien= < 176 WEIGHTS AND MEASURES. II. Swiss MEASURES OF SURFACE. i Square Zoll I -3947 square inches. 100 Square Zoll i Square Fuss = .9688 square foot. 36 Square Fuss i Square Klafter = 34.8768 square feet. 100 Square Fuss i Square Ruthe = 10.7643 square yards 400 Square Ruthen.. i Juchart .8694 acre. 6400 Jucharten i Square Stunde = 5693.52 acres. 350 Square Ruthen i Juchart, of meadow land. 450 Square Ruthen i Juchart, of woodland. III. Swiss MEASURES OF VOLUME. i Cubic Zoll : 1.6476 cubic inches. 1000 Cubic Zoll i Cubic Fuss = .9535 cubic foot. 216 Cubic Fuss i Cubic Klafter = 7.6172 cubic yards. 1000 Cubic Fuss i Cubic Ruthe =35.3166 IV. Swiss MEASURES OF CAPACITY. Dry Measure. i Imi = 1.3206 quarts. 10 Imi i Maass = .4127 bushel. 10 Maass i Malter = 4. 1268 bushels. Liquid Measure. 2 Halbschoppen i Schoppen = 2.6412 gills. 2Schoppen.. i Halbmaass = 1.3206 pints. 2 Halbmaass i Maass = 2.6412 i oo Maass i Saum =33.015 gallons. V. Swiss MEASURES OF WEIGHT. 4 Quntli 2 Loth 1 6 Unzen 100 Pfund... Quntli = 2.2048 drams. Loth .55 ii ounce. Unze 1.1023 ounces. Pfund 1.1023 pounds. Centner = 110.233 pounds, or .9842 cwt. The Pfund is divided into halves, quarters, and eighths. It is also divided into 500 Grammes, and decimally into Decigrammes, Centi- grammes, and Milligrammes. SPAIN. WEIGHTS AND MEASURES. Tables No. 57. The French metric system was established in Spain in 1859. The metre is named the Metro; the litre, Litro; the gramme, Grammo; the are, Area; the tonne, Tonelada. The metric system is established likewise in the Spanish colonies. The old weights and measures are still largely used. SPAIN LENGTH, SURFACE, ETC. 177 I. OLD SPAIN 12 Puntos 12 Lineas FISH MEASURES OF Punto Linea Pulgada Sesma Pies de Burgos Vara Estado Estadal Legua (Castilian) Leeua (Spanish) LENGTH. = .00644 inch. = -07725 inch. -927 inch 6 Pulgadas 2 Sesmas = 5.564 inches. - .9273 foot. = 2.782 feet. = 5.564 feet. -11.128 feet. = 2.6345 miles. = 4.2ic;i miles. 3 Pies de Burgos 2 Varas 4 Varas 5000 Varas 8000 Varas... . i II. OLD SPANISH MEASURES OF SURFACE. i Square Pies 9 Square Pies i Square Vara 1 6 Square Varas i Square Estadal 50 Square Varas . . . . i Estajo 576 Square Estadals. i Fanegada 50 Fanegadas i Yugada = .860 square foot. = .860 square yard. = 13.759 square yards. = 42.997 square yards. = 1.6374 acres. = 81.870 acres. III. OLD SPANISH MEASURES OF CAPACITY. Liquid Measure. i Capo .888 gill. 4 Capos i Cuartillo . 1 1 1 gallon. 4 Cuartillos i Azumbre .444 gallon. 2 Azumbres i Cuartilla .888 gallon. ^ ,.,, C i Arroba Mayor, or Cantara ) 4 Cuartillas...| (for We) }= 3-55'gaH. 16 Cantaras i Mayo =56.832 gallons. The old measure for oil is the Arroba Menor = 2.7652 gallons. Dry Measure. Ions. Ochavillo = -00785 peck. 4 Ochavillos. Racion = O3IA Deck 4 Raciones Quartillo W^J-if. J^/V^\_xJV. = .03 1 4 bushel. 2 Quartillos Medio = .0628 bushel. 2 Medios Almude = .1256 bushel. 12 Amuerzas , Fanega 1.5077 bushels. 12 Fanegas Cahiz = 18.0920 bushels. IV. OLD SPANISH WEIGHTS. Grano .771 grain. 12 Granos Tomin = 9.247 grains. 3 Tomines .... Adarme = 2 7. 7 4 grains. 2 Adarmes .... Ochavo, or Drachma .1268 ounce. 8 Ochavos Onza = 1.0144 ounces. 8 Onzas Marco = 8.1154 ounces. 2 Marcos Libra (Castilia a) = 1.0144 pounds. roo Libras Quintal = 101.442 pounds. 10 Quintals Tonelada = 1014.42 pounds. 12 178 WEIGHTS AND MEASURES. PORTUGAL. The French metric system of weights and measures was adopted in its entirety during the years 1860-63, an d was made compulsory from the ist October, 1868. The chief old measures still in use are, the Libra = 1.012 pounds; Almude, of Lisbon = 3.7 gallons; Almude, of Oporto - 5.6 gallons; Alquiere = 3.6 bushels; Moio = 2.78 quarters. ITALY. The French metric system is used in Italy. The metre is named the Metra; the are, Ara; the stere, Stero; the litre, Litro; the gramme, Gramma; the tonneau metrique, Tonnelata de Mare. The various old weights and measures of the different Italian States are still occasionally used. TURKEY. Length. i Pike or Dra=27 inches, divided into 24 Kerats; i Forsang 3.116 miles, divided into 3 Bern; the Surveyor's Pik, or the Halebi 27.9 inches; and 5*^ Halebis = i reed. Surface. The squares of the Kerat, the Pike, and the Reed. The Feddan is an area equal to as much as a yoke of oxen can plough in a day. Capacity, Dry. The Rottol = 1.411 quarts, contains 900 Dirhems; 22 Rottols = i Killow = 7.762 gallons, or .97 bushel, the chief measure for grain. Liquid. i Oka = 1.152 pints; 8 Oke = i Almud =1.152 gallons; i Rottol -2.5134 pints; 100 Rottols = i Cantar = 31.417 gallons. Weights. The Oke = 2.8342 pounds, divided into 4 Okiejehs, or 400 Dirhems of 1.81 drams; i Rottolo = 1.247 pounds; 100 Rottolos= i Cantar = 124.704 pounds. GREECE AND IONIAN ISLANDS. The French metric system is employed in Greece. The metre is named the Pecheus; kilometre, Stadion; are, Stremma; litre, Litra; gramme, Drachme. i^ kilogrammes = i Mna; i^ Quintals = i Tolanton i y?, Tonneaux = i Tonos = 29.526 cwts. In the Ionian Islands, whilst they were under the protection of Great Britain (1830 to 1864), the British weights and measures were those in use, with Italian names. The foot was named the Piede; the yard, the Jarda; the pole, the Carnaco; the furlong, the Stadio; the mile, the Miglio. The gallon was the Gallone; the bushel, the Chilo; the pint, the Dicotile; the pound avoirdupois, the Libra Grossa; the pound troy, the Libra Sottile. The Talanto consisted of 100 pounds, and the Miglio of 1000 pounds. MALTA. In round numbers, 3^ Palmi = i yard; i Canna = 2 2 / 7 yards. The Salma = 4.964 acres. Approximately, 543 Square Palmi = 400 square feet; 16 Salmi = 71 acres. EGYPT. LENGTH, SURFACE, ETC. 179 i Cubic Tratto = 8 cubic feet; 144 Cubic Palmi = 96 cubic feet; i Cubic Canna = 343 cubic feet. Approximate weights: 15 Oncie=i4 ounces; i Rotolo=i^ pounds; 4 Rotoli = 7 pounds; 64 Rotoli = i cwt. ; i Cantaro = 175 pounds; i Quintal = 199 pounds; 64 Cantari = 5 tons. EGYPT. WEIGHTS AND MEASURES. Tables No. 58. I. EGYPTIAN MEASURES OF LENGTH. Pik, or cubit of the Nilometre ............... 20. 65 inches. Pik, indigenous ................................. 22.37 Pik, of merchandise ........................... 25.51 Pik, of construction ............................ 29.53 6 Palms ........... . ......... i Pik. 24 Kirats .................... i Pik or Draa. 4.73 Piks of construction... i Kassaba in surveying, =11.65 II. EGYPTIAN MEASURES OF SURFACE. i Square Pik - 6.055 square feet. 22.41 Square Piks ...... i Square Kassaba = 15.07 square yards 333.33 Square Kassaba, i Feddan -9342 acre. III. EGYPTIAN MEASURES OF CAPACITY. i Kadah 1.684 pints. 2 Kadahs ........................ i Milwah 6.735 2 Milwahs ........................ i Roobah = 1.684 gallons. 2 Roobahs ........................ i Kelah = 3.367 2 Kelehs ......................... i Webek = 6.734 6 Webeks ........ . i Ardeb = / 4 ' 4 4 S&. \ 6.48 cubic feet. The Guirbah of water (a government measure) is I / I5 cubic metre = 66 2 /z litres, or 11.772 cubic feet. IV. EGYPTIAN MEASURES OF WEIGHT. i Kamhah = .746 grain. 4 Kamhahs ............. i Kerat. 1 6 Kerats ................. i Dirhem = 1.792 drachms. 24 Kerats ................ i Mitkal. 8 Mitkals ............... i Okieh. ioo Rottols ............... i Kantar = 98.207 pounds. 400 Dirhems ............... i Oke = 2.728 36 Okes .................. i Kantar =98.207 180 WEIGHTS AND MEASURES. MOROCCO. Length. The Tomin = 2.81025 inches; the Dra'a-8 Tomins = 22.482 inches. Capacity. The Muhd = 3.08135 gallons; the Saa = 4 Muhds 12.3254 gallons. Weights. The Uckia = 392 grains; the Rotal or Artal = 20 Uckieh = 1. 1 2 pounds; the Kintar= 100 Rotales = 112 pounds. Oil is sold by the Kula = 3.3356 gallons. Other liquids are sold by weight. TUNIS. Length. The Dhraa, or Pike, is the unit of length. The Arabian Dhraa, for cotton goods =19.224 inches; the Turkish Dhraa, for lace = 25. 0776 inches; the Dhraa Endaseh, for woollen goods = 26.4888 inches. The Mil Sah'ari = .9i49 mile. Capacity. For dry goods the Sa= 1.2743 pint; 12 Saa-i Hueba^ 6.8228 gallons. For liquids, the Pichoune = .4654 pint; 4 Pichounes=i Pot =i. 8616 pints; 15 Pots = i Escandeau, and 4 Escandeaux=i Millerole= 13.9623 gallons. ARABIA. The weights and measures of Egypt are used in Arabia. CAPE OF GOOD HOPE. The standard weights and measures are British, with the exception of the land measure. To some extent, the old British and the Dutch measures are in use. The general measure of surface is the old Amsterdam Morgen, reckoned equal to 2 acres; though the exact value is equal to 2.11654 acres. 1000 Cape feet are equal to 1033 British feet. INDIAN EMPIRE. WEIGHTS AND MEASURES. An Act " to provide for the ultimate adoption of an uniform system of weights and measures of capacity throughout British India " was passed in October, 1871. The ser is adopted under the Act as the primary standard or unit of weight, and is a weight of metal in the possession of the Govern- ment, equal, when weighed in a vacuum, to one kilogramme. The unit of capacity is the volume of one ser of water at its maximum density, equiva- lent to the litre. Other weights and measures are to be multiples or sub- multiples of the ser, and of the volume of one ser of water. The following are the weights and measures in common use in India: BENGAL LENGTH, SURFACE, ETC. l8l BENGAL. WEIGHTS AND MEASURES. Tables No. 59. I. BENGAL MEASURES OF LENGTH. T Jow, or Jaub = ^ inch. 3 Jow lUngulee = % 4 Ungulees i Moot = 3 inches. 3 Moots i Big'hath, or Span = 9 2 Big'haths i Hat'h, or Cubit. . . = 1 8 2 Hat'h i Guz i yard. 2 Guz i Danda, or Fathom = 2 yards. T-. , f 2000 yards, or roooDandas i Coss =| J^'^ 4 Coss i Yojan = 4.5454 miles. II. BENGAL MEASURES OF SURFACE. i Square Hat'h = 2.25 square feet. 4 Square Hat'hs i Cowrie i square yard. 4 Cowries ... i Gunda 4 square yards. 20 Gundas i Cottah 80 20 Cottahs i Beegah = { l6o ^ e ^ ards ' or | .3306 acre. For land measure, the following table is used for Government surveys: i Guz 33 lineal inches. 3 Guz i Baus, or Rod = 8^ lineal feet. 9 Square Guz i Square Rod = 68 I / I Q square feet. 400 Square Rods r Beegah = I 3oS square yards, or ^ .025 acre. III. BENGAL MEASURES OF CAPACITY. The Seer is a measure common to liquids and dry goods. It is taken at 68 cubic inches, or 1.962 pints, in volume. But it varies in different localities. 5 Seer= i Palli, and 8 Palli = i Maund, or 9.81 gallons. The Sooli = 3.065 bushels, and 16 Soolis = i Khahoon, or 49.05 bushels. IV. BENGAL MEASURES OF WEIGHT. The Tolah, or weight of a Rupee, 180 grains, is the unit of weight. i Tolah = 1 80 grains. 5 Tolahs i Chittak = 900 16 Chittaks i Seer = 2.057 pounds. 5 Seers i Passeeree = 10.286 8 Passeerees... . i Maund = 82.286 MADRAS. WEIGHTS AND MEASURES. Tables No. 60. I. MADRAS MEASURES OF LENGTH. The English foot and yargl are used. The Guz is 33 inches. The Baum or fathom is about 6% feet. A Nalli-Valli is a little under i^ miles. 7 Nalli-Valli = i Kadam, or about 10 miles. The following are native measures : 1 82 WEIGHTS AND MEASURES. 8 Torah i Vurruh .4166 inch. 24 Vurruh i Mulakoli = 10 inches. 4 Mulakoli i Dumna = 40 II. MADRAS MEASURES OF SURFACE. The English acre is generally known. The native measures are uncer- tain. In Madras and some other districts, the following native measures are used : i Coolie : 64 square yards. 4 z /e Coolies i Ground = 266^/3 square yards. 24 Grounds, or ) r / 6400 square yards, or 100 Coolies ) = \ 1.3223 acres. 1 6 Annas (each 400 yards), i Cawnie. III. MADRAS MEASURES OF CAPACITY. i Olluck = .361 pint. 8 Ollucks i Puddee == 1.442 quarts. 8 Puddees i Mercal == 2.885 gallons. 5 Mercals i Parah =14.426 80 Parahs i Garce = 18.033 quarters. This, though the legal system, is not used. The "customary" Puddee is still in general use; it has, when slightly heaped, a capacity of 1.504 quarts. The Mercal has a capacity of 3.0006 gallons; but, when heaped, it is equal to 8 heaped Puddees. The Seer-measure is the most common; its cubic contents are from 66^4 to 67 cubic inches. IV. MADRAS MEASURES OF WEIGHT. i Tola 1 80 grains. 3 Tolas i Pollum = 1.234 ounces. 8 Pollums i Seer = 9.874 5 Seers i Viss 3.086 pounds. 8 Viss i Maund = 24.686 -Maunds , Candy = { In commerce, the Viss is reckoned as 3^ pounds; the Maund, 25 pounds; and the Candy, 500 pounds. BOMBAY. WEIGHTS AND MEASURES. Tables No. 61. I. BOMBAY MEASURES OF LENGTH. i Ungulee = 9/ l6 inch. 2 Ungulee i Tussoo = i^i inches. 8 Tussoos i Vent'h =9 16 Tussoos i Hat'h = 18 24 Tussoos i Guz =27 The Builder's Tussoo = 2.3625 inches in Bombay; and i inch in Surat. BOMBAY, CEYLON, BURMAH. 183 II. BOMBAY MEASURES OF SURFACE. 34 z /6 Square Hat'h... i Kutty = 9-8175 square yards. 20 Kutties .............. i Fund = 196.35 , [ 3927 square yards, or 2oPund ................ iBeegah=<* r 120 Beegah .............. i Chahur= 97.368 acres. In the Revenue Field Survey, the English acre is used. III. BOMBAY MEASURES OF CAPACITY. i Tippree= .2800 pint. 2 Tipprees ............... i Seer = .5600 4 Seers .................. i Pylee = 2.2401 pints. 1 6 Pylees .................. i Parah = 4.4802 gallons. 8 Parahs .................. i Candy = 35.8415 i Mooda = Another liquid measure is the Seer of 60 Tolas = 1.234 pints. In timber measurement in the Bombay dockyards, a Covit or Candi 12.704 cubic feet. CEYLON. The British weights and measures are used. BURMAH. The English yard, foot, and inch are being adopted; also the English Measures of Capacity. Weights. The Piakthah or Viss is 3.652 pounds, and contains 100 Kyats of 252 grains each. CHINA. WEIGHTS AND MEASURES. Tables No. 62. I. CHINESE MEASURES OF LENGTH. i Fen (line) .141 inch. 10 Fen i Ts'un (punto or inch) = 1.41 inches. 10 Ts'un i Ch'ih (covid or foot) = 14.1 i Ch'ih i Chang (rod) = { '^.75 feet'. " 10 Chang i Yin = 39.17 yards. The Ch'ih of 14.1 inches is the legal measure at all the ports of trade. At Canton, the values of the Ch'ih are as follows: Tailor's Ch'ih 14.685 inches. Mercer's Ch'ih (wholesale) 14.66 to 14.724 inches. Mercer's Ch'ih (retail) 14.37 to 14.56 Architect's Ch'ih 12.7 inches. At Pekin there are thirteen different Ch'ihs. 1 84 WEIGHTS AND MEASURES. ITINERARY MEASURE. 5 Ch'ih (covids) i Pii (pace). 360 Pii i Li - about y$ mile. 250 Li (geographical) i Tii (degree) = 83 miles. II. CHINESE LAND MEASURE. 25 Ch'ih (covids) i Kung (bow) = 30^ square feet. 24 o Kung. . . . , Mou (rood) = { |^ squar >> 100 Mou c i King = 1 6 3/3 acres. The principal land measure is the Mou. III. CHINESE CUBIC MEASURE, AND MEASURES OF CAPACITY. 100 Cubic Ch'ih (covids) i Fang or Ma. i o Ho (gills) i Sheng (pint) = about 2 pints. 10 Sheng i Tou (peck) = 2^ gallons. 5 Tou iHu(bushel)- 12^ Liquids are measured by vessels containing definite weights, as i, 2, 4, and 8 Taels; also large earthen vessels holding 15, 30, and 60 Catties. IV. CHINESE MEASURES OF WEIGHT. i Liang or Tael = i x / 3 ounces. 16 Liang i Chin or Catty = i x / 3 pounds. 100 Chin i Tan or Picul = i COCHIN-CHINA. Length. The Thuoc, or cubit, 19.2 inches, is the chief unit of measure of length. It varies considerably for different places. The Li or mile is 486 yards; 2 Li make i Dam; and 5 Dam make i League = 2. 761 miles. Surface. 9 Square Ngu make i Square Sao = 64 square yards. 100 Square Sao make i Square Mao = 6400 square yards, or 1.32 acres. Weights. The smallest weight is the Ai = . 0000006 grain. The weights ascend by a decimal scale, until 10,000,000,000 Ai are accumulated = i Nen = .8594 pound. The greatest weight is the Quan = 68y^ pounds. Capacity for Grain. i Hao = 6 2 / 9 gallons. 2 Hao = i Shita=i24/ 9 gallons. PERSIA. Length. The Gereh - 2^ inches; 16 Gerehs- i Zer='38 inches. The Kadam or Step = about 2 feet; 12,000 Kadam = i Fersakh = about 4^ miles. Surface and Cubic Measures. These are the squares and cubes of the lengths. Capacity (Dry Goods}. The Sextario = .07236 gallon. 4 Sextarios = i Chenica; 2 Chenicas = i Capicha; 3^ Capichas= i Collothun; 8 Collo- thun = i Artata - 1.809 bushels. PERSIA, JAPAN. 18 5 Liquids are sold by weight. Weights. The Miscal^yi grains; 16 Miscals=i Sihr; 100 Miscals = i Ratel= 1.014 pounds; 40 Sihrs= i Batman (Maund) = 6.49 pounds; 100 Batman (of Tabreez) = i Karvvar = 649.i42 pounds. JAPAN. WEIGHTS AND MEASURES. Tables No. 63. I. JAPANESE MEASURES OF LENGTH. Rin .012 inch. 10 Rin Boo .120 inch. 10 Boo Sun 1.20 inches. 10 Sun Shaku = 23 J 5/ l6 inches. 10 Shaku Jo 9 feet 1 1 s/ l6 inches. 6 Shaku Ken 5 feet ix^i inches. 60 Ken i Cho = 119.4 yards. 3^hu lRi ii^asatr Rough timber is sold by the Yama-Ken-Zaii = 63 Sun. Cloth is measured by the Shaku of 15 inches, with decimal sub-multiples. II. JAPANESE MEASURES OF LAND. i Shaku = .9885 square foot. 36 Square Shaku i Tsubo = 3-954 square yards. 30 Tsubo i Se = 118.615 square yards. 10 Se i Tan = 39.212 square poles. 10 Tan i Cho = 2.451 acres. III. JAPANESE MEASURES OF CAPACITY. i Kei = .0000318 pint. 10 Kei Sat = .000318 pint. 10 Sats Sai = .00318 pint. 10 Sai Shaku = .0318 pint. 10 Shaku Go = .3178 pint. 10 Go Sho = .3973 gallon. 10 Sho To = 3.970 gallons. 10 To Koku = 39.703 gallons. IV. JAPANESE MEASURES OF WEIGHT. Shi .0058 grain. 10 Shi Mo .058 10 Mo Rin .5797 10 Rin Fun = 5.7972 grains. 10 Fun Momme 57.972 ,, 100 Momme Hiyaku-me = .8282 pound. 1000 Momme i Kwam-me = 8.2817 pounds. 1 60 Momme i Kin = i^i 100 Kin i Hiyak-kin 132^ 1 86 WEIGHTS AND MEASURES. STRAITS SETTLEMENTS. The unit measure of length is the yard; land is measured by the acre. The Chupack or quart of 4 Paus = 8 imperial gills ; 4 quarts = i Gantang or gallon = 32 gills. The Kati=i I / 3 pounds; 100 Kati = i Picul = J-SS'/s pounds; 40 Picul= i Koyan = 5333 V 3 pounds. JAVA. Length. The Duim=i.3 inches. 12 Duims=i foot. The Ell = 27.08 inches. Surface. The Djong of 4 Bahu - 7.015 acres. Capacity, for rice and grain. The measures are in fact measures of definite weights, i sack = 61.034 pounds; 2 sacks = i Pecul; 5 Peculs = i Timbang = 5.45 cwts.; 6 Timbang = i Coyau = 32.7 cwts. For liquids : The Kan = .328 gallon; 388 Kans= i Leager = 127.34 gallons. Weights. The Tael= 593.6 grains; i6Taels=i Catty =1.356 pounds: TOO Catties = i Pecul = 135.63 pounds. UNITED STATES OF AMERICA. Length. The measures are the same as those of Great Britain. In Land Surveying, the unit of measurement is the chain, and it is deci- mally subdivided. In City Measurements, the unit is the foot, and it is decimally subdivided. In Mechanical Measurements, the unit is the inch, and it is divided into a hundred parts. Surface. The measures are the same as those of Great Britain. Capacity. The measures of capacity for dry goods and for liquids are the same as the old English measures. The standard U. S. gallon is equal to the old English wine gallon, or 231 cubic inches; it contains 8^3 pounds of pure water at 62 F. Dry Measure. Table No. 64. i gill. = .96945 imperial gill. 4 gills i pint = .96945 imperial pint. 2 pints i quart 1.9388 , pints. 4 quarts i gallon = .96945 , gallon. 2 gallons i peck = 1.9388 , gallons. 4 pecks i bushel -96945 , bushel. 4 bushels i coomb = 3.8777 , bushels. 2 coombs i quarter = .96945 , quarter. 5 quarters i wey or load = 4.8472 quarters. 2 weys i last = 9.6945 ,, quarters. For the Wine and Spirit Measures, and the Ale and Beer Measures, see the Old Measures of Great Britain, page 139. i cord of wood =128 cubic feet = (4 feet x 4 feet x 8 feet). Weights. The Weights are the same as those of Great Britain. (See page 140.) BRITISH NORTH AMERICA, ETC. l8/ There are, in addition, the Quintal or Centner of 100 pounds; and the New York ton of 2000 pounds, which is also used, for retail purposes espe- cially, in most of the States. The old hundredweight and old ton are, for the most part, superseded by the quintal and the New York ton. The wholesale coal and iron ton is 2240 pounds. The French metric system of weights and measures was legalized in 1866 concurrently with the old system. BRITISH NORTH AMERICA. WEIGHTS AND MEASURES. Until the 23d May, 1873, the standard measures of length and surface, and the weights, were the same as those of Great Britain; whilst the measures of capacity were the old British measures for dry goods, for wine, and for ale and beer. At the above-named date a new and uniform system of weights and measures came into force, in which the imperial yard, pound avoirdupois, gallon, and bushel, became the standard units, and the imperial system was adopted in its integrity, with two important exceptions : that the hundredweight of 112 pounds, and the ton of 2240 pounds were abolished; and the hundredweight was declared to be 100 pounds, and the ton 2000 pounds avoirdupois, thus assimilating the weights of Canada to those of the United States. The French metric system of weights and measures has been made permissive concurrently with the standard weights and measures. MEXICO. The weights and measures are the old weights and measures of Spain. CENTRAL AMERICA AND WEST INDIES. WEST INDIES (British). The weights and measures are the same as those of Great Britain. CUBA. The old weights and measures of Spain are in general use. For engineer- ing and carpentry work the Spanish, English, and French measures are in use. The French metric system of weights and measures is legalized, and is used in the customs departments. GUATEMALA AND HONDURAS. The weights and measures are the old weights and measures of Spain. BRITISH HONDURAS. In British Honduras, the British weights and measures are in use. COSTA RICA. The old weights and measures of Spain are in general use. But the introduction of the French metric system is contemplated. 1 88 WEIGHTS AND MEASURES. ST. DOMINGO. The old Spanish weights and measures are in general use. The French metric system is coming into use. SOUTH AMERICA. COLOMBIA. The French metric system was introduced into the Republic in 1857, and is the only system of weights and measures recognized by the govern- ment. In ordinary commerce, the Oncha, of 25 Ibs., the Quintal, of 100 Ibs., and the Carga, of 250 Ibs., are generally used. The Libra is 1. 1 02 pounds. The yard is the usual measure of length. VENEZUELA. The system and practice are the same as those of Colombia. ECUADOR. The French metric system became the legal standard of weights and measures on the ist January, 1858. GUIANA. In British Guiana, the weights and measures are those of Great Britain. In French Guiana or Cayenne, the ancient French system is practised. In Dutch Guiana, the weights and measures of Holland are employed. BRAZIL. The French metric system, which became compulsory in 1872, was adopted in 1862, and has since been used in all official departments. But the ancient weights and measures are still partly employed. They are, with some variations, those of the old system of Portugal. Length. The Line = .0911 inch, and is divided into tenths. The Polle- gada =1.0936 inches. The Pe = 13.1236 inches, or I / 3 metre. The Vara = 1.215 y ar ds; and iy Varas = the geometrical pace =1.82 2 7 yards. The Milha= 1.2965 miles; and 3 Milhas= i Legoa = 3.8896 miles. 6 yards are reckoned equal to 5 Varas. Surface. 64 Square Pollegadas... i Square Palmo == .5315 square foot. 25 Square Palmos i Square Vara = 1.4766 square yards. 4 Square Varas i Square Bra^a =5.9063 ,, 4840 Square Varas i Geira = 1.4766 acres. Capacity (Dry Goods}. The Salamine = .3808 gallon; 2 Salamines = i Oitavo; 2 Oitavo = i Quarto; 4 Quartas=i Alqueiro = .38o8 bushel; 4 Alqueiras = i Fangas; 15 Fangas = i Moio = 2.8560 quarters. Liquids. The Quartilho = .6i4i pint; 4 Quartilhos= i Canada; 6 Cana- das = i Pota or Cantaro; 2 Potas= i Almuda- 3.6846 gallons. PERU, CHILI, BOLIVIA, ETC. 189 Weights. The Arratel= 1.0119 pounds, is divided into 16 Onc.as, and then into 8 Oitavos. 32 Arratels=i Arroba; 4 Arrobas = i Quintal = 129.5181 pounds; and 13^ Quintals = i Tonelada= 15.6116 cwts. There is also the Quintal of 100 Arratels. Ships' freight is reckoned by the English ton = 70 Arrobas. PERU. The French metric system was established in 1860, but is not yet gener- ally used. The weights and measures in common use are : The ounce = 1.014 ounce; the Libra= 1.014 pound; the Quintal = 101.44 pounds; the Arroba = 25. 36 pounds, or 6.70 gallons; the gallon = .74 imperial gallon; the Vara-,927 yard; the square Vara = .859 square yard. CHILI. The French metric system has been legally established; but the old weights and measures are still in general use. These are the same as those of Peru. BOLIVIA. The weights and measures are the same as the old weights and measures of Peru and Chili. ARGENTINE CONFEDERATION. The French metric system has recently been established. The old weights and measures are commonly used : the Castilian standards of the old Spanish system. The Quintal = 101.4 pounds; the Arroba = 25.35 pounds; the Fanega = 1.5 bushels. URUGUAY. The French metric system was established in 1864. The old weights and measures are the same as those of the Argentine Confederation. The weights and measures of Brazil are in general use. PARAGUAY. The weights and measures are the same as the old ones of the Argentine Confederation. AUSTRALASIA. In New South Wales, Queensland, Victoria, South Australia, West Australia, Tasmania, and New Zealand, the legal weights and measures are the same as those of Great Britain. But the old British measures of capacity are also much used. In land measurement, a "section" is an area equal to 80 acres. MONEY. GREAT BRITAIN AND IRELAND. COINS. MATERIAL. WEIGHT. Grains. farthing bronze 43-750 y 2 d. halfpenny do. ... 87.500 4 farthings i penny do. ...145.833 $d. threepenny piece silver 21.818 4^. groat, or fourpenny piece do. ... 29.091 6d. sixpence do. ... 43.636 12 pence i shilling do. ... 87.273 2 shillings i florin do. . . . 1 74.545 2 y 2 s i half-crown do. ... 2 1 8. 1 82 IQS i half-sovereign gold 61.6372 2os i sovereign, or pound sterling do. . . . 1 23. 2 745 The bronze coins are made of an alloy of copper, tin, and zinc; the silver coins contain 92^ per cent, of fine silver, and 7^ per cent, of alloy; the gold coins, 91^/3 per cent, of fine gold, and 8y$ per cent, of alloy. The Mint price of standard gold is ^3, i*js. io^d. per ounce. One pound weight of silver is coined into 66 shillings. The intrinsic value of 22 shillings is equal to i sterling. The intrinsic value of 480 pence is equal to i sterling. FRANCE. MONEY. Copper. COINS. WEIGHT. VALUE IN ENGLISH MONEY. Grammes* s. d. i/ / 100 franc i centime .. . i o 'A- '/SO franc 2 centimes . 2 o o z / 5 '/, franc 5 centimes (sou) 5 o YZ Ac franc 10 centimes (gros sou) . . ,.10 o I Silver. V, franc 20 centimes . .. I 2 franc 5o centimes. .- 2.5 o 4^4 I franc IOO centimes. .. 5 .... o o 9^ more exactly 9.524^. 2 francs ..10 o 1 7 5 francs 25 o 3 ii f6 GERMANY, HANSE TOWNS. IQI Gold. Grammes. "" 5 francs .................. 1-61290 ..................... o 3 n^j 10 francs .................. 3-22580 ..................... o 7 n^ 20 francs (Napoleon)... 6-45161 (99-56 grains). ..o 15 10^ 50 francs ................. 16*12902 ..................... i 19 8 1 / 5 100 francs .................. 32*25805 ..................... 3 19 4 4/ IO The English value is calculated at the rate of 25 francs 20 centimes to ji. The bronze coins consist of an alloy of 95 parts of copper, 4 of tin, i of zinc. The standard fineness of the gold pieces, and of the silver 5-franc pieces is 90 per cent., with 10 per cent, of copper; of the other silver coins, 83.5 per cent.; and of the bronze coins, 95 per cent. GERMANY. MONEY. The following system of currency was established throughout the Ge Empire in 1872: ENGLISH VALUE. s. d. i Pfennig .............. = o .1175 10 Pfennig ............... i Groschen ............ o 1.175 i o Groschen ............ i Mark ................. = o 1 1 ^ i o Marks (gold) .................................. = 9 9^ 20 Marks (gold) ................................... = 19 7 The 2o-mark gold piece weighs 122.92 grains, and the standard fineness of the gold pieces is 90 per cent, of gold. Before 1872, accounts were reckoned in the following currency in North Germany : s. d. 12 Pfennig ............... i Silbergroschen ...... = i i T / 5 30 Silbergroschen ...... i Thaler ................ = 3 o In South Germany: 4 Pfennig ................. i Kreutzer ............. = o y$ 60 Kreutzers ....... ....... i Florin ................. = i 8 HANSE TOWNS. MONEY. The monetary system is that of the German Empire. Hamburg. According to the old monetary system, in which silver was the standard, 12 Pfennig =i Schilling = s/ 6 d.; and 16 Schillings =i Mark Bremen. Old system: 5 Schmaren= i Groot = II / 20 ^.; and 71 Groots = i Rix-dollar = 3 j. 3 3/ s d. The Rix-dollar, or Thaler, was a money of account. Lubec. The old system was the same as that of Hamburg, and, in addition, 3 Marks = i Thaler = 35-. ^d. 1 92 MONEY. AUSTRIA. MONEY. s. d. i Kreutzer (copper) o I / 5 4 Kreutzers (do.) o 4/ s i o Kreutzers (silver) o 2^6 20 Kreutzers (do.) o 4^ # Florin (do.) o 5^ 1 Florin (do.) i n% 2 Florins (do.) 3 11^2 4 Florin piece (gold) 7 n 8 Florin piece (do.) 15 10 100 Kreutzers make i Florin. The 4-florin gold piece weighs 49.92 grains, and the standard of fineness is 90 per cent, of gold. RUSSIA. MONEY. s. d. i Copeck = o .38 100 Copecks i Silver Rouble =32 The copper coins are pieces of ^, ^, i, 2, 3, 5 Copecks. The silver coins are pieces of 5, 10, 15, 20, 25 Copecks, the Half Rouble, and the Rouble; the gold coins are the Three-rouble piece, the Half Imperial of five Roubles, and the Imperial of 10 Roubles. The 5-rouble gold piece weighs 10 1 grains, and the standard of fineness is 91^3 per cent, of gold. Paper currency: i, 3, 5, 10, 25, 50, TOO Roubles. HOLLAND. MONEY. s. d. i Cent = o Vs 100 Cents i Guilder or Florin = i 8 BELGIUM. MONEY. The monetary system is exactly the same as that of France. DENMARK. MONEY. S. d, i Skilling = o .2745 16 Skillings i Mark = o 4.392 96 Skillings, or 6 Marks i Rigsdaler, or Daler = 22 7/20 SWEDEN. MONEY. s. d. i Ore = o .133 100 Ore i Riksdaler = i ij^ NORWAY. MONEY. s. d. i Skilling =? o .444 24 Skillingen i Ort or Mark -- o iojg 5 Ort i Species-Daler =4 5^$ SWITZERLAND, SPAIN, ETC. 193 SWITZERLAND. MONEY. The monetary system of Switzerland is the same as that of France. The Centime is called a Rappe. SPAIN. MONEY. d. i Centimo = 95 100 Centimes i Peseta = i franc, or 9^ The bronze coins are pieces of i, 2, 5, and 10 centimes. The silver coins are pieces of 20 and 25 centimos, and i, 2, and 5 pesetas. The gold coins are pieces of 5, 10, 20, 25, 50, and 100 pesetas. The piece of 5 pesetas is 35-. n^/., English value. The 25 peseta piece is 19^. 9^^., English value. The old monetary system was based on the Real-Vellon, 2~ L />d. English value; it was the 2oth part of the Silver Hard Dollar, ^s. zd. English value, and of the Gold Dollar or Coronilla. The Duro was identical with the American Dollar. PORTUGAL. MONEY. The unit of account is the Rei, of which 18^ Reis make i penny; and 4500 Reis make i sovereign. The Milreis is 1000 Reis, 43. $Vzd. English value. The Corda is the heaviest gold coin, of 10,000 Reis, 2, ^s. 5 *^. English value, and weighs 17.735 grammes. ITALY. MONEY. d. i Centime = .95 100 Centimes i Lira = i franc, or 9^ Copper coins are pieces of i, 3, and 5 Centimes; silver coins, 20 and 50 Centimes, and i, 2, and 5 Lire; gold coins, 5, 10, 20, 50, and 100 Lire. These coins are the same in weight and fineness as the coins of France. TURKEY. MONEY. s. d. i Para = o '/is.s 40 Paras i Piastre.... = o 2.16 100 Piastres i Medjidie, or Lira Turca = 1 8 o The Piastre is roughly taken equal to 2d. sterling. GREECE AND IONIAN ISLANDS. MONEY. 100 Lepta i Drachma = i franc, or ^y z d. The currency of Greece is the same as that of France. In the Ionian Islands, whilst they were under British protection (1830- 1864), accounts were kept by some persons in Dollars, of 100 Oboli = 4^. 2d.; by others in Pounds, of 20 shillings, of 12 pence, Ionian currency; the Ionian Pound being equal to 205-. q.6d. sterling. By other persons accounts were kept in Piastres of 40 Paras = 2 */ 45 d. 13 IQ4 MONEY. MALTA. MONEY. 20 Grani Grano s. d. o */ the Taro... ~ O 1 3^ 12 Tari Scudo - i 8 60 Piccioli Or, Carlino . - o 185 9 Carlini . Taro... O I 'Z/s 1 2 Tari . . SrnHo . - i 8 >h money is in general circulation. The Sovereign =12 Scudi ; 40 Paras 100 Piastres 5 Egyptian Guineas., 1000 Purses 97.22 Piastres Shilling = 7 Tari 4 Grani. EGYPT. MONEY. s. d. Para t o o .0615 Piastre (Tariff ) - o o 2.461 Egyptian Guinea = i o 6.84 Kees, or Purse = 52 10.2 Khuzneh, or Treasury = 5142 10 o English Sovereign. The Egyptian guinea weighs 132 grains, and the standard of fineness is per cent, of gold. Two piastres (current) are equal to one piastre (tariff). MOROCCO. MONEY. s. d. I Flue : O 37/ g6o 24 Flues i Blankeel = o 37/ 40 4 Blankeels i Ounce - o 3.7 10 Ounces i Mitkul = 3 i TUNIS. MONEY. s. d. i Fel - o 35/ 288 3 Fels i Karub = o 35/ 96 1 6 Karubs i Piastre : o 55/5 ARABIA. MONEY. s. d. 80 Caveers i Piastre or Mocha Dollar == 3 5 CAPE OF GOOD HOPE. MONEY. Public accounts are kept in English money; but private accounts are often kept in the old denominations, as follows : s. d. i Stiver = o 3/ 8 6 Stivers i Schilling = o 2^ 8 Schilling i Rix-dollar = i 6 The Guilder is equal to 6d. INDIAN EMPIRE, CHINA, ETC. 195 INDIAN EMPIRE. MONEY. Throughout India, accounts are kept in the following moneys: s. d. i Pie =F o o>y% nominal value. 12 Pies i Anna : = o i^ do. 1 6 Annas i Rupee 20 do. The intrinsic value of the Rupee is is. lo^d.; it weighs 180 grains. The English Sovereign is equal to 10 Rupees 4 Annas. i Lac of Rupees = 100,000 rupees = ,10,000. i Crore of Rupees = 100 lacs = ,1,000,000. In Ceylon, the Rupee is divided into 100 Cents. The gold coin, Mohur, is equal to 15 rupees; it weighs 180 grains, and the standard fineness is 91.65 per cent, of gold. CHINA. MONEY. s. d. i Cash (Le) = o 7/ IOO 10 Cash i Candareen (Fun) = o 7/ IO 10 Candareens i Mace (Tsien) = o 7 10 Mace i Tael (Le'ang) = 5 10 COCHIN-CHINA. MONEY. s. d. i Sapek, or Dong, or Cash = o */ l8 60 Sapeks i Mas, or Mottien si o 3^ 10 Mas i Quan, or String = 2 9^ PERSIA. MONEY. s. d. i Dinar = o J / 8o 50 Dinars i Shahi = o ^ 20 Shahis i Keran = o n}s 10 Kerans i Toman = 9 3^ JAPAN. MONEY. s. a. 10 Rin i Sen = ^ 100 Sen i Yen = 42 There are gold coins of the value of i, 2 and 5 yen, with a standard fineness of 90 per cent. The 5~yen piece weighs 128.6 grains. The silver yen weighs 416 grains, with the same standard of fineness. JAVA. MONEY. The money account of Java is the same as that of Holland. UNITED STATES OF AMERICA. MONEY. s. d. i Cent =o y 2 10 Cents i Dime =05 100 Cents i Dollar = 4 2 196 MONEY. CANADA. BRITISH NORTH AMERICA. MONEY. s. d. i Mil .................. = o Yao sterling. 10 Mils .................. i Cent ................ = o y z do. i oo Cents ................ i Dollar ....... , ....... = 4 i% do. 4 Dollars ....................................... = 20 o currency. Or, i Penny currency o ^ sterling. 12 Pence ................. i Shilling do ..... = o 9 4/ s do. 20 Shillings ............... i Pound do ..... = 16 5^ do. The Dollar of Nova Scotia, New Brunswick, and Newfoundland, is equal to 4s. 2d. sterling. In the Bermudas, accounts are kept in sterling money. MEXICO. MONEY. Accounts are kept in dollars of 100 cents. The dollar is equal to ^s. 2d. sterling. CENTRAL AMERICA AND WEST INDIES. MONEY. WEST INDIES (British). Accounts are kept in English money; and sometimes in dollars and cents, i dollar = 4$. 2d. CUBA. MONEY. The moneys of various nations were in circulation before the current war (1875). But the principal silver currency was the 10 cent and 5 cent pieces of the United States. The gold currency consists of the Ounce, of the value of 1 6 dollars, y z ounce, y^ ounce, Y% ounce. GUATEMALA, HONDURAS, COSTA RICA. The moneys of account are the same as those of Mexico. ST. DOMINGO. Accounts are kept in current dollars (called Gourdes] and cents. The cent= Va^'j an d ioo cents = i dollar = SOUTH AMERICA. MONEY. COLOMBIA, VENEZUELA, ECUADOR. The moneys of account are, the Centavo = %44 co? 806 21 7Q ,. 4.61 7 ^JQ 15 85 speculum metal... 4.6 c 7.4.C Nickel hammered CA T 867 Do cast S4* c 16 828 Brass : cast CQC 8.10 75 copper, 25 zinc, sheet C27 8.4.5 66 ^4. vellow . . . ci8 8 ^o 60 ,, 40 Muntz's metal, . . . JH_- c 1 1 8.20 Brass wire r -30 8 CA8 Manganese JJJ 4.QQ 800 Steel Least and greatest density 4.-5C to 4.0^ 7 72Q to 7 QO4. Homogeneous metal 7.QO4. Blistered steel xRR 7 82^ Crucible steel 488 to 490 7.82 1 ? to 7 8^0 Do. average 4.80 7.84.2 Cast steel . . 4.8Q tO 4.8Q CJ 7 8A4. tO 7 8^1 Do average .. 480 ^ 7 84.8 Bessemer steel . 4.8Q tO 4.QO. 7.8A4. tO 7.8^7 Do average A go 6 7 8?2 Mean for ordinary calculations. . . . 4896 7 852 Iron, wrought : Least and greatest density... Common bar 466 to 487 4.71 7.47 to 7.808 7.cc Puddled slab . . 460 Z tO A7d yy ^ 7 ?^ to 7 DO Various Irons tested by Mr. Kirkaldy Do. average ...468 to 486... 4.77 ...7.5*07.8 7.6i; Common rails 466 to 476 7,4.7 to 7 64. Do. average . ... 4.70 7.C4. Yorkshire iron bar 4.84. 7.758 Lowmoor plates, i)4 to 3 ins. thick Beale's rolled iron 487 4.76 7.808 7.6^2 Pure iron (exceptional), by electro- 1 deposit (Dr. Percy) J 508 8.140 Mean for ordinary calculations 4.80 7.698 OF SOLID BODIES. 203 FAMILIAR META LS (continued reatest densit ) Weight of one cubic foot. Specific Gravity. pounds. 378.25^467.66 468 Water = i. 6.900 to 7.500 7.50 7 2O White GrlTcLV Eglinton hot-blast, ist melting... 2d do. ... I4th do. ... *+ y 435 4.7C 6.969 6 Q7O 470 7-530 6.977t07.II3 7.094 7 217 Mallett 442 ACO Mean, for ordinary calculations.. Tin 462 7.409 7 2O Zinc, sheet Do cast 428 Al8 .ZJ 6.86 671 Antimony Aluminium wrought 167 160 2.67 2 c6 Do. cast Magnesium 108.5 1 165 o 1.74 ... 18.68 18.40 17 60 OTHER METALS. Indium Uranium . 1147.0 IOQ7 O . Tungsten Thallium 742.6 7^ 8 11.91 II 80 Palladium . . Rhodium 660.9 6236 1 0.60 JO OO Osmium Cadmium ... . 542.5 c-27 c 8.70 862 Molybdenum Ruthenium ->J/O 536.2 C3O O 8.60 8.50 6.1 1 . . 6.00 Cobalt Tellurium 381.0 774.. I , Chromium Arsenic J'/T' 1 361.5 3-3O ? 5.80 C -2Q Titanium Strontium 158.4 131 o 2.54 2 IO Glucinum Calcium 98.5 Q4..8 I. 5 8 I.C2 Rubidium Sodium. ^ 60.5 C-2 6 0.97 086 Potassium Lithium 37-0 ES. id Pure -59 Specific Gravity. -3 C2 Zircon PRECIOUS Specific Gravity. A CO > STON Diamoi Boart Garnet 3.60 to 4.20 A OI OO^ 3-50 1 C.O Malachite. Topaz Sapphire 3.98 3 or Tournic Lapis l'< Turquo Jasper, Beryl iline 3.07 206 Emerald izuli Do. Aqua marine.. Amethyst j-yj 2.73 -J Q2 ise ^.yw 2.84 2.6 to 2.7 2.68 2.OQ Onyx, Agate... Ruby. . .... yy* 3-95 3-50 to 3.53 Diamond Opal... 204 VOLUME, WEIGHT, AND SPECIFIC GRAVITY Q'PO'M'R*^ Cubic feet to one ton, solid. Weight of one cubic! foot, solid. Specific Gravity. Specular, or red iron ore Magnetic iron ore cubic feet. ... 6.84 ... 7 CK pounds. ... 327.4 ... 317 5 Water = i. ... 5.251 r OQA Brown iron ore . /.u;> Q ID 2/Ld 6 3 w yi' 3 Q22 Spathic iron ore Q 38 238 8 820 Clydesdale iron ores V- J 11.76 . IQO.I; j.o^y 2 O^ tO 1.^80 Barytes 807 277 A AC Basalt .. 14 7 to 12.0 152 8 to 187 i *y A: to 3.OO Mica 14.0 to 12.3 160.3 t 182.7 .?7 to 2.Q3 Limestone, Magnesian 12.6 178.3 ... 2.86 Do Carboniferous T -J -J 1 68 o 2 60 Marble : Paros 1 > J 12 7 177 I ^.uy 2 84. African 12 8 I7A 6 2 8O Siberian 13.2 170.2 2.73 Pyrenean T -3 2 I7O 2 2 73 Carrara 1 > 13.2 . . 169.6 */ J 2.72 Esrptian, green... I 3 C 1665 267 French 136 I6? 2 26; Florentine, Sienna Trap, touchstone. 14-3 13.2 I57.I I6Q.6 ... 2.52 2.72 Granite Sienite, gneiss 15 2 tO 12 I 147 i to 184 6 2 36 tO 2.96 Do Gray 12 8 tO II 8 1746 to 190 8 2 80 tO 3 06 Porphyry T -2 C tO 131 1 66 5 to 171 5 "* 67 to 2 75 Alabaster, Calcareous 1 JO tw 1 J' L I3.O 172.1 2.76 Do. Gypseous 3*2 i c 6 IAA O 2 31 Chalk Air-dried 14 Q tO 14 I i ^o to 1 1>9 *J* 7 46 to 2 55 Slate ' 13 8 to 126 162 i to 177 7 2 60 tO 2 85 Serpentine . . 12.8 ... I7C.2 ... 2.8l Potter's Stone 12 8 I7A 6 2 80 Schist, Slate 128 ... 1 7/1.6 ... 2.80 Do. Rough IQ Q tO 12 Q 1 12 8 to 173 3 i 81 to 2 78 Lava, Vesuvian ->i o tO 128 i 06 6 to 175 2 171 to 2 8 1 Talc Steatite T "2 -2 1 68 4 ^ 7O Rock Crystal 1 J'J 136 165 2 26; Ouartz 1 J.U 1 3 8 to 133 162 8 to 169 o ^s.u^ 2 6 1 to 271 Do. Crystalline 136 i6c 2 2.6? Do. for paving Do porous for millstones ... 14.4 ... 28 ; ... 155-9 .- 78 6 ... 2.50 I 26 Do. flaky, for do Flint *"O ... 14.1 ... 137 ... 159.0 ... 164. o ... 2.55 2.63 Felspar 1 J./ i3.8 162.1 2.60 Gypsum *>" T c 6 143 A. 2.30 Lias . . l^.U ID O tO 147 **Kr*r 140 3 to 152 8 2.21J to 2.4.; Graphite 16 * I 37. 2 2. 2O Sandstone 17 3 tO I A. 1 I2Q 7 to I ^7 I 2 08 tO 2 52 Tufa volcanic 29 7 to 26 i 75.4 to 86 o 1. 21 tO 1.38 Scoria, do 4.-? ? ci.7 .83 OF SOLID BODIES. 205 SUNDRY MINERAL QTT'RQT'ATSJP'R'Q Cubic feet to one ton, solid. Weight of one cubic foot, solid. Specific Gravity. Glass : Flint ? cubic feet. pounds. l87.0 Water = i. 1 OO Green . 1684. 27O Plate 1684 2 7O Crown .... TCC Q 2 CO St Gobain ^jj-y I cc ? *>*' 2 AQ Common, with base of potash Fine, do. do. Common, with base of soda... Fine, do. do. Soluble 153-4 ... 152.8 ... 152.8 ... I52.I ... 77 Q 2.46 ... 2.45 2.45 ... 2.44 Porcelain China / /'V 14.8 A. 2 -^8 Sevres .... I ^Q 7 2 2A Portland Cement 28 7 to 23 8 i ^y-/ 7o tO QA ^..44 i 25 to i 51 Concrete: P. cement i, and shingle 10 ... 16.1 ... 1 39 2.23 P. cement, rubble, and sand P cement i, and sand 2 1 6.6 to 1 6.0 176 135 to 140 127 2.17 to 2.25 2 OA. Roman cement i,and sand 2 M ortar 18.7 206 120 IOQ 1.92 I 71; Brick 18 i to 16 o 12 A. 7 tO I ^C 3 2 OO to 217 Brickwork 20 4 to 195 1 10 to 115 i 76 to i 84 Masonry, Rubble IQ.A tO I C.6 1 1 C.3 to 14.^ 4. I 85 tO 2 30 Marl 22 4 tO 1 8 Q 99 8 to 1185 I 60 tO I QO Do. very tough T C 1 14.6 2 "34. Potash *jO 17 I 1^1 2 IO Sulphur A/.i 18 o T2/1 7 2 OO Tiles ... 18.0 . . I2A.7 2.OO Rock Salt 171 tO ICQ 131 to 1 40 7 2 IOOtO2 257 Common Salt, as a solid l87 I IQ 7 Q2 Clay.... l87 I IQ 7 Q2 Sand, pure 18 Q 118 c :?* QO earthy j.y 21 I 106 o 7O Earth : Potter's Argillaceous ... 18.9 ... 22 A. ... 118.5 ... QQ 8 ... .90 60 Light vegetable Mud ... 25.7 ... 22 ... 87.3 ... 101 6 ... 1 .40 I 6^ Materials inthebed of the Clyde :- Fine sand and a few pebbles, } laid in a box, loose, not > 26 ^ 87 I.3Q pressed, nearly dry ) Pressed 2A Q2 I 4.8 Mud at Whiteinch, dry, and ) firmly packed, containing > 2* y- 4 Q7 1.56 very fine sand and mica j Wet mud, rather compact and IQ III I.QC firm, well pressed into the box Wet, fine, sharp gravel, well 18 12A. I.QQ pressed Wet, running mud 18 i \22l4 I.Q7 Sharp dry sand deposit, in harbour ... 24.3 ... ... 92 ... 1.48 Port-Glasgow bank (sand), wet, 186 1 2O C I Q3 pressed into a box . . 206 VOLUME, WEIGHT, AND SPECIFIC GRAVITY MINERAL SUBSTANCES (continued}. Materials in the bed of the Clyde: Sand opposite Erskine House, ) wet, pressed ) Alluvial earth, pressed Do. do. loose Plaster: 24 hours after using 2 months after using . . . Coal, Anthracite (see Sect. COAL) Bituminous do. do. Boghead (cannel) do. do. Coke Phosphorus Alum Camphor Melting Ice Cubic feet to one ton, solid. cubic feet. .- 19-3 ... 24 - 33 22.6 .. 25.7 .. 26.2 tO 22.6 30 to 28.1 30 39 to 21.6 20.3 .. 20.9 36.3 39 (Veight of one cubic foot, solid. pounds. . 116 93 .. 67 99.2 - 87.3 .. 85.410 99.1 74.8 to 81.7 74-8 57.4 to 103.5 110.4 107.2 61.7 57.5 .. Specific Gravity. Water = i. .. 1.86 1.49 .. 1 .08 1.59 .. 1.40 1.37 to 1.59 1.20 tO 1.31 1.20 .92 to 1.77 .. 1.72 99 .. .922 .66 COALS. (Delabeche and Playfair.} Welsh : Anthracite Porth Mawr (highest) Llynvi (one of the lowest) Average of 37 samples Newcastle: Hedley's Hartley (highest) ... Original Hartley (one of the lowest) Average of 18 samples Derbyshire and Yorkshire: Elsecar Butterley Stavely Loscoe, soft Average of 7 samples Lancashire: Laffack Bushy Park (highest) Cannel, Wigan (lowest) Average of 28 samples Scotch : Grangemouth (highest) Wallsend Elgin Average of 8 samples Irish : Slievardagh Anthracite Warlich's artificial fuel (Nicoll and Lynn.} South Lancashire and Cheshire Coals, average of 14 samples Cubic feet in a ton. Heaped. cubic feet. 38.4 42.0 42.0 42.7 45.6 45-3 474 47-3 44.9 48.8 47-4 42.6 46.4 45.2 40.1 41.0 42.0 35-7 324 42 Weight of one cubic foot. Solid. Heaped pounds. 85.4 86.7 80.3 82.3 81.8 78.0 78.3 80.8 79-8 79-8 79.6 79.6 84.1 76.8 79-4 80.5 74-8 78.6 99.6 72.2 pounds, 58.3 53-3 53-3 53-i 52.0 49.1 49-8 47-2 474 49-9 45-9 45-9 52.6 48.3 49-7 54-3 54.6 50.0 62.8 69.6 Specific Gravity. Water = i 1-37 .28 315 31 .28 .27 .285 .292 35 23 273 .29 .20 .259 59 OF SOLID BODIES. 207 PEAT. (Dr. Sullivan?) Irish peat (comprising an average amount of water 'from 20 to 25 per cent) : Lightest upper moss peat . . . Average light moss peat Average brown peat Compact black peat Cubic feet per ton, stalked. Weight of one cubic foot, stalked. Weight of one cubic foot, solid. Specific Gravity. cubic feet. . 360.60.. . pounds. ... 6.06 pounds. 62.5to8l.I I3.7t02I.O 20.91025.3 29.71041.7 40.51044.5 45.lto6l.3 53.2to6l.8 ... 66.0 ... 6.9 to 16.2 15.01041.8 25.61056.1 38.7 to 64.2 Water = i. i.o to 1.3 .2I9t0.337 .335 to .405 .476 to .669 .650t0.7I3 .72410.983 .72510.991 ... 1.058 .11 to .26 .24 to .67 .41 to .90 .62 to 1.03 254.20 ...147.00... 131.28 ... 99.36... 2OO.29 .. .188.0 ... 155.5 ...141.75... 5l.2t040.O 8.81 ... 15-13 17.06 ... 22.54 ll.iS ... 11.92 14.40 ... 15.80 43.75 to 56.8 Densest peat Mean of five samples (Another observation?) Average upper brown peat .. Moderately compact lower ) brown turf. . . . I Mean of two classes Condensed peat (Kane and Sullivan?) Excessively light, spongy ) surface peat } Light surface peat Rather dense peat.. Very dense dark brown peat Very dense blackish brown ) compact peat ... \ Exceedingly dense jet black > neat ( Exceedingly dense, dark, ) blackish brown peat... . ( (Karmarsch?) Turfy peat, Hanover Fibrous peat, do. Earthy peat, do. Pitchy peat, do FUEL IN FRANCE. (Claudel.) Pure Graphite onSkfoot Specific Gravity. pounds. Water = i. T/l C "2 -7 2-2 Anthracite 8' r_rj jj ,.5 to 91.0 1.34 to 1.46 >.8 to 84.8 1.28 to 1.36 84.8 1.36 82 3 .... 1.32 Rich coal, with a long flame 7c Dry coal, with a long flame... . Rich and hard coal Smithy coal .... 7C ).8 to 81.1 1.28 to 1.30 .9 to 84.2 1.25 to 1.35 ,.3 to 74.8 1.16 to 1.20 1.6 to 74.2 1. 10 to 1.19 81.7 1.31 72 1 I l6 Lignite . . 75 Do bituminous 72 Do imperfect 6 "Tayet"... Bitumen, red Do. black 66.7 1.07 cj 7 o 8^ Do brown Asphalte . . 66.1 i. 06 208 VOLUME, WEIGHT, AND SPECIFIC GRAVITY \A7f^OriQ Weight of one cubic foot. Specific Gravity. pounds. 842 Boxwood 648 I O4 Do ofHolland 82 1 I 32 C.67 O QI Lignum vitcE .... 4O C to 82 Q ***y* 6l tO I 33 Ebony 7O C >V -O lw *'JJ 113 /'-'O 7; c Do. Black 74.2 I.IQ Oak Heart of 7-3 o I 17 Do. English c8o O Q3 Do European A? o to 61 7 ^'VJ OQ tO QQ Do American Red CA 2 .uy LU .yy 87 41 8 to 63 o 67 to i 01 Rosewood 64 2 I O3 Satin-wood CQ.Q l.Uj O.q6 Walnut, Green C7 A O Q2 Do. Brown 42 4 . . 0.68 Laburnum C7 A O Q2 Hawthorn . . V-4 C.6 7 O QI Mulberry cc c w.yi 0.80 Plum-tree CA 2 *?y O 87 Teak, African 08 Mahogany, Spanish C3 O 0.8; Do St Domingo 46 8 O 7C Do Cuba 34 Q "/i o 56 Do Honduras OH-V 34 Q o. c6 Beech 46 8 to ^ 30 v. 3 w 075 to o 85 Do with 20 per cent moisture . ... CT I o 82 Do cut one year 31.1 41 2 066 Ash C2 A o 84 Do. with 20 per cent, moisture $4.q. 43.7 0.70 Acacia . CT I 082 Do. with 20 per cent, moisture J 1 '* 44 Q 0.72 Holly 47 C o 76 Hornbeam . . . . ^/O 47 C. o 76 Yew ... 4/.} 4O I tO ?O fj 0.74 to 0.81 Birch 44 Q to 46 I o 72 to o 74 Elm .. . 34 3 o cc Do. Green O^*"J 47 ^ 0.76 Do with ''o per cent moisture . yl/1 Q O 72 Yoke-Elm do. do 47. c O.76 Rock-Elm CQ O 0.80 O.74 Do Red pine .... 2Q Q to 43 7 o 48 to o 70 Do. Spruce 2Q Q tO 43 7 o 48 to 0.70 31 1 8 tO 3Q.Q 0.50 to 0.64 Do W^hite pine English 34 3 o cc Do. do Scotch J^fO 34 3 O.C3 Do. do. do. 20 per cent, moisture 30 6 O.4Q Do Yellow pine .... 41 2 066 Do. do American 287 O.46 American Pine-wood, in cord (heaped) 21 O.34 Apple-tree AC C O.73 OF SOLID BODIES. 209 Weight of one cubic foot. Specific Gravity. Pear-tree pounds. AC C O 7 "? Orange-tree TJO AA ^/ j o 71 Olive-tree A2 A. 068 Maple o 6? Do. 20 per cent moisture . . AT 8 o 67 Service-tree 4.1 8 o 67 Cypress, cut one year A\ 2 066 Plane-tree Ao c o 6? Vine-tree 07 A o 60 Aspen-tree J/'*t 3,7 A o 60 Alder-tree. . . ... 3 A. Q o i>6 Do. 20 per cent, moisture J4-V 37 A o 60 Sycamore l68 o so Cedar of Lebanon . . . "3.O 6 tO "3.C C ^/.jy o 4.0 to o ^7 Bamboo JU.U IU j}.} TQ C tO 2 A Q o 31 to o 4.0 Poplar 2A -2 O 3Q Do. White 20 o to "?i 8 o 32 to o ^ i Do. 20 per cent, moisture 2Q O O.A8 Willow "y so 6 O AQ Cork ICQ O 2A Elder pith A 7/1 o 076 INDIAN WOODS. (Berkley.} Northern Teak ^tf/.t cc 0882 Southern Teak A8 O 77O Jungle Teak AI o 658 Blackwood c6 0898 Khair 7-5 I 171 Erroul 63 I 014 Red Eyne .... 68 I OQI Bibla c6 0.8Q8 Poon.. . .. 3Q "7 o 625 Kullum . . . jy AI o6;8 Hedoo 2Q o 62^ COLONIAL WOODS. (Fowke.} JAMAICA: Black heart ebony 74-2 I IQ Lignum vitas 4.0. c; to 7 ^ o ny o 65 to i 17 Small leaf. 73.O I 17 Neesberry bullet-tree .. /J.w 6c c I CK Red bully-tree 62.36 *-**3 I OO Iron wood.. ;; J ' 6l 7 O QQ Sweet wood 60 c ^vy O Q7 Fustic 60 c O Q7 Satin candlewood CQ Q "W o 96 Bastard cabbage bark 5 86 O QA White dogwood ?8.6 O.QA Black do j" >vy c8 o O Q3 Gynip.... ;8o O Q3 14 210 VOLUME, WEIGHT, AND SPECIFIC GRAVITY Weight of one cubic foot. Specific Gravity. COLONIAL WOODS (continued}. JAMAICA (continued']; Wild mahogany pounds. cy.4 O.Q2 Cashaw.... C7.A O.Q2 W^ld orange 53.0 to 56.7 0.85 to 0.91 Sweet do 49-3 0.70 Bullet-tree (bastard) ... 56.1 O.QO Tamarind 54.2 0.87 Do wild 46.8 O.7; Prune. 53.6 0.86 Yellow Sanders "... 53.6 ... ... 0.86 Beech 52.4 0.84 French Oak ... 48.0 ... 0.77 Broad Leaf . . . 48.0 0.77 Fiddle Wood 44. 7 O.7I Prickle Yellow 4.7.O O.6o w.w Boxwood AT. O O OQ Locust-tree 42.4 0.68 Lancewood 4.2.4. . . 0.68 Green Mahogany 4.1.2 0.66 Yacca 3Q ? o 63. Cedar 36.2 0.58 Calabash 34. q ^ 0.^6 Bitter Wood ... . .... 24.? O.CC Blue Mahoe 3-1 7 O ?4. Average of 36 woods of Jamaica C2.I 0.831; NEW SOUTH WALES: Box of Ilwarra 7-2.0 .17 Do. Bastard 6Q.8 1 2 Do. True of Camden ^ 60 5 O.Q7 Mountain Ash .11 Kakaralli 68 6 .IO Iron Bark 64.2 .03 Do. broad-leaved 63.6 . . .02 Woolly Butt ^j ... 63.0 .OI Black Do cc.c 0.8o Water Gum. 636 I.OO Blue Do CJ2.4. 0.84 Cog Wood CQ.Q 0.0,6 Mahogany.. CQ 2 O Q$ Do. swamp 536 j& 0.86 Grav Gum ;8.o O.Q1 Stringy Bark 536 0.86 Hickory 468 o 71; Forest Swamp Oak 4.1 2 0.66 Mean of 18 woods of New South Wales.. BRITISH GUIANA: Sipiri, or Greenheart ... 59.9 ... 65 5 to 68 o 0.96 1.05 to 1.09 Wallaba 64.8 1.04. Brown Ebony 64.2 1.03 Letter Wood 62 36 I.OO Cuamara or Tonka ;: 'J u 6i.7 O.QQ Monkey Pot 58.6 0.04 Mora C7 4 O.Q2 OF SOLID BODIES. 211 COLONIAL WOODS (continued}. BRITISH GUIANA (continued) : Weight of one cubic foot. Specific Gravity. pounds. 56.7 Cabacalli 55-5 54.2 0.89 0.87 Kaieeri-balli Sirabuliballi . .. 52.4 5O. C. 0.84 0.81 Buhuradda Buckati 50.5 c.o. i; 0.8 1 o 81 Houbaballi . Baracara 50.5 480 0.8 1 O 77 White Cedar Locust-tree . 44-3 47.7 0.71 O.7O Cartan Purple Heart 42.4 3Q.Q 0.68 o 64 Bartaballi Crabwood 374 34.3 ;: 4 0.60 O C Silverballi Mean of 22 woods of British Guiana. JT-J '' 46.1 96.7 0.74 I. re WOOD-CHARCOAL (as powder). (Claudel.) Willow Oak 954 Q2.Q 1-53 I 4Q Alder Lime-tree 91.0 QO 4 .^y 1.46 I AC, Poplar Average of 5 charcoals 93-5 IQ.'Z .^.5 1.50 O.63 WOOD-CHARCOAL (in small pieces, heaped). (Claudel.) Walnut Ash 34-3 32.t; 0.55 O C2 Beech Yoke-elm O-4-O 28.7 287 **y> 0.46 o 46 Apple-tree White Oak .... 26.2 215.6 0.42 O.4I Cherry -tree Birch 22.5 . . 22.5 0.36 o 36 Elm Yellow Pine 20.6 17 S o-33 o 28 Chestnut-tree Poplar I 5 .6 IC..O , 0.25 O.24 Cedar Average of 1 3 charcoals 25.3 109.110114.7 15 to 15.6 13.7 to 14.3 12.5 to 13.1 14. 0.405 1.75 to 1.84 0.24 to 0.25 O.22 tO 0.23 0.20 tO 0.21 O.225 Gunpowder . . WOOD-CHARCOAL (as made, heaped). Oak and Beech Birch Pine . L _ 212 WEIGHT AND VOLUME OF A WTTVT A T QlfRCiT A IMPI? Q Weight of one cubic foot. Specific Gravity. (Claudel.) Pearls pounds. ... 169.6 2 72 Coral l67 7 2 60 Ivory . .. IIQ 7 ** f y I Q2 Bone . . . . . 1 1 2 2 to 124 7 I 80 tO 2 OO Wool IOO A. 61 Tendon . 698 12 Cartilage .. 680 ... OQ Crystalline humour 67 3 08 Human body . . 667 O7 Nerve 6d Q OA Wax CQ.Q O Q6 White of whalebone S87 O QA Butter 587 O QA Pork fat C8 7 O QA Mutton fat 5'/ c.7.4. O.Q2 Animal charcoal in heaps CO to C2 o 80 to 081 VEGETABLE SUBSTANCES. Cotton D^ lu D^ . . 1 2 1. 6 QC Flax 1 1 1 6 7Q Starch /V C-3 Fecula . Q-J C Gum Myrrh 848 36 Do Dragon 82 1 2 Do Dragon's blood 74.8 2O Do. Sandarac 680 OQ Do. Mastic 66.7 .07 Resin Jalap . . ... 76 i 22 Do. Guayacum .. . 748 2O Do. Benzoin 68.0 .OQ Do Colophany . . . 667 O7 Amber, Opaque . 680 OQ Do. Transparent . . . 67 l ... I 08 Gutta-percha 60 c O Q7 Caoutchouc v^.-j 58 o ^.y/ O Q3 Grain WTieat, heaped 4.6 7 O 7C, Do Barley, do 36.6 o.co Do Oats do 31 2 O C.O j 1 -^ VARIOUS SUBSTANCES. 213 TABLE No. 66. WEIGHT AND VOLUME OF VARIOUS SUBSTANCES. (Tredgold.} SUBSTANCE. Cubic feet per ton, in bulk. Weight of one cubic foot, in bulk. Lead (cast in pigs) cubic feet. 4 Ibs. ^7 Iron (cast in pigs) 62? 3W 360 Limestone or marble (in blocks) 13 172 Granite (Aberdeen, in blocks) I7.C 1 66 Granite (Cornish in blocks) 164. Sandstone (in blocks) 16 I4.I Portland stone (in blocks) 17 132 Potter's clay 17 I 3O Loam or strong soil 18 126 Bath stone (in blocks) 18 I23.tr Gravel 21 V; IOQ Sand . 23 5 QC Bricks (common stocks dry) 24. Q3 Culm 36 63 \Vater (river) ^6 62 c Splint coal 3Q.C ****J 57 Oak (seasoned) 4? C2 Coal (Newcastle caking) :.. AC CQ Wheat 4.7 48 Barley ... en 38 Red fir cq ... ; 38 Hay (compact, old) 280 8 TABLE No. 67. WEIGHT AND VOLUME OF GOODS CARRIED ON THE BOMBAY, BARODA, AND CENTRAL INDIA RAILWAY. By Colonel J. P. KENNEDY, Consulting Engineer of the Railway. No. of kind. CLASSIFICATION OF GOODS CONVEYED. Cubic feet per ton. Weight per cubic foot. Cubic feet per ton, in bulk (estimated). Unpressed cotton cubic feet. 224. Ibs. .. IO ... cubic feet. ...280 2 Furniture 2OO II 2?O 3 A Half-pressed cotton Cotton seeds ...186... 1 86 ... 12 ... 12 233 233 Wool I4.O ... ... 16 ... ... 17; 6 Fruit and vegetables IOO 22 12$ 7 . EP-O-S ... QO ... . 2C . ..113 Class i. Averages .. 174. .. . 13 .. ... 2iy 214 WEIGHT AND VOLUME OF GOODS. GOODS CONVEYED OVER THE INDIAN RAILWAY (continued}. No. of kind. CLASSIFICATION OF GOODS CONVEYED. Cubic feet per ton. Weight per cubic foot. Cubic feet per ton, in bulk estimated). 8 ... Grass cubic feet. 80 ... Ibs. ... 28 cubic feet. IOO Q 80 28 IOO IO ... Bagging .. 7O 12 87 I I Commissariat stores 7O 87 12 ... Full-pressed cotton .. 7O .. ,. 12 .. / 87 Flax and hemp . 32 14 . Groceries 60 J^ . 17 7C I ^ Grains and seed 60 ... / 5 ll... Twist 60 17 17 Sugar ... / ^ 18 ... Soap .. c6 ,. AO . 7O Firewood 50 7O 20 ... Salt 5 AA . / u 64. 21 Lime * AA 6A 22 ... Dry Fruits CO . AC U 4 ^ Class 2. Averages 60 17 ' * 23 ... 2A Jagree (Molasses) Kupas (Seed cotton) * ... 45 . AC ... 50 ... ... 5 6 Mowra (flowers which produce spirit) Timber 45 ... 45 ... ... 50 ... ... 5 6 c6 27 . Ghee (clarified butter) AO Lj > v CO 28 Oil c6 ... 5^ 2Q .. Piece goods . AO . Lj CO Rape AO c6 11 Beer and Spirits 36 62 AC 12 Coal 28 80 11 . Paper ... 28 . . ... 80 ... T.C Tobacco 28 80 3C 3C . Opium 26 86 36 Machinery 3 T J W . > J L Class 3 Averages AI CA C i 37 .. Cutlery ... 2O ... ... 112 ... . 2C Potash 2O 1 12 2C 3.Q Sand 2O . 112 ... 2O AO Colour ... 18 I2A 22 AI Bricks 17 1 12 21 A2 Stone I c IA8 IQ Metal c . .. AA.1 . . 6* Class 4 Averages 1 1 2O1 . 14 Averages of all classes ...64.4... ...35.4..- ...3o Note. The last column has been added by the author; the quantities are calculated by adding one-fourth to the quantities in the third column, to give approximate estimate of the volume occupied in waggons by the goods, or the space required to load a ton of each kind. Sand, No. 39, lies solid in any situation. WEIGHT AND SPECIFIC GRAVITY OF LIQUIDS. 215 TABLE No. 68. WEIGHT AND SPECIFIC GRAVITY OF LIQUIDS. LIQUIDS AT 32 F. Weight of one cubic foot. Weight of one gallon. Specific Gravity. Mercury Bromine pounds. ..848.7 ... i8;.i pounds. .. 136.0 ... 2Q.7 Water = i. ..13-596 2.966 Sulphuric acid, maximum concentration- Nitrous acid .. 114.9 06.8 .. 18.4 ... ICC .. .84 .cc Chloroform . QS.S . , 1C. 7 ., ,C7 Water of the Dead Sea 77.4 12.4 .24 Nitric acid, of commerce Acetic acid, maximum concentration / / -t .. 76.2 ... 67.4. .. 12.2 ... 10.8 .. .22 1. 08 Milk . 64.^ .. , 10.3 . . I.O^ Sea water ordinary ... ... "H3 04. o? IO. ^ J - I.O26 Pure water (distilled) at 39. i F ;rr -> . 62.4.25 .. IO.O ... .. I.OOO VV^ine of Bordeaux 62.1 Q.Q O.QQ4 Do Burgundy .. 6l Q Q Q .. O.QQI Oil, lintseed .. Vi.V/ ... eg 7 Q.4. O.Q4 Do DODDV 3^./ s8 i Q ^ O Q3 -L-'U. P^JJJJ^ Do rape-seed . .. JU.l ... 57 4. y-j Q 2 O.Q2 Do whale C7 4. Q.2 ., .. O.Q2 Do. olive ">7.I Q.IC O.QII; Do turpentine CA 7 87 . 0.87 Do potato C I 2 8.2 0.82 Petroleum CA.Q . .. 8.8 ... .. 0.88 Naphtha ... c ? i 8 ; 0.8; Ether nitric 6q T- i i.i ... .. i. ii Do sulphurous * ~7'J '' 67.4. 10.8 i. 08 Do. nitrous Do acetic ... 55.6 ... a 6 ... 8.9 ... 8.Q ... 0.89 0.89 Do hydrochloric C4..-2 . 87 ., ... 0.87 Do sulphuric . AA Q 7 2 O.72 Alcohol proof spirit *Ft-y 1:7 A Q.2 .. O.Q2 Do pure 4.Q.-2 7.Q 0.70 . 13.I . 8.5 ... ... 0.85 ^^ood spirit 4.Q Q 80 0.80 H-y-y 216 WEIGHT, ETC, OF GASES AND VAPOURS. TABLE No. 69. WEIGHT AND SPECIFIC GRAVITY OF GASES AND VAPOURS. GASES AT 32 F. AND UNDER ONE ATMOSPHERE OF PRESSURE. Volume of one pound weight. Weight of one cubic foot. Specific Gravity. Vapour of mercury (ideal) Vapour of bromine cubic feet. ... 1.776... 2.236 . 2.3T7 in pounds. ...0.563 ... 0.447 ...0.428 ... 0.378 ...0.245 0.217 ...0.209 ... 0.206 ...0.197 ... 0.1814 ...0.1302 ... 0.12344 ...0.12259 0.089253 ...0.080728 0.078596 ...0.0781 ... 0.0795 o 05022 in ounces. ...9.008 .. 7.156 ...6.846 .. 6.042 ...3.927 .. 3480 .. 1 34.O Air = i. ...6.9740 5.5400 ... 5.3000 4-6978 ... 3.0400 2.6943 2 5860 Chloroform Vapour of turpentine 2.637 A O7 C Acetic ether Vapour of benzine 4.598 ... 4790... 4-777 ... 5.077 ... 5-5I3 ... 7.679 8.101 ... 8.157 ... 11.205 ...12.387... 12.723 ...12.804... 12.580 ...IQ QI3 Vapour of sulphuric ether Vapour of ether (?) 3-302 ...3.IS2 . 2.5563 .. 2.44.OO Chlorine Sulphurous acid 2.902 ...2.083 i-975 ...1.961 ... 1.428 ...1.29165 1.258 ...1.250 ... 1.272 ...0.8035 0.7613 ...0.7139... 0.5658 ...0.0895 2.2470 ... 1.6130 1.5290 ... 1.5186 I.I056 ... 1.0000 0.9736 ...0.9674 0.9847 ... 0.6220 0.5894 ...0.5527 0.4381 . . . 0.0692 Alcohol Carbonic acid (actual). .. . Do. (ideal) Oxygen Air Nitrogen Carbonic oxide Olefiant gas Gaseous steam Ammoniacal gas 21.017 ...22.412 ... 0.04758 ,..0.04462 Light carburetted hydrogen .. Coal-eras (page 4? 8) 28.27 9 178.83 ... 0.03536 ...0.005592 Hvdrogen TABLES OF THE WEIGHT OF IRON AND OTHER METALS. Wrought Iron. According to Table No. 65 of the Weight and Specific Gravity of Solids, the weight of a cubic foot of wrought iron varies, for various qualities, from 466 pounds to 487 pounds per cubic foot, and the average weight, taken for purposes of general calculation, is 480 pounds per cubic foot. This average weight is equivalent to a weight of 40 pounds per square foot, i inch in thickness a convenient unit, which is usually employed in the development of tables of weights of iron for engineering and manufacturing purposes. The extremes of variation from this medium unit, extend from 7/% pound less, to about $/$ pound more than 40 pounds per square foot, or from 2.2 to 1,5 per cent, either way a deviation, the extent of which is of little or no practical consequence, and which, at all events, is comprehended in the percentages allowed in the framing of estimates. The average weight of a cubic inch of wrought iron is .*77 pound, ;-: or one-tenth more than a quarter of a pound. For a round number, when cubic inches are dealt with, it may be, and is usually, taken as .28' pound, which is only four-fifths of i per cent, more than the medium weight, and corresponds to a weight of 483.84 pounds per cubic foot, or to 40.32 pounds per square foot, i inch thick, or to 10 pounds per lineal yard, i inch square. The volume of i pound of wrought iron is 3.6 cubic inches. Steel. The weight of a cubic foot of steel varies from 435 pounds to 493 pounds per cubic foot, and the average weight is about 490 pounds per cubic foot. For convenience of calculation, the average weight is taken in the following tables, as 489.6 pounds per cubic foot, for which the specific weight is 1.02, when that of wrought iron-- i.oo. The weight of a square foot, i inch thick, is 40.8 pounds; of a lineal yard, i inch square, 10.2 pounds; and of a cubic inch, .283 pound. The volume of i pound of steel is 3.53 cubic inches. Cast Iron. The weight of a cubic foot of cast iron varies from 378^ pounds to 467^5 pounds per cubic foot, and the average weight is taken as 450 pounds. The weight of a square foot, i inch thick is, therefore, 37.5 pounds; of a lineal yard, i inch square, 9.375 pounds; and a cubic inch, .26 pound. The specific weight is .9375. The volume of i pound of cast iron is 3.84 cubic inches. The following data, for the weight of iron, are abstracted for readiness of reference : 218 WEIGHT OF METALS. WROUGHT IRON, ROLLED. cubic foot, 480 pounds, or 4.29 cvvts. square foot, i inch thick, 40 pounds. square foot, 3 inches thick, 120 pounds, or 1.07 cwts. 3 square feet, i inch thick, 120 pounds, or 1.07 cwts. lineal foot, i inch square, 3^ pounds, or .03 cwt. cubic inch, say 0.28 pound. 3.6 cubic inch, i pound. lineal yard, i inch square, 10 pounds. lineal foot, 3 inches square, 30 pounds. lineal foot, 6 inches square, 120 pounds, or 1.07 cwts. lineal foot, 3 inches by i inch thick, 10 pounds. lineal foot, ^ inch in diameter,. ... 2 pounds. lineal foot, 2 inches in diameter,... 10.5 pounds. lineal foot, 6^ in. in diameter, about i cwt. CAST IRON. i cubic foot, 450 pounds, or 4 cwts. 5 cubic feet, i ton. i square foot, i inch thick, 37.5 pounds. i square foot, 3 inches thick (^ cub. ft.), 112.5 pounds, or i cwt. 3 square feet, i inch thick, 112.5 pounds, or i cwt. i cubic inch, 0.26 pound. 3.84 cubic inches, i pound. The Table No. 70 contains the weight of iron and other metals for the following volumes : i cubic foot. i square foot, i inch thick, or T / I2 th of a cubic foot. i lineal foot, i inch square, or T / 12 th of a square foot. i cubic inch, or x / I2 th of a lineal foot. A sphere, i foot in diameter. The specific gravity due to the respective weights per cubic foot is also given, and likewise the specific weight or heaviness, taking the weight of wrought iron as i, or unity. The next Table, No. 71, contains the volumes of iron and other metals for the following weights : ton, in cubic feet. cwt., in square feet, i inch thick. cwt., in lineal feet, i inch square. pound, in cubic inches. ton, as a sphere, in feet of diameter. ton, as a cube, in feet of lineal dimension. The next Table, No. 72, contains the weight of i square foot of metals of various thickness, advancing by sixteenths and by twentieths of an inch, up to i inch in thickness. The fourth Table, No. 73, contains the weight of prisms or bars of iron, and other metals, or metals of any other uniform section, for given sectional areas, varying from .1 square inch to 10 square inches of section, advancing by one-tenth of an inch, for i foot and i yard in length. TABLES OF WEIGHT AND VOLUME OF METALS. 219 This table is useful in calculations of the weights of bars of every form, rails, joists, beams, girders, tubes, or pipes, &c., when the sectional area is given. The, table is available for finding the weight of a metal for any sectional area up to 100 square inches, by simply advancing the decimal points one place to the right; or, in round numbers, up to 1000 square inches, by advancing the decimal points two places. For example, to find the weight of wrought iron having a sectional area of 17 square inches: For 1.7 square inches, the weight per foot is 5.67 pounds. For 17 square inches, the weight per foot is 56.7 pounds. For 170 square inches, the weight per foot is 567 pounds. Table No. 70. WEIGHT OF METALS. METAL. Cubic Foot. Square Foot, i Inch Thick. Lineal Foot, i Inch Square. Cubic Inch. Sphere, i Foot Dia- meter. Specific Gravity. Specific Weight. Wrought Iron Cast Iron Ibs. or cwts. 480 or 4.29 450 or 4.02 Ibs. or cwts. 40 or .357 37. 5 or .77C, Ibs. 3-333 3.121; Ib. .278 .260 Ibs. 2 5I 236 Water 7^698 7.217 Wro'ght Iron=i. 1. 000 .9771: Steel 489. 6 or 4.37 40. 8 or .364 3.400 .283 257 7.852 .O2O Copper, Sheet Copper, Hammered Tin 549 or 4.90 556 or 4.96 462 or 4.13 45. 8 or .409 46.3 or .413 38. c, or . 344 3-813 3.861 3.208 .318 .322 .268 287 291 242 8.805 8.917 7.4O9 .144 .158 .962 Zinc 437 or 3.90 36.4 or .325 3.035 .253 229 7.008 .910 Lead . .... 712 or 6.36 59. 3 or . 53O 4.Q44 .412 777 II.4I8 .48? Brass, Cast Brass \Vire. 505 or 4.51 Wl or 4.76 42. i or .375 44. 4 or .396 3-507 ^.701 .292 .308 31 J 264 279 8.099 8.348 .052 . no Gun Metal ... Silver Gold 5 24 or 4.68 655or 5-85 1 200 or 10.72 43. 7 or .390 54. 6 or .488 loo.oor .893 3.639 4-549 8.333 304 379 .604 274 343 628 8.404 10.505 19.24^ .092 .365 2.500 Platinum 1342 or 12. oo in. Son. ocx} 9.320 777 703 21.522 J*~ 2.796 Table No. 71. VOLUME OF METALS FOR GIVEN WEIGHTS. METAL. Cubic Feet to a Ton. Square Feet, i Inch Thick, ,to a cwt. Lineal Feet, i In. Square, to a cwt. Cubic Inches to a Ib. Diameter of a Sphere of i Ton. Side of a Cube of i Ton. Wrought Iron Cast Iron Steel Copper, Sheet Copper, Hammered Tin cubic feet. 4.67 4.98 4.58 4.08 4-03 4 86 square feet. 2.80 2.99 2-75 2-44 2.42 2 QI feet. 33-6 35-8 32.9 29.4 29.0 7.4 Q cubic inches. 3-60 3-84 3-53 3-15 3-H 1 74 feet. 2.07 2.12 2.26 I. 9 8 I. 9 8 2. IO feet. -67 71 .66 .60 59 .69 Zinc Lead 5-13 3 JC ^..yi 3.08 i So 36.8 22 7 3-95 2 47 2.14 1.81 73 47 Brass Cast 4..4J 2 67 V.Q 7.42 2.04 .64 Brass ^Vire 4 2O 2 7.O TO 7 ^.24 2.OO .61 Gun Metal Silver 4.28 3.42 *-J" 2.56 2.O=; 30.8 24.6 3-30 2.64 2.02 1.87 .62 5 1 Gold Platinum 1.8 7 1.67 1. 12 I OO 13-4 12. 1.44 1.29 r -59 1.47 .28 .19 22O WEIGHT OF METALS. Table No. 72. WEIGHT OF i SQUARE FOOT OF METALS. Thickness advancing by Sixteenths of an Inch. THICK- NESS. WRO'T IRON. Specific Wt.=I. CAST IRON. Specific STEEL. Specific Wt.= 1.02. COPPER. Specific wt.=i.i6. TIN. Specific wt.=-962. ZINC. Specific BRASS. Specific wt. = 1.052. GUN METAL. Specific wt. = 1.092. LEAD. Specific inch. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. i/ l6 2.50 2-34 2-55 2.89 2.41 2.28 2.63 2-73 3-71 j 5.00 4.69 5.10 5-79 4.81 4-55 5.26 5.46 7.41 3/i6 7.50 IO.O 7-03 9.38 7.65 10.2 8.68 ii. 6 7.22 9-63 6.83 9.10 7.89 10.5 8.19 10.9 II. I 14.8 5/i6 12.5 ii. 7 12.8 14-5 I2.O 11.4 13.2 13-7 I8. 5 M 15.0 14.1 15-3 17.4 14.4 13-7 15.8 16.4 22.2 7/i 6 16.4 17.9 20.3 16.8 15-9 18.4 I9.I 25-9 # 20.0 18.7 20.4 23-2 19-3 18.2 21. 1 21.9 29.7 9/i6 22.5 21. I 23.0 26.0 21.7 20.5 23-7 24.6 33-4 ^ 25.0 23-5 25-5 28.9 24.1 22.8 26.3 27-3 27-5 25.8 28.1 31.8 26.5 25.O 28.9 30.0 40.8 30.0 28.1 30.6 34-7 28.9 27-3 31-6 32-8 44-5 J 3/i6 32.5 3-5 33-2 37-6 3i-3 29.6 34-2 35-0 48.2 I 5/i6 35-o 37-5 32.8 35-2 35-7 38.3 40-5 43-4 33-7 36.1 31-9 34-1 36.8 39-5 38.2 4I.O 51-9 55-6 I 40.0 37-5 40.8 46.3 38.5 36.4 42.1 43-7 59-3 Thickness advancing by Twentieths of an Inch. inch. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. 05 2.00 1.88 2.04 2.32 1-93 1.82 2. II 2.19 2.96 .IO 4.OO 3-75 4.08 4-63 3-85 3.64 4.21 4-37 5-93 15 6.00 5.63 6.12 6.95 5-78 5-46 6.32 6.56 8.90 .20 8.00 7-50 8.16 9.26 7.70 7.28 8.42 8-74 II.O 25 IO.O 9-38 IO.2 ii. 6 9-63 9-IO. 10.5 10.9 14.8 30 12.0 n-3 12.2 13-9 1 1.6 IO-9 12.6 i3-i 17.8 35 14.0 i3-i 14-3 16.2 13-5 12.7 14.7 15.3 20.8 .40 16.0 15.0 16-3 18.5 i5-4 14.6 16.8 17.5 23-7 45 18.0 16.9 18.4 20.8 17-3 16.4 18.9 19.7 26.7 5 20. o 18.8 20.4 23.2 19-3 18.2 21. I 21.9 29.7 55 22.0 20.6 22.4 25-5 21.2 20.0 23.2 24.0 32.7 .60 24.0 22.5 24-5 27.8 23.1 21.8 25.3 26.2 35-6 .65 26.O 24.4 26.5 30.1 25.0 23-7 27.4 28.4 38.6 .70 28.0 26.3 28.6 32.4 27.0 25-5 29-5 30.6 4i-5 75 3O.O 28.1 3O.6 34.7 28.9 27-3 31-6 32.8 44-5 .80 32.0 30.0 32.6 37-o 30.8 29.1 33-7 35-o 47-5 85 34-o 31.9 34-7 39-4 32.7 30-9 35-8 37-2 50.4 .90 36.0 33-8 36.7 41.7 34-7 32-8 37-9 39-3 53-4 95 38.0 35-6 38.8 44.0 36.6 34-6 40.0 41-5 56.3 1. 00 40.0 37-5 40.8 46.3 38-5 36.4 42.1 43-.7 59-3 Note to Table 73, next page. To find the weight of I lineal foot or I lineal yard of hammered iron, copper, tin, zinc, or lead, multiply the tabular weight for rolled wrought iron of the given dimensions by the following multipliers, respectively : EXACT. APPROXIMATE. Hammered Iron 1.008 i.oi equivalent to I per cent. more. Copper 1.158 1.16 ,, 16 ,, more. Tin 962 96 ,, 4 ,, less. Zinc 91 91 ,, 9 less. Lead 1-483 1.48 ,, 48 ,, more. WEIGHT OF METALS OF A GIVEN SECTIONAL AREA. 221 Table No. 73. WEIGHT OF METALS, OF A GIVEN SECTIONAL AREA, PER LINEAL FOOT AND PER LINEAL YARD. SECT. AREA. ROLLED WROUGHT IRON. Sp. Weight=i. CAST IRON. Sp.Weight=.937S- STEEL. Sp. Weight=i.o2. BRASS. Sp. Weight 1.052. GUN METAL. Sp.Weight=i.o92. i Foot. i Yard. i Foot. i Yard. i Foot. i Yard. i Foot. i Yard. i Foot. i Yard. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. .1 -333 1. 00 .313 .938 340 1.02 351 1.05 .364 1.09 .2 3 .667 1. 00 2.OO 3.00 .625 .938 1.88 2.8l .680 1.02 2.04 3-06 .701 2.IO 3-16 .728 1.09 2.18 3-28 4 1-33 4.OO 1.25 3-75 1.36 4.08 1-43 4.21 1.46 4-37 1.6 7 5-00 I. 5 6 4.69 1.70 5.10 1-75 5.26 1.82 .6 2.OO 6.00 1.88 5-63 2.O4 6.12 2. 1 1 6. 3 I 2.18 6-ls -7 2-33 7.00 2.19 6.56 2.38 7.14 2.46 7.36 2-55 7.64 .8 2.6 7 8.00 2.50 7-5 2.72 8.16 2.81 8.42 2.91 8.74 9 3.00 9.00 2.81 8-44 3-06 9.18 3.16 9-47 3-28 9-83 I.O 3-33 10.0 3-15 9-38 3-40 10.2 3.51 10.5 3-64 10.9 .1 3-67 II. 3-44 10.3 3-74 II. 2 3.86 11.6 4.00 12.0 .2 4.OO 12.0 3-75 11.3 4.08 12.2 4.21 12.6 4-37 I3-I 3 4-33 13.0 4.06 12.2 4.42 13-3 4-5 6 13.7 4-73 14.2 4 4.67 14.0 4-38 I 3 i 4.76 14-3 4.91 14.7 5.10 I 5 > 3 5 5.00 15.0 4.69 14.1 5.26 15.8 5-46 16.4 .6 5-33 16.0 5.00 15.0 5-44 16.3 5.61 16.8 5.82 17-5 7 5.67 17.0 5-31 15-9 5-78 17-3 5-96 17.9 6.19 18.6 .8 6.00 18.0 5.63 16.9 6.12 I8. 4 6.31 18.9 6.55 19.7 9 2.0 6-33 6.67 19.0 20.0 5-94 6.25 17.8 18.8 6.46 6.80 19.4 20.4 6.66 7.01 20.0 21. 6.92 7.28 20.8 21.8 2.1 7.00 21.0 6.56 19.7 7.14 21.4 7.36 22.1 7.64 22.9 2.2 7-33 22.0 6.88 20.6 7.48 22.4 7-72 23.1 8.01 24.0 2-3 7.67 23.O 7.19 21.6 7.82 23-5 8.07 24.2 8-37 25.1 2.4 8.00 24.0 7-5 22.5 8.16 24-5 8.42 25-3 8-74 26.2 2-5 8-33 25.0 23.4 8.50 25-5 8-77 26.3 9.10 27.3 2.6 8.67 26.0 8.13 24.4 8.84 26.5 9.12 27.4 9.46 28.4 2-7 9.00 27.0 8.44 25.3 9.18 27-5 9-47 28.4 9.83 29-5 2,8 9-33 28.0 8-75 26.3 9-52 28.6 9.82 29-5 10.2 30.6 2.9 9.67 29.0 9.06 27.2 9.86 29.6 10.2 30-5 10.6 31.7 3-0 10.0 30.0 9.38 28.1 IO.2 3O.6 10.5 31.6 10.9 32.8 3.1 10.3 31.0 9.69 29.1 10.5 31.6 IO.9 32.6 "3 33.9 3-2 10.7 32.0 10. 30.0 10.9 32.6 II. 2 33-7 11.7 34.9 3-3 II. O 33-0 10.3 30.9 II. 2 33-7 n.6 34.7 12.0 36.0 3-4 "3 34-o 10.6 31.9 ii. 6 34-7 11.9 35-8 12.4 37.1 3-5 11.7 35-0 10.9 32.8 11.9 35-7 12.3 36.8 12.7 38.2 3.6 12.0 36.0 IJ -3 33-8 12.2 36.7 12.6 37-9 I3.I 39.3 3-7 3.8 12.3 12.7 3&o ii. 6 11.9 34-7 35-6 12.6 12.9 37-7 38.8 13.0 13.3 38.9 40.0 13-8 40.4 41.5 3-9 13-0 39-0 12.2 36.6 13-3 39-8 13.7 41.0 I 4 .2 42.6 4.0 13-3 40.0 12.5 37-5 13-6 40.8 14.0 42.1 14.6 43.7 4.1 13-7 41.0 12.8 38.4 13-9 41.8 14.4 43-1 14.9 44-8 4.2 I4.O 42.0 I 3- * 39-4 14-3 42.8 14.7 44.2 15.3 45-9 4-3 14-3 43- 13.4 40.3 14.6 43-9 15.1 45-2 15.7 46.9 4.4 14.7 44.0 13.8 4L3 15.0 44-9 15.4 46.3 16.0 48.0 4-5 15.0 14.1 42.2 15.3 45-9 15.8 47-3 16.4 49.1 4.6 15-3 46.0 14.4 43- * 15-6 46.9 16. i 48.4 16.7 50.2 4-7 15.7 47.0 14.7 44.1 16.0 47-9 16.5 49-4 17.1 51.3 4.8 16.0 48.0 15.0 45-o 16.3 49.0 16.8 50-5 17.5 52.4 4-9 16.3 49.0 15.3 45-9 16.7 50.0 17-2 51.6 17.8 53-5 5-0 16.7 50.0 15.6 46.9 17.0 51.0 17-5 52.6 18.2 54-6 222 WEIGHT OF METALS. Table No 73 (continued}. SECT. AREA. ROLLED WROUGHT IRON. Sp. Weight^ i. CAST IRON. Sp.Weight=.937S. STEEL. Sp. Weight=i.o2. BRASS. Sp. \Veight= i . 052. GUN METAL. Sp.Weight=i.o92. i Foot. i Yard. i Foot. i Yard. i Foot. i Yard. i Foot. i Yard. i Foot. i Yard. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. 5- 1 17.0 51.0 15-9 47-8 17-3 52.0 17.9 53.7 18.6 55-7 5-2 17-3 52.0 16.3 48.8 17.7 53-0 18.2 54-7 18.9 56.8 5-3 17.7 53-o 16.6 49-7 18.0 54- 1 8.6 55-8 19-3 57-9 5-4 iS.o 54-0 16.9 50.6 18.4 55- 18.9 56.8 19.7 58.9 5-5 18.3 55-o 17.2 51.6 18.7 56. i9-3 57-9 20. o 60.0 5.6 18.7 56.0 J 7-5 52.5 19.0 57- 19.6 58.9 20.4 61.1 5-7 19.0 57.0 17.8 53-4 19.4 58. 20.0 60.0 20.8 62.2 5.8 19-3 58.0 18. i 54-4 19.7 59-2 20.3 61.0 21.1 63.3 5-9 6.0 19.7 20. o 59-0 60.0 18.4 18.8 55-3 56.3 20.1 20.4 60.2 61.2 20.7 21.0 62.1 63-1 21-5 21.8 64.4 65-5 6.1 20.3 61.0 19.1 57-2 20.7 62.2 21.4 64.2 22.2 66.6 6.2 20.7 62.0 19.4 58.1 21. 1 63.2 21.7 65.2 22.6 67.7 6-3 21.0 63.0 19.7 59-1 21.4 64.3 22.1 66.3 22.9 68.8 6.4 21.3 64.0 20. o 60.0 21.8 65-3 22.4 67-3 23-3 69.9 6.5 21.7 65.0 20.3 60.9 22.1 66.3 22.8 68.4 23-7 70.9 6.6 22. 66.0 20. 6 61.9 22.4 67-3 23.1 69.4 24.0 72.0 6.7 22.3 67.0 20.9 62.8 22.8 68.3 23-5 70.5 24.4 73-i 6.8 22.7 68.0 21.3 63.8 23.1 69.4 23-9 71-5 24.8 74-2 6.9 23.0 69.0 21.6 64.7 23-5 70.4 24.2 72.6 25-1 75-3 7.0 23-3 70.0 21.9 65.6 23.8 71-4 24.6 73-6 25-5 76.4 7-i 23-7 71.0 22.2 66.6 24.1 72.4 24.9 74-7 25.8 77-5 7.2 24.0 72.0 22.5 67.5 24.5 73-4 25-3 75-7 26.2 78.6 7-3 7-4 24-3 24-7 73-o 74.0 22.8 23.1 68.4 69.4 2 4 .8 25-2 74-5 75-5 25.6 26.O 76.8 77-9 26.6 26.9 79-7 80.8 7-5 7.6 25.0 25-3 75-o 76.0 23-4 23-8 70.3 7i.3 25-5 25.9 76.5 77-5 26.3 26.7 78.9 80.0 27.3 27-7 81.9 83-0 7-7 25-7 77.0 24.1 72.2 26.2 78.5 27.0 81.0 28.0 84.1 7.8 26.0 78.0 24.4 73i 26.5 79.6 27.4 82.1 28.4 85.2 7-9 26.3 79.0 24.7 74.1 26.9 80.6 27.7 83-1 28.8 86.3 8.0 26. 7 80.0 25.0 75-o 27.2 81.6 28.1 84.2 2 9 .I 87.4 8.1 27.0 81.0 25-3 75-9 27-5 82.6 28.4 85.2 29-5 88.5 8.2 27-3 82.0 25.6 76.9 27-9 83.6 28.8 86.3 29.9 89.5 8-3 27-7 83.0 25-9 77-8 28.2 84.7 29.1 87-3 30.2 90.6 8.4 28.0 84.0 26.3 78.8 28.6 85.7 29.5 88.4 3O.6 91.7 8-5 28.3 85.0 26.6 79-7 28.9 86.7 2 9 .8 89.4 30-9 92.8 8.6 28.7 86.0 26.9 80.6 29.2 87.7 30.2 90.5 3L3 93-9 8.7 29.0 87.0 27.2 81.6 29.6 88.7 30-5 9i.S 31-7 95-0 8.8 29-3 88.0 27-5 82.5 29.9 89.8 3-9 92.6 32.0 96.1 8.9 29.7 89.0 27.8 83-4 3-3 90.8 31.2 93- 6 32.4 97.2 9.0 30.0 90.0 28.1 84-4 30.6 91.8 31-6 94-7 32.8 98.3 9-1 30-3 91.0 28. 4 85-3 30-9 92.8 3i-9 95-7 33-1 99-4 9.2 30.7 92.0 28.8 86.3 31-3 93-8 32.3 96.8 33-5 100.5 9-3 31.0 93- 29.1 87.2 31.6 94-9 32.6 97-8 33-9 101.6 9-4 31-3 94.0 29.4 88.1 32.0 95-9 33-o 98.9 34-2 102.7 9-5 31-7 95-0 29.7 89.1 32-3 96.9 33-3 99-9 34-6 103.7 9.6 32.0 96.0 30.0 90.0 32.6 97-9 33-7 IOI.O 34-9 104.8 9-7 32.3 97.0 30-3 90.9 33-0 98.9 34-0 102.0 35-3 105.9 9-8 32.7 98.0 30.6 91.9 33-3 IOO.O 34-4 103.1 35-7 107.0 9-9 33-o 99.0 30-9 92.8 33-7 IOI.O 34-7 104.2 36.0 108.1 10.0 33-3 100.0 31-3 93-8 34-o 102.0 35-i 105.2 36.4 109.2 See note at foot of page 220. RULES FOR WEIGHT. 223 RULES FOR THE WEIGHT OF IRON AND STEEL. The following rules for finding the weight of wrought iron, cast iron, and steel, are based on the data contained in Tables No. 70 and 71. RULE i. To FIND THE WEIGHT OF IRON OR STEEL, when the volume in cubic feet is given. Multiply the volume by 4.29 for wrought iron, 4.02 for cast iron, 4.37 for steel. The product is the weight in hundredweights. RULE 2. When the volume in cubic inches is given, multiply the volume by .278 (or .28) for wrought iron, .26 for cast iron, .283 for steel. The product is the weight in pounds. RULE 3. When the quantity is reduced to square feet, one inch in thickness, multiply the area by 40 for wrought iron, 37^ for cast iron, 40.8 (or 41) for steel. The product is the weight in pounds. Or, multiply the area by 357 f r wrought iron, .335 for cast iron, .364 for steel. The product is the weight in hundredweights. RULE 4. When the sectional area in square inches, and the length in feet, of a bar or prism are given, multiply the sectional area by the length, and by 3 x /s for wrought iron, 3^6 for cast iron, 3.4 for steel. The product is the weight in pounds. For large masses, multiply the sectional area by the length, and divide the product by 672 for wrought iron, 717 for cast iron, 659 for steel. The quotient is the weight in tons. RULE 5. When the sectional area in square inches, and the length in yards, of a bar or prism, are given, multiply the sectional area by the length, and by 10 for wrought iron, 9-375 f r cast iron, 10.2 for steel. The product is the weight in pounds. 224 WEIGHT OF METALS. RULE 6. To FIND THE SECTIONAL AREA OF 'A BAR OR PRISM OF IRON OR STEEL, when the length and the total weight are given. Divide the weight in pounds by the length in feet, and by 3 x / 3 for wrought iron, 3^6 for cast iron, 3.4 for steel. The quotient is the sectional area in square inches. RULE 7. To FIND THE LENGTH OF A BAR, PRISM, OR OTHER PIECE OF UNIFORM SECTION OF IRON OR STEEL, when the total weight and the sectional area are given. Divide the weight in pounds by the sectional area in square inches, and by 3 T / 3 for wrought iron, 3 J$ for cast iron, 3.4 for steel. The quotient is the length in feet. In applying the last rule to calculate the length of wire of a given size, for a given weight, say i cwt. of wire, the sectional area of the wire is found, in the usual way, by multiplying the square of the thickness or diameter, d, by .7854. Then, by the rule, the length in feet of i cwt. of iron wire is equal to i2 _ 42^78 5 23-4 27-3 31.2 35-2 28.7 ^fie .29 4-30 8.59 12.9 17.2 21.5 25.8 30.1 34-4 38.7 26.1 tf .41 4.69 9.38 14.1 18.8 23-4 28.1 32-8 37-5 42.2 23.9 J 3/i6 52 5.08 10.2 15-2 20.3 25-4 30.5 35-5 40.6 45-7 22.1 7 /B .64 5.47 IO.9 16.4 21.9 27.3 32-8 43-9 49.4 20.5 W/x6 .76, 5.86 ii. 7 17.6 23-4 29-3 35-i 41.0 46.9 52.7 I9.I I .88 6.25 12.5 18.8 25.0 3i-3 37-5 43-8 50.0 56.2 17.9 2 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet X .500 1.6 7 3.33 5.00 6.67 8-33 IO.O 11.7 13-3 15.0 6 7 .2 5/i6 .625 2.08 4.17 6.25 8.33 10.4 12.5 14.6 I6. 7 18.8 53-8 H 750 2.50 5-00 7.50 IO.O 12-5 15.0 !7-5 2O. O 22.5 44-8 7/i6 875 2.92 5-83 8-75 11.7 I 4 .6 17-5 20.4 23-3 26.3 38.4 X 1. 00 3-33 6.67 10.0 13-3 I6. 7 20.0 23-3 26.7 30.0 33-6 9/i 6 !3 3-75 7.50 "3 15.0 18.8 22.5 26.3 30.0 33-8 29.9 H .25 4.17 8-33 12.5 16.7 20.8 25.0 29.2 33-3 37-5 26.9 n/ie .38 4o8 9.16 13-8 18.3 22.9 27-5 32.1 36.7 41.2 24.4 * .50 5.00 10. 15.0 20.0 25.0 30.0 35-o 40.0 45- 22.4 J 3/i6 .63 5-42 10.8 16.3 21.7 27.2 32.5 37-9 43-3 48.8 20.7 H 75 5-83 11.7 23-3 29.2 35.0 40.8 46.7 52.5 19.2 *5/*fi .88 6.25 12.5 18.8 25.0 3i-3 37-5 43-8 50.0 56.3 17.9 I 2.00 6.67 13-3 20. o 26.7 33-3 40.0 46.7 53-4 60.0 16.8 230 WEIGHT OF METALS. WEIGHT OF FLAT BAR IRON. 2 l /% INCHES WIDE. THICK- NESS. SECT. AREA. LENGTH IN FEET. Length to weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. 1" 'Ml 797 1.77 2.21 2.66 3-54 4-43 5-3 1 6.64 7-97 7.08 8.85 10.6 8.85 11.7 13-3 10.6 13-3 15-9 12.4 15-5 18.6 14.2 17.7 21.2 15-9 19.9 23-9 6 3 .2 50.6 42.2 7/i6 93 3-10 6.20 9-3 12.4 15-5 18.6 21.7 24.8 27.9 36.1 X i. 06 3-54 7.08 10.6 14.2 17.7 21.3 24.8 28.3 31-9 31-6 9/i6 .20 3-98 7-97 12.0 15-9 20. o 23-9 27.9 31-9 35-8 28.1 H 33 4-43 8.85 13-3 17.7 22.1 26.6 31.0 35-4 39-8 25-3 J1 /i6 .46 4.87 9-74 14.6 19-5 24.4 29.2 34-i 39-o 43-8 23.0 X 59 5-31 10.6 15-9 21.2 26.6 31-9 37-2 42.5 47-8 21. 1 *3/i6 74 5-76 "5 17.3 23.0 28.8 34-5 40-3 46.0 51.8 I 9 .8 7 /* .86 6. 20 12.4 24.8 31.0 37-2 43-4 49.6 55-8 18. i *5/i6 .98 6.64 13-3 19.9 26.6 33-2 39-8 46.5 53-i 59-8 16.9 1 2.13 7.08 14.2 21.3 28.3 35-4 42-5 49.6 56.7 63.8 15-8 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X .563 1.88 3.75 5.63 7-50 9-4 11.3 13.1 15.0 16.9 59-7 5/i6 .703 2.34 4.69 7.03 9.38 ii. 7 14.1 16.4 18.8 21. 1 47.8 M .844 2. 8 1 8.44 11.3 14.1 16.9 19.7 22.5 25-3 39-8 7/i6 .984 3.28 6*56 9.84 13.1 16.4 19.7 23.0 26.3 29-5 34-1 /z 3-75 7.50 "'3 15.0 18.8 22.5 26.3 30.0 33-8 29.9 9/i6 1.27 4.22 8.44 12.7 I6. 9 21. 1 25.3 29-5 33-8 38.0 26.5 $6 I.4I 4.69 9.38 14.1 18.8 23-4 28.1 32.8 37-5 42.2 23-9 T '55 1.69 5.16 5.63 10.3 "3 16.9 20.6 22.5 25.8 28.1 3-9 33-8 36.1 39-4 41-3 45-o 46.4 50.6 21.7 19.9 J 3/i6 1.83 6.09 12.2 18.3 24.4 30.5 36-6 42.7 48.8 54-9 18.4 % 1.97 6.56 I 3- I 19.7 26.3 32.8 39-4 45-9 52.5 59-1 17.1 '5/16 2. 1 1 7-03 14.1 21. 1 28.1 35-2 42.2 49-2 56-3 63.3 15-9 I 2.25 7.50 15-0 22.5 30.0 37-5 45- 52-5 60.0 67.5 14.9 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 594 1.98 3-96 5-94 7.92 9.90 11.9 13-9 I 5 .8 I 7 .8 56.6 5/i6 .742 .891 2.47 2.97 4-95 5-94 7.42 8.91 9.90 ii. 9 12.4 14.8 14.8 17.8 17-3 20.8 19.8 23-8 22.3 26.7 45-3 37-7 7/i6 1.04 3-46 6-93 10.4 13-9 17-3 20.8 24.2 27.7 31.2 32.3 % 1.19 3-96 7.92 11.9 15.8 I 9 .8 23.8 27.7 31-7 35-6 28.3 9/i6 1-34 4.45 8.91 13-4 17.8 22.3 26.7 31.2 35-6 40.1 25.2 H 1.48 4-95 9.90 14.8 19.8 24.7 29.7 34-6 39-6 44-5 22.6 1.67 5-44 10.9 16.3 21.8 27.2 32.7 38.1 43-5 49.0 20.6 X 1.78 5-94 11.9 17.8 23.8 29.7 35.6 41.6 47-5 53-4 18.9 *3/i6 i-93 6.43 12.9 19-3 25.7 32.2 38.6 45- 51.5 57-9 17.4 A 2.08 6-93 13-9 20.8 27.7 34-6 41.6 48.5 55-4 62.3 16.2 *5/i6 2.23 7.42 14.8 22.3 29.7 37-1 44-5 Si-9 59-4 66.8 15-1 I 2.38 7.92 i 15.8 23.8 31-7 39-6 47-5 55-4 63-3 71-3 14.2 FLAT BAR IRON. 231 WEIGHT OF FLAT BAR IRON. z z INCHES WIDE. THICK- NESS. SECT. AREA. LENGTH IN FEET. Length o weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X .625 2.08 4.17 6.25 8-33 10.4 12.5 14.6 I6. 7 18.8 53-8 5/i6 .781 2.60 5.21 7 .8l 10.4 13.0 15.6 18.2 20.8 23-4 43- H 7/i6 .938 .09 3-13 3-65 6.25 7.29 9.38 10.9 I2 -5 14.6 15.6 18.2 _0 21.9 21.9 25.5 25.0 29.2 32'- 8 35-8 30-7 /* 25 4.17 8.33 12.5 16.7 20.8 25.0 29.2 33-3 37-5 26.9 K .41 56 4.69 5.21 9.38 10.4 I4.I I 5 .6 18.8 20.8 23.4 26.0 28.1 31.3 32.8 36.5 37-5 41.7 42.2 46.9 23-9 21.5 .72 5-73 ii-5 17.2 22.9 28.6 34.4 40.1 45-8 51.6 19.6 % .88 6.25 12.5 18.6 25.0 31.3 37.5 43.8 50.0 56.3 18.0 I 3/i6 2.03 6-77 13.5 20.3 27.1 33-8 40.6 47.4 54-2 60.9 16.5 6 2.19 7.29 14.6 21.9 29.2 36.5 43-7 51.0 58.3 65.7 15-4 2-34 7.81 15.6 23-4 31.3 39-0 46.9 54.7 62.5 70.3 14-3 I 2.50 8.33 16.7 25.0 33-3 41.7 50.0 58.3 66.7 75-0 13-4 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X .656 2.19 4.38 6.56 8.75 10.9 I3-I 15.3 17-5 19.7 51.2 5/i6 .820 2.73 5.47 8.20 10.9 13-7 16.4 I9.I 21.9 24.6 4 I.O 7/i6 .984 15 3.28 3.83 6.56 7.66 9.84 ii-S i3-i 15-3 16.4 I9.I 19.7 23.0 23.0 26.8 26.2 30.6 29-5 34-4 34-2 29-3 ^ 31 4.38 8-75 13-1 17-5 21.9 26.3 30.6 35-0 39-4 25.6 9/16 .48 4.92 9.84 14.8 19.7 24.6 29-5 34-5 39-4 44-3 22.8 * .64 5-47 10.9 16.4 21.9 27-3 32.8 38.3 43-8 49.2 2O.5 "/i6 .81 6.02 12.0 18.1 24.1 30.2 36.1 42.1 48.1 54-1 18.6 % 97 6.56 I3-I 19.7 26.3 32.8 39-4 45-9 52.5 59-1 17.1 I 3/i6 2.13 7.11 14.2 21.3 28.4 35-5 42.7 49.8 56.9 64.0 15.8 ft 2.30 7.66 15.3 23.0 30.6 38.3 45-9 536 61.3 68.9 14.7 iS/i6 2.46 8.20 16.4 24.6 32-8 41.0 49-2 57-4 65.6 73-8 13.7 I 2.63 8.75 17.5 26.3 35-0 43-8 52.5 61.3 70.0 78.8 12.8 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X .688 2.29 4.58 6.87 9.17 "5 13.8 16.1 I8. 3 20.6 48.9 5/i6 .859 2.86 5.73 8-59 II-5 14-3 17.2 20. 22.9 25.8 39-1 H .03 3-44 6.88 10.3 13-8 17.2 20.6 24. 27-5 30-9 32.8 7/,6 .20 4.01 8.02 12.0 16.0 20.1 24.1 28. 32.1 36.1 27.9 X .38 4.58 9.17 13-8 18.3 22.9 27.5 32. 36.7 41-3 24-4 9/i6 55 5.16 10.3 15-5 20.6 25.8 30.9 36. 41-3 46.4 21.7 H 72 5-73 ii-S 17.2 22.9 28.6 34-4 40. 45-8 51.6 19.5 "/i6 .89 6.30 12.6 18.9 25.2 31-5 37-8 44. 50.4 56.7 17.8 % 2.06 6.88 13.8 20.6 27-5 34-4 41.3 48. 55-o 61.9 16.3 x 3/i6 K 2.23 2.41 7-45 8.02 14.9 16.0 22.3 24.1 29.8 32.1 37-2 40.1 44-7 48.1 52.1 56.1 59-6 64.2 67.0 72.2 15.0 14.0 *S/i6 2.58 8-59 17.2 25.8 34.4 43- 51.6 60. i 68.8 77.3 13.0 I 2-75 9.17 18.3 27.5 36.7 45-8 55-o 64.2 73-3 82.5 12.2 232 WEIGHT OF METALS. WEIGHT OF FLAT BAR IRON. ?. INCHES WIDE. THICK- NESS. SECT. AREA. LENGTH IN FEET. Length o weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X .719 2.40 4-79 7.19 9.58 12.0 14.4 16.8 19.2 21.6 46.7 5/i6 .898 3.00 6.00 9.00 12.0 I5.O 18.0 21.0 24.0 27.0 37-4 H .08 3-59 7.19 10.8 14.4 18.0 21.6 25.2 28.8 3 2 -3 31.2 7/i6 .26 4.19 8-39 12.6 16.8 21.0 25.2- 29.4 33.5 37-7 26.7 X . 44 4-79 9.58 14.4 19.2 24.0 28.8 33-5 38.3 43-1 23-4 9/i6 .62 5-39 10.8 16.2 21.6 27.0 32.3 37-7 43-i 48.5 20.8 k .80 6.00 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54-o 18.7 "Ii6 .98 6-59 13.2 19.8 26.4 33-0 40.5 46.1 52.7 59-3 17.0 H 2.16 7.19 14.4 21.6 28.8 36.0 43-i 50-3 57-5 6 f7 15.6 J 3/i6 2-34 7-79 I 5 .6 23-4 3M 39-o 46.7 54-5 62.3 7p."i 14.4 7 A 2.52 8.39 16.8 25.2 33-5 42.0 50-3 58.7 67.1 75-5 13-4 '5/16 2.70 8.98 18.0 27.0 35-9 45- 53-9 62.9 71.9 80.9 12.4 I 2.88 9-58 19.2 28.8 38.3 48.0 57-5 67.1 76.7 86.3 11.7 3 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X .750 2.50 5.00 7-50 IO.O 12.5 15 o 17-5 20.0 22.5 44.8 5/i6 .938 3-13 6-25 9.38 12.5 16.7 18.8 21.9 25.0 28.1 35-8 & 13 3-75 7.50 H3 15.0 18.8 22.5 26.3 3O.O 33-8 29.9 7/i6 31 4.38 8.75 I3-I 17.5 21.9 26.3 30.6 35-o 39-4 25.6 y* 50 5-00 IO.O 15.0 20.0 25.0 30.0 35-o 40.0 45- 22.4 9/i6 .69 6? "3 16.9 22.5 28.2 33-8 39-4 4S-o 50.6 19.9 H .88 12.5 18.8 25.0 31-3 37-5 43-8 50.0 56.3 17.9 II /i6 2.06 6! 88 13-8 20.6 27-5 34-4 41-3 48.1 55-o 61.9 16.3 % 2.25 7-50 15.0 22.5 30.0 37-5 45-o 52.5 60.0 67.5 14.9 '3/16 2.44 8.13 16.3 24.4 32-5 40.7 48.8 56.9 65.0 73-i 13-8 % 2.63 8-75 17.5 26.3 35- 43-8 52.5 61.3 70.0 78.8 12.8 *5/i6 2.81 9-38 18.8 28.1 37-5 46.9 56-3 65.6 75-0 84.4 12.0 I 3.00 10. 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 II. 2 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X .813 2.71 5-42 8.1 3 10.8 13.6 16-3 19.0 21.7 24.4 41-3 5/i6 1.02 3-39 6.77 10.2 13-5 16.9 20.3 23.7 27.1 30.5 33-1 y$> 1.22 4.06 8.13 12.2 16.3 20.3 24.4 28.4 32.5 36.6 27.5 7/16 1.42 4-74 9 . 4 8 14.2 19.0 23.7 28.4 33-2 37-9 42.7 23-6 A- 1.63 5-42 10.8 16-3 21.7 27.1 32.5 37-9 43-3 48.8 20.7 9/i6 1.83 6.09 12.2 18.3 24.4 30.5 36.6 42.7 48.7 54.8 18.4 '/S 2.03 6.77 13-5 20.3 27.1 33-9 40.6 47-4 54-2 60.9 16.5 2.23 7-45 14.9 22-3 29.8 37-2 44-7 52.1 59-6 67.0 15.0 % 2.44 8.13 I6. 3 24.4 32-5 40.6 48.8 56.9 65.0 73- l 13-7 i3/i6 2.64 8.80 I 7 .6 26.4 35-2 44.0 52.8 61.6 70.4 79.2 12.7 H 2.84 9.48 19.0 28.4 37-9 47-4 56.9 66.4 75-8 85-3 n.8 1 5/i6 3-05 10.2 20.3 30.5 40.6 50.8 60.9 71.1 81.2 91.4 II. I 3-25 10.8 21.7 32.5 43-3 54-2 65.0 75-8 86.7 97-5 10.3 1 FLAT BAR IRON. 233 WEIGHT OF FLAT BAR IRON. INCHES WIDE. THICK- NESS. SECT. AREA. LENGTH IN FEET. Length to weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 875 2.92 5.83 8.75 ii. 7 14.6 17-5 20.4 23-3 26.3 38.4 5/i6 1.09 3.65 7.29 10.9 14.6 18.2 21.9 25.5 29.2 32.8 30-7 H 'Si 4.38 8. 75 I3-I 17-5 21.9 26.3 30.6 35.0 39-4 25.6 7/i6 1.53 5.10 10.2 15-3 20.4 25-5 30.6 35-7 40.8 45-9 21.9 X i-75 5.83 ii. 7 17-5 22.3 29.2 35-0 40.8 46.7 52.5 19.2 9/i6 1.97 6.56 13-1 19.7 26.3 32.8 39-4 45-9 52.5 59-1 I7.I # 2.19 7.29 14.6 21.9 29.2 36.5 43-7 51.0 58.3 65.6 15-4 /x6 2.41 8.02 16.0 24.1 32.1 40.1 48.1 56.1 64.2 72.2 14.0 # 2.63 8.75 17-5 26.3 35-o 43-8 52.5 61.3 70.0 78.8 12.8 I 3/i6 2.84 9.48 19.0 28.4 37-9 47-4 56.9 66.4 75-8 85.3 II.9 $ 3-o6 IO.2 20.4 30.6 40.8 51.0 61.2 7i-5 81.6 91.9 II. O '5A6 3-28 IO.9 21.9 32.8 43-8 54-7 65.6 87.5 98.4 10.2 I 3-50 II.7 23-3 35-o 46.7 58.3 70.0 8i!7 93-3 105.0 9.6O INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X .938 3.13 6.25 9.38 12.5 15.6 18.8 21.9 25.0 28.1 35-8 5/i6 I.I7 3.91 7-8l II.7 15.6 19.5 23.4 27.3 31.3 35-2 28. 7 3/ I.4I 4.69 9.38 I4.I 18.8 23.4 28.1 32.8 37-5 42.2 23-9 7/x6 1.64 5-47 10.9 16.4 21.9 27.3 32.8 38.3 43-7 49-2 20.5 K 1.88 6.25 12.5 18.8 25.0 31.3 37-5 43-8 50.0 56.3 17.9 9/i 6 2. 1 1 7.03 I4.I 21. 1 28.1 35.3 42.2 49.2 56.3 63.3 15-9 S^ 2-34 7.81 I 5 .6 23.4 31.2 39-1 46.9 54-7 62.5 70.3 14-3 n/ I 6 2.58 8.59 17.2 34-4 43-0 51.6 60.2 68.8 77-3 13-0 X 2.81 9.38 18.8 28.1 37-5 46.9 56-3 65.6 75- 84-4 I2.O J 3/i6 3.05 10.2 20.3 30-5 40.6 50.8 60.9 71.1 81.3 91.4 II. /x6 3.28 3-52 IO.9 II-7 21.9 23-4 32-8 35-2 43-8 46.9 %l 65.6 70.3 76.6 82.0 87.5 93-7 98.4 105.5 10.2 9-5 6 I 3-75 12-5 25-0 37-5 50.0 62.5 75-0 87.5 IOO.O 112.5 8.96 4 INCHES WIDE. inches. sq. in. Ibs. lbs.1 Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X I.OO 3-33 6.67 10. 13-3 I6. 7 20.0 23.3 26.7 30.0 33.6 5/i6 1-25 4.17 8-33 12.5 20.8 25.0 29.2 33-3 37.5 26.9 H 1.50 5.00 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45- 22.4 7/i6 1-75 5-83 ii. 7 17.5 23.3 29.2 35-0 40.8 46.7 52.5 I 9 .2 X 2.OO 6.67 13.3 20.0 26.7 33.3 40.0 46.7 53-3 60.0 16.8 9/i6 2.25 7-50 15.0 22.5 3O.O 37.5 45-0 52.5 60.0 67.5 14.9 "As 2.50 2-75 8-33 9.17 3 25.0 27-5 33-3 36.7 41.7 45-8 50.0 55-o 64.2 66.7 73-3 75-o 82.5 13.4 12.2 X 3-00 10.0 20.0 3O.O 40.0 50.0 60.0 70.0 80.0 90.0 II. 2 x 3/x6 3.25 10.8 21.7 32-5 43-3 54-2 65.0 75.8 86.7 97-5 10.3 % 3-50 ii. 7 23.3 35-0 46.7 58.4 70.0 81.7 93-3 105.0 9.60 '5/i6 3-75 12.5 25.O 37-5 50.0 62.5 75.o 87.5 IOO.O 112.5 8.96 I 4.00 13-3 26. 7 40.0 53-3 66.7 80.0 93-3 106.7 120.0 8.40 234 WEIGHT OF METALS. WEIGHT OF FLAT BAR IRON. 4" INCHES WIDE. THICK- NESS. SECT. AREA. LENGTH IN FEET. Length to weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 1. 06 3-54 7.08 10.6 14.2 17.7 21.3 24.8 28.3 31-9 31-6 5/i6 i-33 4-43 8.85 13-3 17.7 22.1 26.6 31.0 35-4 39-8 25.3 H i-59 5-3i 10.6 15-9 21.3 26.6 31-9 37-2 42.5 47.8 21. 1 7/16 1.85 6.20 12.4 24.8 31.0 37.2 43-4 49.6 55-8 18.1 l /2 2.13 7.08 14.2 21.3 28.3 35-4 42.5 49.6 56.7 63-8 15.8 t 2-39 2.66 7-97 8.85 15-9 17.7 23-9 26.6 31.9 35.4 39-8 44-3 47.8 53-1 55-8 62.0 63.7 70.8 71.7 79-7 14.1 12.7 /i6 2.92 9-74 19-5 29.2 39.0 48.7 58.4 68.2 77-9 87.7 11.5 H 3-19 10.6 21.3 3i-9 42.5 53-1 63.8 74-4 85.0 95-6 10.5 *3/i6 3-45 H.5 23.0 34-5 46.0 57.6 69.1 80.6 92.1 103.6 9-9 ft 3-72 12.4 24.8 37-2 49.6 62.0 74-4 86.8 99.2 in. 6 9.0 *S/x6 3.98 13-3 26.6 39-8 53.1 66.4 79-7 93-0 106.2 "9-5 8.4 I 4-25 14.2 28.3 42.5 56.7 70.8 85.0 99-2 "3-3 127-5 7-9 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X $.13 3-75 7.5 II-3 15.0 18.8 22.5 26.3 30.0 33-8 29.9 5/i6 I.4I 4.69 9.38 I4.I 18.8 23-4 28.1 32.8 37-5 42.2 23.9 H 1.69 5-63 "3 16.9 22.5 28.1 33.8 39-4 45- 50.6 19.9 7/i6 1.97 6.56 13-1 19.7 26.3 32.8 39-4 45-9 52.5 59-1 I7.I A. 2.25 7.50 15.0 22.5 30.0 37-5 45- 52.5 60.0 67.5 14.9 9/i6 2-53 8-44 16.9 25-3 33-8 42.2 50.6 59 I 67.5 75-9 13-3 H 2.8l 9.38 18.8 28.1 37-5 46.9 56-3 65.6 75> 84.4 12.0 3.09 10.3 20.6 30.9 41-3 51.6 61.9 72.2 82.5 92.8 10.9 X* 6 3.38 II-3 22.5 33-8 45- 56-3 67.5 78.8 90.0 101.3 9-95 13/16 3-66 12.2 24.4 36.6 48.8 60.9 73-1 85.3 97-5 109.7 9.19 % 3-94 '3- I 26.3 39-4 52-5 65.6 78.8 91.9 105.0 118.1 8-53 J 5/i6 4.22 14.1 28.1 42.2 56.3 70.3 84.4 98.4 112.5 126.6 7.96 I 4.50 15.0 30.0 45- 60.0 75-0 90.0 105.0 120.0 i35-o 7.46 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. 5/i6 I.I9 1.48 3.96 4-95 7.92 9.90 II.9 14.8 15.8 19.8 19.8 24.8 23.8 29.7 27.7 34.6 31.7 39.6 35.6 44.4 28.3 22.6 X 1.78 5-94 11.9 I 7 .8 23.8 29.7 35-6 41.6 47.5 53-4 18.9 7/i6 2.08 6-93 13-9 20.8 27.7 34-7 41.6 48.5 55-4 62.3 16.2 X 2.38 7.92 15.8 23.8 31-7 39-6 47-5 55.4 63.3 7L3 14.2 9/i6 2.6 7 8.91 17.8 26.7 35-6 44.6 53-4 62.3 71-3 80.2 12.6 H 2.97 9.90 19.8 29.7 39-6 49-5 59-4 69.3 79.2 89.1 II.3 /i6 3-27 10.9 21.8 32.7 4-3.5 54-5 65-3 76.2 87.1 98.0 10.3 * 3-56 11.9 23-8 35-6 47-5 59-4 7i-3 83.1 95- o 106.9 9.4 13/16 3-86 12.9 25-7 38.6 5L5 64-3 77.2 90.1 102.9 115.8 8.7 % 4.16 13-9 27.7 41.6 55-4 69-3 97.0 uo.8 124.7 8.1 '5/16 4-45 14.8 29.7 44-5 59-4 74.2 89/1 103.9 118.8 133.6 7-5 1 4-75 15.8 31.7 47.5 63.3 79-2 95. 1 10. 8 126.7 142.5 7-i FLAT BAR IRON. 235 WEIGHT OF FLAT BAR IRON. 5 INCHES WIDE. THICK- NESS. SECT. AREA. LENGTH IN FEET. Length to weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 1.25 4.17 8-33 12.5 16.7 20.9 25.0 29.2 33-3 37-5 26.9 5/i6 1.56 5.21 10.4 IS.6 20.8 26.1 31-3 36.5 41.7 46.9 21-5 ft 1.88 6.2 5 12-5 18.8 25.0 31-3 37-5 43-8 50.0 56.3 17.9 7/16 2.19 2.50 7.29 8-33 14.6 I6. 7 21.9 25.0 29.2 33-3 36.5 41.7 43-8 50.0 51.0 58.3 58,3 66.7 65.6 75-0 15-4 13-4 9/i6 2.81 9.38 18.8 28.1 37-5 46.9 56.3 65.6 75-o 84.4 12.0 X 3-13 10.4 20.8 3i-3 41.7 52.1 62.5 72.9 83-3 93-8 10.8 "/i6 3-44 "3 22.9 34-4 45-8 57-3 68.8 80.2 91.7 103.1 9-77 K 3-75 12.5 25.0 37-5 50.0 62.5 75-0 87.5 IOO.O 112.5 8.96 *3/i6 4.06 13-5 27.1 40.6 54-2 67.7 81.3 94-8 108.3 121.9 8.27 % 4-38 14.6 29.2 43-8 58.3 72.9 87-5 102. 1 116.7 J3I-3 7.68 *S/*6 4.69 15.6 31.3 46.9 62.5 78.i 93-8 109.4 125.0 140.6 7.17 I 5.00 16.7 33.3 50.0 66.7 83-3 100.0 II6.7 133-3 150.0 6.72 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 5/i6 1.64 4.38 5-47 8.75 10.9 If- 1 16.4 17-5 21.9 21.9 27.3 26.3 32-8 30.6 38-3 35-o 43-8 39-4 49.2 25.6 20.5 # 1.97 6.56 13-1 19.7 26.3 32.8 39-4 45-9 52-5 59- * I7-I 7/i6 # 2.JO 2.63 7.66 8.75 iS-3 i7.5 23.0 26.3 30.6 35-o 38.3 43.8 45-9 52-5 53-6 61.3 70.0 fs.l 14.6 12.8 9/i6 2-95 9-84 19.7 29.5 39-4 49.2 59-1 68.9 78.8 88.6 II.4 X 3 .28 10.9 21.9 32.8 54-7 65.6 76.6 87.5 98.4 10.3 "/i6 3-61 12.0 24.1 36.1 48.1 60.2 72.2 84.2 96.3 108.3 9-3 1 X 3-94 !3-! 26.3 39-4 52.5 65.6 78.8 91.9 105.0 118.1 8.55 *3/i6 4.27 14.2 28.4 42.7 56.9 71.1 85.3 99-5 "3-7 128.0 7.88 h 4-59 '5-3 30.6 45-9 61.3 76.6 91.9 107.2 122.5 137.8 7-3 1 "5/16 4.92 16.4 32.8 49.2 65.6 82.0 98.4 114.8 W-3 147-7 6.83 I 5-25 '7-5 35-o 52.5 70.0 87.5 105.0 122.5 140.0 157-5 6.40 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 1.38 4.58 9.17 13.8 18-3 22.9 27-5 32.1 36.7 41.3 24-5 5/i6 1.72 5-73 II.5 17.2 22.9 28.6 34-4 4O.I 45-8 51.6 19-5 3 7/i6 2.06 2.41 8.02 16.0 20.6 24.1 27.5 34-4 40.1 41-3 48.1 48.1 64.2 61.9 72.2 16.4 14.0 X 2-75 9.17 18.3 27.5 3 6 -7 45-8 55-o 64.2 73-3 82.5 12.2 9/i6 3-09 10.3 20.6 30.9 41-3 51.6 61.9 72.2 82.5 92.8 10.9 H 3-44 11.5 22.9 34-4 45-8 57-3 68.8 80.2 91.7 103.1 9-77 ^ 3-78 4-13 12.6 13.8 25.2 27.5 37-8 4i-3 50.4 55-o 63.0 68.8 i$2.5 88.2 96.3 100.8 IIO.O "3-4 123.8 8.14 *3/i6 4-47 14.9 29.8 44-7 59-6 74-5 89.4 104.3 119.2 I34.I 7-52 H 4.81 16.0 32.1 48.1 64.2 80.2 96.3 112.3 128.3 144.4 6.98 *S/i6 5.16 17.2 34-4 51.6 68.8 85.9 103.1 120.3 137-5 154-7 I 5-50 18.3 36.7 55-o 73-3 91.6 no.o 128.4 146.7 165.0 6 ii 236 WEIGHT OF METALS. WEIGHT OF FLAT BAR IRON. INCHES WIDE. THICK- NESS. SECT. AREA. LENGTH IN FEET. Length to weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. # 1.44 4-79 9.58 14.4 19.2 24.0 28.8 33-5 38.3 43- i 23-4 5/i6 X 1. 80 2.16 5-99 7.19 12.0 14.4 18.0 21.6 24.0 28.8 30.0 35-9 35-9 43-i 41.9 50-3 47.9 57.5 64.7 18.7 15.6 7/i6 X 2.52 2.88 9^58 16.8 19.2 25.2 28.8 33-5 38.3 41.9 47-9 5-3 57-5 ft! 67.1 76.7 ! 13.4 II.7 9/i6 3- 2 3 10.8 21.6 32-3 43-1 53-9 64.7 75-5 86.2 97.0 IO.4 H 3-59 12.0 24.0 36.0 48.0 60.0 71.9 83-9 95-8 107.8 9-35 "/i6 3-95 13.2 26.4 39-5 52.7 65-9 79.1 92.2 105.4 118.6 8.50 1? 4-3 1 14.4 28.8 43-i 57-5 71.9 86.3 100.6 115.0 129.4 7-79 J 3/i6 4.67 15.6 31.2 46.7 62.3 77-9 93-4 109.0 124.6 140.2 7.19 % 5-3 16.8 33-5 50.3 | 67.0 83-9 100.7 117.4 134.2 150.9 6.68 J S/i6 5-39 18.0 35-9 53-9 71.9 89.8 107.8 125.8 143-7 161.7 6.22 1 5-75 19.2 38-3 57-5 76.7 95-8 115.0 134-2 153-3 172.5 5-83 6 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. . Ibs. Ibs. feet. X 1.50 5.00 IO.O 15.0 2O. O 25.0 30.0 35-0 40.0 45-0 22.4 5/i6 1.88 6.25 12-5 _ O O 25.0 31-8 37.5 43.8 5O.O 56.3 17.9 H 2.25 7-50 15.0 22^5 3O.O 37-5 45. 52.5 60.0 67-5 14.9 7/i6 2.63 8.75 17-5 26.3 35-0 43-8 52.5 61.3 7O.O 7 8.8 12.8 k 3.00 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 II. 2 9/i6 3-38 "3 22.5 33-8 45-0 56-3 67.5 78.8 9O.O IOI.3 IO.O H 3-75 12.5 25.0 37-5 50.0 62.5 75-o 87.5 1 00.0 II2.5 8.96 n/i6 4.13 13-8 27-5 41-3 55-0 68.8 82.5 96.3 I IO.O 123.7 8.1 5 X 4-5 15.0 30.0 45- 60.0 75-o 90.0 105.0 120.0 135-0 7-47 13/16 4.88 16.3 32.5 48.8 65.0 81.3 97-5 113.7 130.0 146.3 6.90 ft 5.25 17-5 35-0 52.5 70.0 87-5 105.0 122.5 140.0 157.5 6.40 15/16 i 5.63 6.00 18.8 20.0 37-5 40.0 56-3 60.0 80! o 93-8 100.0 112.5 120.0 131.3 140.0 150.0 1 60.0 168.7 l8o.O 5-97 5-6o INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 1.63 5-42 10.8 16.3 21.7 27.2 32.5 37-9 43-3 49.0 20.7 5/i6 2.03 6.77 13-5 20.3 27.1 33-9 40.6 47-4 54-2 60.9 16.5 3 A 2.44 8.13 16.3 24.4 32.5 40.6 48.8 56.9 65.0 73- I I 3 .8 7/i6 2.84 9-47 18.9 28.4 37-9 47-4 56.8 66.3 75-8 85-2 14.8 # 3-25 10.8 21.7 32.5 43-3 54-2 65.0 75-8 86.7 97-5 10.3 9/i6 3.66 12.2 24.4 36.6 48.8 60.9 73-1 85-3 97-5 109.7 9-2O X 4.06 13-5 27.1 40.6 54-2 67.7 94-8 108.3 121.9 8.27 "/t6 4-47 14.9 29.8 44-7 59-6 74-5 89*4 104.3 119.2 i34-i 7-52 H 4.98 I6. 3 32-5 48.8 65.0 81.3 97-5 113-8 130.0 146.3 6.89 J 3/i6 5.28 I 7 .6 35-2 52.8 70.4 88.0 105.6 123.2 140.8 158.4 6.36 % 5.68 19.0 37-9 5 6 .9 75-8 94.8 113-8 132.7 151-7 170.6 5-91 15/X6 6.09 2O.3 40.6 60.9 81.3 101.6 121.9 142.8 162.5 182.8 5.51 I 6.50 21.7 43-3 65.0 86.7 108.3 130.0 I5L7 173-3 195.0 5-29 FLAT BAR IRON, 237 WEIGHT OF FLAT BAR IRON. 7 INCHES WIDE. THICK- NESS. SECT. AREA. LENGTH IN FEET. Length to weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X i-75 5-83 ii. 7 17.5 23.3 29.2 35-o 40.8 46.7 5 2 -5 19.2 5/i6 2.19 7.29 14.6 21.9 29.2 36.5 43-8 51.0 58-3 65.6 !5-4 H 2.63 8.75 17-5 26.3 35-o 43-8 52.5 61.3 70.0 78.8 12.8 7/i6 3-06 10.2 20.4 30.6 40.8 51-0 61.3 71.5 81.7 91.9 II. O */2 3-50 11.7 23-3 35-0 46.7 58.3 70.0 81.7 93-3 105.0 9.60 9/i6 3-94 13-1 26.3 39-4 52.5 65.6 78.8 91.9 105.0 118.1 8-53 "/i6 4.38 4.81 14.6 16.0 29.2 32.1 43-8 48.1 58-3 64.2 72.9 80.2 87.5 96.3 102. 1 II2.3 116.7 128.3 I3L3 144.4 7-68 6.98 X 5.25 17-5 35-0 52.5 70.0 87.5 105.0 122-5 140.0 157.5 6.40 J 3/i6 5.69 19.0 37-9 56.9 75.8 95-0 113.8 132.7 I5I.7 170.6 5-91 ^ 6.13 20.4 40.8 61.3 81.7 102. 1 122.5 142.9 163-3 183.8 5-49 i5/i6 6.56 21.9 43-8 65.6 87.5 109.4 131-3 I53-I 175.0 196.9 5.12 I 7.00 23-3 46.7 70.0 93-3 II6.7 140.0 163.3 186.7 2IO.O 4.80 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 1.88 6.25 12.5 18.8 25.0 31-3 37-5 43-8 50.0 56-3 17.9 5/16 2-34 7.81 15.6 23.4 31-3 39.1 46.9 54-7 62.5 70.3 14-3 7/16 2.81 3-28 9.38 10.9 18.8 21.9 28.1 32.8 37.5 43.8 46.9 54-7 56-3 65-6 7 6.'6 75-0 87-5 84.4 98.4 II.9 10.2 X 3-75 12.5 25.0 37-5 50.0 62.5 75-0 87-5 100.0 112.5 8.96 r 4.22 4-69 14.1 15.6 28.1 31.3 42.2 46.9 56.3 62.5 70.3 78.1 84-4 93-8 98.4 109.4 112.5 125.0 126.6 140.6 7.96 7.17 "/i6 5.16 17.2 34-4 51.6 68.8 85.9 103.1 120.3 137.5 154.7 6. 5 2 K 5.63 18.8 37-5 56.3 75-0 93-8 112.5 131-3 150.0 168.8 5-97 J 3/i6 6.09 20.3 40.6 60.9 8i-3 101.6 121.9 142.2 162.5 182.8 5-51 7 /s 6.56 21.9 43-8 65.6 87.5 109.4 131.3 I53-I 175.0 196.9 5-12 *S/x6 7-03 23-4 46.9 70.3 93-8 117.2 140.6 164.1 187.5 2IO.9 4.78 I 7-50 25.0 50.0 75-0 100.0 125.0 150.0 175.0 200.0 225.0 4.48 8 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. 5/i6 2.00 2.50 6.67 8-33 13.3 16.7 20.0 25.0 26.7 33-3 33-3 41.7 40.0 50.0 46.7 58.3 a? 60.0 75-o 16.8 13-4 H 3.00 IO.O 2O. O 30.0 40.0 50.0 60.0 70.0 80.0 90.0 II. 2 7/i6 /2 3.50 4.OO II.7 13-3 23.3 26.7 35-0 40.0 46.7 53.3 58.3 66.7 70.0 80.0 8l. 7 93-3 93-3 106.7 105.0 I2O.O 9.60 8.40 9/i6 4-5 15.0 30.0 45-o 60.0 75-0 90.0 105.0 120.0 135-0 7-47 H 5-00 16.7 33-3 50.0 66.7 83.3 100.0 II6.7 133-3 150.0 6.72 t 5-50 6.00 I8. 3 20. o 36.7 40.0 55.0 60.0 73.3 80.0 91.7 100.0 I IO.O I2O.O 128.3 140.0 146.7 1 60.0 165.0 ISO.O 6.ii 5.60 '3/i6 6.50 21.7 43-3 65.0 86.7 108.3 130.0 I5L7 173-3 195.0 5-J7 H 7.00 23.3 46.7 70.0 93-3 116.7 I4O.O 163.3 186.7 2 IO.O 4.80 x 5/i6 7.50 25.0 50.0 75-0 100.0 125.0 150.0 175.0 200.0 225.0 4.48 1 8.00 26.7 53-3 80.0 106.7 133.3 160.0 186.7 213.3 24O.O 4.20 2 3 8 WEIGHT OF METALS. WEIGHT OF FLAT BAR IRON. 9 INCHES WIDE. THICK- NESS. SECT. AREA. LENGTH IN FEET. Length to weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 2.25 7.50 15.0 22.5 30.0 37.5 45-o 52.5 60.0 67-5 14.9 S/i6 2.8l 9.38 18.8 28.1 37-5 46.9 i 56.3 65.6 75-0 84.4 II.9 X 3.38 u-3 22.5 33-8 45-0 56.3 67.5 78.8 9O.O 101.3 10.0 7/i6 3-94 26.3 39-4 52.5 65.6 78.8 91.9 105.0 118.1 8-53 A. 4-5 15.0 30.0 45-0 60.0 75-0 90.0 105.0 I2O.O I 35-o 7-47 9/i6 5-06 16.9 33-8 50.6 67.5 84.4 101.3 118.1 135-0 151.9 6.64 X 5.63 1 8. 8 37.5 56-3 75-0 93-8 112.5 131.3 150.0 168.8 5-97 6.19 20.6 41.3 61.9 82.5 103.1 123-8 144.4 165.0 185.6 5-43 % 6.75 22.5 45-o 67.5 90.0 112.5 157.5 iSo.O 202.5 4.98 J 3/i6 7 7 :& 24.4 26.3 48.8 52.5 7! 8 97-5 105.0 121.9 I3I-3 146.3 104.5 170.6 183.8 195-0 210.0 219.4 236.3 4-59 4.26 J 5/i6 8.44 28.1 56-3 84.4 112.5 140.6 168.8 196.9 225.01253.1 3.98 I 9.00 30.0 60.0 90.0 I2O.O 150.0 180.0 210.0 24O.O 27O.O 3.73 10 INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 2.50 8-33 16.7 25.0 33-3 41.7 50.0 58.3 66. 7 75-0 13.4 5/i6 3-13 10.4 20.8 31.3 - 4L7 52.1 62.5 72.9 83.3 93-8 10.7 X 3-75 12.5 25.0 37.5 50.0 62.5 75.0 87.5 IOO.O 112.5 8.96 7/i6 4-38 14.6 29.2 43.8 58.3 72.9 87.5 IO2.I 116.7 131.3 7.68 # 5.00 I6. 7 33-3 50.0 66.7 83.3 IOO.O II6.7 133.3 150.0 6.72 t 5-63- 6.25 18.8 20.8 37-5 41.7 56.3 75-0 83-3 93-8 104.2 II2.5 125.0 145.8 150.0 166.7 168.8 187.5 I'll / l6 6.88 22.9 45.8 68^8 91.7 114.6 137.5 160.4 183.3 206.3 4.89 H 7-50 25-0 50.0 75-0 IOO.O 125.0 150.0 175-0 200.0 225.0 4.48 J 3/i6 8.13 27.1 54-2 81.3 108.3 135.4 162.5 189.6 216.7 243-8 4.14 H 8-75 29.2 58.3 87.5 116.7 145.8 175.0 204.2 233-3 262.5 3.84 z 5/i6 9-40 31.3 62.5 93-8 125.0 156.3 187.5 218.8 25O.O 281.3 3-58 I IO.O 33-3 66.7 loo.o 133.3 166.7 2OO.O 233-3 266.7 300.0 3-36 ii INCHES WIDE. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. X 2-75 9.17 18.3 27-5 36.7 45-8 55.0 64.2 73-3 82.5 12.2 5/i6 3-44 II.5 22.9 34-4 45-8 57-3 68.8 80.2 91.7 103.1 9-77 H 4-13 13-8 27.5 4L3 55-0 68.8 82.5 96.3 IIO.O 123.8 8.15 7/i6 4.81 16.0 32.1 48.1 64.2 80.2 96.3 II2.3 128.3 144.4 6.98 X 5.50 18.3 36.7 55-0 73-3 91.7 IIO.O 128.3 146.7 165.0 6.ii 9/16 6.19 20.6 41.3 61.9 82.5 103.1 123.8 144.4 165-0 185.6 5-43 X 6.88 22.9 45-8 68.8 91.8 114.6 137-5 160.4 183-3 206.3 4.89 I1 /i6 7.56 25.2 50.4 75-6 100.8 126.0 151.3 176.5 201.7 226.9 4-44 X 8.25 27-5 55-0 82.5 IIO.O I 37-5 165.0 192.5 220.0 247.5 4.07 13/16 8-94 29.8 59-6 89.4 119.2 149.0 178.8 208.5 238.3 268.1 3.76 n 9-63 32.1 64.2 96.3 128.3 160.4 192.5 224.6 256.7 288.8 3-49 '5/i6 10.4 34-4 68.8 103.1 137-5 171.9 206.3 240.6 275.0 309.4 3.26 I II. 36.7 73-3 IIO.O 146.7 183.3 220.0 256. 7 293-3 330.0 3.06 SQUARE IRON. 239 Table No. 75. WEIGHT OF SQUARE IRON. SIDE. SECT. AREA. LENGTH IN FEET. Length to weigh I CWt. I 2 3 4 5 6 7 8 9 incMes. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. /4 .0156 052 .104 .156 .208 .260 313 365 .417 .469 2154 3/i6 .0351 .0625 .117 .208 234 .417 :gj .468 -833 .584 1.04 .701 1.25 .818 1.46 935 1.6 7 1.05 960.0 537.6 5/i6 .0977 .326 .651 977 1.68 1-95 2.28 2.60 2-93 343-8 3^ .141 .469 .938 1.41 1.88 2.34 2.81 3.28 3-75 4.22 238-3 7/i6 .191 .638 1.28 1.91 2.55 3-19 3.83 4.46 5.10 5.74 176.0 25 .833 1.6 7 2.50 3-33 4.17 5.00 5.83 6.6 7 7-50 J34-4 9/x6 .316 1. 06 2. II 3.16 4.22 5-27 6.33 7.38 8.44 9.49 106.3 391 1.30 2.60 3.91 5.21 6.51 7.81 9.11 10.4 II.7 85.9 u/ l6 .473 I. 5 8 3.15 4-73 6.30 7.88 9.45 II. O 12.6 14.2 71.0 ^ .563 1.88 3-75 5-63 7-50 9.38 11.3 I3.I 15.0 16.9 59.7 I 3/i6 .661 2.20 4.40 6.61 8.80 II. O 13.2 15-4 1 6. 6 I 9 .8 50.8 ^8 .766 2-55 5.10 7.66 10.2 12.8 15.3 17.9 20.4 23.0 43-9 T 5/i6 .879 2-93 5.86 8-79 ii. 7 14.7 17.6 20.5 23-4 26.4 38.2 I 1. 00 3-33 6.67 10. 13.3 16.7 20. o 23.3 26.7 30.0 33-6 I /i6 Iff .13 .27 3.76 4.22 7-53 8-44 "3 12.7 15.1 16.9 18.8 21. 1 22.6 25.3 26.3 29-5 30.1 33.8 38.0 29.7 26.5 I 3/i6 .41 4.70 9.40 14.1 18.8 23-5 28.2 32.9 37-6 42.3 23.8 'X .56 5.21 10.4 J5-6 20.8 26.0 31-3 36.5 41-7 46.9 21.5 I 5/i6 .72 5-74 ii. 5 23.0 28.7 34-4 4O.2 45-9 51-7 19-5 I^g .89 12.6 18.9 25.2 31-5 37-8 44.1 50.4 56.7 17.8 I 7/i6 2.07 6.89 13-8 20.7 27.6 34-5 4L3 48.2 55-i 62.0 16.2 J K 2.25 7-50 15.0 22.5 30.0 37-5 45-0 52.5 60.0 67-5 14.9 I 9/i6 2.44 8.14 16.3 24.4 32.6 40.7 48.8 57-0 65-1 73.2 13.8 I/^ 2.64 8.80 17.6 26.4 35-2 44-0 52.8 61.6 70.4 79.2 12.7 I IJ /i6 2.88 9.60 19.2 28.8 38.4 48.0 57-6 6 7 .2 76.8 86.4 11.7 I* 3.06 10.2 20.4 30.6 40.8 51.0 61.3 71.4 81.6 91.9 II. I 3/i6 3-29 II. O 21.9 32.9 43-8 54-8 65-7 7 6.7 87.6 98.6 10.2 1% 3.52 11.7 23-4 35-2 46.9 58.6 70.3 82.0 93-8 105.5 9.56 I I 5/i6 3.75 12.5 25.0 37-5 50.1 62.6 75-1 87.6 100. 1 II2.6 8-95 2 4.00 13.3 26.7 40.0 53-3 66.7 80.0 93-3 106.7 120.0 8.40 2 1 A 4-52 15.1 30.1 45-2 60.2 75-3 90-3 105.4 120.0 135.5 7-43 2# 5.06 16.9 33-8 50.6 67.1 84.4 101.3 118.1 135-0 I5L9 6.64 5.64 18.8 37-6 56.4 75-2 94-o II2.8 131.6 150.4 169.2 5.96 2^ 6.25 20.8 41.7 62.5 83-3 10.4 125.0 145.8 166.6 187.5 5.38 2/6 6.89 23.0 45-9 68.9 91-9 114.9 137-8 160.8 183.9 206.7 4.99 2/^ 7.56 25.2 50.4 75-6 100.8 126.1 151.3 176.5 201.7 226.9 4-44 2^ 8.27 27.6 55.1 82.7 no. 2 137-8 165-3 192.9 22O.4 248.0 4.06 3 9.00 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 27O.O 3-73 3/ 10.6 35-2 70.4 105.6 140.8 176.0 211.3 246.5 281.7 316.9 3-17 3^s 12.3 40.8 81.7 122.5 163.3 204.2 245.0 285.8 326.7 367.5 2-73 32^ 14.1 46.9 93-8 140.6 187.5 234.4 281.3 328.1 375-o 42L9 2.38 4 16.0 53-3 106.7 1 60.0 213.3 266.7 320.0 373-0 426.0 480.0 2.10 4X 18.1 60.2 120.4 180.6 240.8 301.1 361.2 421.5 481.7 541.9 .86 4^ 20.3 67-5 135.0 202.5 270.0 337-5 405.0 472.5 540.0 607.5 .66 4^ 22.6 75-2 150.4 225.6 300.8 376.1 451-3 526.5 601.7 676.9 49 5 25.0 83-3 166.7 250.0 333-3 416.7 500.0 583.3 666.7 750.0 34 5X 2 7 .6 91.9 183.8 275.6 367.5 459-4 551.3 643-1 735-o 826.9 .21 5/^ 30.3 100.8 201.7 302.5 403.3 504.2 605.0 705.8 806.7 907.5 .11 5/^ no. 2 220.4 330.6 440.8 *\ ^ I.O 661.3 77L5 881.7 991.8 .02 6 36.0 120.0 240.0 360.0 480.0 600.0 720.0 840.0 960.0 1080 933 240 WEIGHT OF METALS. Table No. 76. WEIGHT OF ROUND IRON. DlAM. SECT. AREA. LENGTH IN FEET. Length to weigh I CWt. I 2 3 4 5 6 7 8 9 inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. feet. 1 A .0123 .041 .082 .123 .164 .205 245 .286 .327 -368 2738 3/i6 .0276 .092 .184 .276 .368 .460 .552 .644 .736 .828 1217 X .0491 .164 .327 .491 .655 .818 .982 1.15 1.31 1.47 684.4 5/i6 .0767 .256 S" .767 1. 02 1.28 i-53 1.79 2.04 2.30 438.1 X .110 368 .736 1. 10 1.47 1.84 2.21 2.58 2.94 3-31 305.4 7/i6 .150 .501 1. 00 1.50 2.00 2.51 3.01 3.51 4.01 4-51 224.0 >/2 .196 654 1. 21 1.96 2.62 3-27 3-93 4.58 5-23 5.89 I7I.4 9/z6 .248 .828 1.66 2.49 3-31 4.14 4-97 5.80 6.63 7.46 135-5 H 37 I. O2 2.05 3-7 4.09 5-II 6.14 7.16 8.18 9.20 109.5 /i6 371 1.24 2.48 3-71 4-95 6.19 7.42 8.66 9.90 II. I 90.6 y .442 1.47 2-94 4.42 5.89 7.36 8.83 10.3 11.8 13.3 76.0 J 3/i6 .518 1-73 3-46 5-19 6.91 8.64 10.4 12. 1 13-8 70.5 # .601 2.OO 4.01 6.01 8.02 10. 12.0 14.0 16.0 18.0 55-9 *5/i6 .690 2-30 4.60 6.90 9.20 n-5 13-8 16.1 18.4 20.7 48.7 I .785 2.62 5-24 7-85 10.5 13.1 15-7 18.3 20.9 23-6 42.8 I Vi6 .887 2. 9 6 5-91 8.87 11.8 14.8 17.7 20.7 23.6 26.6 37-9 '# 994 3-31 6.63 9-94 13-3 16.6 19.9 23-2 26.5 29.8 33-8 I 3/i6 .11 3.69 7.38 ii. i 14.8 18.5 22.2 25.8 29-5 33-2 30.3 i* 23 4.09 8.18 12.3 16.4 20.5 24-5 28.6 32.7 36.8 27-3 I 5/i6 35 4-51 9.02 13-5 18.0 22.6 27.1 31-6 36-1 40.6 24.9 IN .48 4-95 9.90 14.9 19.8 24.8 29.7 34-6 39-6 46.6 22.7 I 7/i6 .62 5.08 IO.2 16.2 20.3 25-9 32.5 35-5 40.6 48.7 20.7 '# 77 5.89 ii. 8 17.7 23.6 29-5 35-3 41.2 47.1 53-0 19.0 I 9/i6 1.92 6-39 12.8 19.2 25.6 32.0 38.4 44-7 5I-I 57-5 17-5 I# 2.07 6.91 13.8 20.7 27.7 34-6 41.5 48.4 55-3 62.9 16.2 I/i6 2.24 7.46 14.9 22.4 29.8 37-3 44-7 52.2 59-6 67.1 15-0 MC 2.41 8.02 16.0 24.1 32-1 4O.I 48.1 56.1 64.1, 72.2 13-9 1*3/16 2.58 8.60 17.2 25.8 34-4 43-0 51-6 60.2 68.8 77-4 13.0 1$ 2.76 9.20 18.4 27.6 36.8 46.0 55-2 64.4 73-6 82.8 12.2 1*5/16 2-95 9.83 19.7 29-5 39-3 49.1 59-0 68.8 79.6 88.4 II.4 2 3-14 10.5 20.9 31-4 41.9 52.4 62.8 73-3 83.8 94-3 10.7 2^ 3-55 1 1.8 23-6 35-5 47-3 59-1 70.9 82.8 94.6 106.4 9-47 2# 3-98 J 3-3 26.5 39-8 53-0 66.3 79-5 92.8 1 06.0 "9-3 8.44 2^ 4-43 14.8 29-5 44-3 59-i 73-8 88.6 103-3 118.1 132.9 7-59 2^ 4.91 16.4 32.7 49.1 65.5 81.8 98.2 II4-5 130.9 147-3 6.84 2^ 5-4i 18.0 36-1 54-1 72.2 90.2 108.2 126.2 144.3 162.3 6.21 # 5-94 19.8 39-6 59-4 79-2 QQ.O 118.8 138.5 158.4 178.2 5.66 2% 6.49 21.6 43-3 64.9 86.6 108.2 129.8 151-5 I73-I 194.8 5.18 3 7.07 23-6 47.1 70.7 94-3 117.8 141.4 164.9 188.5 212. 1 4-75 3X 3K 8.30 9.62 27.7 32.1 64.1 83.0 96.2 110.4 128.3 138.3 160.4 165-9 192.4 193.6 224.5 221.2 256.6 248.9 288.6 4-05 3-49 3^ II. 33-5 73-6 110.4 147-3 164.1 220.9 257-7 294-5 331-3 3-04 4 12.6 41.9 83-8 125-7 167.6 209.4 251-3 293.2 335-0 377-0 2.67 4# 14.2 47-3 94.6 141.9 189.1 236.4 283.7 33i.o 378.3 425-6 2-37 4^ 15.9 53-o 106.0 159.0 212. 1 265.1 319-1 37I-I 424.1 477-1 2. II 4^ 17.7 59-i 118.1 177.2 236.3 295-3 354-4 413.5 472.5 531.6 .90 5 19.6 65-5 130.9 196.4 261.8 327.3 392-7 458.2 523.6 589.1 71 5# 21.7 72.2 144-3 216.5 288.6 360.8 432.9 505.1 577-3 649.4 55 S 11.97 13.46 13-4 8/ 2 56.7 .689 3.378 5.067 6.756 8.444 10.13 11.82' 13.50 15.20 n,8 9 63.6 .893 3.786 5.680 7.572 9-46 11.36 13-25 15.14 17.04 10.6 9 1 A 70.9 2. 1 10 4.220 6.329 8.440 IO -55 12.66 14-77 16.88 18.99 9.48 10 78.5 2.338 4.676 7.012 9-352 11.69 14.03 16.37 18.70 21.04 8.56 10^ 86.6 2-577 4-754 7.731 10.31 12.89 15.46 18.04 19.02 23.19 7.76 II 95- 2.828 5.656 8.485 11.31 14.14 16.97 19.80 22.62 25-46 7-07 II# 103.9 3.088 6.176 9-265 12.35 J 5-44 18.53 21.62 24.70 27.80 6-47 12 113.1 3.366 6.732 10.10 13.46 16.83 20.20 23-56 26.93 30.29 5-94 12^ 122.7 3.656 7.312 10.96 14.62 18.28 21.91 25-59 29.25 32.90 5.48 13 132.7 3-95 7.900 11.85 15.80 19-75 23.70 27.65 31.60 35-15 5-o6 '3# I43-I 4.260 8.520 12.78 17.04 21.30 25-56 29.82 34.08 38.34 4-70 H 153-9 4.581 9.162 13-74 18.32 22.90 26.49 32.07 36.65 41.23 4-37 H^ 165.1 4.915 9.830 14.74 19.66 24-58 28.49 34.41 39.32 44-24 4.07 15 176.7 5-259 10.52 15.78 21.04 26.30 31.46 36.81 42.08 47-33 3-8o i5K 188.7 5.616 11.23 16.85 22.46 28.08 32.70 39-31 44.92 50.54 3-56 16 201. 1 5-984 11.97 17-95 23.93 29.92 35-90 41.89 47-88 53-86 3-34 i6# 213.8 6.364 12.73 19.09 25.46 31.82 38.18 44-55 50.92 57-28 3-14 17 227.0 6-755 13-51 20.27 27.02 33-78 40.53 47-29 54.04 60.80 2.96 i7# 240.5 7-159 14.32 21.48 28.64 35-80 42.95 50.11 57.28 64-43 2-79 18 254-5 7-573 15-15 22.72 30.29 37-86 45-44 53-01 60.60 68.16 2.64 19 283.5 8.438 16.88 25-32 33-75 42.19 50.63 59-03 67-52 75-94 2-37 20 314.2 9-35 18.70 28.05 37-40 46.75 56.10 65-45 74-80 84.15 2.14 21 346.4 10.31 20.62 30.93 41.23 51-54 61.85 72.16 82.47 92.78 1.94 22 380.1 11.31 22.63 33-94 45-25 56.57 67.88 79.19 90.51 101.8 1.77 23 4I5-5 12.37 24-73 37.10 49.46 61.83 74-19 86.56 93-92 111.3 1.62 24 452-4 13.46 26.93 40-39 53.86 67.32 80.78 94-25 107.7 121.3 1.49 16 242 WEIGHT OF METALS. Table No. 77. WEIGHT OF ANGLE-IRON AND TEE-IRON, i FOOT IN LENGTH. NOTE. When the base or the web tapers in section, the mean thickness is to be measured. THICK- NESS. SUM OF THE WIDTH AND DEPTH IN INCHES. 'X *H *Jf 1% 2 2/8 2# 2/8 2/2 2^ 2% inches. y f 3/i6 5/i6 lbs.i 57 .81 1.04 1.24 Ibs. .62 .89 1.15 i-37 Ibs. .68 97 1-25 1.50 Ibs. 73 3 1.63 Ibs. .78 I-I3 I. 4 6 I. 7 6 Ibs. .83 1. 21 l'.8 9 Ibs. .88 1.29 1.67 2. 02 Ibs. .94 1.37 1.77 2.15 Ibs. .99 1.45 1.88 2.28 Ibs. 1.04 1:11 2.41 Ibs. 1.09 1. 60 2.08 2-54 2^ 3 3/8 3# 3/8 3/2 3/8 3^ 3^8 4 4# 3/i6 5/i6 H 7/1 6 I.I4 1.68 2.19 2.67 3-13 3-57 i. 20 1.76 2.29 2.80 3-28 3-75 1-25 1.84 2.40 2-93 3-44 3-93 1.30 1.91 2.50 3-06 3-59 4.11 1-45 1.99 2.60 3-19 3-75 4.29 1.41 2.07 2.71 3.32 3.91 4.48 1.46 2.15 2.81 3-45 4.06 4.66 i.5i 2.23 2.92 3.58 4.22 4.84 1.56 2.30 3-02 3.71 4.38 5.02 1.62 2.38 3-J3 3-84 4-53 5.20 1.72 2-54 3-33 4.10 4.84 5-56 4^ 4^ 5 5X 5/2 5K 6 6X 6/2 6% 7 3/i6 X 5/i6 H 7/i 6 * 9/i6 2.70 3-54 4-36 5-i6 5-92 7.38 2.85 3-75 4.62 5-47 6.29 7.08 7.85 3.01 1 78 6.65 7-50 8.32 3-16 4.17 I' 14 6.09 7.02 7.92 8.79 3-32 4-38 5-40 6.41 7.38 8.33 9.26 3-48 4-58 5.66 6.72 7-75 8.75 9.73 3.63 4.79 5.92 7-03 8. ii 9.17 IO.2O 3-79 5.00 6.18 7-34 8.48 9.58 10.66 3-95 5-21 6-45 7.66 8.84 IO.OO 11.13 4.10 5-42 6.71 7-97 9.21 10.42 ii. 60 4.26 5.63 6.97 8.28! 9-57 10.83 12.07 7X 7/2 7X 8 8K 8/2 S% 9 9# 9/2 9^ X 5/i6 H 7/i6 K 9/i 6 # 5-83 7-23 8-59 9-93 11.25 12.54 13.80 6.04 7-49 8.91 10.30 11.67 13.01 14.32 6.25 7-75 9.22 10.66 12.08 13.48 14.84 6.46 8.01 9-53 11.03 12.50 I3.94 15-36 6.67 8.27 9-84 "39 12.92 14.41 15-89 6.88 8-53 10. 16 11.76 13-33 14.88 16.41 7.08 8.79 10.47 12.12 13-75 15.35 16.93 7.29 9-05 10.78 12.49 14.17 15.82 17-45 7.50 9.31 11.09 12.85 14.58 16.29 17.97 7.71 9-57 11.41 13.22 15.00 16.76 18.49 7.92 9.83 11.72 13.58 15-42 17.23 19.01 10 ioX ii n/ 2 12 12^ 13 13/2 14 14^ 15 N 7/i6 9/i6 # # 12.03 13-95 15-83 17.70 19.53 23-13 12.66 14.67 16.67 18.63 20.57 24.38 13.28 15.40 17-50 19.57 2I.OI 25-63 13-91 16.13 18.33 20.51 22.66 26.88 H-53 16.86 19.17 21.44 23.70 28.13 17-59 2O.OO 22.38 24-74 29-37 18.31 20.84 23.3I 25.78 30.63 19.04 21.67 24.25 26.83 31.88 19.77 22.50 25-I9 27.87 33.13 20.50 23-34 26.12 28.91 34.38 21.22 24.17 27.06 29-95 35.63 12 "# 13 I3K 14 15 16 17 18 19 2O ^ I I 23.70 28.13 32.45 36.67 24.74 29-37 33.91 38.33 25.78 30.63 35.36 40.00 26.83 31.88 36.82 41.67 27.87 33-13 38.28 43-33 29-95 35.63 41.19 46.67 32.03 38.13 44.12 50.00 34.12 40.63 47.02 53-33 36.20 4LI3 49.95 56.67 38.28 43-63 52.87 60.00 40.36 46.13 55-78 63.33 WROUGHT-IRON PLATES. 243 Table No. 78. WEIGHT OF WROUGHT-IRON PLATES. SECT. AREA IN SQUARE FEET. Number THICK- NESS. AREA, when i foot of sq. ft. in wide. I 2 3 4 5 6 7 8 9 i ton. inches. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. sq. feet. X 3-oo 10. 20.0 30.0 40.0 50.0 60.0 70.0 80.0 9O.O 224.0 5/i6 3-75 12.5 25.0 37-5 50.0 62.5 75.0 87.5 IOO.O II2.5 179.2 X 4-5 15.0 3.0 45.o 60.0 75-0 90.0 105.0 I2O.O 135-0 149.3 7/i6 5.20 6.00 17-5 20.0 35-0 40.0 60.0 70.0 80.0 87.5 IOO.O 105.0 120.0 122.5 140.0 140.0 1 6O.O 157-5 iSo.O 128.0 II2.O 9/i6 6.75 22.5 45-o 67.5 90.0 112.5 135-0 150.0 ISO.O 202.5 99-67 i& 7-5 25.0 50.0 75-0 IOO.O 125.0 I5O.O 175-0 2OO.O 225.0 89.60 XI /i6 8.25 27.5 55-0 82.5 IIO.O 137.5 165.0 192.5 220.0 247-5 81.45 3/ 9.00 30.0 60.0 90.0 I2O.O 150.0 iSo.O 2IO.O 24O.O 270.0 74.67 13/16 9-75 32-5 65.0 97-5 I3O.O 162.5 195-0 227.5 260.0 292.5 68.92 ^ 11.50 35-o 70.0 105.0 140.0 175-0 2IO.O 245.0 280.0 3I5.O 64.00 I 5/i6 11.25 37-5 75-o 112.5 I5O.O 187.5 225.0 262.5 300.0 337-5 59-73 I I2.OO 40.0 80.0 120.0 1 6O.O 2OO.O 24O.O 280.0 32O.O 360.0 56.00 I J /i6 12-75 42.5 85.0 127.5 I7O.O 212.5 255.0 297-5 340.0 382.5 52-71 I/^ 13.50 45-o 90.0 135-0 1 80.0 225.0 270.0 3 I 5" 360.0 405.0 49.78 I 3/i6 14.25 47-5 95-o 142.5 I9O.O 237.5 285.0 332.5 380.0 427.5 47.16 I X I5.O 50.0 IOO.O 150.0 200.0 250.0 300.0 350.0 400.0 450.0 44.80 l|^ 16.5 55-o IIO.O 165.0 220.0 275.0 330.0 385-0 440.0 495-0 40.73 i/4 18.0 60.0 I2O.O iSo.O 24O.O 300.0 360.0 420.0 480.0 540.0 37-33 i|/ 21.0 70.0 140.0 210.0 280.0 350.0 42O.O 490.0 560.0 630.0 32.00 2 24.0 80.0 1 6O.O 24O.O 32O.O 4OO.O 480.0 560.0 640.0 720.0 28.00 cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 2*A 30 .893 1.79 2.68 3-57 4.46 5.36 6.25 7.14 8.04 23-40 3 36 .07 2.14 3-21 4-29 5.36 6.64 7.50 8.57 9.64 18.67 42 .25 2.50 3-75 5-oo 6.25 7.50 8.75 10.00 11.25 16.00 4 4 8 43 2.86 4-29 5-7i 7.14 8.57 IO.OO 11.43 12.86 14.00 54 .61 3.21 4.82 6-43 8.04 9.64 11.25 12.86 14.46 12.44 5 2 60 79 3-57 5-36 7.14 8.93 10.71 12.50 14.29 16.07 11.20 5> 66 1.96 3-93 5-89 7.86 9.82 11.79 13.75 I5.7I 17.68 IO.I8 6 72 2.14 4.29 6-43 8.57 10.71 12.86 15.00 17.14 19.29 9-33 7 84 2.50 5.00 7-50 10.00 12.50 15.00 17.50 20.00 22.50 8.00 g 96 2.86 8.57 11.43 10.29 17.14 20.00 22.86 25-71 7.00 9 108 3.21 6.43 9.64 12.86 16.07 19.29 22.50 25.7I 28.93 6.22 10 120 3-57 7.14 10.71 14.29 12.86 21.43 25.00 28.56 32-14 5.60 ii I 3 2 3-93 7.86 11.79 I5-7I 19.64 23.57 27.50 31-43 35.36 5.09 12 144 4.29 8-57 12.86 17.14 21-43 25.7I 30.00 34-29 38.57 4.67 13 I 5 6 4.64 9.29 13-93 23.21 27.86 32.50 37-14 41.79 4.31 1 68 5.00 10.00 15.00 20.00 25.OO 3O.OO 35-oo 4O.OO 45.00 4.00 15 180 5.36 10.71 16.07 21.43 26.79 32.14 42.86 48.21 3-73 244 WEIGHT OF METALS. Table No. 79. WEIGHT OF SHEET IRON. AT 480 LBS. PER CUBIC FOOT. According to Wire-gauge used in South Staffordshire (Table No. 17). THICKNESS. AREA IN SQUARE FEET. Number of sq. ft. in i ton. I 2 3 4 5 6 7 8 9 B.W.G. inch. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. sq. ft. 32 .OI25 .500 .00 1.50 2.00 2.50 3.00 3.5 4.00 4-5 4480 31 .0141 .562 13 1.69 2.25 2.8l 3.38 3.94 4.50 5.06 3986 30 .0156 .625 .25 1.88 2.50 3-13 3-75 4.38 5.00 5-63 3584 29 .0172 .688 .38 2.06 2-75 3-44 4-13 4.81 5.50 6.19 3256 .28 27 .0188 .0203 .750 .813 .50 .63 2.25 2.44 3.00 3.25 4.06 Jjg 4.00 5.25 5.69 6.00 6.50 6.75 7-31 2987 2755 26 .O2I9 .875 75 2.63 3-50 4.38 5-25 6.13 7.00 7.88 2560 25 .0234 .938 .88 2.81 3-75 4.69 5-63 6.56 7.50 8-44 2 3 88 24 .0250 .00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 2240 23 .0281 13 2.25 3.38 4-50 5.63 6.75 7.88 9.00 10. 1 1982 22 .0313 25 2.5O 3-75 5.00 6.25 li 8-75 IO.O n-3 1792 21 0344 .38 2-75 4.13 6.88 9-63 II. O 12.4 1623 2O 375 50 3.00 4-5 6.00 7.50 9.00 10.5 12.0 i3-5 1493 19 .0438 75 5-25 7.00 8.75 10.5 12.3 I4.O 1280 18 .0500 2.OO 4.OO 6.00 8.00 IO.O I2.O 14.0 16.0 iS.'o 1 120 17 0563 2.25 4-5 6.75 9.00 II-3 13-5 15.8 iS.o 20.3 996 16 .0625 2.50 5.00 7-5 IO.O 12.5 I5.O 17-5 20.0 22.5 8 9 6 15 .0750 3.00 6.00 9.00 I2.O 15.0 18.0 21.0 24.0 27.0 747 .0875 3-50 7.00 10.5 14.0 17.5 21. 24-5 28.0 31.5 640 13 .1000 4-OO 8.00 12. 16.0 20.0 24.0 28.0 32.0 36.0 560 12 .1125 4-50 9.00 13-5 18.0 22.5 27.0 31-5 36.0 40.5 498 II .1250 5.00 IO.O 15.0 20. o 25.0 30.0 35-0 40.0 45-o 448 10 .1406 5.63 H'3 16.9 22.5 28.1 33-8 49-4 45' 50.6 398 9 1563 6.2 5 12.5 16.8 25.0 31-3 37-5 43-8 50.0 56.3 358 8 .1719 6.88 13-8 20.6 27-5 34-4 41.3 48.1 55-0 61.9 326 7 1875 7-50 15.0 22.5 30.0 37-5 52.5 60.0 67.5 299 6 .2031 8.13 16.3 24.4 32.5 40.6 4&8 56.9 65.0 72.1 276 5 .2188 8.75 17-5 26.3 35-0 43-8 52.5 61.3 70.0 78.8 256 4 2344 9-38 18.8 28.1 37-5 46.9 56-3 65.6 75.0 84-4 239 3 .2500 IO.O 20. o 30.0 40.0 50.0 60.0 70.0 80.0 90.0 224 2 .2813 11.25 22.5 33-8 45-0 56-3 67.5 78.8 90.0 101.3 199 I 3125 12.5 25.0 37-5 50.0 62.5 87.5 100.0 112.5 179 IRON SHEETS. 245 Table No. 80. WEIGHT OF BLACK AND GALVANIZED IRON SHEETS. (MORTON'S TABLE, FOUNDED UPON SIR JOSEPH WHITWORTH & Co.'s STANDARD BIRMINGHAM WIRE-GAUGE.) NOTE. The numbers on Holtzapffel's wire-gauge are applied to the thicknesses on Whitworth's gauge. Gauge of Black Sheets. Approximate number of square feet in i ton. Gauge of Black Sheets. Approximate number of square feet in i ton. Wire- Gauge. Thickness. Black Sheets. Galvanized Sheets. Wire- Gauge. Thickness. Black Sheets. Galvanized Sheets. No. inch. square feet. square feet. No. inch. square feet. square feet. I .300 187 I8 5 17 .060 933 876 2 .280 200 197 18 .050 1 120 1038 3 .260 215 212 19 .040 I4OO 1274 4 .240 233 229 20 .036 1556 1403 5 .220 254 250 21 .032 1750 1558 6 .200 280 275 22 .028 2000 1753 7 .180 311 34 23 .024 2333 2004 8 .I6 5 339 33i 2 4 .022 2545 2159 9 .150 373 363 25 .020 2800 2339 10 135 415 403 26 .018 3 III 2553 ii .120 467 452 2 7 .016 35 2808 12 .110 509 491 28 .014 4000 3 I22 r 3 .095 589 566 29 .013 4308 3306 H .085 659 630 30 .012 4667 3513 15 .070 800 757 31 .OIO 5600 4017 16 .065 862 813 32 .009 6222 4327 246 WEIGHT OF METALS. Table No. 81. WEIGHT OF HOOP IRON. I FOOT IN LENGTH. According to Wire -gauge used in South Staffordshire. WIDTH IN INCHES. X X % i x# x# x* '# B. W. G. inches. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. 21 0344 .0716 .0861 .100 "5 .129 .144 .158 .172 20 0375 .0781 .0938 .109 .125 .141 .156 .172 .188 19 .0438 .0911 .109 .128 .146 .164 .182 .2OO .219 18 .0500 .104 .125 .146 .167 .188 .208 .229 .250 17 0563 .117 .141 .164 .188 .211 234 .258 .281 16 .0625 .130 .156 .182 .208 234 .260 .286 313 15 .0750 .156 .188 .219 .250 .281 313 344 375 .0875 I8 3 .219 .256 293 329 366 .402 .438 13 .1000 .208 .250 .292 333 375 .416 458 .500 12 .1125 234 .281 .328 375 .422 .469 .516 563 II .1250 .260 313 365 .417 .469 .521 573 .625 10 .1406 293 352 .410 .469 527 .586 645 703 9 .1563 326 391 456 .522 587 .652 .717 783 8 .1719 358 430 .501 573 .644 .716 .788 859 7 .1875 391 .469 547 .625 703 .781 .859 .938 6 .2031 423 .508 593 677 .762 .836 931 1.02 5 .2188 .456 547 .638 729 .820 .912 1O.O 1.09 4 2344 .488 .586 683 .781 879 977 10.7 I.I7 WIDTH IN INCHES. rp X# x# i# 2 *x *y* 2^ 3 B.W.G. inches. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. 21 0344 .197 .201 .215 .229 .258 .287 315 344 20 0375 .203 .219 .224 .250 .281 .313 344 375 19 .0438 238 257 274 .292 .328 365 .400 437 18 .0500 .271 .292 .312 333 375 .417 .458 .500 17 5 6 3 305 328 351 375 .422 .469 .516 563 16 .0625 339 .365 391 .417 .469 .521 573 .625 15 .0750 307 .438 .469 .500 .562 .625 .687 750 14 .0875 475 .512 549 .658 .804 875 13 .1000 543 .584 .626 .667 75 ^833 .917 1. 00 12 .1125 .609 .656 .703 750 .842 .938 3 13 II .1250 .677 729 .781 833 937 .04 *5 .25 IO .1406 .762 .820 879 .938 i. 06 17 .29 .16 9 1563 .848 .913 978 .04 17 30 43 .56 8 .1719 931 .OO .07 15 .29 43 58 .72 7 .1875 1.02 .09 17 25 .41 56 72 .88 6 .2031 1. 10 .19 27 35 52 .69 1.86 2.03 5 .2188 I.I9 .28 37 .46 .64 .82 2.00 2.19 4 2344 1.27 37 .46 .56 .76 95 2.15 2-35 WARRINGTON IRON WIRE. 247 Table No. 82. WEIGHT AND STRENGTH OF WARRINGTON IRON WIRE. TABLE OF WIRE MANUFACTURED BY RYLANDS BROTHERS. NOTE. The Wire-Gauge is that of Rylands Brothers. Size on Wire- Gauge. Diameter. Weig 100 Yds. htof i Mile. Leng i Bundle of 63 Ibs. thof iCwt. Breakin An- nealed. g Strain. Bright. Specific Density, the aver- age den- sity of iron =i. . average inch. milli- metres. Ibs. Ibs. yards. yards. Ibs. Ibs. iron = i. 7/o # 12.7 193-4 3404 33 58 10470 15700 .9852 6 /o 15/32 II.9 170.0 2991 37 66 92OO 13810 S/o 7/i6 II. I 148.1 2606 43 76 8O2O 12000 4/o 13/32 10.3 127.6 2247 49 88 6910 10370 3/c r 8 9-5 108.8 1915 58 103 5890 8835 .9852 /o n/32 8.7 91.4 1609 69 123 4960 7420 O .326 8-3 82.1 1447 77 136 4450 6678 I .300 7.6 69.6 1227 9 o 161 3770 5655 2 .274 7.0 58.1 1022 108 193 3HO 4717 3 .250 (i) 6.4 48.4 8 5 I 130 232 26l8 3927 .9852 4 5 .229 .209 5-8 5-3 40.6 33-8 714 595 III 276 332 2197 1830 3295 2740 6 .191 4-9 28.2 495 223 397 1528 2290 7 .174 4.4 23-4 412 269 479 1268 I9OO 8 .159 4.0 19.6 344 322 573 1060 1558 9 .i 4 6 3-7 16.5 290 382 680 893 1340 10 J 33 3-4 13-7 241 460 819 741 1 1 10 10/2 I25(i) 3-2 12. 1 213 52i 927 654 980 .9852 II .117 3-o 10.6 186 595 1059 573 860 12 .ioo( T V) 2.6 8.0 142 783 1393 436 650 13 .090 2-3 6.3 no 1006 1790 339 509 14 .079 2.0 4.8 85 1305 2322 261 390 15 .069 .8 3-7 65 1715 3052 199 299 16 .0625 (^) 5 2.9 5i 2188 3894 156 233 9378 17 .053 3 2.2 38 2900 5160 118 176 18 .047 .2 1-7 30 3687 6560 93 138 19 .041 .O i-3 2 3 4847 8620 70 105 20 .036 9 I.O 18 5985 1 1 120 54 Si 21 . 03125^) .8 .8 14 ! 7574 I4I52 43 64 1.0843 22 .028 7 .6 n 9893 18486 33 49 Mem. This Table of the weight and strength of Warrington wire is given by permission of Messrs. Rylands Brothers; and it is said to be based on very accurate measurements of sizes and weights. The last column is added by the author, to show that the density of the wire is stationary for diameters of from ^ inch to % inch, and probably somewhat smaller diameters ; but that, contrary to current opinions of the density of wire, the density becomes greater when the diameter is reduced to x / 32 inch. 248 WEIGHT OF METALS. Table No. 83. WEIGHT OF WROUGHT-IRON TUBES, BY INTERNAL DIAMETER. LENGTH, I FOOT. Thickness by Holtzapffel's Wire-Gauge. THICK- NESS. W. G. 4 5 6 7 -238 .220 .203 .180 INCH. # 9/i6 K 7/i6 H 5/i6 X i5/6 4 / 7/ 3 2/ 13/64 3/i6& INT. DIAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. DM. inches. % 4.91 4-05 3.27 2. 5 8 1.96 1-43 .982 .905 -795 .698 575 X 5-73 4-79 3-93 3-15 2-45 1.84 1.31 1.22 i. 08 963 .811 X 6.54 5-52 4-58 3-72 2-95 2.25 1.64 1.53 i-37 1.23 1.05 7.36 6.26 5-24 4-3 3-44 2.66 1.96 1.8 4 1.66 1.50 1.28 X 9.00 7-73 6-55 5-44 4-40 3-48 2.62 2.46 2.24 2.03 1-75 I 10.6 9.20 7.86 6-59 5.40 4-30 3.27 3-09 2.81 2. 5 6 2.23 J X 12.3 10.7 9.17 7-73 6.38 5.11 3-93 3-71 3-39 3-09 2.70 i/5 13-9 12.2 10.5 8.88 7.36 5-93 4.58 4-33 3.96 3-62 3- 1 7 i# 15.6 13-6 n.8 IO.O 8-34 6.75 5-24 4.96 4-54 4.15 3-64 2 17.2 I5- 1 *3-* II. 2 9-33 7-57 5.89 5.58 5.12 4.68 4.11 2 X 18.8 16.6 14.4 12.3 10.3 8.38 6.55 6.20 5.69 5-21 4-58 2/^j 20.5 18.0 15-7 13-5 "3 9.20 7.20 6.83 6.27 5-75 5-05 2 X 22.1 19-5 17.0 14.6 12.3 IO.O 7.85 7.45 6.84 6.28 5.52 3 r 23-7 21.0 18.3 15.8 13-3 10.8 8.51 8.07 7-42 6.81 6.00 27.0 23-9 20.9 18.0 15.2 12.5 9.82 9.32 8-57 7-87 6.94 4 30.3 26.9 23-6 20.3 17.2 14.1 n. i 10.6 9.72 8.94 7.88 4> 33-5 29.8 26.2 22.6 19.1 15.8 12.4 ii. 8 10.9 IO.O 8.82 5 36.8 32.8 28.8 24-9 21. 1 17.4 13.7 13.1 12.0 ii. i 9-77 5^2 40.1 35-7 3i-4 27.2 23.1 19.0 15.1 14-3 13.2 12. 1 10.7 6 43-4 38.7 34-o 29-5 25.0 20.7 16.4 15-6 14-3 13.2 11.7 6X 46.6 41.6 36.7 31-8 27.0 22.3 17.7 16.8 14.3 12.6 7 49-9 44-6 39-3 34-1 29.0 23-9 19.0 18.0 i6!6 15.3 13-5 7K 53-2 47-5 41.9 36.4 30.9 25.6 20.3 19-3 17.8 16.4 14-5 8 56.5 50.4 44-5 38-7 32.9 27.2 21.6 20.5 18.9 17.4 15-4 9 63.0 56-3 49-7 43-2 36.8 3-5 24.2 23.0 21.2 19.6 17-3 10 69.5 62.2 55-o 47-8 40-7 33-8 26.8 25-5 23-5 21.7 19.2 ii 76.1 68.1 60.2 52.4 44-7 37-o 29.5 28.0 23.8 21. 1 12 82.6 74-0 65.5 57-0 48.6 40.3 32.1 30.5 28.1 25-9 23.0 13 89.2 80.0 70.7 61.6 52.5 43-6 34-7 33-0 30.4 28.1 24.9 95-7 85.8 75-9 66.2 56.5 46.8 37-3 35-5 32.7 30.2 26.7 15 102.3 91.7 81.2 70.7 60.4 50.1 39-9 38-0 35-o 32.3 28.6 16 108.8 97-6 86.4 75-3 64-3 53-4 42-5 40.5 37-3 34-4 30.5 17 II5-4 103-5 91.6 79-9 68.2 56.7 45-2 43-0 39-6 36.6 32.4 18 121.9 109.3 96.9 84-5 72.2 59-9 47-8 45-5 41.9 38.7 34-3 19 128.5 115.2 102. I 89.1 76.1 63-2 5-4 48.0 44-2 40.8 36.2 20 135-0 121. 1 107.3 93-6 80.0 66.5 53-0 5-4 46.5 42.9 38.0 21 141-5 I27.O 112. 6 98.2 83-9 69.7 55-6 52-9 48.8 45-1 39-9 22 148.1 132.9 117.8 102.8 87.9 73-o 58.3 55-4 5 1 - ! 47.2 41.8 23 154-6 138.8 123.1 107.4 91.8 76-3 60.9 57-9 53-4 49-3 43-7 24 161.2 144.7 128.3 II2.0 95-7 79-6 63-5 60.4 55-7 51.5 45-6 26 174.3 156.5 138.8 121. 1 103.6 86.1 68.7 65-4 60.3 55.7 49-3 28 187.4 168.3 149.2 130.3 111.4 92.7 74-o 70.4 64.9 60.0 53-1 30 200.4 ISO.O 159.7 139-5 119-3 99.2 79-2 75-4 69-5 64.2 56.8 32 213-5 I9I.8 170.2 148.6 127.1 105.7 84-4 80.4 74.1 68.5 60.6 34 226.6 203.6 180.6 157.8 I 35- 112.3 89-7 85-4 78.7 72.8 64-4 36 239-7 215.4 191.1 167.0 142.9 118.8 94-9 90.4 83-4 77.0 68.1 WROUGHT-IRON TUBES. Table No. 83 (continued}. LENGTH, i FOOT. Thickness by Holtzapffel's Wire-Gauge. 249 THICK- NESS. W. G. 8 9 10 ii 12 13 14 15 16 jr 18 .165 .148 134 .120 .109 095 083 .072 .065 .058 .049 INCH. 11 / 646. 9/6 4 / ft 6 - 3/ 3 a/ 5/64^. Vi6 b. 3/6 4 /. INT. DIAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. inches. 1 A .501 .423 364 318 .267 .219 .181 .149 .130 .Ill 0895 1 A .717 .610 539 .472 .410 343 .290 243 215 .187 154 Y% 934 797 .714 .625 553 .468 .398 337 .300 263 .218 g 1.15 1.58 I.OO .890 1.24 779 1.09 .841 .507 .718 431 .620 .385 555 339 49 i .282 .410 i 2.OI !'.? i-59 1.41 1.27 1.09 935 .808 725 643 .538 l /i 2-45 2.17 1.94 1.72 i-55 i-34 997 .895 795 .667 l */2 2.88 2.55 2.29 2.04 1.84 i-59 i-37 1.19 1.07 .946 795 1 H 3-31 2.94 2.64 2.35 2.12 1.84 i-59 i-37 .24 1. 10 923 2 3-74 3-33 3.00 2.66 2.41 2.08 1.81 1.56 .41 1.25 .05 2 X 4.17 3-72 3-35 2.98 2.6 9 2-33 2.02 1-75 58 1.40 .18 i 2j^5 4.61 4.10 3-29 2.98 2.58 2.24 1.94 75 i-55 .31 2 ^ 5-04 4-49 4-05 3-61 3-26 2.83 2.46 2.13 .92 1.71 44 3 5-47 4.88 4.40 3-92 3-55 3-o8 2.68 2.31 2.09 1.86 57 3^ 6-33 5.65 5.10 4-55 4.12 3-58 3-" 2.69 2-43 2.16 .82 4 7.20 6-43 5.80 5.18 4.69 4.07 3-55 3-07 2.77 2.47 2.08 4tYz 8.06 7.20 6.50 5-8i 5.26 4-57 3.98 3-45 3.11 2-77 '2.34 5 8-93 7.98 7.21 6.44 5-83 5-07 4.42 3-83 3-45 3-07 2-59 5K 9-79 8-75 7.91 7.06 6.40 5-57 4-85 4.20 3-79 3.38 2.85 6 10.7 9-53 8.61 7.69 6.97 6.07 5-29 4-58 4-13 3.68 6^ "5 10.3 9-31 8.32 7-55 6.56 5-72 4.96 4-47 3-98 136 7 12.4 ii.i IO.O 8.95 8.12 7.06 6.16 5-33 4.81 4.29 3.62 13-3 10.7 9.58 8.69 7.56 6-59 5-15 4-59 3-88 8 14.1 \2.6 11.4 10.2 9.26 8.06 7-03 6.09 5-49 4.90 4-13 9 15-8 14.2 12.8 "5 10.4 9-05 7.90 6.84 6.17 5-5 4-65 10 17.6 15-7 14.2 12.7 "5 IO.O 8.77 7.60 6.85 6. ii 5- 16 ii 19-3 17-3 15.6 14.0 12.7 II. 9.64 8-35 7-53 6.72 5.67 12 21.0 18.8 I 5 2 13-8 12.0 10.5 9.10 8.21 7-33 6.19 13 22.7 20.4 18.4 16.5 15.0 13-0 11.4 9.86 8.89 7-93 6.70 H 24-5 21.9 19.8 17.7 16.1 14.0 12.2 10.6 9-57 8-54 7.22 15 26.2 23-5 21.3 19.0 17.2 15.0 *3- * 11.4 10.3 9-15 7-73 16 27.9 25.0 22.7 20.3 18.4 16.0 14.0 12. 1 10.9 9.88 8.24 i 17 29.6 26.6 24.1 21.5 19-5 17.0 14.9 12.9 11.6 10.4 8.76 18 3M 28.1 25-5 22.8 20.6 18.0 13-6 12.3 II. 9-27 19 33-i 29.7 26.9 24.0 21.8 19.0 i6!6 14.4 13.0 n.6 9.78 20 34-8 31.2 28.3 25-3 22.9 20.0 17-5 I5.I 13-7 12.2 10.3 21 36.6 32.8 29.7 26.5 24.1 21.0 18.3 15-9 14.3 12.8 10.8 22 38.3 34-3 27.8 25.2 22.0 19.2 16.6 15.0 13-4 11.3 23 40.0 35-9 32.5 29.1 26.4 23.0 20.1 17.4 15-7 I4.O ii. 8 2 4 41.8 37-4 33-9 30-3 27.5 24.0 20.9 18.1 16.4 14.6 12.6 26 45-2 40.5 36.7 32-8 29.8 26.0 22.6 19.7 17.7 I 5 .8 13-4 28 48.7 43-6 39-5 35-3 32.1 28.0 24-4 21.2 19.1 17.0 14.4 30 52.1 46.7 42.3 37-8 34-4 3O.O 26.1 22.7 20.5 18.3 15-4 32 55-5 49-8 45-1 40.4 36.7 32.0 27.9 24.2 21.8 19-5 16.5 34 59-0 52.9 48.0 42.9 39-0 34-0 29.7 25.8 23.2 20.7 17.5 36 62.4 56.0 50.8 45-4 41-3 36.0 31-4 27-3 24.6 21. 9 18.6 250 WEIGHT OF METALS. Table No. 84. WEIGHT OF WROUGHT-IRON TUBES, BY EXTERNAL DIAMETER. LENGTH, i FOOT. Thickness by Holtzapffel's Wire-Gauge. THICKNESS. W. G. 7 8 9 IO II 12 13 14 15 INCH* .180 .165 .148 134 .I2O .109 095 083 .072 3/i6 & /64 b. 9/64 / 9/64 b. /s b. 7/64 3/3*/ S/64/ S/6 4 b. EXT. DIAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs Ibs. Ibs. Ibs. I inch. 1.55 1.44 1-32 22 .11 .02 .900 797 .700 1/^5 I. 7 8 1.66 $! 39 .26 .16 1.0 3 .906 1/4 2. 02 1.88 1.71 57 .42 30 1.15 .01 .888 \y% 2.25 2.09 1.90 74 58 45 1.27 .12 983 I /^ 2.49 2.31 2.10 , 92 f > '3 59 1.40 ! .23 1.08 iH 2.72 2.52 2.29 2.09 .i >9 3 1-52 -34 1.17 2.96 1 2.74 2.48 2. 27 2.05 .j 7 1-65 i -45 1.27 ifc 3-19 2.96 2.68 2. 45 2.21 2. 02 1-77 | -56 1.36 2 3-43 3- J 7 2.87 2. 62 2.' ;6 2.16 1.90 .67 i-45 2/^ 3-67 3-39 3-06 2. 80 2.52 2.30 2. 02 .78 2% 3-90 3.60 3-26 2. 97 2.68 2-44 2.14 .88 1.64 2 H 4.14 3-82 3-45 3- 15 2.83 2-59 2.27 99 1.74 "2]/z 4-37 4.04 3-65 3- 32 2-99 2-7 3 2-39 2. IO 1.83 2/4 4.61 4-25 3-84 3- 50 3- J 5 2.8 7 2.52 2.21 i-93 2^ 4.84 4-47 4.03 3- 67 3-3 1 3.02 2.64 2.32 2.02 2/S 5.08 4.68 4-23 3-85 \6 3.16 2.77 2-43 2. 1 1 3 5-32 4.90 4.42 4.02 3-62 3.30 2.89 2-54 2.21 3/4^ 5-79 5-33 4.81 4-37 3-94 3-5 9 3-H 2.75 2.40 3/^5 6.26 5-76 5-20 4- 72 4-25 3-8 7 3-39 2.97 2-59 3^ 6-73 6.19 5.58 5-7 4-f 7 4.16 3- 6 4 3.19 2-77 4 7.20 6.63 5-97 5- 43 4-* & 4-44 3-89 3-40 2. 9 6 4 f 7.67 7.06 6.36 5- 78 5-20 4-73 4-13 3.62 3-15 8.14 7-49 ; 7-45 6. 13 5-f ,i 5.01 4.38 3-84 3-34 4^ 8.61 7.91 7-13 6. 48 5-* >2 5-30 4-63 4.06 3-53 5 9.08 8.35 7-52 6. 83 6.13 5.58 4.88 4.27 3-72 9-56 8.79 7.91 7- 18 6.44 5-87 5- 1 " 4-49 3-90 1/2 10.0 9.22 8.30 7- 53 6.76 6.15 5.38 4.71 4.09 10.5 9-65 8.68 7- 88 7.07 6.44 5.63 4-93 4.28 6 II. 10. 1 9.07 8. 23 7-39 6-73 5.87 5-H 4-47 6/4f 11.4 10.5 '< 9.46 8. 58 7.70 7.01 6.12 5.36 4.66 6^ 11.9 10.9 j 9.85 8-93 8.02 7-30 6-37 5.58 4-85 6^ 12.4 11.4 10.2 9- 28 8-33 7.58 6.62 5-79 5-3 7 12.9 ii. 8 i 10.6 9-63 8.64 7.87 6.87 6.01 5.22 13-3 12.2 ! II. 9-99 8.96 8.15 7.12 6.23 5-41 7^ 13.8 12.7 11.4 10.3 9.27 8.44 7-37 6-45 5.60 73^ 14.3 ii. 8 10.7 9-59 8.72 7.62 6.66 5-79 8 14.7 13.5 12.2 II. 9.90 9.01 7.86 6.88 5.98 THICKNESS. W. G. 4 5 6 7 8 9 INCH. .3125 .281 .238 .220 .203 .180 .165 .148 S/i6 9/ 32 'S/64/ 7/32 13/64 '3/16 b. "/6 4 b. 9/64 / EXT. DIAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. 7 inch. 21.9 19.8 16.9 I 5 .6 14.5 12.9 ii. 8 10.6 7'A 23-5 21.3 18.1 1 6,8 15-5 13-8 12.7 11.4 8 25.2 22.7 19-3 17.9 16.6 14.7 J3-5 12.2 8/^ 26.8 24.2 20.6 19.1 17.6 15-7 14.4 I2. 9 9 i 28.4 25.7 21.8 20.2 18.7 16.6 15-3 13-7 30.1 27.1 23.1 21.4 19.8 17.6 16.1 14-5 10 31.7 28.6 24-3 22.5 20.8 18.5 17.0 15-3 LIST OF TABLES OF CAST IRON, STEEL, ETC. 251 LIST OF TABLES OF THE WEIGHT OF CAST IRON, STEEL, COPPER, BRASS, TIN, LEAD, AND ZINC. The following Tables are devoted to the specialities of manufacture in Cast Iron, Steel, and other metals, embracing the utmost range of dimen- sions to which objects in the several metals are executed in the ordinary course of practice. Thus, whilst it is customary for certain classes of Cylinders in Cast Iron steam cylinders, for example to be constructed according to given internal diameters, other classes are constructed according to diameters given externally, as the iron piers of railway bridges. Two distinct tables accord- ingly have been composed, showing the weights of Cylinders of various thicknesses, and of diameters as measured internally and externally. The weights of Copper Pipes and Cylinders are only calculated for in- ternal diameters, as it is not the practice to construct them to given external diameters. Brass Tubes, on the contrary, are calculated only for external diameters, as they are not ordinarily made to given internal diameters. TABLE No. 85. Weight of Cast-iron Cylinders, i foot in length, advanc- ing, by internal measurement, from i inch to 10 feet in diameter, and from Y^ inch to 2^ inches in thickness. TABLE No. 86. Weight of Cast-iron Cylinders, i foot in length, advanc- ing, by external measurement, from 3 inches to 20 feet in diameter, and from 3/ l6 inch to 4 inches in thickness. TABLE No. 87. Volume and weight of Cast-iron Balls, when the diameter is given; from i inch to 32 inches in diameter, with multipliers for other metals. TABLE No. 88. Diameter of Cast-iron Balls, when the weight is given ; from y?, pound to 40 cwts. TABLE No. 89. Weight of Flat Bar Steel, i foot in length ; from ^ inch to i inch thick, and from y z inch to 8 inches in width. TABLE No. 90. Weight of Square Steel, i foot in length ; from ^ inch to 6 inches square. TABLE No. 91. Weight of Round Steel, i foot in length; from y% inch to 24 inches in diameter. TABLE No. 92. Weight of Chisel Steel: hexagonal and octagonal, i foot in length; from ^ inch to i^ inches diameter across the sides. Oval-flat, from ^ x ^ inch to i^ x ^ inch. TABLE No. 93. Weight of one square foot of Sheet Copper; from No. i to No. 30 wire-gauge, as employed by Williams, Foster, & Co. TABLE No. 94. Weight of Copper Pipes and Cylinders, i foot in length, advancing, by internal measurement, from ^ inch to 36 inches in diameter, and from No. oooo to No. 20 wire-gauge in thickness. 252 WEIGHT OF METALS. TABLE No. 95. Weight of Brass Tubes, i foot in length, advancing, by external measurement, from ^ inch to 6 inches in diameter, and from No. 3 to No. 25 wire-gauge in thickness. TABLE No. 96. Weight of one square foot of Sheet Brass; from No. 3 to No. 25 wire-gauge in thickness. TABLE No. 97. Size and weight of Tin Plates. TABLE No. 98. Weight of Tin Pipes, as manufactured. TABLE No. 99. Weight of Lead Pipes, as manufactured. TABLE No. 100. Dimensions and weight of Sheet Zinc. (Vielle-Mon- tagne.) CAST-IRON CYLINDERS. Table No. 85. WEIGHT OF CAST-IRON CYLINDERS. BY INTERNAL DIAMETER, i FOOT LONG. 253 INT. THICKNESS IN INCHES. DlAM. X 5/x6 H 7/i6 l /2 9/i6 H /* X % I inches. Ibs. Ibs.. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. I 3-07 4-03 5.06 6.17 7.36 8.63 9.97 II.4 12.9 16.1 19.6 1^ 4.30 5-56 6.90 8.32 9.82 11.4 14.8 16.6 20.4 24.5 2 5-52 7.09 8.74 10.5 12.3 14.2 i6ii 18.1 20.3 24-7 29.5 2) 6.75 8.63 10.6 12.6 14.7 16.9 19.2 21.5 23-9 29.0 34-4 a 7.98 10.2 12.4 14.8 17.2 19.7 22.2 24.9 27.6 33-3 39-3 9.20 II.7 14-3 16.9 19.6 22.4 25.3 28.3 31-3 37-6 44-2 4 10.4 II.7 13.2 14.8 16.1 18.0 19.1 22.1 22.1 24-5 28.0 28.4 3L5 31-6 35-o 38.7 41.9 46.2 49.1 54-0 5 2 12.9 16.3 19.8 23.4 27.0 30.7 34-5 38.4 42-3 50.5 58.9 5^ 14.1 17.8 21.6 25-5 29.5 33-5 37-6 41.8 46.0 54-8 63-8 6 15-3 19.4 23.5 27.7 32.0 36.2 40.7 45-i 49-7 68.7 THICKNESS IN INCHES. X 7/i6 Yz 9/i6 H /i6 X H i i iX inches. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. 6 23-5 27.7 32.0 36.2 40.7 45.1 49-7 59-1 68.7 78.7 89.0 6J/2 25-3 29.8 34-4 39-0 43.7 48.5 53-4 63-4 73-6 84.2 95-1 7 27.2 32.0 36.8 41.8 46.8 51.9 57-i 67.7 78.5 89.7 IOI.2 I* 29.0 30.8 39-3 41.7 44-5 47-3 49-9 52.9 HI 60.8 64.4 71.9 76.2 83.5 88.4 95-3 100.8 107.4 H3.5 S)4 32.7 38'.4 44.2 50.0 55-9 62.0 68.1 80.5 93-3 106.3 II9.7 10 36.4 38.2 40.0 40.5 42.7 44-8 47.0 46.6 49.1 5L5 54-o 52.8 55-6 58.3 61.1 59.0 62.0 68^2 65.4 68.8 72.1 75-5 71.8 75-5 79-2 82.8 84.8 89.1 93.4 97-7 98.2 103.1 108.0 112.9 in. 8 117.4 122.9 128.4 125.8 I 3 8] I 144.2 II 41.9 49.1 56.5 63-9 71.2 78.9 86.5 102.0 117.8 133-9 150.3 H# 43-7 5L3 58.9 66.6 74.5 82.3 90.2 106.3 122.7 139.4 156.5 12 45 - 6 53-4 61.4 69.4 77-5 85.6 93-9 1 10. 6 127.6 145.0 162.6 13 49-2 57-7 66.3 74-9 83-6 92.4 IOI.2 119.2 137-5 156.0 174.9 14 52.9 62.0 71.2 80.4 89.7 99.1 108.6 127.8 147-3 167.1 187.2 15 56.6 66.3 76.1 85-9 95-9 105.9 116.0 136.4 I57-I 178.1 199.4 16 60.3 70.6 81.0 9L5 IO2.O II2.6 I2 3>3 145.0 166.9 189.1 2II.7 17 64.0 74-9 85.9 97.0 108.2 119.4 130.7 153-6 176.7 200.2 224.0 18 67.7 79-2 90.8 102.5 114.3 126. i 138.1 162.2 186.5 211. 2 236.2 THICKNESS IN INCHES. X 7/i 6 K H X # i 1/8 'X 1/8 1^ inches. cwt. cwt. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 18 .604 .707 .811 .02 1.23 45 1.67 1.89 2. 1 1 2.34 2.56 19 .637 .746 .855 .08 1.30 52 2.22 2.46 2.70 20 21 .670 703 .784 .823 .898 .942 '3 .19 1.36 1-43 .60 .68 I'.ll 1.93 2^18 2.33 2.44 2. S 8 2.70 2.96 22 736 .861 .986 .24 1.49 .76 2.02 2.28 2-55 2.82 3.09 2 3 .769 .900 1.03 .29 1.56 83 2. IO 2.38 2.66 2.94 3.22 24 .802 939 1.07 35 1.63 .91 2.1 9 2. 4 8 2-77 3.06 3.35 25 835 977 1. 12 .40 1.69 99 2.28 2.58 2.88 3-18 3-48 254 WEIGHT OF METALS. Table No. 85 (continued}. BY INTERNAL DIAMETER, i FOOT LONG. INT. THICKNESS IN INCHES. DlAM. H 7/i6 1 A H H I i# i# 1/8 'X inches. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 26 .868 .02 .16 .46 1.76 2.06 2.37 2.68 2.99 3-30 3.62 27 .901 05 .21 51 1.82 2.14 2.45 2.77 3-09 3.42 3-75 28 934 .09 25 57 1.89 2.22 2.54 2.87 3-20 3-54 3-88 29 .967 *3 29 1.96 2.29 2.63 2.97 3-31 3-66 4.01 30 .998 .1? 34 .68 2.02 2-37 2.72 3-07 3-42 3-78 4.14 32 .06 25 43 79 2.15 2.52 2.89 3-27 3-64 4.02 4.41 34 13 32 5i .90 2.29 2.6 7 3-07 3-46 3-86 4.26 4-67 36 .20 .40 .60 2.01 2.42 2.8 3 3.24 3-66 4.08 4-5 4.94 38 .26 47 .69 2.12 2-55 2.98 3.42 3-86 4-30 4-75 5.20 40 33 55 77 2.23 2.68 3-14 3-59 4-05 4-52 4-99 5-47 42 39 63 .86 2-34 2.81 3-29 3-77 4-25 4-74 5-23 5-73 45 49 75 99 2.50 3.01 3-52 4-03 4-55 5-07 5-59 6.13 48 59 .86 2.12 2.66 3.21 3-75 4-3 4-85 5-40 5.96 6.52 THICKNESS IN INCHES. H X g i 1/8 i* I# */* Ig 2 2X inches. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 48 2.66 3.21 3.75 4.30 4.85 5-40 5.96 6.52 7.63 8.77 9.91 51 2.82 3-40 3.98 4.56 5-14 5-73 6.32 6.91 8.09 9.29 10.5 54 2.99 3.60 4.21 4.82 5-44 6.06 6.69 7.31 8.55 9.82 II. I 57 3-!5 3.80 4.44 5-09 5-73 6.38 7-05 7.70 9.01 10.4 11.7 60 3-32 4.00 4.67 5-35 6.03 6. 7I 7.41 8.10 9-47 10.9 12.3 63 3-48 4.19 4.90 5.61 6-33 7.04 7.78 8-49 9-93 11.4 12.9 66 69 3-64 3-8i 4.39 4-59 1:11 5.88 6.14 6.62 6.92 7-37 7.70 8.14 8.51 8.89 9.28 10.4 10.9 11.9 I2 -5 13-5 14.1 72 3-97 4.78 5-59 6.40 7.21 8.03 8.87 9.67 U-3 13.0 14.7 75 4.14 4-98 5.82 6.66 7-5 1 8.36 9.24 10. 1 ii. 8 J 3-5 J 5-2 78 4-3 5.18 6.05 6.93 7.81 8.69 9.60 10.5 12.2 14.0 I 5 .8 81 4.46 5.38 6.28 7.19 8.10 9.02 9-97 10.9 12.7 14.6 !6. 4 84 4-63 5-57 6.51 7-45 8.40 9-35 10.3 "3 13.2 J5- 1 17.0 87 4-79 5-77 6.74 7.72 8.69 9.67 10.7 ii. 6 13-6 15.6 17.6 90 4.96 5-97 6.97 7.98 8.99 IO.O ii. i 12.0 I4.I 16.1 18.2 93 5.12 6.17 7.20 8.24 9.29 10.3 11.4 12.4 14-5 16.7 18.8 96 5.28 6.36 7-43 8.51 9.58 10.7 ii. 8 12.8 15.0 17.2 19.4 99 5-45 6.56 7.66 8.77 9.88 II. 12.2 13.2 15-5 17.7 20. o 102 5.61 6.76 7.89 9-03 IO.2 "3 I2 -5 13.6 20. 6 105 5-78 6.95 8.12 9.29 10.5 11.7 12.9 14.0 16 4 i8.'8 21.2 108 5-94 7-15 8.36 9.56 10.8 12.0 !3-3 14.4 16^8 19-3 21.8 in 6.10 7-35 8-59 9.82 ii. i 12.3 13.6 14.8 17-3 19.8 22.3 114 6.27 7-55 8.82 IO. I 11.4 12.6 14.0 15-2 17.8 20.3 22.9 117 6-43 7-74 9-05 10.4 11.7 I 3 .0 14-3 15.6 18.2 20.9 23-5 120 6-59 7-94 9.28 10.6 12.0 13.3 14.7 16.0 18.7 21.4 24.1 CAST-IRON CYLINDERS. Table No. 86. WEIGHT OF CAST-IRON CYLINDERS. BY EXTERNAL DIAMETER, i FOOT LONG. 255 EXT. DlAM. THICKNESS IN INCHES. 3/i6 X 5/i6 X 7/z6 X 9/i6 # X K i inches. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. 3 5.18 6.75 8.25 9.65 II. 12.3 13-5 14.6 16.6 18.3 19.6 6.10 7.98 9.78 II.5 13.2 14.7 16.2 17.6 20.3 22.6 24.5 4 x 7.02 9.20 "3 I3.3 15.3 17.2 19.0 20.7 24.0 26.9 29.5 7-94 10.4 12.9 15.2 17.5 19.6 21.7 23.8 27.7 3 1 -! 34-4 5 8.86 11.7 14.4 17.0 19.6 22.1 24.5 26.9 3L5 35.4 39-3 3% 9-78 12.9 15.9 18.9 21.8 24-5 27.3 29.9 35-2 39.7 44.2 6 10.7 14.1 17.5 20.7 23-9 27.0 30.0 33-o 38.9 44-o 49.1 6)4 11.6 19.0 22.5 26.0 29.5 32.8 36-1 42.6 48.3 54-0 7 12.5 16.6 20.5 24.4 28.2 3L9 35.6 39-1 46.4 52.6 58.9 7/ 13-5 17.8 22.1 26.2 30.3 34-4 38.3 42.2 50.1 56.9 63.8 8 14.4 19.0 23-6 28.1 32.5 36.8 41.1 45-3 53.8 61.2 68.7 &)4 15-3 20.3 25.1 29.9 34-6 39-3 43-8 48.3 57-5 65-5 73.6 9 16.2 21.5 26. 7 31.8 36.8 41.7 46.6 51-4 61.3 69.8 78.5 9/2 17.2 22.7 28.2 33-6 38.9 44-2 49-4 54-5 65.0 74-i 83-5 10 18.1 23-9 29.7 35-4 41.1 46.6 52.1 57-5 68.7 78.4 88.4 ii 19.9 26.4 3 2.8 39-i 45-4 5L5 57-6 63.7 76.0 87.0 98.2 12 21.8 28.8 35-9 42.8 49-7 56.5 63.2 69.8 83-4 95-6 108.0 13 23.6 31 3 38.9 46.5 54-o 61.4 68.7 75-9 90.7 104.2 117.8 25-5 33-8 42.0 50.2 58.3 66.3 74-2 82.1 98.0 II2.8 127.6 15 27-3 36.2 45- * 53-8 62.6 71.2 79-7 88.2 105.4 121.3 137-4 16 29.1 38.7 48.1 57-5 66.9 76.1 94-3 112.7 129.9 147-3 17 31.0 41.1 51.2 i 61.2 71.1 81.0 9 o.'8 100.5 I2O.O 138.5 18 32.8 43-6 54-3 i 64.9 75-4 85-9 96.3 106.6 127.4 147.1 166.9 19 34-6 46.0 57-3 68.6 79-7 90.8 101.8 II2.8 134-7 155-7 176.7 20 36.5 48.5 60.4 72.3 84.0 95-7 107.3 118.9 142.0 164.3 186.5 21 38-3 50.9 63.5 75-9 88.3 100.6 112.9 125.0 149.4 172.9 196.4 22 40.2 53-4 66.5 79.6 92.6 105-5 118.4 131.2 156.7 181.5 206.2 23 42.0 55-8 69-6 83.3 96.9 110.5 123.9 137.3 164.0 190.1 215.0 2 4 43-8 58.3 72.7 87.0 IOI.2 ii5-4 129.4 143.4 I7I.4 198.7 225.8 2 5 45-7 60.8 75.7 ! 90.7 105.5 120.3 135-0 149.6 178.7 207.2 235.6 26 47-5 63-2 78.8 94.3 109.8 125.2 140.5 155.7 I86.I 215.8 245-4 27 49-4 65.7 81.9 98.0 II4.I 130.1 146.0 161.8 193.4 224.4 255-3 28 5 1 - 2 68.1 85.0 j 101.7 II8.4 135-0 I 5 I -5 1 68.0 200. 7 233-0 265.1 29 53-o 70.6 88.0 105.4 122.7 139-9 I 57.o 174.1 208. 1 241.6 274.9 30 54-9 73-o 91.1 109.1 127.0 144.8 162.6 180.2 215.4 250.2 284.7 31 56-7 75-5 94-2 II2.8 I3I-3 149-7 168.1 186.4 222.7 258.8 294-5 32 58.6 77-9 97.2 116.4 135-6 154.6 173.6 192.5 230.1 267.4 304.3 33 60.4 80.4 100.3 1 20. i 139-9 *59-5 179.1 198.7 237.5 276.0 314.2 34 62.2 82.8 103.4 123.8 144.2 164.5 184.7 204.8 244.8 284.6 324.0 35 64.1 85.3 106.4 127.5 148.5 169.4 190.2 210.9 252.2 293.1 333-8 36 65.9 87.8 109.5 131.2 152.7 174.3 195.7 217.1 259.5 301.7 343-6 38 69.6 92.7 115.6 138-5 161.3 184.1 206.8 229.3 274.3 318.9 363-2 40 73-3 97-6 121. 8 145-9 169.9 193-9 217.8 241.6 289.0 336.1 382.9 42 77.0 102.5 127.9 153-3 178.5 203.7 228.8 253-9 303-7 353-3 402.5 45 82-5 109.8 137.1 164.3 I9I.2 218.5 245-4 272.3 325.8 379-1 432.0 48 88.0 117.2 146.3 175.4 203.8 233-2 262.0 290.7 347-9 404.8 461.4 51 93-6 124.6 155.5 186.4 216.5 247.9 278.6 309.1 370.0 430.6 490.9 54 99.1 I 3 I -9 164.7 197.5 229.2 262.6 295.1 327.5 392.1 456.4 520.3 57 104.6 139-3 173.9 208.5 241.8 277.4 3"-7 345-9 414.2 482.1 549-8 60 1 10. 1 146.6 183.1 219.6 254-5 292.1 328.3 364-3 436.3 507.9 579-3 2 5 6 WEIGHT OF METALS. Table No. 86 (continued}. BY EXTERNAL DIAMETER, i FOOT LONG. EXT. THICKNESS IN INCHES. DlAM. 3/i 6 X 5/i 6 K 7/16 # 9/i6 H X 7 /s i ft. in. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 5 3 03 44 1.71 2.06 2.39 2-74 3.08 3-42 4.09 4-77 5-43 56 .08 50 1. 80 2.16 2.50 2.8 7 3-22 3.58 4.29 5.00 5 9 *3 55 1.88 2.26 2.62 3.00 3-37 3.75 4.49 5-23 5*96 60 .18 .61 1.96 2.36 2.74 3-14 3-52 3-91 4.69 5-46 6.22 63 23 67 2.05 2.45 2.85 3-27 3-66 4.08 4.88 5.69 6-49 66 .28 73 2.13 2.55 2.97 3-40 3.81 4.24 5.08 5-92 6.75 69 33 .78 2.21 2.65 3.09 4.41 5.28 6.15 7.01 7 o 38 .84 2.29 2.75 3.20 3.66 4.10 4.57 5-47 6.38 7.28 76 48 95 2.46 2.95 3-43 3-92 4-39 4.90 5-87 6.84 7.80 80 58 2.07 2.62 3.15 3-67 4.19 4.69 5.23 6.26 7-30 8-33 THICKNESS IN INCHES. i% i* I# # *# 2 2X *y 2 2^ 3 3/2 4 inches. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 6 .481 .520 557 592 .652 .701 .740 .761 6 l /2 53 575 .618 .657 .789 .838 .872 .906 7 579 630 .678 723 805 .876 .938 .982 03 05 .629 .685 738 .789 .882 .964 1.04 1.09 15 .18 8 .678 .740 799 .855 959 .05 .14 .20 27 32 1.38 S^2 .727 794 859 .921 .04 .14 23 31 39 45 i-53 9 777 .849 .919 .986 .11 23 33 .42 51 58 1.69 1-75 .826 .904 .980 1.05 19 31 43 53 63 71 1.84 i-93 10 .875 959 .04 .12 .27 .40 53 .64 75 1.84 1.99 2.10 ii 974 .07 .16 .25 .42 .58 73 .86 99 2.IO 2.30 2.46 12 .07 .18 .28 38 57 75 .92 2.08 2.23 2 -: 57 2.61 2.81 13 17 29 .40 51 73 93 2.12 2.30 2.47 2.(. >3 2.92 3.16 14 27 .40 .52 64 1.88 2.10 2.32 2.52 2 71 2.8 9 3-22 3- Si 15 37 65 78 2.03 2.28 2.52 2.74 2 95 3.] 6 3-53 3-86 16 47 .62 77 .91 2.19 2-45 2.71 2.96 3 19 3-42 3-84 4.21 17 57 73 .89 2.04 2-34 2.6 3 2.91 3.18 3 44 3.68 1 4.14 4-56 18 1.66 1.84 2.01 2.17 2.49 2.81 3.H 3-40 3 .68 3-95 4-45 4.91 20 1.86 2.06 2.25 2-43 2.80 3.16 3-50 3-83 4.16 4-47 5.06 I' 61 22 2.06 2.27 2.49 2.70 3- 11 3.51 3-90 4.27 4 .64 5.00 5-68 6.32 24 2.26 2-49 2-73 2. 9 6 3-41 3.86 4.29 4.71 5 .12 5-52 6.29 7.01 27 2-55 2.82 3-9 3-35 3-87 4.38 4.88 5-37 5 85 6.31 7.21 8.06 30 9 2.85 3-14 3-44 3-48 3-8i 3-82 4.18 3-75 4.14 4-54 4-33 4-79 5-25 4.91 5-44 5-47 6.06 6.66 6.03 6.68 7-34 6-57 7.29 8.01 7.10 7.89 8.68 8.13 9-05 9-97 9.12 10.2 II. 2 39 42 3-74 4-3 4.14 4-47 4-54 4.90 4-93 5-33 lit 6.49 7.01 7-25 7.84 8.00 8.66 8.74 9.46 9-47 10.3 10.9 ii. 8 12.3 13-3 4-33 4-79 5-26 5-72 6.64 7-54 8-43 9.31 10.2 ii .1 12.7 14.4 48 4.62 5.12 5.62 6.12 7.10 8.07 9.02 9-98 10.9 ii .8 13-7 15-4 51 4.92 5-45 5.98 6.51 7-56 8-59 9.61 10.6 11.6 12.6 14.6 16.5 54 1? 5.22 S'.sl 6.44 6-35 6.71 7.07 6.91 7-30 7.70 8.02 8.48 8.94 9.12 9.64 10.2 IO.2 10.8 11.4 "3 "' 12.6 12.4 III 13-4 14.2 15-0 15-5 16.4 17-3 17-5 18.6 I9 .6 CAST-IRON CYLINDERS. 257 Table No. 86 (continued}. BY EXTERNAL DIAMETER, i FOOT LONG. EXT. DlAM. THICKNESS IN INCHES. 2 ol/ I/ 93/ I/ ft. in. Z A 2> 2% 3 3/2 4 cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. M 6.10 6.40 6.77 7.09 7-43 7-79 8.09 8.48 9.40 9.86 10.7 II. 2 I2.O 12.6 13-3 13-9 14.5 1 S- 2 15.8 16.6 18.3 59 6.70 7.42 8.15 8.88 10.3 II. 8 13.2 14.6 15-9 60 7.00 7-75 8.51 9.27 10.8 I2. 3 13-8 15.2 63 7.29 8.08 8.88 9.67 II. 2 12.8 14.4 15-9 66 7.58 8.41 9.24 10. 1 11.7 13-3 14.9 16.6 69 7.88 8.74 9.60 10.5 12.2 13-9 15-5 17.2 70 8.17 9.07 9.96 10.9 12.6 14.4 16.1 17.9 76 8-77 9.72 10.7 ii. 6 13-5 15-4 17-3 19.2 80 9-36 10.4 11.4 12.4 14-5 16.5 18.5 20.5 86 9-95 II. 12. 1 13.2 15-4 19.7 21.8 90 10.5 ii. 7 12.9 14.0 I6. 3 18.6 ' 20.8 23.1 96 ii. i 12.3 13-6 14.8 17.2 19.6 22.0 24.4 100 11.7 13.0 14.3 15.6 18.1 20.7 23.2 25.7 10 6 12.3 13-7 15-0 16.4 19.1 21.7 24.4 27.1 II 12.9 14-3 15-7 17.2 20.0 22.8 2 5 .6 28.4 ii 6 13-5 15.0 I6. 5 17.9 20.9 23-8 26.7 29.7 12 14.1 15.6 17.2 18.7 21.8 24.9 27.9 31.0 I 3 15-3 16.9 18.6 20.3 23-7 27.0 30-3 33.6 140 16.5 18-3 20.1 21.9 25-5 29.1 32-7 36.3 15 o ll'l 19.6 21-5 23-5 27-3 31.2 35- 38.9 160 17 o 180 20. o 21.2 20.9 22.2 23-5 23.0 24.4 25-9 25.0 26.6 28.2 29.2 31.0 32.9 33-3 35-4 37-5 37-4 !39-8 42.2 41.5 1 190 22.4 24.8 27-3 29.8 34-7 39-6 44-5 49-4 i 200 23-6 26.1 28.8 31-4 36.5 41.7 46.9 52.0 17 2 5 8 WEIGHT OF METALS. Table No. 87. VOLUME AND WEIGHT OF CAST-IRON BALLS. GIVEN THE DIAMETER. Diameter. Contents. Weight. Diameter. Contents. Weight. Diameter. Contents. Weight. inches. cubic inches. pounds. inches. cubic inches. pounds. inches. cubic feet. cwts. I .524 .136 8 268.1 69.8 19 2.078 8.35 I ^2 1.77 .460 Sj/2 321.5 83-7 20 2.424 9-74 2 4.19 I.O9 9 38L7 99-4 21 2.806 11.28 2 /^2 8.18 2.13 448.9 116.9 22 3.227 12.97 3 I4.I 3.68 r> 10 523-6 136.4 23 3-688 O O 14.82 3/^ 22.5 5-85 inches. cubic feet. cwts. 24 4.188 16.83 4 33-5 8-73 II .403 1.62 25 4.736 19.03 4/^ 47-7 12.4 12 .524 2.10 26 5-327 21.40 5 65-5 17.0 13 .666 2.68 27 5-963 23.96 87.1 22.7 14 .832 3-34 28 6.651 26.72 6 113.1 29-5 15 1.023 4.11 29 7.390 29.69 6^ 143.8 37-5 16 1.241 4.99 30 8.181 32.87 7 179.6 46.8 17 1.489 5.98 31 9.027 36.27 7^ 220.9 57.5 18 1.767 7.10 32 9.930 39-90 Note. To find the weight of balls of other metals, multiply the weight given in the table by the following multipliers : For Wrought Iron 1.067, making about 7 per cent. more. Steel i. 088 ,, 9 ,, Brass 1.12 ,, 12 Gun Metal 1.165 ,, i6> ,, Table No. 88. DIAMETER OF CAST-IRON BALLS. GIVEN THE WEIGHT. Weight. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. pounds. inches. pounds. inches. pounds. inches. cwts. inches. % 1-54 14 4.68 80 8.37 8 18.73 I 1.94 16 4.89 90 8.71 9 19.48 2 2-45 18 5-9 IOO 9.O2 10 20.17 3 4 2.OO 3.08 20 25 5-27 5.68 cwts. T inches. 9f+ >7 12 14 21.44 22.57 5 3-32 28 5-9 l% 37 10.72 16 23.60 6 3-53 30 6.04 2 11.80 18 24-54 7 3-72 40 6.64 3 i3-5i 20 25.42 8 3-89 5 7.16 4 14.87 2 5 27.38 9 4.04 56 7-43 5 16.02 30 29.10 10 4.19 60 7.60 6 17.02 35 30.64 12 4-45 70 8.01 7 17.91 40 32.03 WEIGHT OF FLAT BAR STEEL. 259 Table No. 89. WEIGHT OF FLAT BAR STEEL. i FOOT LONG. WIDTH n 4 INCHES. THICKNESS # % # ft I iX K# IM inches. Ib. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. X 5/i 6 .425 53 1 -m .640 .800 743 .929 850 1. 06 1.06 i-33 1.28 i-59 1.49 1.86 y& .638 .798 .960 i. ii 1.28 1.91 2.23 7/i6 744 .931 .12 1.30 1.49 1.86 2.23 2.60 $ .850 i. 06 .28 1.49 1.70 2.13 2-55 2.98 9/i6 1.20 44 1.67 I.9I 2-39 2.87 3-35 i-33 .60 1.86 2.12 2.66 3-19 3-72 6 .76 2.04 2-34 2.92 3-5 1 4.09 3^ .92 2.23 2-55 3-83 4.46 J 3/i6 2.41 2. 7 6 3-45 4.14 4-83 ?8 2.60 2. 9 8 3-72 4.46 5-21 J 5/i6 3-19 3-98 4-78 5-58 I 3-40 4-25 5- 10 5-95 WIDTH n 4 INCHES. THICKNESS 2 2* 2/2 2^ 3 3X 3/2 4 inches. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. X 1.70 1.91 2.13 2-34 2-55 2. 7 6 2.98 3-40 5/i6 2.13 2-39 2.66 2.92 3-45 3-72 4-25 N 7/x6 2.98 2.87 3-35 3-19 3-72 3-51 4.09 3-83 4.46 4.14 4-83 4.46 5.21 5.10 5-95 X 3-40 3-83 4-25 4.68 5.10 5-53 5-95 6.80 9/i 6 3-83 4-30 4.78 5.26 5-74 6.22 6.69 7.65 # 4-25 4-78 5-3 1 5.84 6.38 6.91 7-44 8.50 4.68 5.26 5-84 6-43 7.01 7.60 8.18 9-35 ^ 5.10 5-74 6.38 7.01 7.65 8.29 8-93 IO.2 x 3/i6 5-53 6.22 6.91 7 .60 8.29 8.98 9.67 II. I % 6.69 7-44 8.18 8-93 9.67 10.4 II.9 i5/! 6 6.38 7.17 7-97 8-77 9-5 6 10.4 II. 2 12.8 I 6.80 7.65 8.50 9-35 10.2 II. I II.9 13.6 WIDTH it i INCHES. THICKNESS 4X 5 5/2 6 6/2 7 7/2 8 inches. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. X 3-82 4.26 4.68 5.10 5-52 5.96 6.38 6.80 5/i6 4.78 5-32 5-84 6.38 6.90 7.44 7-97 8.50 N 5-74 6.38 7.02 7.66 8.28 8.92 9-56 10.2 7/i6 6.70 7-44 8.18 8.92 9.66 10.4 II. 2 II.9 ^2 7.66 8.50 9-3 6 10.2 II. I 11.9 12.8 13-6 9/i6 8.60 9.56 10.5 lI -5 12.4 13-4 14-3 15-3 4 9.56 10.6 11.7 12.8 13-8 14.9 15-9 17.0 10.5 11.7 12.9 14.0 15.2 16.4 17-5 18.7 ^ "5 12.8 14.0 15.3 16.6 17.9 I9.I 2O.4 3/x6 12.4 13.8 15.2 16.6 18.0 19-3 20.7 22.2 ^ 6 13-4 14.9 16.4 17.9 19.4 20.8 22.3 23-8 H-3 15-9 17-5 19.1 20.8 22.4 23-9 2 5 .6 I 15-3 17.0 18.7 20.4 22.1 23.8 25.5 27.2 260 WEIGHT OF METALS. Table No. 90. WEIGHT OF SQUARE STEEL. i FOOT IN LENGTH. Size. Weight. Size. Weight. Size. Weight. Size. Weight. inches. pounds. inches. pounds. inches. pounds. inches. pounds. /8 053 'V.6 3.06 1% IO.4 3*/4 35-9 3 A6 .119 I 3-40 I I3 /i6 II. 2 1% 41.6 % .212 I Vi6 3.83 1/8 II. 9 31/4 47-8 5 /i6 333 1/8 4-3 I I5 /i6 12.8 4 54-4 tf .478 I Vi6 4-79 2 13-6 4^ 61.4 7 /x6 .651 '# 5-3i 2/8 15-4 4^ 68.9 % .850 I Vi6 5.86 *X 17.2 4& 76.7 9 /!6 i. 08 iH 6-43 23/8 19.2 5 85.0 & i-33 I 7 /i6 7-03 *% 21.2 5^ 93-7 */* 1.61 '# 7.65 2/8 23-5 5^ 102.8 # 1.92 I 9 /i6 8.30 2% 25-7 5?4 112.4 13 A6 2.24 Ijl 8.98 2/8 28.2 6 122.4 /8 2.60 l"/i6 9-79 3 30.6 Table No. 91. WEIGHT OF ROUND STEEL. i FOOT IN LENGTH. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. Weight. inches. pounds. inches. pounds. inches. pounds. inches. cwts. / .0417 7 /i6 5 .l8 4 42.7 12 3-433 3 A6 939 % 6.01 48.3 l2/ 2 3-729 k .167 9 /i6 6.52 4/ 54-6 13 4.029 5 /x6 .260 y 7.05 4^4 60.3 J 3>2 4-345 375 "/.6 7.61 5 66.8 14 4.682 7 /i6 5 11 8.18 5/4 73-6 14^ S-0 1 3 % .667 13 /i6 8.77 80.8 15 c. 364 9 /x6 845 1^ 9.38 5/4 88.3 I 5 I A c. 7 28 1.04 I I5 /i6 10. 6 96.1 16 6.103 I / 6 1.27 1.50 2 10.7 12.0 inches. cwts. J 7 6.471 6.868 < 3 A6 1.76 2/ I 3 .6 6/^ 7 .007 .168 7.302 7^ 2.04 2^/8 I tC. I / 18 7.724 'V.6 2-35 2/ 2 rf>. 7 8 34 1 .526 i9 8.607 I 2.67 2/8 18.4 8)^ 723 20 9-537 I x /i6 3.00 2 %i 20.2 9 I-93 1 21 10.52 1^ 3.38 2J/Z 22.0 9^3 2.152 22 n-54 I 3 /x6 3.76 3 i 24.1 10 2-385 23 12.62 ll^ 4.17 28.3 io/ 2.629 24 13-73 I S /i6 4.60 3/ 32.7 II 2.884 13/8 5-05 3^ 34-2 II/ 3-150 WEIGHT OF CHISEL STEEL. 26l Table No. 92. WEIGHT OF CHISEL STEEL HEXAGONAL, OCTAGONAL, AND OVAL-FLAT. i FOOT IN LENGTH. HEXAGONAL SECTION. OCTAGONAL SECTION. Diameter across the Sides. Sectional Area. Weight. Length to weigh Sectional Area, Weight. Length to weigh I CWt. I CWt. inches. square inches. pounds. feet. square inches. pounds. feet. M .1217 .414 245 .1164 39 6 268 % .2165 73 6 138 .2070 .704 151 y* .3383 i.J;5 88.3 .3 2 36 1. 10 9 6 -5 X .4871 1.66 6l-3 .4659 1.58 67 % .6631 2.25 45 .6342 2.16 49-3 i .8661 2.94 34-5 .8284 2.82 37-7 i# 1.096 3-73 27-3 1.048 3.56 3 '# 1-353 4.60 22.5 1.294 4.40 24 13/S 1.637 5-57 18.3 1.566 5-32 20 i* 1.949 6.63 15-3 1.864 6.34 16.8 OVAL-FLAT SECTION. inch. inch. 24 x 3/ 8 .2510 .853 119 I x ^ .4463 1-52 67 1% x^ .6974 2-37 43 Table No. 93. WEIGHT OF ONE SQUARE FOOT OF SHEET COPPER. To Wire-Gauge employed by Williams, Foster, & Co. Specific Weight taken as 1.16 (Specific Weight of Wrought Iron = i). Thickness. Weight of i Square Foot. Thickness. Weight of i Square Foot. Thickness. Weight of i Square Foot. Wire- Inch Wire- Inch Wire- Inch Gauge. No. (approxi- mate). pounds. Gauge. No. (approxi- mate). pounds. Gauge. No. (approxi- mate). pounds. I .306 14.0 II .123 5^5 21 .0338 i-55 2 .284 13.0 12 .109 5.00 22 .0295 i-35 3 .262 12. J 3 .0983 4-5 23 .0251 i-i5 4 .240 II. 14 .0882 4.00 2 4 .0218 I.OO 5 .222 IO.I5 15 .0764 3-5 25 .0194 .89 6 .203 9-30 16 655 3.00 26 .0172 79 7 .186 8.50 17 .0568 2.60 27 0153 .70 8 .168 7.70 18 .0491 2.25 28 o!35 .62 9 J 53 7.00 !9 0437 2.00 29 .0122 56 10 .138 6.30 20 .0382 i-75 3 .OIIO 5o 262 WEIGHT OF METALS. Table No. 94. WEIGHT OF COPPER PIPES AND CYLINDERS, BY INTERNAL DIAMETER. LENGTH, i FOOT. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). Specific Weight=i.i6 (Specific Weight of Wrought-Iron = i). THICK- NESS. W. G. 0000 ooo oo I 2 3 4 5 6 7 INCH. 454 29/64 .425 *7/6 4 / .380 340 11/32 .300 .284 259 .238 <5/6 4 / .220 7/32 / .203 13/64 .180 3/i 6 b. INT. DIAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. inches. y& 3M4 2.8 4 2-33 1.92 i-53 1.41 1. 21 1.05 934 .809 .66 7 1 A 3-49 2.91 2.44 1.99 1.84 1. 60 1.27 1. 12 .941 y% 4-54 4-13 3-49 2-95 2-45 2.27 2.00 1.77 i. 60 i-43 1. 21 l /2 5-23 4-78 4.06 3-47 2.91 2.71 2-39 2.13 i-93 1-73 1.49 H 5-93 5-42 4.64 3-99 3-37 3-H 2. 7 8 2.50 2.26 2.04 I. 7 6 y* 6.63 6.07 5.22 4-5 3.83 3-57 3. r 7 2.86 2.60 2 -35 2.03 y& 7.32 6.71 5-79 5.02 4.29 4.00 3-57 3.22 2-93 2.66 2. 3 I 1 8.02 7.36 6-37 5-53 4-74 4-43 3.96 3-57 3.26 2.97 2.58 i# 8.71 8.00 6-95 6.05 5-20 : 4.86 4-35 3-94 3.60 3.28 2.8 5 i^ 9.40 8.65 7-52 6-57 5-65 \ 5-29 4-75 4-3 3-93 3.58 3-13 \y^ IO. I 9-3 8.10 7.08 6. ii ; 5-72 5-14 4.66 4.26 3.89 3-40 l& 10.8 9-94 8.68 7.60 6.57 \ 6.16 5-53 5.02 4.60 4.20 3-68 1% "5 10.6 9.26 8.12 7.02 ! 6.59 5-93 5-39 4-93 4.51 3-95 ify( 12. 1 II. 2 9-83 8.63 7.48 7.02 6.32 5-75 5-27 4.82 4.22 iH 12.8 11.9 10.4 9-15 7-93 7-45 6.71 5.60 5.12 4-5 2 13-5 12.5 II. 9.66 8-39 7-88 7.11 6.47 5-93 5-43 4-77 2% 14.2 13.2 ii. 6 10.2 8.84 8.31 7-5 6.83 6.27 5-74 5-04 2X 14.9 13.8 12. 1 10.7 9-30 8.74 7.89 7.19 6.60 6.05 5-32 15.6 14.5 12.7 II. 2 9-75 9-17 8.29 7.56 6.94 6.36 5-59 2>f I6. 3 13-3 II.7 10.2 \ 9.60 8.68 7.92 7.27 6.67 5.86 2% 17.0 15.8 13-9 12.2 10.7 10.0 9.07 8.28 7.60 6.97 6.14 17.7 16.4 14-5 12.8 II. I 10.5 9-47 8.64 7-94 7.28 6.41 2,7/% 18.4 17.1 15.0 *3'3 11.57 [ 10.9 9.86 9.00 8.27 7-59 6.68 3 I9.I 17.7 I 5 .6 13-8 12. j "-3 10.3 9-3 6 8.61 7.90 6-95 3X 20.4 19.0 16.8 14.8 12.9 12.2 ii. i 10. 1 9-27 8.52 7-50 21.8 20.3 17.9 15-9 13-9 I 3- * ii. 8 10.8 9-94 9-13 8.04 3K 23.2 21.6 19.1 16.9 I 4 .8 13-9 12.6 11.5 10.6 9-75 8-59 4 24.6 22.9 20.2 17.9 15-7 14.8 13-4 12.3 II-3 10.4 9-13 4# 25-9 24.2 21.4 19.0 16.6 15.6 14.2 13.0 12. II. 9.67 4/i 27-3 25-4 22.5 20. o 17-5 16.5 15.0 13-7 12.7 ii. 6 IO.2 4^ 28.7 26.7 23-7 21.0 18.4 17.4 15.8 14.4 13-3 12.2 10.8 5 30.1 28.0 24-8 22.1 19-3 18.2 16.6 i5-i 14.0 12.8 "3 s# 3 J -5 29-3 26.0 23.1 20. 2 19.1 17-3 15-9 I 4 .6 13-5 11.9 sK 32*8 30.6 27. I 24.1 21. 1 20.0 18.1 16.6 I4.I 12.4 5^ 34-2 31-9 28.3 25-2 22.1 20.8 18.9 17-3 16.0 14.7 12.9 6 35-6 33-2 29-5 26.2 23.0 21.7 19.7 18.0 16.6 15-3 13-5 WEIGHT OF COPPER PIPES AND CYLINDERS. Table No 94 (continued}. LENGTH, i FOOT. Thickness by Holtzapffel's Wire- Gauge (Table No. 13). Specific Weight- 1. 16 (Specific Weight of Wrought Iron = i). 26 3 THICK- NESS. 8 9 10 ii 12 13 14 15 16 17 18 19 20 W. G. INCH. ix 1 ? 5 * .148 134 9/64-5 .120 .109 7/6 4 095 .083 5/6 4 / .072 .065 .058 .049 3/6 4 / .042 3/64 ^ 035 INT. DlAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. inches. ft .581 .491 .422 357 .310 .254 .210 173 .150 .129 .IO4 .086 .068 1 832 1.08 .716 .941 .626 .830 540 .722 .476 .641 543 .282 391 249 .348 .217 305 .178 253 .149 .213 .121 175 i-33 1.17 1.03 .904 .807 .687 .588 .500 447 393 .327 .277 .228 8 1.58 i-39 1.62 1.24 1.44 1.09 1.27 .972 I.I4 .831 975 .714 .840 .610 .719 .644 .481 570 .402 .476 341 .281 334 y& 2.09 1.84 1.65 i-45 1.30 1. 12 .966 .828 743 .658 550 .468 .387 i 2-34 2.05 1.85 1.63 1.47 1.26 1.09 .938 .842 .746 .625 .532 .440 1/8 2.59 2.27 2.05 1.82 1.63 I.4I .22 05 .940 834 .699 .596 493 2.84 2.49 2.26 2.00 1. 80 i-55 34 .16 .04 .922 774 659 547 j3^ 3-9 2.72 2.46 2.18 1.97 1.70 47 27 .14 .01 .848 723 .600 i*5 3-34 2-94 2.67 2.36 2.13 1.84 .60 .38 .24 .10 .922 .787 653 i# 3-59 3-17 2.87 2-55 2.30 1.99 .72 .48 34 .19 997 .851 .706 ;| 3-84 4.09 3-39 3-62 3-07 3-28 2-73 2.91 2.46 2.63 2.13 2.27 85 97 59 .70 43 53 .27 36 .07 15 915 .978 759 .812 2 4-34 3-84 3-48 3-09 2.79 2.42 2. IO .81 63 45 .22 1.04 .865 2^ 4-59 4.07 3.69 3-27 2. 9 6 2.56 2.23 1.92 73 54 2 9 .11 .919 2^ 4.84 4.29 3*89 3-45 3.12 2.71 2-35 2.03 83 63 38 17 .972 2^ 5-09 4-52 4.09 3-64 3-29 2.85 2. 4 8 2.14 93 7i 45 23 03 2> 5-34 4-74 4-30 3-82 3-45 3.00 2.60 2.25 2.03 .80 53 30 .08 2 ^ 5-59 4-97 4-5 4.00 3-62 2-73 2.36 2.13 .89 .60 .36 .13 2 ^ 5-19 4.71 4.18 3-79 3'-28 2.86 2.47 2.22 98 .68 43 .18 2% 6.09 5-42 4.91 4-37 3-95 3-43 2.98 2.58 2.32 2.07 75 49 .24 3 6-34 5.66 5-" 4-55 4.12 3-57 3-n 2.69 2. 4 2 2.16 .82 55 29 3/< 6.85 6.ii 5-52 4.91 4-45 3-86 3-36 2.91 2.62 2-33 1.96 .68 .40 3X 7-35 6.56 5-93 5.28 4-78 4-15 3-6i 3.12 2.82 2.51 2. II .81 5 1 3^ 7.85 7.01 6-33 5-64 5-" 4-44 3-87 3-34 3.01 2.68 2.26 94 .62 4 8.35 7.46 6.74 6.01 5-44 4-73 4.12 3-56 3-21 2.86 2. 4 I 2.06 73 4X 8.85 7.91 7.14 6-37 5-77 5.02 4-37 3-78 3.41 3-04 2.56 2.19 84 4/^ 9-35 8.36 7.55 6.74 6. 10 5-30 4.62 4.00 3.61 3.21 2.71 2.32 94 4* 9-85 8.81 7.96 7.10 6-43 5-59 4.87 4.22 3.80 3-39 2.86 2-45 2.05 5 10.4 9.26 8.36 7.46 6.77 5.88 4-44 4.00 3-56 3.01 2-57 2.16 5X 10.9 9.71 8-77 7.83 7.10 6. 17 5-38 4.66 4.20 3-74 3-15 2.70 2.27 50 11.4 11.9 10.2 10.6 9.18 9.58 8.19 8.56 6.46 6-75 & 4.88 5-09 4.40 4-59 3-92 4.09 3-30 3-45 2.83 2.96 2.38 2.48 6 12.4 II. I 9-99 8.92 I'.lg 7.04 6.14 5-31 4-79 4.27 3-6o 3-09 2.58 264 WEIGHT OF METALS. Table No. 94 (continued). LENGTH, i FOOT. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). Specific Weight^ 1. 16 (Specific Weight of Wrought- Iron = i). THICK- NESS. W. G. oooo ooo 00 o i 2 3 4 5 6 7 454 .425 .380 340 .300 .28 4 259 ^238 .220 203 .180 INCH. 26 /6 4 Hf- 11/32 T 9/6 4 / k/ 7/3* / 13/64 3/i6 b. INT. DlAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. inches. & l /z 38.4 35-8 31-8 28.3 21.3 19-5 18.0 14.6 7 41. 1 38.3 34-i 30.3 26.6 25-1 22.8 20.9 19-3 17.8 15.7 43-9 40.9 36-4 32.4 28.4 22.4 20.6 16.8 8 2 46.6 43-5 38.7 34-5 30.3 28.6 26.0 23-8 22.O 20.2 17.9 9 52.1 48.7 43-3 38.6 33-9 32.0 29.1 26.7 24.6 22.7 20.1 10 57-7 53-8 47-9 42.7 37-5 35-5 32.3 29.6 27-3 25-2 22.2 ii 63-2 59-o 52.5 46.8 41.2 38.9 35-4 32.5 30.0 27-7 24.4 12 68.7 64.2 57-2 51.0 44-8 42.4 38.6 35-4 32.7 30.1 26.6 13 74-2 69-3 61.8 55.1 48.5 45-8 41.7 38.3 35-3 32-6 28.8 14 79-7 74-5 66.4 59-2 52.1 49-3 44-9 41.2 38.0 35- i 31.0 15 85.2 79-6 71.0 63-4 55-8 52-7 48.0 44.1 40.7 37-6 33-2 16 90.7 84.8 75-6 67-7 59-4 56.2 5 1 - 2 46.9 43-4 40.0 35-4 17 96.3 90.0 80.2 7 !.8 63.0 59-6 54-3 49-8 46.0 42.5 37-5 18 101.8 95- i 84.9 76.0 66.7 63. i 57-4 52.7 48.7 45-o 39-7 19 107.3 100.3 89.5 80. i 70.3 66.5 60.6 55-6 51.4 47-4 41.9 20 II2.8 105.5 94.1 84.2 74-o 70.0 63-7 58.5 54-o 49-9 44.1 21 22 23 118.3 123.8 129.3 110.7 115.8 120.9 98.7 103.3 107.9 88.3 92.5 96.6 77-6 81.3 84-9 73-4 76.9 80.3 66.9 70.0 73-2 61.4 64-3 67.2 56.7 a? 52.4 54-9 57-3 46.3 48.5 50.7 24 134.8 126.1 II2.6 100.6 88.6 83-8 76.3 70.1 6 4 .7 59-8 52.9 26 146.0 136.4 121. 8 1 08. 8 95-9 90.7 82.6 75-9 70.1 64.7 57-2 28 157.2 146.7 131.0 117.1 103.1 97-6 89.0 81.7 75-4 69.7 61.6 30 168.4 157.1 140.2 125.4 110.4 104.5 95-3 87.5 80.8 74-6 66.0 32 179-6 167.4 149.5 133-6 117.7 111.4 101.6 93-3 86.2 79-6 70.4 34 190.7 177.7 141.9 125.0 118.3 107.9 99-i 91-5 84-5 74-7 36 201.9 1 88.0 167.9 150.1 132-3 125.2 114.2 104.9 96.9 89.5 79.1 WEIGHT OF COPPER PIPES AND CYLINDERS. Table No. 94 (continued}. LENGTH, I FOOT. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i). 265 THICK NESS. W. G. 8 9 10 ii 12 13 M 15 16 17 18 19 20 INCH. ^165 9 l f 134 9/6 4 ^ .120 .109 7/6 4 095 .083 S/64/ 072 5/6 4 ^ .065 .058 .049 .042 .035 INT. DIAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. inches. 6>^ 13-4 12.0 10.8 9.65 8.75 7.61 6.64 5-75 5.19 4.62 3-90 3-34 2.8o 7 14.4 12.9 u.6 10.4 9.42 8.19 7.14 6.19 5.58 4.97 4-20 3-6o 3-oi 7/^ 15-4 13-8 12.47 II. I 10. 1 8.77 7.65 6.63 5.98 5-33 4.49 3.85 3-23 8 16.4 14.7 13.2 11.8 10.74 9-34 8.1 5 7.06 6.37 5.68 4-79 4.10 3-43 9 18.4 I6. 5 14.9 13-3 12. 1 10.5 9.16 7-94 7.16 6.38 5-39 4.61 3-86 10 20.4 18.2 16.5 14.8 13-4 11.7 10.2 8.81 7-95 7.08 5.12 4.28 ii 22.4 20.0 18.1 16.2 14.7 12.8 II. 2 9.69 8.74 7-79 6 58 5.63 4.70 12 24.4 21.8 19.8 17.7 16.0 14.0 12.2 10.6 9.53 8.49 7.18 6.14 5.13 I 3 26.4 23-6 21.4 19.1 17.4 15.1 13.2 11.4 10.3 9.20 7-77 6.65 5-55 H 28.4 25.4 23.0 20. 6 18.7 16.3 14.2 12.3 ii. i 9.90 8-37 7.16 5.98 15 16 30-4 32.4 27.2 29.0 24.6 26.3 22.1 23-5 20.0 21-3 17.4 18.6 15-2 16.2 13.2 14.1 11.9 12.7 10.6 8.96 9.56 6.40 6.82 17 34-4 30.8 27.9 25.0 22.7 19.7 17.2 14.9 13-5 12. 1 10.2 8.69 7-27 18 36.4 32-6 29-5 26. 4 24.0 20.9 18.2 14-3 12.7 10.7 9.20 7.69 19 38.4 34-4 31.2 27.9 25.3 22.0 19.2 16.7 I5-I 13-4 "3 9.71 8.12 20 40.4 36-2 32.8 29.3 26.6 23.2 20.2 17.6 15-9 I4.I 11.9 10.2 8.54 21 42.4 38.0 34-4 30.8 27-9 24-3 21.3 18.4 16.6 14.8 12.5 10.7 8.96 22 44-4 39-8 36.0 32.3 29-3 25-5 22.3 19-3 17.4 15-5 II. 2 9-39 23 46.4 41.6 37.7 33.7 30.6 26.7 23.3 20.2 18.2 16.2 13-7 ii. 8 9.81 24 48.5 43-4 39-3 35-2 31-9 2 7 .8 24-3 21. 1 19.0 16.9 14.3 12.3 10.2 26 52.5 47.0 42.6 38.1 34-6 30.1 26.3 22.8 20.6 18.4 15.5 13.3 II. I 28 56.5 50.6 45-8 41.0 37-2 32.4 28.3 24.6 22.2 19.8 16.7 14-3 II.9 30 60.5 54-2 49.1 43-9 39-9 34-7 30-3 26.3 23-7 21.2 17.9 15-3 12.8 32 64-5 57-8 52.3 46.8 42-5 37-0 32.3 28.1 25.3 22.6 19.1 16.3 13.6 34 68.5 61.4 55-6 49.8 45-i 39-4 34-4 29.8 26.9 24.0 20.3 17.4 14.5 36 72.5 65.0 58.8 52.7 47-8 41.7 36.4 31.6 28.5 25.4 21.5 18.4 15.3 266 WEIGHT OF METALS. Table No. 95. WEIGHT OF BRASS TUBES, BY EXTERNAL DIAMETER. LENGTH, I FOOT. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). Specific Weight = 1.1 1 (Specific Weight of Wrought Iron=i). THICK- NESS. W. G. 15 16 17 18 iQ 20 21 22 23 24 25 INCH. .072 5/6 4 ^ .065 /*/ .058 1/16 b. .049 3/6 4 / .042 3/6 4 b. 035 '/32/ .032 '/3 .028 i/ 3a b. .025 i.6/6 4 .022 I -4/6 4 .020 1-3/64 DlAM. Ibs. Ibs. lb. lb. lb. lb. lb. lb. lb. lb. lb. inches. # 037 35 .031 .029 .026 .024 3/i6 .087 .079 .072 .062 058 .052 .047 .042 039 K .130 "5 .IO2 .088 .081 .072 .065 .058 053 5/i6 .201 .I8 7 .172 .150 .132 "3 .104 .092 .083 .074 .068 H 255 234 .214 .186 163 .138 .128 H3 .102 .090 .082 7/16 .306 .281 .256 .221 193 .164 151 133 .120 .106 .097 1 A 358 .329 .298 257 .224 .189 .174 154 .138 .122 .III 9/i6 .411 376 340 293 254 215 .197 .174 .156 .138 .126 K .463 .423 .382 .328 .285 .240 .221 .194 .174 154 .141 W/i6 tf 515 .567 .470 5!7 .424 .467 .364 399 346 .265 .291 .244 .267 215 235 .192 .211 .170 .186 155 .170 J 3/T6 .620 564 .509 435 376 -316 .290 255 229 .202 .184 % .6 7 2 .611 55 1 .471 .407 342 .314 .276 .247 .218 .199 J 5/i6 .724 .658 593 .506 437 .367 337 .296 .265 234 .213 I 777 .706 635 542 .468 393 360 .316 283 .250 .228 Itf .881 .801 .719 613 529 443 .407 357 320 .282 257 I* Ifc .986 1.09 .896 .991 .804 .888 .684 755 59 .651 494 545 453 .500 .398 439 356 392 346 .286 3*5 iX 1.20 1.09 .972 .827 .712 596 .546 479 429 378 344 W. G. 9 IO ii 12 13 14 15 16 17 18 19 INCH. .148 9/6 4 / 9/6 4 b. .120 #* .109 7/6 4 .095 3/3/ 083 5/6 4 / .072 S/6 4 b. .065 '/*/ .058 1/16 b. .049 3/64/ .042 3/6 4 b. DlAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. inches. I# 1.90 1.74 I. 5 8 1-45 1.28 13 .986 .896 .804 .684 590 1/8 2. 1 1 1-93 1.76 1. 60 1.41 25 .991 . 99 I .888 755 .651 I# 2-33 2.13 1.94 1.76 i-55 37 1.20 1.09 972 i 2 l .712 iH 2-54 2.32 2.12 1.92 1.69 49 1.30 1.18 i. 06 .898 773 i# 2. 7 6 2.52 2.30 2.08 1.83 .61 1.40 .28 .14 .969 834 i# 2.97 2.71 2.47 2.24 1.97 73 I-5I 37 23 .04 895 2 3-19 2.91 2.6 5 2.39 2. IO 85 1.61 47 3i .11 95 6 2^ 3-40 3.10 2.8 3 2.55 2.24 97 1.72 56 39 .18 1.02 , 2X 3-62 3-30 3.01 2.71 2.38 2.09 1.82 .66 .48 25 1. 08 2^ 3.83 3-49 3-19 2.86 2.52 2.21 i-93 75 56 33 I.I4 2^ 4.04 3-69 3-37 3.02 2.66 2-33 2.03 85 .65 .40 1.20 WEIGHT OF BRASS TUBES. 267 Table No. 95 (continued). LENGTH, I FOOT. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). Specific Weight 1. 1 1 (Specific Weight of Wrought Iron=i). THICK- NESS. W. G. 3 4 5 6 7 8 9 10 ii 12 13 INCH. Tf. 238 <5/6 4 / .220 .203 13/64 .180 3/i6 b. ii ^b .148 .134 9/64 b. .120 .109 7/6 4 .095 3/32 DIAM. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. inches. 2 5.24 4.87 4-55 4.24 3-80 3-52 3-19 2.91 2.65 2-39 2.10 2% 5-62 5-22 4.87 4-54 4.07 3-76 3-40 3.10 2.83 2-55 2.24 2 % 5-99 5-57 5.19 4-83 4-33 4.00 3-62 3-30 3.01 2.71 2.38 2-y& 6-37 5-5i 5-13 4-59 4.24 3.83 3-49 3-19 2.86 2.52 2/2 6-75 6.26 5-83 5-42 4.85 4.48 4.04 3.69 3-37 3.02 2.66 2% 7.12 6.60 6.14 5-72 5-12 4.72 4.26 3-88 3-55 3.18 2.79 2 7-5 6-95 6-47 6.01 5.38 4.96 4-47 4.07 3-73 3-34 2-93 7.88 7-30 6.79 6.31 5-64 5.20 4.69 4.27 3-9i 3-50 3-07 3 8.25 7.64 7.11 6.60 5.90 5-44 4.90 4-46 4.09 3-66 3-21 3X 9.01 8-33 7-75 7-19 6.43 5-92 5-49 4-85 4-43 3.98 3.48 3# 9.76 10.5 9.02 9.72 8-39 9-03 7.78 8-37 6-95 7-47 688 6.07 6.65 5-24 5-63 4.78 5.12 4-30 4.61 3.76 4.04 4 ii. 3 10.4 9-67 8.96 8.00 7^6 7.24 6. 02 5-46 4-93 4.31 4X 12.0 11. i 10.3 9-55 8.52 7-83 7.82 6.41 5.80 5-25 4-59 4-^2 12.8 ii. 8 10.9 10. 1 9.04 8.31 8.41 6.80 6.15 4.87 4^ 13-5 12.5 ii. 6 10.7 9.56 8-79 8-99 7.19 6.49 5! 88 5 14-3 13.2 12.2 "3 10. 1 9.27 9-57 7-58 6.83 6. 20 5-42 5X 15.0 13-9 I2. 9 11.9 10.6 9-75 IO.2 7-97 7.17 6.51 5-69 5 JJ 203 SD C 15 x ii 200 168 SDX )) )> 189 SDXX )) 210 SDXXX ,, J; 2 3 I S D XXXX jj 252 DC I 7 X I2# IOO 9 8 DX 5> 5J J? 126 DXX >> M 147 DXXX )> )) J) 168 DXXXX J> J) JJ 169 WEIGHT OF TIN AND LEAD PIPES. 269 Table No. 98. WEIGHT OF TIN PIPES. As manufactured. I FOOT IN LENGTH. Diameter THICKNESS. Diameter THICKNESS. Externally 3/32" inch. J^ inch. X inch. inches. Ibs. Ibs. inches. Ibs. * .148 *% 5-4 % .384 .472 *% 5.67 ti .620 .78 7 m 6.30 I .856 T.I03 3 6 -93 & 1.095 I.4I7 i l A 7.56 5# 1.328 1.732 3^ 8.19 & 1.564 2.047 2 1. 802 2.362 Table No. 99. WEIGHT OF LEAD PIPES. As manufactured. WEIGHT AND THICKNESS OF PIPE. Bore. Length. Calcu- Calcu- Calcu- Calcu- Weight. lated Thick- Weight. lated Thick- Weight. lated Thick- Weight. lated Thick- ness. ness. ness. ness. inches. feet. Ibs. inch. Ibs. inch. Ibs. inch. Ibs. inch. H 15 14 .097 16 .112 18 .124 22 .146 y% 5) 17 .101 21 .121 30 .140 N 24 .112 28 .147 32 .l8l 36 215 i 5) 36 .136 42 .156 56 .2OO 64 .225 ijf 12 36 139 42 .l6o 4 8 .l8o S 2 193 T-Yz 4 8 .I 5 6 56 .179 72 .224 8 4 .257 ify 72 .199 8 4 .228 9 6 .256 2 jj 72 .I 7 8 84 .204 9 6 .231 112 .266 2^ IO 84 .200 9 6 .227 112 .26l 3 112 .218 140 275 3/2 I 3 225 160 273 4 ,, 170 257 22O 327 4/2 170 .232 220 295 4?/ | s/ l6 inch thick. Weight per lineal foot 22.04 Ibs. 4/ 23.25 A 3/i ,, }) 2/1. /1C .. 5 2* ( 36 * j'^ 2/0 WEIGHT OF METALS. Table No. 100. ENGLISH ZINC GAUGE. (London Zinc Mills.} Ap- 7 Ft. X 2 Ft. 8 Ins. 7 Feet X 3 Feet. 8 Feet X 3 Feet. 8 Gauge No proxi- mate Weight per Sq. Foot. i,oooths of an Inch. Approximate ,,, , Number asLjSss Appro Weight Der Sheet. ximate Number of Sheets in loCwts. Appro Weight per Sheet. ximate Number of Sheets in icCwts. Gauge No Nearest Birmingha Wire Gau ozs. Ibs. ozs. Ibs. ozs. Ibs. ozs. I *x .004 2 IO 427 I 41 2 3% .006 3 J 3 294 2 38 3 3% .007 4 IS 227 3 37 4 4 3 A .008 6 4 180 4 34 5 s% .010 7 9 148 5 3 1 6 6% .Oil 7 14 142 8 14 126 10 2 Ill 6 30 7 7 3 /4 .013 9 i 124 10 3 no II IO 96 7 29 8 9 .015 10 8 107 ii 13 95 13 8 83 8 28 9 10 .017 ii ii 96 13 2 85 15 75 9 27 10 ntf .019 13 7 8 3 !5 2 74 17 4 65 IO 25 ii 13 .021 15 3 74 I 7 I 66 19 8 57 ii 24 12 15 .025 17 8 64 19 II 57 22 8 5 12 23 13 17 .028 22 5 50 25 8 44 13 22 J 4 *9 .031 24 15 45 28 8 39 J 4 21 15 22 .036 28 14 39 33 o 34 15 20 16 2 5 .041 3 2 13 34 37 8 30 16 19 17 28 .046 36 12 30 42 o 27 17 18 18 31 .051 40 II 28 46 8 24 18 J 9 35 059 45 15 24 52 8 21 19 17 20 39 .065 5i 3 22 58 8 r 9 20 16 21 43 .072 56 7 20 64 8 *7 21 15 Sheets thicker than above are rolled to Birmingham Wire Gauge Table No. IOOA. "VM" ZINC GAUGE. (Vieille-Montagne). Approximate Thickness. Approximate Weight per Sqi are Foot. 36 Ins. X 72 Ins. 36 Ins. X 84 Ins. 36 Ins. X 96 Ins. Gauge. Approximate Weight of Sheets. 1> M "hV Q. J-. c Approximate Weight of Sheets. &?* s.5! Approximate Weight of Sheets. >? =5, D.I. c Thou- , 3 '3 5 * aj sandths W yj-4) <" W < W w of an .H*~*3.5 flj O " JJlrf V Q Inch. HP Lbs. Ozs. Drms. Lbs. Ozs. Drms. Is 1 Lbs. Ozs. Drms. r3j3 Lbs. Ozs. Drs. !- about about about i 0.004 0. 100 _ 2 5 > Nos. i and 2 are only rolled to order and special dimensions. 2 .006 .141 3 4 1 3 .007 .171 3 15 4 6 14 249 5 ii 213 5 14 8 187 4 .008 .209 4 13 5 6 IO 204 6 I 175 3 8 153 5 .010 .247 5 ii 6 6 6 172 7 7 J 47 8 8 8 129 6 .Oil .291 6 ii 7 8 6 146 8 i 7 126 10 8 no 7 .013 337 7 12 8 ii 8 126 10 12 108 ii IO 95 8 .015 .386 8 14 9 15 12 no ii i 6 95 13 5 83 9 .018 450 10 5 ii 9 10 95 13 8 9 81 15 7 8 71 10 .020 .500 ii 7 12 14 86 15 3 73 17 2 8 64 ii .023 -580 13 5 X 4 15 10 74 T-7 7 9 63 19 15 8 55 12 .026 .660 15 2 17 o 4 65 19 I 3 10 56 22 II 49 13 .029 .740 I 15 19 14 57 22 3 ii 50 25 6 8 43 14 .032 .820 I 2 12 21 I 8 S 2 24 9 12 45 28 2 39 15 .038 95 I 5 12 24 7 8 45 28 8 12 39 3 2 10 34 16 043 i. 080 I 8 12 2 7 13 8 39 3 2 7 12 34 37 2 30 17 .048 I.2IO I ii II 31 2 6 35 36 5 7 30 8 8 27 18 053 1.340 i 14 II 34 8 6 40 4 7 27 46 8 24 19 .058 1.470 2 i II 37 I 4 6 29 44 3 7 25 So 8 8 22 20 063 I. 600 2 4 IO 4 1 3 4 27 48 i 2 23 54 15 20 21 .O7O 1.780 2 8 12 45 13 8 24 53 7 12 21 61 2 18 22 .077 1.960 2 12 14 50 7 2 22 58 6 19 67 5 16 23 .084 2.I4O 3 I I 55 3 2 2O 64 6 5 73 9 8 15 24 .091 2.320 3 5 3 59 13 16 18 69 12 11 79 12 8 14 25 .098 2.500 3 9 5 64 7 IO 17 75 3 9 15 85 15 8 13 26 .105 2.680 3 7 69 i M 16 80 10 3 92 8 12 FUNDAMENTAL MECHANICAL PRINCIPLES. FORCES IN EQUILIBRIUM. SOLID BODIES. Parallelogram of Forces. When a body remains at rest whilst being acted on by two or more forces, it is said to be in a state of equilibrium, and so also are the forces. Thus, if the forces P/, Q ^, R r, Fig. 86, acting on the body p q r, keep it at rest, they are in equilibrium, and any two of them balance the third. The lines of force, if produced, meet at one point o within the body, and if a parallel- R ogram be constructed having two adjacent sides proportional to and parallel tO tWO Of the forces respec- Fig. 86. Equilibrium of Forces. tively, to represent them in magni- tude and direction, the diagonal of the parallelogram will represent the third force in magnitude and direction. Let the lines OP, OQ, Fig. 87, represent the forces P/, Q^ in magni- tude and direction, and com- _p plete the parallelogram by drawing the parallels p R, Q R, iv _ ^ and draw o R. Then o R re- ^*x^ presents in magnitude and direction the resultant of the tWO forces; and RO taken in Fig. 8 7 .-Parallelogra m of Forces. the opposite direction repre- sents the third force Rr, Fig. 86. If it be applied in this direction to the point o, as indicated by a dot line o R', it would balance the other two. This construction is called the parallelogram of forces, and is applicable to any three forces in equilibrium. Three forces in equilibrium may also be represented by a triangle, or half a parallelogram. For example, the triangle OPR represents by its three sides the forces o P, o Q, o R, the side P R being substituted for o Q. Three forces in equilibrium must be in the same plane. When the directions of three forces holding a body at rest, and also the magnitude of one of them, are known, the magnitudes of the other two can be determined by constructing a parallelogram in the manner above exem- plified, and measuring the lengths of the sides and the diagonal. 2/2 FUNDAMENTAL MECHANICAL PRINCIPLES. Polygon of Forces. Equilibrium may subsist between more than three forces, which need not necessarily be in the same plane; and they can, like those already illustrated, be developed in direction and magnitude by diagram. Thus, let the point o, Fig. 88, representing a solid body, be kept at rest by a number of forces, OP, o Q, o R, o s, o T. Find the equivalent diagonal o/ for the first two forces ; then construct the parallelogram and diagonal o r for the resultant of o/ and the third force OR; and again the parallelogram and diagonal o s for the resultant of o r and the fourth force o s. The last resultant o s represents in one the four distributed forces OP, OQ, OR, os, and it balances the fifth force o T equal and opposite to it. As o s and o T are in the same straight line, their resultant is of course nothing. The several forces thus dealt with may be combined into a polygon of forces. Draw o P, Fig. 89, parallel and equal to o P, Fig. 88, p Q parallel and equal to o Q, Q R parallel to o R, R s parallel to os; then, finally, so, completing the polygon, will be parallel and equal to OT, Fig. 88, the last of the series. Professor Mosely illustrates the polygon of forces by the united action of a number of bell-ringers, pulling by a number of ropes joined to a single rope. The polygon constructed as in Fig. 90, shows successively by corresponding letters, the individual contributions of the bell-ringers, combined into one vertical force. Again, equilibrium may be estab- lished between a number of forces acting in the same plane, applied to different points in a body, or system of bodies. For example, let the forces P o, Q o, R o, s o, and T o, be applied to several points, o, o, o, o, o, on a flat board ABC, Fig. 91, by means of cords and weights; it will settle into a position of equilibrium, when the opposing forces arrive at a balance between themselves. An axis or pivot may be established at any point, M, on the surface of the board, without disturbing the equilib- rium, and it may be viewed as a centre of motion round which the forces tend to turn the board, some in one direction, the others the opposite way, balancing each other. The effect of each force to turn the body about the centre is represented by the product of its magnitude by the leverage, or perpendicular distance of its direction from the centre; draw these perpendiculars, and multiply each force by its perpendicular or leverage, then the resulting products will be divisible into two sets, tending to turn the board in opposite directions. The sum of the first set of products is equal to the sum of the second set, as is proved by the fact of equilibrium. Moments of Forces. The product of a force by the perpendicular dis- tance of its direction from any given point, is called the moment of the Fig. 88. Equilibrium of more than Three Forces. Fig. 89. Polygon of Forces. FORCES IN EQUILIBRIUM. 273 Fig. 90. Bell-ringers] Polygon of Forces. force about that point; and the equilibrium above discussed, in connection with Fig. 91, is the result of the equality of moments. The law of the equality of moments may be thus set forth: If any number of forces acting anywhere in the same plane, on the same body or connected system of bodies, be in equilibrium, then the sum of the forces tending to turn the system in one direction about any point in that plane, is equal to the sum of the mo- ments of those forces tending to turn the system in the other direction. Such balanced forces may be transferred to a single point, and placed about it, as in Fig. 88, parallel to their directions as they stand; and they will continue in equilibrium, holding the point at rest. A polygon of the forces p q r s t within Fig. 90, may be constructed similarly to Fig. 89. Though the principle of the polygon of forces be sufficient to test the equilibrium of a system of forces acting at one point, yet the principle of the equality of moments, in addition, is necessary to establish the equilibrium of a system applied to different points. The two principles conjointly are necessary, and they are sufficient, as conditions of equilibrium. The Catenary. When a chain, or a rope, or a flexible series of rods, is suspended by its extremi- ties, supporting weights distributed along its length, in a state of rest, it forms a polygon of forces in equilibrium, as in Fig. 92. If all the forces except those which act on the extremities of the chain, be combined into a resultant, then the two extreme sides being produced, will meet the direction of the resultant at one point. Thus, in the polygon, Fig. 92, loaded with weights, w, w, &c., the vertical resultant of these weights w' w", passing through their common centre of gravity, intersects at w" the two extreme sections PA, P'B, when produced downwards. Similarly, in the catenary, Fig. 93, which is the curve assumed by a rope or other flexible medium uniformly loaded and suspended by the two 18 Fig. 91. Equality of Moments. FUNDAMENTAL MECHANICAL PRINCIPLES. extremities, if tangents be drawn to the extremities A, B, of the curve, meeting at w", they represent the directions of the forces sustaining the curve at Fig. 92. The Catenary. Fig. 93. The Catenary. those points, and they intersect at the same point w", the vertical line G w" passing through the centre of gravity of the curve. Let the weight of the Fig. 94. Centrifugal Forces in Equilibrium. Fig. 95. Parallel Forces in Equilibrium. curve be represented by G w", and complete the parallelogram M N, then w" M and w" N represent in force and direction the tension at the points B and A. Centrifugal Forces in Equilibrium. When a cylindrical vessel is exposed to a uniform internal pressure, as the pressure of steam within a boiler, for example, the pressure is balanced by the resistance of cohesion of the material of the boiler. Let A B c D, Fig. 94, be the section of a cylindrical boiler, the radial pressure of the steam may be represented by the arrows, which are equal and opposite in direction. The tension on the metal in resisting the internal pressure at any particular section B D, is equal to the sum of the pressures resolved into the direction at right angles to B D, or parallel to AC, according to the triangles, or half-parallelograms of force attached to each oblique arrow. The total vertical pressure thus obtained by the resolution of forces is equal to the total vertical pressure which FORCES IN EQUILIBRIUM. 275 would be exerted on the sectional line B D if it be supposed to be a rigid diaphragm across the boiler, which is easily calculated. If the radial pressure be, for example, 100 Ibs. on each square inch of surface, then the total pressure, or the tension on the two sides at B and D, would be TOO x BD on each inch of length of the two sides; that is to say, if the diameter B D be equal to 60 inches, the tension on the two sides would be 60 x 100 = 6000 pounds for each inch of length. A similar argument applies to the tension on the rim of a revolving fly- wheel. Parallel Forces. Systems of parallel forces are particular cases of the foregoing. Let A, B, c, D, E, F, Fig. 95, be a system of parallel forces in equili- brium; and MN a line perpendicular to them in the same plane, and cut by them at the points a, b, c, d, e, f. They may act at any points in their lines of direction without disturbing the equilibrium, and they may be sup- posed to be applied at those points in the line M N. Then, the sum of the moments of the three forces A, B, c, acting in one direction, with respect to any point M as a centre, is equal to the sum of the moments of the forces D, E, F, opposed to them. Further, the sum of the simple forces A, B, c, irrespective of their moments, is equal to that of the forces D, E, F, so that the fact of their being in equilibrium resolves itself into a case of action and reaction, for the two equivalent forces representing the two opposing sums, act in the same straight line in opposite directions. When three parallel forces balance each other, acting on a straight line, two of them must be opposed to the third; and the third must act between the other two, being equal and opposite to their resultant. Let A, B, c, Fig. 96, be three such forces applied to the line E G F, at the points Fi s- 96. Three Parallel Forces in E,G,F respectively; then, with respect to the point G, the moment of the force B is nothing, because it passes through that point and has no leverage on it. There remain the moments of the extreme forces, A and c, which are equal to each other, that is to say AXEG = CXFG. From this it follows, by proportion, that A : c : : F G : E G, and that the extreme forces are to each other inversely as their distances from the middle force. In general, of three parallel forces acting in equilibrium on an inflexible line, the first in order is to the third as the distance of the third from the second, is to that of the first from the second. The sum of the first and third is equal to the second; and when the distances or leverages are equal, the first and third forces are equal to each other. If the position of the line E F be inclined to the direction of the three forces, and changed to E' F', Fig. 96, the forces A, B, c, continue in equilibrium; 276 FUNDAMENTAL MECHANICAL PRINCIPLES. Fig. 97. Parallelepiped of Forces. for the perpendicular lines G E and G F continue, as before, to be the lever- ages of the extreme forces A and c, on the central point G. When three forces not in the same plane act on one point, there cannot be equilibrium between them. Two of these may be reduced to their resultant by parallelogram, and this resultant reduced with the third force to a final resultant For example, let the lines o P, OQ, OR, Fig. 97, represent in magnitude and direction three forces not in one plane acting on the point o. By parallel- ogram, o s is the resultant of the two forces o P, o Q, and o T is the final resultant of o s and the third force o R. That is to say, o T is the resultant of the three given forces. If parallelograms be formed from each two of the three forces, they form, when duplicated, a parallelepiped of forces, of which the diagonal is found by the final resultant o T, and the principle of the parallelepiped of forces may be thus defined: If three forces be represented in magnitude and direction by three adjacent edges of a parallelepiped, their resultant is represented in magnitude and direction by the adjacent diagonal of the solid. There must be at least four forces to produce equilibrium about a point, when the forces are not in the same plane. The triangle OST, Fig. 97, comprises in its three sides the resultant of the first and second forces, the third force, and the resultant of the three. If the first resultant o s be replaced by the two lines o Q and Q s, which represent the first and second forces, they form the four-sided figure o Q s T, the polygon of the four equilibrating forces. A greater number of forces than four acting on a point may be reduced in like manner. FLUID BODIES. The characteristic property of fluids is the capability of transmitting the pressure which is exerted upon a part of the surface of the fluid, in all directions, and of the same intensity: the same pressure per square inch or per square foot. The pressure of water in a vessel, caused by its own gravity, increases in proportion to the depth below the surface; and the pressure on a horizontal surface, say, the bottom, is equivalent to the weight of the superincumbent column of water, and the intensity of the pressure is independent of the form of the vessel. The same rule applies when the pressure is from below upwards. The same rule also applies when the surface is either vertical or inclined, and the mean height of the superincumbent column of water is measured by the depth of the centre of gravity of the given surface below the surface of the water. The water in open tubes communicating with each other, when in a state of equilibrium, stands at the same level in the tubes, whatever may be the relative diameters of the tubes. MOTION. GRAVITY. 277 The height of the superincumbent column of water is called the head of water. The buoyancy, or the upward force with which water presses a body immersed in it, from below upwards, is equal to the weight of water dis- placed by the body, or of a quantity of water equal in volume to the sub- merged body, or submerged portion of a body. The resultant of the upward pressure passes through the centre of gravity of the water displaced; and also, when the floating body is at rest, through the centre of gravity of the body. This resultant line is called the axis of floatation, and the horizontal section of the body at the surface of the water is the plane of floatation. Bodies float either in an upright position or in an oblique position. A body floats with stability, when it strives to maintain the position of equili- brium, and when it can only be moved out of this position by force, and will return to it when the force is withdrawn. The metacentre is the point at which the axis of floatation intersects the axis of a symmetrical body, as a ship, when inclined. If the metacentre lies above the centre of gravity of the ship, the ship floats with stability; if below, the ship is unstable; if the centres coincide, which they must do in a cylinder or a sphere, for example, the body floats indifferently in any position. For the weight, volume, and pressure of water and air, see Water and Air as standards of measure, page 124. MOTION. The motion of a body is uniform, when the body passes through equal spaces in equal intervals of time. Velocity is the measure of motion, and is expressed by the number of feet or other unit of length moved through per second or other unit of time. Motion is accelerated, when the body moves through continually increased spaces in equal intervals of time, like a railway train starting from a station. Motion is retarded, when the body moves through continually decreased spaces in equal intervals of time, like a railway train arriving at a station. The acceleration and retardation are uniform, when the spaces moved through increase or decrease by equal successive amounts, like a body falling by the action of gravity, or, on the contrary, projected upwards in opposition to gravity. GRAVITY. When bodies fall freely near the surface of the earth, the motion, as already said, is uniformly accelerated; equal additions of velocity being made to the motion of the body in equal intervals of time. During the ist second of time, the body, starting from a state of rest, falls through 16.095 f eet > or > say 16.1 feet; during the ad second, it falls through three times 16.1 feet; during the 3d second, it falls through five times 16.1 feet, and so on. The body having, in the ist second, fallen through 1 6. i feet, from a state of rest, with a motion uniformly accelerated, it would move through 32.2 feet in the next second, with the velocity thus acquired, without any additional stimulus from gravity; that is to say, the velocity acquired at the end of the ist second is 32.2 feet per second. During the 2d second, it in fact acquires an additional velocity of 32.2 feet per second, making up, at the end of this second, a final velocity of 64.4 2?8 FUNDAMENTAL MECHANICAL PRINCIPLES. feet per second. In like manner the body acquires an additional velocity of 32.2 feet per second during the 3d second, making a final velocity of three times 32.2 feet, or 96.6 feet per second. And so on. Each of these additional velocities is acquired in falling through 16.1 feet additional to the space fallen through in virtue of the velocity acquired at the beginning of each second. The relations of height fallen, velocity acquired, and time of fall, are simply exhibited in the following manner : During the successive seconds the heights fallen through are consecutively as follow: time, i, i, i, i second, height of fall, 16.1, 16.1 x 3, 16.1 x 5, 16.1 x 7 feet. And reckoning the totals from the commencement of the fall, total times, i, 2, 3, 4 seconds, total height of fall, 16.1, 16.1 x 4, 16.1 x 9, 16.1 x 16 feet. or 16.1, 1 6. i x 2 2 , 16.1 x 3 2 , 16.1 x 4 2 feet._ or 16.1, 64.4, 144-9; 2 57- 6 feet. Showing that the total height fallen is as the square of the total time. Again, during the successive seconds, the successive additional velocities acquired are: time, i, i, i, i second. velocities acquired, 32.2, 32.2, 32.2, 32.2 feet per second. And the total or final velocities acquired, reckoning from the commence- ment of the fall, are : total times, i, 2, 3, 4 seconds. final velocities, 32.2, 32.2 x 2, 32.2 x 3, 32.2 x 4 feet per second, or 32.2, 64.4, 96.6, 128.8 feet per second. Showing that the velocity acquired is in direct proportion to the time of the fall. The above relations of time, height, and velocity are brought together for comparison, thus : time as, i, 2, 3, 4, &c. velocity acquired as, i, 2, 3, 4, &c. height of fall as, i, 4, 9, 16, &c. or as i, 2 2 , 3 2 , 4 2 , &c. Showing that, whilst the velocity increases simply with the time, the height of fall increases as the square of the time, and as the square of the velocity. The force of gravity is expressed by the velocity communicated by gravity to a body falling freely in a second, namely, 32.2 feet per second, and is symbolized by g. The foregoing relations of time, velocity, and height of fall, are comprised in the six following propositions with their answers applicable to the condition of a body falling freely. They are much used in mechanical calculations. i and 2. Given the time, to find the velocity and the height. 3 and 4. Given the velocity, to find the time and the height. 5 and 6. Given the height, to find the time and the velocity. GRAVITY. 2/9 RULES FOR THE ACTION OF GRAVITY. Putting / = the time of falling in seconds, z> = the velocity in feet per second, ^ = the height of fall in feet, and g = gravity or 32.2; then, RULE i. Given the time of fall, to find the velocity acquired by a falling body. Multiply the time in seconds by 32.2, the product is the final velocity in feet per second. Or # = 32.2 / ...................................... ( i ) RULE 2. Given the time of fall, to find the height of the fall. Multiply the square of the time in seconds by 16.1. The product is the height of fall in feet. Or *=i6.i / 2 ..................................... (2) RULE 3. Given the velocity, to find the time of falling. Divide the velocity in feet per second by 32.2. The quotient is the time in seconds. Or RULE 4. Given the velocity, to find the height of fall "due" to the velocity. Square the velocity in feet per second, and divide by 64.4. The quotient is the height of fall in feet. Or RULE 5. Given the height of fall, to find the time of falling. Divide the height in feet by 16.1, and find the square root of the quotient. The root is the time in seconds. Or J. 16.1 (5) RULE 6. Given the height of fall, to find the velocity due to the height. Multiply the height in feet by 64.4, and find the square root of the product The root is the velocity in feet per second. Or, multiply the square root of the height in feet by 8.025; tne product is the velocity in feet per second. Note. It is usual to take the integer 8 only for the multiplier. Symbolically, these operations are expressed as follows : 32.2 or in a round number v = 8 \J h (6) The above rules are applicable, inversely, to the motion of bodies pro- jected upwards and uniformly retarded by gravity. The height to which a body projected vertically upwards by an initial impulse, will ascend, is equal to the height through which the body must fall in order to acquire the initial velocity, and the same rule (4) applies in these two cases. 280 FUNDAMENTAL MECHANICAL PRINCIPLES. The following table, No. 101, gives the velocity acquired by a falling body in falling freely through a given height. Table No. 102 gives, conversely, the height of fall due to a given velocity. Table No. 103 gives the fall and the final velocity due to a given time of falling freely. Table No. 101. VELOCITY ACQUIRED BY FALLING BODIES, DUE TO GIVEN HEIGHTS OF FALL. \/ h. Height of Fall. Velocity in Feet per Second. Height of Fall. Velocity in Feet per Second. Height of Fall. Velocity in Feet per Second. Height of Fall. Velocity in Feet per Second. feet. ft. per sec. feet. ft. per sec. feet. ft. per sec. feet. ft. per sec. .01 .803 3-o 13.90 2 3 38.49 50 56.74 .02 I.I4 3-5 15.01 2 4 39.31 IOO 80.25 03 i-39 4.0 16.05 2 5 40.12 150 98.28 .04 1.61 4-5 17.03 26 40.92 200 ">5 05 i. 80 5-o 17.99 27 41.70 300 139.0 .06 1.97 5-5 18.82 28 42,47 400 160.5 .07 2.12 6.0 19.66 29 43-22 500 179.9 .08 2.27 6-5 20.46 3 43-95 600 196.6 .09 2.41 7.0 21.23 3i 44.68 700 212.3 .1 2-54 7-5 21.97 32 45-39 800 226.9 .2 3.20 8.0 22.69 33 46.10 90O 240.7 3 4.40 8-5 23.40 34 46.79 IOOO 253-8 4 5-07 9.0 24.07 35 47-47 1500 310.8 5 5.68 9-5 24-73 36 48.15 2000 358.9 .6 6.22 10 25.38 37 48.81 2500 401.2 7 6. 7 I ii 26.62 38 49-47 3000 439-5 .8 7 .l8 12 27.80 39 50.11 3500 474-7 9 7 .6l I 3 28.93 40 50.75 4000 57-5 I.O 8.03 14 30.03 4i 51-38 4500 538.3 1.2 8.79 15 31.08 42 52.01 500O 567.4 1.4 9-5 16 32.10 43 52.62 6000 621.6 1.6 10.15 i7 33-09 44 53-23 7OOO 671.4 1.8 10.77 18 34.05 45 53.83 8000 717.8 2.0 "35 19 34.98 46 54-43 90OO 761.3 2.25 12.04 20 35.89 47 55-02 IOOOO 802.5 2.50 12.69 21 36.77 48 55-6o 2-75 I3-3 1 22 37.64 49 56-17 GRAVITY. 28l Table No. 102. HEIGHT OF FALL DUE TO GIVEN VELOCITIES. 64.4' Velocity in Feet per Second. Height of Fall. Velocity in Feet per Second. Height of Fall. Velocity in Feet per Second. Height of Fall. Velocity in Feet per Second. Height of Fall. ft. per sec. feet. ft. per sec. feet. ft. per sec. feet. ft. per sec. feet. 25 .OOIO 19 5-6l 4 6 32.9 73 82. 7 50 .0039 20 6.21 47 34-3 74 85.0 75 .0087 21 6.85 48 35-8 75 87.4 I.OO .Ol6 22 7.52 49 37-3 80 99.4 1-25 .024 23 8.21 50 38.8 85 112. 2 1.50 035 24 8.94 5 1 40.4 90 125.8 J-75 .048 25 9.71 52 42.0 95 I4O.I 2 .062 26 10.5 53 43- 6 100 155-3 2 -5 .097 27 u-3 54 45-3 i5 I7I.2 3 .I4O 2'8 II. 2 55 47.0 no 187.9 3-5 .190 29 I3-I 56 48.7 115 205.4 4 .248 3 14.0 57 50-4 I2O 223.6 4-5 .314 3i 14.9 58 52.2 I 3 262.4 5 .388 32 15-9 59 54-1 140 304.3 6 559 33 16.9 60 55-9 150 349-4 7 .761 34 17.9 61 57-8 J 75 475-5 8 994 35 I9.O 62 59-7 200 621 9 1.26 36 2O. I 63 61.6 300 1397 10 i-55 37 21.3 64 63.6 400 2484 ii 1.88 38 22.4 65 65.6 500 3882 12 2.24 39 23.6 66 67.6 600 559 13 2.62 40 24.9 67 69.7 700 7609 14 3-4 4i 26.1 68 71.8 800 9938 J 5 3-49 42 27.4 69 73-9 900 12578 16 3-98 43 28.7 70 76.! IOOO 15528 i7 4.49 44 30.1 7i 78.3 18 5-3 45 31-4 72 80.5 282 FUNDAMENTAL MECHANICAL PRINCIPLES. Table No. 103. HEIGHT OF FALL AND VELOCITY ACQUIRED, FOR GIVEN TIME OF FALL. Time of Fall. Height of Fall. Velocity acquired in Feet per Second. Time of Fall. Height of Fall. Velocity acquired in Feet per Second. Time of Fall. Height of Fall. Velocity acquired in Feet per Second. seconds. feet. ft. per sec. seconds. feet. ft. per sec. seconds. feet. ft. per sec. I 16.1 32.2 12 2 3 l8 386.4 23 8517 740.6 2 64.4 64.4 13 2721 418.6 24 9 2 73 772.8 3 144.9 96.6 14 3I5 6 450.8 25 IO062 805.0 4 257.6 128.8 I 5 3623 483.0 26 10884 837.2 5 402.5 161.0 16 4122 5 I 5- 2 27 H737 869.4 6 579- 6 193.2 17 4653 547-4 28 12622 901.6 7 788.9 225.4 18 5217 579-6 2 9 13540 933-8 8 1030 257.6 I 9 5812 611.8 30 14490 966.0 9 i34 289.8 20 6440 644.0 31 15473 998.2 10 1610 322.0 21 7IOO 676.2 3 2 16487 1030 ii 1948 354-2 22 7792 708.4 ACCELERATED AND RETARDED MOTION IN GENERAL. The same rules and formulas that have been applied to the action of gravity are applicable to the action of any other uniformly accelerating force on a body, the numerical constants being adapted to the force. If an accelerating or retarding force be greater or less than gravity; that is to say, than the weight of the body, the constants 16.1, 32.2, and 64.4 are to be varied in the same proportion. To do this, multiply the constant by the accelerating force, and divide the product by the weight of the body. Let f be the accelerating force, and w the weight of the body, then the constant becomes 16.1 / 32.2 / 64.4 / LJL or * _, or ** J WWW .. (a) and substituting this in the formulas (i) to (6) for gravity, the following general rules and formulas are arrived at for the action of uniformly accel- erating or retarding forces. The rules are written for accelerating forces, but they apply by simple inversion to retarding forces also. GENERAL RULES FOR ACCELERATING FORCES. The accelerating force and the weight of the body are expressed in the same unit of weight; and the space in feet, the time in seconds, and the velocity in feet per second. In the following rules the time during which a body is acted on by an accelerating force is called the time; the velocity acquired at the end of the ACCELERATED AND RETARDED MOTION. 283 time is called the final velocity; the space traversed by the body during the time is called the space; the accelerating force is called the force. t = the time. v the final velocity. s -the space. / = the force. w - the weight. RULE 7. Given the weight, the force, and the time; to find the final velocity. Multiply the force by the time and by 32.2, and divide by the weight. The quotient is the final velocity. Or .._ RULE 8. Given the weight, the force, and the time; to find the sface. Multiply the force by the square of the time and by 16.1, and divide by the weight. Or . = 1*1/1* ................................... (8) w RULE 9. Given the weight, the final velocity, and the force; to find the time. Multiply the final velocity by the weight, and divide by the force, and by 32.2. The quotient is the time. Or "5=7 ...................................... < RULE i o. Given the weight, the final velocity, and the force; to find the space. Multiply the weight by the square of the velocity, and divide by the force, and by 64.4. The quotient is the space. Or w v 1 / \ s - ....... RULE ii. Given the weight, the force, and the space; to find the time. Multiply the weight by the space, and divide by the force; find the square root of the quotient, and divide it by 4. The last quotient is the time in seconds. Or f= T , , w s RULE 12. Given the weight, \he force, and the space; to find the final velocity. Multiply the space by the force, and divide by the weight; find the square root of the quotient, and multiply by 8. The product is the final velocity. Or f s t**\ w RULE 13. Given the weight, the space, and \htfinal velocity; to find the force. Multiply the weight by the square of the final velocity, and divide by the space, and by 64.4. The quotient is the force. Or (I3) 284 FUNDAMENTAL MECHANICAL PRINCIPLES. RULE 14. Given the weight, time, and final velocity ; to find the force. Multiply the weight by the velocity, and divide by the time, and by 32.2. Or IV V Note i. When the accelerating or retarding force bears a simple ratio to the weight of the body, the ratio may, for greater readiness in calculation, be substituted in the quantities (a) representing the modified constants, for the force and the weight. Suppose the accelerating force is a tenth part of the weight, then the ratio is i to 10, and 16.1 - = i.6i, 10 and these quotients may be substituted for 16.1, 32.2, and 64.4 respectively in the formulas for the action of gravity (i) to (6), to fit them for direct use in dealing with an accelerating force one-tenth of gravity, the height h in those formulas, of course, being taken to mean space traversed. Note 2. The tables, Nos. 101-103, pages 280-282, for the relations of the velocity and height of falling bodies, may be employed in solving questions of accelerating force generally. Example. A ball weighing 10 Ibs. is projected with an initial velocity of 60 feet per second on a level bowling-green, and the frictional resistance to its motion over the green is i Ib. What distance will it traverse before it comes to a state of rest? By rule 10, the distance traversed. Again, the same result may be arrived at, according to Note i, by multiplying the constant 64.4, in rule 4, for gravity, by the ratio of the force and the weight, which in this case is ^, and 64.4x^ = 6.44. Substituting 6.44 for 64.4 in that rule and formula, the formula becomes * = /-= 1^ = 559 feet, 6.44 6.44 the distance traversed, as already found. But the question may be answered more directly by the aid of the table for falling bodies (No. 102, page 281). The height due to a velocity of 60 feet per second, is 55.9 feet; and it is to be multiplied by the inverse ratio of the accelerating force and the weight of the body, or i2, or 10; that is, 55.9 x 10 = 559 feet, the distance traversed. If the question be put otherwise What space will a ball move over before it comes to a state of rest, with an initial velocity of 60 feet per GRAVITATION ON INCLINED PLANES. 285 second, allowing the friction to be i-ioth the weight of the ball? The answer may be given, that the friction, which is the retarding force, being i-ioth of the weight, that is of gravity, the space described will be 10 times as much as is necessary for gravity, supposing the ball to be projected vertically upwards to bring the ball to a state of rest. The height due to the velocity is 55.9 feet; then 55.9 x 10 -559 feet, the space described by the ball. ACTION OF GRAVITY ON INCLINED PLANES. If a body freely descend an inclined plane by the force of gravity alone, the velocity acquired by the body when it arrives at the foot of the plane, is that which it would acquire by falling freely through the vertical height. Or, the velocity is that " due " to the height of the plane. The time occupied in making the descent is greater than that due to the height, in the ratio of the length of the plane, or distance travelled, to the height. The time is therefore directly in proportion to the length of the plane, when the height is the same. The impelling or accelerating force by gravitation acting in a direction parallel to the plane, is less than the weight of the body, in the ratio of the height of the plane to its length. It is, therefore, inversely in proportion to the length of the plane, when the height is the same. The time of descent, under these conditions, is inversely in proportion to the accelerating force. If, for instance, the length of the plane be five times the height, the time of making freely the descent on the plane by gravitation is five times that in which a body would freely fall vertically through the height; and the impelling force down the plane is I / 5 th of the weight of the body. Problems on the descent of bodies on inclined planes are soluble by the aid of the rules 7 to 14, for the relations of accelerating forces. But, as a preliminary step, the accelerating force is to be determined, by multiplying the weight of the descending body by the height of the plane, and dividing the product by the length of the plane. For example, let a body of 15 pounds weight gravitate freely down an inclined plane, the length of which is five times the height, the accelerating force is 1 5 -=- 5 = 3 pounds. If the length of the plane be 100 feet, the height is 20 feet, and the velocity acquired in falling freely from the top to the bottom of the plane would be, by rule 12, 20 =35.776 feet per second. The time occupied in making the descent is, by rule n, ? = X. Radius of oscillation = si : ? r- ( 3 ) distance of centre of gravity from axis. If the axis of suspension be in the vertex or uppermost point of a plane figure, and the motion be edgewise, then, In a right line, or straight rod, the radius of oscillation is two-thirds of the length of the rod. In a circle suspended at the circumference, the radius of oscillation is three-fourths of the diameter. In a rectangle suspended by one of its angles, it is two-thirds of the diagonal. In a parabola suspended by the vertex, it is five-sevenths of the axis plus one-third of the parameter. In a parabola suspended by the middle of its base, it is four-sevenths of the axis plus half the parameter. But, if the oscillation of the plane figure be sidewise, then, In an isosceles, or equal-sided triangle, it is three-fourths of the height of the triangle. In a circle it is five-eighths of the diameter. In a parabola it is five-sevenths of the axis. In a sector of a circle suspended by the centre, it is three-fourths of the radius multiplied by the length of the arc, and divided by the length of the chord. In a cone it is four-fifths of the axis, plus the quotient obtained by dividing the square of the radius of the base by five times the axis. In a sphere it is two-fifths of the square of the radius divided by the sum of the radius and the length of the cord by which the sphere is suspended, plus the radius and the length of the cord. For example, in a sphere 1 6 inches in diameter, suspended by a cord 25 inches long, the radius of oscillation is 2 x 8 2 8 + 25 = 0.78 + 33 = 33.78 inches, 5(8 + 25) or 0.78 inch below the centre of the sphere. It may be noted that the depression of the centre of oscillation below the centre of the sphere, namely, 0.78 inch, is signified in the first quantity in this equation. The Pendulum. A "simple pendulum" is the most elementary form of oscillating body, consisting theoretically of a heavy particle attached to one end of a cord, or an inflexible rod, without weight, and caused to vibrate on an axis at the other end, or the centre of suspension. If an ordinary pendulum be inverted, so that the centre of oscillation shall become the centre of suspension, then the first centre of suspension will become the new centre of oscillation, and the pendulum will vibrate in the FUNDAMENTAL MECHANICAL PRINCIPLES. same time as before. This reciprocal action of the pendulum is a property of all pendulous bodies, and it is known as the reciprocity of the pendulum. The time of vibration of an ordinary pendulum depends on the angle or the arc of vibration, and is greater when the arc of vibration is greater, but in a very much smaller proportion; and if this arc do not exceed 4 or 5, that is to say, from 2 to 2^ on each side of the vertical line, the time of vibration is sensibly the same, however the length of the arc may vary within that limit. This property of a pendulum, of equal times of vibration, is known as isochronism. To construct a pendulum such that the time of vibration shall be the same whatever the magnitude of the angle of vibration may be, it is neces- sary to cause the pendulum to vibrate, not in a circular arc, but in a cycloidal curve. For this object the pendulum is suspended by a flexible thread or rod, which oscillates between two cycloidal surfaces, on which it alternately laps and unlaps itself; these are generated by a circle of which the diameter is equal to half the length of the pendulum. By means of the circle o B, Fig. 98, for example, of which the diameter is half the length of the pendulum, describe the right and left cycloidal curves OCA, OC'A', on the horizontal line A A'; and draw the tangent c B c', touching the cycloids at the middle of their lengths. The half-lengths o c, o c', are equal to twice the diameter of the generating circle OB, and consequently equal to the length of the pendulum, which Fig. 98. Cycloidal Pendulum. will vibrate in equal times, on the centre of suspension o, between the entire half- lengths o c, o c', or in any shorter path. The curve c P c' thus described by the pendulum, is itself a cycloidal curve, and is a duplicate of the other cycloids. Though a cycloidal motion of the pendulum is necessary to render it isochronous for all angles of vibration, yet taking very small arcs of the cycloidal path on either side of the vertical line, they do not sensibly differ from the circular arcs which would be described by an ordinary pendulum of the same length (o P) swinging freely. Hence the reason that the ordinary pendulum vibrates in equal times when its vibrations do not exceed 4 or 5 in extent. The length of the pendulum vibrating seconds at the level of the sea in the latitude of London is 39.1393 inches, nearly a metre; at Paris it is 39.1279; at Edinburgh it is 39.1555 inches; at New York, 39.10153 inches; at the equator it is 39.027 inches, and at the pole it is 39.197 inches. Generally, if the force of gravity, or the length of the seconds pendulum at the equator be represented by i, the gravity, or the length of pendulum at other latitudes will be as follows : Length of Seconds Pendulum. At the equator i.ooooo 30 latitude 00141 45 0028 3 ,, 52 00357 60 00423 90 (the pole) 00567 THE PENDULUM. 293 According to these ratios, the force of gravity, and the length of the seconds pendulum, at the pole, are Vijetii greater than at the equator; there being a difference of length of between a fourth and a fifth of an inch. The following are the relations of the lengths of pendulums and the times of their vibrations, that is to say, of such as vibrate through equal angles, or of which the total angle of vibration does not exceed 4 or 5: The times of vibration of pendulums are proportional to the square root of the lengths of the pendulums. Conversely, the lengths of pendulums are to each other as the squares of the times of one vibration, or inversely as the squares of the numbers of vibrations in a given time. The length of the seconds pendulum at London, 39.1393 inches, may be taken as a datum for calculation applicable to pendulums of different lengths, and to different times of vibration. RULE 4. To find the time of vibration of a pendulum of a given length. Divide the square root of the given length in inches by the square root of 39.1393, or 6.2561. The quotient is the time of a vibration in seconds. Or ' in which / is the given length of pendulum in inches, and / the time of vibration in seconds. RULE 5. To find the number of vibrations per second of a pendulum of given length. Divide 6.2561 by the square root of the length in inches. The quotient is the number of vibrations per second. For the number of vibrations per minute. Divide 375.366 by the square root of the length in inches. The quotient is the number of vibrations per minute. Or -^ (per second); ....................... (5) minute) . ...................... ( 5 j v// in which n is the number of vibrations. RULE 6. To find the length of a pendulum when the time of a vibration is given. Multiply the square of the time of one vibration in seconds by 39.1393. The product is the length of the pendulum in inches. Or /=/ 2 x 39.1393 (6) RULE 7. To find the length of a pendulum when the number of vibrations per second is given. Divide 39.1393 by the square of the num- ber of vibrations in a second. The quotient is the length of the pendulum in inches. When the number of vibrations per minute is given. Divide 140,900 by the square of the number of vibrations in a minute. The quotient is the length of the pendulum in inches. Or /_ 39.1393 . ( 7 ) n 2 (per second) 3 " 7- 140,900 .- ( 7 ) n 2 (per minute) 294 FUNDAMENTAL MECHANICAL PRINCIPLES. A pendulum may be shortened and yet vibrate in the same time as before, by the action of a second weight fixed on the pendulum rod above the centre of suspension. Here the upper weight counteracts the lower, and there is only the balance of gravitating force due to the preponderance of the lower weight available for vibrating both masses. The mass being thus increased while the gravitating force is diminished, a longer time is required for each vibration when the length of pendulum remains unaltered, or the pendulum may be shortened so that the time of the vibrations con- tinues the same. By varying the height of the upper weight above the centre of suspension, and thus varying the level of the common centre of gravity, the period of vibration is varied in proportion. RULE 8. To find the weight of the upper bob of a compound pendulum necessary to vibrate seconds, when the weight of the lower bob is given, and the respective distances of the bobs from the centre of suspension. Multiply the distance in inches of the lower bob from the centre of suspen- sion by 39.1393, and from the product subtract the square of that distance (i); again, multiply the distance in inches of the upper bob from the centre of suspension by 39.1393, and add the square of that distance (2); multiply the lower weight by the remainder (i), and divide by the sum (2). The quotient is the weight of the upper bob. Or in which D and d are the respective distances of the lower and upper bobs from the centre of suspension, and W, w, their respective weights. Thus, by means of a second bob, pendulums of small dimensions may be made to vibrate as slowly as may be desired. The metronome, an instrument for marking the time of music, is constructed on this principle, the upper weight being slid and adjusted on a graduated rod to measure fast or slow movements. THE CENTRE OF PERCUSSION. If a blow is struck by an oscillating or revolving body moving about a fixed centre, the percussive action is the same as if the whole mass of the body were concentrated at the centre of oscillation. That is to say, the centre of percussion is identical with the centre of oscillation, and its position is found by the same rules as for the centre of oscillation. If an external body is so struck that the mean line of resistance passes through the centre of percussion, then the whole force of percussion is transmitted directly to the external body; on the contrary, if the revolving body be struck at the centre of percussion, the motion of the revolving body will be absolutely destroyed, so that the body shall not incline either way, just as if every particle separately had been struck. CENTRAL FORCES. When a body revolves on an axis, every particle moves in a circle of revolution, but would, if freed, move off in a straight line, forming a tangent to the circle. The force required to prevent the body or particle flying from the centre is called centripetal force, and the tendency to fly from the centre is centrifugal force. These forces are equal and opposite examples of action and reaction and are classed as central forces. CENTRAL FORCES. 2Q5 Centrifugal force varies as the square of the speed of revolution. It varies as the radius of the circle of revolution. It varies as the mass or the weight of the revolving body. Let c be the centrifugal force in pounds, w the weight of the revolving body in pounds, r the radius of revolution or gyration in inches, m the mass of the body = , in which ^ = 32.2 or gravity; and v the linear or circum- e> ferential velocity in feet per second; then _ m v 2 _w v 1 r 32.2 r That is to say, the centrifugal force of a revolving body is equal to the weight of the body multiplied by the square of the linear velocity, divided by 32.2 times the radius of gyration. If the height due to the velocity be substituted for the velocity in the above equation, the height h being equal to - , then 64.4 _2WV 2 '_2. W k 64.4 r r and c : w : : 2 h : r. That is to say, the centrifugal force is to the weight of the body as twice the height due to the velocity is to the radius of gyration. From the first equation the following rules for revolving bodies are deduced, for finding one of the four elements when the other three are given: namely, the centrifugal force, the radius of gyration, the linear velocity, and the weight. RULE i. For the centrifugal force. Multiply the weight by the square of the speed, and divide by 32.2 times the radius of gyration. The quotient is the centrifugal force. Or (!) 32.2 r RULE 2. For the linear velocity. Multiply the centrifugal force by the radius of gyration, and by 32.2, and divide by the weight; and find the square root of the quotient. The root is the velocity. Or RULE 3. For the weight. Multiply the centrifugal force by the radius of gyration, and by 32.2, and divide by the square of the velocity. The quotient is the weight. Or (3) RULE 4. For the radius of gyration. Multiply the weight by the square of the velocity, and divide by the centrifugal force, and by 32.2. The quotient is the radius of gyration. Or wv* , ; r- (4) 32.2 c 296 FUNDAMENTAL MECHANICAL PRINCIPLES. Note. When the velocity is expressed as angular velocity, in revolutions per unit of time, it is to be reduced to linear or circumferential velocity by multiplying it by the radius of gyration and by 6.2832; or v 6.2832 v' r, in which v' is the angular velocity. By substitution and reduction in equation (i), the following equation in terms of the angular velocity is arrived at: 0.8156 c^wrv" 2 , ....................................... (5) from which is found .. .(6) . 0.8156 That is to say, the centrifugal force is equal to the weight multiplied by the radius of gyration and by the square of the angular velocity, and by 1.226. MECHANICAL ELEMENTS. The function of mechanism is to receive, concentrate, diffuse, and apply power to overcome resistance. The combinations of mechanism are num- berless; but the primary elements are only two, namely, the lever and the inclined plane. By the lever, power is transmitted by circular or angular action; that is to say, by action about an axis; by the inclined plane, it is transmitted by rectilineal action. The principle of the lever is the basis of the pulley and the wheel and axle; that of the inclined plane is the basis of the wedge and the screw. For the present, frictional resistance and the weight of the mechanism are not considered; the terminal resistance is called the weight; and the elemental mechanisms are to be treated as in a state of equilibrium, in which the power exactly balances the weight without actual movement. The action, or work done, will be subsequently treated. THE LEVER. The elementary lever is an inflexible straight bar, turning on an axis or fixed point, called the fulcrum; the force being transmitted by angular motion about the fulcrum, from the point where the power is applied to the i point where the weight is raised, or other resistance overcome. There are _ three varieties of the lever, according a as the fulcrum, the weight, or the power is placed between the other two, but the action is, in every case, re- Fig. 99 . Lever. ducible to that of three parallel forces in equilibrium (page 275). First. The power is applied at one end a, of the lever a b c, Fig. 99, and transmitted through the fulcrum, b, to the weight at the other end c. The moments of the power and the weight about the fulcrum are equal, or power x a b = weight x b c. That is, the product of the power by its distance from the fulcrum is equal THE LEVER. 297 to the product of the weight by its distance from the fulcrum. Conse- quently power : weight : : b c : a b, that is, the power and the weight are to each other inversely as their respective distances from the fulcrum. The ratio of the length of the power end of the lever to the length of the weight end is called the leverage of the power. The respective lengths, Fig. 99, being 7 feet and i foot, the leverage is 7 to i, or 7. The three varieties of the lever are grouped together in Figs. 100, 101, and 1 02. In each case, the lever is supposed to be 8 feet long and divided into feet. The leverage, in the first, is 7 to i, or 7; in the second, 8 to i, or 8; in the third, Y% to i, or y% : showing that, in the first case, the power balances seven times its own amount; in the second case, eigh times its amount; in the third case, only Fig. 100. Lever, ist kind. Fig. 101. Lever, ad kind. Fig. 102. Lever, 3d kind. one-eighth of itself, because it is nearer to the fulcrum than the weight. In each case the moments of the power and the weight about the fulcrum are equal, for, in each case, The pressures exerted at the extremities of the lever act in the same direction, and the sum of them is equal and opposite to the intermediate pressure, whether it be that of the fulcrum, the weight, or the power ( ). From this the pressure on the fulcrum may be found. If it be in the middle, the pressure is equal to the sum of the power and the weight, that is, 60 + 420 = 480 Ibs. in the example above; if at one end, it is equal to the difference of them, that is, it is 480 - 60 = 420 Ibs. when the weight is in the middle, and it is 60-7^ = 52^ Ibs. when the power is in the middle. From the equation for the equality of moments, orPxL =Wx/, in which L and / are the respective distances of the power and the weight from the fulcrum, rules may be formed for finding any one of the four quantities, when the other three are given. RULE i. To find the power. Multiply the weight by its distance from the fulcrum, and divide by the distance of the power from the fulcrum. The quotient is the power. 298 FUNDAMENTAL MECHANICAL PRINCIPLES. Or, divide the weight by the leverage. The quotient is the power. Or P = d) RULE 2. To find the weight. Multiply the power by its distance from the fulcrum, and divide by the distance of the weight from the fulcrum. The quotient is the weight. Or, multiply the power by the leverage. The product is the weight. Or TTT -t JLj / x RULE 3. To find the distance of the power from the fulcrum. Multiply the weight by its distance from the fulcrum, and divide by the power. The quotient is the distance of the power from the fulcrum. Or i-/ W/ (3) RULE 4. To find the distance of the weight from the fulcrum. Multiply the power by its distance from the fulcrum, and divide by the weight. The quotient is the distance of the weight from the fulcrum. Or / PL /= -w (4) If the weight of the lever be included in such calculations, its influence is the same as if its whole weight or its mass were collected at its centre of gravity. Thus, if the lever of the first kind, Fig. 100, weighs 30 Ibs., and its centre of gravity be at the middle of its length, the weight of the lever co-operates with the power, at a mean distance of 3 feet from the fulcrum. By equality of moments (P x 7) x (30 x 3) = W x i = 420 Ibs. x i, and P x 7 = 420 - 90 = 330 Ibs.; therefore P, the power at the end of the lever required to balance the Fig. 103. Inclined Lever. Fig. 104. Inclined Lever. weight, is only 330-^-7 = 47.1 Ibs. in co-operation with the weight of the lever, as compared with 60 Ibs., without reckoning the aid from this source. When the lever is inclined to the direction of the forces, as in Fig. 103, THE LEVER. 299 equilibrium, or the equality of moments, may nevertheless be maintained. Drawing the horizontal line a' b c f through the fulcrum, to meet the ver- ticals through the power and the weight at a' and d, the moments of the power and the weight are to be estimated on the horizontal lengths a' b, b fc, as before intimated at (a), page 307, to express equality of moments. For example, the length of the plane is 24 feet and the height 2 feet; the weight is 120 Ibs., the power 10 Ibs. Then, the work done in raising the weight up the whole of the incline is 240 Ibs., thus (P) 10 Ibs. x 24 feet = (W) 120 Ibs. x 2 feet. (240 foot-pounds) (240 foot-pounds). The power is here supposed to be applied in a direction parallel to the plane. If applied in a direction at an angle to the plane, as in Fig. 122, page 308, it is to be resolved into its components, parallel and perpendicular to the plane. Draw the line b c parallel to the incline; then the power applied, b c, is equivalent to the force actually expended b c, and to the pressure without motion c c', The consumption of power is expressed by the pro- duct of its parallel equivalent, b c, into the length of the plane. Taking, for example, as above, the weight, 120 Ibs., and the active power, 10 Ibs., represented by the parallel force b c ; then the amount of the horizontal force, or the power applied, b c, is found by proportion, thus A C : A B : : b c : b c; that is, the parallel and horizontal forces are to each other as the base to the length of the incline. WORK IN MOVING BODIES. 315 WORK DONE WITH THE WEDGE. Supposing the wedge driven by a constant pressure through a distance equal to its length, the work done by the power is expressed by the power into the length, and the work done on the weight is expressed by the pro- duct of the weight into the breadth of the wedge. By equality of work, as before stated, in expressing equality of moments. If the wedge be driven for only a part of its length, the work done by the power is in the proportion of the part of the length driven; and the work done on the weight is similarly in the proportion of the part of the breadth by which the resisting surfaces are separated. WORK DONE WITH THE SCREW. In one revolution of the screw, the weight is raised through a height equal to the pitch of the thread, whilst the power acts through the circum- ference of the circle described by the point at which it is applied to a lever. The products of the power and the weight by the spaces described by them are equal, or Px6.28 r = Wx/, as before stated (page 311) to express equality of moments. WORK DONE BY GRAVITY. The work done by gravity on a falling body is equal to the weight of the body multiplied by the height through which it falls. WORK ACCUMULATED IN MOVING BODIES. The quantity of work stored in a body in motion is the same as that which would be accumulated in it by gravity if it fell from such a height as would be sufficient to give it the same velocity; in short, from the height clue to the velocity. (See GRAVITY, page 277). The accumulated work expressed in foot-pounds, is equal to the height so found in feet, multiplied by the weight of the body in pounds. The height due to the velocity is equal to the square of the velocity divided by 64.4, and the work and the velocity may be found directly from each other, according to the following rules : RULE i. Given the weight and velocity of a moving body, to find the work accumulated in it. Multiply the weight in pounds by the square of the velocity in feet per second, and divide by 64.4. The quotient is the accumulated work in foot-pounds. Or, putting U for the work, v for the velocity, and w for the weight, U = !^... .. (i) 64.4 Or, secondly : Multiply the weight in pounds by the height in feet due to the velocity. The product is the accumulated work in foot-pounds. Or, putting h for the height, U = wxh ....................................... ( i a) 316 FUNDAMENTAL MECHANICAL PRINCIPLES. WORK DONE BY PERCUSSIVE FORCE. If a wedge be driven by blows or strokes of a hammer or other heavy mass, the effect of the percussive force is measured by the quantity of work accumulated in the striking body. This work is calculated by the preceding rules, from the weight of the body and the velocity with which the blow is delivered, or directly from the height of the fall, if gravity be the motive power. The useful work done through the wedge is equal to the work delivered upon the wedge, supposing that there is no elastic or vibrating reaction from the blow, just as if the work had been delivered by a constant pres- sure equal to the weight of the striking body, exerted through a space equal to the height of the fall, or the height due to its final velocity. Of course, in order to give effect to the constant pressure on the wedge, now imagined to be brought into action, the pressure would require to be applied to the resisting medium through some combination of the mechanical elements. But where elastic action intervenes, a portion of the work delivered is uselessly absorbed in elastically straining the resisting body; and the elastic action may be, in some situations, so excessive as to absorb the whole of the work delivered. In this case, there would not be any useful work done. These remarks, applied to the action of a blow on a wedge, are applicable equally to the action of a blow of the monkey of a pile-driver upon a pile. If there be no elastic action, the work delivered being the product of the weight of the monkey by the height of its fall, is equal to the work done in sinking the pile: that is, to the product of the frictional and other resistance to its descent by the depth through which it descends for one blow of the monkey. Supposing that the pile rests upon and is absolutely resisted by a hard unyielding obstacle, the work done becomes wholly useless, and consists of elastic or vibrating action j or it may be that the head of the pile is split open. HEAT. THERMOMETERS. The action of Thermometers is based on the change of volume to which bodies are subject with a change of temperature, and they serve, as their name implies, to measure temperature. Thermometers are filled with air, water, or mercury. Mercurial thermometers are the most convenient, because the most compact. They consist of a stem or tube of glass, formed with a bulbous expansion at the foot to contain the mercury, which expands into the tube. The stem being uniform in bore, and the apparent expansion of mercury in the tube being equal for equal increments of temperature, it follows that if the scale be graduated with equal intervals, these will indi- cate equal increments of temperature. A sufficient quantity of mercury having been introduced, it is boiled to expel air and moisture, and the tube is hermetically sealed. The freezing and the boiling points on the scale are then determined respectively by immersing the thermometer in melting ice and afterwards in the steam of water boiling under the mean atmospheric pressure, 14.7 Ibs. per square inch, and marking the two heights of the column of mercury in the tube. The interval between these two points is divided into 180 degrees for Fahrenheit's scale, or 100 degrees for the Centigrade scale, and degrees of the same interval are continued above and below the standard points as far as may be necessary. It is to be noted that any inequalities in the bore of the glass must be allowed for by an adaptation of the lengths of the graduations. The rate of expansion of mercury is not strictly constant, but increases with the temperature, though, as already referred to, this irregularity is more or less nearly compensated by the varying rates of expansion of glass. In the Fahrenheit Thermometer, used in Britain and America, the number o on the scale corresponds to the greatest degree of cold that could be artificially produced when the thermometer was originally introduced. 32 ("the freezing-point") corresponds to the temperature of melting ice, and 212 to the temperature of pure boiling water in both cases under the ordinary atmospheric pressure of 14.7 Ibs. per square inch. Each division of the thermometer represents i Fahrenheit, and between 32 and 212 there are 180. In the Centigrade Thermometer, used in France and in most other countries in Europe, o corresponds to melting ice, and 100 to boiling water. From the freezing to the boiling point there are 100. In the Reaumur Thermometer, used in Russia, Sweden, Turkey, and Egypt, o corresponds to melting ice, and 80 to boiling water. From the freezing to the boiling point there are 80. 3l8 HEAT. Each degree Fahrenheit is f of a degree Centigrade, and y of a degree Reaumur, and the relations between the temperatures indicated by the different thermometers are as follows : C. being the temperature in degrees Centigrade. R. do. do. Reaumur. F. do. do. Fahrenheit. That is to say, that Centigrade temperatures are converted into Fahrenheit temperatures by multiplying the former by 9 and dividing by 5, and adding 32 to the quotient; and conversely, Fahrenheit temperatures are converted into Centigrade by deducting 32, and taking -fths of the remainder. Reaumur degrees are multiplied by |- to convert them into the equivalent Centigrade degrees; conversely, -fths of the number of Centigrade degrees give their equivalent in Re'aumur degrees. Fahrenheit is converted into Reaumur by deducting 32 and taking -|ths of the remainder, and Reaumur into Fahrenheit by multiplying by f , and adding 32 to the product. Tables No. 104, 105 contain equivalent temperatures in degrees Centigrade for given degrees Fahrenheit, from o F., or zero on the Fahrenheit scale, to 608 F. ; and conversely, the temperature in degrees Fahrenheit correspond- ing to degrees Centigrade, from o C., or zero on the Centigrade scale, to 320 C. EQUIVALENT TEMPERATURES. 319 Table No. 104. EQUIVALENT TEMPERATURES BY THE FAHRENHEIT AND CENTIGRADE THERMOMETERS. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. -17.78 + 38 + 3-34 + 7 6 + 24.45 + 114 + 45.56 + I 17.23 39 3-90 77 25.00 i*5 46.11 2 16.67 40 4-45 78 25.56 116 46.67 3 i6.n 4i 5.00 79 26.12 117 47-23 4 I5-56 42 5-56 80 26.67 118 47.78 5 15.00 43 6.ii 81 27.23 119 48.34 6 14-45 44 6.67 82 27.78 I2O 48.90 7 13.90 45 7-23 83 28.34 121 49-45 8 13-34 46 7-78 84 28.89 122 5O.OO 9 12.78 47 8-34 85 29-45 I2 3 50.56 10 12.23 48 8.89 86 30.00 I2 4 5 I.II ii 11.67 49 9-45 87 30-55 I2 5 5I-67 12 ii. ii 5 10.00 88 31.11 126 52.23 J 3 10.56 5 1 10.56 89 31.67 127 52.78 14 10.00 52 II. II 90 32.22 128 53-34 15 9-45 53 11.67 9i 32.78 129 53-90 16 8.89 54 12.23 9 2 33-33 130 54.45 17 8-34 55 12.78 93 33.80 O O s 131 55-00 18 7.78 56 13-34 94 34-45 132 55.56 J 9 7-23 57 13.90 95 35- i33 56.11 20 6.67 58 14.45 96 35.56 i34 56.67 21 6.ii 59 15.00 97 36.11 r 35 57-23 22 5-56 60 15.56 98 36.67 136 57.78 2 3 5.00 61 1 6. 1 1 99 37.23 137 58.34 24 4-45 62 16.67 IOO 37-78 138 58.90 25 3-90 63 17.23 IOI 38.34 139 59-45 26 3-34 64 17.78 102 38-9 140 60.00 27 2.78 65 18.34 103 39-45 141 60.56 28 2.23 66 18.89 104 40.00 142 6i.n 2 9 1.67 67 *9-45 10 5 40.56 i43 61.67 30 i. ii 68 20.00 106 41.11 144 62.23 31 0.56 69 20.56 107 41.67 i45 62.78 3 2 o.oo 70 21. II 108 42.23 146 63-34 33 + 0.56 7i 21.67 109 42-78 i47 63.90 34 I. II 72 22.23 no 43-34 148 64.45 35 1.67 73 22.78 in 43-9 149 65.00 36 2.23 74 23.34 112 44-45 J5 65-56 37 2.78 75 23.90 H3 45.00 *f* 66.ii 3 20 HEAT. Table No. 104 (continued). FAHRENHEIT AND CENTIGRADE. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. + I 5 2 + 66.67 + 193 + 89.45 + 234 + 112.23 + 275 + 135.00 J 53 67.23 I 94 9O.OO 235 112.78 276 135.56 i54 67-78 195 90.56 2 3 6 113-34 277 136.11 i55 68.34 196 91.11 237 113.90 2 7 8 136.67 156 68.90 I 9 7 91.67 2 3 8 114-45 279 137.23 i57 69-45 198 92.23 239 115.00 280 137.78 158 70.00 199 92.78 240 II5-56 28l 138.34 i59 70.56 200 93-34 241 n6.ii 282 138.90 160 71.11 201 93-90 242 116.67 283 139-45 161 71.67 2O2 94-45 243 117.23 284 140.00 162 72.23 20 3 95.00 244 117.78 285 140.56 163 72.78 204 95.56 245 118.34 286 I4I.II 164 73-34 20 5 96.11 246 118.90 287 141.67 165 73-9 2O6 96.27 247 119-45 288 142.23 166 74-45 207 97-23 248 120.00 289 142.78 167 75.00 208 97-78 249 120.56 290 143.34 168 75-56 2O9 98.34 250 121. II 2 9 I I33-90 169 76.11 210 98.90 2 5 T 121.67 292 144.45 170 76.67 211 99-45 252 122.23 293 145.00 171 77-23 212 100.00 2 53 122.78 294 145.56 172 77.78 2I 3 100.56 254 123.34 295 146.11 i73 78.34 214 101.11 2 55 123.90 296 146.67 i74 78.90 215 101.67 256 124-45 297 I47.23 *75 79-45 216 102.23 257 I25.OO 298 147.78 176 80.00 217 102.78 2 5 8 125.56 299 148.34 177 80.56 218 103.34 259 126.11 300 148.90 178 8i.ii 219 103.90 260 126.67 3 OI 149-45 179 81.67 220 104.45 26l 127.23 3 02 150.00 180 82.23 221 105.00 262 127.78 303 ISO'S 6 181 82.78 222 105.56 263 128.34 34 I5I.II 182 83-34 223 106.11 264 128.90 305 151.67 183 83.90 224 106.67 265 129.45 36 152.23 184 84-45 225 107.23 266 130.00 37 152.78 185 85.00 226 107.78 267 130.56 3 08 153-34 186 85-56 227 108.83 268 I3I.II 309 I53.90 187 86.11 228 108.90 269 131.67 3 IO 154.45 188 86.67 229 109.45 270 132.23 3 11 155-00 189 87.23 230 I IO.OO 271 132.78 312 155.56 190 87.78 2 3 I 110.56 272 133.34 3i3 156.11 191 88.34 2 3 2 III. II 273 133.90 $*4 156.67 192 88.90 233 111.67 274 134.45 3 J 5 I57-23 EQUIVALENT TEMPERATURES. 321 Table No. 104 (continued}. FAHRENHEIT AND CENTIGRADE. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. + 3l6 + I57-78 + 357 + 180.56 + 398 + 203.34 + 439 + 226.11 3*7 158.34 358 181.11 399 203.90 440 226.67 3i8 158.90 359 181.67 400 204.45 441 227.23 3i9 !59-45 360 182.23 401 205.OO 442 227.78 320 160.00 361 182.78 402 205.56 443 228.34 321 160.56 362 183.34 403 206. II 444 228.90 322 i6i.n 363 183.90 404 206.67 445 229.45 3 2 3 161.67 364 184.45 405 207.23 446 23O.OO 3 2 4 162.23 365 185.00 406 207.78 447 230.56 325 162.78 366 185-56 407 208.34 448 231.11 326 163-34 367 i86.n 408 208.90 449 231.67 327 163.90 368 186.67 409 209.45 45 232.23 328 164.45 369 187.23 410 210.00 45 1 232.78 329 165.00 37o 187.78 411 210.56 452 233-34 33 165-56 37i 188.34 412 211. II 453 233.90 33i i66.n 372 188.90 4i3 211.67 454 234-45 332 166.67 373 189.45 414 212.23 455 235.00 333 167.23 374 190.00 4i5 212.78 456 235-56 334 167.78 375 190.56 416 213-34 457 236.11 335 168.34 376 191.11 4i7 213.90 458 236.67 336 168.90 377 191.67 418 214-45 459 237.23 337 169.45 378 192.23 419 2I5.OO 460 237.78 338 170.00 379 192.78 420 215.56 461 238.34 339 170.56 380 193-34 421 2l6.II 462 238.90 340 171.11 38i 193.90 422 216.67 463 239-45 34i 171.67 382 194-45 423 217.23 464 240.00 342 172.23 383 195.00 424 217.78 465 240.56 343 172.78 384 I95-56 425 218.34 466 241.11 344 173-34 385 196.1 1 426 218.90 467 241.67 345 173.90 386 196.67 427 219-45 468 242.23 346 174-45 387 197.23 428 220.OO 469 242.78 347 175.00 388 197.78 429 220.56 470 243-34 348 I75-56 389 198.34 43 221. II 47i 243.90 349 176.11 39 198.90 43i 221.67 472 244.45 350 176.67 39i I99.45 432 222.23 473 245.00 35i 177.23 392 200.00 433 222.78 474 245-56 352 177-78 393 200.56 434 223.34 475 246.11 353 178.34 394 201. II 435 223.90 476 246.67 354 178.90 395 201.67 43 6 224.45 477 247.23 355 179-45 396 202.23 437 225.OO 478 247.78 356 180.00 397 202.78 438 225.56 479 248.34 21 322 HEAT. Table No. 104 (continued}. FAHRENHEIT AND CENTIGRADE. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. Degrees Fahr. Degrees Centigrade. + 480 + 248.90 + 513 + 267.23 + 546 + 285.56 + 579 + 303.90 481 249.45 5.4 267.78 1 547 286.11 5 80 304.45 482 250.00 515 268.34 548 286.67 581 305.00 483 250.56 516 268.90 549 287.23 582 305.56 4 8 4 251.11 517 269.45 550 287.78 583 306.II 485 251.67 518 270.00 55 1 288.34 584 306.67 4 86 252.23 5 J 9 270.56 SS 2 288.90 585 307.23 487 252.78 520 271.11 553 289.45 586 307.78 4 88 2 53-34 521 271.67 554 290.00 587 308.34 489 253-90 522 272.23 555 290.56 588 308.90 49 254-45 523 272.78 556 291.11 589 309.45 491 255.00 524 273.34 557 291.67 59 310.00 492 255-5 6 525 273.90 558 292.23 59i 310.56 493 256.11 ; 526 274.45 559 292.78 592 3II.II 494 256.67 527 275.00 560 293-34 593 311.67 495 257-23 528 275.5 6 561 293.90 594 312.23 496 257.78 529 276.11 562 294.45 595 312.78 497 258.34 530 276.67 563 295.00 596 3J3-34 498 258.90 53i 277.23 564 295.56 597 313-90 499 259-45 S3 2 277.78 565 296.11 598 314.45 500 260.00 533 278.34 566 296.67 599 3!5.oo So 1 260.56 534 278.90 567 297.23 600 3I5.5 6 502 261.11 535 279-45 568 297.78 601 316.11 503 261.67 536 28O.OO 569 298.34 602 316.67 54 262.23 537 280.56 57o 298.90 603 317-23 505 262.78 538 28l.II 57 1 299.45 604 317.78 506 263.34 539 281.67 572 300.OO 605 318.34 507 263.90 540 282.23 573 300.56 606 318.90 508 264.45 54i 282.78 574 301.11 607 319.45 509 265.00 542 283.34 575 301.67 608 320.00 5 10 265.56 543 283.90 576 302.23 5" 266.11 544 284.45 577 302.78 512 266.67 545 285.00 i 578 303.34 EQUIVALENT TEMPERATURES. 323 Table No. 105. EQUIVALENT TEMPERATURES BY THE CENTIGRADE AND FAHRENHEIT THERMOMETERS. Degrees Cent. Degrees Fahr. Degrees Cent. Degrees Fahr. Degrees Cent. Degrees Fahr. Degrees Cent. Degrees Fahr. -20 - 4.0 4-21 + 69.8 + 62 + 143.6 + 103 + 217.4 19 2.2 22 71.6 6 3 1454 104 219.2 18 0.4 23 73-4 6 4 147.2 105 221.0 I? + 1.4 24 75-2 65 149.0 106 222.8 16 3-2 25 77.0 66 150.8 107 224.6 15 5.0 26 78.8 67 152.6 108 226.4 14 6.8 27 80.6 68 154.4 109 228.2 13 8.6 28 82.4 69 156.2 no 230.0 12 10.4 29 84.2 70 158.0 III 231.8 II 12.2 30 86.0 7i 159-8 112 233-6 10 I4.O 31 87.8 72 161.6 H3 2354 9 I 5 .8 32 89.6 73 163.4 H4 237.2 8 I 7 .6 33 91.4 74 165.2 H5 239.0 7 194 34 93-2 75 167.0 116 240.8 6 21.2 35 95.0 76 168.8 117 242.6 5 23.0 36 96.8 77 170.6 118 244.4 4 24-8 37 98.6 78 172.4 119 246.2 3 26.6 38 100.4 79 174.2 120 248.0 2 28.4 39 102.2 80 176.0 121 249.8 I 30.2 40 104.0 81 177-8 122 251.6 32.0 4i 105.8 82 179.6 I2 3 2534 + I 33-8 42 107.6 83 181.4 124 255.2 2 35-6 43 109.4 84 183-2 125 257.0 3 37-4 44 III. 2 85 185.0 126 258.8 4 39-2 45 II3.0 86 1 86.8 127 260.6 5 41.0 46 II4.8 87 1 88.6 128 262.4 6 42.8 47 II6.6 88 190.4 129 264.2 7 44.6 48 118.4 89 192.2 130 266.0 8 46.4 49 120.2 90 194.0 I 3 l 267.8 9 48.2 50 122.0 9i 195.8 132 269.6 10 50.0 51 ; 123.8 92 197.6 133 271.4 ii 51.8 52 125.6 93 199.4 134 273.2 12 53.6 53 1274 94 201.2 135 275.0 13 1 55-4 54 129.2 95 203.0 136 276.8 H 57-2 55 I3I.O 96 204.8 137 278.6 15 59.0 56 132.8 97 206.6 138 280.4 16 60.8 57 134.6 98 208.4 139 282.2 17 62.6 58 136.4 99 210.2 140 284.0 18 19 64.4 66.2 138.2 140.0 100 IOI 2I2.O 213.8 141 142 285.8 287.6 20 68.0 61 I4I.8 102 215.6 H3 28 9 .4 324 HEAT. Table No. 105 (continued}. CENTIGRADE AND FAHRENHEIT. 1 K-^rees Kal.r. IVgivcs Cent Degrees Degrees Cent. Degrees. Fahr. Cent 1 ' _;rees Fahr. + 144 + 291.2 + I8 9 + 372.2 + 234 + 453-2 + 279 + 534-2 145 293.0 I 9 374-0 235 455-o 280 536.0 146 294.8 191 375-8 2 3 6 456.8 28l 537-8 U7 296.6 192 377-6 237 458.6 282 539-6 148 298.4 '93 379-4 460.4 283 541.4 149 300.2 194 381.2 239 462.2 284 543-2 150 302.0 195 383.0 240 464.0 285 545.0 I 5 I 303.8 ,96 384-8 241 465.8 286 546.8 152 305.6 197 j86.6 242 467.6 287 548.6 i53 3074 198 388.4 243 469-4 288 550.4 154 309.2 199 390.2 244 471.2 289 552.2 i55 31 i.o 200 392.0 245 473-o 290 554-o 156 312.8 201 393-8 246 474-8 291 555-8 i57 3H.6 202 5-6 247 476.6 292 - 557-6 158 3l6.4 203 7-4 2 4 8 478.4 293 5594 i59 318.2 204 399-2 249 480.2 294 561.2 100 320.0 205 401.0 250 482.0 295 563.0 161 321.8 206 402.8 251 483.8 2 9 6 564.8 162 323.6 207 404.6 252 485.6 297 566.6 163 325.4 208 406.4 253 487.4 2 9 8 568.4 164 327.2 20 9 408.2 254 489.2 299 570.2 165 329.0 210 410.0 255 491.0 300 572.o 166 330-8 21 I 411.8 2 5 6 492.8 301 573-8 167 332.6 212 413.6 257 494-6 302 575-6 168 3344 213 415.4 258 496.4 303 577-4 169 336.2 214 417.2 259 498.2 34 579-2 170 338.0 215 419.0 260 500.0 35 581.0 171 339-8 216 420.8 26l 501.8 306 582.8 172 341.6 217 422.6 262 503.6 307 584.6 173 343-4 218 424.4 26 3 505.4 3 08 586.4 174 345.2 219 426.2 264 507.2 309 588.2 175 176 347-0 348.8 220 221 428.0 265 429.8 266 509.0 510.8 310 590.0 591.8 177 350.6 222 431.6 267 512.6 312 593-6 - 178 352.4 22 3 433-4 268 514.4 313 595-4 179 354-2 224 435-2 269 516.2 314 597.2 1 80 356.0 225 437-0 270 518.0 315 599-o 181 357-8 226 438.8 271 519.8 316 600.8 182 359.6 227 440.6 272 521.6 317 602.6 183 361-4 228 442.4 273 5234 318 604.4 184 363.2 229 444.2 274 525.2 319 606.2 185 365.0 2 3 446.0 527.0 320 608.0 1 86 366.8 231 447.8 276 528.8 187 368.6 449-6 277 530.6 188 370-4 4514 2 7 8 53-4 AIR-THERMOMETERS. 325 AIR-THERMOMETERS. Air-thermometers, or gas-thermometers, though inconvenient because bulky, are, by reason of the great expansiveness of air, superior to such as depend upon the expansion of liquids or solids, in point of delicacy and exactness. In any thermometer, whether liquid or gas, the indications depend jointly upon the expansion by heat of the fluid substance, and that of the tube which holds it. The expansion of mercury is scarcely seven times that of the glass tube within which it expands, and the exactness of its indications are interfered with by the variation in the expansiveness of glass of different qualities. In the gas-thermometer, on the contrary, the expansiveness of the gas is 160 times that of the glass, and the inequalities of the glass do not sensibly affect the indications of the instrument. Gas-thermometers, or, as they are commonly called, air-thermometers, are designed either to maintain a constant pressure with a varying volume of air, or to maintain a constant volume of air while the pressure varies. In the first case, Fig. 119, the thermometer consists of a reservoir A, to be placed in the substance of which the temperature is to be ascertained; a tube df, connected at a suitable distance by a small tube ab to the reservoir; a tube cd, open above, through which mercury is introduced into the instrument; a stop-cock r to open or close a communication ist, between the tube dfand the atmosphere; 2d, between the base of the tube cd and the atmosphere; 3d, between the two tubes df, cd-, 4th, between both these tubes and the atmosphere. The tube df, which is carefully gauged, answers the purpose of the gradu- ated tube of the mercury-thermometer, and receives the air driven over by expansion from the reservoir, at the same time that it is maintained at or near the temperature of the surrounding atmosphere. Thus the air is divided between the reservoir A and the tube df, of which the air in the former is at the Fig. 119. Air-Thermometer. temperature of the substance under observation, and that in the latter is at the temperature of the atmosphere. These two portions of air support the same pressure, which can at all times be approximated to that of the atmosphere by means of the cock r, through which the mercury is allowed to escape until it arrives at the same level in the two tubes. By means of a formula embracing the respective volumes of the two portions of air and the temperature of the atmosphere, the temperature of the substance under observation is determined. But it is apparent that, when applied as a pyrometer to the measurement of high temperatures higher, that is to say, than the boiling point of mercury (676 F.) the greater part of the air passes by expansion into the tube df, leaving but a small remainder in the reservoir A. A serious objection to this is that the proportion of air which passes over into the tube df for a new increase of temperature is very small, and is with difficulty measured with sufficient precision. The second form of air thermometer, in which the pressure varies whilst the volume remains the same, was used by M. Regnault in his researches. 326 HEAT. The temperature is measured by means of the increased elastic force of the inclosed air, and the instrument is both more convenient and more precise than that in which the volume varies, for at all temperatures the sensibility of the instrument is the same. At high temperatures the apparatus is liable to distortion under the pressure of the inclosed air; but this may be pre- vented, if needful, by introducing air of a lower than atmospheric pressure at an ordinary temperature, even so low as one-fourth of an atmosphere; for, although the apparatus is less sensitive in proportion as the first supply of air is of less density and pressure, yet withal it is sufficiently sensitive. The thermometer, as employed by M. Regnault, is shown in Fig. 120. Two glass tubes, df, cd, about half-an-inch bore, are united at the base by a stop-cock r. The tube cd is open above, and df is con- nected to the reservoir A by a small tube ab. The cover of the boiler in which the reser- voir is inclosed is shown at B, and the tubes are protected from the heat of the boiler by the partition CD. By means of a three-way connection, g, and tube //, the connecting tube ab communicates with an air pump, by means of which the apparatus may be dried, and air or other gas supplied to it. The first thing to be done is to completely dry the apparatus, and for this object, a little mercury is passed into the tube & 133 1 30 .00740 .00722 Palladium '. . . . . /9 2 3 Z /Trv ^^oo IOOOO . j-j^ 120 .00667 Platinum I /TTA, .08^70 .103 .OO?7l / "O7 From o to 300 C. (32 F. to 572 F.) n / o to 100 C. C PP er ioto 3 ooC. f o to 100 C. Iron -so n s~t Vita Vssx VS 4 6 .17182 .18832 .11821 .206 .226 .142 .0115 .00418 .00788 \ o to 300 C. igs-ssg V68i I /no3 Vxo8 9 .14684 .08842 .09183 .176 .106 .in .00326 .00589 .OO204 LINEAR EXPANSION OF SOLIDS BY HEAT. Table No. 108 (continued}. 337 GLASS. Expansion between 32 and 212 F. in common fractions. Expansion between 32 and 212 F. in a length = 100. Expansion between 32 and 212 F. in a length of 10 feet. Expansion for i F. in a length of zoo feet. Flint Glass i/ inch. inch. French Glass, with lead 71248 i/ 08*720 .0974 UO 54 1 Glass tube, without lead /i i47 i/ OOT *7 C .105 Glass of St. Gobain.. 71090 i/ "y 1 /^ Barometer tubes (Smeaton). . . . Glass tube (Roy) /II22 7 1 175 I/ 08333 O*7 *7 C C .107 .IOO .OO594 555 Glass rod, solid (Roy) Glass (Dulong and Petit) Do. (o to 200 C.) /I289 71237 Vxrfx i/ 7755 .08083 .08613 00484 .0931 .0970 .103 1 1 A .00517 00539 .00574 Oo6 3 2 Do. (o to 300 C.) 71032 I/ Q o, 10108 . j. J.^. 121 006 74 7907 Ice.., , O'Z'Z'Z u ooo STONES. Initial Temperature. Final Temperature. Expansion in a length = 100. Expansion for i F. in a length of loo feet. Granite 4C F 220 F length = TOO. 2Ol6 inch. O2OO Do. . to J Af IOO O4l6 Clay-slate T-D 46 87 041 6 Do 46 ), then 12.387 x 14.7=0x493, whence the coefficient, , for air is a = -36935, or and the formula (b) becomes, for air, ^ + 461 . 2.7074'" 2.7074 Table No. 114. OF COEFFICIENTS OR CONSTANTS, a, IN THE EQUATION (b) FOR THE RELATIONS OF THE VOLUME, PRESSURE, AND TEM- PERATURE OF GASES; NAMELY, V p = a (t + 461). Name of gas. Volume of one pound of gas, at 32 F., under one atmosphere. Value of coefficient a Hydrogen cubic feet. 178 S-z 522200 or */ o Gaseous steam A /u.tj^ TQ QT 7 OJ^ UU J U1 70.1875 O ^ Q 2 7 2 Or x / KS Nitrogen *af3F*j 12 722 o 27o 27 or */ f. Olefiant gas 1 2 ^80 Wt O/Vo/J U1 72.6359 O27co6 or J / &P.P. Air 12 *87 o 26031? or J /o ~ Oxvsen **S^f 1 1 2O5 Wt o w yoo> vi 72.7074 O 23AO6 Or X /o nn-ir Carbonic acid (ideal)* 8 1^7 **TOOV*J ** 7 2 '9935 O 24222 Or I A TTT>i Do. do. (actual) r**3l 8 101 o 241^1; or I A toon Ether vapour* ... A. 7 77 O IA2A6 Or */tmne Vapour of mercury* T-/ / / I 776 o 0^206 or I / T Q Q^S i. / /V * The densities are computed by Rankine for the ideal condition of perfect gas. 35 HEAT. that is to say, the volume of one pound of air, multiplied by the pressure per square inch, is equal to the absolute temperature divided by the constant 2.7074. To adapt the formula (&) for other gases, the respective coefficients, or constants, are found in the same manner, in terms of the volume of one pound of each gas, at 32 F., under one atmosphere of 14.7 Ibs. per square inch. They are given in table No. 114. 6. The volume of one pound of air at any pressure and any temperature is deduced as follows: 4. I . f. T (8) 2- 7074 P RULE 8. To find the volume of one pound of air, of a given temperature and pressure. Divide the absolute temperature by the pressure in Ibs. per square inch, and by 2.7074. The quotient is the volume in cubic feet. For the ordinary case when the pressure is constant at 14.7 Ibs. per square inch, the formula (8) becomes, by substituting and reducing, RULE 9. To find the volume of one pound of air under 14.7 Ibs. pressure per square inch, at a given temperature. Add 46 1 to the temperature, and divide the sum by 39.80. The quotient is the volume in cubic feet. 7. The pressure of one pound of air of any volume, and at any tempera- ture, is found as follows : /+ 461 ................................. (10) ..... * 2.7074V RULE 10. To find the pressure of one pound of air, of a given temperature and volume. Divide the absolute temperature by the volume and by 2.7074. The quotient is the pressure in Ibs. per square inch. 8. The temperature of one pound of air of any volume and pressure is found as follows : ^2.7074 V/-46i ........................... (n) RULE 1 1. To find the temperature of one pound of air, of a given volume and pressure. Multiply the volume by the pressure in pounds per square inch, and also by 2.7074; subtract 461 from the product. The remainder is the temperature. 9. The density of air is inversely as the volume, and is expressed by an inversion of the formula (8), for the volume; thus, putting D for the density, or the weight in pounds of one cubic foot of air RULE 12. To find the density of air, at a given temperature and pressure. Multiply the pressure in pounds per square inch by 2.7074, and divide by the absolute temperature. The quotient is the density, or weight in pounds of one cubic foot. VOLUME, DENSITY, AND PRESSURE OF AIR. 351 Table No. 115. VOLUME, DENSITY, AND PRESSURE OF AIR AT VARIOUS TEMPERATURES. Temperature. Volume of one pound of air at constant atmospheric pressure, 14. 7 Ibs. per square inch. Datum Volume at 62 F. = i. Density, or weight of one cubic foot of air at atmospheric pressure. Pressure of a given weight of air having a constant volume. Datum Atmospheric pressure at 62 F. = i. Fahrenheit. cubic feet. comparative volume. pounds. pounds per square inch. comparative pressure. H.583 .881 .086331 12.96 .881 32 12.387 943 .080728 13.86 -943 40 12.586 .958 079439 14.08 .958 50 I2.4O 977 .077884 14,36 977 62 13.141 I.OOO .076097 14.70 I.OOO 70 13.342 1.015 .07495 14.92 1.015 80 13.593 1.034 .073565 15.21 1.034 90 13-845 1.054 .072230 15.49 1.054 IOO 14.096 1.073 .070942 15-77 1.073 I2O 14.592 I. Ill .068500 16.33 I. Ill I4O 15.100 1.149 .O6622I. 16.89 1.149 160 15-603 1.187 .064088 17.50 1.187 180 16.106 1.226 .062090 18.02 1.226 200 16.606 1.264 .O602IO 18.58 1.264 210 16.860 1.283 .059313 18.86 1.283 212 16.910 1.287 59 I 35 18.92 1.287 22O 17.111 1.302 .058442 19.14 1.302 230 17.362 1.321 .057596 19.42 1.321 240 17.612 1.340 056774 19.70 1.340 250 17.865 1-359 055975 19.98 1-359 260 18.116 1-379 .055200 20.27 1.379 270 18.367 I.398 054444 20.55 1.398 280 18.621 1.417 .053710 20.83 1.417 290 18.870 1.436 052994 21. II 1.436 300 19.121 r "455 .052297 21.39 L455 3 20 19.624 J .493 050959 21.95 L493 340 20.126 !-532 .049686 22.51 1.532 360 20.630 I -57o .048476 23.08 1-570 3 80 21.131 i. 608 047323 23.64 i. 608 4OO 21.634 1.646 .046223 24.2O 1.646 425 22.262 1.694 .044920 24.90 1.694 45 22.890 1.742 .043686 25.6l 1.742 475 23-518 1.789 .042520 26.31 1-789 500 24.146 1.837 .041414 27.01 1-837 5 2 5 24.775 1.885 .040364 27.71 1.885 550 25-403 I 933 039365 28.42 1-933 575 26.031 1.981 .038415 29.12 1.981 600 26.659 2.029 .037510 29.82 2.029 352 HEAT. Table No. 115 (continued). Temperature. Volume of one pound of air at constant atmospheric pressure, 14. 7 Ibs. per square inch. Datum Volume at 62 F. = i. Density, or weight of one cubic foot of air at atmospheric pressure. Pressure of a given weight of air having a constant volume. Datum Atmospheric pressure at Fahrenheit. cubic feet. comparative volume. pounds. pounds per square inch. comparative pressure. 650 70O 750 800 27.915 29.172 30.428 3L685 2.124 2.22O 2.315 2.4II .035822 .034280 .032865 .031561 3I- 2 3 32.63 34-04 35-44 2.124 2.22O 2.315 2.4II 850 QOO 95 IOOO 32.941 34.197 35-453 36.710 2.507 2.6O2 2.698 2.793 .030358 .029242 .O282O6 .027241 36.85 38.25 39.66 4I.O6 2.507 2.6O2 2.698 2-793 1500 2000 2500 3OOO 49.274 61.836 74.400 86.962 3-749 4.705 5.661 6.618 .020295 .Ol6l72 .013441 .OII499 55-12 69.17 83.22 97.28 3-749 4.705 5.661 6.618 Note to Rules 8, 10, 1 1, 12. The coefficients or constants for other gases, in the application of the preceding five formulas and rules, are given in table No. 114. The table No. 115 contains the volume, density, and pressure of air at various temperatures from o to 3000 F., starting from 62 F. and 14.7 Ibs. per square inch respectively as unity for the proportional volumes and pres- sures. The second column of the table, containing the volumes of one pound of air at different temperatures, was calculated by means of the formula (9), page 350. The third column, of comparative volumes, the volume at 62 F. being = i, was calculated by means of formula (2), page 347. The fourth column, of density, contains the reciprocals of the volumes in column 2, but it is calculable independently by means of formula (12), page 350. The fifth column, of pressures, due to the temperatures, was calculated by means of formula (7), p. 348. The sixth column contains these pressures expressed comparatively, the atmospheric pressure, 14.7 Ibs. per square inch, being taken as i. SPECIFIC HEAT. The specific heat of a body signifies its capacity for heat, or the quantity of heat required to raise the temperature of the body one degree Fahrenheit, compared with that required to raise the temperature of a quantity of water of equal weight one degree. The British unit^of heat is that which is required to raise the temperature of one pound of water one degree, from 32 F. to 33 F.; and the specific heat of any other body is expressed by the quantity of heat, in units, necessary to raise the temperature of one pound weight of such body one degree. The specific heat of water at 32 F. is represented by i, or unity, and there are very few bodies of which the specific heat equals or exceeds that of water. Specific heats are, therefore, almost universally expressible by fractions of a unit. SPECIFIC HEAT OF WATER. 353 It is necessary to fix a standard of temperature, such as the freezing point, for the datum of specific heat, as the specific heat of water is not exactly the same at different parts of the scale of temperatures, but increases in an appreciable degree, as well as in an increasing ratio, as the tempera- ture rises. For temperatures not higher than 80 or 90 F., the quantity of heat required to raise the temperature of water one degree is sensibly constant ; at 86 F., it is not above one-fifth per cent, in excess of that at the freezing-point. At 212 F., it is about 1^3 per cent, in excess of that at 32 F. Above 212 F., it increases more rapidly; at 302, it is 2^ per cent, more than at 32, and at 402, it is 4.^4 per cent. more. The average specific heat of water between the freezing and the boiling points is 1.005, or one-half per cent, more than the specific heat at the freezing point. It follows from the increasing specific heat of water, as the temperature rises, that the consumption of heat in raising the temperature is slightly greater expressed in units than in degrees of temperature. To raise, for example, one pound of water from o to 100 C., or from 32 to 212 F., there are required 100.5 ^- units, or 180.9 F. units, of heat. The specific heats of water in the solid, liquid, and gaseous state are grouped as follows : Ice o. 504 Water i.ooo Gaseous Steam 0.622 showing that in the solid state, as ice, the specific heat of water is only half that of liquid water ; and that, in the gaseous state, it is a little more than that of ice, or barely five-eighths of that of liquid water. The specific heat of all liquid and solid substances is variable, increasing sensibly as the temperature rises, and the specific heats of such bodies, as tabulated, are not to be taken as exact for all temperatures, but rather as approximate average values, sufficiently near for practical purposes. The specific heat of the same body is, however, nearly constant for temperatures under 212 F. The specific heats of such gases, on the contrary, as are perfectly gaseous, or nearly so, do not sensibly vary with density or with temperature. For the same body, the specific heat is greater in the liquid than in the solid state. For example : Liquid. Solid. Water (specific heat) i.ooo 0.504 Bromine o.m 0.084 Mercury 0.0333 0.0319 M. Regnault has verified, by numerous experiments, the conclusion arrived at by previous experimentalists, that, for metals, the specific heats are in the inverse ratio of their chemical equivalents. Consequently the products of the specific heats of metals, by their respective chemical equivalents, are a constant quantity. The same rule holds good for other groups of bodies of the same composition, and of similar chemical constitu- tion. The specific heat of alloys is sensibly equal to the mean of those of the alloyed metals. The following are the specific heats of water for various tempera- 23 354 HEAT. tures from o to 230 C., or 32 to 446 F., by the air-thermometer, calculated by means of Regnault's formula : c= i + 0.00004 /+ 0.0000009 / 2 ; (i) in which c is the specific heat of water at any temperature /, the specific heat at the freezing point being = .1. Table No. 116. SPECIFIC HEAT OF WATER. Temperature. Units of Heat required to raise the temperature from the freezing point to the given temperature. Specific Heat at the given temperature. Mean Specific Heat between the freezing point and the given temperature. Centigrade. Fahrenheit. Cent, units. Fahr. units. Freezing point=i. 32 0.000 0.000 I.OOOO IO 50 10.002 18.004 1.0005 1.0002 20 68 20.010 36.018 1. 0012 1.0005 3 86 30.026 54-047 1.0020 1.0009 40 104 40.051 72.090 1.0030 I.OOI3 50 122 50.087 90.157 1.0042 I.OOI7 60 140 60.137 108.247 1.0056 1.0023 70 158 70.210 126.378 1.0072 .0030 80 I 7 6 80.282 144.508 1.0089 0035 90 194 90.381 162.686 I.OI09 .0042 IOO 212 I00.5OO 180.900 I.OI30 .0050 no 230 II0.64I 199.152 I-OI53 .0058 I2O 248 120.806 217.449 I.OI77 .0067 130 266 130.997 235-79 1 1.0204 .0076 140 284 I4I.2I5 254.187 1.0232 .0087 150 302 151.462 272.628 1.0262 .0097 160 3 20 161.741 291.132 1.0294 .OIO9 170 338 172.052 309.690 1.0328 .0121 1 80 356 182.398 328.320 1.0364 I33 190 374 192.779 347.004 I.040I .0146 200 39 2 2O3.200 365.760 1.0440 .0160 210 410 213.660 384.588 1.0481 .0174 220 428 224.162 403.488 1.0524 .0189 2 3 446 234.708 422.478 1.0568 .0204 THE SPECIFIC HEAT OF AIR AND OTHER GASES. The specific heat, or capacity for heat, of permanent gases is sensibly constant for all temperatures, and for all densities. That is to say, the capacity for heat of each gas is the same for each degree of temperature. For air, M. Regnault proved that the capacity for heat was uniform for temperatures varying from -30 C. to + 225 C. ( - 22 to 437 F.); thus the specific heat for equal weights of air, at constant pressure, were as follows : SPECIFIC HEAT OF AIR, ETC. 355 Air between -30 and + 10 C Specific heat, 0.2377 Do. 10 and + 100 C Do. 0.2379 Do 100 and 4- 225 C Do. 0.2376 Average 0.2377 The temperature is then without any sensible influence on the specific heat of air; neither has the pressure, so far as it has been subjected to experiment from one to ten atmospheres any influence on the magni- tude of the specific heat. The specific heat of gases is to be observed from two points of view : ist, When the pressure remains the same, and the gas expands by heat. 2d, When the volume remains the same, and the pressure increases with the temperature. There is a striking difference in the specific heat, or capacity for heat, according as it is measured under an increasing volume, or an increasing pressure. When the temperature is raised one degree, under constant pressure, with increasing volume, the gas not only becomes hotter to the same extent as when the volume remains the same and the pressure alone is increased, but it also expands I / 493 d part of its volume at 32 F., and thus absorbs an additional quantity of heat in proportion to the work done by expansion against the pressure. It follows that the specific heat of a gas at constant pressure is greater than that of the same gas under a constant volume; and though the former alone has been made the subject of direct experiment, the latter being of a difficult nature for experimenters, yet the latter, which is properly the specific heat, is easily deducible from the former on the principle of the mechanical theory of heat. When the volume of a gas is enlarged by expansion against pressure, the work thus done in expanding the gas may be expressed in foot-pounds by multiplying the enlargement of volume in cubic feet by the resistance to expansion in pounds per square foot. Having thus found the work done in foot-pounds, it may be divided by Joule's equivalent, 772, and the quotient will be the expression of that work in units of heat. It becomes latent, or insensible to the thermometer, and is called the latent heat of expansion. It constitutes an expenditure of heat in addition to the heat that is sensible to the thermometer, and that raises the temperature. The sum of these two quantities of heat is that which has been observed in the gross by experimentalists, and which gives the specific heat at constant pressure. It follows that, when the specific heat at constant pressure is known, the specific heat at constant volume may be arrived at by subtracting the pro- portion of heat devoted to the enlargement of the volume from the total heat absorbed at constant pressure. The remainder is the proportion of heat necessary and sufficient to elevate the temperature when the volume remains unaltered, from which the specific heat at constant volume is deduced by simple proportion; thus As the total heat absorbed at constant pressure, Is to the proportion of heat absorbed at constant volume, So is the specific heat at constant pressure To the specific heat at constant volume. For example, the specific heat of air at constant pressure and with in- 356 HEAT. creasing volume has been observed to be .2377, that of water being i. Let one pound of air at atmospheric pressure, and at 32 F., having a volume equal to 12.387 cubic feet, be expanded by heat to twice its initial volume, the pressure remaining the same. The absolute temperature, which is 32 + 461 = 493 F., will be doubled, and the indicated temperature will be 32 + 493 = 525 F. Thus, 493 degrees of heat are appropriated, and if the capacity for heat of the air were the same as that of water, 493 units of heat would be expended in the process of doubling the volume. But, as the specific heat is only .2377, or less than a fourth of that of water, the expen- diture of heat is just 493 x .2377 = 117.18 units, and this quantity comprises the fraction of heat consumed in displacing the atmosphere and overcoming its resistance through a space of 12.387 cubic feet additional to the original or initial volume of the same amount. Now, the work thus done is equal to 12.387 cubic feet x 2116.4 Ibs. pressure per sq. foot = 26,216 foot-pounds; and dividing this by 772, Joule's equivalent, the work of enlarging or doub- ling the volume is found to be equivalent to 33.96 units of heat. Deduct- ing these 33.96 units from the gross expenditure, which is 117.18 units, the remainder, 83.22 units, is the proportion of heat required to raise the temperature through 493 degrees, under an increasing pressure simply, without increasing the volume; and this remainder is the datum from which the proper specific heat of air is to be deduced. The distribution of heat thus detailed may be concisely exhibited thus : Units. To double the temperature without adding to the volume.... 83.22 To double the volume, in addition 33-96 To double the temperature and double the volume at con- stant pressure 117.18 Now, as before stated, the specific heat at constant volume bears the same ratio to that at constant pressure, as the respective quantities, or units of heat, absorbed, do to each other, or as 83.22 and 117.18; and it is found by simple proportion to be .1688 ; thus 117.18 : 83.22 : : .2377 : .1688. The proper specific heat of air is then .1688, in raising the temperature without enlarging the volume, and it bears to the so-called specific heat of air, at constant pressure and with expanding volume, the ratio of i to 1.408. This ratio, i to 1.408, deduced by means of the mechanical theory of heat, is practically identical with the ratio experimentally arrived at by M. Masson from the fall of temperature of a quantity of compressed air, which was liberated and allowed to expand back until it regained its initial pres- sure. The ratio he deduced from his inverse experiment was i to 1.41; which is the ratio of i to SPECIFIC HEAT OF AIR, ETC. 357 It may be added, by way of explanation, and to enforce the distinction, that though the pressure of a gas under constant volume rises with the temperature, a phenomenon which is analogous, at first sight, to that of the volume of a gas at constant pressure increasing with the temperature, yet there is no expenditure of work in simply raising the pressure in the former case, while the volume remains unaltered; whereas, in the latter case, there is an expenditure in increasing the volume, as has already been shown. To generalize the foregoing process, by which the specific heat of air at constant volume has been deduced from the specific heat for constant pressure; and to show its applicability for finding the specific heat of all gases at constant volume : Given / = the initial temperature of the gas, in degrees Fahrenheit. t' = the final temperature to which the gas is raised. V = the initial volume of the gas, under one atmosphere of pres- sure, in cubic feet. v the final volume of the gas, heated under constant pressure. h the specific heat of the gas under constant pressure. Put h' the specific heat of the gas under constant volume. H = the total heat expended at constant pressure, in units of heat. H' = the total heat expended at constant volume. ,, H" = the fractional quantity of heat expended in increasing the volume, at constant pressure; or the latent heat of expansion. To find the value of h'; then by proportion, H : H' : : h : h', NowH' = H-H", TT/ TT _ TT// And g = TT , and, by substitution, H H (a) Again, H = (/'-/)x/;, And H" = (V-z>) x 14.7 x 144^- 772 *(V-r)x 2.742; And H-ir h (/'-/)- 2. 74 2 (v-p) ~ir~ *(/'-/) Substituting this value in equation (a) above, ,,,_h (h(t'-f)- 2.742 (V-g)). *(''-') or = , Whence the following rule : RULE i. To find the specific heat of a gas at constant volume, when the specific heat at constant pressure is given together with the initial and final temperatures due to given initial and final volumes under an atmosphere of 358 HEAT. pressure. Multiply the difference of the initial and final temperatures by the specific heat at constant pressure. Also, multiply the difference of the initial and final volumes by 2.742. Find the difference of these two pro- ducts, and divide it by the difference of the temperatures. The quotient is the specific heat of the gas at constant volume. Applying the rule to the example of one pound of air at atmospheric pressure, and at 32 F., doubled in volume by heat; ^ = .2377, /'-/=493, and V-^=i2.38y cubic feet. Then h > _ (- 2 377 x 493) ~ (2.742 x I2 -3^7) _ l68g 493 the specific heat of air at constant volume, as already found. The comparative volumes of other gases are given in table No. 69, page 216, of the Weight and Specific Gravity of Gases and Vapours. THE SPECIFIC HEAT OF GASES FOR EQUAL VOLUMES. The specific heats of equal volumes of gases are deducible from their specific heats proper, which are for equal weights. The greater the density, the less is the volume, and the greater the weight of gas that is necessary to equal in volume a lighter gas; it is greater, in fact, in propor- tion to the density. Hence the following rule : RULE 2. To find the specific heat of a gas for equal volumes of the gas and of air. Multiply the specific heat of the gas, that is, the specific heat for equal weights of the gas and air, by the specific gravity of the gas. The product is the specific heat for equal volumes. Note. The specific heat for equal volumes may be found for constant pressure, and for constant volume, in terms respectively of the specific heat of equal weights at constant pressure and constant volume. TABLES OF THE SPECIFIC HEAT OF SOLIDS, LIQUIDS, AND GASES. The annexed table, No. 117, contains the specific heats of a number of solids, classified for convenience of reference, into Metals, Stones, Precious Stones, Sundry Mineral Substances, Woods. It appears from the tables that the metals, generally speaking, have the least specific heat: ranging from bismuth, having a specific heat of .031, to iron, which has a specific heat of from .11 to .13, and iridium, which has the greatest specific heat, namely, .189. Stones show a specific heat of about .20, or a fifth of that of water. Precious stones average less than that. Of the sundry mineral substances, glass, sulphur, and phosphorus average about a fifth of the specific heat of water, and coal and coke a fourth. Woods average a half of the specific heat of water. SPECIFIC HEAT OF SOLIDS. 359 It is a useful practical conclusion, as Dr. Rankine remarks, that the average specific heat of the non-metallic materials and contents of a furnace, whether bricks, stones, or fuel, does not greatly differ from one-fifth of that of water. Of the liquids specified in the table No. 118, it appears that all, with the exception of bromine, which has a specific heat of i.m, have less specific heat than water. Olive oil has the lowest, only .31 ; alcohol averages .65, and vinegar, .92. The table No. 119 of the specific heat of gases, contains, in the second column, their specific heat, for equal weights, at constant pressure, as determined by M. Regnault. The third column contains the specific heat, for equal weights, at constant volume, calculated by means of the Rule i, above. The fourth and fifth columns contain the specific heat of gases, for equal volumes, at constant pressure, and at constant volume, arrived at by means of the Rule 2, above. There is a remarkable nearness to equality in the specific heat for equal volumes of air, oxygen, hydrogen, carbonic oxide, and nitrogen. It may be noted, in particular, that hydrogen, though it has fourteen times the specific heat of air for equal weights, and has barely a fourteenth of the density of air, has no more specific heat than air, for equal volumes. Table No. 117. SPECIFIC HEAT OF SOLIDS. (Authority, Regnault, when not otherwise stated.) METALS, from 32 to 212 F. Bismuth .03084 Lead Platinum, sheet .03 243 Do. spongy 03293 Do. 32 F. to 212 F (Petit and Dulong) Do. 32 F. to 572 F. (300 C.) .0355 Do. at 212 F. ( 100 C.) (Pouillet} .0335 Do. at 5 7 2F. ( 300 C.) .03434 Do. at 932 F. ( 5ooC.) Do. at 1292 F. ( 700 C.) Do. at 1832 F. (1000 C.) Do. at 2192 F. (1200 C.) .03818 Gold .03 244 Mercury, solid Do. liquid Do. 59 to 68 F. (15 to 20 C.) 029 Do. 32 to 212 F (Petit and Dulong) .033 Do. 32 to 572 F. (300 C.) .035 Tungsten .03636 Antimony .05077 Do. 32 to 212 F (Petit and Dulong) .0507 Do. 32 to 572 F. (300 C.) .0547 Tin, English . . Do. Indian .05623 Water at 32= i. 360 HEAT. METALS (continued}. Cadmium .05669 Silver .05701 Do. 32 to 212 F (Petit and Dulong) .0557 Do. 32 to 572 F. (300 C.) .0611 Palladium -59 2 7 Uranium .06 1 9 Molybdenum .07218 Brass -939i Cymbal metal .086 Copper .095 1 5 Do. 32 to 212 F (Petit and Dttlong) .094 Do. 32 to 572 F. (300 C.) .1013 Zinc -9555 Do. 32 to 212 F (Petit and Dulong) .0927 Do. 32 to 572 F. (300 C.) .1015 Cobalt .10696 Do. carburetted .11714 Nickel 10863 Do. carburetted .11192 Wrought iron .11379 Do. 32 to 212 F (Petit and Dulong) .1098 Do. 32 to 392 F. (200 C.) .115 Do. 32 to 572 F. (300 C.) .1218 Do. 32 to 662 F. (350 C.) .1255 Steel, soft .1165 Do. tempered . 1 1 7 5 Do. Haussman . 1 1 848 Cast iron, white . 1 2983 Manganese, highly carburetted . 1441 1 Iridium .1887 STONES. Brickwork and masonry (Rankine) about . 2 o Marble, gray . 20989 Do. white . 2 1 585 Chalk, white -21485 Quicklime .2169 Dolomite (Magnesian limestone) .21 743 PRECIOUS STONES. Sapphire -21732 Zircon -14558 Diamond .14687 SUNDRY MINERAL SUBSTANCES. Tellurium .05 155 Iodine .0541 2 Selenium -0837 Bromine .0840 Phosphorus, 50 to 86 F 1887 Water at 32 =i. SPECIFIC HEAT OF SOLIDS. 361 SUNDRY MINERAL SUBSTANCES (Continued). Phosphorus, 32 to 212 F 25034 Glass 19768 Do. flint 19 Do. 32 to 2 1 2 F (Petit and Dulong) . 1 77 Do. 32 to 572 F 19 Sulphur 20259 Do. crystallized, natural Do. cast for two years Do. cast for two months .1803 Do. cast recently .1844 Chloride of lead 06641 Do. zinc I3 6j 8 Do. manganese I 4 2 55 Do. tin *4759 Do. calcium .16420 Do. potassium I 7 2 95 Do. magnesium .19460 Do. sodium 214 to .230 Perchloride of tin 10161 Protochloride of mercury .06889 Nitrate of silver *435 2 Do. barytes 15228 Do. potass 23875 Do. soda 27821 Sulphate of lead 08 7 23 Do. barytes 11285 Do. potash .1901 Carbonaceous : Coal 24111 Charcoal 2415 Coke of cannel coal .20307 Do. pit coal 20085 Coal and coke, average (Rankine) .20 Anthracite, Welsh 20172 Do. American .201 Graphite, natural .20187 Do. of blast furnaces -497 Animal black 26085 Sulphate of lime J 965 9 Magnesia .22159 Soda 23115 Ice 54 WOODS. Turpentine .467 I Pear tree 500 i Oak 570 Fir 650 Water at 32= i. 362 HEAT. Table No. 118. SPECIFIC HEAT OF LIQUIDS. Mercury 0333 Olive oil (Laplace and Lavoisier) . . . .3096 Sulphuric acid, density 1.87 ... .3346 Do. do. 1.30 ... .6614 Benzine, 59 to 68 F 3932 Turpentine, 4160 Do. density .872 (Despretz] ... .4720 Ether, oxalic 4554 Do., sulphuric, density 0.76 (Daltori) ... .6600 Do. do. do. 0.715 (Despretz} ... .5200 Essence of juniper 477 Do. lemon 4879 Do. orange 4886 Hydrochloric acid 6000 Wood spirit, 59 to 68 F 6009 Chloride of calcium, solution 6448 Acetic acid, concentrated 6581 Alcohol 6588 Do. density 0.793 (Dalian) ... .6220 Do. do. 0.81 ... .7000 Vinegar 9200 Water, at 32 F i.oooo Do. at 212 F 1.0130 Do. from 32 to 212 F 1.0050 Bromine .. i.ino Water at 32 =i. FUSIBILITY OR MELTING POINTS OF SOLIDS. 363 Table No. 119. SPECIFIC HEAT OF GASES. Water at 32 F. = I. GAS. SPECIFIC HEAT FOR EQUAL WEIGHTS. SPECIFIC HEAT FOR EQUAL VOLUMES. At constant pressure. At constant volume. (Real speci- fic heat.) At constant pressure. At constant volume. Sulphurous acid water = i. o.i553 0.1568 0.2164 0.2182 0.2377 0.2440 0.2479 0.3694 3.4046 0-3754 0.4008 o.45 T 3 0.4750 0.5061 0.5080 0.5929 water = i. 0.1246 0.1438 0.1714 0-1559 0.1688 0.1740 0.1768 0.2992 2.4096 0-3499 0.3781 0.4124 0.3643 0.4915 0.3911 0.4683 air = .2377, as in col. 2 0.3489 0.8310 0.3308 0.2412 0.2377 0.2370 0.2399 0.3572 0.2356 I.OII4 1.2184 0.7171 0.2950 2.3776 0.2994 0.3277 air = .1688, as in col. 3. 0.2799 0.7621 O.262O 0.1723 0.1688 0.1690 0.1711 0.2893 0.1667 0.9427 1.1490 0.6553 0.2262 2.3090 0.2305 0.2588 Vapour of chloroform Carbonic acid Oxygen Air Nitrogen Carbonic oxide Olefiant gas Hydrogen Vapour of Benzine Acetic ether Vapour of alcohol Gaseous steam .... Vapour of turpentine Ammoniacal gas Light carburetted hydrogen FUSIBILITY OR MELTING POINTS OF SOLIDS. The metals are solid at ordinary temperatures, with the exception of mercury, which is liquid down to - 39 F. Hydrogen, it is believed, is a metal in a gaseous form. All the metals are liquid at temperatures more or less elevated, and they probably vaporize at very high temperatures. Their melting points range from 39 degrees below zero of Fahrenheit's scale, the melting, or rather the freezing, point of mercury, up to more than 3000 degrees, beyond the limits of measurement by any known pyrometer. Certain of the metals, as potassium, sodium, iron, platinum, become pasty and adhesive at temperatures much below their melting points. Potassium and sodium, which melt at temperatures between 130 and 200 F., can be moulded like wax at 62 F. Two pieces of iron raised to a welding heat, are softened, and readily unite under the hammer; and pieces of platinum unite at a white heat. The melting points of alloys do not follow the ratios of those of their constituent metals, so that it is impossible to infer their melting points from these data. A remarkable instance of the absence of this relation is afforded in the fusible metal consisting of five parts of lead, three of tin, and eight of bismuth, which melts at 212 F., the heat of boiling water, though the 3^4 HEAT. melting point, if it were an average of those of the component metals, would be about 520 F. The addition of bismuth to mixtures of lead and tin has the effect of lowering the melting points. According to Professor Rankine, the melting point of ice is lowered by pressure, at the rate of 0.0000063 F. for each pound of pressure on the square foot. An atmosphere of pressure being 2116 Ibs. per square foot, the lowering of the melting point per atmosphere of pressure, is o.ooooo63 x 2116 o.oi33 Fahrenheit. To lower the melting point one degree, a pressure of 75 atmospheres would be required. In the case of water, antimony, and cast iron, and probably other sub- stances, the bulk of the substance in the solid state exceeds that in the liquid state, as is evidenced by the floating of ice on water, and of solid iron on molten iron. The volume of water is to that of ice at 32 F., as i to i. 088; that is to say, that water, in freezing at 32 F., expands nearly 9 per cent. The following table, No. 120, contains the melting points of metals, metallic alloys, and other substances: Table No. 120. MELTING POINTS OF SOLIDS. VARIOUS SUBSTANCES (Pouillet, Claudel, &c.) MELTING POINTS. Sulphurous acid - 148 F. Carbonic acid . . - i o 8 Bromine +9-5 Turpentine 14 Hyponitric acid 1 6 Ice 32 Nitro-glycerine 45 Tallow 92 Phosphorus 112 Acetic acid 113 Stearine 109 to 120 Spermaceti 120 Margaric acid 131 to 140 Wax, rough 142 ,, bleached 154 Stearic acid 158 Iodine 225 Sulphur 239 MELTING POINTS OF SOLIDS. Table No. 120 (continued']. 365 METALS. MELTING POINTS. Pouillet, Claudel. Wilson. Mercury Fahrenheit degrees. -39 + 136 194 446 54 608 680 810 1692 (very pure) 1832 2156 (very pure) 2282 1922 to 2012 2OI2 2192 2282 2372 to 2552 2732 2912 Fahrenheit degrees. 101 144 208 356 442 442 507 561 6l 7 773 1150 full red heat, full red heat. 1873 1996 2016 2786 [Fusible in highest heat of forge. I Not fusible in forge > fire, but soften and 1 agglomerate. Only fusible before the oxyhydrogen blow-pipe. Rubidium Potassium . Sodium Lithium . Tin Cadmium Bismuth Thallium Lead Zinc Antimony Bronze Aluminium . . , Calcium . ... Silver Copper . Gold, standard Gold Cast Iron white srav 2d melting... with manganese... Steel Wrought Iron French Hammered Iron, English Malleable Iron Cobalt Nickel Manganese Palladium Molybdenum Tungsten Chromium Platinum Rhodium Iridium Ruthenium Osmium 366 HEAT. Table No. 120 (continued}. ALLOYS OF LEAD, TIN, AND BISMUTH. MELTING POINTS. No. i. 2. 3- 6. 7- i 8. 2 9- 3 10. 4 ii. 5 12. 6 14- 3 15. 2 16. i 17. 2 18. 3 Tin 33 33 3) 33 33 33 33 Lead 33 33 33 33 33 , 25 10 5 3 2 3 4 3 2 I I 5 Lead Holtzapffel. 5" 482 44 1 340 378 38i 320 310 292 254 236 202 Claudel. 4 66 367 372 33 33 Tin i Bismuth i SUNDRY ALLOYS OF TIN, LEAD, AND BISMUTH. MELTING POINTS. Lead 2 Tin 5 Bismuth Ure IQQ I 2 x yy 2OI I 4. Claudel 2OI c 5 8 Ure 212 2 c . . . Claudel 212 T c 246 i I 286 I 1 -7 A 2 I OOT- 774 T r Holtzapffel 360 a I ' { Claudel 385 7Q2 1. I CC2 ALLOYS FOR FUSIBLE PLUGS. Softens at Melts at 2 Tin 2 Lead . . . ^6qF 772 Y 2 6 o^j *- 11 2 o i * - 1 7.8?. 2 7 01* 111 1 A O^O ^88 2 8 7.QC */ 406 to 410 oyj /a LATENT HEAT OF FUSION OF SOLID BODIES. 367 LATENT HEAT OF FUSION OF SOLID BODIES. When a solid body is exposed to heat, and ultimately passes into the liquid state under the influence of the heat, the temperature of the body rises until it attains the point of fusion, or melting point. The temperature of the body remains stationary at this point until the whole of it is melted; and the heat meantime absorbed, without affecting the temperature, is said to become latent, as it is not sensible to the thermometer. It is, in fact, the latent heat of fusion, or the latent heat of liquidity, and its function is to separate the particles of the body, hitherto solid, and change their condition into that of a liquid. When, on the contrary, the liquid is solidified, the latent heat is disengaged. M. Person gave the following law as the result of his experiments on the latent heat of fusion of non-metallic substances : Let c be the specific heat of the substance in the solid state, and c' its specific heat in the liquid state; / the temperature of fusion, or melting point, by Fahrenheit's scale, and / the latent heat. Then the latent heat of fusion of one pound, in British thermal units, is /^'-.(/+2 5 6) (i) RULE. To find the latent heat of fusion of a non-metallic substance. Subtract the specific heat of the substance in the solid state from its specific heat in the liquid state, and multiply the remainder by the tempera- ture of fusion or melting point by the Fahrenheit scale, plus 256. The product is the latent heat of fusion in heat-units. Table No. 121. LATENT HEAT OF FUSION OF SOLID BODIES. Person. Non-metallic. Melting Point. Specific Heat. Latent Heat in heat-units, of i pound. Liquid. Solid. Ice 32 F. 83 97 112 120 142 239 591 642 I.OOOO 5550 .7467 .2045 .2340 .4130 3319 .5040 345 .4077 .1788 .2026 .2782 .2388 142.6 73 120 9 148 175 17 H3 85 Chloride of calcium. . Phosphate of soda Phosphorus Spermaceti Wax Sulphur Nitrate of soda Nitrate of potass .... Metallic. Tin 442 442 507 6l 7 773 1873 .0637 .0642 .0363 .0402 .0562 .0567 .0308 .0314 .0956 .0570 25.6 25-6 22.7 9.86 50.6 37-9 Cadmium Bismuth Lead Zinc . . . Silver 368 HEAT. EXAMPLE. To find the latent heat of fusion of ice, the specific heat of ice, c= 0.504, and that of water c' = i ; /= 32 F. Then the latent heat -= (i - 0.504) (32 + 256) = 0.496 x 288 = 142.86 Do., by M. Person's experiment = 142.65 Difference o. 2 1 showing that the latent heat effusion of one pound of ice is 142.86 units. The table No. 121 contains the latent heat of fusion of several metals and other bodies, according to M. Person. On an inspection of the table, it appears generally that the latent heat of fusion of non-metallic bodies is greater for those which have the lower melting points, and that, for metals, the proportion lies rather the other way. The greatest latent heat of fusion belongs to wax, which has 175 units per pound, and the least to phos- phorus and lead, which have only 9 and 9.86 units respectively per pound weight. BOILING POINTS OF LIQUIDS. When a cold liquid, contained in a vessel open to the air, is subjected to heat, the temperature of the liquid is raised, and a quantity of vapour is emitted from the surface of the fluid, the pressure of which gradually increases until it becomes equal to the pressure of the atmosphere. When this pressure is reached, the aggregation of the vapour becomes visible in the interior of the liquid, and the vapour rises to the surface and escapes. This is ebullition^ or evaporation, or vaporization. When the liquid has attained to the state of ebullition, the temperature ceases to rise, and remains stationary, and it so remains until the whole of the liquid is evapo- rated. This phenomenon of stationary temperature is analogous to that which attends the fusion of solids into liquids. The proper boiling point of water, under one atmosphere of pressure, is 100 C., or 212 F. It is affected to some extent by the nature of the vessel which contains it and the presence of other objects in the vessel. In a glass retort, for example, water boils with jolts and small explosions, and the temperature of the water at which ebullition takes place is from two to three degrees higher than when it is evaporated in an iron vessel. Sulphuric acid behaves similarly under ebullition, and the explosions are as much more violent as the liquid has greater cohesiveness, and as it acts chemically upon the matter of the vessel in which it boils. A few pieces of metal thrown into the glass vessel arrest the explosive ebullition, and the temperature of the liquid falls to the same level as in the metallic vessel. The boiling point of liquids is not altered by the presence of foreign bodies mechanically in mixture with them, such as sand, sulphate of lime, and carbonate of lime. But it is always greater when matters are present in chemical combination with the liquids. All the soluble salts have this effect when dissolved in water; but, on the contrary, it has been proved experimentally, That the vapour produced at the surface of saline solutions is the steam of pure water, And that at atmospheric pressure the temperature of the steam formed is invariably 212 F., whatever be the nature of the dissolved salt, or of the vessel containing the solution. It further appears that at higher pressures BOILING POINTS OF LIQUIDS. 369 Table No. 122. BOILING POINTS OF LIQUIDS UNDER ONE ATMOSPHERE OF PRESSURE. Fahrenheit. Sulphuric ether IOO Sulphuret of carbon 118 4 Ammonia 1 4O Chloroform T A O Bromine I4C Wood spirit *-*TJ I ^O Alcohol A D W 17-2 Benzine... 176 Water x i v 212 Average sea- water 211 2 Saturated brine * ^O" 6 226 Nitric acid 248 Oil of turpentine 2 I C Phosphorus 3*-J C CA Sulphur DJT- C.7O Sulphuric acid / w CQO Linseed oil . . Oy^ CQ7 Mercury DV / 648 Table No. 123. BOILING POINTS OF SATURATED SOLUTIONS OF SALTS UNDER ONE ATMOSPHERE. NAME OF SALT. BOILINC , POINT. Quantity of salt which saturates 100 parts of water. Chlorate of potash Centigrade. 1 04. 2 Fahrenheit. 2iq.6 per cent. 61 c Chloride of barium I 04 A 220 o " A O 60 I Carbonate of soda . . I O4 6 220 3 48 c Phosphate of soda ICK.C. w -o 222 O V**3 1112 Chloride of potassium AW 3O 108.3 227 O i 3.4 CQ A Chloride of sodium (common salt) . . . Hydrochlorate of ammonia 108.4 I 14 2 227.2 237 6 OV'^r 41.2 88 9 Neutral tartrate of potash I 14 67 2^8 A 2Q6 2 Nitrate of potash 1 1 C Q 2AO 6 }} C.I Chloride of strontium A * O'y 117. Q 2AA. 2 JOD' * H7. c Nitrate of soda 121 O 2 ZO O 224 8 Acetate of soda I2A 37 .^3^. w 2?5 8 2OQ.O Carbonate of potash I -3C O jj*" 27C O 20^.0 Nitrate of lime X OJ' W I r j o / >w 2Q4 O 362 2 Acetate of potash 160 o 2-26 O 70S 2 Chloride of calcium I 7Q 5 OO^' 7CC.I fjr**' 32?. O Nitrate of ammonia x / VO 1 80 O JJJ' A 2 r6 O unlimited O J^' 3/0 HEAT. the temperatures of steam formed from sea-water are the same as those of steam generated from fresh-water under equal pressures. Table No. 122 contains the boiling points of liquids under one atmos- phere of pressure. It shows that the boiling points vary from 100 F. for sulphuric ether to 648 F. for mercury. Linseed oil boils at 597 F., and the great elevation of this temperature, as representing more or less approxi- mately the boiling points of oils and fats generally, explains the capacity of these substances when heated for cooking meat immersed in them. It is shown that sea-water boils at 2 13. 2 F. under one atmosphere, and that saturated brine does not boil until the temperature rises to 226 F. The boiling point of sea-water is raised in proportion to its concentration as brine. Table No. 123 contains the boiling points of saturated solutions of various salts in water, under one atmosphere, according to the experiments of M. Legrand. They vary from 220 to 356 F. They present a striking diversity, even among salts having the same base. BOILING POINTS OF LIQUIDS AT VARIOUS PRESSURES. The boiling points of liquids rise as the pressure increases under which they are evaporated, and they contrast strikingly in this respect with the melting points of solids, which are practically constant under all pressures. It has already been stated that, to lower the melting point of ice only one degree Fahrenheit, 75 atmospheres of pressure were required. On the contrary the boiling point of water is raised 75 degrees by an augmentation of less than three atmospheres above the atmospheric pressure. Table No. 124 contains a comparative statement of the pressures of the vapours of water and other liquids, at temperatures varying from o C., or 32 F., to 222 C., or 432 F. in fact, their boiling points for various pres- sures the results of experiments by Regnault. The table No. 124 shows a great diversity of pressure of saturated vapours for given temperatures. At the temperature of 2 12 F., for example, at which water boils under one atmosphere of pressure; in other words, at which the pressure of the vapour of boiling water is 14.7 Ibs. per square inch, the pressures of the saturated vapours of the several liquids are as follows : per square inch. Vapour of water at 212 F pressure, 14.7 Ibs. Do. alcohol 32.6 Ibs. Do. ether 95.17 Ibs. Do. chloroform,, 45.54 Ibs. Do. turpentine 2.61 Ibs. The relations of the vapour of water or steam are fully considered in a subsequent section. LATENT HEAT AND TOTAL HEAT OF EVAPORATION OF LIQUIDS. Liquids, in the course of being transformed into vapour on the applica- tion of heat, absorb a certain quantity of heat which remains latent in the vapour, and is, on the contrary, restored to sensibility, and communicated to other bodies when the vapours are condensed into liquids. The following BOILING POINTS OF SATURATED VAPOURS. 371 Table No. 124. BOILING POINTS OF SATURATED VAPOURS UNDER VARIOUS PRESSURES, OR THEIR CORRESPONDING TEMPERATURES AND PRESSURES. Regnault. Pressure per square inch of the vapour of the following liquids: TEMPERATURE. Water. Alcohol. Ether. Chloroform. Turpentine. Centigrade. Fahrenheit. Ibs. Ibs. Ibs. Ibs. Ibs. 32 .089 .246 3-53 .041 10 5 .178 .466 5-54 2.52 045 20 68 -337 -8 5 I 8.60 3-68 .083 30 86 .609 1.52 12.32 5-34 135 40 104 i. 06 2-59 17.67 7.04 .217 50 122 1.78 4.26 24-53 10.14 333 60 I4O 2.88 6-77 33-47 14.27 .520 70 158 4-51 10.43 44.67 18.88 .810 80 I 7 6 6.86 15-72 57-01 26.46 1.18 90 194 10.16 23.02 75-41 35-03 1.74 100 212 14.70 32.60 95-17 45-54 2.61 no 2 3 20.80 45-50 120.9 58.42 3-62 116 240.8 25-37 137.0 I2O 248 29.88 62.05 4-97 130 266 39-27 83.80 6.71 I 3 6 276.8 46.87 I4O 284 52-56 109.1 8.94 150 3O2 69.27 140.4 11.70 I5 2 305-6 73-07 147-3 160 320 89.97 147.3 13.10 170 338 II5-3 I9-I3 1 80 356 146.0 23.70 190 374 182.6 29.30 200 392 226.1 36.09 2IO 410 277.1 43-54 22O 428 336.4 52.04 222 431.6 349-3 53-74 BOILING POINTS UNDER ONE ATMOSPHERE. i st. According to table No. 122. Water. Alcohol. Sulphuric ether. Chloroform. Turpentine. 212 F 173 F 100 F. 140 ] F. 3JS F. 2d. By interpolation in the above table. 212 173 94 142 335 table, No. 125, gives, on the authority of Despretz, Favre and Silbermann, and Regnault, the latent heat of evaporation of several vapours under one atmosphere. The total heat of evaporation, reckoned from 32 F., is added in the last column. It is calculated for each liquid by multiplying its 372 HEAT. boiling point less 32, by its specific heat, to find the quantity of heat in units required to raise it to the boiling point, and adding this product to the latent heat. Table No. 125. LATENT HEAT AND TOTAL HEAT OF EVAPORATION OF LIQUIDS UNDER ONE ATMOSPHERE. Liquid. Boiling Point. Latent Heat of Evaporation. Total Heat, reckoned from 32 F. Sulphuric ether Fahrenheit. 100 Units of heat. 175 Units of heat. 2IO.4 Wood spirit I ^O A.1 ^ CAC Q Acetic ether - 1 J w i6c T-/ J IQI OT-J'? Alcohol pure . .... A ^0 17-2 j.y A 374 4.6l 7 Water . x / o 212 O 1 T- o6c 2 ij.WA. ^ I 146 I Essence of lemon 2Q7 y w j" 6 126 2CC ? Oil of turpentine 31 e 124. J J'O 256 6 LIQUEFACTION AND SOLIDIFICATION OF GASES. Professor Faraday succeeded in liquefying, and even solidifying, many gases, and it is probable that all gases are susceptible of being solidified, and that they might be so condensed if sufficiently low temperatures and sufficiently strong vessels could be produced. At - 112 F., and under a pressure less than one atmosphere, Faraday reduced the following gases to the liquid or the solid state : Chlorine, cyanogen, ammonia, hydrosulphuric acid, arseniated hydrogen, hydriodic acid, hydrobromic acid, and carbonic acid. The following gases were solidified at the annexed temperatures : Cyanogen -31 F. Hydriodic acid - 60 Carbonic acid - 72 Oxide of chlorine - 76 Ammonia - 103 Sulphurous acid - 105 Sulphuretted hydrogen -123 Hydrobromic acid - 126 Protoxide of nitrogen - 148 The following gases were not solidified, even at a temperature of -i66F.: Olefiant gas. Fluosilicic acid. Protophosphuretted hydrogen. Fluoboric acid. Hydrochloric acid. Arseniated hydrogen. The following gases gave no sign of even approaching liquefaction, even at - 166 F., and with many atmospheres of pressure: LIQUEFACTION AND SOLIDIFICATION OF GASES. 373 Hydrogen at - 166 F., and 27 atmospheres. Oxygen - 166 and 27 Do -140 and 58 Nitrogen - 166 and 50 Nitric oxide - 166 and 50 Carbonic oxide - 166 and 40 Coal gas - 166 and 32 The greatest known degree of cold, - 166 F., or - 110 C., was pro- duced for these experiments by Professor Faraday. As to the method of production, see SOURCES OF COLD. According to the results of more recent experiments, hydrogen has been subjected to a pressure of 8000 atmospheres without making any sign of condensation. SOURCES OF COLD. The production of cold the abstraction of heat is a curious subject of inquiry. When a salt is dissolved in water, cold is produced. When a liquid vaporizes, the heat, latent and sensible, necessary for the production of the vapour, is abstracted from some other body in contact with the liquid, and cold is produced. When spirits of wine or aromatic vinegar, for example, is thrown on the body, a sense of cold immediately results from the vaporization of the liquid which draws heat from the body. If air is allowed to expand, there is a reduction of temperature, and a transla- tion of heat from neighbouring bodies. Again, in hot climates, water is successfully cooled in porous vessels, through the pores of which the water passes to the exterior, and is vaporized, and the cooling process is accele- rated by a current of air directed upon the vessel, which quickens the vaporization. Siebe's ice-making machine, invented originally by Jacob Perkins in 1834, is based on the principle of producing cold by the evaporation of a volatile fluid ether by preference. The fluid is placed in an air-tight vessel, and evaporated in vacuo, the vacuum being formed by means of a pump, which, in its continued efforts to reduce the pressure, promotes the evaporation of the fluid at a low temperature. A temperature 50 below the freezing point may be effected; but in place of an unprofitably low temperature, the cooling action is distributed through the mass of salt water employed as the freezing medium, the salt water retaining its fluidity below 32 F., and circulating in the refrigerator around and between the ice-moulds, which are filled with fresh water. The water in the moulds is successively frozen, and replaced by fresh moulds filled with water. Carre's cooling apparatus is based on the fact that water, when cold, absorbs a large quantity of ammoniacal gas, which, when the water is heated, escapes, and is condensed in a cold vessel. On the contrary, when the water just heated becomes cold, a vacuum is formed, and excites a rapid evaporation of the ammonia into the vessel of cooled water, when it is again absorbed. The heat necessary for the evaporation of the ammonia is extracted from the water surrounding the vessel in which the liquid ammonia is contained, and the water consequently is frozen. FRIGORIFIC MIXTURES. For the production of intense cold, mixtures of various salts and acids in 374 HEAT. various proportions with water are very effective. But more intense degrees of cold are produced with snow or ice. Table No. 126 contains the ordinary mixtures for the artificial produc- tion of cold, known as freezing mixtures. The first part of the table com- prises mixtures of salts and acids with each other and with water; the second part, mixtures of salts and acids with snow or ice. The blanks in the third column of the table indicate that the thermo- meter sinks to the degrees named in the second column, but never lower, whatever may be the initial temperature of the materials when mixed. The vessels containing the mixtures should be cooled before the elements are put into them. If the materials of the mixtures enumerated in the first part of the table be mixed at a higher temperature than that given in the table, namely, 50 F., the fall of temperature is greater. Thus, if the most powerful of these mixtures, No. n, be made at the temperature 80 F., it will sink the thermometer to + 2, making a fall of 78 degrees, as against 71 degrees in the table. The third part of the table contains frigorific mixtures partly selected from the other parts, and combined so as to extend the cold to the extreme degree, -91 F. The materials should be cooled previously to being mixed to the initial temperature, by mixtures taken from previous parts of the table. Table No. 126. FRIGORIFIC MIXTURES. FIRST PART. Proportional mixtures of Salts and Acids with Water. Mixtures. Fall of Temperature. Degrees of cold produced. i Nitrate of ammonia i ) Fahrenheit. Fahr. Water T | from +50 to + 4 4 6 2. Muriate of ammonia 5 Nitrate of potash 5> from + 50 to + 10 AO Water T6J 3 Muriate of ammonia c \ Nitrate of potash 5 Sulphate of soda if from + 50 to + 4 4 6 Water T6 4. Sulphate of soda 3 ) Diluted nitric acid 2} from + 50 to - 3 53 5. Nitrate of ammonia Carbonate of soda 4 from + 50 to 7 ^7 Water )] 6 Phosphate of soda A / Q ) Diluted nitric acid 7 Sulphate of soda ...4} . 8 ) from + 50 to- 12 62 Hvdrochlonc acid . ,} from + 50 to o 5 8 Sulphate of soda . . J ) C } Dilute sulphuric acid 47 from + 50 to + 3 47 FRIGORIFIC MIXTURES. Table No. 126 (continued}. 375 Mixtures. Fall of Temperature. Degrees of cold produced. 9. Sulphate of soda 6 \ Fahrenheit. Fahr. Muriate of ammonia U Nitrate of potash *\ from + 50 to- 10 60 Dilute nitric acid 4J 10 Sulphate of soda 6 i Nitrate of ammonia sl from + 5 o to 14 64 Dilute nitric acid 4J 1 1 Phosphate of soda T^ / . Q ) Nitrate of ammonia 6\ from + 50 to 2 1 71 Dilute nitric acid ' 4) 1 T- / SECOND PART. Proportional mixtures of Salts and Acids with Snow or Ice. Mixtures. Fall of Temperature. Degrees of cold produced. 12. Muriate of soda (common salt). Snow, or pounded ice *} Fahrenheit. from any temp, to - 5 Fahr. 1 3. Muriate of soda 2 ) Muriate of ammonia do. do. to- 12 Snow or pounded ice 5 14 Muriate of soda / IO i Muriate of ammonia 5 Nitrate of potash sf do. do. to - 1 8 Snow, or pounded ice 4J 15. Muriate of soda jfl Nitrate of ammonia . B do do. to 25 Snow, or pounded ice '') 1 6. Dilute sulphuric acid 1 } Snow 3} from + 32 to -23 55 1 7 Muriatic acid tr ) Snow 1) from + 32 to- 27 59 1 8 Dilute nitric acid 1 ) Snow 77 from + 3 2 to -30 62 i o Muriate of lime . ... / ) r ) Snow I/ from + 3 2 to -40 72 20. Crystallized muriate of lime Snow . 1} from + 3 2 to -50 82 2 1 Potash A) Snow 3J from + 32 to -51 83 O J 376 HEAT. Table No. 126 (continued}. THIRD PART. Mixtures partly selected from the foregoing series, and combined so as to increase or extend the cold to the greatest extremes. Mixtures. Fall of Temperature. Degrees of cold produced. 22 Sea salt . . r\ Fahrenheit. Fahr. Muriate of ammonia ) I from- 5 to- 1 8 Q Nitrate of potash j Snow, or pounded ice 23 Sea salt 5 "I Nitrate of ammonia i t > from- 1 8 to- 25 Snow or pounded ice 24 Phosphate of soda X^ J c } Nitrate of ammonia . . 3 1 - from o to 34 Dilute nitric acid l) 34 25. Phosphate of soda *r / <2 } Nitrate of ammonia n from 34 to 50 16 Dilute mixed acids 1 26. Snow 3 ) > from o to 46 46 Dilute nitric acid 2 I 27. Snow X) Dilute sulphuric acid 3 1 - from 10 to ^6 46 Dilute nitric acid 3J q.\j 28. Snow O J T ) Dilute sulphuric acid T | from -10 to -60 5o 2 Q Snow Muriate of lime 1 from + 20 to - 48 68 30. Snow 1 ) Muriate of lime . 4/ from+ 10 to- 54 64 31. Snow T- ) 2 \ Muriate of lime 3f from -i 5 to -68 S3 32. Snow... O ) T ) Crystallized muriate of lime 33. Snow 1} from oto-66 66 Crystallized muriate of lime M. Snow... 3} 8 ) from -40 to- 73 33 Dilute sulphuric acid . . . TO} from -68 to -91 23 >W J COLD BY EVAPORATION. M. Gay-Lussac directed a current of air, dried or desiccated by being passed through chloride of calcium, upon the bulb of a thermometer wrapped in moist cambric. The temperature was lowered from 10 to 26 F., according to the temperature of the current of air, which varied from 32 to 77 F. It is presumed that the surrounding temperature was the same as that of the current. The following are the falls of temperature for currents of air of given temperatures : COLD BY EVAPORATION. 377 Temperature of current, Fahrenheit, 32, 41, 50, 59, 68, 77. Fall of temperature, do. io.5, 13, 16, i9.5, 23, 26.$. The most intense cold as yet known was produced by Professor Faraday in the course of his experiments on the liquefaction and solidification of gases, from the evaporation of a mixture of solid carbonic acid and sulphuric ether under the receiver of an air-pump. For the following pres- sures, measured in inches of mercury, and given also in pounds per square inch, he obtained the corresponding temperatures subjoined : Inches of mercury 28.4, 19.4, 9.6, 7.4, 5.4, 3.4, 2.4, 1.4, 1.2. Lbs. per square inch . 14.0, 9.5, 4.6, 3.6, 2.7, 1.7, 1.2, 0.7, 0.6. Temperatures, Fahr. . - 107, - 112, - 121, - 125, - 132, - 139, - 146, - 161, - 166. Showing that when a perfect vacuum was nearly approached, an intense cold, measured by- 166 F., was attained by the evaporation of a mixture of solid carbonic acid and sulphuric ether. STEAM. When steam is generated in a boiler, the water is heated till it arrives at the temperature of ebullition, and the elevation of temperature is sensible to the thermometer; next, the water is converted into steam by an additional absorption of heat, which is not measured by the thermometer, and is therefore called latent heat. The heat is not, in fact, latent, but is appro- priated in converting water into steam, of the same temperature. The pressure, as well as the density, of steam which is generated over water in a boiler rises with the temperature; and, reciprocally, the tempera- ture rises with the pressure and density. There is only one pressure and one density for each temperature ; and thus it is that steam, produced in a boiler over water, is always generated at the maximum density and maxi- mum pressure corresponding to its temperature. In such condition steam is said to be saturated, being incapable of vaporizing more water into the same space, unless the temperature be raised. Saturation is therefore the normal condition of steam generated in contact with a store of water, and the same density and the same pressure are always to be found in conjunc- tion with the same temperature. In consequence, saturated steam over water stands both at the condensing point and at the generating point; that is, it is condensed if the temperature falls, and more water is evaporated if the temperature rises. But, supposing the whole of the water to be evaporated, or that a body of saturated steam is isolated from water, in a space of fixed dimensions, if an additional quantity of heat be supplied to the steam, the state of satura- tion ceases, the steam becomes superheated, and the temperature and the pressure are increased, whilst the density is not increased. Steam, thus surcharged with heat, approaches to the condition of a perfect gas. PHYSICAL PROPERTIES OF STEAM. RELATION OF THE TEMPERATURE AND PRESSURE OF SATURATED STEAM. The results of the experimental observations of M. Regnault on the temperature and pressure of saturated steam, whose observations have superseded in practice those of previous experimentalists, show that the temperature rises more slowly than the pressure. For example, the pressures being advanced at equal intervals of 5 Ibs. per square inch, thus : i Ib. 6 Ibs. ii Ibs. 16 Ibs. 21 Ibs. 26 Ibs. 31 Ibs. 36 Ibs., the temperatures in Fahrenheit degrees are 102. I, I 7 0.2, I 97 .8, 2l6. 3 , 2 3 0.6, 2 4 2.3, 2S2.2, 2 6o. 9 , PHYSICAL PROPERTIES OF STEAM. 379 which advance by the following diminishing differences, 68.i, 2 7 .6, i8.s, i 4 .3, ".7, 9-9> 8-7, Without quoting the formula employed by M. Regnault for calculating the pressures due to the temperatures in French measures, it will suffice to give, in a subsequent table (No. 127), the relative pressures in inches of mercury and in pounds per square inch, based on his formula, for low temperatures ranging from 32 F. to 212 F., as given by Claudel. To define the relation of the temperature and pressure of saturated steam for the higher temperatures comprised in the observations of M. Regnault. the late Mr. W. M. Buchanan arranged a simple formula which applies with accuracy to temperatures ranging from 120 F. to 446 F., the higher limit of the range of Regnault's observations. These limits correspond to pressures of from 1.68 Ibs. to 445 Ibs. per square inch. The formula is as follows:- in which / is the pressure in Ibs. per square inch, and / is the temperature of saturated steam in degrees Fahrenheit, as observed by means of an air- thermometer. TOTAL HEAT OF SATURATED STEAM. The constituent or total heat of steam consists of its latent heat, in addition to its sensible heat. The latent heat of saturated steam at o C, the freezing point, was experimentally determined by Regnault to be equal to 606. 5 C.; or such that the total heat of one pound of saturated steam at o C. would be capable of raising the temperature of 606.5 I DS - f water one degree. At higher temperatures, the total heat of saturated steam was found to increase uniformly between the temperatures o C. and 230 C., at the rate of .305 C. for each increment of temperature of i; and, therefore, if the temperature in Centigrade degrees be multiplied by .305, and 606.5 be added to the product, the sum will express the total heat of saturated steam at the given temperature measured from o C. ; or H = 606. 5 + .305 /(Centigrade), ............... ( 2 ) in which H = the total heat of saturated steam of any temperature f C. This formula is adapted to the Fahrenheit scale, by taking the total heat at 32 F. equal to 6o6.5 C. x 9/ s = 1091. 7 F. For any other temperature / F., the total heat is equal to 1091. 7 F. + .305 (^-32). The first quantity in this expression, namely 1091. 7, is slightly too much, for whilst Regnault found that the total heat of steam at 100, his starting point, was 636.67 C., it was calculated by his formula (No. 2 above) to be 637 C. The above-named quantity should therefore be reduced to 1091.16, and the formula for the total heat of steam in terms of Fahrenheit degrees will stand thus:- H = 1091.16 + .305 (/- 32), or H= 1081.4 +.305 *; .............................. (3) that is, the total heat of saturated steam of any given temperature in Fahrenheit degrees is equal to io8i.4 plus the product of the temperature by .305, supposing that the water from which the steam is generated is supplied at the temperature 32 F. 380 STEAM. The expression of the total heat represents units of heat when the weight of the steam is one pound. Supposing that the water to be evaporated is supplied at any higher temperature than 32 F., the total heat to be expended in evaporating it is found by deducting the difference of temperature from the total heat as found by the formula ( 3 ). Or, the formula may be modified by deducting the difference of temperature from the first quantity, 1081.4. For example, if water be supplied at the ordinary temperature, 62 F., which is 30 degrees above 32 F., then 1081.4-30=1051.4 will be the proper first quantity. For these and the other cardinal temperatures of water, 100 F. and 212 F., the four equations for the total heat of steam raised from water at the respective temperatures are as follow: (Initial temperature 32 F.), H= 1081.4 + .305^ (3) ( Do. 62 R), H = io 5 i.4 + . 3 o5/ (4) ( Do. 100 F.), H=ioi 3 . 4 + .3o5/ (5) ( Do. 212 R), H= 900.5 + . 305^ (6) In the reduction for the last equation ( 6 ), 32 has been deducted from 2i2.9, and not from 212, in order to take into account the item .9, being the extra specific heat of water at 212 F., compared with that of water at 32 F. LATENT HEAT OF SATURATED STEAM. As the total heat is increased .305, which is less than a third of a degree, whilst the sensible heat, or temperature, rises i, and the sensible heat thus rises faster than the total heat, the latent heat must be reduced as the temperature rises, by as much as .305 is less than i, or by i - .305 = .695, for each degree of temperature, and the latent heat for any temperature t C. is expressed by the quantity 6o6.5 - .6g$t. There is a modifying element, namely, the specific heat of the water, which slightly increases with the temperature, and which requires the fraction .695 / to be proportionally increased. The equation of Clausius, in which this slight variation is allowed for, is L=6o7 -.708 / (Centigrade), ( 7 ) where L = the latent heat due to the temperature / C. To adapt this formula to the Fahrenheit scale, take 9/s tns f 607 = io92.6 F., and substitute (t - 32) F. for t C.; then the formula becomes L= 1092.6 - .708 (t 32); or L= 1 1 15. 2 -.708 /; (8) that is, the latent heat of saturated steam at any given temperature in Fahrenheit degrees is equal to 1115.2 less the product of the temperature by .708, supposing that the water which is converted into steam is supplied at 32 F. APPROPRIATION OF THE CONSTITUENT HEAT OF SATURATED STEAM AT 212 F. To trace the appropriation of all the heat that goes to the formation of a pound of steam, in the sensible and the latent state, in terms of thermal units, as well as of dynamic units, or foot-pounds, take one pound of water at 32 F., and convert it into saturated steam at 212 F., the first CONSTITUENT HEAT OF SATURATED STEAM. 381 instalment of heat is the sensible heat, and it is required for elevating the temperature of the water to 2 1 2, through 180; in other words, to increase the molecular velocity, and slightly expand the liquid, which appropriates 180.9 units of heat, equivalent to i8o.9x 772, or 139,655 foot-pounds. Secondly, latent heat is applied in overcoming the molecular attraction, and separating the particles; that is to say, in the formation of steam, which appropriates 892.9 units of heat, equal to 689,318 foot-pounds. Thirdly, latent heat is applied in repelling the incumbent pressure, whether of the atmosphere or of the surrounding steam; that is to say, in raising a load of 14.7 Ibs. per square inch, or 2116.4 Ibs., on a square foot, through a cubic space of 26.36 cubic feet, being the volume of one pound of saturated steam. The work thus done is equal to 2116.4. x 26.36, or 55,788 foot-pounds, or its equivalent, 72.3 units of heat. In strictness, there is the initial volume of the pound of water to be deducted from this total volume, to show the exact volume generated ; but it is relatively very small, and is inconsiderable. The second of the above appropriations of the heat was found by sub- tracting the sum of the first and third, which are both arrived at by direct observation, from the total heat. The first appropriation of heat is thus seen to be the sensible heat, and the second and third together constitute the latent heat. The third, it may be added, is simply an expression of the mechanical labour necessary to disengage 26.36 cubic feet of steam, and force it into space against an atmospheric pressure of 2116.4 Ibs. per square foot. The appropriation of the heat expended in the generation of one pound of saturated steam at 212 F., from water supplied at 32 R, may be exhibited thus: To GENERATE ONE POUND OF STEAM AT 212 F. Units of heat. The sensible heat: 1. To raise the temperature of the water from 3 2 to 2 1 2 F., 180.9 The latent heat : 2. In the formation of steam ... 892. 93^ 3. In resisting the incumbent atmospheric pressure of 1 4. 7 Ibs. per square inch, or 2116.4 Ibs. per square foot, 72.265 965.2 Total or constituent heat, 1 1 46. i Mechanical equivalent in foot-pounds. 139^55 689,346 55,788 745 I 34 884,789 VOLUME AND DENSITY OF SATURATED STEAM. The density of steam is expressed by the weight of a given constant volume, say, one cubic foot; and the volume is expressed by the number of cubic feet in one pound of steam. The density and volume, which are the reciprocals of each other, have not yet been accurately ascertained by direct experiment. They are, however, determinable in terms of the pressure, temperature, and latent heat of steam, all of which have been experimentally ascertained, by means of the mechanical theory of heat. 382 STEAM. Mr. Brownlee has deduced a simple expression for the density of saturated steam in terms of the pressure, as follows : or, log D-.94I Iog/-2.5i9, ......................... (10) in which D is the density, and p the pressure in Ibs. per square inch. In this expression, p-w is the equivalent of p, as employed by Dr. Rankine ; and it is simpler to handle. The equation signifies that the logarithm of the pressure is to be multiplied by .941, and that 2.519 is to be subtracted from the product ; the remainder is the logarithm of the density, from which the density is found by means of a table of logarithms. The results presented by the above formula are very accurate; they do not differ from those obtained in terms of the temperature and the latent heat, for pressures of from i Ib. to 250 Ibs. per square inch, by more than one-seventh per cent. The volume being the reciprocal of the density, then, putting V for the volume, or log = 2.519 -.941 log/; ......................... (12) that is, that if the logarithm of the pressure in Ibs. per square inch be multiplied by .941, and the product be deducted from 2.519, the remainder is the logarithm of the volume, in cubic feet, of one pound of saturated steam. The nearness of the power, .941, to unity, indicates that the density of saturated steam varies nearly as the pressure, but in a lower ratio; and that the volume of saturated steam varies, for short intervals, nearly in the inverse ratio of the pressure. For example, the pressures per square inch being i, 2, 4, 8, 16, 32, 64, 128 Ibs., the densities, or weights per cubic foot, which are inversely as the volumes, are .0030, .0058, .0112, .0214, .0411, .0789, .1516, .2911 Ibs., being in the ratios of i, 1-93, 3-73, 7-13, 13-7, 29.3, 50.5, 97. RELATIVE VOLUME OF SATURATED STEAM. The relative volume of saturated steam is expressed by the number of volumes of steam produced from one volume of water, the volume of water being measured at the temperature 62 F. The relative volume is found by multiplying the volume, in cubic feet, of one pound of steam by the weight of a cubic foot of water at 62 F., which is 62.355 Ibs. Or, it may be found directly in terms of the pressure, by multiplying the second member of the formula (n) by 62.355. Thus, putting n for the relative volume, GASEOUS STEAM. 383 = 62.355 x 33.36 or, log = 4.31388 -(.941 *log/); .................. (14) that is, if the logarithm of the pressure in Ibs. per square inch be multiplied by .941, and the product be deducted from 4.31388, the remainder is the logarithm of the relative volume. GASEOUS STEAM. When saturated steam is superheated, or surcharged with heat, it advances from the condition of saturation into that of gaseity. The gaseous state is only arrived at by considerably elevating the temperature, supposing the pressure remains the same. Steam thus sufficiently superheated is known as gaseous steam, or " steam-gas," as Dr. Rankine has named it. The test of perfect gaseity is the uniformity of the rate of expansion with the rise of temperature; and, whereas, during the first few degrees which follow the temperature of saturation, the rate of expansion is notably greater than that of air, the rate diminishes at still higher temperatures, and ultimately becomes uniform, like that of the expansion of permanent gases. Dr. C. W. Siemens, experimenting on the expansion of isolated steam, generated at 212, and superheated and maintained at atmospheric pressure, found that expansion proceeded rapidly until the temperature rose to 220, and less rapidly up to 230, or 18 above the saturation point; above which it expanded uniformly, as a permanent gas. Up to 230, the expansion was five times as much as that of air. Messrs. Fairbairn and Tate found that for steam generated at low temperatures of saturation, under 150 F., the rate of expansion when the steam was heated was nearly uniform. At 175 F., the expansion for the first five degrees averaged more than three times that of air; above that point, it was nearly the same. For steam generated at the high temperature of 324 F., for a total pressure of 95 Ibs. per square inch, the rate of expansion up to 331 was nearly three times that of air; and for the next 25 degrees, one-sixth greater. M. Regnault concluded from his experiments that saturated steam was nearly gaseous at temperatures below 60 F. It may be gathered from these observations that saturated steam of ordinary temperatures may be made gaseous by superheating it to the extent of from 10 to 20 degrees. It is thought that the rapidity of expan- sion by heat, near the boiling point, is to be accounted for by the supposed insensible moisture of steam in the saturated condition, as generated from water, being evaporated and contributing to increase the quantity of steam without raising the temperature. This argument is plausible; but it might be argued, on the contrary, that in the converse process, of abstracting heat from superheated steam, the accelerated reduction of volume when it ap- proaches the point of saturation, is due to incipient condensation, which would be absurd. It may be inferred, further, that saturated steam of very low temperatures, under 150 or 100 F., is gaseous. 384 STEAM. TOTAL HEAT OF GASEOUS STEAM. Regnault found that the total heat of gaseous steam increased, like that of saturated steam, uniformly with the temperature; and at the rate of .475 for each degree of temperature, under a constant pressure. A formula for the total heat of gaseous steam may be constructed on the basis of that for saturated steam, by a modification of the constants; and for the adjust- ment of these, take the two steams at a low temperature, as 40 F., where they are identical in constitution, both being gaseous. Then, by formula ( 3 ) for saturated steam, the total heat at this temperature is 1081.4 + (.305 x 40)= io93.6 F. Substituting for the second quantity in this equation, the quantity (.475 x 40), and reducing, then 1074.6 + (.475 x 40) = io93.6 F. Whence the general formula for the total heat of gaseous steam, produced from water at 32 F., H' = 1074.6 + .475 /, (15) FT being the total heat, in Fahrenheit degrees, and / the temperature ; that is, that to the constant 1074.6, is to be added the product of the tempera- ture by .475, to find the total heat. By this formula it is found that the total heat of gaseous steam at 212 F., and at atmospheric pressure, is 1175.3 F., which is 29.2 degrees, or 2% per cent, more than that of saturated steam. SPECIFIC HEAT OF STEAM. The specific heat of saturated steam is .305, that of water being unity; or, it is 1.281, that of air being unity. It may be noted that .305 is the quantity by which the total heat of saturated steam is increased for each degree of temperature (see formula 3); so that equal intervals of tempera- ture correspond to equal quantities of heat. The expression, .305, for specific heat, is taken in a compound sense, comprising the changes both of volume and of pressure which take place in the production of saturated steam. The specific heat of gaseous steam is .475, under constant pressure, as found by Regnault. It is upwards of a half more than that of saturated steam. It is identical with the increase of total heat for each degree of temperature (formula 15). THE SPECIFIC DENSITY OF STEAM. The specific density of gaseous steam has been found by M. Regnault to be .622, that of air being i. That is to say, that the weight of a cubic foot of gaseous steam is about five-eighths of that of a cubic foot of air, of the same pressure and temperature. The specific density of saturated steam is usually taken at the same value as that of gaseous steam, as an approximation to the actual value. Thus approximated, it is only correct at very low temperatures, for the specific density increases, though not rapidly, with the temperature, inso- much that though it is practically the same as that of gaseous steam at 100 F., it becomes .643 at 212 F.; and at 303 F., with 70 Ibs. absolute PROPERTIES OF SATURATED STEAM. 385 pressure per square inch, it becomes .664, or two-thirds of that of air. At 358.3 R, with 150 Ibs. pressure, it is .681. (See table No. 129.) DENSITY OF GASEOUS STEAM. The density or weight of a cubic foot of gaseous steam is expressible by the same formula as for that of air (page 350), except that the multiplier or coefficient is less in proportion to the less specific density, thus : TV _ 2 -7Q74/ x -622 _ 1.684 / / T6 x ~ ~'' in which D' is the weight of a cubic foot of gaseous steam,/ the total pressure per square inch, and t the temperature by Fahrenheit. TABLES OF THE PROPERTIES OF SATURATED STEAM. The first table, No. 127, of the properties of saturated steam of tempera- tures ranging from 32 to 2 1 2 F. is adapted from a table prepared by Claudel, partly based on Regnault's formulas, and partly on the assumption that the specific density of saturated steam is uniformly .622, or about five-eighths that of air at the same temperature. As already mentioned, the specific density increases, in fact, slightly with the temperature, and this deviation from uniformity explains the small discrepancies between the weights of steam as given in table No. 127, and those as given for temperatures below 212 in the next table. The table No. 128 gives the properties of saturated steam for pressures of from i Ib. per square inch to 400 Ibs. per square inch, the temperatures ranging from 102 to 445 F. The first column contains the ascending total pressures in Ibs. per square inch. The second column, of tempera- tures, was calculated from the pressures by means of the formula ( i ) : / = - 2 93 8 - 16 _ -371.85. 6- 1993544 -log/ The third column, of the total heat of saturated steam, by formula ( 3 ) : H= 1081.4 + . 305 /. The fourth column, of the latent heat of saturated steam, by formula ( 8 ) : L= 1115.2 - 708 /. The fifth column, of the density of saturated steam, by formula (10): log D = . 94 i log/- 2.519. The sixth column, of the volume of saturated steam, was calculable by finding the reciprocals of the densities, or by formula (12) : log -2.519 -.941 log/. The seventh column, of the relative volume of saturated steam, by the formula (14): log n = 4.31388 - (.941 x log/). The table No. 129 contains the comparative densities and volumes of air and saturated steam for pressures up to 300 Ibs. total pressure per square inch, and temperatures to 4 17. 5 F., together with the specific density of saturated steam. 25 3 86 STEAM. Table No. 127. PROPERTIES OF SATURATED STEAM FROM 32 TO 212 F. PRES 5URE. Total heat Volume of one TEMPERATURE. Inches of Lbs. per reckoned from water at 32 F. VV eight of 100 cubic feet. pound of vapour. mercury. square inch. Fahrenheit. inches. Ibs. Fahrenheit. Ibs. cubic feet. 3 2 ... ... .l8l... ... .089... ... I09I.2 ... ... .031 ... ...3226 35 .204 .100 IO92.I 034 2941 40 .248 .122 1093.6 .041 2439 45 ... .299... ... .147... ... I0 9 5.I ... ... .049... ...2041 5o .362 .178 1096.6 059 1695 55 .426 .214 1098.2 .070 1429 60 ... ... .517... ... .254... ... 1099.7 ... ... .082... ...1220 65 .619 304 IIOI.2 .097 1031 70 733 .360 II02.8 .114 877.2 75 ... .869... ... .427... ... 1104.3 ... .134... ... 746.3 80 1.024 503 II05.8 .156 641.0 35 1.205 592 II07.3 .182 549-5 QO . 1.410... 607 1 108.9 .212 ... 471 7 y T^ w yo' T- / / 95 1.647 .809 IIIO-4 .245 408.2 100 1.917 .942 IIII.9 283 353-4 105 ... ... 2.229... ... 1.095... ... III3.4 -.. ... .325 ... ... 307.7 no 2 -579 1.267 III5.0 "373 268.1 115 2.976 1.462 IIl6.5 .426 234-7 120 ... ... 3.430... ... 1.685... ... IIlS.O ... ... .488... ... 204.9 I2 5 3-933 1.932 III9.5 554 180.5 130 4-59 2.215 II2I.I .630 158.7 135 . 5.174... ... 2.542... ... II22.6 ... ... .714... ... 1 40. i 140 5.860 2.879 II24.I .806 124.1 145 6.662 3-273 II25.6 .909 IIO.O 150 ... ... 7-548... ... 3.708... ... II27.2 ... ... .022 ... ... 97.8 J 55 8-535 4.193 II28.7 145 87-3 1 60 9.630 4-73 1 II30.2 333 75-o 165 ... ...10.843... ... 5.327... II3I.7 . ... .432... ... 69.8 170 12.183 5.985 H33.3 .602 62.4 175 13-654 6.708 II34.8 774 5 6 -4 180 ... ...15.291... ... 7.511... ... 1136.3 ... ... 1.970 ... ... 50.8 185 17.044 8.375 II 3 7.8 2.181 45-9 190 19.001 9-335 H39.4 2.411 41-5 195 ... ...21.139... ...10.385... ... II40.9 ... ... 2.662 ... ... 37.6 200 23.461 11.526 II42.4 2-933 34-i 205 25-994 12.770 H43-9 3-225 31.0 210 ... ...28.753... ...14.126... ... 1145.5 3-543 . ... 28.2 212 29.922 14.700 II46.I 3-683 27.2 PROPERTIES OF SATURATED STEAM. Table No. 128. PROPERTIES OF SATURATED STEAM. 387 Total pressure per square inch. Temperature in Fahrenheit degrees. Total heat, in Fahr. degrees, from water at 32 F. Latent heat, Fahrenheit degrees. Density, or weight of one cubic foot. Volume of one pound of steam. Relative volume, or cubic feet of steam from one cubic ft. of water. Ibs. Fahr. Fahr. Fahr. Ibs. cubic feet. Rel. vol. I 1 02. 1 III2.5 1042.9 .0030 330.36 20600 2 126.3 III9.7 1025.8 .0058 172.08 10730 3"- ... I4I.6 ... ... II24.6 ... ... IOI5.O ... ... .0085 ... ... 117.52... ... 7327 4 I53.I II28.I 1006.8 .0112 89.62 5589 5 162.3 II30.9 1000.3 0138 72.66 4530 6... ... 170.2 ... ... 1133.3 . ... 994-7... ... .0163 ... ... 6l.2I ... ... 3816 7 176.9 H35-3 990.0 .0189 52.94 3301 8 182.9 1137.2 985.7 .0214 46.69 2911 9... ... 188.3... ... 1138.8 ... ... 981.9... ....0239... ... 41-79- ... 2606 10 193.3 1140.3 978.4 .0264 37.84 2360 ii 197.8 1141.7 975-2 .0289 34.63 2157 12 ... 2O2.O Q72 2 O^T A. 31 88 IQ88 13 205.9 1144.2 969.4 .0338 29.57 1844 14 209.6 H45-3 966.8 .0362 27.61 1721 IA 7 .. ... 2I2.O ... ..ii 46. i n6 c, 2 ... .0380 ... 2636 1642 *+ / * 15 2I3.I 1146.4 ... yv^^.^i 964.3 .0387 25.85 1611 16 216.3 1 147.4 962.1 .0411 24.32 1516 17 2IQ 6 1148.3 ... QCQ 8 O/13C 22 06 . 1432 * yv* .. . ... V3V-" > ... ^^,.yw ... T^ O 18 222.4 1149-2 957-7 .0459 21.78 1357 19 225.3 1150.1 955-7 .0483 20.70 1290 20 ... ... 228.0... ... 1150.9... ... 953.8... ....0507... ... 1972... ... 1229 21 230.6 1151.7 95L9 0531 18.84 1174 22 233.1 1152.5 950.2 .0555 18.03 1123 23.. -.235.5... ... 1153.2... ... 948.5 ... ... .0580... ... 17.26... ... 1075 24 237.8 II53-9 946.9 .0601 16.64 1036 25 240.1 1154.6 945-3 .0625 15.99 996 26.. ...242.3... ... 1155.3... ... 943-7 ... .0650 ... ... 15.38... ... 958 27 2444 1155.8 942.2 .0673 14.86 926 28 246.4 1156.4 940.8 .0696 14-37 895 29.. ... 248.4... ... 1157.1 ... ... 939.4... ....0719... ... 13.90... ... 866 30 250.4 1157.8 937-9 .0743 13.46 838 31 252.2 1158.4 936.7 -.0766 I3.05 813 Art 2C/L I ...i 158.9 ... 0780 . 12.67 ' 780 33 255.9 1159.5 934-0 .. . ,\J 1 <_>y .. . .0812 12.31 l^-y 767 34 257.6 1 1 60.0 932.8 -0835 11.97 746 2C 2CQ 7 ... 1 1 60. 5 ... 031.6 ... .0858 ... 1 1.65 726 3 260.9 1161.0 930-5 .0881 11-34 * * " / ~ W 707 37 262.6 1161.5 .0905 11.04 688 38 264.2 . ...i 162.0 ... 928.2 ... 10 76 671 39 265.8 1162.5 927.1 .0952 ... A >-. / \j . . . 10.51 w/ * 655 40 267.3 1162.9 926.0 .0974 10.27 640 41 .. ...268.7... ... 1163.4... ... 924.9... ....0996... ... 10.03... ... 625 42 270.2 1163.8 923-9 .1020 9.8l 6n 43 271.6 1164.2 922.9 .1042 9-59 598 44. 273 O 1164.6 ... .1065 ... 585 45 274.4 1165.1 920.9 .1089 9 9 .?s" j ^ 572 46 275.8 1165.5 919.9 .1111 9.00 561 47 .. ... 277.1 ... ... 1165.9... ... 919.0... ....1133... ... 8.82... ... 550 48 278.4 1166.3 918.1 .1156 8.65 539 49 279.7 1166.7 917.2 .1179 8.48 529 50... ... 281.0... ... 1167.1 ... ... .1202 ... ... 8.31 ... ... 518 3 88 STEAM. Table No. 128 (continued). Total pressure per square inch. Temperature in Fahrenheit degrees. Total heat, in Fahr. degrees, from water at 32 F. Latent heat, Fahrenheit degrees. Density, or weight of one cubic foot. Volume of one pound of steam. Relative volume, or cubic feet of steam from one cubic ft. of water. Ibs. Fahr. Fahr. Fahr. Ibs. cubic feet. Rel. vol. 51 282.3 1167.5 9154 .1224 8.1 7 509 52 283.5 1167.9 9H.5 .1246 8.04 500 53." ...284.7... ...II68.3... ... 913.6 ... ... .1269 ... .. 7.88 ... ... 491 54 285.9 II68.6 912.8 .1291 774 482 55 287.1 1169.0 912.0 I3H 7 .6l 474 56... ...288.2... ... 1169.3 ... ... 9II.2 ... ....1336... .. 748 ... ... 466 57 289.3 1169.7 910.4 .1364 7.36 458 58 290.4 II70.0 909.6 .1380 7-24 45 l 59- ... 291.6 ... ... 1170.4... ... 908.8 ... ....1403 ... .. 7.12 ... ... 444 60 292.7 II70.7 908.0 .1425 7.01 437 61 293.8 II7I.I 907.2 .1447 6.90 43 62... ...294.8... ... II7I.4... ... 906.4 ... ... .1469... .. 6.81 ... ... 424 63 295.9 II7I.7 905.6 H93 6.70 417 64 296.9 II72.0 904.9 .1516 6.60 411 65... ... 298.0... ... II72.3 ... ... 904.2 ... ....1538... ... 6.49 ... ... 405 66 299.0 II72.6 903.5 .1560 6.41 399 67 300.0 II72.9 902.8 .1583 6.32 393 68 3OO.Q . , 1 17"? 2 .. QO2.I . ... .1605 ... ... 6.23 ... ... 388 69 301.9 II73-5 ;/ 901.4 .1627 v. j 6.15 383 70 302.9 II73.8 900.8 .1648 6.07 378 71... 3 3.9. " ... II74.I ... ... 900.3 ... ... .1670 ... ... 5.99 ... . 373 72 304.8 H74.3 8 99 .6 .1692 368 73 305.7 II74.6 898.9 .1714 5-83 363 74-.. ...306.6... ... 1174.9... ... 898.2 ... ... .1736 ... 5.76 ... 359 75 3 7.5 II75.2 897.5 1759 5.68 353 76 308.4 II754 896.8 .1782 5.61 349 77... ...309.3... ... II75.7 ... ... 896.1 ... ... .1804... ... 5.54 ... .- 345 78 310.2 II76.0 895.5 .1826 5.48 34 r 79 3II.I II76.3 894-9 .1848 5.41 337 80 3I2O... 1 176. t\ SQA i ... .1869 ... c ->c 333 O\J. . * ... 0^4.^) ... 81 312^8 II76.8 . 893.7 .1891 5.29 329 82 3I3.6 II77.I 893.1 1913 5.23 325 83,.. ... 3J4-5 ... II774... ... 892.5 ... ....I935". ... 5.17 ... ... 321 84 3I5.3 II77.6 892.0 1957 5.11 318 IL 316.1 ... 316.9... II77.9 ... II78.I ... 891.4 ... 890.8 ... .1980 ... .2002 ... 5.05 ... 5.00 ... 3H 87 317.8 II78.4 890.2 .2O24 4.94 308 88 318.6 II78.6 889.6 .2044 4.89 305 80 2TO A 1 178 Q 889.0 ... 2O67 . . . 4.84 . 2QJ oy... ... i i / o.y . . .... 4*\~>\j j . 9 320.2 II79.I " 888.5 .2089 4-79 298 9i 321.0 II79-3 887.9 .2111 4-74 295 92... ...321.7... ... II79.5 .-" ... 887.3 ... ....2133... ... 4.69 ... ... 292 93 322.5 II79.8 886.8 .2155 4.64 289 94 323.3 IlSo.O 886.3 .2176 4.60 286 9f 1180.3 ... 885.8 ... .2198 ... A er 283 5... 96 324*8 II80.5 885.2 .2219 4.51 281 97 325.6 1 1 80.8 884.6 .2241 4.46 278 98... ... 326.3 ... ... IlSl.O ... ... 884.1 ... ... .2263 ... ... 4.42 ... ... 275 99 327.1 1181.2 883.6 .2285 4-37 272 100 327.9 Il8l.4 883.1 .2307 4-33 270 IOI... ...328.5... ... 1181.6... ... 882.6 ... ....2329... ... 4.29 ... ... 267 PROPERTIES OF SATURATED STEAM. 389 Table No. 128 (continued}. Total pressure per square inch. Temperature in Fahrenheit degrees. Total heat, in Fahr. degrees, from water at 32 F. Latent heat, Fahrenheit degrees. Density, or weight of one cubic foot. Volume of one pound of steam. Relative volume, or cubic feet of steam from one cubic ft. of water. Ibs. Fahr. Fahr. Fahr. Ibs. cubic feet. Rel. vol. 102 329.1 1181.8 882.1 .2351 4.25 265 103 329-9 1182.0 881.6 2373 4.21 262 104... ...330.6... .. 1182.2 ... .. 88I.I ... ...2393... .. 4.18 ... .. 260 105 331-3 1182.4 880.7 .2414 4.14 257 1 06 331-9 1182.6 880.2 2435 4.11 255 I O7 332 6 .. 1182.8 ... .. 87Q.7 .. . . 24.^6 . . 4. O7 2C3 j.\-Y 108 jj^"*- 333-3 1183.0 T/y/ 879-2 ^TO W .2477 4.04 251 109 334-0 1183-3 878.7 .2499 4.00 249 I IO 334. 6 . 1183 ; .. .. 878.3 .. . .2S2I . 3 Q7 247 in jjf"*' 335-3 ..ii '-'JO 1183.7 * **/ *-'* J ' 877.8 "3 * -2543 3-93 245 112 336.o 1183.9 877-3 .2564 3-90 243 I I -3 336 7 1 184.1 ... .. 876.8 .. . 2;86 . . 386 24.1 i i j. . . 114 jj v -'-/ 337-4 1184.3 **/ ^^-' 876.3 2 .2607 ... J.JV/ 3.83 239 H5 338.0 1184.5 875.9 .2628 3.80 237 116... ...338.6... .. 1184.7 ... .. 875.5 . ... .2649... ... 3.77 -.. 235 117 339-3 1184.9 875.0 .2652 3.74 233 118 339-9 1185.1 874-5 .2674 3.71 231 119,.. ...340.5 ... ..1185.3... .. 874.1 ... ....2696... ... 3.68 ... ... 229 120 34I-I 1185.4 873.7 .2738 3.65 227 121 341.8 1185.6 873.2 .2759 3.62 225 122... ... 342.4... .. 1185.8 ... ... 872.8 ... ....2780... ... 3.59 ... ... 224 123 343-0 1186.0 872.3 .2801 3.56 222 124 343-6 1186.2 871.9 .2822 3-54 221 125... ...344.2 ... .. 1186.4... ... 871.5 ... ....2845... ... 3.51 ... ... 219 126 344-8 1186.6 871.1 .2867 3-49 217 127 345-4 1186.8 870.7 .2889 3'46 215 128... ... 346.0... .. 1186.9... ... 870.2 ... ... .2911 ... ... 3.44 ... ... 214 I2 9 346.6 1187.1 869.8 -2933 3-41 212 I 3 347-2 1187.3 869.4 -2955 3.38 211 131... ...347-8... ...1187.5... ... 869.0 ... ....2977... ... 3.35 ... 209 I 3 2 348.3 1187.6 868.6 .2999 3-33 2o8 133 348.9 1187.8 868.2 .3020 3-3 1 206 I 34. 34.Q.H .. ... 1188.0 ... 867 8 3O4.O . 3 2Q 2o S 135 350.1 1188.2 8674 .... jw-pj . . .3060 ... j^y 3-27 &w 3 203 136 350.6 1188.3 867.0 .3080 3-25 202 177.. .. 3 O m O O ON M c^ T^ \o N O ro ^ ON CO O roO ON M M M M W r|- too ^OO ro f dry air re- Ib. of vapour, ed mixture. Volume at 62 F. M CO Tf o co N oo M 2 O ro -xi-00 ro rf C^ O OO J>- MMM ON to r~- rocO ro ^" N O M ON o M-a *ii ^ CO Tf CN^- too ^ O O O Th too to O w \o IV bfl J CO rt- M 00 O oo M o o to O ON ^ vO to toco M i>- ro t- ^ d M M tlis * M MO MO d t-GO ro 00 roOs^-ON * o to 1| g-s i M M ro too 2 ON ON ON ON ON 300000 00 ON M M Tf ON ON O O O to t^CO O M ro toO JJ|S 3 fa O . i H . CO CO 00 rO ro M ON 1>O t* ^0 t-ONM o* 15 3 2 22 5 * cg' S ro ^ to t^ ON OO O ^ ON M M ^f O OO M M M M M CV| tOCO ro t^ N t- ro ON ^ 10 in . *2* " fa 5 . ^ M ON ON M t^ 3&g 1 .t; o to 10 T}- N ~ M N ro M O 00 ^t ro Tj- to tOO J^- M oo to M r^. 00 CO ON O O M CO M M M CM -P fl .CO M CO O f^ 00 M oo O ON 100 M M f CO 00 "3- ii I .- ro i>- *-f ro ^f ro ro rf too O O O to t MMM CM M U") CM T^ M N M M CO M IOO O M J>. . turated ixture. ^ O O CNCO r^ CO ON t^ to W O to to ^f ro M OO CO ON TJ- M M M o ON M o co t-o o 3 S t^ r- r- t-0 o o o 3 8 IM JM* ro O ON to ro N t^ tco ON Ja O ONOO t^-O CO t- O M CO to to Tf ro M O ON ONO ^1- M ONOO r-o tOCO rf- O O CM to ro M 00 l^ t^ t- f t^O O O O O O O 1 cS ^ M Tf M ON ON - o o 5 o o o^ 5-^=5 O M CN to CO tooo M TJ-OO M M M CM CM to fOO CO ro T ^ - _ 1 2-S . 1 5 1 & S ft O O O O ro t^oo i> M co ON O O ON to 10 COCO O ro ro 11 MMMMM ad 35T o | ON O M t^OO . CO O a ^ i>. sHi? ro M ro ON CM to too OO ON ONO O H * MMM 111" 1 . o 2 a to O to O os ro ro ^" "^t" in fe m o m O m too o r- t^ O m o m o 00 00 ON ON O to o m AIR AND AQUEOUS VAPOUR. 397 .s-s O^.-grtS vO 10 N OC i> w i-O O ^ CO O O 10 O OO^-fO-*rJ- M\OOOM>O t^^OO-^- ON M covO 00 M vO M M ro rOOO O CN ||j O M OOOOVOOOOM \o O NCXJO ' vo Tf M O 00 t^-vo 3 M M M M CONNMM M J> coxO ro vOi>>OMOO t^ Tf rj-vo O o" d d d d s^Us 1-S 3?a2 O^gog OVOMVOM coco roco ON rj- CN m M IOI>.COOI-H MMwroro 33 ^vh' II 3.2 s ci O 06 r"- CO* CO i>-CO voco* O vd N M 06 M* 3 vO t^-OO ON O MM TJ-VO t^ ON M co vooO M ^J- t^ O N M MMMMM MMWMN CO^OCOTfTj- co <* N w i>> N vo M N N vq vq co rj-co o ON coco o .5 vO O co vovo t^vO vo W CO cJ vovo vd co O"N M N O^ O CNCOCOCOCO COCOCOCOM WMO ONOO vO vo co voM- i>- d ONOO* M 3 vovo t^-OO ON O M cj vovO OO O M ^ ** O covo O C4 MMMMM MNttNCM COCOCO^i-Tj- ra tu mMTfCNt^ MVOVOMM cocoO^OCO 10 10 Tt- CO co M CNVO t^OO ^i>-Oco^}- "3-COM r-00 IO^CONM OCO f^vo -^J- COMOCOVO Tj-N CO O t^ Tj- vo VO ON ONVO vo ONMVOMM MQvoNO VOOMC^N OvOM-^-to coOcOvocooOOOQcoO M OO vO co O t*- ^ O vO N t > co t* N O IO '^"^j"''^"'^" COCOCONM MMOOO CNWIOCOM CS^OMM Mcoiococo iOCOMQ O^^fOco Ot^ t^OO M VO co C4 ^-OO tovO t-. CO ON O M co ^ vO t^ ON M ^- vO ON M vovO 1000 loco M M vo OO vo w M ON O VO M O O* ON ON ,OO lOMONtoto M ONCO N vO M t-- co -t>> t^CNMCOco COMONVO vo N vo N ON COCO co M COcOM^t-t^ r^QONco vO ON a vooO wr^Mf^ VOOO M vo vo vovo O VO O ONt^VOCOCO COVOt^.MJ>. lOVO l>-od ON O M N TJ- rf OvoOvoO voovoOvo OvoOvoO voOvoON C) N co co ^" Tj~ vo vovO vO *> fOO CO ON ON O O M M COMBUSTION. The combustible elements of fuel are carbon, hydrogen, and sulphur. There are other elements in fuel nitrogen, water, and solid incombustible matter which do not take part in combustion. Fuel is burned with atmospheric air, of which the oxygen combines with the combustible matter, whilst the nitrogen remains neutral. The combining proportions of the elements concerned in or about combustion are given in table No. 131 : Table No. 131. COMPOSITION AND COMBINING EQUIVALENTS OF GASES CONCERNED IN THE COMBUSTION OF FUEL. (OLD NOMENCLATURE.) Gases. Elements of the Gases. Combining Equivalents. By Weight. By Measure. ELEMENTS : Oxvsren Equivalents. Oxygen, i Hydrogen, I Carbon, i Sulphur, i Nitrogen, i Carbon, 2 Hydrogen, 4 Carbon, 4 Hydrogen, 4 Oxygen, 23 Nitrogen, 77 Oxygen, i Carbon, i Oxygen, 2 Carbon, i Oxygen, i Hydrogen, i Oxygen, 2 Sulphur, i 8 One Volume = Q Hvdrosren Carbon, 6 Sulphur 16 Q Nitrogen . 14. n COMPOUNDS : Light Carburetted Hy- ( drogen, ( 12 4 24 4 8 26.8 8 6 16 6 8 ,6 16 = 16 - 28 - - 1 ' i Olefiant Gas, \ Atmospheric Air (me- \ chanical mixture), ) Carbonic Oxide, 34-8 = 14 = 22 = 9 = 32 ) m 5 ~; a D (ideal) B D (ideal) a | D (ideal) i i i CEP ipproximately - n Carbonic Acid Aqueous Vapour or ( Water ( - n = a Sulphurous Acid, j The volume of one pound of the principal gases at 62 F., under one atmosphere is as follows: CHEMISTRY OF COMBUSTION. 399 GAS AT 62 F. ONE POUND. Cubic feet. Oxygen, ................................................ 11.887 Hydrogen, ............................................. 1 90.000 Nitrogen, .............................................. J 3- 5 o i Air...... ...... . ......................................... 13-141 Carbonic Acid, ....................................... 8.594 Aqueous Vapour, as Gaseous Steam, ............. 21.125 Sulphurous Acid, .................................... 5.848 The source of oxygen, as the supporter of the combustion of fuel, is atmospheric air, which consists of oxygen and nitrogen in mechanical combination, in the proportion of 8 to 26.8; or i Ib. of oxygen to 3.35 Ibs. of nitrogen; or, by volume, i cubic foot of oxygen to 3.76 cubic feet of nitrogen. For every pound of oxygen employed in combustion, 4.35 Ibs. of air are consumed; or, by measure, for every cubic foot of oxygen employed in com- bustion, 4.76 cubic feet of air are consumed. For the combustion of one pound of hydrogen, of carbon, and of sulphur, therefore, the quantities of air chemically consumed are as follows : One Pound. Hydrogen consumes ............. 34.8 Ibs., or 457 cubic feet, of air at 62. ' 6 lbs " OT '5 do - do - 5-8 lbs., or 76 do. do. Sulphur consumes: ............... 4.35 lbs., or 57 do. do. The process of their combustion is indicated in the following tablets COMBUSTION OF HYDROGEN. Elements. Process. Products. ' ' I nitrogen, 26.8 pounds, ... 2 6. 8 pounds nitrogen, 35-8 35-8 35-8 COMPLETE COMBUSTION OF CARBON. i pound carbon,.. carbon, i pound,.. 1 , x i 6 pounds air { ^ 2 ' 66 PA 1 lir ''" \ nitrogen, 8.94 pounds, ...8.94 pounds nitrogen. 12.6 12.6 12.6 COMBUSTION OF SULPHUR. i pound sulphur... sulphur, i pound ) -j . oxygen, i pound } 2 P OUnds sul P hurous aci lir ''" \ nitrogen, 3.35 pounds. ..3. 35 pounds nitrogen. 5-35 5-35 5-35 4366 C = the weight of carbonic acid ( a ) 9H + 100= .o9H = the weight of steam (b ) 28+- 100 = .02 S =the weight of sulphurous acid ( c ) To this is to be added the weight of atmospheric nitrogen separated from the oxygen chemically consumed, and the weight of the constituent nitrogen, N, of the fuel. The quantity of atmospheric nitrogen is 3.35 times, by weight, that of the oxygen consumed; and, Pound. Pounds. For i carbon there are 2.66 x 3.35 = 8.93 nitrogen. For i hydrogen there are 8 x 3.35 = 26.8 do. For i sulphur there are IX 3-35 3-35 do. Multiply each of these quantities by their respective percentages of combus- tible, and divide by 100; the sum of the quotients is the weight of nitrogen separated from the atmospheric oxygen consumed. To this is to be added the constituent nitrogen of the fuel: 8.93 C-T-IOO = . 0893 C. 26.8 H-r- loo- .268 H. 3.35 S -MOO = .0335 S. N-T- 1 00 = .01 N. Thus, the total weight of nitrogen is equal to (.0893 C + .268 H + .0335 S + .oi N) (d) Add together the total weights of carbonic acid, steam, sulphurous acid, and nitrogen above noted, and put w for the total weight of the gaseous products of combustion, then o/ = . 03660 + . 09 H + . 02 $ + (.0893 C + .268 H + .0335 S + .oi N); orze/= .126 C + .358 H + .053 S + .oi N ( 2 ) RULE 2. To find the total weight of the gaseous products of the complete combustion of one pound of a fuel. Let the elements be expressed as per- centages of the fuel; multiply the carbon by 0.126, the hydrogen by 0.358, the sulphur by 0.053, and the nitrogen by .01, and add together those four products. The sum is the total weight of the gases in pounds. Note. The weight, in pounds, of the carbonic acid, separately, may be found from the quantity (a), above; that of the steam from (), that of the sulphurous acid from (c) t and that of the nitrogen from (d). 2. By Volume. Multiply the weight of each gaseous product, (), (), ( or 3 P er cent. In the second stage, forming carbonic acid... 10,092, or 70 Heat evolved by complete combustion... 14,544, or 100 TABLE OF THE HEATING POWERS OF COMBUSTIBLES. The experimental results of MM. Favre and Silbermann are adopted with some slight revision, recommended by M. Peclet, in table No. 133, column 5. The weight of oxygen, column 2, is calculated from the known equivalents and weights of the elements, as given in table No. 131, page 398. HEATING POWERS OF COMBUSTIBLES. 405 The weight of air, column 3, is 4.35 times the weight of oxygen, column 2, and the volume of air at 62, column 4, is 13.14 times the weight, column 3. The equivalent evaporative power, columns 6 and 7, is expressed by the weight of water evaporable at 212 by one pound of combustible first, if supplied at 62 F., by dividing the total heat of combustion, column 5, by 1116, which is the total heat of atmospheric steam raised from water supplied at 62; second, if supplied at 212 F., by dividing by 966, the total heat of atmospheric steam raised from water supplied at 2 1 2. Table No. 133. TOTAL HEAT EVOLVED BY COMBUSTIBLES AND THEIR EQUIVALENT EVAPORATIVE POWER, WITH THE WEIGHT OF OXYGEN AND VOLUME OF AIR CHEMICALLY CONSUMED. Combustibles. Weight of oxygen consumed per Ib. of combustible. Quantity of air consumed per pound of combustible. Total heat of combustion of i pound of combustible. Equivalent evaporative power of i pound of combustible, under one atmosphere, at 212. cubic pounds pounds i pound weight. Ibs. Ibs. feet at units. of water of water 62. at 62. at 212. Hvdrosen . . 8.0 34 8 4.C7 62 O"?2 rr 6 64.. 2O Carbon, making ) carbonic oxide... ) i-33 O^' 5-8 T- J / 7 6 ^^j^O^ 4,452 JD' W 4.0 4.6l Carbon, making ) carbonic acid.... J 2.66 n.6 152 14,500 13.0 15-0 Graphite . . 2.66 ii. 6 I C2 I A. OAO 12.58 1/4. C "2 Carbonic oxide o.57 2.48 A J* 33 *4fo\tfJ,\t 4,325 3 .88 1 4-DO 4.48 Light carburetted ) hydrogen J 4.0 17.4 229 23,513 2I.O7 24-34 Bi-carburetted hy- j drogen,orolefiant > 3-43 15.0 196 2i,343 19.12 22.O9 eras ... 1 Sulphuric ether .... 2.60 n-3 149 16,249 14.56 16.82 Alcohol 2.78 12. 1 J 59 12,929 11.76 I3-38 Turpentine. . 3-29 14.3 188 19^534 17-5 2O.22 Sulphur I.OO 4-35 57 4,032 3-6l 4.17 Wax 3.24 I4.I 185 18,893 16.93 19.56 Olive oil 3.03 I 3 .2 173 18,796 16.84 19.46 Tallow 2 -95 12.83 169 18,028 16.15 1 8. 66 ( Supplementary. ) Coal, of average ) composition J 2.46 10.7 141 14,133 12.67 14.62 Coke, desiccated... 2.50 IO.9 J 43 !3,55 12.14 14.02 Wood, desiccated.. 1.40 6.1 80 7,792 6. 9 8 8.07 Wood-charcoal, ) desiccated f 2.25 9.8 129 12,696 11.38 i3-i3 Peat, desiccated... 1-75 7.6 IOO 9,95 1 8.91 10.30 Peat-charcoal, ) desiccated J 2.28 9.9 129 12,325 11.04 12.76 406 COMBUSTION. From the table, it appears that when carbon is not completely burned, and becomes carbonic oxide, it produces less than a third of the heat yielded when it is completely burned. For the heating power of carbon an average of 14,500 units will be adopted. The heating power of hydrogen is about four and a quarter times that of carbon. The calculation for the heating power of a combustible may be reduced to a simple formula. Let C, H, O, and S represent, as before, the per- centages of carbon, hydrogen, oxygen, and sulphur, in 100 parts. The elements of the heat evolvable are as follows : Of the carbon, 14,500 C -f- 100. Of the hydrogen, 62,032 ( H - Q)-MOO. Of the sulphur, 4,032 S^ioo. The quantity Q is a deduction made from the hydrogen to satisfy the constituent oxygen of the fuel: being an eighth of the weight of the oxygen. The total evolvable heat is 14,500 C + 62,032 ( H - Q) + 4032 S ,or IOO 14,500 C + (14,500 x 4.28 (H - Q) ) + (14,500 x .28 S) ; 100 or, putting h for the total heat, =145 (C + 4.28 (H-Q) + 0.288) (6) 8 RULE 4. To find the total heating power of one ponnd of a combustible, of which the percentages of the constituent carbon, hydrogen, oxygen, and sulphur are given. From the hydrogen deduct one-eighth of the oxygen, and multiply the remainder by 4.28; multiply the sulphur by 0.28; add the two products to the carbon; and multiply the sum by 145. The final product is the total heating power of one pound of the combustible, in units of heat. Note. The item of sulphur as a combustible may be ignored in cal- culations for ordinary purposes. Dividing the second member of the formula (6) by 1116, the total heat of steam at 212 raised from water at 62; or by 966 if the water be supplied at 212; the quotients express the equivalent evaporative power of the combustible. Putting e for the evaporative power, in pounds of water per pound of combustible, ^ = o. I3 (0 + 4.28 (H-Q) + 0.288), (7) 8 when the water is supplied at 62; and W 3 r AC. "tO 1 1 02 sr/ 1.4.2 O'T-J *-y* Average 81. o^ ^.2^ IO.I7 I.cc Caking, from Garesfield, Newcastle.. Do., from South Hetton, Durham 87.95 83.27 5-24 5-17 5-42 9.04 i-39 2.52 Average 85 61 5 2O 723 i 06 /*o j-.yv/ Total average 80 80 5 A C 8 8s 4QO 40 .yw WEIGHT AND COMPOSITION OF BRITISH AND FOREIGN COALS. BY MESSRS. DELABECHE AND PLAYFAIR, 1847-50. An extensive series of analyses and of trials of British coals were con- ducted by Sir Henry Delabeche and Dr. Lyon Playfair, at the College for Civil Engineers, Putney, in the years 1847-50, to the order of the govern- ment. The results of their investigations were published in three Reports on Coals suited to the Royal Navy, in the years 1849, 1850, 1851. Samples of 98 British coals were analyzed and tried for their evaporative performance, namely: 37 Welsh coals, 18 Newcastle coals (Hartley dis- trict), 7 Derbyshire and Yorkshire coals, 28 Lancashire coals, 8 Scotch coals Total, 98 coals. In addition to these there were analyzed and tried, one sample of anthracite from Ireland, six patent fuels, and 24 foreign coals. The chief results of these analyses and trials, compiled from the reports, are averaged and embodied in table No. 136, together with deductions as to the total heat of combustion of the fuels. The specific gravity, and the weight and bulk, of the coals, are given in columns 2, 3, 4, 5; and the chemical composition in columns 6, 7, 8, 9, 10, n. The quantity of coke produced from each coal is given in column 12. The total heat of com- bustion is given in units of heat in column 13, and also in equivalent evaporative efficiency in columns 14, 15, when the water is supplied at 62 and at 212 R, and evaporated at atmospheric pressure. These columns, 13, 14, 15, have been calculated by means of formulas ( 6 ), ( 7 ), ( 8 ), page 406. The evaporative efficiency found by the trials is given in column 1 6. 414 FUELS. COAL. a ^ < O P^H U M H 3 o m H w O P3 S O 2 u S fa o ^ S ^ O c^ u K w o 5 w I 5 S I <5 ^+- Q W ^5 a " S. si^u} Aq \ziz UIQ.IJ poo jo punod auo JO JSAVOd SAIJBJOdBAg 3 i^co TfO n Q O 10 ON tx. COLO - 00 |!l ! ^ ON O ONOO tx tx, tx. CO ON at of Combustion pound of coal. Equivalent evapo- rative power from 212. rt g LO CO CO l-O tx. N O 00 LO \O COOO tx vq oo vq oo LTI M d HH i J5 *O ON co c^ LO g co co rl rJ rJ CO LO OO LO LO M 1-1 ^f co O O ON -I| 1^1 CO O O OO ^J- .S OO OO OO ON i-i 88 coco rf CO Q O O oo O o co co O co O "^r 1 M SJBOO aq; UIGJJ paonpoad 3^03 C CO i-i ONOO Th ft -i O vO ON | jjtg tt 1 1 1 a 1 i *j i-i tx. LOOO CO g ON tx>O oo O ft ^ io O O co CO _ _, rj- ON tx cooo LO c^ covO d 4-1 LO ONOO co ON oo -o-S .S'" ON oo Th ON tx. iooo vo CJ 10 TJ- M a. z co Tt -, ^ ^ vd vo O O ON M I-H iovO -< ^ ci d ON ft ""* .*; z 00 LO H-, O Q -^ ON co ^t- co O -. co M C^ ff OO O N CO O cooo co CO 1 S ON M Th M >-i -y tx co ON covo Z% ix. O co 10 ON LO Tj- N TJ- co LO -4- - 1 3 CO M 00 O CO g tx 1-1 vo ON LO co "3- O d d 0000 i- ^8 CO LO CO LO CO vo VO VO so co ci d\ txoo' ex co oo t-v.tx.rx 1 ii j tx.CO^-C4 LO tx. 5 I I CO M 3-$^r 1 M ' H- 00 W tx, O _c co ON rx ON O O CO LO^ II * 1 ill t/i O covO ^hvO co vo gill CO "c N CO* O ONOO" "Q^ OO txOO tx. Jx. ON ON tx ON Av\viS oypsdg LOVO N CO O >-i LO ON txvO CO M M M 0* ONO tx. ON (^ LO ! H . S Averaged groups. Welsh, 37 samples Newcastle, 18 Derbyshire & Yorkshire, 7 Lancashire, 28 Scotch, 8 Average of British samples Anthracite, Ireland ca CO c I-J Oj "S '5 O 'J^ wT S H - BRITISH AND FOREIGN COALS. 415 There are very great variations in the chemical composition and properties of coals. In British coals, the constituents vary in quantity as follows : Carbon, from about 70 to 91 per cent, of the gross weight. Hydrogen, from 3^ to nearly 7 per cent. Oxygen, from about */>, to 20 per cent. Nitrogen, from a mere trace to 2 J / s per cent. Sulphur, from nothing to 5 per cent. Ash, from 1 / s to 15 per cent. Coke, from 49 to 93 per cent. The average composition of British coals deduced from the table, is as follows : Carbon about 80 per cent. Hydrogen 5 Nitrogen i */ 5 Sulphur ii^ Oxygen 8 Ash 4 Fixed carbon, or coke . 100 61 The foreign coals, from Van Diemen's Land, and from Chili, had only from 63 to 66 per cent, of constituent carbon, with 28 per cent, of oxygen and ash. Welsh Coals. It may be noted here that Mr. G. J. Snelus, in 1871, made an analysis of Llangennech coal, 1 of which the particulars are sub- joined, with those of a few other coals, extracted from the Reports of Dela- beche and Playfair, for comparison. " The Ebbw Vale coal," it is said, " may be taken to represent the Monmouthshire steam coals ; and Powell's Duffryn represents the Merthyr and Aberdare coals, highly esteemed for locomotives and ocean steamers." There is a close correspondence between the analyses of Llangennech coal made in 1848 and in 1871. Class of Coal, and Date of Analysis. Carbon. Hydrogen. Nitrogen. Sulphur. Oxygen. Ash. Coke. Ebbw Vale, 1848 Powell's Duffryn, 1848... Llangennech, 1848 Llangennech, 1871 Graicrola 1848 per cent 89.78 88.26 85-46 8 4 . 97 84 8? per cent. tJI 4.20 4-26 ^ 8d per cent. 2.16 1-45 1.07 1-45 4i per cent. 1.02 1.77 ,2 9 .42 Af per cent. % 2.44 3-50 7 10 p. cent. * 3.26 6.54 5-40 I. ^O p. cent. 77-5 84-3 83.7 86.7 85.5 1 See Appendix to the Report of the Judges, Mr. F. J. Bramwell and Mr. W. Menelaus, on the Trials of Portable Steam Engines at Cardiff in 1872 ; for a full Report on the Coal used in the Trials of Steam Machinery by the Royal Agricultural Society. 416 FUELS. COAL. PATENT FUELS. The patent fuels tried by Delabeche and Playfair consisted of mixtures of bituminous or tarry matter with small bituminous coal. They had an average of 83.4 per cent, of constituent carbon, and 5 per cent, of hydrogen, with less than 3 per cent, of oxygen, and 6 per cent, of ash. Three patent fuels produced an average of 74 per cent, of coke. Warlich's patent fuel was the richest in carbon, of which it contained 90 per cent.; of hydrogen, 5.56 per cent; of ash, 2.91 per cent. It yielded 85 per cent, of coke; and it evaporated, by trial, 10.36 Ibs. of water, per pound of fuel, reckoned from water supplied at 2 1 2. WEIGHT AND BULK OF BRITISH COALS. The average specific gravity of coal, as by the table No. 136, is 1.279; it varies from 1.20 to 1.39. The average weight of coal is 80 Ibs. per cubic foot, solid; the weight varies from 78 to 86 Ibs. The average weight is 50 Ibs. per cubic loot, heaped; the weight varying from 45 to 58 Ibs. The average bulk of one ton, heaped, of coal, is 44^ cubic feet; the bulk varying from 38 to 49 cubic feet. The average specific gravity of patent fuels is 1.167; tne average weight is 73/^ Ibs. per cubic foot, solid, and 65 Ibs. per cubic foot, heaped. The bulk of one ton, heaped, is 34^2 cubic feet. These averages show the advantage of the patent fuels in point of com- pactness, over coals; for though they are the lighter fuel, they occupy less space per ton than coals, on account of the regular forms in which the blocks are manufactured, and the facility for stowing them without much interspace. HYGROSCOPIC WATER IN BRITISH COALS. The hygroscopic water in coal, apart from what is chemically combined with it, varies considerably. In the analyses of Delabeche and Playfair, in which the specimens were dried at 212 F., it varied from 0.61 to 9.31 per cent, of the weight of the coal. The following are examples : HYGROSCOPIC WATER. Powell's Duffryn coal 1.13 per cent. Mynydd Newydd 0.61 ,, Pentrefelin 0.70 Park End coals, Lydney 2.78 Ebbw Vale 1.34 Resolven 1.55 Pontypool i. 60 Grangemouth coal 6.42 Broomhill coal 9.3 1 Wallsend Elgin 2.49 Fordel splint 8.40 Warlich's patent fuel 0.92 Bell's patent fuel 0.90 Wylam's patent fuel 1.38 Hartley coals 6.19 to 10.17 Steamboat Wallsend 1.14 Andrew's House, Tanfield 6.58 BRITISH COALS. 417 HYGROSCOPIC WATER. Cannel coal, Wigan i.oi per cent. Stavely 8.54 Vancouver's Island 7.21 Chirique , 9.11 Sydney, New South Wales 3.25 Juan Fernandez 6.00 It appears from this that the Welsh coals and the patent fuels contained the least proportion of hygroscopic water. TORBANEHILL OR BOGHEAD COAL. The Boghead coal is a special mineral found on the estate of Torbane- hill, Linlithgowshire. Its colour varies from dark snuff-brown to brownish- black. It is exceedingly hard; the fracture is slaty and conchoidal. When struck with a hammer, it gives a woody sound. Its specific gravity varies from 1.155 to 1.260, the average being 1.189. In composition, Boghead coal occupies the opposite end of the scale to anthracite, having a comparatively small percentage of carbon, and a large excess of hydrogen. According to Dr. Penny's analysis of the coal, dried at 2 1 2, the composition is as follows : per cent. Carbon 63.94 Hydrogen 8.86 Nitrogen 0.96 Sulphur o. 3 2 Oxygen 4.70 Ash .. 21.22 100.00 As the oxygen amounts to only 4.7 per cent., it leaves free a surplus of 8^ per cent, of hydrogen, to form hydrocarbons with the constituent carbon, when the coal is distilled ; and it is found that coal of the above composition yields 67 per cent, of volatile matter, and 31 per cent, of ash. The composition varies in different specimens, as may be observed in the following analyses of four specimens taken from the pit at different dates, table No. 137: Table No. 137. COMPOSITION OF BOGHEAD COAL. COMPOSITION. COAL. gravity. Fixed carbon. Volatile matter. Sulphur. Ash. Water. COKE. per cent. per cent. per cent. per cent. per cent. per cent. Brown, 1849.... I-I55 II-3 71.0 o-3 16.8 0.6 28.1 Do., 1851.... 1.160 7-1 71.00 0.2 21.2 o-5 28.3 Black, 1851.... i. 218 9.25 62.70 0-35 26.5 i. 20 35-75 Do., 1853.... 1.188 10.52 67.11 0.32 21.0 1.05 3i-5 2 Averages 1.180 9-54 67.95 0.29 21.4 0.84 30-94 27 41 8 FUELS. COALS. From the table it appears that the fixed carbon averages only 9% per cent.; and that, including ash, the coke averages only 31 per cent., the vola- tile matter exceeding two-thirds of the whole weight of the coal. When distilled at comparatively low temperatures, Boghead coal affords large quantities of paraffin, paraffin oil, &c. : a discovery made by Mr. Young. AMERICAN AND FOREIGN COALS. BY PROFESSOR W. R. JOHNSON, 1843-44. The results of an investigation of the qualities of American coals, at the Navy Yard, Washington, for the Navy Department of the United States, conducted by Professor W. R. Johnson, were published in " A Report to the Navy Department of the United States, on American Coals," in 1844. Thirty-nine samples of coal, and three samples of coke, were tried; and the general results are given in table No. 138. The constituents, so far as the analyses extended, were in the following proportions : Volatile matter, other ) T / . T , than moisture, } 2 ^ to 34# Per cent., average 16.17 per cent. Hygrometric moisture,., o to 3^ Sulphur, o to 2^ Fixed carbon, 53 to 91 Earthy matter, 4^ to 15 About 100.00 Coke (fixed carbon and earthy matter), 82.50 per cent. The proportions of volatile matter, fixed carbon, ash, and coke, were for the three classes of American coal as follows : Volatile Fixed . , ~ , matter. carbon. Ash " Coke ' Anthracites, 3.97 ... 88.54 ... 6.28 ... 94.82 Free burning bituminous coals,. ...15.11 ... 73.21 ... 10.27 83.48 Bituminous caking coals, 29.43 ... 58.29 ... 10.90 ... 69.19 Averages, 16.17 73-35 9-*5 82.50 WEIGHT AND BULK OF AMERICAN COALS. The specific gravity of American coal varies from 1.283 to 1-610, and it averages 1.400. The weight of solid coal varies from 80 to 100 pounds per cubic foot, and it averages 87^ pounds. The weight of heaped coal varies from 45 to 56 pounds per cubic foot, and it average? 51 24 pounds. The bulk of one ton of heaped coal varies from 49^/3 to 40 cubic feet, and it averages 43^ cubic feet. AMERICAN AND FOREIGN COALS. 419 Table No. 138. AMERICAN COALS: AVERAGE WEIGHT, BULK, AND COMPOSITION, 1843. (Compiled from the Report of Professor W. R. Johnson.) WEIGHT, BULK, COKE, AND ASH. COAL. Specific gravity. WEIGHT AND BULK. Coke produced from coal. Ash and clinkers left by combus- tion. One cubic foot, solid. One cubic foot, heaped. Bulk of one ton, heaped. Anthracites 1.500 1.358 1.342 I.3I8 pounds. 93-78 84.93 83.90 82.39 pounds. 53.05 32.13 52.84 49.28 49.31 cubic feet. 42.35 69.76 42.42 45.71 45-5 1 per cent. 94.82 83.68 69.01 65.27 per cent. 8.60 14-94 11.27 8. 4 8 7. 9 8 Cokes Free-burning bituminous Bituminous caking Foreign and Western Average of the three ) classes of American > coals 1 I.4OO 87.54 5L72 43-49 82.50 9.42 COMPOSITION. COAL. COMPOSITION, IN PERCENTAGES OF THE TOTAL WEIGHT. Moisture. Volatile matter, other than moisture. Sulphur. Fixed carbon. Earthy matter. Anthracites, Pennsylvania per cent. I.I9 i-37 1.56 2.50 per cent. 3-97 15.11 29-43 32.68 per cent. O.04 0.42 I.OI O.24 per cent. 88.54 73.21 58.29 57-42 per cent. 6.28 14.94 IO.27 10.90 7.85 Coke, two samples from Mid- } lothian and NeiFs Cumber- > land coal, Virginia j Free-burning bituminous, Mary- ) land and Pennsylvania j Bituminous caking, Virginia Foreign and Western bituminous. Average of the three classes ) of American Coals J i-37 16.17 0.49 73-35 9-15 420 FUELS. COAL. FRENCH COALS. French coals are divided into five classes, according to their behaviour in the furnace : i st. Bituminous caking coals (houilles grasses marechales). 2d. Bituminous hard coals (houilles grasses et dures), differing from the first by having less fusibility; the coke is more dense than that of the first, and is best for blast furnaces. 3d. Bituminous coals, burning with a long flame (houilles grasses a longues flammes] ; they are still less fusible or caking than the preceding, and are best for boiler and other furnaces. They are known by the designation flenu, and are similar to Lancashire cannel coal. 4th. Dry coals, with a long flame (houilles seches a longues flammes]. The coke has not much coherence. These coals are burned on grates; they are less durable than the foregoing. 5th. Dry coals, with a short flame (houilles seches a courtes flammes\ These coals burn with some difficulty, and are used chiefly for burning bricks, and in lime-kilns ; in breweries for drying malt, and for domestic fires. Anthracites are classed by themselves. The coal, as it comes from the mine, large and small together, is known as tout-venant "as it comes." In the market, the coal from a mine is dis- tinguished, according to the size of the pieces, into, ist, le gros, round coal; 2d, la gaillette, coal of medium size, in pieces 5 or 6 inches in diameter, which is separated by screening from the third sort; 3d, le memt, slack, which is subdivided into three kinds : gailletin, the size of nuts ; tete de moineau, smaller than gailletin literally the size of a sparrow's head; and fine, which is again distinguished mtofine menue&n&fine pottssier, coal dust. UTILIZATION OF THE SMALL COAL. The menu, or small coal, is made into briquettes, or rectangular blocks; being agglomerated by means of tar, and compressed into moulds, as has already been pointed out in describing English patent fuels, with some slight differences of treatment, ist. The small coal is mixed with pitch, and compressed in moulds to form blocks. These blocks have great durability, and do not deteriorate by exposure to air. 2d. When the slack is derived from rich bituminous coals it is filled into cast-iron moulds, which are so closed that nothing but gas can escape from them. The moulds are heated in a furnace to upwards of 900 F., where they remain from half an hour to three hours, according to the quality of the coal. By the action of the heat the coal becomes a kind of paste, and tends to swell; but it is on the contrary powerfully compressed by the moulds. 3d. For the slack of dry coals a certain proportion of the slack of bituminous coal is mixed with it, to give cohesive power. COMPOSITION OF FRENCH COALS. The table No. 139 contains the specific gravity and composition of a number of French coals. For the first section, comprising Regnault's analyses, compiled from a table by M. Peclet, the samples were dried at a temperature of 120 C, or about 250 F.; and the loss of weight, repre- FRENCH COALS. 421 senting moisture, varied from 1.36 to 1.60 per cent. The quantity of nitrogen was in general very small in the anthracites; and in the other coals it was from 1.50 to 2.0 per cent. The united weights of oxygen and nitrogen have therefore been taken by M. Peclet to represent the quantity of oxygen, in calculating the heating power, according to the principle already explained, page 403. Table No. 139. MEAN DENSITY, COMPOSITION, AND HEATING POWER OF FRENCH COALS. COALS. Regnault, 22 samples. Marsilly, 79 samples. Specific gravity. Quan- tity of coke. Composition. Hydro- gen in excess. Heating Power. Carbon. Hydro- gen. Oxygen and ni- trogen. Ash. (Regnault. ) Anthracites. .498 .319 293 .303 .362 .265 293 .197 .289 .280 per cent. 88.83 74.81 67.54 60.86 54-72 71-5 81.79 86.58 80.75 74.66 per cent. 86.17 88.56 87-73 82.94 76.48 83.85 86.38 86.65 86.50 84.94 per cent. 2.6 7 4.88 5.08 5-35 5.23 5-19 4.51 4 .l8 4-52 5.15 per cent. 2.85 4.38 1 16.01 8.09 5.46 5-23 5-39 7.02 per cent. 8.56 2.19 I.S4 3-08 2.28 2.8 5 3-66 3-95 3-52 2-93 per cent. 2-43 4.27 4-3 4-15 3-09 4.24 3-82 3.52 3.85 4.22 units of heat 14,038 15,525 I5>422 14,622 13,041 14,884 14,931 14,787 14,976 15,003 Bituminous hard coals.. Bituminous caking coals Bitum. coals, long flame Dry coals, long flame. . . (Marsilly. ) Mons Basin. Mons Centre Basin Charleroi Basin Valenciennes Basin Calais Basin.. Average. I.3IO 74-20 85.02 4 . 4 8 6.87 3-46 3-79 14,723 Note. The averages are here deduced from averages ; being averages of averages, and are to be accepted as approximate, not necessarily exact results. For the second section, the samples were dried by exhaustion in the receiver of an air-pump during from twelve to twenty-four hours. It appears from the table that the average composition of French coals is as follows: Carbon, 85 percent. Hydrogen, 4^ Oxygen arid nitrogen, 7 Ash, 3}^ Sulphur, ? 100 The average specific gravity is 1.310, giving a weight of 81.68 Ibs. per cubic foot solid. According to Peclet the weight of heaped coal from different mines is as follows, in table No. 140; to which are added the weight of one cubic foot heaped, and the volume of one ton heaped. 422 FUELS. COAL. Table No. 140. WEIGHT AND VOLUME OF FRENCH COALS, HEAPED. MINE. Weight of one hectolitre, heaped. Weight of one cubic foot, heaped. Volume of one ton, heaped. Labarthe.. kilogrammes. 88 pounds, tc o cubic feet. 4O 7^ Auvergne and Blanzy 87 00 >v CA ? H- w- / 41 22 Combelle 86 c^.7 41. 7O Latauoe . 85 C5.I 42. IQ Saint- Etienne 84 C2X 42.6Q Decise 8? qi 8 43 21 Mons "O 80 D A- ^O.O ro** * 44.83 Creusot 70 40 3 4^.30 Averages of bituminous coals 84 52.5 42.75 Anthracite . ... oo ^6.2 4O.OO An abstract of a resume of analyses of French and other coals and lignites, by MM. Scheurer-Kestner and Charles Meunier-Dollfus, is given in table No. 141, together with the observed heat of combustion. The figures have reference to pure fuel, from which the ash has been separated, in terms of the gaseous constituents only. Table No. 141. FRENCH AND OTHER COALS AND LIGNITES. ANALYSIS OF GASEOUS CONSTITUENTS AND OBSERVED HEAT OF COMBUSTION. (Scheurer-Kestner and Meunier-Dollfus.} The fuel is assumed to be dry and pure without any ash. Designation of Combustible. Gaseous Elements. Heat of combustion oi i pound pure (observed). Carbon. Hydro- gen. Oxygen and nitrogen. COAL. Ronchamp 3 samples per cent. 88.59 Sl.IO 90.60 78.58 87.02 84.45 83.94 91.08 92.49 96.66 9 T -45 82.67 per cent. 4.69 4-75 4.IO 5-23 4.72 4.21 4.43 3^3 4.04 1-35 4-5 5-o7 per cent. 6.72 14.15 5-30 16.19 8.26 11.32 11.63 5-9 3-47 1.99 4-05 12.26 units. 16,416 15.320 16,994 14,985 l6,40O 16,663 16,290 15,804 1 6, 1 08 14,866 15*651 14,438 Sarrebruck 7 do Creusot 4 do. Blanzy Montceau Do Anthracitic Angin Denain . English : Bwlf Do Powell-Duffryn Russian : Grouchefski anthracite. . . . Do. Miouchi, bituminous Do. Goloubofski, flaming INDIAN COALS. Table No. 141 (continued}. 423 Gas eous Elem snts. Heat of Designation of Combustible Carbon. Hydro- gen. Oxygen and Nitrogen. Combustion of i Ib. pure (observed). LIGNITES. Rocher bleu per cent. 72.Q8 per cent. 4. 04. per cent. 22 08 units. 1 1 67O Manosque bituminous 7O ^7 544 23 QQ T -2 2 C 3 Do dry / w ' j / 66 ii *Hh 4 8"; ^o-yy 28 84 i o>^oo 12 ?84 Bohemia bituminous w. jj. 76 s8 8 27 I C T C "^O^T- 14 26"? Russian Xoula / " 0" 7-2 72 V^J 6 OQ A D* * 3 2O IQ T-2 8"?7 Lignite, passing to fossil wood / O'/ ^ 66.1; I vy 4 72 *\j. j.y 28 77 OJ^O/ II 444 Fossil wood, passing to lignite '"3 -> 67.6O 4. CC 27 85 I I."?6o T-'OO * / / I 2 C w.y^j. 3OO Scotland : From .200 40 3O CO.7O l *3 O 33 I 13 To .316 Co i c 3 V ^/ W 40 8 c. w< oo 1^7 '*O 14 ^7 Average . . . .260 C.4.OO H-^-'-'O 46 oo **Jl I.IO Xi fO / 4 OO Newcastle : From 1.23 62.7O 37.30 o 06 0. 2O To 1. 31 72 3O 27 7O i Sc; I 3 QI Average . o x 1 28 / ^'O^ 66 oo ^ / / w 34 OO * w o I OO ^O'V- 1 4 oo O i t" v -' Average of English coals. . 1.28 66 66 ?? -J-? 112 3 66 OO'O O An official memorandum was addressed to the Indian government, in January, 1867, by Dr. Oldham, superintendent of the geological survey of India, containing the results of analysis of eighty-one samples of Indian coal: showing the volatile matter, the fixed carbon, and the ash. These results are given in table No. 143, and a column is prefixed showing the percentage of coke, which is arrived at by adding that of the ash to that of the fixed carbon. For comparison, the results of a similar analysis of English coals saleable at Calcutta, are added. The distinguishing characteristic of the Indian coal is the great propor- tion of ash it contains, varying from i^ per cent, though in only one instance, to 59 per cent., and averaging, for 81 samples, 23 per cent. The English coal saleable at Calcutta has only an average of 2.7 per cent. The following are the average compositions of Indian and of English coal at Calcutta, deduced from table No. 143: Indian coals. English coals, per cent. per cent. Coke, 70.2 ... 70.8 Fixed carbon, 47.3 ... 68.1 Volatile matter, 29.6 ... 29.2 Ash, 22.9 ... 2.7 showing, notwithstanding the great excess of ash in the composition of the Indian coals, that the quantity of volatile matter is about the same as in the English coals, about 29 per cent. In the absence of a full chemical analysis, it is impossible to say how much of this consists of carbon, INDIAN COALS. 425 oxygen, hydrogen, and nitrogen individually; and therefore the heating power of the volatile matter cannot be estimated. Table No. 143. COMPOSITION OF INDIAN COALS, 1867. Compiled from a Report by Dr. Oldham. Locality. Coke (sum of fixed carbon and ash). Fixed carbon. Volatile matter. Ash. Kurhurbali Field : From per cent. 7^.2 per cent. ^O.Q per cent. 12.6 per cent. 4 .8 To 87.4 7^.1 24.8 ^0.2 Average . . , 7Q.Q 62.8 2O.2 I7.I Rajmahal Hills: From C4.4 2S.2 28.8 i*^ To y*r*f 71.2 1:7.6 44-8 37-6 Average . 60.8 44.2 -20. -2 16.6 Ranigunj Field: From CQ.O 3Q.2 2EJ.6 1.75 To 7^O 6? 8 38.7 -2C.2 Average / J' w 6c o **o*** co.o 3C.O I^.O Sherria Field: From \O-w CC.4 o w< 30.8 18.0 1.7 To OJ'T- 86.0 68.4 44.6 28.8 Average 60.0 <;6.3 3I.O 12.7 Central India (Pench River) : From ^2. 30.3 I4.O 2.2 To 86 o 61 6 CA o 48.7 Average 6* 6 47.4 32.8 18.2 Madras (Godavery River) "0-" 81.0 23 2 IQ.O ^7.8 Total averages of Indian coals. 70.2 47-3 29.6 22.9 English coal, saleable at Calcutta : Averages 70 8 68.1 2Q.2 2.7 It may be added that Dr. Oldham, in 1859, analyzed two specimens of anthracitic coal from Kotlee, in the Punjab, and found their composition as follows : Carbon. per cent. No. i 90.5 No. 2 90.0 Volatile matter. per cent. ... 4.0 .. 6.0 Ash. per cent. 5-5 4.0 Much of the Indian coal is peculiarly liable to disintegration from ex- posure to the atmosphere, particularly in the hot seasons. Coal from the new Chanda coalfields is reported to have fallen to so small pieces, after a short period of exposure, as to have become unfit as fuel for locomotives. 426 FUELS. COAL. COMBUSTION OF COAL. When coal is exposed to heat in a furnace, the carbon and hydrogen, associated in various chemical unions, as hydrocarbons, are volatilized and pass off. At the lowest temperature, naphthaline, resins, and fluids with high boiling points are disengaged; next, at a higher temperature, volatile fluids are disengaged; and still higher, olefiant gas, followed by common gas, light carburetted hydrogen, which continues to be given off after the coal has reached a low red heat. As the temperature rises, pure hydrogen is also given off, until finally, in the fifth or highest stage of temperature for distillation, hydrogen alone is discharged. What remains after the distil- latory process is over, is coke, which is the fixed or solid carbon of coal, with earthy matter, the ash of the coal. The hydrocarbons, especially those which are given off at the lowest temperatures, being richest in carbon, constitute the flame-making and smoke-making part of the coal. When subjected to degrees of heat much above the temperatures required to vaporize them, they become decom- posed, and pass successively into more and more permanent forms by precipitating portions of their carbon. At the temperature of low redness, none of them are to be found, and the olefiant gas is the densest type that remains, mixed with carburetted hydrogen and free hydrogen. It is during these transformations that the great body of smoke is made, consisting of precipitated carbon passing off uncombined. Even olefiant gas, at a bright red heat, deposits half its carbon, changing into carburetted hydro- gen; and this gas, in its turn, may deposit the last remaining equivalent of carbon at the highest furnace heats, and be converted into pure hydrogen. Throughout all this distillation and transformation, the element of hydro- gen maintains a prior claim to the oxygen present above the fuel ; and until it is satisfied, the liberated carbon remains unburned. SUMMARY OF THE PRODUCTS OF DECOMPOSITION IN THE FURNACE. Reverting to the statement of the average composition of coal, page 415, it was found that the fixed carbon or coke remaining in the furnace after the volatile portions of the coal are driven off, averages 6 1 per cent, of the gross weight of the coal. Taking it, for round numbers, at 60 per cent., the proportion of carbon volatilized in combination with hydrogen, will be 20 per cent. making up the total of 80 per cent, of constituent carbon in average coal. Of the 5 per cent, of constituent hydrogen, i part is united to the 8 per cent, of oxygen, in the combining proportions to form water, and the remaining 4 parts of hydrogen are found partly united to the volatilized carbon, and partly free. These particulars are embodied in the following summary of the condi- tion of the elements of 100 pounds of average coal, after having been decomposed, and prior to entering into combustion : COMBUSTION OF COAL. 427 100 Pounds of Average Coal in the Furnace. Composition. Ibs. Ibs. Decomposition. Carbon j v^arilized 20 1 f 6 fi ^ed carbon. Hydroeen 5 I 2 * hydrocarbons and free hydrogen. Nitrogen I T / S Ash 4 About loo J nitrogen. 4 ash. IOO showing a total useful combustible of 86^ per cent., of which 26^ per cent, is volatilized. Whilst the decomposition proceeds, combustion pro- ceeds, and the 25^ per cent, of volatilized portions, and the 60 per cent, of fixed carbon, successively, are burned. It may be added that the sulphur and a portion of the nitrogen are dis- engaged in combination with hydrogen, as sulphuretted hydrogen and ammonia. But these compounds are small in quantity, and, for the sake of simplicity, they have not been indicated in the above synopsis. QUANTITY OF AIR CHEMICALLY CONSUMED IN THE COMPLETE COMBUSTION OF COAL. Take coal of average composition. Then, applying the rule i, page 400,. the carbon C = 80, the available hydrogen (H Q) = 4, and the sulphur S= 1.25, and C + 3 (H - O) + .4 S = 80 + 1 2 + .5 - 92.5, o and 92.5 x 1.52 = 140.6 cubic feet of air at 62, the quantity chemically consumed by one pound of average coal. To find the proportions in which this quantity of air is appropriated for the volatilized and the fixed portions of the coal, as above divided, for loo Ibs. of the fuel: FOR THE VOLATILIZED PORTION Hydrogen 4 Ibs. x 457 = 1828 cubic feet. Carbon 20 Ibs. x 152 = 3040 Sulphur i% Ibs. x 57 = 71 4939 cubic feet. FOR THE FIXED PORTION Carbon 60 Ibs. x 152 = 9120 Total useful combustible, 85^ Ibs. *4>59 showing that 14,059 cubic feet of air at 62 are required for the complete combustion of 100 Ibs. of coal of average composition. It is equivalent 428 FUELS. COAL. to 140.6 cubic feet of air for one pound of coal, as already found, or in round numbers, 140 cubic feet, of which there are required, For the volatilized portions 50 cubic feet, or 36 per cent. For the fixed portion 90 or 64 140 100 The weight of this quantity of air, dividing the volume by 13.14, is 10.7 Ibs. The following table, No. 144, gives the composition of, and the quanti- ties of air chemically consumed in the complete combustion of, British coals of the highest, the lowest, and average heating powers, placed together for comparison. It appears from the table that the quantity of air chemically consumed in the combustion of one pound of British coal varies, according to the composition of the coal, from 116 to 163 cubic feet at 62: Table No. 144. COMPARATIVE STATEMENT OF COMPOSITION, HEAT OF COMBUSTION, AND AIR CHEMICALLY CONSUMED BY BRITISH COALS - OF THE HIGHEST, LOWEST, AND AVERAGE QUALITY. Air chemically /" 1 Total heat of consumed in Coal (Selected from Delabeche and Carbon. Hydrogen. Oxygen. Sulphur. combustion of one pound the complete combustion of Playfair's Report). of coal. one pound of coal. per cent. per cent. per cent. per cent. units. cubic feet at 62. Warlich's patent fuel. 90.02 5-56 1.62 16,495 I6 3 Ebbw Vale 80.78 e 1C O. 7Q I. O2 l6,22I 161 Haswell Wallsend... v y* 1 w 83.47 3' *J 6.68 O .7 8.1 7 O.O6 15,502 153 Coal of average com- ) position j 8O.OO 5.00 8.00 *'*S !4>i33 140 Ince Hall, Pember- \ ton five feet (low- > 68.72 4.76 18.63 I -35 JI >5 2 5 116 est British) J Chirique, Chili (low- ) est foreign) J 38. 9 8 4.01 13-38 6.14 7,349 74 GASEOUS PRODUCTS OF THE COMPLETE COMBUSTION OF COAL. The quantity of the gaseous products is found by rules 2 and 3, pages 401, 402. Take, for example, the case of coal of average composition. i. By weight. The percentages of carbon, hydrogen, sulphur, and nitrogen are respectively 80, 5, 1.25, 1.20. Then, by rule 2, page 402, the weight of the gaseous products, taken collectively, of the combustion of one pound of coal, is (.126 x 80) + (.358 x 5) + (.053 x i. 25) + (.01 x 1.20)= 11.94 pounds. COMBUSTION OF COAL. 429 The weights of the gases individually are given by the expressions a, b, c, d, page 401, as follows: Pounds. Per cent. Carbonic acid .............................................. 0366x80 =2.93 or 24.5 Steam. ....................................................... 09 x 5 = .45 or 3.8 Sulphurous acid ........................................... 02x1.25 = .025 or 0.2 Nitrogen = (.0893 x 80) + (.268 x 5) + (.0335 x 1.25) + (.01 x 1.20) = 8.536 or 71.5 11.94 100.0 2. By volume. The total volume is found by rule 3, page 402; thus: (1.52 x 80) + (5. 52 x 5) + (.567 x i. 25) + (.135 x i. 20) -150.07 cubic feet. The volumes in detail are, by the expressions e, f, g, h, page 401, as follows : Carbonic acid ...................................... . ......... 315x80 = 25.2 or 17 Steam ......................................................... 1.9 x5 = 9.5 or 6 Sulphurous acid ............................................. 117x1.25 = 0.15 trace Nitrogen = (i.2o6x 80) + (3.618 x 5) + (.45 x 1.25) + (.135 x 1.20) = 115.29 or 77 150.14 100 showing that the 12 pounds of gaseous products have a volume of 150 cubic feet at 62, equal to 12^ cubic feet per pound. The element of nitrogen is nearly three-fourths by weight, and fully three-fourths by volume, of the total quantity of gaseous products. The relatively larger volume of the gaseous products at the higher tem- perature at which they enter the chimney, is found by the formula ( 2 ), page 347, repeated at page 402. If the final temperature be 500 R, the final volume of the gaseous products for one pound of average coal is, 150 5 + 4l = 276 cubic feet; 62 + 461 or nearly double the volume at 62. At 585 temperature, the volume would be exactly double, or 300 cubic feet; and at 1108 F. it would be just three times the normal volume at 62. SURPLUS Am. The quantity of surplus air which passes off with the producik of cbfrit ' bustion into the chimney, is to be added to that of these products to find the total weight or volume of the gases in the chimney, as already stated, A , * page 402. Taking the case of coal of average composition, suppose that the quantity of surplus air is equal to that which is chemically consumed by the fuel; then it amounts to 140 cubic feet by volume, or 10.7 pounds by weight, for one pound of coal consumed; adding these to the weight and the volume of the products of combustion above found, there is Cubic feet at 62. Weight in Ibs. Gaseous products of combustion per Ib. of coal, ...150 ...... 12 Surplus air ...140 ...... 10.7 Total escaping gases, .................. 290 ...... 22.7 430 FUELS. COKE. When the quantity of surplus air is less than that which is chemically consumed, the volume and weight to be added to those of the products of combustion, are less than 140 cubic feet and 10.7 pounds respectively, in the same proportion. The total quantity of escaping gases, therefore, produced by the com- bustion of one pound of average coal, varies according to the proportion of surplus air From 150 cubic feet to 290 cubic feet, at 62; From 276 to 533 at 500; From 12 pounds to 22.7 pounds in weight. It is here assumed that the maximum quantity of surplus air does not exceed the quantity of air chemically consumed. TOTAL HEAT OF COMBUSTION OF BRITISH COALS. The total heat of combustion of coal of average composition, having 80 per cent, of carbon, 5 per cent, of hydrogen, 8 per cent, of oxygen, and 1.25 per cent, of sulphur, is, by rule 4, page 406, 145 (80 + 4.28 (5 - 8 /s ) + (o.28 x 1.25) ) = 14,133 units. The heating power, expressed in pounds of water evaporable under one atmosphere by one pound of the fuel, is, by rule 5, p. 406, as follows: When the water is supplied at 62 the total evaporative power is 0.13 (80 + 4.28 (5 - 8 / 8 ) + (o.28 x 1.25)) = 12.67 pounds of water. When the water is supplied at 212 the evaporative power is 0.15 (80 + 4.28 (5 - 8 / 8 ) + (o.28 x 1.25) ) = 14.62 pounds of water. The total heat of combustion of British coals is given in table No. 136. page 414; and for contrast in table No. 144, above. COKE. The quantity of residuary coke in various coals, was found by laboratory analysis as follows (see previous tables) : (Excluding anthracites.) COKE. COKE. per cent. per cent. English coals 50 to 72 Average 61.4 American coals 64 to 86 76.4 French coals 53 to 76 64.5 Indian coals 521084 70.2 Anthracite coke scarcely deserves the name; it is without cohesion, and pulverulent. The best coke, from bituminous coal, is clean, crystalline, and porous; and it is formed in columnar masses. It has a steel-gray colour, possesses a metallic lustre, with a metallic ring when struck, and is so hard as to be capable of cutting glass. QUALITY OF COKE. 431 The quality of coke obviously depends, in a great measure, on the propor- tions of the constituent hydrogen and oxygen of the coal from which it is made, which regulate the degree of fusibility of the coal when exposed to heat. Taking, for example, the particulars of the coke produced from the French coals, table No. 139, and arranging the averages for each kind of coal in the order of the quantity of hydrogen in excess, the nature of the coke produced, as described by M. Peclet, was as follows : AVERAGES. Hydrogen. Oxygen and Nitrogen. Hydrogen in excess. Nature of the Coke. Anthracites per cent. 2.67 per cent. 2 8q per cent. 2.4. 1 ? pulverulent Dry coals, long flame C.23 "a 16.01 *"T-O 3OQ in fragments Bituminous coals long flame 51 e 8 6^ A T C porous Bituminous hard coals 'OJ 488 A 1& 4- A 427 porous Bituminous caking coals ; 08 e 6; / A 1Q very porous D' w D' W D *t'O^ J Showing a series of five coals, with an ascending series of hydrogen in excess, from 2.43 to 4.30 per cent. The nature of the cokes advances correspondingly from pulverulent or powdery, to very porous or excessively fused and raised. The first is, in fact, a failure as a coke, and the second, with 3.09 per cent, of hydrogen in excess, barely coheres, being in frag- ments; the third and fourth, with about 4.20 per cent, of hydrogen in excess, produce a porous and cohesive coke, and the fifth an excessively porous coke, bright, but comparatively light for metallurgical operations. From this it appears that a coal having less than 3 per cent, of hydrogen in excess, is unfit for coke-making; and that, for the manufacture of good coke, coal containing at least 4 per cent of free hydrogen is required. It is not clear in what manner the presence of free hydrogen operates in fusing the substance of the coal; unless, probably, that the hydrogen being in combination with carbon in various proportions to form tar and oils, softens the fixed carbon, and forms a pasty mass, which is raised like bread by the expansion of the confined gases and vapours seeking to escape. The increasing proportions of volatilized matter which is raised by heat, successively, from anthracites, bituminous, and caking coals, are clearly exemplified by the analyses of American coals in table No. 138, page 419; and they evidently have relation to an increase of the hydrogen in excess above that required to form water with the constituent oxygen. They are as follows: AMERICAN COALS. VOLATILIZED MATTER. per cent. Anthracite 5.16 Free burning bituminous coals 16.48 Bituminous caking coals 3-99 COKE PRODUCED. per cent. 94.82 83.68 69.01 The increasing volatilized matter explains, as above suggested, the increas- ing porosity and bulk of the coke yielded by the respective coals. 432 FUELS. COKE. ANTHRACITIC COKE. By a process recently introduced at Swansea by Messrs. Penrose & Richards, and described by Mr. W. Hackney, 1 anthracite has been success- fully used as the basis of a coke, manufactured from the following mixture: Anthracite, 60 per cent; Bituminous coal, 35 per cent; Pitch, 5 per cent. The materials are crushed and mixed together through a disintegrator. The yield of coke is 80 per cent of the weight of the charge. The coke is steel-gray in colour, and so hard as to scratch glass easily; and it is about 23 per cent, heavier than the best Welsh coke. It burns in a common fire, or in a blast furnace, without any sign of crumbling. QUANTITY OF COKE YIELDED BY COAL. The quantity of coke produced, on the large scale, from coal varies from 60 to 80 per cent, in weight The following are examples of the yield : COALS. COKE PRODUCED. Andrew's House, Tanfield 65 per cent of the coal. Bristol 60 to 63.5 Kilsyth 60 Mons 77 to 80 Seraing 67 In general, the yield of good coke is about two-thirds, or 66 per cent, of the coal. The whole of the coke matter in coal cannot be extracted from it, on the large scale; a portion of it is burned off. Thus, Seraing coal, from which 67 per cent, of coke was made, on the large scale, yielded 80 per cent of coke, by laboratory analysis. WEIGHT AND BULK OF COKE. Coal expands in volume in the coking process, insomuch that the volume of the resulting coke is greater by from 10 to 30 per cent, than that of the coal from which it is made. Tanfield coke has 1 1 per cent, more volume than the coal from which it is made; and as the specific gravity of the coal is 1.26, that of the coke is 0.74, calculated as follows: 1.26 x 5 = 0.74. I.I I X IOO The weight and volume of Tanfield coal, and of coke made from it, are as follows: Specific Weight of i cubit Weight of i cubic Volume of i gravity. foot, solid. foot, heaped. ton, heaped. Tanfield coal... 1.26 78.57103. 52.19105. 42.92 cubic feet Do. coke 0.74 46.14 30.00 74.66 Mickley coke weighs 28 Ibs. per cubic foot, heaped, and measures in bulk 80 cub. ft. per ton. Gas coke weighs from 12)^ cwt. to 1 5 cwt. per chaldron. 1 In a paper read at the meeting of the Iron and Steel Institute, 1875, published in Engineering, November 12, 1875. COMPOSITION OF COKE. 433 The American cokes, from Midlothian, Va., and Cumberland, Md., averaged' a weight of 32.13 Ibs. per cubic foot, heaped, and a volume of 69.8 cubic feet per ton. The coke used for smelting furnaces in France weighs, ordinarily, 25 Ibs. per cubic foot, heaped, and measures, in bulk, about 90 cubic feet per ton. Of the Seraing coking coal, and the coke produced from it, the weight and bulk are as follows, assuming that the coal is the same as "average New- castle coal, with which it is almost identical in chemical composition : Weight of i cubic Volume of i ton, foot, heaped. heaped. Seraing coal 50 Ibs. 45 cubic feet. Do. coke 31 72 From the foregoing particulars it may be gathered that coke of good quality weighs from 40 to 50 Ibs. per cubic foot solid, and about 30 Ibs. per cubic foot, heaped; and that the average volume of one ton is 75 cubic feet, varying from 70 to 80 cubic feet per ton. COMPOSITION OF COKE. For all purposes, the less ash there is in coke the more valuable it is. Pure coke, if such there be, consists entirely of carbon. But in practice, coke consists of carbon, sulphur, and ash. The purest coke known is Ramsay's Garesfield coke. The composition, as ascertained by analysis by Dr. Richardson, is as follows : Carbon 97.6 percent. Sulphur 0.85 Ash 1.55 IOO.OO The composition of Durham coke varies within the following limits: Carbon 85 to 92 per cent. Sulphur i| to 2 Ash 4toi2 Dr. Muspratt gives the results of nineteen analyses of coke of the usual qualities supplied to manufacturers; they are here given in table No. 145, arranged in the order of the percentages of carbon, the first in the list being Ramsay's coke above-mentioned. For the service of locomotives on railways, coke, besides being dense and hard, should not contain more than 6 per cent, of ash to insure its passing as coke of good quality; with 9 per cent, of ash, it is of mediocre quality; with 12 per cent, of ash, it is decidedly bad coke. The washing of coal destined for the formation of coke has already been described. Its effect in removing the earthy matter and in improving the quality of the coke has already been referred to. Suppose a coal which, in its ordinary condition, yields a coke containing from 10 to 15 per cent, of ash, the effect of previously washing the coal would be to reduce the quantity of ash in the coke to from 4 to 6 per cent. 434 FUELS. COKE. Table No. 145. COMPOSITION OF COKES. Arranged from data given by Dr. Muspratt. No. OF COKE. COMPOSITION. No. OF COKE. COMPOSITION. Carbon. Sulphur. Ash. Carbon. Sulphur. Ash. per cent. per cent. per cent. per cent. per cent. per cent. I 97.60 0.85 i-55 II 92.70 .60 5-70 2 96.42 0.83 2-75 12 92.44 -56 6.00 3 95-5 1 1.64 2.85 J 3 91.49 . 4 6 7-05 4 94.67 1.07 4.26 14 9I.l6 19 7-65 5 94-3 1 0.72 4-97 15 90-53 .01 8.46 6 94.21 0.69 5.10 z.6 89.87 .78 8.35 7 94.08 0.88 5-4 17 89.69 .96 8-35 8 93-54 0.76 5-7o 18 85.85 2.08 12.07 9 93-4i 0.79 5.80 19 84.82 0.78 14.40 10 93-05 1.58 5-37 Average of ) 19 cokes j 93-44 1.22 5-34 MOISTURE IN COKE. Coke is capable of absorbing from 15 to 20 per cent, of its weight of water. It has been found to absorb as much as 8 per cent, of water on its way from the ovens to its destination in uncovered waggons. Directly exposed to rain, it may absorb as much as 50 per cent, of its weight of water; the most part of which is afterwards quickly evaporated, leaving from 5 to 10 per cent, in the coke. Loss OF COMBUSTIBLE MATTER IN THE CONVERSION OF COAL INTO COKE. Peclet quotes the experience with Alais coal, a bituminous hard coal, having the average composition of the coals used for the manufacture of coke at Seraing : Per cent. Carbon, 89.27 Hydrogen...... 4.85 Oxygen and nitrogen, 4.47 Ash, 1.41 100.00 The yield of coke is 67 per cent, and deducting the ash, 1.41 per cent, there remains 65.59 per cent, as carbon in the coke. The total loss of combustible matter in parts of the coal is then Per cent. Carbon, 89.27-65.59 = 23.68 Hydrogen, 4.85 The heat of combustion of the lost carbon and hydrogen is, by rule 4, page 406, COMBUSTION OF COKE. 435 145 (23.68 + 4.28 (4.85 -) ) = 6o 9 6 units, o showing a loss of 40 per cent, of the total heating power of the coal, which is 15,606 units. AlR CONSUMED IN THE COMPLETE COMBUSTION OF COKE. The quantity of air chemically consumed in the complete combustion of coke is found by means of rule i, page 400. Take, for example, coke of average composition, having 93.44 per cent, of carbon, and 1.22 per cent, of sulphur. By the formula, the volume of air at 62 chemically consumed is 1.52 (93.44 + (1.22 x 0.4) ) = i-52 x 93.93 = 142.8 cubic feet. To find the weight of this quantity of air, divide the volume by 13.14, and the quotient is the weight, 10.87 Ibs. Similarly, the air chemically consumed by the best and worst cokes, in table No. 145, is found, and the quantities for the three cokes are here placed together for comparison : Carbon. Sulphur. Ash. QUANTITY OF AIR CHEMICALLY CONSUMED. per cent, per cent. per cent. Volume at 62. Weight, No. i ............ 97.60 0.85 1.55 148.9 cubic feet. 11.33 Ibs. No. 19 .......... 84.82 0.78 14.40 128.9 )> 9-8 1 ?? Average coke... 93.44 1.22 5.34 142.8 10.87 GASEOUS PRODUCTS OF THE COMBUSTION OF COKE. The combustible elements of coke carbon and sulphur produce car- bonic acid and sulphurous acid. These, together with the nitrogen of the air chemically consumed, constitute the products of combustion. By rule 2, page 401, the weight of these products is as follows, for coke of average composition with 93.44 per cent, of carbon and 1.22 per cent. of sulphur : PRODUCTS. Pounds. Per cent. Carbonic acid, .......................... 93.44 x .0366 =3. 42 or 28.4 Sulphurous acid, ........................ 1.22 x .02 =0.24 or 2.0 Nitrogen, ............ (93-44 x .0893) + (1.22 x .0335)^8.38 or 69.6 Total weight, 1 2.04 or 100.0 showing a total weight of 12 Ibs. of gaseous products for one pound of average coke the same weight as was found for average coal (page 428). The volume of the gaseous products, at 62 F., is found from the per- centages of the combustibles by the data (e) (g) (h\ page 401, respec- tively as follows : PRODUCTS. Cubic feet at 62. Per cent. Carbonic acid, 93-44 x -3 J 5 = 2 9-43 or 20 -6 Sulphurous acid, 1.22 x .117 0.14 or o.i Nitrogen, (93.44 x 1.206) + (1.22 x .45) = 113.25 or 79.3 Total volume, 142.82 or 100.0 Showing a total volume, as at 62, of about 143 cubic feet of gaseous products for one pound of average coke. 436 FUELS. LIGNITE AND ASPHALTE. HEATING POWER OF COKE. The heating power of coke is calculated directly from the quantity of constituent carbon, if the sulphur be neglected. Taking, for example, the first and the last samples of coke, of which the analyses are given in table No. 145, with coke of average composition, the percentages of constituent carbon are as follows, to which are added the heating powers of one pound of the fuels, calculated by rules ( 4 ) and ( 5 ), page 406 : COKE. Constituent Total Heating Carbon. Power. per cent. units of heat. No. i 97- 6c No. 19 84.82 Average coke 93-44 12,300 Total Evaporative Power from water at 212. 14.64 pounds. 12.72 14.02 TEMPERATURE OF COMBUSTION OF COKE. The temperature of combustion of carbon, which is the combustible matter of coke, was found, page 407, to be 4877 F. when completely burned. It may therefore be assumed that the temperature of combustion of coke is under 5000 F. LIGNITE AND ASPHALTE. Lignite, or as it is occasionally called, brown coal, though it is often found of a black colour, belongs to a more recent formation the tertiary than coal. It is in fact an imperfect coal. Brown lignite is sometimes of a woody texture, sometimes earthy. Black lignite is either of a woody texture, or it is homogeneous, with a resinous fracture. Some lignites, more fully developed, are of a schistose character, with pyrites in their composi- tion. The coke produced from various lignites is either pulverulent, like that of anthracite, or it retains the forms of the original fibres. Lignite is less dense than coal. The table No. 146 contains the composition of lignites of various quali- ties, including the hygrometric moisture. Table No. 147 contains the results of analyses and other particulars of lignites and of asphalte, according to Regnault. See also table No. 141, page 423. Table No. 146. DENSITY AND COMPOSITION OF VARIOUS LIGNITES, INCLUDING HYGROMETRIC MOISTURE. Locality and description. Specific gravity. Carbon. Hydro- gen. Oxygen. Nitro- gen. Ash. Mois- ture. Meissner red brown, woody Rheinhardswalde gray or "J black, with abundance of > resin ) 1. 12 I-I3 p. cent. 51.24 58.78 p. cent. 4.17 4.04 p. cent. 52.33 20.80 p. cent. 0.17 0.15 p. cent. 0.80 5-94 p. cent. 10.30 10.28 Meissner brilliant black, ~) fracture fibrous, lustre > vitreous J 1.32 70.0 3-19 17-59 0.12 547 3.63 Hirschberg-brownish black, ) in tree-like masses ) i-35 60.30 4.86 20.17 0.12 3-17 n-39 COMPOSITION OF LIGNITE AND ASPHALTE. 437 II B.S Oxygen Nitroge I *S u s^ 31 *o ^j M 00 N ON NO to O COVO > rt CO t OO !> w CO O CO lOVO '3 c^O h-T M" O O ON-^- "vO~ M CO CO N M 10 ON to to OO ^ NLOOOO fO ON M ON ON( N ^ CO ON M NO !> !> 00 00 w M M M (NMMM O t^ ON !> VO N 00 g ONOO 10OOOO O vO^oO w 10 lOCONOON l^ Tj-NcO )-i 10 ^ ^t 1 >O 1O T^ 1O t>> I> ON t 00 O fO 1>-OO M . g o | Q . Q Q 3. S3 3 ooo 6 W M tOLOONHH ON M 00* ONOO VO M TJ- MO^OO r^ t^-t>.ro r>. 10 10 J>-00 O NO 10 ONVO M N CO N "-I M M MMO 00 rf ON 10 NO 00 ^NO M" O^ 1000 00 10 10 10 xo O o o\ tood N M M 10 ON NO O O M *O to xo 10 t> O\ M 00 N 00 q w oo M O\ O -4" ON g -3 O 10 N O >.!>. tO ON n- to to OO M i^ to oo 10 i>.vo M M M O ite Imp ignite Bituminous lignite. Asphalte gn t li ect erfe P 438 FUELS. WOOD. ASPHALTE. Asphalte, like lignite, has a large proportion of hydrogen. It has less than 9 per cent, of oxygen and nitrogen, and thus leaves 8^ per cent, of free hydrogen, and it accordingly yields a porous coke. The average composition of perfect lignite and of asphalte may be taken in whole numbers as follows : Lignite. Asphalte. Carbon 69 per cent. 79 per cent. Hydrogen 5 9 Oxygen and nitrogen 20 9 Ash 6 3 IOO IOO Coke, by laboratory analysis 47 9 The lignites are distinguished from coal by the large proportion of oxygen in their composition from 13 to 29 per cent., which goes far to neutralize the hydrogen, so that for the first and second lignites the free hydrogen is less than 3 per cent. For the third bituminous lignite the free hydrogen amounts to nearly 6 per cent., and the varied effect of the proportion of free hydrogen is visible on the nature of the coke of lignite, as was found in the case of the coke of coal. Thus, With 2.64 per cent, of free hydrogen, the coke is pulverulent. 1.75 like wood charcoal. 5.76 ,, raised and porous. The small yield of coke from asphalte only 9 per cent. though the constituent carbon amounts to 79 per cent., is evidently caused by the great amount of free hydrogen volatilizing a large proportion of the carbon. TOTAL HEATING POWER OF LIGNITE AND ASPHALTE. The total heating power of lignite and asphalte, in units of heat, and their equivalent evaporative powers in water from 212, under one atmosphere, are as follows : Fuel. Heating power. Total evaporative power in water from 212 per pound of fuel. Perfect lignite units of heat. 11,678 pounds. 12 IO Imperfect lignite Bituminous lignite ... 9,834 ... I A AA.Q IO.I8 IA 06 Asphalte 16 6c c I 7 2A It may be observed, with reference to the lignites noted in table No. 141, that the more perfectly converted lignites possess the greatest heating power. There is a fine distinction between lignite passing to fossil wood, and fossil wood passing to lignite; their heating powers are nearly equal to each other, and both are less than the heating powers of the perfect lignites. HYGROMETRIC MOISTURE IN WOOD. 439 WOOD. Wood, as a combustible, is divisible into two classes: ist. The hard, compact, and comparatively heavy woods, as oak, beech, elm, ash; 2d. The light-coloured, soft, and comparatively light woods, as pine, birch, poplar. In France, firewood is classed as fresh wood (bois neuf), carried by land or water to its destination; raft wood (bois flotte], floated to its destination; and peeled wood (bois pelard), or oak stripped of its bark. According to M. Leplay, green wood, when cut down, contains about 45 per cent, of its weight of moisture. In the forests of Central Europe, wood cut down in winter holds, at the end of the following summer, more than 40 per cent, of water. Wood kept for several years in a dry place retains from 15 to 20 per cent, of water. Wood which has been thoroughly desiccated, will, when exposed to air under ordinary circumstances, absorb 5 per cent, of water in the first three days; and will continue to absorb it, until it reaches from 14 to 16 per cent., as a normal standard. The amount fluctuates above and below this standard, according to the state of the atmosphere. M. Violette found that, by exposing green wood to a temperature of 212 F., it lost 45 per cent, of its weight, which accords with the observa- tion of M. Leplay. He further found that by exposing small prisms of wood half an inch square and eight inches long, cut out of billets that had been stored for two years, to the action of superheated steam, for two hours, they lost from 1 5 to 45 per cent, of their weight, according to the temperature of the steam, which varied from 257 F. to 437 F. (125 C. to 225 C). The following are the particulars for four woods : Loss OF WEIGHT. TEMPERATURE OF DESICCATION. Oak. Ash. Elm. Walnut. per cent. per cent. per cent. per cent. 125 C. or 257 F. 15.26 14.78 I5.32 !5.55 150 302 17.93 16.19 I7.O2 17.43 175 347 32.13 21.22 36.94 21.79 200 392 35.80 27-5 1 33.38 (?) 41-77 (?) 225 437 44.31 33.38 40.56 36-56 The hardest wood, oak, lost, according to this statement, more weight than the softer woods. The observations queried appear to have been errors of observation. At a temperature of 200 C., or 392 F., wood becomes visibly altered, and the alteration, or decomposition, may likely commence at a lower temperature; and it may be that the losses of weight are not entirely due to a reduction of hygrometric water. A higher tempera- ture than 2 1 2 F. appears to be necessary to disengage all the water. Ordinary firewood contains, by analysis, from 27 to 80 per cent, of hygrometric moisture. 440 FUELS. WOOD. COMPOSITION OF WOOD. M. Chevandier, in 1844, published the results of analysis of five woods, beech, oak, birch, poplar, and willow. The woods were reduced to powder, and desiccated at a temperature of 140 C., or 284 R, before being sub- mitted to analysis. The results of analysis are given in table No. 148. Table No. 148. COMPOSITION OF WOODS. (From Analysis by M. Eugene Chevandier, 1844.) WOODS. COMPOSITION. Carbon. Hydrogen. Oxygen. Nitrogen. Ash. Beech per cent. 49-36 49.64 50.20 49-37 49.96 iposing tl S - 1 ? 49.96 51.24 49-5 5i-54 per cent. 6.01 5-92 6. 20 6.21 5-96 le Fagots 6.12 6. 02 6.22 6.09 6.26 per cent. 42.69 4I.l6 41.62 41.60 39-56 40.38 4I.IO 40.17 40.43 36.21 per cent. 0.91 1.2 9 I-I5 0.96 0.96 5 .OO 05 .OO .41 per cent. I.OO 1.97 0.81 1.86 3-37 1.77 1.90 1.32 2.98 4-57 Oak Birch Poplar Willow Twigs and Branches con Beech Oak Birch . Poplar. . . .... Willow Average of woods 49.70 50.46 6.06 6.14 41.30 39-65 05 .11 i. So 2.50 Average of fagots There is a remarkable nearness to identity in the composition of these woods, and also in the composition of the trunk and the branches. The results show that the composition of woods is practically as follows : Carbon 50 per cent. Hydrogen 6 Oxygen 41 Nitrogen i Ash... 2 100 Showing that there is only 56 per cent, of combustible matter, that there is a large quantity of oxygen, nearly sufficient to neutralize the whole of the hydrogen, and that there is only 2 per cent, of ash. The above-mentioned analysis is corroborated by the analysis of M. Violette, who ascertained the composition of different parts of the same tree, desiccated at a temperature of 80 C. or 176 F., with the results given in the following table No. 149: COMPOSITION OF WOOD. 441 Table No. 149. COMPOSITION OF THE VARIOUS MEMBERS OF ONE TREE. (From Analysis by M. Violette.) MEMBERS OF THE TREE. COMPOSITION. Carbon. Hydrogen. Oxygen and Nitrogen. Ash. Leaves per cent. 45.01 52.50 48.36 48.85 49.90 46.87 48.00 46.27 48.92 49.08 49-32 50-37 47-39 45.06 per cent. 6.97 7.31 6.60 6-34 6.61 5-57 6.47 5-93 6.46 6.O2 6.29 6.07 6.26 5-4 per cent. 40.91 36.74 44-73 41.12 43-36 44.66 45- I 7 44-75 44.32 48.76 44.11 41.92 46.13 43-5 per cent. 7.12 3-45 0.30 3-68 0.13 2.90 0-35 2.66 0.30 i.i3 0.23 1.64 O.22 5-oi 0111 i f bark . . Small branches.... | wood , , . , f bark Medium branches j ^ f bark . Large branches. ..| wood ; rr, i ( bark . . Trunk < " { wood fbark.. Large root.. . < i ( wood.. . Medium root i bark ; " \ wood Small roots with bark Averages : Leaves 45.01 49.00 48.66 45.06 6.97 6.21 6.45 5-4 40.91 43.00 44.64 43-50 7.12 2.58 0.25 5-oi Bark . Wood.. .. Small roots with bark Here it appears that the composition of the wood is about the same throughout the tree, and that of the bark also; that the wood and the bark have about the same proportion of carbon, 49 per cent., but that the bark has more ash than the wood. The leaves and the small roots have less carbon than the wood, only 45 per cent.; and more ash, 7 and 5 per cent. The leaves when dried at 100 C. lost 60 per cent, of water, and the branches 45 per cent. COMPOSITION OF ORDINARY FIREWOOD. The respective percentages of the constituent elements of stacked wood in its ordinary state are, of course, reduced in amount when the water is taken into account. Thus, in the following analysis of ordinary firewood, containing 25 per cent, of moisture, the carbon constitutes only 37.5 per cent, of the fuel : 44: FUELS. WOOD. per cent. Hygrometric water 25 Carbon 37.5 Hydrogen 4.5 Oxygen 30.75 Nitrogen 0.75 Ash 1.5 IOO.OO WEIGHT AND BULK OF WOOD. The density of a large number of woods has already been given in table No. 65, page 208. These values can, in most instances, only be given as approximate, for the density changes with the hygrometric condition of the wood. The specific gravity varies from 1.35, that of pomegranate, to 0.24, that of cork wood. The density of the ligneous fibre of which wood is formed, has been ascertained by M. Violette, from a great number of observations. The samples of wood were reduced to powder in a mortar, and dried at a temperature of 100 C. He found that the fibre of all woods had the same density, and that its specific gravity was 1.50. It is said that the quantity of intersticial space in a closely-packed pile of Table No. 150. OF THE WEIGHT AND BULK OF WOODS IN FRANCE. Woods in ordinary state of dryness. Weight of one cubic foot, heaped. Bulk of one ton, heaped. Firewood pounds. 21. Q tO 23 4 cubic feet. IO2 3 tO Q^ 7 Wood for charring, hard and soft, cut up Do. do. hard wood, cut up... Oak, cut up 18.8 23-4 22 A. tO 23 7 IIQ.I 95-7 TOO to 04. 4 Do. branches. *.<} \.\j ^^. i 17 -2 132 A Do. small branches A / 'O IQ.8 1 13. 1 Beech, cut up 237 Q4 A Do. branches *3* / IQ O V4-H- Il8 Do. small branches j-y.^ IQ 6 1 14. 2 Yoke-elm, cut up j.y.v- 23 I 07 o Do. branches 186 I2O.3 Do small branches IQ. C 1 14.6 Birch, cut up 21. 1 106. i Do. branches f 16.8 133.3 Fir 16.0 1 40. 1 Alder, cut up. 18 3 122.4 Willow, cut up 18.0 124.7 Aspen, cut up ... . 17. I3I.4 Pine in the United States 21. 106.0 Averages 20. o 1 14.0 COMBUSTION OF WOOD. 443 wood, consisting of uncloven stems, is 30 per cent, of the gross bulk ; for cloven stems, the intersticial space amounts to from 40 to 50 per cent. A cord of pine wood, that is, of pine wood cut up and piled, in the United States, measures 4 feet by 4 feet by 8 feet, and has a volume of 128 cubic feet. Its weight, in ordinary condition, averages 2700 Ibs.; or 21 Ibs. per cubic foot. A "corde" of wood, in France, has a volume of 4 cubic metres, or 141 cubic feet. Firewood is measured, in France, by the vote, of which the volume is 2 cubic metres, or 2 s feres. As the length of the billets is 1.14 metres, or 3.74 feet, the half-zwV, or sfere, measures 1.14 metres x 0.88 metre x i metre, equal to i cubic metre, or 35.3 cubic feet; and the vote is equal to 70.6 cubic feet in bulk. The weight of the vote of firewood, in Paris, is from 700 to 750 kilogrammes, or from 1544 to 1653 Ibs., averaging 1600 Ibs. The vote of wood for making charcoal, in the forests of the Ardennes, weighs 1324 Ibs. ; it consists of one-fourth oak and beech, one-fourth poplar and willow, and one-half elm. The hard wood for charring, of the forests of the Meuse, weighs 1653 Ibs. per vote. The above and other particulars given by M. Chevandier are collected and arranged in table No. 150, showing the weight and bulk of ordinarily dry wood. QUANTITY OF AIR CHEMICALLY CONSUMED IN THE COMPLETE COMBUSTION OF WOOD. In terms of the average percentages of carbon, hydrogen, and oxygen, in wood, page 440, the quantity of air consumed is, by the rule i, page 400, 1.52 (50 + 3 (6 - ) )= 1.52 x 52.625 = 80 cubic feet, 8 or 80 4- 13. 14 = 6.09 Ibs. GASEOUS PRODUCTS OF THE COMBUSTION OF WOOD. For one pound of dry wood the products are, by the expressions ( a ), (b\(c\ page 401, PRODUCTS. Pounds. Per cent. Carbonic acid, 50 x .0366 = 1.83 or 21.7 Steam, 6 x .09 =0.54 or 6.4 Nitrogen,.... (50 x .0893) + (6 x .268) + (i x.oi) =6.08 or 71.9 Total weight of products, 8.45 100.0 says 8*4 Ibs. weight of products. The volumes of the products at 62 are, by the expressions (e), (/), (/&), page 401, PRODUCTS. Cubic feet. Per cent. Carbonic acid, 50 x .315 = 15.75 or 14.4 Steam, 6x1.9 =11.40 or 10.4 Nitrogen, (50 x 1.206) + (6 x 3.618) = 8a'.bi or 75.2 Total volume of products for i Ib. of wood, 109.16 100.0 being about 13 cubic feet per Ib. weight of gaseous products. 444 FUELS. WOOD-CHARCOAL. TOTAL HEAT OF COMBUSTION OF WOOD. The total heat of combustion of dry wood is, by rule 4, page 406. 145 (50 + 4.28 (6 - II) ) - 145 x 53.745 = 7792 units, which is a little more than half, or 54^ per cent., of that of coal, and is equivalent, by rule 8, page 406, to the evaporation of 0.15 x 53.745 = 8.07 Ibs. of water at 212. When the wood holds 25 per cent, of water, there is only 75 per cent, or three-quarter pound of wood-substance in one pound; and the total heat of combustion is 75 per cent, of 7792 units, or 5844 units, which is only 41 per cent, of that of average coal. Similarly, the equivalent evaporative power is reduced to 6.05 Ibs. of water at 212, of which the equivalent of a quarter of a pound is appropriated to the vaporizing of the contained moisture. TEMPERATURE OF THE COMBUSTION OF WOOD. It is found, in the manner already shown, page 407, that 2.136 units raise the temperature of the products i F. The total heat of combustion, 7792 units -^2.136 = 3648 F.; and 3648 + 62 = 3710 F., is the temperature of combustion. When the wood holds 25 per cent, of water, the weight of the direct products is 75 per cent, of 8.45 Ibs., or 6.34 Ibs.; and the total heat of combustion is 5844 units, of which 1116 (total heat of steam) ^4 = 279 units, are appropriated to evaporate a quarter of a pound of water from 62, leaving 5844 - 279 = 5565 units of heat available for raising the temperature of the gases. To raise the direct products one degree of temperature, there are required Units. 2.136 x 24 = ............................................. 1. 602 The evaporated water, as gaseous steam, ) Total for i F ....................... 1.721 Then, 5565 -^1.721 =3234 F., the temperature of combustion. It is only 88.6 per cent, of the temperature for perfectly dry wood. In order to obtain the maximum heating power from wood as fuel, it is the practice, in some works on the Continent, as glass works and porcelain works, where intensity of heat is required, to dry the wood-fuel thoroughly, even using stoves for the purpose, before using it. WOOD-CHARCOAL. When wood is exposed to heat it is at first desiccated and afterwards carbonized. Under temperatures up to 300 F., the wood is simply desic- cated. Under temperatures over 300 the gaseous elements are driven off, until at 650 the wood yields a charcoal which is black, solid, and brittle. The gases are not completely driven off except under much higher temper- atures. Wood charcoal, completely converted, is black, solid, brittle, and friable; it preserves the form and structure of the wood from which it is made. Though easily pulverized, it makes a very hard powder. THE CARBONIZATION OF WOOD. 445 The following are the results of the experiments of M. Violette on the carbonization of wood. He experimented on black elder-wood, formed into prisms 2.4 inches long and 0.4 inch in diameter, made up in sets of twenty prisms. Each set was dried separately at a temperature of 300, in a current "of superheated steam, to which it was subjected during two hours. The carbonization was effected by the same medium, at least up to 662 F.; and in crucibles placed in a furnace, at higher temperatures. The temperatures arrived at when in the furnace were checked by the melting of small pieces of various metals placed in the crucible with the samples. The table No. 151 gives the weight of the products obtained as the result of carbonization at the given temperatures : Table No. 151. YIELD OF CHARCOAL FROM BLACK ELDER WOOD, CARBONIZED AT DIFFERENT TEMPERATURES. (By M. Violette.) Temperature of Carbonization. Weight of gross product from dry wood. Observations. Cent. Fahr. per cent. 150 3 02 IOO \ 160 320 9 8 170 338 94-55 1 80 356 88.59 190 374 81.99 These products are only wood more and 200 392 77.10 ) more altered, but they are not charcoal. 2IO 410 73-14 They are called, in France, brulots. 22O 428 67.50 2 3 446 55-37 240 464 50.79 250 482 49.67 t 260 500 40.23 270 518 37-14 280 536 36.16 ( Brown charcoal (charbon roux). Commences ( to be friable. 290 554 34-09 3 00 572 33.61 3 IO 590 32.87 320 608 32.23 330 626 3 z -77 340 644 3L53 Very black charcoal. 35 662 29.26 432 810 18.87 Melting point of antimony. Charcoal very hard. 1023 1873 18.75 Do. silver. Do. do. 1 100 2OI2 18.40 Do. copper. Do. do. 1250 2282 17.94 Do. gold. Do. do. 1300 2372 17.16 Do. steel. Do. do. 1500 2732 17.31 Do. iron. Do. do. ? 15.00 Do. platinum. Do. do. 446 FUELS. WOOD-CHARCOAL. From this table, it appears that charcoal, properly so-called, is not formed until a temperature of 536 F. is reached. From 536 to 644 F., brown charcoal (charbon ronx), from 36 to 31^ per cent., is formed. Beyond 644 F. the charcoal is black, and the yield diminishes with the increase of temperature, until, at the unknown temperature of melting platinum, it becomes just 15 per cent, of the weight of the dried wood from which it is produced. Brown charcoal is flexible, unctuous, and soft to the touch; black char- coal is rigid, brittle, and harsh to the touch. COMPOSITION OF CHARCOAL. The composition of these charcoals varies with the temperatures at which they are produced, as may be seen by the annexed table, No. 152, showing the results of analysis of some of the charcoals obtained : Table No. 152. COMPOSITION OF CHARCOAL PRODUCED AT VARIOUS TEMPERATURES. (By M. Violette.) Composition of the Solid Product. 1 Carbon Temperature for a given of Carbonization. Oxygen, weight of Carbon. Hydrogen. Nitrogen, Ash. wood. and Loss. Centigrade. Fahrenheit. per cent. per cent. per cent. per cent. per cent. 15 3 02 47-5 1 6.12 46.29 0.08 47-5 1 2OO 392 51.82 3-99 43-98 0.23 39.88 250 482 6 5-59 4.81 28.97 0.63 32.98 300 572 73-24 4-25 21.96 0-57 24.61 35 662 76.64 4.14 18.44 0.61 22.42 432 810 81.64 4.96 I5-24 1.61 15.40 1023 1873 81.97 2.30 14.15 i. 60 !5-3 1 1 00 2012 83.29 1.70 *3'79 1.22 !5-3 2 1250 2282 88.14 1.42 9.26 1. 2O 15.80 1300 2372 90.81 1.58 6.49 I.I5 15-85 1500 2732 94-57 0.74 3.84 0.66 16.36 Melting point of platinum }- 96.52 0.62 0.94 1-95 14.47 From this table, it is evident how materially a higher temperature operates in driving off the injurious gases, oxygen and nitrogen injurious, that is, in reducing the heating power; though the useful gas, hydrogen, is likewise driven off which is a loss for heating power. The carbon and the ash, the solid constituents, on the contrary, are proportionally increased. At the same time, there is an absolute loss of carbon, though less in degree than the diminution of the gases, as the temperature rises. The rate of diminution of the absolute quantity of carbon for a given weight of wood, is arrived at COMPOSITION AND YIELD OF WOOD-CHARCOAL. 447 by multiplying the percentages of constituent carbon in the second table (No. 152) by the relative percentages of gross products in the first table (No. 151), for a given temperature, as given in the last column of the second table. Here, the absolute quantity of carbon, which is 47.5 per cent, in the dry wood in its natural state, at 150 C, is reduced to 15.4 per cent., or one-third, at 432 C. or 810 F.; and beyond this temperature, however great the heat may be, there is practically no further diminution of the carbon; that is to say, that no more carbon is driven off by raising the temperature, the gaseous elements alone being driven off. It is remarkable that the proportion of ash found by M. Violette is only from i to 2 per cent., whilst the ash of the original wood averages i^ per cent.; for, it is naturally supposed that the whole of the ash should be concentrated in the charcoal, and should average 7^ per cent, suppos- ing that the yield of charcoal is one-fifth the weight of wood. The ash- element must have been withdrawn with the gases that escaped during the process of carbonization. It is found that, practically, the ash of the char- coal of commerce amounts to from 7 to 8 per cent. According to M. Sauvage, the charcoal manufactured in the forests is composed as follows : Carbon, 79 per cent. Hydrogen, free 2 Hydrogen, oxygen, and nitrogen, 1 1 Ash, 8 100 YIELD OF CHARCOAL BY LABORATORY ANALYSIS. M. Violette ascertained that the greater or less rapidity with which carbonization is effected influence materially the quantity of the yield. He obtained by slow carbonization twice as much charcoal as by rapid carbonization, at the same temperature; but he does not give the details of the experiments by which he arrived at this conclusion. He further found that when wood was carbonized in close vessels hermetically sealed, the yield was decidedly greater than in open vessels, thus : Temperature of Carbonization. In Open Vessels. In Closed Vessels. yield. yield. 1 60 C. or 320 F 97 per cent 97.4 per cent 340 C. or 644 F 29 79.1 The charcoal obtained at 180 C., in a close vessel, was brown, friable, and similar to that produced at 280 C. in an open vessel (table No. 151); though differently constituted, as the former held a greater proportion of gaseous matter, and also more ash than the latter. Finally, M. Violette ascertained by experiments, similarly conducted in open vessels with superheated steam, the quantity of charcoal for various woods and other ligneous substances, dried, in the first place, at 150 C. or 302 F., and then exposed to a temperature of 300 C. or 572 F. These are arranged in table No. 153, in the order of the quantities yielded: 448 FUELS. WOOD-CHARCOAL. Table No. 153. YIELD OF CHARCOAL FROM VARIOUS WOODS, DRIED AT 150 C., OR 302 R, AND CARBONIZED AT 300 C, OR 572 F. (By M. Violette. ) WOOD. Weight of Charcoal. WOOD. Weight of Charcoal. Cork per cent. 62.80 Apple tree per cent. M.6o Decayed Willow C2.I 7 Elm -24. en Wheat straw 46.00 Hornbeam . 34.44 Oak 46.00 Alder 34.4O Yew . 46.06 Birch 34.17 Beech 44. 2 c; Plum tree 34.O6 Pine 41.48 Maple 33. 7^ Poplar (leaves) 4O.QCJ Willow 33 74 Do (roots) 4O.QO Black elder 33 6l Wild pine 4O.7^ Ash . 33.28 Larch .... 4O.3I Pear tree. 31.88 Chestnut tree . 36.06 Lime tree 31.8s Cherry tree 2C.C-2 Poplar (stem) 31.12 Aspen 34.87 Sweet chestnut tree... 30.86 It appears from this table that cork, the lightest of woods, yields the largest percentage of charcoal, about 63 per cent.; and that poplar and sweet chestnut tree yield the lowest, about 31 per cent. But there does not appear to be any definite relation between the density of the wood and the quantity of the yield. CARBONIZATION OF WOOD IN STACKS YIELD OF CHARCOAL. Wood has been carbonized, from the remotest times, in heaps on the ground; and this process is still generally followed on the Continent. The stack or pile is covered with a mixture of earth and powdered charcoal, or with turf. A few openings are left in the covering to admit air to the interior, as well as a larger opening at the summit. When the stack is lit it burns rapidly, and so soon as flame appears at the chimney it is partially damped down by a turf. The progress of carbonization is indicated by the colour of the smoke, and when, finally, the mass becomes incandescent, it is covered up with earth and allowed to cool. By this process, the charcoal obtained usually amounts in weight to from 17 to 20 per cent, of the wood, and to more than this in the larger heaps. From 25 to 30 per cent., in volume is obtained in the small heaps, and from 30 to 34 per cent, in the larger heaps. The charring requires from sixty to eighty hours to produce a good quality of charcoal. In the departments of the Ardennes and the Meuse, in France, according to M. Sauvage, the following are the particulars of the yield of charcoal from wood. In the case of the Ardennes, the wood prepared for carbon- ization is a mixture of one-fourth oak and beech, one-fourth poplar and MANUFACTURE OF WOOD-CHARCOAL. 449 willow, and one-half elm. In the example from the Meuse, hard wood is used. The billets are about 30 inches in length; they are piled on end, in three tiers. The stack contains from 60 to 90 cubic metres, or from 80 to 120 cubic yards: Ardennes. Meuse. Mixed wood. Hard wood. Weight of a cubic metre of wood,. . . 662 Ibs. 827 Ibs. '3 to 145 lbs - ' ? 6 " Yi b d votL' ) ubic .. me . tr . e . of .. wood : } io ^ to "* cub - ft - <* to '4 **. ft - Percentage of yield, in weight, ....... 20 to 22 per cent. 21 per cent. Weight of a cubic metre of char- ) ., coal (heaped), ..................... } 44 lbs ' 53 lbs. It is obvious, from the small percentages of yield, averaging 21 per cent. for the mixed woods and the hard wood, that much of the substance of the wood is lost, which would by a better system of carbonization be yielded as charcoal. According to the table No. 153, the maximum yield obtainable from the mixed woods is 38 per cent.; and from the hard woods upwards of 40 per cent. MANUFACTURE OF BROWN CHARCOAL. The best method of making brown charcoal consists in heating the wood to be charred in a close vessel, by means of superheated steam introduced into the vessel. The required temperature is thus readily obtained, and a homogeneous product is yielded. This process was introduced by Messrs. Thomas & Laurens, and is successfully employed in France and Belgium in the production of brown charcoal for the manufacture of gunpowder, principally for fowling-pieces. DISTILLATION OF WOOD. The distillation of wood in close vessels affords evidence of the increased yield of charcoal obtainable by more careful treatment than in the open- air stack. In France, the wood is distilled in large iron cylinders or retorts capable of holding about 180 cubic feet of wood, as piled; and the opera- tion is completed in from seven to eight hours. By this process, 28 per cent, of charcoal is obtained, with the products of distillation in addition. But 12*4 per cent, of wood is consumed as fuel, making a total of 112^4 parts of wood for a yield of 28 parts of charcoal; which reduces the avail- able yield to 25 per cent, of the whole quantity of wood consumed, as against 2 1 per cent, in the open-air stacks of hard wood. There is a gain, in addition, in reduced cost of labour, and in the value of the yield of pyroligneous acid. The gases are directed into the furnace to aid as fuel in heating the retorts. CHARBON DE PARIS (ARTIFICIAL FUEL). Charbon de Paris, or Paris charcoal, is a mixture of two parts of powdered charcoal with one part of gas-tar, formed by powerful compression into 450 FUELS. WOOD-CHARCOAL. round pieces 4 inches long and 1% mcn i n diameter, and submitted to a high temperature. It takes fire easily, and burns slowly until it is entirely consumed, without making flame or smoke; it makes from 20 to 22 per cent, of ash. WEIGHT AND BULK OF WOOD-CHARCOAL. It does not appear that the density of wood-charcoal, as manufactured, has been accurately determined. M. Violette determined the density of the matter of the charcoal of black alder, reduced to impalpable powder, so as to extinguish the pores. It varied according to the temperature of carbonization, as shown in table No. 154: Table No. 154. ABSOLUTE DENSITY OF THE CHARCOAL OF BLACK ALDER, DRIED AT 212 R, AS POWDER. (By M. Violette.) Temperature of Carbonization. Specific gravity. Temperature of Carbonization. Specific gravity. Centigrade. Fahrenheit. Centigrade. Fahrenheit. 150 302 1.507 330 626 1.428 170 338 1.490 350 662 1.500 IQO 374 1.470 432 810 1.709 2IO 410 1-457 IO23 1873 1.841 230 446 .416 1250 2282 1.862 250 482 .413 I5OO 2732 1.869 270 5i8 .402 ( Fusing point } 290 554 .406 ? 1 of the plati- > 2.0O2 3 IO 590 .422 ( num retort, j The table shows that the density at 302 R, and of course at inferior temperatures, is that of the natural wood, dried at 100; that the density of the charcoals produced at from 302 R to 518 R was reduced by the increasing temperature from 1.507 to 1.402. The table shows briefly as follows : 1. That the density of the charcoal at 302 R, is that of the natural wood dried at 212, namely 1.507. 2. The density of the charcoals produced at from 302 F. to 518 R was reduced from 1.507 to 1.402. 3. The density at temperatures above 518 R increases with the temper- ature until it reaches 2.000, or double that of water, at the melting point of platinum. The specific gravity, weight, and bulk of various charcoals are given in table No. 65, page 211, and they are here abstracted for reference supple- mented by the weight and bulk of the charcoal of the Ardennes and the Meuse, derived from the data, page 449 : MOISTURE IN WOOD-CHARCOAL. 451 WOOD CHARCOAL. Specific gravity. Weight of a cubic foot. Bulk of one ton. As powder IXOO pounds. Q-3 CT cubic feet. 2/1 O In small pieces, heaped O.AOZ yj'j 2C.7 8S c As manufactured, heaped O.22Z "O'O I4..O ->V.J 160.0 Ardennes heaped O.2OI 12.? 180 o Meuse, heaped O.2AI "'0 I tJ.O 1 4.0 O *J"? m.y.\s MOISTURE IN WOOD-CHARCOAL. Charcoal absorbs moisture with avidity. The charcoal of commerce is usually exposed to the atmosphere, and open to rain- and it contains generally from 10 to 12 per cent, of moisture. Charcoal fresh made, from different woods, was exposed by M. Nau, for twenty-four hours, to an atmosphere loaded with moisture, and the weights of water they absorbed during that time are given in the following table, No. 155: Table No. 155. MOISTURE ABSORBED BY CHARCOALS DURING TWENTY-FOUR HOURS. (ByM. Nau.) Wood from which the Charcoal was made. Moisture Absorbed. Wood from which the Charcoal was made. Moisture Absorbed. White beech per cent. o 8 Horse chestnut per cent. 6 06 Ash . . 4 06 Elm 6 60 Oak 4 28 Alder 7 Q'? Birch 4AQ Scotch fir / -yo 8 20 Larch 4 to Willow 8 20 Maple J^ 4 80 Italian poplar 8 50 Pine Z.IA Fir 8. QO Red beech J- *-*+ 5 -20 Black poplar r6 "?o o w Showing a capacity for absorption varying from 0.8 to 16 per cent. It is certain that the period of exposure was not sufficiently long to saturate the charcoals. For charcoals have been known to absorb increas- ing quantities of moisture during three months. M. Violette made some observations on the capacity for moisture of charcoals which had been prepared from black alder at various temper- atures. The samples were exposed in a room the air of which was satur- ated with moisture. Observations were made every eight days, and they lasted three months until the charcoals ceased to absorb more moisture. The results show that charcoal is less absorbent the higher the temperature at which it is produced. The ordinary black charcoals, produced at tern- 452 FUELS. PEAT. peratures of from 480 to 750 F., are capable of absorbing from 5 to 7 per cent, of water; and taking the extreme observations, the absorption ranges from 21 to 2.2 per cent, between the extreme temperatures. At the lower temperatures, of course, the charcoal was only partially converted. Am CONSUMED IN THE COMPLETE COMBUSTION OF DRY WOOD-CHARCOAL. According to the analysis of M. Sauvage, page 447, there is 79 per cent. of carbon, and 2 per cent, of free hydrogen, in forest-charcoal. By formula (i), page 400, the volume of air at 62 F. chemically consumed in the complete combustion of one pound of charcoal, is I -5 2 (79 + (3 x 2 ) ) = I2 9 cubic feet of air at 62. The weight of the air is 129 -=- 13.14 = 9.8 pounds. GASEOUS PRODUCTS OF THE COMPLETE COMBUSTION OF DRY WOOD-CHARCOAL. The gaseous products of combustion consist of carbonic acid, steam, and nitrogen, and the total weight of them is found by formula ( 2 ), page 401, as follows: (79 x o.i 26) + (2 x 0.356) - 10.66 pounds. The total volume of the gases, as at 62, by formula (3), page 402, is, (79 x z -5 2 ) + ( 2 x 5-5 2 ) = I 3 I cubic feet at 62. HEAT EVOLVED BY THE COMPLETE COMBUSTION OF WOOD-CHARCOAL. The total heating power of dry wood-charcoal, having 79 per cent, of carbon and 2 per cent, of free hydrogen, is by formula (6), page 406: 145 (79 + (4. 28 x 2) )= 12,696 units of heat. The total evaporative efficiency is, by formula (8), page 406, - I 5 (79 + (4- 28 x 2 ) ) = I 3- I 3 pounds of water, evaporated from 212, under one atmosphere. For charcoal containing moisture the heating power is less, and may be estimated in the manner already adopted in the case of coke. PEAT. Peat is. the organic matter, or vegetable soil, of bogs, swamps, and marshes, decayed mosses or sphagnums, sedges, coarse grasses, &c., in beds varying from i or 2 feet to 20, 30, or 40 feet deep. The peat near the surface, less advanced in decomposition, is light, spongy, and fibrous, of a yellow or light reddish -brown colour; lower down, it is more compact, of a darker brown colour; and, in the lowest strata, it is of a blackish brown, or almost a black colour, of a pitchy or unctuous feel, having the fibrous texture nearly or altogether obliterated. Peat, in its natural condition, generally contains from 75 to 80 per cent, of its entire weight, of water. The constituent water occasionally amounts COMPOSITION OF PEAT. 453 to 85 or even to 90 per cent., in which case the peat is of the consistency of mire. It shrinks very much in drying; and its specific gravity varies from .22 or .34 to i. 06, the surface peat being the lightest, and the lowest peat the densest. Detailed particulars of the weight and specific gravity of peat are given at page 207. Table No. 156. CHEMICAL COMPOSITION OF IRISH PEAT, TAKEN AS PERFECTLY DRY. (Sir Robert Kane.) DESCRIPTION AND LOCALITY OF PEAT. Specific Gravity. Carbon. Hydrogen. Oxygen. Nitrogen. Ash. per cent. per cent. per cent. per cent. p. cent. I. Light surface, Philipstown. . . . .405 57.5 2 6.83 32.23 1.42 1.99 2. Rather dense, do. .669 S8.S6 5-91 31.40 85 3.30 3. Light surface, Wood of Allen 335 58.30 6-43 3L36 1.22 2.74 4. Compact and dense, do. 655 56.34 4.81 30.20 74 7-90 5. Light fibrous, Ticknevin , 6. Light fibrous, Upper Shannon .500 .280 58.60 58.53 6.55 5-73 30.50 32.32 1.84 93 2.6 3 2.47 7. Very dense, compact, do. *S3 59.42 5-49 30.50 1.64 2.97 Averages S28 58 18 ? 06 jj 21 I 23 3A<2 Table No. 157. COMPOSITION OF SUNDRY PEATS, INCLUDING MOISTURE. FIRST, EXCLUSIVE OF MOISTURE. .DESCRIPTION OF PEAT. Moisture. Carbon. Hydrog. Oxygen. Nitrog. Sulphur. Ash. Coke. Good air-dried Poor air-dried Dense, from Gal way per cent. per cent. 59-7 59-6 59-5 per cent. 6.0 4-3 7-2 per cent. 31 29 24.8 percent. .8 2.3 per cent. .8 p. cent. 2-4 6-3 5-4 p. cent. 44-3 Averages 59-6 5-8 29.6 3 4.7 Good air-dried Poor air-dried Dense, from Galway SECON 24.2 29.4 29-3 D, INCL 45-3 42.1 42.0 USIVE 4.6 3-1 F Mois 24 21 17.5 TURE. .1 .0 .6 .2 1.8 4.4 3-8 31-3 Averages 27.8 43-i 4.3 21.4 3-3 When wet, peat is masticated, macerated, or milled, so that the fibre is broken, crushed, or cut. The contraction in drying is much increased by this treatment; and the peat becomes denser, and is better consolidated than when it is dried as cut from the bog. Peat so prepared is known as condensed peat; and the degree of condensation varies according to the natural heaviness of the peat. Peat from the lowest beds is but little condensed ; but peat from the middle and upper beds is condensed, when dry, to from two to three times its natural density. So effectively is peat consolidated and condensed by the simple process of breaking the fibres 454 FUELS. PEAT. whilst wet, that no merely mechanical force of compression is equal in efficiency to mastication. The table No. 156 contains the results of chemical analysis of Irish peat of various qualities by Dr. Kane; the samples were desiccated before being submitted to analysis. Mr. A. M'Donnell gives the composition of average " good air-dried " peat and "poor air-dried" peat, analyzed by Dr. Reynolds, as in table No. 157; to which are added an analysis of dense peat from Galway, made by Dr. Cameron : From the above tables, it appears that sulphur is rarely found in Irish peat, and that the average composition of the peat is as follows : Perfectly Dry. per cent. Carbon 59 Hydrogen 6 Oxygen 30 Nitrogen i % Sulphur ? Ash . . 4 Including 25 per cent, of moisture. per cent. 22.; I Moisture 100 75 IOO Ordinary air-dried peat contains from 20 to 30 per cent, of its gross weight of moisture. If dried in air in the most effective manner, it contains at least 15 per cent, of moisture; and even when dried in a stove, it seldom holds less than 7 or 8 per cent. The peats named in table No. 156 were subjected to distillation, when they yielded water, tar, charcoal, and gas, in the proportions shown in table No. 158: Table No. 158. PRODUCTS OF DISTILLATION OF IRISH PEAT. Description and Locality of Peat. Water. Crude Tar. Charcoal. Gas. Nos. i and 2, Philipstown per cent. 23 6 per cent. 2.O per cent. 7.7. S per cent. 36 Q 3) Wood of Allen 4, do. 3 2 '3 ^8.1 3-6 2.8 o / o 39-i 32.6 O^'y 25.0 26. Z n 5, Ticknevin 3^.6 2.Q 71. 1 72.7 J? 6, Upper Shannon 38.1 4.4 21.8 3^-7 )) 7) 5? 21.8 i-5 19.0 57-7 Averages . 71.4. 2.8 2Q.2 36.6 O^'T- The tar, when re-distilled, yielded water, paraffine, oils, charcoal, and gas, The water yielded chloride of ammonium, acetic acid, and wood-spirit. PEAT-CHARCOAL. TAN. 455 Heating Power of Irish Peat. In peat of average composition, as given above, the heating power is by rule 4, page 406, perfectly dry, .................... 145 (59 + 4.28 ( 6 ~y) ) = 995 J units of heat; ' (44 + 4-8 (4.5 -1) ) = 743S units of heat. Deduct for evaporating the moisture, % lb., supplied at 62; ni6-^4 ..................... = 279 do. do. Effective heating power ............... 7156 do. do. The total evaporative power of i lb. of fuel, evaporating at 212, is as follows : Perfectly dry Containing 25 per cent, of moisture. When water is supplied at 62, divisor in6... 8.91 Ibs. 6.41 Ibs. Do. do. 212, do. 966... 10.30 Ibs. 7.41 Ibs. British and foreign peats are very much like Irish peat in composition ; the principal variation takes place in the proportion of ash. PEAT-CHARCOAL. fc The charcoal of ordinary dried peat is very porous, and, in general, light and fragile; but the charcoal of condensed peat is dense and solid. It burns easily but slowly : small incandescent pieces separated from the fire continue to burn until the whole of the carbon disappears. Good peat yields from 30 to 40 per cent, of charcoal; and the charcoal when perfectly dry consists generally of from 85 to 90 per cent, of carbon, and 10 to 15 per cent, of ash. The heating power of one pound of peat -charcoal, containing 85 per cent, of carbon, by rule 4, page 406, is, 145 x 85 = 12,325 units of heat; equivalent to the evaporation, at 212, of 11.04 Ibs. of water supplied at 62, or of 12.76 Ibs. supplied at 212. The temperature of combustion is that of carbon, 4877 F. In France, the peat-charcoal of Essonne contains 18.2 per cent, of ash. In the Ardennes, Bar peat carbonized in ovens yields 44 per cent, of char- coal; but this contains one-third volatile matter, one-fourth ash, and only 43 per cent, of carbon. TAN. Tan, or oak-bark, after having been used in the processes of tanning, is burned as fuel. The spent tan consists of the fibrous portion of the bark. According to M. Peclet, five parts of oak-bark produce four parts of dry tan; and the heating power of perfectly dry tan, containing 15 per cent, of ash, is 6100 English units; whilst that of tan in an ordinary state of dryness, containing 30 per cent, of water, is only 4284 English units. The weight of water evaporated at 212 by one pound of tan, equivalent to these heat- ing powers, is as follows : FUELS. STRAW, LIQUID FUELS. Perfectly dry. Water supplied at 62 5.46 Ibs. 3.84 Ibs. 212 6.31 4.44 STRAW. The average composition of wheat-straw is as follows: Water 14.23 per cent. Organic or combustible matter; consisting of ) g carbon, hydrogen, oxygen, and nitrogen., j 7 '3 Ash 7.47 100.00 Chemists have not, so far as the author has learned, thought it worth while to record the proportions of the organic elements. But it may be supposed that the composition of straw is similarly proportioned to that of peat. The weight of pressed straw is from 6 to 8 Ibs. per cubic foot. LIQUID FUELS. Petroleum is a hydro-carbon liquid which is found in abundance in America and Europe. According to the analysis of M. Sainte-Claire Deville, the composition of fifteen petroleums from different sources was found to be practically constant. The average specific gravity was .870. The extreme and the average elementary composition were as follows: Carbon 82.0 to 87.1 per cent. Average 84.7 per cent. Hydrogen 11.2 to 14.8 13.1 Oxygen 0.510 5.7 2.2 IOO.O The total heating and evaporative powers of one pound of petroleum having this average composition are, by rules 4 and 5, page 406, as follows : Total heating power .......... = 145 (84.7 + 4.28 (13.1 - ^) ) = 20, 240 units. 8 Evaporative power : evaporating at 2 1 2, water supplied at 62 = = 18.13 Ibs. Do. do. do. 212= 20.33 Ibs. Petroleum-Oils are obtained in great variety by distillation from petro- leum. They are compounds of carbon and hydrogen, ranging from C IO H 24 to C 32 H 64 ; or, in weight, from I 7I ' 42 carbon 1 to I 73 ' 77 carbon n \ 28.58 hydrogen j tO ( 26.23 hydrogen IOO.OO IOO.OO COAL-GAS. 457 The specific gravity ranges from .628 to .792. The boiling point ranges from 86 to 495 F. The total heating power ranges from 28,087 to 2 6,975 units of heat: equivalent to the evaporation, at 212, of from 25.17 Ibs. to 24.17 Ibs. of water supplied at 62, or from 29.08 Ibs. to 27.92 Ibs. of water supplied at 212. Schist-Oil, like petroleum, consists of carbon, hydrogen, and oxygen; but there is less hydrogen and more oxygen, as may be seen from the following analysis by St.-Claire Deville: From From Vagnas Schist. Autun Schist. Carbon 80.3 79.7 Hydrogen 11.5 zi.8 Oxygen 8.2 8.5 100.0 100.0 Pine-wood Oil, analyzed by the same chemist, contains 87.1 per cent, of carbon, 10.4 per cent, of hydrogen, and 2.5 per cent, of oxygen. COAL-GAS. Mr. Vernon Harcourt made an analysis of coal-gas, 1 one pound of which COAL-GAS. (Mr. V. Harcourt.) Carbon. Hydrogen. Oxygen. TotaL Olefiant gas. per cent. io-5 39-7 5-9 1.9 per cent. 1-7 13.2 8.1 per cent. 7-9 5-o per cent 12.2 5 2 -9 13.8 6.9 8.1 5-8 -3 Marsh gas. Carbonic oxide Carbonic acid Hydrogen Nitrogen Oxveen . . . 58.0 23.0 IOO.O had a volume of 30 cubic feet at 62 F. The heating power, calculated for the three elements, is as follows: Units. Carbonic oxide 13. 8 per cent x 4,3254-100= 597 Carbon 50.2 x 14,500-^ 100 = 7,279 Hydrogen 23.0 x 62,000-7-100= 14,260 Total heat of combustion 22,136 This is equivalent to the evaporation of 19.84 Ibs. of water from 62 at 1 See a paper on "Petroleum and other Mineral Oils, applied to the Manufacture of Gas," by Mr. Owen C. D. Ross, in the Proceedings of the Institution of Civil Engineers, vol. xl. page 150. 45 8 FUELS. COAL-GAS. 212 F., or to 22.92 Ibs. from and at 212. The heating power of i cubic foot is 738 units, equivalent to the evaporation of .66 Ib. or .76 Ib. of water. Mr. F. W. Hartley 1 tested the heating power of gas manufactured by the South Metropolitan Gas Company, London. By means of his gas calorimeter he determined the heating power of one cubic foot of gas at 60 F., under 30 inches of mercury, to be 622.15 units, equivalent to the evaporation of .56 pound of water from 60 at 212, or .64 pound from and at 212. Taking the means for these two gases, the heating power of i cubic foot at 62 F. is equivalent to the evaporation of .70 pound of water from and 1 Report on the Gas Section of the International Electric and Gas Exhibition at the Crystal Palace, 1882-83, P a g e 2 3- It ma 7 ^ ere he stated that according to the results of tests of several "Instantaneous Gas Waterheaters," made by Mr. D. K. Clark, in the same con- nection, much more heat was generated than was deduced from Mr. Hartley's deter- minations. It is due to Mr. Hartley to state that he was cognizant of the usual presumption that the potentiality of gas as a heating power is greater than what is deduced from such experiments. "The problem," he says, " of determining with ease and certainty when a quantity of two to three cubic feet can be had, the absolute calorific value of a combustible gas, within about ]/ 2 per cent, of the truth, is, I think, completely solved; and the fact that the calorific power indicated for the gas in question is much below that which ordi- nary coal-gas is presumed to possess, in no degree disturbs my belief in the accuracy of the results which I have now the honour to submit." Report, page 23. APPLICATIONS OF HEAT. This section on the applications of heat comprises the principles of the transmission of heat through solid bodies. The consideration of the application of the heat of furnaces for the generation of steam in boilers will be taken up in the section on steam-boilers. In the present section, the subjects dealt with relate to the heating and evaporation of water by steam, the condensing of steam by water, the heating of air by hot water and by steam, the warming and ventilation of buildings, distillation, cooling, drying, blast-furnaces, and cognate subjects. TRANSMISSION OF HEAT THROUGH SOLID BODIES FROM WATER TO WATER THROUGH SOLID PLATES OR BEDS. With a view to educe the general principles of the transmission of heat through solid bodies, M. Peclet made a series of experiments on the trans- mission of heat through plates of metal, heated on one side by heated water, and cooled on the other side by water at a low temperature. He found from experiments made with wrought iron, cast iron, copper, lead, zinc, and tin, that when the fluid in contact with the surface of the plate was not changed by artificial means, the rate of conduction of metals was not only the same for different metals, but also for different thicknesses of the same metal. Correctly ascribing this uniformity of performance to the presence of a stagnant film of water adhering to the surfaces of the plates, which, by its inferior conductivity, negatived in a greater or less degree the conduc- tivity of the plates themselves, he made a new series of experiments with lead plates, in which the water was thoroughly circulated over the surface; and he found that the quantity of heat transmitted through the plates was inversely proportional to the thickness. Having by this means settled the constant for lead, he adopted the results of Depretz's experi- ments on the conducting power of metals (see page 331), and calculated the constants for other metals from these data. He, further, made a series of experiments on the conducting or transmissive power of " bad conduc- tors" of heat, between two surfaces of water: stone, and wood and other vegetable substances, which were incased in two thin coats of copper, to prevent the absorption of water by them. M. Peclet lays down the elementary law of the transmission of heat as follows : The flow of heat which traverses an element of a body in a unit of time is proportional to its surface, and to the difference of temperature of the two faces perpendicular to the direction of the flow; and is in the inverse 460 APPLICATIONS OF HEAT. ratio of the thickness of the element. This law, he maintains, is rigorously deduced from the nature of the motion of heat, and he embodies it in the following formula: M = (/-/-)-J.j (i) in which / and f are the temperatures of the surfaces, C the quantity of heat transmitted per hour for one degree of difference of temperature through one unit of thickness, and E the thickness. That is to say, using English measures: if the difference of temperatures in degrees Fahren- heit be multiplied by the constant C for the given material, one inch thick, and divided by the thickness in inches, the quotient is the quantity of heat in English units passed through the plate per square foot per hour. The quantities of heat transmitted through plates or beds of metals and other solid bodies, one inch in thickness, for i F. difference of tempera- ture per hour, as determined by M. Peclet, are given in table No. 159, being the values of the constant C in formula (i). The conditions are, that the surfaces of the conducting material must be perfectly clean, that they be in contact with water at both faces of different temperatures, and that the water in contact with the surfaces be thoroughly and con- Table No. 159. QUANTITIES OF HEAT TRANSMITTED FROM WATER TO WATER THROUGH PLATES OR BEDS OF METALS AND OTHER SOLID BODIES i INCH THICK, PER SQUARE FOOT, FOR i F. DIFFERENCE OF TEMPERATURE BETWEEN THE Two FACES PER HOUR. Selected from M. Peclet's tables, and converted for English measures. Substance. Quantity of heat. Substance. Quantity of heat. Substance. Quantity of heat. Gold units. 62O Fir across fibre units. 7 A Coke powd units. 80 Platinum Silver . 604 co6 Fir, along the fibre /'T I l6 Iron filings Cotton wool 1.26 22 Copper . . OV" crc Caoutchouc -L'O^ I l6 Calico 6 Z AO Iron 000 22^ Gutta-p ercha 'y I 37 Carded wool ,q.v 2C Zinc "^0 22^ Glass *-'O I 6 q6 Eider-down oo -7 I Tin Sand *voy 2 l6 Canvas o x A2 Lead 112 Brick powder'd 112 \Vhite wntincf- Marble Plaster 24 2 6 Chalk, do. Ashes of wood .69 r -2 paper Gray paper un- 34 Terra cotta 4.8 Wood-charcoal, oo sized .10 Oak, across fibre 1.69 powdered.... .63 stantly changed. M. Peclet found that when metallic surfaces became dull, the rate of transmission of heat through all metals became very nearly the same. Mr. James R. Napier made experiments with experimental boilers of iron and copper of various thicknesses, over a gas flame, and he found only a HEATING AND EVAPORATION OF LIQUIDS. 461 small difference in evaporating power of about a twentieth or a thirtieth in favour of the copper: results which are corroborative of M. Peclet's deductions. Professor Rankine states that in all experiments of this kind the con- dition of the heating surface is important, whether smooth or rough, and whether perfectly clean or incrusted to any extent. But, the rate of transmission of heat through metallic plates also differs very much according to the substances in contact with the plate, between which the heat is transmitted : as between water or steam and water, or between water and air, or gaseous matter and water, and so on. Mr. Thomas Craddock, at an early period, proved that the rate of cooling by transmission of heat through metallic surfaces, was almost wholly dependent upon the rate of circulation of the cooling medium over the surface to be cooled, and that water was enormously more efficient than air for the abstraction of heat. He suspended a tube filled with hot water having a thermometer suspended in the water. The water was cooled from a temperature of 180 to 100 F., in still air, in 25 minutes, and in still water in one minute. Again, when he moved the tube filled with hot water, by rapid rotation, at the rate of 40 miles per hour through air, it lost as much heat in i minute as it did in still air in 1 2 minutes. In water, at a velocity of two miles per hour, as much heat was abstracted in half a minute as was absorbed in one minute when at rest in the water. Mr. Craddock concluded that the circulation of the cooling fluid becomes of greater importance as the difference of temperature on the two sides of the plate becomes less. HEATING AND EVAPORATION OF LIQUIDS BY STEAM THROUGH METALLIC SURFACES. Mr. John Graham heated water in a square wooden cistern, having a double iron bottom, into which steam of 16^ Ibs. per square inch, abso- lute pressure, having a temperature of 218 F., was admitted. When the water in the cistern stood at 60 F., the steam was admitted, and the following were the successive temperatures at equal intervals of time, as- reported by Mr. Graham : Time from the commencement. seconds. O IO 20 40 50 60 70 80 90 IOO It was found to be difficult to raise the temperature above 210 F. The increased activity in the rise of temperature towards the end, was no doubt ie Temperature it. of the water. Fahrenheit. 60 Increments of temperature. Fahrenheit, o , IOO 4 : 1*8 34 m" 16 108 6 210 4. 462 APPLICATIONS OF HEAT. due to the increased movement in the water as it approached the boiling point. To show the rate of the passage of heat with respect to the mean difference of the temperatures of the steam and the water during each interval of ten seconds, the mean temperature of the water during each interval is given in the second column below, the difference of these mean temperatures and that of the steam in the third column, the increments of temperature in the fourth column; and, in the last column, the rise of temperature per degree of difference of temperature is given : Times. Mean tempera- tures of the water. Difference of temperatures of the water and the steam. Increments of temperature. Increments of temperature per degree of difference. seconds. Fahrenheit. Fahrenheit. Fahrenheit. Fahrenheit. IO 80 138 40 .290 2O 117 101 34 .336 30 I 4 6 72 24 333 40 166 52 16 .308 5 178 40 9 .225 60 187 3 1 9 .290 70 i95 2 3 6 .261 80 199-5 18.5 3 .162 90 203-5 14-5 5 345 IOO 208 10 4 .400 The quantity of heat transmitted per degree of difference of temperature is in proportion to the increments of temperature in the last column. Though irregular, they are, taken together, practically uniform per degree of difference of temperature. At the same time, the quantity per degree in the middle stages appears to be slightly reduced as the total difference of temperature is reduced. M. Clement found that a sheet of copper, i metre square and about Y% inch thick, when heated on one face by steam of 212 R, and cooled on the other face by water at 8 2. 4 R, making an excess of temperature of i29.6, condensed 20.5 Ibs. of steam per square foot per hour, equivalent to 20.5-4- i29.6 = o.i6o Ibs. of steam condensed per square foot per degree of difference of temperature per hour. The total heat of steam at 212 R is io95.6 above 82.4, and for 20.5 Ibs. of steam there are 1095.6 x 20.5 - 22,460 units of heat, and 22, 460 -f- 129.6= 173 units of heat, which is the quantity of heat passed through the plate per square foot per degree of difference of temperature per hour. M. Peclet gives the performance of a copper boiler with double bottom for boiling beet-root juice by steam of three atmospheres, or 275 R, admitted into the bottom. The area exposed to steam amounted to 25.82 square feet. A quantity of juice weighing 1984.5 Ibs. was delivered into the copper at a temperature of 39 F., and heated to 212 F., through 173 HEATING AND EVAPORATION OF LIQUIDS. 463 in sixteen minutes, equivalent to a rise of (173 x 60 -f- 16) = 649 of tempera- ture per hour. The total heat transmitted per square foot per hour was 649 x 1984.5 -r 25.82 = 49,880 units of heat. The mean temperature of the juice was (2 12 + 39) -^ 2 = 126, and the mean difference of temperature was 275- 126= 149; then the total heat trans- mitted per square foot per degree of difference of temperature per hour was 49,880+ 1 49 = 335 units of heat. The total heat of the steam was 1071 above 126, the mean temperature of the juice, and the quantity of steam condensed per square foot per degree per hour was =.313 lb. of steam. 1071 M. Peclet quotes the results of experiments made by Laurens and Thomas on the heating power of steam operating through coils of pipe. In the first experiment, the pipe was 137.8 feet long and 1.36 inches in diameter externally, presenting 48.20 square feet of surface. Steam of three atmospheres, or 275 F., was freely admitted into the pipe, and it raised the temperature of 882 Ibs. of water from 46 F. to 212 through 166 in four minutes, equivalent to a rise of (166 x 60 -4- 4) = 2490 in an hour. The mean temperature of the water was (212 + 46) -r- 2 = 129, and the difference of temperature was 275 - 129 = 146. Hence the total heat transmitted per square foot per hour was 2490 x 882 -f- 48.20 = 45564 units of heat, and the total heat per square foot per degree per hour was 45564 -f 146 = 312.1 units of heat. The total heat of the steam was 1068 above 129, and the quantity of steam condensed per square foot per degree per hour was, therefore, '* = .292 lb. of steam. 1068 Next, 551.25 Ibs. of water was evaporated from 212 by the same steam, in ii minutes, being at the rate of 3007 Ibs. per hour, or 62.38 Ibs. per square foot per hour. The total heat of the atmospheric vapour was 966 above 2 1 2, and the heat transmitted per square foot per hour was 966 x 62.38 = 59,710 units of heat. The difference of temperature was 275-212 = 63, and the quantity of heat passed per square foot per degree per hour was 59,7 10 H- 63 = 948 units of heat. The quantity of steam condensed per square foot per degree per hour was ?4? = . 981 lb. of steam. 900 464 APPLICATIONS OF HEAT. In another experiment, two coils of steam-pipe 49.2 feet long, and 1.34 inches in diameter, presenting a surface of 34.52 square feet, with steam of two atmospheres, or 25o.4 F., evaporated 1587.6 Ibs. of water at 212 per hour; or 46 Ibs. per square foot per hour, with a difference of tem- perature (250.4 - 212) = 38.4 F. The quantity of heat passed per square foot per hour was, 966 x 46 = 44,430 units, and, per degree per square foot per hour, was, 44,430 -f- 38.4 = 1 157 units. The quantity of steam condensed per degree per square foot per hour was .. 966 The following are the results of experiments by M. P. Havrez in heating water by steam with a coil of copper pipe, given in Engineering, vol. vi. The coil was 14.21 feet long, and 1.57 inches in diameter; superficial area, 5.85 square feet. The pipe was incrusted to some extent: i. 67 Ibs. 300 F. 232.65 Ibs. from 68 to 212 2-89,, 319.4 232.65 to 122, 4min.; 212, 10 min. 3. 89 319.4 217.80 104 to 212 3 7^ Actual weight of steam Weight per square foot Weight per i F. differ- condensed. per hour. ence of temperature. 1. 41.25 Ibs. 42.25 Ibs. .264lb. 2. 44.00 45.00 .271 3. 28.60 39.00 .270 Averages, 42.08 .268 It may be noted that the water was heated to 122, and then to 212, in the second and third experiment, at the following rates : 2d Experiment, to 122 at 13. 5 per minute. To 212 at 15 per minute. 3d do. 6.0 20 In continuation of the experiments, with the same pressure of steam, portions of the water were evaporated : Water evaporated in 10 minutes. Ebullition. Evaporated per And per degree. I. 2. 3- 9.9 Ibs. 15-95 7-7 soft, violent, very soft, 10. 1 5 Ibs. 16.35 7.89 .115 lb. 174 .084 11.46 .126 There are, as Engineering remarks, inconsistencies in these results. The scale, no doubt, impeded the activity of the heat. The results of experiments by M. Havrez, with a cast-iron boiler having a double bottom, are also given by Engineering. The boiler was 18.5 inches in diameter, and 13.5 inches deep, and had a jacketted surface of 6.576 square feet: HEATING AND EVAPORATION OF LIQUIDS. 465 Total pressure and temperature of steam. 1. 67 Ibs. 300 F. 2. 67 300 Water heated. 229.7 Ibs. 237-6 from 80 to 212, in 25 minutes. 68 to 212, in 22 Actual weight of steam condensed. 3i.24lbs. i 34.10 Averages, Per square foot per hour. 10.14 Ibs. 14.10 12.12 And per degree differ- ence of temperature. .066 lb. .088 077 In evaporating the water at 212, from 8.8 to 11.44 Ibs. of steam was condensed in 10 minutes; being at the rate of 8.02 to 10.4 Ibs. per square foot per hour; or .091 to .118 lb. per square foot per degree of difference of temperature per hour. The quantities of heat, in M. Havrez's experiments, transmitted per square foot per degree of difference of temperature per hour are found from the quantities of steam condensed, as follows : Heat utilized from i lb. of steam. Coiled pipe ; heating water : ist Experiment, 2d Do. 3d Do. Steam condensed per square foot per degree per hour. 1005 units 1071 1053 Averages, Coiled pipe ; evaporating water : 961 units ist Experiment, 2d Do. 3d Do. 967 967 Averages, Cast-iron boiler ; heating water :- ist Experiment, 2d Do. 1059 units 1063 Averages, Cast-iron boiler ; evaporating water : 961 units x ist Experiment, 2d Do. Averages, .264 lb. .271 .270 .iiSlb. 174,, .084 126 .066 lb. .088 .077 .091 lb. .118,, .105 Heat transmitted per square foot per degree per hour. 265. 3 units. 290.2 284-3 280.0 1 10.5 units. 168.2 81.2 120.0 69.90 units. 93-54 81.72 87.45 units. 113.40 100.43 Mr. William Anderson 1 gives the results of experiments on the power of sugar-clarifiers in heating water. They were 6 feet 6^ inches in diameter, and 2 feet 6 inches deep; containing a copper pan 18 inches deep, bolted into cast-iron steam-jackets, with a working capacity of 450 gallons, and having a heating surface of 52.58 square feet. The 1 "On the Aba-el- Wakf Sugar Factory, Upper Egypt." Proceedings of the Institution of Civil Engineers, vol. xxxv., 1872-73. 30 466 APPLICATIONS OF HEAT. average results of three experiments in heating water to 212 are as follows : Mean duration of the experiments 24 minutes. Mean initial temperature of water 67 F. Mean steam pressure above atmosphere ...42.1 Ibs., 289 F. Mean weight of condensed steam 742 Mean weight of water heated 4558 units. Units of heat in condensed steam 742 x 990 734,580 Heat spent in heating copper 840 x 145 x. 095= 11,571 cast iron 2828 x 145 x. 129= 52,900 wrought iron 567 ,,xi45x.ii3= 9,200 water 4558 x 145 =660,910 734,671 units. Units of heat per square foot per difference of i per hour in heating water 2 10.2 Loss in heating clarifier, radiation, &c u.i per cent. The mean temperature of the water was - = 140, and the heat utilized per pound of steam was io62(=ii7o + 32- 140). Then, 2ICX2 = .i981b. of steam, 1062 condensed per square foot per degree per hour. In other experiments, with a smaller clarifier, similar in construction, of 1 2 gallons of capacity, the trials were carried further, and the rate of boiling was ascertained, both for water and for sirup, the latter consisting of a solution of 9 Ibs. of molasses and 4 Ibs. of sugar in 90 Ibs. of water, equal to juice at about 8 Beaume. The quantities of heat passed through the metal were as follows : Water. Juice, units. units. In heating, per square foot per difference of i F. per hour 260 219 In evaporating, 606 521 showing a greatly accelerated passage of heat when evaporating, 2 y$ times as much as in only heating the water: also, that the addition of 14^ per cent, of sugar reduced the efficiency of the surface by about 1 5 per cent. Mr. Anderson made similar trials to test the efficiency of the concen- trators, for their evaporating powers. It is only necessary to state here that each of the concentrators consists of a copper tray 23 feet long by 6 feet wide, % of an inch thick, heated by a steam-boiler beneath it, and forming part of it. The boiler is 12^ inches deep, flat-bottomed, and stayed to the tray at 6 inches pitch. The heating surface of the tray is increased by 495 upright hollow nozzles of brass, screwed into it, very thin, and slightly taper; average external diameter 2^ inches, vertical projection 4^ inches. The tray is inclosed by a sheet-iron cover. The heating surface of the tray consisted of 138 square feet horizontal surface, and 187 feet of vertical surface, together, 325 square feet. By experiment, it was found that surfaces similar to those of the tray performed as follows : HEATING AND EVAPORATION OF LIQUIDS. 467 In heating water to the boiling point, 5.8 Ibs. effective pressure per square inch, 228 F., per square foot per i F. difference per hour 368 units. In evaporating 660 Here the passage of heat for evaporation was 1.8 times as much as in heating without evaporation. Applying this ratio to the performance of the tray itself, Mr. Anderson calculates that the efficiency of the tray by experiment was For heating 271 units. For evaporation 49 1 The obviously superior efficiency of the model is accounted for by its having been fully charged with steam from the factory boilers; "whilst in the actual tray the generator was evidently unequal to the work." The mean pressure in the generator was 47 Ibs. effective, with the temperature 294 F.; and that in the tray was 5.8 Ibs., temperature 228 F. The total heat of the first steam was 1171 from 32 F., or 975 above 228; and the quantity of steam condensed per square foot per degree per hour was Condensed. For heating 271 975 = 0.278 pound of steam. For evaporating 491-975 = 0.504 Mr. F. J. Bramwell, in discussing Mr. Anderson's paper, gave particulars of similar experiments made by him with a jacketed copper pan, having a working capacity of 100 gallons, and a heating surface of 25 square feet. The pan had been at work for eight or nine years, and probably was incrusted on the steam side. He tried the performance of the pan with steam successively of 5 Ibs., 10 Ibs., 15 Ibs., and 20 Ibs. effective pressure, raising the temperature of the water from 58 to 212, and evaporating it. In the first experiment, with 5 Ib. steam, he found that the rates of trans- mission of heat per square foot per degree of difference per hour, taking observations every five minutes, in raising the temperature to 200, were successively 161, 151, 176, 160 units of heat, whilst in heating from 200 to 212 the rate advanced to 327 units; and, when ebullition commenced, to 427 units. The observed rates at the different pressures are subjoined for comparison: Effective pressure Initial temper- Average rate of trans- Average rate of transmission, of steam. ature of water. mission up to 212. evaporating at 212 F. 5 Ibs 58 F 427 units. 10 Ibs 58 1 86 units 435 15 Ibs 58 45 8 20 Ibs 58 205 488 Averages, 196 45 2 From these results it appears that the rate of transmission for evapora- tion is more than double the rate for heating; and the detailed observa- tions, at 5 Ibs. pressure, show a marked acceleration of transmission when the water was within 12 of the boiling point. These experiments confirm those of Mr. Anderson, and it is very probable that the greater agitation 468 APPLICATIONS OF HEAT. and quicker circulation of the water as it neared the boiling point, and whilst boiling, was the cause of the increased rate of transmission of the heat. The average rates above given show that, per square foot per degree per hour, the quantities of steam condensed were : Condensed. In heating up to 212, 201 Ib. In evaporating at 212, 463 The various results of performance above detailed are numbered and collected in table No. 160, and the averages for copper-plate surfaces, copper-coil surfaces, and cast-iron surfaces are given in the lower part of the table. Table No. 160. RESULTS OF PERFORMANCE OF COILED PIPES AND BOILERS IN HEATING AND EVAPORATING WATER BY STEAM. AUTHORITY. APPARATUS. Steam condensed per square foot, for i F. difference of tem- perature per hour. Heat transmitted per square foot, for i c F. difference of tem- perature per hour. Heating. Evapo- rating. Heating. Evapo- rating. i. Clement .... 2. Peclet Copper plate Ibs. .l6o .313 .292 .268 .077 .198 .278 .2OI Ibs. .981 1. 2O .126 .105 54 463 units. 173 335 312 280 82 210 271 3 68 196 units. Copper boiler. 94 8 1120 120 100 49 1 660 45 2 3. Laurens 4. Do 5. Havrez 6. Do 7. Anderson... 8. Do. ... 9. Do. ... 10. Bramwell... Copper coil 2 Do do Copper coil Cast-iron boiler Copper clarifier Copper concentrator.. ( Copper concentra- ) \ tor (model) J Copper pan Averages for c Nos. 2, 7, 8, Averages for Nos. 3, 4... :opper- plate surface, ) 9, 10. f .248 .292 .077 483 1.090 .105 2 7 6 3 I2 82 534 1034 100 copper-pipe surface, ) Cast-iron plate s urface, No. 6 Note. Nos. i and 5 are omitted from the averages as the information is incomplete, and for No. 5, the results are not consistent. It appears that the efficiency of copper-plate surface for evaporation is double its efficiency for heating water ; for copper-pipe surface the efficiency is more than three times as much; and for cast-iron plate surface, a fourth more. That the efficiency of pipe-surface is a fifth more than that of plate- surface for heating, and more than twice as much for evaporation. COOLING OF HOT WATER IN PIPES. 469 That, in round numbers, copper-plate surface condenses half a pound of steam, copper pipe condenses a pound of steam, and cast-iron plate-surface a tenth of a pound, per square foot per degree of difference of temperature per hour, for evaporation. That the quantity of heat transmitted is at the rate of about 1000 units per pound of steam condensed. These are the results to be expected when the surfaces are in good condition. COOLING OF HOT WATER IN PIPES. M. Darcy states that the water from the artesian wells at Crenelles passed underground through cast-iron pipes of from 6% to 10 inches diameter, for a length of 2530 yards, in 8^ hours, equivalent to an average velocity of 3 inches per second, discharging about 50 gallons per minute. The water was cooled from 80 to 69.$ F., or io.5; being at the rate of i.24 per hour. The loss of heat amounted to 307,600 units per hour, which passed through 16,424 square feet of surface: at the rate of 18.7 units per square foot per hour, for a mean temperature of 75. When at rest in the pipe, the water was cooled at the rate of 10 F. in 7 hours, or 21.6 units per square foot per hour. Taking the temperature of the ground at 62, the mean difference of temperature was 13 F., and the heat trans- mitted per square foot per degree per hour was 18.7 + 13 = 1.44 units when the water was in motion, and 21.6-^13 = 1.66 units when the water was at rest. Taking the results of experiments by Mr. Tredgold on the rate of cool- ing of water in pipes, in air, as corrected by Mr. Hood, a cast-iron pipe 30 inches long, 2*4 inches in diameter internally, and % inch thick, was filled with water at 152 F. It exposed a surface of 2 square feet, with a surrounding temperature of 67 F.; and the quantity of water, including an equivalent for the heated iron, was 172 cubic inches, or 6 Ibs. weight. The water was cooled at a nearly uniform rate, from 152 to 140 F., in the following times, to which are added the cooling and the units of heat passed per minute : State of cast-iron surface. Cooled ? F. in ^.^ ^Sf^fm^ute. 1. Ordinary brown (rusty), 15 minutes ... o.8 F....0.8 x 3 = 2.4units. 2. Black varnished, T 4-53 ...0.83 ...0.83x3 = 2.5 3. White, two coats of lead paint, 15.33 ...0,78 ...0.78x3 = 2.34 To reduce these results to the general standard for comparison : Heat passed off Mean tem- perature of water. Mean difference of temperature of water and air. per square foot per hour. per square foot per degree of differ- ence per hour. Fahr. Fahr. units. units. I 146 . . 7Q 2.4 x 60= 144 1.823 2 ..146 . 70 2 C X 60= I HO . I.QOO 1. , ..146 / 7 . 70 w *Jf w 2.^4 X 60=140.4 y wv - . 1.778 From other experiments by Tredgold, hot water was cooled in vessels made of tinned plate, sheet iron, and glass from 180 to 159 F., in a room at 56, showing an average excess of temperature of 114 F. They con- 4/0 APPLICATIONS OF HEAT. tained 2.2 Ibs. of water, including an equivalent for the metal. The results were as follows : g r Area of Time to cool Cooled per Heat passed per square surface. 30 F. minute. foot per minute. square feet. minutes. Fahr. units. 4. Tinplate, ...... 55 ...... 46 ...... o.65 ....... 65x2.2- -55 = 2.60 5. Sheet iron,.. .533 ...... 29 ...... i.c>3 ...... 1.03 x 2.2 - .533 ^4.26 6. Glass, ......... 500 ...... S 1 /^ ...... -94 ....... 94 x 2.2 - .50 = 4.14 The heat passed off per hour was, 4. 156.0 units per square foot, and 1.37 units per degree of difference. 5- 2 55-6 2 - 2 4 >? >, 6. 248.4 2.18 To group the experimental results adduced for the transmission of heat from hot water in iron pipes and vessels to the external air : Per square foot per degree difference of temperature per hour. 2^ inch cast-iron pipe, ^ inch thick, naked, ..... 1.82 units. Sheet-iron vessel, ........................................ 2.24 Mean, 2.03 COOLING OF HOT WORT ON METAL PLATES IN AIR. The results of experiments on the cooling of wort at Trueman's brewery are recorded in Engineering, vol. vi. Two coolers, no feet by 25 feet, made of thin copper, No. 15 wire-gauge, or T /i 3 inch thick, were supported on open joists, and air was free to circulate above and below the coolers. The total cooling surface amounted to 5500 square feet. The wort was run over the coolers in a thin stream, of which 50 barrels of 360 Ibs. each were cooled from 212 to no F. per hour. The total heat passed off by evaporation and by conduction through the metal was 50 x 360 x (212- iio) = 1,836,000 units per hour; being at the rate of 334 units per square foot per hour. When the wort was left to stand on the coolers, from 2 to 2 ^ inches deep, it was cooled 140 in from six to eight hours. Taking 10 Ibs. per square foot as the weight of the water, the quantity of heat passed off was 140 x 10 . - = 200 units per square foot per hour. The mean temperature of the wort was, in the first case, - -1^ - 161 and in the second case 212- = 142. The mean differences of tem- perature, taking that of the air at 62, were 99 and 80, and the heat passed off per square foot per degree of difference of temperature per hour was For the flowing wort, 334^99 = 3-37 units. For the still wort, 200-^-80 = 2.50 COOLING OF HOT WORT BY COLD WATER. 471 COOLING OF HOT WORT BY COLD WATER IN METALLIC REFRIGERATORS. From the instructive discussion of the principles of brewery engineering in Engineering, vol. vi., the following particulars are derived of the perform- ance of tubular refrigerators, in which cold water is passed through thin metallic tubes, which are surrounded by the wort to be cooled. The water and the wort are moved in opposite directions in such a manner that the cold water, on its entrance into the refrigerator, meets the cooled wort just before it leaves the refrigerators, and the warmed water passes away from the refrigerator where the hot wort enters. The following are parti- culars of the performance in five experiments : Table No. 161. RESULTS OF PERFORMANCE OF METALLIC REFRIGERA- TORS IN COOLING HOT WORT WITH COLD WATER. WORT. WATER. Area of cool- ing surface of refrigerator. Specific gravity. Quantity passed through per hour. Initial tempera- ture. Final tempera- ture. Cooled down. Quantity passed through per hour. Initial tempera- ture. Final tempera- ture. Warmed up. Square feet. Barrels. Fahr. Fahr. Fahr. Barrels. Fahr. Fahr. Fahr. i. 881 -i 33-9 212 7 2 I40 61.1 6S I6 9 104 2. 514 I.IO4 36.1 155 59 9 6 75-5 54 IOO 46 3- 5H 1. 088 36.6 IQI 59 132 99-5 54 100 4 6 4. 514 1-035 47-3 193 59 134 90.7 54 IOO 46 5- 5H I.OI8 48.0 I 7 8 59 119 102.0 54 IOO 4 6 Note i. A barrel contains 36 gallons, or 360 Ibs. of water. 2. The temperature of the air in Nos. 2 and 4 was 44 F., and in Nos. 3 and 5, 40. Dealing with the data of this table, the following are the mean tempera- tures and differences of temperature of the wort and the water, with the quantities of heat transmitted per unit of surface, temperature, and time : Mean temperatures Heat transmitted per square foot per degree per hour. No. of experiment. Mean differ- ence of temperature. Of wort. Of water. Measured by reduction of temperature Measured by increase of temperature of wort. of water. Fahr. Fahr. Fahr. units. units. I I 4 2 H7 25 78 IO4 2 107 77 30 81 81 3 125 77 4 8 7i 67 4 126 77 49 9 1 59 5 118.5 77 41-5 96 79 Averages, 83-4 78 472 APPLICATIONS OF HEAT. To show how the quantities of heat in the last two columns are calculated, take the first example. The quantity of wort passed through per hour was 33.9 barrels of, say, 360 Ibs. each, neglecting the extra specific gravity; cooled down through 140 F., the cooling surface was 88 1 square feet, and the mean difference of temperature of the wort and the water was 25 F. Then, 2?.o x -?6o x 140 r , z J - = 78 units of heat, 881 x 25 passed from the wort, per square foot per i F. difference of temperature per hour. Again, 61.1 barrels of water were warmed up through 104 F. Then, 61.1 x 160 x 104 r , ^ *= 104 units of heat, 881 x 25 absorbed by the water, per square foot per i F. per hour, and similarly for the other examples. There is an inconsistency in the excess of heat taken up by the water, as calculated, above that which was passed from the wort in the first example, indicating that there was an error of observation. For the second example, the quantities are equal. The remaining observations show, reversely, that more heat passed from the wort than was taken up by the water. The averages of all the examples show that 83.4 units of heat were passed from the wort, and 78 units were absorbed by the water, per square foot per i F. difference of temperature per hour. It is well to note, as observed in Engineering, that the rate at which the wort parts with its heat increases generally as the specific gravity is less. This acceleration points to the conclusion, that if water be substituted for wort, the rate of transmission would be 100 units per square foot per i F. difference per hour; although, conversely, the cooling water would absorb only 80 units. The difference, 20 units, would be passed off by radiation and conduction. CONDENSATION OF STEAM IN PIPES EXPOSED TO AIR. Tredgold found, by experiment, that steam of an absolute pressure of 17.5 Ibs. per square inch, temperature 221 F., produced one cubic foot of water per hour by condensation in iron pipes exposing 182 square feet of surface in a room at 60 F. The difference of temperature was 161, and the condensation per square foot per hour was .352 Ib. of water; or, per degree of difference of temperature, .0022 Ib. Experiments made in 1859 by M. Burnat, on the efficiency of coating for cast-iron steam pipes, afford valuable data in this connection. 1 The pipes were 4.72 inches in diameter externally, and ^ inch thick; they were arranged in five groups of four pipes each, each group presenting an aggre- gate surface of 58^ square feet. The groups were placed at 40 inches apart, and inclined at an angle of i in 20, in a large unheated hall free from air-currents. The pipes of the first group were covered with straw laid lengthwise to the thickness of 0.6 inch, bound with straw rope laid closely round it. The second group were left bare as they came from the 1 Reported in Proceedings oj 'he Institution of Civil Engineers, vol. xli., 1874-75. CONDENSATION OF STEAM IN PIPES. 473 foundry. In the third group, each pipe was laid in a pottery pipe, with an air-space between the two, and coated with a mixture of loamy earth and chopped straw, covered with tresses of straw. In the fourth group, the pipes were covered with cotton waste to a thickness of an inch, wrapped in cloth bound with string. In the fifth group, the pipes were coated with a com- position of clay and cow's hair to a thickness of 2.36 inches. Finally, trials were made with the second group of pipes by coating them with some old felt which had been treated with caoutchouc; and a second trial of the fifth group, after the composition had received a coat of white paint. The pipes were supplied with steam of from 16^ Ibs. to 30 Ibs. absolute pressure per square inch; and each experiment lasted from 40 to 56 minutes. The results of the experiments are given in the annexed table No. 162. Table No. 162. RESULTS OF EXPERIMENTS ON THE CONDENSATION OF STEAM IN CAST-IRON PIPES. (M. Burnat.) Temperatures. Steam condensed per square foot of external surface of pipes per hour. Absolute pres- sure of steam per square inch. Steam. Air. Differenc'", Straw coat, Bare, Pottery coat, Waste coat, Plaster coat, ISt. ad. 3d. 4 th. 5th. Ibs. Fahr. Fahr. Fahr. lb. lb. lb. lb. lb. 16-5 2l8.0 4 6. 4 i7i.6 139 .496 .170 .217 -254 I6. 5 218.0 33-3 184.2 .152 .485 .166 .205 .262 18.4 223.4 33-7 189.7 .164 -555 .186 .229 .287 18.4 223.4 27.1 196.4 .182 571 .264 .287 344 22. 233-2 4i.5 191.7 .246 -576 .258 .244 .320 22. 233-2 36.5 196.7 .164 .158 .250 22. 233-2 36.1 197.1 .162 557 .178 .260 22.0 233-2 28.9 204.3 .2OI .586 .264 .328 .346 25-7 241.6 43-3 198.4 .244 -645 .301 375 -389 25.7 241.6 36.5 205.1 .274 .28 5 3 6 9 29.4 249.1 43-3 205.8 .252 .721 .270 342 379 29.4 249.1 30.6 218.4 .225 .621 .250 .328 .336 Averages, 22.0 233-1 36.5 196.6 .2OO .581 .229 .286 324 When the plaster coat 'of the fifth group was painted white, an average of 0.307 lb. of steam was condensed per square foot of pipe per hour; and the second group, with the felt coating, condensed 0.313 lb. of steam per square foot per hour. From these data the following constants have been derived, for an absolute pressure of steam of 22 Ibs. per square inch; for the quantity of steam con- densed, and the quantity of heat passed off, per square foot of external surface of pipe per hour for i F. difference of temperature. The quantity of heat transmitted per pound of steam is the difference of the total and sensible heats of the steam, or (1152.5 -f 32) - 233.1 = 951.4 units: 4/4 APPLICATIONS OF HEAT. Steam Condensed CoNomoN o, S F ACB. ?Sf JSST Payoff. Ib. units. Bare, or uncovered pipe 00300 2.812 Coated with straw 00102 0.968 Cased in pottery pipes, with air space 00115 1.108 Coated with cotton-waste, i inch thick 00146 1.384 Coated with old felt 00159 I -5 I 5 Coated with plaster of loamy earth and hair 00165 1.568 The same, painted white 00156 1.486 The most effective coat for the prevention of condensation was the straw coat, and that it had the effect of reducing the loss by condensation to one- third of that which took place with the naked pipe. With the naked pipe, 2.812 units of heat were transmitted per square foot per degree per hour. In experiments by Mr. B. G. Nichol, a wrought-iron pipe 3^ inches in diameter outside, ^ inch thick, and lagged to half an inch thick with felt and spun yarn, condensed steam at 245 F. at the rate of .262 Ib. per square foot per hour, in an external temperature of 60, equivalent to 1.26 units of heat per square foot per i difference of temperature. According to M. Clement's experiments, the quantities of steam given in the second column below, were condensed per square foot of pipe-surface per hour, in a temperature of 7 7 F. Assuming that the steam condensed was of 20 Ibs. absolute pressure, the difference of temperature was 151 F., and the weight of steam condensed per i F. is given in the third column. c. TD . Steam condensed per square foot per hour, total. per i F. Bare cast-iron pipe, horizontal 328 Ib. .002 1 7 Ib. Blackened do. do 308 .00204 Bare copper pipe, do 267 .00177 Blackened do. do 308 .00204 Do. do. upright 359 .00238 Here it appears that the blackened surfaces of iron and of copper were equally active; and that the upright pipe condensed more steam than the same pipe laid horizontally. Mr. Grouvelle found that, in a temperature of 60 F., a square foot of pipe heated by steam condensed 0.328 Ib. of steam per square foot per hour. Assuming that the steam was of 20 Ibs. absolute pressure, the differ- ence of temperatures was 228 60 = 168 F.; and 0.0020 Ib. of steam was condensed per square foot per i F. Summarizing the several results above for bare cast-iron pipes: Difference of temperature. Steam condensed per square foot per hour, total. per i F. Tredgold.... 161 F. 0.352 Ib. .0022 Ib. Burnat 196.6 0.581 .0030 Cle'ment 151 0.328 .00217 Grouvelle 168 0.328 .0020 Average, say, for steam 1 of 20 Ibs. absolute > 169 0.400 .00235 Sa 7 x /42o pressure j CONDENSATION OF VAPOURS IN PIPES. 475 To find the quantity of heat dissipated by the condensation of J /42o lb. of steam: the difference of the sensible and total heats of one pound of steam of 20 Ibs. absolute pressure, is the latent heat, 954 units; and 954^-420 = 2.27 units, the heat dissipated per square foot of surface per i F. difference of temperature per hour. To compare the condensing power of still air with that of still water, and referring to the contents of table No. 160, page 468, in the absence of records of experiments made under exactly the same conditions, it may be inferred that the rate of condensation in thin pipes in air is to that in water below the boiling point, per unit of surface, temperature, and time, as 2.26 units to 312 units, or as i to 138. M. Peclet takes the ratio as i to 200, though so high a ratio is scarcely warranted by the evidence. Condensation of Steam in a Boiler Exposed in Open Air. Messrs. Fox, Head, & Co., Middlesborough, made comparative experiments with a steam-boiler on their premises, in two conditions naked, and covered with non-conducting cement. From an account of the experiments in Engineering, vol. vi., it appears that steam of 50 Ibs. absolute pressure per square inch was maintained, and that the effect of removing the covering was that one cubic foot of water converted into steam was condensed by 50 square feet of exposed boiler-surface per hour. This is equivalent to 1.25 Ibs. of steam per square foot per hour. The weather was fine, and taking the temperature of the open air at 62 F., that of the steam was 298 - 62 = 236 above the atmospheric temperature; and the rate of con- densation per square foot per degree of difference of temperature per hour was, i. 25-^236 = .0053 lb. The latent heat of one pound of the steam was 904 and 904 xi.25 units of heat transmitted per square foot per hour. The quantity of heat transmitted per square foot per degree of difference of temperature per hour was, 1130-^ 236 = 4.79 units. This is more than three times as much as was found to be transmitted in the still air of a room. CONDENSATION OF VAPOURS IN PIPES OR TUBES BY WATER. The condensation of vapours by the application of cold water or air, is in principle the same as the heating of water or air by steam ; and the same proportions for condensing surface, when steam is to be condensed, are applicable in the two cases. The surface-condenser of a steam-engine is a case in point. To educe the constant quantity of heat transmitted per unit of surface, temperature, and time, close analysis of the indicator-diagram would be required, to follow exactly the variations of pressure and temperature of the condensing steam. From the investigations of M. Audenet, 1 of the action of the surface-condensers on board the transport ship Dives, it appears that 500 English units of heat were transmitted per square foot per i F. difference of temperature per hour. These condensers were arranged in three groups 1 Proceedings of the Institution of Civil Engineers, vol. xxxix., 1874-75, p. 399. 476 APPLICATIONS OF HEAT. of tubes, successively traversed by the water. For the condensers, arranged in two groups, on board the Rochambeaii, the constant was only from 220 to 240 English units. A valuable series of experiments on the surface-condensation of steam was made, in 1875, by Mr. B. G. Nichol, at the Ouseburn Engine Works, New- castle. 1 A brass tube, ^ inch in diameter outside, and No. 18 wire-gauge in thickness, was inclosed in an iron pipe 3^ inches in diameter outside, y inch thick, and 5 feet 5 ^ inches long between the ends. The brass tube exposed an external condensing surface of 1.0656 square feet. Steam was admitted into the pipe, and was condensed by cold water passed through the tube. The pipe was lagged with felt and wrapped with white spun yarn to a diameter of 4^ inches. It was tested for the radiation of heat from its external surface, which had an area, including the ends, of 5.48 square feet; the inner tube having been sealed up during the test. It was found that steam of an average temperature of 245 F. was condensed in the pipe at the rate of 1.4375 Ibs. per hour, equivalent to .262 Ib. per square foot of surface. The heat transmitted was (total heat 1154 + 32) - 245 = 941 units per pound of steam condensed; and it was (941 x .262) = 246.5 units per square foot per hour. The external temperature in the workshop was 60 F. ; the difference of internal and external temperatures was 245 60 = 195; thence the radiation per degree of difference of temperature was 246.5 ^-195 = 1. 26 units per square foot. The temperature oif the steam introduced for experiment into the pipe, was about 255 F., for a total pressure of 32.5 Ibs. per square inch, and the initial temperature of the condensing water was 58. Two series, of three experiments each, were made with the pipe in a vertical and in a horizontal position. The following are the principal results of the six experiments : Vertical Position. Horizontal Position. I2 3 456 Steam condensed per square foot of tube per hour, 52.32, 78.18, 84.34, 67.8, 104.6, 121.3 pounds. Condensing water passed through tube per square foot per hour, 659, 2272, 3184, 633, 2505, 3390 pounds. Condensing water per pound of steam condensed, 12.6, 29, 37.7, 9.3, 24, 27.9 pounds. Velocity of water through the tube in feet per minute, 81, 278, 390, 78, 307, 415 feet. Final temperature of condensing water, 140, 9 3 .5, 85, 165, 101, 94 .5 F. Rise of temperature of condensing water, 82, 35-5, 27, 107, 43, 36-5 F. 1 An excellent account of these experiments was published in Engineering, of December 10, 1875, fro which the principal data are derived for this notice. WARMING AND VENTILATION. 477 Vertical Position. Horizontal Position. 123 456 Mean temperature of condensing water, 99, 75-7, 7i-5, m.5, 79-5, 76.2 F. Mean difference of temperature of steam and condensing water, Heat transmitted from steam, reckoned from its temperature, per square foot per hour, 45,960, 68,670, 74,040, 59,650, 91,950, 106,700 units. Heat transmitted from steam, reckoned from its temperature, per square foot per hour, per i F. difference of temperature, 2 95> 3 8 3> 4oi ? 422, 530, 600 units. Heat absorbed by the water per square foot per hour, 54,038, 80,656, 85,968, 67,731, 107,715, 123,735 units. Heat absorbed by the water, per square foot per hour, per i F. difference of temperature, 346, 449, 466, 479, 621, 696 units. The condensing tube acted more efficiently in the horizontal position than in the vertical position : a result the reverse of what was found by M. Clement, condensing in air (p. 474). There is a large excess of heat as carried off by the water, above the heat as calculated from the quantity of steam condensed. In this calculation, it is assumed that the condensed steam left the pipe at the temperature of the steam; but very probably the water was reduced within the pipe more nearly to the temperature at which it was discharged about 200 F. It appears, further, that the efficiency of the condensing surface was very much increased by an increase of velocity of the water through the tube. When other vapours, as those of alcohol, are to be condensed, it may be assumed for purposes of general comparison, that the weight of vapour that may be condensed per unit of surface, temperature, and time, will be inversely as the total heat of the vapour. The total heat of vaporized alcohol, by table No. 125, page 372, is 461.7 units, which is about 4/ 10 ths of that of steam at one atmosphere; and the relative weights of steam and alcoholic vapour, at this pressure, that may be condensed per unit of surface, temperature, and time, are as 461.7 to 1146.1, or as i to 2.5 nearly. WARMING AND VENTILATION. VENTILATION. Mr. Hood finds that in winter from 3^ to 5 cubic feet of air per head per minute are sufficient, under ordinary conditions, for the proper ventila- tion of apartments; and in summer, from 5 to io cubic feet per minute. With these proportions the wholesomeness and purity of the atmosphere are maintained. These proportions agree with those deduced by M. Peclet; according to 4/8 APPLICATIONS OF HEAT. his deductions, 3^ cubic feet of air per head per minute is the minimum that should be provided, in ordinary circumstances. When the ventilation takes place by numerous apertures from below upwards, from 4 to 6^ cubic feet maintains the air of the room sufficiently pure. In peculiar cases, as in hospitals, from 30 to 60 cubic feet of air per bed per minute are admitted. Ventilation is produced by natural draft, or by artificial draft produced by mechanical means. The second method will be considered in a subsequent section. With respect to the first, the ascensional force is measured, as it is with hot water, page 484, by the difference in weight of two columns of air of the same height, the height being measured by the total difference of level between the inlets for warm air and the outlets into the atmosphere. The difference of weight is ascertained from the difference of the temperatures of the ascending warmer air and the external atmosphere, by the aid of table No. 115, page 351; or for inter- mediate temperatures, by the formulas (9), page 350, and (2), page 347. The reasoning that is applied to the question of the circulation of water- columns, page 485, is applicable to that of air-columns. Suffice it for the present to reproduce the following table, No. 163, by Mr. Hood, showing the rate of discharge through a ventilating opening one foot square, for various heights and differences of temperature, calculated by a rule like that for water at page 485; and subjected to a reduction of one-fourth the calculated quantities, to comprise the necessary corrections for the contaminations, chiefly carbonic acid, which go to increase the specific gravity of the current, for frictional resistance, and for the resistance of angular deviations : Table No. 163. AIR DISCHARGED THROUGH A VENTILATOR PER SQUARE FOOT OF OPENING, FOR VARIOUS HEIGHTS AND DIFFERENCES OF TEMPERATURE. Excess of Temperature of the Room above that of the External Air, Height of Ventilator in Fahrenheit degrees. 5 10 15 20 25 30 feet. cubic feet. cubic feet. cubic feet. cubic feet. cubic feet. cubic feet. 10 116 I6 4 200 235 260 284 15 142 2O2 245 284 318 348 20 164 232 285 330 3 68 404 25 184 260 318 3 68 410 450 30 2OI 284 347 403 45 493 35 218 306 376 436 486 53i 40 235 329 403 465 5i8 570 45 248 348 427 493 55 1 605 50 260 367 45 5i8 579 635 The velocity of the draft having been found for any particular case, together with the quantity of air to be supplied per minute, the sectional area of the air passages, inlet and outlet, may be simply calculated from those data. VENTILATION OF MINES BY HEATED AIR. 479 " In all methods of ventilation," says Mr. Hood, "it is advisable to make the aggregate area of the openings that admit the fresh air larger than the aggregate openings for the efflux of the vitiated air. This becomes necessary notwithstanding the increase of volume which takes place in the heated and vitiated air. If the opposite course be adopted, and the eduction-tubes be larger than the induction-tubes, then a counter-current takes place in the hot-air or ventilating tubes, and the cold air descends through them; but by making the induction-tubes numerous, and of a large total area, the velocity of the entering current is reduced, and unpleasant drafts are avoided. It is also expedient to divide the entering current as much as possible; for by so doing, it prevents the dangerous effects of cold draughts, when the entering current is colder than the air of the room; and when it is hotter than the air of the room it prevents the air from rising too rapidly towards the ceiling, and therefore distributes it more equally throughout the apartment. Provided the aggregate openings for the admission of cold air be not less in size than those for the emission of the heated air, the quantity of air which enters a room depends less upon the size or number of the open- ings which admit the fresh air than upon the size of those by which the vitiated air is carried off." In very hot weather and with crowded assemblies, the draft is assisted in theatres and some other large buildings, by heating the air in the upper part of the ventilating tube, which materially accelerates the upward current, and increases the influx of fresh air. The heat of the large gasa- lier in the centre of the house near the ceilings of theatres is thus utilized for ventilation. Another mode of accelerating the draft is to conduct the spent air into the lower part of a vertical shaft, where a furnace is maintained in active combustion, and a very hot column of air is maintained. VENTILATION OF MINES BY HEATED COLUMNS OF AIR. Reserving for a subsequent section the consideration of mechanical venti- lation, the ventilation of a mine by the assistance of a furnace placed at the bottom of the upcast shaft is effected by the heating of the ascending column of air and other gases discharged from the mine, just before entering the shaft, by burning fuel. The furnace should be as low down as possible, so as to afford the longest column of heated air that may be got, since the velocity of draft increases as the square root of the height of the column. The furnace should be so constructed that all the air from the mine should pass freely under and over the grate. The grate may be six feet in length from front to back, but only the first four feet of bar-surface are covered with fuel; and with air-space round the arch, the radiant heat of the furnace is economized. There is a great loss of heat by lateral conduction through the rock and the walls of the shaft. When shafts are dry and bricked throughout, a temperature of 200 F. is the greatest that can be had economically. Even in such shafts the loss of heat laterally often amounts to a fifth; and in shafts which are wet and unwalled, the loss amounts occa- sionally to four-fifths of the whole of the heat communicated. According to Mr. Mackworth, 100 F. should be a sufficiently high temperature for good ventilation; it is relatively economical, and does not do much injury to machinery. With a powerful furnace, and in the absence of obstructions, 480 APPLICATIONS OF HEAT. the greatest velocity of the current is 30 feet per second; but when there is machinery in the shaft, the velocity seldom exceeds 10 feet per second. The first object in ventilation is to produce a slow perceptible motion in the whole of the air of the mine. At a velocity of 30 feet per minute, the flame of a candle is just perceptibly deflected. The air should not, if possible, be made to travel faster; for the resistance of the sides, and leak- ages, increase rapidly as the velocity is increased. The air should be heated uniformly, but slightly; and that it may not be impeded, the furnace-drift should be 5 or 10 fathoms in length, and should rise at an inclination of i in 4. One pound of coal of average composition, when completely burned, is capable of raising, in round numbers, 600,000 cubic feet of air i F. in temperature. At Hetton Colliery, where there are three furnaces, of which one is 9 feet wide, and two are 8 feet wide, one pound of coal raises the temperature of 11,066 cubic feet of air 62 F., equivalent to the raising of 1 1, 066 x 62 = 686,092 cubic feet i in temperature. One of the best examples of furnace-ventilation is, or was, to be found at Morfa Colliery, South Wales. The furnace is 6 feet 2 inches wide, at the base of a shaft 10 feet in diameter, and 60 fathoms deep: it delivers 62,000 cubic feet of air per minute, raised to a temperature of 198 F. by the combustion of 5^ Ibs. of coal. The average temperature of the ascending column at a depth of 25 yards down the shaft was observed to be 1 88, just before coals were charged on the grate; two minutes after charging the temperature was 196; three minutes after charging, 196; and eight minutes after, 191. The "drag" or draft was 3^ Ibs. per square foot, not including the shafts. The useful effect was, therefore, 62,000 x 3. s ., O i > ^ = 6.58 horse-power, 33,000 or 6^ horse-power, as estimated by Mr. Mackworth; from which he infers that i^ horse-power was obtained by one pound of coal per minute. This, reduced to the ordinary form for comparison, is equivalent, for (5.25 Ibs. x 60 = ) 315 Ibs. of coal consumed per hour, to 48 Ibs. of coal per horse-power per hour. At Hetton Colliery it is found, by a similar calculation, that 40 Ibs. of coal was consumed per horse-power per hour, in a shaft 150 fathoms deep. It is stated that a consumption of one pound of coal per minute for furnace-ventilation is sufficient for a mine employing 300 men, in the hottest summer day. In collieries at Wrexham, the waste-steam of the engine is employed to heat the air in the upcast shaft. A cage, consisting of 150 gas-pipes united at top and bottom by hollow cast-iron rings, is placed in the lower part of the shaft, or in the return drift, the exhaust steam is condensed in the cage, and a temperature of 80 F. is thereby maintained. 1 COOLING ACTION OF WINDOW GLASS. Mr. Hood states that one square foot of window glass will cool 1.28 cubic feet of air (say at 62 F.) i F. per minute, or 76.8 cubic feet per 1 The data contained in the above notice of the ventilation of mines are derived from a lecture by Mr. H. Mackworth, reported in the Colliery Guardian, in 1858. HEATING ROOMS BY HOT WATER. 481 hour, per degree of difference of temperatures of the internal and external air. One unit of heat will raise the temperature of 55^ cubic feet of air at 62 F. by i F., from which it follows that heat is transmitted through window-glass from the air of a room to the external air, at the rate of 1L- = 1.40 units, 55-5 per square foot per degree of difference of temperature per hour. The relative cooling influence of wind, or air in motion, on glass, was tested by exposing the bulb of a thermometer, which was raised to a maxi- mum temperature of 120 F., to a current of air at 68, moving at various velocities. The time required to cool the thermometer 20, varied inversely as the square root of the velocity. HEATING ROOMS BY HOT WATER. The effect of hot water in heating air is a function of the respective specific heats. The average specific heat of water between 32 and 212 F. is 1.005 The specific heat of air is ; .2377 Ratio of densities of water and air at 62 & i to 819.4 Ratio of the volumes of water and air raised i F. by equal quantities of heat (i to 819.4^.2377). i to 3465. From this it appears that one cubic foot of water will, by parting with i F. of heat, raise the temperature of 3465 cubic feet of air at 62 by i F.; or one unit of heat will raise 55}^ cubic feet of air at 62 by i F. Mr. Hood estimates, from experiments made by Tredgold, that the water contained in an iron pipe of 4 inches diameter internally and 4^/2 inches externally, loses 0.851 F. of heat per minute when the excess of its temperature is 125 F. above that of the surrounding air, and that one foot in length of the pipe will heat 222 cubic feet of air one degree per minute when the difference of temperature is 125 F. This estimate is too low, as it is based upon too high a value for the specific heat of air, namely, .2767. If the quantity be increased in the inverse ratio of the assumed and the actual specific heat of air, the volume of air raised i by one foot length of four-inch pipe, when the excess of temperature is 125 F., will be 222 x ' 2 7 ? = 258 cubic feet. .2377 Assuming that the rate of cooling of a hot-water pipe is proportional to the excess of temperature, it would follow from the observation above recorded that when the temperature of the pipe is 147 F. above that of the air in the room, it falls i in a minute. Let / = the temperature of the pipes, f = the required temperature of the room, t" = the temperature of the external air, V = the volume of air in cubic feet to be warmed per minute, and /= the length of the pipe in feet Then, according to the preceding data, 31 482 APPLICATIONS OF HEAT. 222 (t-f) f -t" /=.56V using Mr. Hood's divisor 222. But .rt'-t' (O t-t' .. (i a) using the divisor 258. Whence the rule: RULE. To find the length of four-inch pipe required for heating the air in a building. Multiply the volume of air in cubic feet to be warmed per minute, by the difference of temperature in the room and the external temperature, and by 0.56 (Mr. Hood), or by 0.50 (the author), and divide Table No. 164. LENGTH OF FOUR-INCH PIPE TO HEAT 1000 CUBIC FEET OF AIR PER MINUTE. Temperature of the Pipe, 200 F. EXTERNAL TEMPERA- TURE. TEMPERATURE OF THE ROOM. 45 5o 55 60 65- 7o 75 80 85 90 Fahrenheit. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. 10 126 J 5o 174 2OO 229 259 292 328 367 409 12 119 142 166 192 220 25 1 28 3 318 357 399 14 112 135 159 184 212 242 274 309 347 388 16 105 127 J5 1 I 7 6 204 233 265 300 337 378 18 9 8 120 143 168 195 225 2 5 6 290 328 368 20 91 112 135 160 I8 7 216 247 28l 3i8 358 22 83 i5 128 152 179 207 2 3 8 271 308 347 24 7 6 97 120 144 170 199 22 9 262 298 337 26 6 9 90 112 136 l62 190 220 253 288 327 28 61 82 104 128 154 181 211 243 279 3i7 30 54 75 97 1 20 145 173 202 234 269 307 32 47 67 8 9 112 137 164 193 22 5 259 296 34 40 60 81 IO4 129 155 184 215 249 286 36 3 2 52 73 9 6 120 147 175 2O6 239 276 38 2 5 45 66 88 112 138 166 196 230 266 40 18 37 58 80 104 129 157 I8 7 220 255 42 10 30 50 72 95 121 148 I 7 8 210 245 44 3 22 42 64 87 112 i39 168 200 235 46 15 34 56 79 I0 3 130 159 I9O 225 48 7 27 48 70 95 121 i5 181 214 50 19 40 62 86 112 140 171 204 52 ii 32 54 77 I0 3 131 161 194 HEATING ROOMS BY HOT WATER. 483 the product by the difference of the internal temperature and that of the pipes. The quotient is the length of pipe in feet. Mr. Hood. Author. Note. For three-inch pipes, use the multiplier 0.75, or 0.67. For two-inch pipes, do. do. 1. 12, or i. oo. The table No. 164, composed by Mr. Hood, shows the length of four- inch pipe required to heat 1000 cubic feet of air per minute, when the temperature of the pipe is 200 F. Total Quantity of Air to be Warmed per Minute. In habitable rooms the Table No. 165. LENGTH OF FOUR-INCH PIPE REQUIRED TO WARM ANY BUILDING. (fl Building. Length of Pipe per 1000 cubic feet. Temperature maintained. Remarks. Churches and large ) public rooms J feet. 5 Fahrenheit. 55 {In very cold weather. If the air is regularly changed, from 50 to 70 Dwelling-rooms I 2 6c per cent, more pipe is required. Do 14. v j 7O Halls, shops, waiting- ) rooms &c j 10 55 Do. do. Work-rooms, manu- ) factories, &c J 12 6 60 5 to 55 Do. do. Schools and lecture- ) rooms. ... .. J 8 6 to 7 60 55 to 58 Drying-rooms for wet \ linen, &c. When V empty... . I 150 to 180 120 Do., when filled 80 Drying-rooms for cur- \ ing bacon, drying > paper, leather, hides ) Greenhouses and con- ) servatories J 20 35 70 55 In coldest weather. Graperies and stove- ) houses j 45 65 to 70 Do., do. Do. do. Pineries, hot-houses, ) and cucumber pits . J 5o 55 70 to 75 80 Do., do. Note to Table. The lengths of pipe are only suitable for buildings on the usual plan and of ordinary proportions. 4 8 4 APPLICATIONS OF HEAT. total quantity is equal to from 3^ to 5 cubic feet per minute for each person, plus the equivalent of i ^ cubic feet for each square foot of glass. For conservatories, forcing-houses, and like buildings, the quantity of air to be warmed is i ^ cubic feet per square foot of glass per minute. The radiation of heat from frames and sashes made of metal is as great as from glass. The surfaces of these are to be included in the calculation. For wood frames, deduct one-eighth from the gross area of surface. Approximate Rules for the Length of Four-inch Pipe required to Warm any Building. Rules are deduced by Mr. Hood from the results of experi- ence, and they are generally useful in practice. The multipliers are collected in the table No. 165. Proper Diameter of Pipe. The four-inch pipe is of the best size for all horticultural purposes. For most other purposes, smaller pipes may generally be more advantageously employed. Loss by Sinking Heating Pipes in Trenches. When pipes are placed in trenches covered with grating, the loss of heat, as estimated by Mr. Hood, amounts to from 5 to 7 per cent., which passes into the ground. Motive Power of Water in Circulation through Heating Pipes. The ascensional force is measured by the difference in weight of the two columns of water of the same height, ascending and descending from and to the boiler. The difference of weight is ascertained from the difference of the average temperatures of the columns from which the respective densities are deduced by the aid of table No. 109, page 339. The following table showing the difference of weight of two columns of water one foot high at various temperatures, which is calculated by Mr. Hood by Dr. Young's formula, and gives practically the same results as Rankine's formula, table No. 109, page 339. Table No. 166. DIFFERENCE OF WEIGHT OF Two COLUMNS OF WATER, EACH ONE FOOT HIGH, AT VARIOUS TEMPERATURES. Assumed actual Temperatures from 170 to 190 F. i Difference of Temperature of the two Columns. Diameter of Pipe. Difference of weight per square inch. i Inch. 2 Inches. 3 Inches. 4 Inches. Fahrenheit. grains. grains. grains. grains. grains. 2 1.5 6-3 14-3 25.4 2.028 4 3.1 12.7 28.8 5 1 - 1 4.068 6 4-7 I9.I 43-3 76.7 6.108 8 6-4 25.6 57-9 102.5 8.160 10 8.0 32.0 72.3 128.1 10.200 12 9.6 38.5 87.0 154.1 12.264 14 II. 2 101.7 180.0 14.328 1.6 12.8 51.4 116.3 205.9 16.392 18 14.4 57-9 131.0 231.9 18.456 20 16.1 64.5 145-7 258.0 20.532 HEATING ROOMS BY HOT WATER. 485 The velocity of circulation is that of a falling body due to the difference of height of two columns of water of equal weights or pressures on the base, and it varies as the square root of the difference of height. The velocity may be found by the aid of table No. 85, page 280. The difference of height is proportional to the difference of volumes, table No. 109; and if the mean height be increased in the same proportion, the increase will be the height from which the velocity is to be calculated. For example, let the mean height be 10 feet, and the difference of average temperatures of the two columns 10 F., say between 170 and 180. The respective volumes are as 1.0269 an ^ I -3 I > io feet x 1.0269 - 10.04 feet. Then 10.04 - 10 = .04 f ot > the difference of height; and the velocity due to this height is 1.61 feet per second, or 96.6 feet per minute. If the height be 20 feet, the difference is .08 foot, for which the velocity due is 136.20 feet per minute. In practice, of course, the velocities due are not attained, nor, at least in the more complex forms, nearly attained. The actual velocities are, in some cases, not more than a half or even a ninth of the velocities due to gravity. Quantity of Coal Required to Heat the Pipes. Mr, Hood gives the fol- lowing table, No. 167, showing the quantities of coal consumed in heating 100 feet of pipe for various differences of temperatures. These quantities are based on the results of experiments by Rumford and others in heating water with coal as fuel, and are no doubt approximately correct. Table No. 167. COAL CONSUMED PER HOUR TO HEAT 100 FEET OF PIPE. For given differences of temperature of the pipe and the air. Diameter of Pipe. Difference of Temperature of the Pipe and the Air in the Room in Fahrenheit Degrees. ISO H5 140 135 I 3 125 120 "5 no 105 100 95 9 85 80 inches. 4 3 2 I Ibs. 4-7 3-5 2-3 i.i Ibs. 4.5 3-4 2.2 I.I Ibs. 4-4 3-3 2.2 I.I Ibs. 4.2 3-i 2.1 I.O Ibs. 4.1 3-o 2.0 I.O Ibs. 3-9 2.9 1-9 0.9 Ibs. 3-7 2.8 1.8 0.9 Ibs. 3-6 2.7 0.9 Ibs. 3-4 2-5 oi Ibs. 3-2 2.4 1.6 0.8 Ibs. 3-i 2.3 i-5 jo.; Ibs. 2.9 2.2 1.4 0.7 Ibs. 2.8 2.1 1.4 0.7 Ibs. 2.6 2.0 J -3 0.6 Ibs. 2.5 1.2 0.6 Boiler-Power. One square foot of boiler-surface exposed to the direct action of the fire, or three square feet of flue-surface, will suffice, with good coal, for heating, in round numbers, 50 feet of pipe. Mr. Hood fixes the proportion at 40 feet of four-inch pipe for all purposes. The usual rate of combustion of coal is about 10 Ibs. or n Ibs. of coal per square foot of fire-grate, and at this rate, 20 square inches of grate suffice for heating 40 feet of four-inch pipe. 486 APPLICATIONS OF HEAT. Four square feet of boiler-surface exposed to the direct action of a good fire are capable of evaporating one cubic foot of water per hour. The best form of boiler for heating purposes is shown in Fig. 126 annexed. It is Fig. 126. Boiler for heating purposes. Fig. 127. Boiler for heating purposes. generally made of wrought-iron plates rivetted together. Another good form is shown in Fig. 127. French Practice. M. Claudel states that to warm a factory 13 metres wide by 3.25 metres high (43 feet by 10.5 feet), a single line of hot-water pipe 6^ inches in diameter along the room appears to be sufficient, the temperature in the pipe being from 170 to 180 F. He adds that, in prac- tice, the water being at 180 F., and the air at 60 F., making a difference of 120 F., it is convenient to reckon from 1.5 to 1.75 square feet of water- heated surface as equivalent to one square foot of steam-heated surface, and to allow from 8 to 9 square feet of hot- water pipe-surface per 1000 cubic feet of room. M. Grouvelle affirms that four square feet of cast-iron pipe-surface, whether heated by steam or by water at 80 or 90 C., or 176 to 194 F., will warm 1000 cubic feet of workshop, maintaining a temperature of 60 F. Steam is condensed at the rate of 0.328 Ib. per square foot per hour. Perkins 1 System. This system consists of the continuous circulation of water through endless wrought-iron tubes of ^-inch bore and i inch out- side diameter, proved under a pressure of 200 atmospheres. The tempera- ture of the water at the upper part of the circuit, varies from 300 to 400 F., corresponding to pressures of from 4^ to 15 atmospheres. The tubes become red-hot in the furnace. The length of tube in the furnace is a sixth of the total length of the circuit. Twenty feet of length are allowed for heating 1000 cubic feet of capacity. Taking the mean diameter ^ inch, this gives four square feet of surface per 1000 cubic feet. Though the heater is apparently water-tight, the larger sizes are subject to a loss of about a pint of water in eight or ten days, which is restored by means of a force-pump. M. Gaudillot, in France, manufactures heaters on this system with tubes of from 1.20 to 1.60 inches in external diameter. They support a pressure of 40 atmospheres very well. HEATING ROOMS BY STEAM. To find the length of pipe required for heating a room by steam, the temperature of the steam, which varies with the pressure, and may be found in table No. 128, page 387, is to be employed for the value of t in the formulas ( i ) and ( i a\ page 482. The length of pipe required for heating by steam, is of course less than that required with water, as the temperature HEATING ROOMS BY STEAM. 487 is much higher. Taking a standard absolute pressure of steam of 20 Ibs. per square inch, the temperature is 228; and if the room is to be heated to 60, the difference is 168, and the formula ( i a), page 482, becomes ' f" (2) - 336 RULE. To find the length of fotir-inch pipe required for heating the air in a building by steam of 20 Ibs. absolute pressure per square inch. Multiply the volume of air in cubic feet to be warmed per minute, by the difference of the external and internal temperatures, and divide the product by 336. The quotient is the length of pipe in feet. Note. For three-inch pipes use the divisor ............ 252 For two-inch pipes ............ 168 For one-inch pipes ............ 84 The boiler for a steam-heating apparatus should be capable of evapor- ating as much water per hour as the pipes would condense in the same time. Mr. Hood recommends that six square feet of direct surface of boiler should be provided to evaporate a cubic foot per hour. Now, adopt- ing the mean weight of steam of 20 Ibs. absolute pressure condensed per square foot of pipe per degree of difference of temperature per hour, namely .00235 Mb., the quantity of pipe-surface that would form a cubic foot of condensed water per hour, taking the weight of this volume of water at 62.4 Ibs., would be, per i difference of temperature, 62.4 -f- .00235 = 26,550 square feet. For a difference of 1 6 8 the required surface would be 26,550 -T- 168= 158 square feet, say 160 square feet. Four square feet of direct boiler-surface, or its equivalent of flue-surface, should, therefore, be provided for every 160 square feet of steam-pipe con- taining steam of 20 Ibs. absolute pressure per square inch, and maintaining a temperature of 60 F. in a room. The following lengths of pipe are required to present 160 square feet of surface : Length for Length for i square foot. 160 square feet. 4-inch pipe, ^ inch thick, ............. 10.2 inches, 136 feet. 3 ............. 13- J 73 2 >, ............. 18.3 244 * rt ............. 36.6 4 88 French practice. According to M. Grouvelle, one square metre of pipe- surface, heated by steam, sufficed to heat and maintain at 15 C., or say 60 F., a room with ordinary proportions of walls and windows, such as a library or an office, of from 66 to 70 cubic metres of capacity, or a work- shop of from 90 to 100 cubic metres. If the workshop is to be maintained at a high temperature, a square metre of surface is allowed for 70 cubic metres. The Exchange at Paris is sufficiently heated by one square metre 488 APPLICATIONS OF HEAT. for 67 cubic metres. The allowance of one square metre for 70 cubic metres is equivalent to 4.35 square feet per 1000 cubic feet of capacity; or to 5.11 lineal feet of four-inch pipe per 1000 feet. For heating workshops, 8 metres wide by 3 metres high, having 260 square feet of section, with a window-surface one-sixth of the total surface, engineers in France allow an iron pipe of 16 inches in circumference, or 5 inches in diameter, passing once through the shop, presenting 1.33 square feet of surface per foot run, or 5.2 square feet per 1000 cubic feet, the same as has just been calculated. According to the observations of M. Peclet on steam-heating apparatus, particularly in a large factory, for a maximum difference of 36 F. between the interior and exterior temperatures, it was necessary to reckon on a delivery of 26 units of heat per hour per square foot of wall of 13 or 14 inches in thickness, and 30 units of heat per square foot of glass. HEATING BY ORDINARY OPEN FIRES AND CHIMNEYS. M. Claudel says that the quantity of heat radiated into an apartment from a fireplace is about one-fourth of the total heat radiated by the combustible. The heat radiated into an apartment from wood when burned amounts to only 6 or 7 per cent, of the total heat of combustion. For coal and for coke, the heat thus utilized amounts to about 13 per cent. In burning wood, ordinary chimneys draw about 1600 cubic feet of air per pound of fuel; and better constructed chimneys about 1000 cubic feet. A sectional area of from 50 to 60 square inches is sufficient for the chim- neys of ordinary apartments. For apartments designed to hold a great number of persons, a section of 400 square inches, say 32 by 13 inches, is usually employed. From experiment it appears that the proportions of fuel required to heat an apartment are as 100 for ordinary fire-places, 63 for metal stoves, and from 13 to 1 6 for apparatus similar to stoves, with open fires. HEATING BY HOT AIR AND STOVES. Sylvester's cockle-stove is constructed of wrought-iron, % inch thick, formed with an arch and two sides, closed at the ends, through one of which the furnace-mouth is made. The furnace is formed of fire-brick within the case, and the products of combustion are drawn off by flues below the furnace. The case is inclosed in fire-brick, with about 5 inches clear space for the circulation of the air to be heated. The air is introduced through the brickwork at the lower part of the sides, through numerous iron tubes, which are laid to within an inch clear of the sides of the case, and cause the fresh air to impinge upon the heated surface. The air thus Fig. 12 8.-syivesters cockle-stove. brou g ht in P asses over the entire surface of the cockle into the upper part of the envelope, whence it is led away through any required number of pipes to the different rooms to be warmed. The ends of these exit pipes are placed within an inch HEATING BY OPEN FIRES, HOT AIR, AND STOVES. 489 above the top of the case. One of these cockle-stoves is illustrated by Fig. 128; the wrought-iron case is 5 feet square by 5 feet high. There are from 150 to 200 air pipes 2 inches in diameter, or 2 inches square, at the sides. The grate contains about 5 square feet of area, and the flues at the bottom are 9 by 6 inches. From the results of Mr. Sylvester's experiments with a smaller cockle- stove, it was found that with a consumption of 5 Ibs. of coal per hour, and a heating surface of 17 square feet, the temperature of 344,600 cubic feet of air was raised 56 F. in twelve hours, with 60 Ibs. of coal; being equivalent to the heating of 95,000 cubic feet of air i F. per square foot of surface per hour; or to the heating of 321,626 cubic feet of air i F. by one pound of coal. It thus appears that each square foot of cockle-surface is equal to 7 square feet of hot-water pipe. French Practice. From results obtained by M. Peclet, it is ascertained that when the flue-pipes of stoves, conveying hot products of combustion, heat directly the air of a room, the quantities of heat passed off per square foot per hour for i F. difference of temperature, vary according to the material of the pipe as follows : Cast-iron, 3.65 units of heat. Sheet-iron, 1.45 ,, Terra-cotta, 0.4 inch thick, 1.42 It may be noted that the great difference here observable between cast and wrought iron in passing heat from a flue to the outer air, does not exist when the pipe is occupied by steam or hot water. If an excess of temper- ature equal to 800 F. be assumed, as between the inside and outside of the pipe, the quantities of heat given off per square foot per hour would be, For cast-iron, 3.65 x 450= 1642 units of heat. For sheet-iron, 1.45x450= 652 For terra-cotta, 1.42x450= 639 Yet, in practice, the same surface is allowed for cast and for sheet iron; at the rate of one square foot for 328 cubic feet of space to be heated. The diameters of stove-pipes vary from 4 to 8 inches. The air thus heated receives a degree of humidity from a vase full of water placed on the stove. The water so dissipated amounts to a little more than 2^ pints per day for a room of from 2500 to 3000 cubic feet of capacity. House-stoves placed in the Room to be Warmed. M. Claudel says that inside stoves are employed in schools and hospital-wards; they consist of an upright column, square or cylindrical, from 5 to 7 feet high, inclos- ing the furnace; surmounted by a pipe which rises vertically, and is then carried nearly horizontally through the apartment to a chimney. The column is inclosed in an outer casing of sheet iron or brickwork, with an interspace into which the external air is admitted, and from the upper part of which the air passes into the room. The temperature of the furnace does not exceed from 1100 to 1300 F. In practice, it is found conven- ient to assume that the products of combustion leave the stove at a temper- ature of about 950 F., or 500 C., that they are completely cooled in their course, that the temperature of the room is 60 F., and that the quantity of 4QO APPLICATIONS OF HEAT. heat emitted is the same as if the pipe had an average temperature of 480 F., or 250 C. A heating surface of from 20 to 30 square feet is allowed per pound of coal burned per hour, not reckoning the surface of the stove. Large grates are preferred, with slow combustion. House-stoves placed Outside the Room to be Heated. The useful effect of these stoves may be taken at from 60 to 70 per cent, of the heating power of the fuel. The surface of grate should be 15 square inches per pound of coal consumed per hour. The heating surface is two square metres per kilogramme of coal, or per two kilogrammes of wood : equivalent to 10 square feet per pound of coal per hour. From 2^ to 3}^ pints of water are consumed per 1000 cubic metres; or i pint for from 1000 to 1400 cubic feet of space. The spent air of the room is passed off into a chimney. HEATING OF WATER BY STEAM IN DIRECT CONTACT. The heating of water by steam, when the elements are brought into direct contact, is practically instantaneous. The author made experiments on this subject by admitting steam at 90 Ibs. effective pressure from a locomotive-boiler into a body of cold water contained in a cylindrical reservoir, 3 feet 6 inches in diameter, and 15 feet long, made of ^-inch iron plate, having a total capacity of 144 cubic feet. The steam was con- veyed from the boiler to the reservoir by a i-inch pipe, from which it was freely discharged into a 2-inch iron pipe, open at the extremity, laid in the water along the bottom of the reservoir. The reservoir lay horizontally, without any covering, in a factory. Fifty-five and a half cubic feet, or 3464 Ibs. of cold water at 60, were delivered into the reservoir, and the water was heated by the steam blown into it to a pressure of 85 Ibs. effective per square inch, in two hours, with a temperature of 328 F., or through 328-60 268 F. The quantity of heat communicated in two hours was, therefore, 3464 x 268 = 928,352 units, at the rate of 464,176 units per hour. Taking the initial temperature of the steam of the boiler, 331 F., and the mean temperature of the heated water * - - = 194; the mean difference 2 of temperature was 331 - 194= 137, and the quantity of heat communi- cated per i F. of difference per hour was, 464176 -3388 units of heat. To communicate the whole of this quantity X)f heat through the surface of a pipe at the rate of 300 units per foot per i F. per hour, there would have been required 3388 -f- 300 = 1 1.3 square feet of surface. It is probable, as a matter of fact, that, though the 2-inch pipe was open to the water at the end, the most of the steam was condensed within the pipe before it could reach the end. The surface of the pipe had about 8 square feet of area. There was, of course, a loss of heat by radiation from the surface of the reservoir; but it is not material to the purpose of this notice. EVAPORATION IN OPEN AIR. 491 EVAPORATION (SPONTANEOUS) IN OPEN AIR. So-called " spontaneous " evaporation from water exposed to air proceeds at all temperatures, when the conditions are suitable. The total rate of evaporation is in proportion to the extent of the surface exposed to the air. An increase of the temperature of the liquid is attended by an increase of the rate of evaporation, though not in direct proportion. The rate of evapora- tion is greater when the air is in motion over the surface of the water than when it is at rest. The rate of evaporation is also greater in proportion as the air is dryer, or the less the moisture previously existing in the air; and on the contrary, when the air is saturated with moisture, the evaporation is reduced to nothing. When the atmosphere is perfectly dry, the rapidity of evaporation is proportional to the pressure of the vapour due to the temperature of the water, for which reference may be made to tables No. 127, page 386, and No. 130, page 396. This law was discovered by Dr. Dalton, who gives the following illustration : At the temperatures 212, 180, 164, 152, 144, 138, The pressures are 30, 15, 10, 7^, 6, 5 inches of mercury ; And the weights of water evaporated at these temperatures are proportional to 30, 15, 10, 7^, 6, 5. But the atmosphere impedes the diffusion, and, consequently, the genera- tion of vapour; although, ultimately, the full charge of saturated vapour due to the temperature is absorbed by it. When vapour is present in the air, which it usually is to a greater or less degree, the pressure of this vapour is to be deducted from that of the vapour due to the temperature of the water; and the residual force is the active "evaporating force." Dr. Dalton found that with the same evaporating force, thus determined, the same rapidity of evaporation is maintained, whatever be the tempera- ture of the air. But when a current of air blows over the surface of the water, the rapidity of evaporation is greater than when the air is still, because the air in motion sweeps away the vapour as it rises, and a continuous supply of compara- tively dry air is secured. With the same evaporating force, a strong wind will double the production of vapour, compared with the quantity produced in a still atmosphere. Dr. Dalton's experiments were made with an evaporating surface of 6 inches in diameter, in still air and in wind, and he gives a table of the rates of evaporation in grains per minute, for temperatures up to 85 F., on the assumption that the air is perfectly dry. 1 The following table, No. 168, is calculated to show the rate of evaporation, in pounds per square foot per hour, extended up to 212 F., and for three states of the air: when still, when there is a gentle wind, and when there is a brisk wind. The pressures are given in inches of mercury, and are those adopted by Dr. Dalton, which, for the purpose of the table, do not materially vary from those given in table No. 127. 1 Memoirs of the Literary and Philosophical Society of Manchester, vol. v. p. 579. 492 APPLICATIONS OF HEAT. Table No. 168. "SPONTANEOUS" EVAPORATION OF WATER IN STILL AIR AND IN WIND, ASSUMING THE AIR TO BE PERFECTLY DRY, FOR TEMPERATURES FROM 32 TO 212 F. (Founded on Dr. Dalton's tables.) Tem- perature of the water. Pressure of the vapour. Water evaporated per square foot of surface per hour. Tem- perature of the water. Pressure of the vapour. Water evaporated per square foot of surface per hour. Air still. Gentle wind. Brisk wind. Air still. Gentle wind. Brisk wind. Fahr. inches of Ibs. Ibs. Ibs. Fahr. inches of Ibs. Ibs. Ibs. mercury. mercury. 32 .200 .0349 .0448 55 125 3-79 .6619 .8494 043 35 .221 .0386 0495 .0608 I 3 4-34 .7580 .9727 .194 40 .263 0459 .0589 .0723 !35 5.00 .8730 1. 121 .376 45 .316 55 2 .0708 .0869 140 5-74 1.003 1.286 579 5o 375 655 .0841 .1032 i45 6-53 I.I40 1.463 .796 55 443 .0774 0993 .1218 150 7.42 1.296 1.663 2.043 60 .5 2 4 .0917 .1175 .1441 J 55 8.40 1.467 1.882 2.310 62 .560 .0979 1255 .1540 1 60 9.46 1.652 2.120 2.602 65 .616 .1076 .1381 .1694 165 10.68 1.865 2-394 2.938 70 .721 1257 .l6l6 .1983 170 12.13 2.118 2.719 3-336 75 .851 .1486 .1907 .2341 i75 13.62 2.378 3-53 3.746 80 1. 000 .1746 .2241 2751 180 15-15 2.646 3-395 4.167 35 1.17 .2043 .2622 .3218 185 17.00 2.969 3.8io 4.676 90 1.36 2 375 .3048 3745 190 19.00 3.318 4.258 5.226 95 1.58 .2760 3541 .4346 !95 21.22 3.706 4.758 5.837 100 1.86 . 3 2 4 8 .4169 .5116 200 23.64 4.128 5.298 6.502 105 2.18 .3807 .4886 5996 205 26.13 4-5 6 3 5-856 7.187 no 2-53 .4418 .5670 6959 210 28.84 5.034 6.464 7-933 IJ 5 2.92 .5IOO .6544 .8030 212 30.00 5-239 6.724 8.252 I2O 3-33 .5815 7463 .9160 It appears from the table that the rates of evaporation, for each of the three conditions of the air, when perfectly dry, are in simple proportion to the pressure of the steam ; and, as affected by the stillness or the motion of the air, they are for still air, as i, a gentle wind, 1.28, a brisk wind, It is to be understood that the temperature 212 is a limiting tempera- ture, which cannot be actually reached without displacing the air entirely. It is also to be remarked that, though Dr. Dalton lays down the proposi- tion that the rapidity of the evaporation is the same, whatever may be the temperature of the air, yet it is clear that water is evaporated more rapidly when a warm current blows over it than when it is traversed by a cold current. Such increase of evaporation is probably the result of the reflex action of heat imparted by the air to the superficial water, the " evaporative DESICCATION. 493 force " of which is increased by the rise of temperature due to the heat abstracted from the air. The cooling of air by passing it over or through water is a well-known expedient. In India, the air of apartments is cooled by passing it, as it enters, through and over the " tatta," a bamboo frame or trellis, over which water is suffered to trickle. From what has been stated, with respect to "mixtures of gases and vapours," page 392, it appears that the condition of " saturation," attributed to the mixture of vapour and air, properly belongs to the vapour itself, as vapour, when it has arrived at its maximum density and pressure for the temperature of the air. Use of the Table No. 168. Dr. Dalton gives the solution of the problems based upon the original tables, of which the first is here rendered into a rule in relation with the table No. 168. RULE. To find the quantity of water exposed to air that would be evaporated per square foot of surface per hour, at a given temperature of air, with a given dew-point. Subtract the tabulated weight of water corresponding to the dew-point from the weight corresponding to the temperature of the air; the remainder is the weight of water that would be evaporated per square foot of surface per hour. The weights of water are to be selected from the 3d, 4th, or 5th columns, according to the state of the wind. To find the dew-point, Dr. Dalton used a very thin glass vessel, into which he poured cold water, of which he noted the temperature. If the vapour in the atmosphere was instantly condensed on the glass, he changed the water for warmer water, and so proceeded until he ascertained the proper temperature the dew-point when he could just perceive a slight dew deposited on the glass. The dew-point may be found with much greater precision by means of hygrometers, described at page 393. Dr. Pole has constructed an empirical formula which roughly represents the results of Dr. Dalton's experiments. Let T = the temperature of the atmosphere in degrees Fahr., t= the dew-point, V = the velocity of the wind in miles per hour, E = the rate of evaporation in inches per day from a water surface, and A = a numerical coefficient. Then, m A(ioo- V) The value of A = 80 for high or summer temperatures, and A 100 for low or winter temperatures. Dr. Pole remarks that Dalton's tables do not provide for cases where the temperature of the water differs materially from that of the air; and that, probably, in such cases, T should be made to represent the temperature of the water-surface, and not that of the air. 1 DESICCATION. The drying of wet or moist materials, by means of currents of air, is based on the principles already announced which regulate the evaporation of water from the surface. If a current of air be saturated with moisture 1 Minutes of Proceedings of the Institution of Civil Engineers, vol. xxxix. page 36. 494 APPLICATIONS OF HEAT. or vapour, its efficiency for drying out moisture from bodies with which it comes in contact, is exactly nothing. To act as a dryer, in other words, to assist in evaporating and carrying off moisture, it must be either perfectly dry, or, at the least, sub-saturated ; and inasmuch as its capacity, in the conven- tional language already explained, for absorbing moisture in the state of vapour, of course increases with its temperature, it is obvious that the higher the temperature of the air, the greater is its efficiency. If, then, the air-current be surcharged with heat, it stimulates evaporation in two forms by imparting a portion of its heat to the wet or moist surface, which is utilized in the evaporation of the moisture, and by tolerating the presence of a greater quantity of moisture in mixture with it, which is carried away as it rises from the surface by the current. The drying, or vaporization of moisture, by such means, involves, of course, a lowering of the temperature of the air, or the moist body, or both; and the problem arises: What is the initial temperature of dry air required? The first problem for solution is twofold: Given the final temper- ature at which the saturated mixture is to be discharged, what is the quantity of dry air required for a given weight of vapour in saturated mix- ture? and to what initial temperature is the air required to be raised in order to supply heat for the evaporation of the given weight of steam? The answer is to be found in the sub-section on the " Properties of Satur- ated Mixtures of Air and Aqueous Vapour," with table No. 130, page 394. But, in ordinary practice, the artificially heated air-current does not arrive at the condition of saturation before it is discharged; and a large surplus of air is therefore to be provided, the proportional amount of which varies with the circumstances under which the current is applied. The standard of perfect efficiency is presented in the table No. 130. M. Peclet notices a process employed by M. Montgolfier for drying the skins of grapes after having been pressed, by means of a forced current of air. It was found that, in autumn, 5340 cubic feet of air, moving at a velocity of about 1 6 feet per second, were required for the evaporation of one pound of water from the pressed grapes. Let the initial temperature of the air be assumed at 64 R, then, by the table No. 130, a volume of dry air equal to 2526 cubic feet would have sufficed to evaporate one pound of water; but if, as is probable, the air had already been loaded with half the quantity of moisture it could carry, then at least double the tabular quantity would have been necessary, or 2526 x 2 = 5 05 2 cubic feet, which is nearly equal to the quantity actually employed. In the design of a drying-chamber, it is of the first importance that the air-current should be admitted at the highest point of the chamber and discharged at the level of the floor. The reverse process, of admitting it at or near the floor, and discharging it at the upper part, is vicious practice. In the latter case the circulation is imperfect, for the hot air seeks the most direct route to the points of egress; in the former case the hot air is uniformly distributed, and if the points of discharge are properly placed, the descending current is applied equally over the area of the chamber. In a drying-chamber, noticed by M. Peclet, for drying vermicelli, at Saint Ouen, having a capacity of upwards of 6000 cubic feet, and heated by air of from 86 to 104 F., the following were the results of heating by ascend- ing and by descending currents : DESICCATION. 495 DISCHARGE OF CURRENT. VERMICELLI PRODUCED. First Quality. Second Quality. Fermented. Above (mean of 5 trials) 540 Ibs 400 Ibs 4.5 Ibs. Below 3,200 143 86.0 These results prove decisively the superiority of the descending current. The consumption of fuel with the descending current was also much the less. Dry ing-house for Calico (Peclet). In a drying-house used by M. Rene Duvoir, the pieces of calico were suspended from bars ranged horizontally across the upper part of the house. Air heated to 250 F. was admitted through a number of openings from a brick flue at the floor, regulated by dampers, from which it rose to the upper part, and thence descended to the floor, where it was discharged. The external temperature was 77 F., and the temperature of the discharged current in the chimney was 100 F. In six hours, 150 pieces of calico, holding 2490 Ibs. of water, were dried, with a consumption of 706 Ibs. of coal, corresponding to an evaporation of 3.52 Ibs. of water per pound of coal. The quantity of air heated to 250 F., for this duty, was 1,943,000 cubic feet, the weight of which, at the rate of 13.52 cubic feet to the pound (by Rule 9, page 350), was 143,713 Ibs. Then 143713 x .2377 = 34160 units of heat for i F. eleva- tion of temperature, and for 250 - 77 = 173, the total elevation of temper- ature, the total heat consumed was 34160 x 173 = 5,909,256 units, being at the rate of 5'99> 2 5 = 8370 units per pound of coal. 706 The water evaporated per pound of coal was 3.52 Ibs., for which 1067 units of heat, reckoned from 77 F., were absorbed per pound of water; and for 3.52 Ibs., 1067 x 3.52 = 3756 units of heat, was the quantity of heat utilized for evaporation per pound of coal, being 45 per cent, of the total heat communicated to the air. The temperature at which the air was discharged being 100, it was 23 above the external temperature, or the loss by the excess was ^- x 100 = 13 per cent, of the whole heat communicated to the air. The distribution of the heat communicated to the air was, therefore, approximately as follows : In evaporating moisture, 45 per cent. Carried off by the air, 13 Loss by radiation and conduction, 42 100 It is easy to show that the air when discharged was not nearly saturated. At 1 00 F., the temperature of discharge, the proportions of moisture and air in one pound of a saturated mixture, by table No. 130, are as .283 to 6.641, or as i to 23.5. In 143,713 Ibs. of air, therefore, when in a state of saturation, there would have been 1 43, 713 -=-23. 5 = 6, 115 pounds of moisture. 496 APPLICATIONS OF HEAT. But there was only 2490 Ibs. of moisture in the air, or about two-fifths of the proportion for saturation. The single good feature in this drying-house, is the extraction of the spent current at the level of the floor. No provision was made to effect the distribution of the heat uniformly through the room; and there can be no doubt that the condition of sub-saturation of the air was, for the most part, the result of the absence of such provision. Drying Linen. The maximum evaporative performance of coal in drying linen, does not exceed an evaporation of 3 Ibs. of water per pound of fuel; and it is sometimes as low as 1.36 Ibs. Drying Various Stuffs. According to M. Rouget de Lisle, in one pound of wet cloth, after having been wrung or pressed, or passed through the hydro-extractor, there remained the respective quantities of water as follows : Water Left in One Pound of Flannel. Calico. Silk. Linen. When twisted, 2.00 Ibs. i.o Ib. .95 Ib. .75 Ib. When pressed, i.oo .6 .5 .40 When passed through the ) , hydro-extractor, j In these instances, the centrifugal machine was 26 inches in diameter, and made from 500 to 600 turns per minute. M. Penot made experiments on drying-houses at Mulhouse, and found that one pound of coal evaporated from 1.02 to 2.86 Ibs. of water: the latter under favourable circumstances. According to M. Royer, in a drying-house 31^ feet long, 26 feet wide, and 62% feet high, the heating surface of the stove amounted to 758 square feet, with a consumption of 55 Ibs. of coal per hour. There were during these trials, lasting fifteen days each, evaporated successively 2.37, 2.53, and 2.18 Ibs. of water per pound of coal. Mr. J. R. Napier, in drying stuffs by air heated to 240 F., with a descending draft, evaporated 3 Ibs. of water per pound of coal. Drying Stuffs by Contact with Heated Metallic Surfaces. M. Clement applied a piece of calico, weighing 2^ Ibs., holding an equal weight of water, to a plate of copper of the same extent, heated by steam at 212 F. ; and it was dried in one minute. The evaporation was effected at the rate of 1.42 Ibs. of water per square foot per hour. When stuffs are dried by passing them over cast-iron cylinders, heated by steam internally, it appears from experiments made by M. Royer, that in drying calico which held its weight of water, 74 Ibs. of water were evaporated by the condensation of 102 Ibs. of steam. In other experiments made with a machine of six cylinders, the efficiency in drying was only two-thirds of that attained in the first-described experiment. The experi- ments were made in winter in a place which was imperfectly closed. Drying Grain. It is reported in the Engineer, that Messrs. Crighton & Co., Abo, dried 450 Ibs. of grain, extracting 15 per cent, of its weight, or 67^ Ibs. of water, by the consumption of 18 Ibs. of birchwood, being at the rate of 3.75 Ibs. of water per pound of wood. Drying Wood. In the forges of Lippitzbach, Carinthia, according to M. Leplay, wood is piled and dried in close chambers by burning a part of HEATING OF SOLIDS. 497 the wood, averaging a fourth of the total quantity. The furnaces are below the floor, between which and the furnaces a space is provided for the circulation of the products of combustion under the floor. Air in consider- able quantity is admitted to and mixed with the products of combustion to moderate their temperature to 350 F.; when the current passes into the upper chamber amongst the wood to be dried. On this system, there is considerable loss of heat by radiation and by the excessive dilution of the products of combustion with air. At the Neuberg factory, the products of combustion circulate in a species of stove constructed of thin masonry, and pass thence through cast-iron pipes by which the air is heated for drying the wood. On this system the wood consumed does not exceed an eighth of the total quantity. The limit of temperature at which wood should be dried ought not to exceed 340 or 350 F. M. Leplay states that the wood to be dried contains 40 per cent, of water; whence it appears that one pound of the fresh wood evaporates 1.20 Ibs. of water in the first of the above-described processes; and in the second process, 2.80 Ibs. of water. In a system of drying-furnace recently adopted in France for wood, peat, &c., a chamber 62 feet long and 14 feet wide is employed. The wood, in billets, is loaded into waggons, having a capacity of about 100 cubic feet each, on rails. Each waggon-load successively is introduced at one end and withdrawn at the other, whilst the mixture of hot gases and air is introduced at the other end, and passes to the end at which the waggons are introduced. A temperature of 270 F. is maintained at the middle of the chamber. The maximum temperature is 320 F. The wood remains sixty-four hours in the chamber, and the usual quantity of moisture it contains, from 20 to 25 per cent, is evaporated by the combustion of i^ cords of wood for every 16 cords to be dried; that is, i^ Ibs. are burned to dry 16 Ibs., evaporating 4 Ibs. of water; being at the rate of 2.66 Ibs. of water per pound of fresh wood. HEATING OF SOLIDS. Cupola Furnace. M. Peclet estimates that, in melting pig iron in an ordinary cupola, by the combustion of 30 per cent, of its weight of coke, 14 per cent, only of the heat of combustion is actually utilized. This estimate is based on the result of an experiment by Clement, showing that to heat and melt i pound of pig iron, 504 English units of heat are necessary. Plaster Ovens. He also states that to dry plaster, the heat of combustion of 7 per cent, of its weight in wood is absorbed, whereas the actual consumption of wood amounts to from 9 to 14 per cent., showing that from 50 to 80 per cent, of the total heat generated is utilized. Metallurgical Furnaces. Dr. Siemens states that, in an ordinary reheating furnace, employed in metallurgical operations, one ton of coal is consumed in heating 1^3 tons of wrought iron to the welding point, 2700 F.; whilst he estimates, in terms of the specific heat of iron, .114, and the heating power of coal, 14,000 units of heat, that a ton of coal is capable of heating up 39 tons of iron. From this it appears that only 4^ per cent, of the whole heat generated is appropriated by the iron. Similarly, he estimates that 32 498 APPLICATIONS OF HEAT. barely i % per cent, of the whole heat generated is utilized in melting pot- steel, in ordinary furnaces ; whilst, in his regenerative furnaces, a ton of steel is melted by the combustion of 12 cwts. of small coal, showing that 6 per cent, of the heat produced is utilized. Blast-Furnace. Mr. J. Lothian Bell 1 has formed detailed estimates of the appropriation of the heat of Durham coke in the Cleveland blast- furnaces ; from which the following abstract has been prepared : Durham coke, it is assumed, consists of 92.5 per cent, of carbon, 2.5 per cent, of water, and 5 per cent, of ash and sulphur. To produce i ton of pig-iron, there are required 1 1 cwts. of limestone, and 49 cwts. of calcined iron-stone; the iron-stone consists of 18.6 cwts. of iron, 9 cwts. of oxygen, and 21.4 cwts. of earths. There is formed 7.26 cwts. of slag, of which i.i cwt. is formed with the ash of the coke, and 6.16 cwts. with the limestone. There are 21.4 cwts. of earths from the iron-stone, less .74 cwt. of bases taken up by the pig-iron and dissipated in fume; say, 20.66 cwts. Total of slag and earths, 27.92 cwts. Mr. Bell assumes that 30.4 per cent, of the carbon of the fuel, which escapes in a gaseous form, is carbonic acid; and that, therefore, only 51.27 per cent, of the heating power of the fuel is developed, and the remaining 48.73 per cent, leaves the tunnel-head undeveloped. He adopts, as a unit of heat, the heat required to raise the temperature of 1 1 2 Ibs. of water i Centigrade. Distribution of the heat generated in the blast-furnace for the production of i ton of pig-iron: UNITS. PER CENT. Evaporation of water in coke, and chemical action, in smelting, 48,354 54-i Fusion of pig-iron, 6,600 7.4 Fusion of slag, I 5,3S^ > I 7- 2 Expansion of blast, 3,700 4.1 Appropriated for the direct work of the furnace, 74,010 82.8 Loss by radiation through the walls, 3,600 4.0 Carried away by tuyere-water, i ,800 2.0 Sensible heat of gaseous products, 10,000 11.2 Waste, 1 5,400 17.2 Total heat generated in the furnace, 89,410 100.0 The undeveloped heat of the fuel amounts proportionally to 89,410 x = 84,980 units. Add to this, the sensible heat of the gaseous products, 10,000 units, and the sum, 94,980 units, is disposed of as follows: 1 The Journal of the Iron and Steel Institute, 1872, 1875. Tne abstract given in the text affords but a meagre notion of the variety and extent of Mr. J. Lothian Bell's investi- gations, the value and importance of which are highly and justly appreciated by manufac- turers of iron. HEATING OF SOLIDS. 499 Distribution of the waste and undeveloped heat of the fuel required for the production of i ton of pig-iron. UNITS. PER CENT. Generation of steam for blast-engine and various pumps connected with the work, 28,080 29.6 Heating the blast to 905 F., 11,920 12.5 Appropriated for direct work, 40,000 42. i Loss by radiation from the gas tubes, 3320 3.5 Loss of heat escaping by the chimneys, 21,660 22.8 (temperature, 770 R, from boilers) ( Do. 640 F., from stoves) Radiation at boilers and stoves, 25 per cent, 16,240 17.1 Waste, 41,220 43.4 Loss of gases from blast-furnaces, in charging, 5 per cent, 4,740 5.0 Sundry, 9,020 9. 5 Total waste and undeveloped heat, 94,980 100.0 For the performance of the duty according to these analyses, Mr. Bell states that 19.08 cwts. of carbon, or 20.62 cwts. of coke, are required, per ton of iron produced from ore yielding 41 per cent, of iron. In a furnace having 18,000 cubic feet of capacity, 80 feet high, i ton of No. 3 pig-iron was produced with 21^ cwts. of ordinary Durham coke, from Cleveland iron-stone. In recent years, by raising the temperature of the blast to 485 C., or 905 F., the consumption of coke, with a furnace 48 feet high, was reduced to 28 cwts. per ton of iron. With a cold blast, more than 60 cwts. would probably have been required. It is stated, that at Barrow works, where the Siemens-Cowper regenerative stove is employed for heating the blast to 1100 F., the quantity of coke consumed is 20.08 cwts. per ton of iron. THE STRENGTH OF MATERIALS. The strength of materials is measured by the resistance which they oppose to alteration of form, and ultimately to rupture, when subjected to force, pressure, load, stress, or strain. The exigencies of scientific precision have caused the general substitution of the word "stress" for the good old engineer's word "strain," as expressive of force, though "strain" may still be employed to express alteration of form. 1 Stress is applied in five recognized modes : i st. Tensile stress, tending to draw or pull the body asunder. The immediate effect is elongation. 2d. Compressive stress, tending to crush it. The immediate effect is compression. 3d. Shearing stress, tending to cut it through. The immediate effect is lateral compression, elongation, and deflection. 4th. Transverse or lateral stress, tending to bend it and break it across, the force being applied laterally, and acting with leverage. The immediate effect is lateral deflection. 5th. Torsional stress, tending to twist it asunder, the force acting with leverage. The immediate effect is angular deflection. Mr. Callcott Reilly aptly reduces the varieties of stress to three kinds of simple stress: Tensile stress, compressive stress, and shearing stress. These are the ultimate forms of stress; they are combined in transverse stress, and the third is substantially the form of torsional stress, where the strain is applied over a very short length. Or, where torsional stress is applied over a considerable length, the tensile form of stress is combined with shearing stress. When stress is applied gradually to a solid body, the strain, or alteration of form, is proportional to the intensity of the stress, so long as the inherent elastic force of the body is not overbalanced by the stress so long, that is to say, as the alteration of form remains within the elastic limit, the stress, at the same time, remaining within the limit of elastic strength. When the elastic limit is turned and exceeded, the body begins to yield under gradually accumulating stress, and the strain or alteration of form becomes proportionally greater and greater with the intensity of the stress, until, finally, rupture or breakage takes place. 1 As Dr. Pole says, in his lectures on Iron as a Material of Construction, this word strain " appears to convey its idea so clearly that there must be little chance of expunging it from the practical mechanic's vocabulary." Mr. Stoney employs it exclusively in his work on The Theory of Strains. Mr. Kirkaldy uses the word "stress "in his reports of his experiments on the strength of materials. WORK OF RESISTANCE OF MATERIAL. 5OI When a body is loaded in excess of the elastic limit, without breakage, it returns, when unloaded, towards its normal form, but it fails to regain it. It is, in so far, deformed, and it has acquired a permanent set or a set. There are five data of importance to be observed in the measurement of the strength of materials. i st. The limit of elasticity, or the elastic limit. 2d. The greatest stress which the material is capable of sustaining within the elastic limit, or the elastic strength. 3d. The strain, or alteration of form elongation, compression, deflec- tion, or torsion within the elastic limit. 4th. The total extent of the strain, or alteration of form, with the set, before rupture takes place. 5th. The greatest stress which the material is capable of supporting before rupture takes place ; or, the absolute strength. The first and second data are of prime importance; the others are sub- sidiary. For, in practice, it is necessary, in order to insure the permanency of a structure, that its proportions should be such that, under the maximum stress to which any piece is to be subjected, it should not be strained be- yond the elastic limit of its strength. WORK OF RESISTANCE OF MATERIAL. Under a Quiescent Load. Since the intensity of the elastic resistance in- creases uniformly with the total space through which the action of the stress takes effect, it may be represented by the triangular space ABC, Fig. 129, in which AB is the total space de- scribed, and B c is the measure of the stress applied. Suppose the stress BC= 10,000 Ibs., and the space AB = i inch, then the stress, 10,000 Ibs., which has been applied, operates through a space of i inch, and has been opposed by an elastic resistance which commenced at A, and increased uniformly, from o at A, to 10,000 Ibs. at B. The intensity of the resistance at different points along the space AB, is measured by the ordinates of the triangle parallel to the base through the given points; and if the space be divided into four parts, for example, at the points a, b, c, the values of the ordinates a a, b b', cc', or the intensities of the resistance at the points of elongation a, b, c, are respectively 2500, 5000, and lg v 7500 Ibs. If an indefinitely great number of ordi- nates be drawn, they will occupy the whole area of the triangle, and the average length of the ordinates will be half the base B c, equivalent to 5000 Ibs. Hence a ready means of calculating the quantity of work necessary to strain a piece to its elastic limit: Multiply half the elastic strength in pounds by the space in feet described by the resistance or by the stress. The product is the work expended. If R = the elastic strength in pounds, 502 THE STRENGTH OF MATERIALS. s = the space described by the load or stress in feet, and w - the work done, the rule is formulated thus : w^y 2 ^s (i) For example, using the above data, if 10,000 be the elastic strength, and i inch or .0833 foot be the space described, then the work done in strain- ing the piece to the limit of its elastic strength is YZ (10,000 x .0833) = 416. 7 foot-pounds. Under a Load suddenly applied. In these calculations of stress and work, it is assumed that the stress is applied gradually, so that no appreciable velocity and momentum be generated as the stress is applied. If, on the contrary, a weight equal to the total load be applied suddenly and all together, the momentary deflection under the load amounts to twice the permanent deflec- tion, or twice that which is effected by the load when gradually applied, supposing that the total deflection -D' does not exceed the elastic limit. Let AB, Fig. 130, Fig. 130. Deflection under be the deflection caused by the gradual application of a weight w, and AB' the momentary deflection caused by the sudden application of it. Draw the ordinates BC and B'C' to measure the resistance at the points B and B', and complete the triangle A B'C'. Through c draw the vertical D D'. Then the rectangle A B' D' D measures the work done by the load in falling through the height AB', and the triangle A B'C' measures the work of resistance to deflection. These are equal to each other; and as B'C' must be twice B'D', so AB' is twice AB; that is to say, the momentary deflection under a load suddenly applied is twice the steady deflection under the same load very gradually applied. It follows that in proportion to the rapidity with which loads are applied, as when railway trains run upon a bridge, of course in the absence of per- cussive action, the deflection is greater than that due to the same load at rest on the bridge, and increases with the speed of transit. But it does not amount to twice the deflection due to a quiescent load, though it approaches to this limit as the speed increases. Under Stress by Percussion. When a solid material is exposed to percus- sive stress, as, for instance, when a heavy weight falls upon a beam transversely, the work of resistance is measured by the product of the weight by the total fall the total fall being equal to the height of fall above the beam plus the deflection. To exemplify percussive action within the elastic limit, reproduce Fig. 130 on the same scale, in thick lines, in Fig. 131, with the same letters of reference, and let A' A be the height of fall above the beam, and B' B" the additional deflection under the weight. A'B" is the total fall, and AB" the total deflection. Draw the horizontal line B"C", and produce AC' and DD' to ^ c " meet it at c" and D". Then the rectangle A'B^D^D'" measures the work for the total fall, and the triangle D' Fig. 131. Deflection under ABC measures the work of resistance to deflection; and these quantities of work are equal to each other. The scale of the diagram above the normal level of the beam, A D,. TRANSVERSE STRENGTH OF HOMOGENEOUS BEAMS. 503 is the same as that of the portion below A D, for the sake of simplicity of illustration, and for ready comparison of the squares representing quantities of work. COEFFICIENT OF ELASTICITY. The elasticity of a bar of any solid material subjected to a direct tensile or a direct compressive force, within the elastic limits, is measured by a constant fraction of the length per unit of force per unit of sectional area. The unit of force and area is usually taken as one pound per square inch, but it is sometimes taken as one ton per square inch. E is used to symbolize the denominator of the fraction. For example, if a bar of iron be extended V^.oooth part of its length per ton of stress per square inch of section, 12,000 E' The bar would therefore be stretched to double its normal length by a force of 12,000 tons per square inch, if the material were perfectly elastic. The supposition, though imaginary, is convenient; and the coefficient of elasticity is usually denned as the weight which would stretch a perfectly elastic bar of uniform section to double its length. It is represented by E, which may be employed to express pounds, tons, or any other measure of weight. The coefficient of elasticity may also be expressed in terms of the length in feet of a bar of the given material, the weight of which would be equal to the force required to stretch it to twice its normal length. For example, a i-inch square bar of iron weighing 33^ pounds per lineal foot would require to be (12,000 x 2240 -r 3^3 = ) 8,064,000 feet in length to stretch it at the upper end to twice the normal length, and this is another expression for the coefficient of elasticity. The same methods of expressing the coefficient of elasticity are applied to the elastic resistance to compression. That is, the coefficient, in weight, is expressed by the denomination of the fraction of its length by which a bar is compressed per unit of weight per square inch of section. TRANSVERSE STRENGTH OF HOMOGENEOUS BEAMS. Tensile resistance is selected as the basis of the following formulas, which are constructed on the assumption that, within elastic limits, extension is equal to compression, under equal stresses; and their strict application is confined to the calculation of stress, strain, and strength, within elastic limits. At the same time, they are practically applicable for calculating ultimate strength. i. SYMMETRICAL SOLID BEAMS. Let a homogeneous beam, AB, Fig. 132, of rectangular section, be freely supported horizontally at both ends, at a and b. Bisect the depth cd at o y and draw the horizontal line n op. Let the beam be loaded by the weight W, applied at the middle, cd, of the beam. Then the beam is deflected under the load, and the upper half, above the line n op, is com- pressed, and the lower -half is extended, in such a manner that, having regard to the vertical section cd, the compression and elongation respectively increase uniformly from zero at the central point, 0, to a maximum at the upper and lower surfaces, c and d. The proportional increase may be 504 THE STRENGTH OF MATERIALS. represented by the triangles oef and ogh, formed by the lines eh an intersecting at the central point o\ in which the graduated shortening and lengthening of the fibres at any given height are represented by horizontal lines drawn across the tri- angles. The horizontal com- pressive stress in the upper half, and the ten- sile stress in the lower half of the beam, with respect to the section cd, likewise increase uni- formly from the central point o, where they are zero, to the upper and lower surfaces at c and d. Such is the ordinary theory of transverse stress in a rectangular beam, and it is assumed that, throughout the whole length of the beam, as at the section at the middle, there is no horizontal stress in the central line nop t with respect to any vertical section. This line is therefore called the neutral line or neutral axis; and it is a line of demarcation between the directly horizontal compressive stress above and the tensile stress below. Further, the sum of the compressive stress above the neutral axis is equal to the sum of the tensile stress below, and each may be replaced by its re- sultant stress at the resultant centre without affecting the equilibrium. If the cross section of the beam, Fig. 133, No. i, be divided into a number of strips of equal thickness, the moment of stress for each strip in the upper and in the lower group, with respect to the neutral axis n op, may be calcu- lated, and the sum of the moments for each group divided by the sum of the Fig. 132. Transverse Stress on a Rectangular Beam. fig- 133, No. i. Longitudinal Resistance in Loaded Beams. - !33> No. 2. Diagonal Resistance in Loaded Beams. Fig- I 33> No. 3. Combined Resist- ance in Loaded Beams. resistances, when the quotient will be the resultant radius. But this calcu- lation may be saved by drawing the diagonals eh and/^", when it is apparent that the shaded triangles formed by them exhibit the relative quantities of stress for each strip, and that the resultant lines of stress, m m and //, pass through the centres of gravity of the triangles, each at a distance from the neutral axis equal to two-thirds of the half-depth of the beam. (Fig. 133,1.) These, the moments of the normal stresses or resistances due to the abso- TRANSVERSE STRENGTH OF HOMOGENEOUS BEAMS. 505 No. i. Let c B d, a triangular lute horizontal compression and extension of the beam, are supplemented by diagonal resistances, by which each of them is augmented 75 per cent. To elucidate the origin of diagonal resistance, it may be observed that the upper and lower portions of a beam above and below the neutral axis may be considered as two individual members of a frame, united at their surface of contact the neutral axis. Figs. 134, No. i, be frame, fixed at cd and loaded at B. The pieces c c 1 and d d' of the upper and lower members are respectively extended and com- pressed, when the load is applied, to the lengths c c" and dd". If the members of the frame are placed parallel to each other, in close contact, as in Figs. 134, No. 2, extension and compres- sion take place as before. Let, . now, the two members be united in the line nop, No. 3, and so consolidated as to form a semi- beam; the extension and the compression partially neutralize each other: at the neutral line nop they are absolutely neutral- ized, and the amounts of exten- sion and of compression are represented by the triangles c' c" o and d' d" o. The structure is thus, in a certain sense, crip- pled ; and the extension and Figs. 134. Stress in a Loaded Beam. compression, instead of being rectilineal, are curvilineal, and the semi-beam is deflected, as in No. 4. The counteraction here pointed out is necessarily exerted diagonally, at an angle of 45 with the neutral line; as in the line o'c' t Fig. 135, at 45 with the transverse section c'd'. The diagonal forces, as applied to the d' Fig. 135. Diagonal Stress in a Beam. o y Fig. 136. Diagonal Stress in a Beam. transverse section c'd' strained into the position c"d", are represented by diagonals at 45 drawn from points in the upper half-depth c" o to the neutral line oo' t for tensile resistance; and from points in the lower half- 506 THE STRENGTH OF MATERIALS. depth od", for compressive resistance. Reproduce the upper half-depth to a larger scale, in Fig. 136. Draw the perpendicular c'n, and thence the perpendicular ns; the diagonal extension is measured by nc", and the horizontal component sc" is the horizontal extension at the upper side of the beam, due to the length of the portion c'c'". These extensions are also measures of the diagonal force, and its horizontal component. Take next the horizontal line t u, at any other position in the half-depth. The diagonal vu becomes, when strained, vu', at 45, and, by the same con- struction as before, the extension of the diagonal is measured by z u', whilst its horizontal component is x u 1 . It is easily deduced from the similarity of the construction, that the forces measured by the horizontal components xu' and sc" are equal to each other. Similarly, the horizontal components of the diagonal forces acting throughout the whole depth of the section c' o, are equal to each other. They may, therefore, be represented in their entirety by the rectangle c'syo, of which the length c' s is equal to half the length c' c" of the triangle c' c" o\ and the areas of force represented by the rectangle and the triangle, are equal to each other. By a similar argument, the diagonal compressive resistance in the lower section of the semi-beam, may be analyzed. It is the simple converse of the diagonal tensile resistance in the upper section. The horizontal resistance due to diagonal stress, is represented diagram- atically, on the cross section, by Fig. 133, No. 2, in which the shaded area inclosed by the verticals e'g' and/'/#' represents, in its upper half, the tensile stress; and in its lower half, the compressive stress; for which the resultant lines of stress, m m and //, are each at a distance from the neutral axis equal to half the half-depth of the beam. The two elements of resistance (No. i and No. 2) are combined in Fig. 133, No. 3, showing in deep shad- ing the combined areas of stress of uniform intensity; the amount of which is equal to that of the semi-rectangle. This investigation for a semi-beam is applicable for a beam supported at the ends, and loaded at the middle, like Fig. 132, the tensile and compressive stresses being inverted. The tensile and compressive stresses act at a resultant radius, measured from the neutral line, of ((- + -)-=- 2 =) or .5833, taking the half-depth as i. As \ 3 2 / 12 5775 is the geometrical radius of gyration of the semi-rectangle on the neutral axis, when the half-depth =i (see page 289), the moment of resist- ance will, for simplicity, be taken as .5775, when the area of the semi- rectangle and the half-depth are each represented by unity. Then the total moment of either resistance, tensile or compressive, with reference to the neutral axis, is expressed by the product of half the sectional area of the beam by half the depth, and by .5775, and by the extreme tensile or compressive stress per unit of sectional area. That is to say, by, ................ (a) in which = the breadth, ^=the depth, both in inches; and j=the extreme tensile stress per square inch, to which the extreme compressive stress is taken as equal. The sum of the moments of the tensile and the com- pressive resistances is, therefore, practically twice the moment ( a ) round the neutral axis; or .1444 or W = .7854 b d*s ~~7 ' (16) Let the ratio of the breadth to the depth = <:, then b = cd, and 2. FLANGED OR HOLLOW BEAMS OF SYMMETRICAL SECTION. Hollow or flanged beams may be generally described as beams of over- hung section. In such beams the diagonal resistance to flexure is only excited in the vertical portions of the section. Figs. 138 are examples of overhung sections; and the neutral axis passes through the centre of gravity. No. i. No. 2. No. 3. No. 4. Figs. 138. Symmetrical Flanged Beams. Sections. i. Hollow-rectangular or Double-flanged Sections. Nos. i, 2, 3. When the depth is considerable compared with the thickness of the flanges, calculate for the flanges and for the web separately. The separation of the web from the flanges is shown in Fig. 139, for the flanged beam, No. i, and it is to be done in the same manner for the hollow beam, No. 3. The moment of resistance of one flange is sensibly equal to the product of its sectional area multiplied by the distance d" between the centres of the flanges, and by the tensile strength per square inch; or to FLANGED BEAMS OF SYMMETRICAL SECTION. in which / is the depth or thickness of a flange, and a its sectional area. The web is treated as a rectangular beam of the depth d' ', the distance between the centres of the flanges. This is greater than the actual depth, as between the flanges; but the excess is compensated by the metal rilled in at the angles. Putting t' for the thickness of the web, the moment of resistance is, by (), page 506, .2888 t'd" 2 s= 2888 a" d" s, (g) \ o / in which a" is the sectional area of the web. The sum of the moments (f) and (g) is equal to the moment of the weight ; or, W / 4 whence, W/-4 ad" s+ 1.155 a" d" s = d" s (4 a+ 1.155 a Fig. 139. Calculation of Strength of Beam. (18) That is to say: When the depth is considerable compared with the thickness of the flanges multiply the sectional area of one flange by 4 ; and multiply the sectional area of the web by 1.155. Add the products together, and multiply the sum by the reputed depth of the beam, and by the tensile strength per square inch, and divide the product by the span. The quotient is the breaking weight. Note. The reputed depth of the beam, and also that of the web, are taken, for calculation, as the total depth minus the thickness of one flange. 2d Method. In some cases the strength of the flanges only is calculated, W/ when the web is comparatively slight. Then, = a d" s, or W / = 4 a d" s; 4 ad's _ 4 bid's (2o) That is to say: When the strength of the web is neglected, the breaking weight is equal to four times the sectional area of one flange by the distance apart between the centres of the flanges, and by the ultimate tensile strength per square inch, divided by the length. In applying the formula to the hollow beam, No. 3, Fig. 138, t is taken as the sum of the thicknesses of the sides. When the thickness of the flanges is considerable com- pared with the depth of the beam, No. 4, Figs. 138, and Fig. 140. In the double-flanged section, Fig. 140, cal- culate the strength of the web for the whole depth, as indicated in sectioning. For the lateral flange portions, the average stress is less than s in the ratio of the total depth d d" to the reputed depth d", or it is s and the net area of one flange, or of Fig. 140. Calculation of Strength of Beam. 512 THE STRENGTH OF MATERIALS. the unshaded parts, Fig. 140, being put equal to a', the moment of resist- ance of one flange is, , ,,, a t $ 7// tCL / i \ ~ X ~ S ...................... The sum of the moments of resistance of the flanges and the web is equal to the moment of the weight, or (21 in which / = the thickness of the web, and ^= rpi ' d2 ' d ~ + depth of the section. 1.155 * ' d 2 }; and w = (4 1.155 (22) That is to say : When the thickness of the flanges is considerable compared with the depth of the beam. Multiply the net sectional area of one flange, cal- culated for its width minus the thickness of the web, by the square of the reputed depth, and by 4, and divide by the total depth. Multiply the thickness of the web by the square of the total depth, and by 1.155. Add together the quotient and the product, and multiply the sum by the tensile strength per square inch, and divide the product by the span. The quotient is the breaking weight. Note. The reputed depth is equal to the total depth minus the thick- ness of one flange. Double-headed rails, as No. 4, Fig. 138, will be speci- ally treated. 2. Annular Section, Figs. 141, No. i. The hollo wness of the section deprives it of a large proportion of the diagonal resistance exerted in the No. i. No. 2. No. Figs. 141. Symmetrical Hollow Beams. Sections. solid circular section; otherwise the strength might have been calculated from the section, Fig. 143, in which the material of the annular section is collected about the vertical centre line. 1 The lateral portions of the 1 This mode of aggregation of the section is employed in Mr. Edwin Clark's work on the Britannia Bridge, page ill; and also by Mr. Baker in his excellent work on the Strength of Beams, page 26. Mr. Baker very properly points out the fallacy of the ordinary mode of calculating the transverse strength of a beam of annular section, which does not take cognizance of the loss of "resistance to flexure" in a hollow beam. FLANGED BEAMS NOT SYMMETRICAL IN SECTION. 513 section, end, c' pd\ Fig 142, are by their position subject to diagonal stress, and they are reproduced in darker shading in Fig. 143. The breaking strength may be approximated to by, in the first place, reducing the breadth of the overhung portions, and calculating the strength of the re- duced section, on the principle to be explained in treating of beams of unsymmetrical sections. Figs. 142, 143. Annular Section of Beams. Fig. 144. Thin Annular Section of Beam. zd Method. When the section is thin, as in Fig. 144, the matter of the section may be assumed to be collected at the outer circumference, for which the radius of gyration is .7071, when the radius of the section is i; whence, rating the stress s exerted throughout the section as the maximum stress, the resultant point of resistance, V, of the half-section is ?7 I = .50, i when the radius is i ; or the distance of the centres of resistance, V, V, is half the diameter. The sectional area is equal to the product of the circumference by the thickness, or to 3.14 dxt; and the half-section = 1.57 dt. The moment of the half-section is 1.57 dt^y^ ^=.785 d*t, and = .785 d 2 ts; whence, 4 Hollow Elliptical Sections. These sections, Nos. 2 and 3, Fig. 141, page 512, may be treated on the same principle as the annular sections, No. i, Figs. 141, and Fig. 143. 2d Method. When the section is thin the breaking strength is, by adapting formula ( 23 ), putting b = the breadth, and d -= the depth, W=*.i4?_fL, or, W= 1.57 (24) 3. FLANGED BEAMS WHICH ARE NOT SYMMETRICAL IN SECTION. For such beams, of which the sections, Figs. 145, are examples, it is necessary to ascertain the quantity of longitudinal tensile resistance, and the distance apart of the resultant centres of tensile and compressive stress, for a given section; and to multiply these together to obtain the moment of resistance of the section; whence the ultimate transverse strength may be calculated. The first operation is to find the neutral axis 33 514 THE STRENGTH OF MATERIALS. of the section; and as the ultimate longitudinal resistance in the web is greater than that of a flange, the neutral axis does not pass through the centre of gravity of the section. But, if the area of the flange be reduced in proportion to the potential or ultimate unit-resistance in the web to that No. i. No. 2. No. _J No. 4. IT 3 No. 5. Figs. 145. Sections of Unsymmetrical Beams. of the flange, or as 1.73 to i, the neutral axis will pass through the centre of gravity of the reduced section. RULE. To find the neutral axis of a beam of unsymmetrical section. Divide the section, as reduced, into its simple elements, and assume a datum-line from which the moments of the elements are to be calculated. Multiply the area of each element by the distance of its own centre of gravity from the datum-line, to find its moment. Divide the sum of these moments by the total reduced area; and the quotient is the distance of the centre of gravity of the reduced section, or of the neutral axis of the whole section, from the datum-line. For example, the _L section, No. i, Figs. 145, and shown in Fig. 146 annexed, is 12 inches deep, 12 inches wide, and i inch thick throughout. Extend the web, c d, to the lower surface at d' and d", leaving 5 ^ inches of web, a d' and d" b, on each side. Reduce this width in the ratio of 1.73 to i, or to (5.5 -- 1.73 = ) 3.2 inches, and set off d' a' and a" b' each equal to 3.2 inches. Then the reduced flange a' b' is (6.4+1=) 7.4 inches wide, and the reduced section consists of the two rectangles a' b' and cd. Assume any datum-line, as ef at the upper edge of the section, and bisect the depths of the rectangles, or take the intersections of their diagonals at g and h, for their centres of gravity. The distances of these from the datum-line are 5 ^ and 1 1 ^ inches respectively, and the areas of the rectangles are ii x i = ii square inches, and 7.4 x i = 7.4 square inches. By the rule, FLANGED BEAMS NOT SYMMETRICAL IN SECTION. 515 Upper rectangle, n x $%= 60.5 Lower do 7.4x11^ = 85.1 18.4 x 7.91 = 145.6 Showing that the centre of gravity of the reduced section, being the neutral axis of the whole section, is 7.91 inches below the upper edge, in the line ii. The centre of gravity of the entire section, it may be added, is 8.63 inches below the upper edge, or .72 inch lower than that of the reduced section. The neutral axes of the other sections, Figs. 145, found by the same process, are marked on the figures. The section of a flange rail, No. 5, which is very various in breadth, may be treated in two ways : either by preparatorily averaging the projections of the head and the flange into rectangular forms; or, by taking it as it is, and dividing it into a con- siderable number of strips parallel to the base, for each of which the moment with respect to the assumed datum-line is to be found. The first mode of treatment is approximate ; the second is more nearly exact. Ultimate Strength of Beams of Unsymmetrical Section. Resuming the _L section, Fig. 146, for which the neutral axis has been ascertained, to find the tensile resistance, divide the portion below the neutral axis /, Fig. 147. with the reduced width of flange, d b', into parallel strips, say ^ inch deep, ED w rid}' b Figs. 146, 147, 148. Beam of Unsymmetrical Section. as shown, and multiply the area of each strip by its mean distance from the neutral axis for the proportional quantity of resistance at the strip. Divide the sum of the products, amounting to 31.3, by the extreme depth belov, the neutral axis in this instance 4.09 inches, and multiply the quotient b\ 1.73 s, the ultimate tensile resistance at the lower surface. The final pro- duct is the total tensile resistance of the section; or, 31.3 x i.73-y_ 4.09 = 13.24 s total tensile resistance. Again, multiply the area of each strip by the square of its mean distance from the neutral axis, and divide the sum of these new products, amount- ing to 104.64, by the sum of the first products. The quotient is the dis- tance of the resultant centre of tensile stress, d', from the neutral axis. Or, the resultant centre is, 104.64 = 3.34 inches below the neutral axis. 516 THE STRENGTH OF MATERIALS. This process is, in fact, the process for finding the centre of gravity of all the tensile resistances. By a similar process for the upper portion, in compression, the sum of the first products is found to be the same as for the lower part, and is 31.3. But the maximum compressive stress at the upper surface is greater than the maximum tensile stress at the lower surface, in the ratio of their dis tances from the neutral axis; or it is 1.73 s x ZlZ_ = 3.34 j and 4.09 3 T -3 x 3-34 s _ ^.24 s, total compressive resistance, 7.91 which is the same as the total tensile resistance, in conformity to the general law of the equality of tensile and compressive stress in a section. The sum of the products of the areas of the strips, divided by the squares of their distances respectively from the neutral axis, is 164.90, and the resultant centre, c, is 4iz_ = - 2 y i nc hes above the neutral axis. 3i-3 The sum of the distances of the centres of stress or of resistance from the neutral axis, (3.34 + 5.27 =)8.6i inches, is the distance apart of these centres, as represented by a central line, c'd', in Fig. 147. Abbreviated calculation, As the upper part of the section is a rectangle, detailed calculation is not necessary, for its resultant centre is known to be at two-thirds of the height, or (7.91 x 2 / 3 = ) 5.27 inches above the neutral axis. The average resistance, too, is half the maximum stress, namely, that at the upper edge, which is, as above explained, 3.34 ^ per square inch. The area of the rectangle is (7.91 x i = )7.9i square inches; and 7.01 x 3.^4 s - - 2_^_ - 13.24 s the compressive resistance, y been calculated. e: the moment of tensile resistance is 13.24^x8. 114 s, and it is equal to ; whence W = 1 14 s x 4 ; or, in a general form, * as has already been calculated. To resume: the moment of tensile resistance is 13.24^x8.61 inches 4 ( 25 ) W = the breaking weight, in tons. S = the total tensile resistance of the section, in tons. d z = the vertical distance apart of the centres of tension and compres sion, in inches. /=the span, in inches. Strength of the Beam, Fig. 147, inverted. When inverted, the maximum tensional resistance of the beam at the lower surface c, in Fig. 148, is 1.73 s. The area of the rectangle ic is 7.91 square inches, and i-l? ''3 _ 5 g^ ^ total tensile resistance; which is only about half the tensile resistance offered by the beam in its first position, Fig. 147. The breaking weight is, therefore, also only about FORMS OF BEAMS OF UNIFORM STRENGTH. 517 a half; and the reason why it is calculated that the beam bears double the breaking weight in the first position that it does in the second is, that the rectangular portion, c i, is expected to oppose at least twice as much resist- ance, when above, to compression as it does, when below, to tension. If the effective resistance to compression were only equal to the resistance to tension, the beam would have the same ultimate strength in both positions. RULE. To find the Ultimate Strength of a Homogeneous Beam of Unsym- metrical Section, i. Reduce the section to a section of uniform potential resistance, as explained at page 514. 2. Find the position of the neutral axis of the reduced section by the previous rule. 3. Divide the section into thin strips parallel to the neutral axis ; if such division has not already been made, for No. 2. 4. Multiply the areas in square inches of the strips, under tensional stress, by their mean distances respectively (that is, the distances of their centres of gravity) from the neutral axis. 5. Multiply the same areas by the squares of their mean distances respectively from the neutral axis. 6. Divide the sum of the first products by the extreme dis- tance of the surface in tension from the neutral axis, and multiply the quotient by the ultimate tensile resistance per square inch; the product is the total tensile resistance of the section. 7. Divide the sum of the second products ( 5 ) by the sum of the first products ( 4 ); the quo- tient is the distance of the resultant centre of tension from the neutral axis. 8. Multiply the ultimate tensile resistance by the distance of the neutral axis from the upper surface, and divide the product by the distance of the neutral axis from the lower surface; the quotient is the maximum compressive stress at the upper surface. 9. Make the calculations 4, 5, and 7 for the strips under compression. 10. The sum of the distances of the resultant centres from the neutral axis is the distance apart of the centres, n. Find the breaking weight by formula (25); that is, multiply the total tensile resistance of the section in tons by the distance apart of the resultant centres of tension and compression in inches, and by 4; and divide the product by the span in inches. The quotient is the breaking weight in tons at the middle. Note to Rule.l\. is assumed that the ability of the beam to resist com- pression is sufficient to insure that fracture shall take place by tension. Beams of comparatively slender dimensions laterally are liable to cant under the thrust of compression if not supported laterally, and canting occasionally takes place in beams which are tested experimentally. But beams, when in their destined places, are in general so supported that any liability to canting is removed. Elastic Strength of Beams of Unsymmetrical Section. The elastic strength is approximately deducible from the ultimate strength, according to the ordinary ratio of one to the other, ascertained experimentally. The elastic strength and deflection of a homogeneous beam of any section is the same, whether in its normal position or turned upside down. FORMS OF BEAMS OF UNIFORM STRENGTH. A beam is said to be of uniform strength when its capability of resistance to transverse stress under a given load, applied in a given manner, is the same at all parts of its length. 5 i8 THE STRENGTH OF MATERIALS. r SEMI-BEAMS OF UNIFORM STRENGTH LOADED AT THE END. By a semi-beam is meant a beam fixed at one end and free at the other; as it represents the half of a beam supported at both ends. The moments of stress due to the weight on the end, at any section of the beam, increase directly as the distance of the section from the end of the beam. Let cb, Fig. 149, be a rectangular beam, fixed at the base cd, and loaded at the end b. Draw the dia- gonal straight line bd, then the ordinates to the triangle b cd, represent proportionally the moment of stress at all parts of the length; and the moments of stress vary directly as the length. Now, the ultimate moments of resistance at any section are as the square of the depth, when the breadth is uniform; and it follows, conversely, that the depth of the beam, of uniform strength, must vary as the square root of the distance from the end b. Take, for instance, the sec- tion c' d' at the half-length of the beam; the moment of stress at cd is to that of the stress at c'd', as i to ^ ; and the required depth at the origin cd, is to the depth at c' d', as */ i to */ }4, or as i to .707. The depth c' c", equal to .707, would be the depth of a beam of uniform strength, at that section. The depth for uniform strength at any other section may be calculated in the same way ; and the form of the lower side of the beam, of uniform strength, is that of a parabola, b c" d, of which the vertex is at the end b. With respect to transverse resistance, then, the semi-beam would be equally strong if the lower portion b b' d were removed. The semi-beam, rectangular in section, of uniform strength, fixed at one end, and loaded at the other end, having the breadth constant, may therefore be moulded in depth to any of the parabolic outlines, Figs. 150. Fig. 149. Stress in a Semi-beam loaded at the end. No. i. No. 2. No. 3. Figs. 150. Semi-beams loaded at one end. When the depth of the semi-beam, rectangular in section, is constant, the breadth is in simple proportion to the distance from the end of the beam, as in Fig. 151, and the beam is triangular in plan. When the section of the semi-beam is double-flanged, or is hollow rectangular, and the breadth is constant, the flanges are assumed to be of a constant sectional area. Leaving out of the calculation the strength of the vertical SEMI-BEAMS OF UNIFORM STRENGTH. 519 web, and calculating only for the flanges, the moment of resistance at any section is as the depth, and the form of the beam is triangular, as in Fig. 152, which shows a semi-beam with double flanges. If the strength of the vertical web be taken into the calculation, the form of the beam is intermediate between the triangular and the parabolic. Fig. 151. Fig. 152. Fig. 153- Semi-beams loaded at one end. When the section of the semi-beam is double-flanged, or is hollow-rectangular, and the depth is constant, calculating only for the flanges, Fig. 153, their sectional area increases uniformly with the distance from the end, and if their thickness be uniform, they are triangular in plan, as shown. If the web be taken into the calculation, it is calculated as a solid semi- beam rectangular in section, and the thickness should increase as the dis- tance from the end. The web would, therefore, be triangular in plan. When the section of the semi-beam is circular, the moment of resistance varies as the cube of the diameter, and the cube of the diameter is therefore as the distance from the end; or, inversely, the diameter is as the cube root of the distance, and the outline of the semi-beam may be formed by the revolution of a cubic parabola on its axis, Fig. 154. When the section of the semi-beam is annular; when the thickness is uniform and small in proportion to the diameter, the square of the diameter varies as the distance from the end, or the diameter varies as the square root of the distance, and the semi-beam is formed by the revolution of a parabola on its axis. If the thickness varies with the diameter, the diameter varies as the cube root of the distance from the end, and the semi-beam is cubic-parabolic, like Fig. 154. When the section of the semi-beam is elliptical, the sections being similar at all points of the length, the cube of the depth varies as the distance from the end, or the depth varies as the cube root of the distance, and the elevation of the beam is cubic-parabolic, like Fig. 154. When the section of the semi-beam is hollow- elliptical, the beam being of similar sections throughout. When the thickness is uniform, and is small in proportion to the depth, the Square Of the depth varies as the distance Fig. 154. Semi-beam loaded at one end. from the end, or the depth varies as the square root of the distance, and the side elevation of the beam is parabolic. 520 THE STRENGTH OF MATERIALS. If the thickness varies with the depth, the depth varies as the cube root of the distance from the end, and the beam is cubic-parabolic in side elevation, like Fig. 154. SEMI-BEAMS OF UNIFORM STRENGTH UNIFORMLY LOADED. The moment of stress due to the weight when uniformly distributed increases as the square of the distance from the end of the beam, as will be shown in the following case: When the semi-beam is rectangular in section, and its breadth is constant. Suppose the load equally divided and distributed as a great number of weights, W, W", W", &c., Fig. 155; and suppose the beam to be divided into an equal number of corresponding sections at c' , c", c'" , &c. The loads supported by the successive sections are W, 2 W, 3 W 3 &c. ; the distances of the centres of gravity of these loads, from the respective sections, are as W~ W W Figs. 155, 156. Semi-beams uniformly loaded. i, 2, 3, Sz:c. Therefore, the moments of stress at the successive intersec- tions, c', c", c", &c., are as i 2 , 2 2 , 3 2 , &c., or as the square of the distance from the end. But the moments of resistance at the intersections are as the squares of the depths at c',c",c"', &c.; and so the square of the depth is as the square of the distance, or the depth is as the distance from the end. The beam is therefore triangular in elevation. When the semi-beam is rectangtdar in section, and has the depth constant, Fig. 156. As the depth is constant, the breadth must increase as the square of the distance; and it may be, in outline, of the form of two parabolas b <:, b c', back to back, touching each other at their vertices at b ; the axes being perpendicular to the length. When the section of the semi-beam is hollow-rectangular, or is double-flanged; and the breadth is constant. Calculating the strength of the upper and lower members, or flanges, only, and supposing the thickness to be uniform, the moment of resistance is as the depth; the depth is, therefore, as the square of the distance from the end, and is of the form of a parabola, Fig. 157, of which the vertex is at b, and the axis is perpendicular to the length. Calculating the strength of the vertical webs or rib only, the beam would be triangular in side elevation. Combining the webs and the flanges in the calculation, the form of the beam would be intermediate between the parabolic and the triangular. 2d. When the depth is constant. Calculating for the flanges only, the thickness being uniform ; the breadth of the flanges is as the square of the distance from the end, Fig. 158, the same as in Fig. 156. BEAMS SUPPORTED AT BOTH ENDS. 521 If the vertical web or rib, of uniform thickness, be included in the calcula- tion, it does not materially modify the form of the flange. When the section of the semi-beam is circular. The moment of resistance is as the cube of the diameter, and the moment of stress is as the square of the length; therefore the cube of the diameter is as the square of the Figs. 157, 158. Semi-beams uniformly loaded. length, or the diameter is as the cube root of the square of the length, or as the ^ power, or .666 power of the length. The solid is formed by the revolution of a semi-cubic parabola on its axis. When the section of the semi-beam is annular, the thickness being uniform and small compared to the diameter. The moment of resistance of any section is as the square of the diameter. The square of the diameter is, therefore, as the square of the length, or the diameter is as the length, and the semi-beam is triangular or conical in elevation, Fig. 159. When the thickness diminishes with the diameter, the moments of resist- ance of sections are as the cubes of the dia- meters, and the diameter varies as the ^ power, or .666 power of the length, as with a solid circular section, and the form is derived from the revolution of a semi-cubic parabola on its axis. When the section of the semi-beam is elliptical. The moment of resistance of a section is as the cube of the depth, and the form is the same as that of a circular beam. When the section of the semi-beam is hollow- elliptical. The form is the same as that of a beam of annular section. BEAMS OF UNIFORM STRENGTH SUPPORTED AT BOTH ENDS. The forms of beams supported at both ends, and loaded at the middle, are simply doubles of the forms of semi-beams, or such as are fixed at one end and unsupported at the other end. In the beam of rectangular section, for example, A B, Fig. 160, the diagonal lines, ca and cb, from the top at the middle to the supports at each end, are simply doubles of the diagonal b d, in the semi-beam, Fig. 149, and represent the graduated moment of bending stress from the middle, where it is a maximum, to the ends, where it vanishes; and the parabolic curves ca and cb, meeting base to base at the middle cd, form the outline of the rectangular beam of uniform strength, when the breadth is constant. The beam, rectangular in section, of uniform strength, loaded at the middle, Fig. 159. Annular Semi-beam uniformly loaded. 522 THE STRENGTH OF MATERIALS. and having breadth constant, may therefore be moulded according to any of the parabolic forms, Figs. 161, 162, 163, having the axes horizontal, and the vertices at the points of support. Fig. 160. Stress in rectangular beam supported at both ends. Fig. 161. Beam loaded at the middle. When a rectangular beam, with a constant breadth, is loaded uniformly; referring to formula (5), page 508. When the weight is constant, together with the breadth b, and the length /, the square of the depth, d^, varies as Figs. 162, 163. Beams loaded at the middle. the products, m n, of the segments, m and , of the length of the span at any point of the length. Or, the depth varies as the square root of the product of the segments, and the form of the beam, Fig. 164, is a semi- ellipse. It may be a complete ellipse, Fig. 165. Figs. 164, 165. Beams uniformly loaded. For a rectangular beam, with a constant depth, and loaded at the middle, the form of the breadth, Fig. 166, is a double of Fig. 151, page 519; con- sisting of two triangles, in plan, united at their base. Fig. 166. Beam loaded at the middle. Fig. 167. Beam uniformly loaded. When a rectangular beam, with a constant depth, is uniformly loaded; referring to the formula ( 5 ) above noticed, the variables are the breadth b and the product mn, and the breadth varies as the product of the BEAMS SUPPORTED AT BOTH ENDS. 523 segments of the length of the span at any point of the length. The form of the breadth in plan is therefore that of two parabolas having their vertices at the middle and meeting at the points of support, Fig. 167. A hollow-rectangular or double-flanged beam with a constant breadth, and loaded at the middle, consists of the double of Fig. 152, page 519; being two triangles united at their base, at the middle, Figs. 168, 169. In this and the three following cases, the resistance of the flanges only is calculated; and the flanges are supposed to be of uniform thickness. Figs. 168, 169. Beams loaded at the middle. When a hollow-rectangular or double-flanged beam, with a constant breadth? is uniformly loaded; the depth varies as the product of the segments of the beam at any point in the span; and the side of the beam, Fig. 170, is of the form of a parabola, having its axis at the middle. The resistance of the flanges only is here calculated. Fig. 170. Beam uniformly loaded. Fig. 171. Beam loaded at the middle. A hollow-rectangular or a double-flanged beam, with a constant depth, and loaded at the middle, consists of the double of Fig. 153, page 519; the flanges being of uniform thickness, and forming two triangles in plan, joined at their base at the middle of the beam. Their form, Fig. 171, is the same as that of a rectangular beam, with a constant depth, Fig. 166. When a hollow-rectangular or a double-flanged beam, with a constant depth, is uniformly loaded, the form of the flanges in plan is the same as that of a rectangular beam (Fig. 167), consisting of two parabolas, having their vertices at the middle of the beam, Fig. 172. Fig. 172. Beam uniformly loaded. Fig. 173. Beam loaded at the middle. When the section of the beam is circular, and the load is at the middle, the form is the double of Fig. 154, page 519, consisting of the revolutions of two cubic parabolas, base to base, at the middle, Fig. 173. 524 THE STRENGTH OF MATERIALS. When a beam of circular section is uniformly loaded, the cube of the diameter varies as the product of the segments of the length of the span at any point in the length; and the radius varies as the cube root of the product of the segments. When the section of the beam is annular, and the load applied at the middle, the form is that produced by the revolution of two parabolas, base to base, with their vertices at the ends of the beam. The thickness is supposed to be inconsiderable. When the annular beam is uniformly loaded, the square of the diameter varies as the product of the segments of the length of the span at any point in the length; and the radius varies as the square root of the product of the segments. The form of the beam is that produced by the revolution of an ellipse on one of its axes. When the section of the beam is elliptical^ and the load applied at the middle, the form is that of two cubic parabolas joined base to base. When the beam of elliptical section is uniformly loaded, the form is that of an ellipse. When the section of the beam is hollow-elliptical, and the load applied at the middle, and the thickness is uniform, and is small in proportion to the depth, the form of the beam is that of two parabolas, united base to base, having their vertices at the points of support. If the thickness varies with the depth, the forms are cubic parabolas. When the hollow beam of elliptical section is uniformly loaded, the form of the beam is elliptical. BEAMS OF UNIFORM STRENGTH UNDER A CONCENTRATED ROLLING LOAD. Reverting to formula (5), page 508, it signifies that the breaking weight varies inversely as m x n, or the product of the segments of the length of the span, at any point of the length; but if the weight be constant, the moment of stress at any point is as the product m x n, and therefore, also, the moment of resistance of a beam of uniform strength varies as m x n, at all points of its length. Hollow-rectangular, or flanged beam, with a constant breadth, under a con- centrated rolling load. Calculating the resistance of the booms or flanges only, the depth varies as m x n, and is according to the form of a parabola, of which the axis is vertical, when the upper or lower side is horizontal, Figs. 174, 175; or of two parabolas, on the same axis, meeting at the points of support. Under these conditions, the sectional area of the flanges is constant. Figs. 174, 175. Flanged Beams under a Concentrated Rolling Load. Hollow -rectangular or flanged beam, with a constant depth, or parallel flanges, tinder a concentrated rolling load. The breadth varies as m x n, and the flanges, supposed to be of uniform depth, are of the form of two parabolas on the same axis passing through the middle, Fig. 172, page 523. SHEARING STRESS IN BEAMS AND PLATE-GIRDERS. 525 Stress in the curved flange, Figs. 174, 175. Mr. Stoney gives a simple means of finding the stress diagramatically. Let A B, Fig. 176, represent the horizontal stress, which is uniform throughout the length. Draw A c parallel to the tangent of the curve at the given point, and B c perpendicular to A B. Then A c is the maximum longitudinal stress at the given point, and A B and B c are its horizontal and vertical components. It follows that the sec- tion of the curved flange should increase as it approaches the points of support in proportion to A c, or the secant of the angle A. SHEARING STRESS IN BEAMS AND PLATE-GIRDERS. Shearing stress in beams is caused by the vertical pressure of the load. A conception of this stress is easily formed on reflecting that the weight of the beam and its load tends to force it downwards at the abutment, whilst the abutment, by its upward pressure, tends to force upwards the part of the beam which rests on it. The stress thus caused tends to a vertical rupture, or slicing off of the loaded end of the beam, called shearing stress. The same kind of stress acts with various intensity in the portion of the beam between the abutments, and it is the duty of the web to resist the shearing stress. In a beam supported at one end, and loaded at the other end, the vertical shearing stress is equal to the weight, at every point of the length. In the same beam, uniformly loaded, the shearing stress increases uniformly from the end, where it is nothing, to the abutment, where it is equal to the weight. A diagram indicating the gradations of stress would have the form of a triangle. In the same beam, uniformly loaded, and also weighted at the end, the shearing stress is represented by a compound diagram, Fig. 177, in which the triangle a b c represents the graduated shearing stress due to a uniform load, in a beam of the length ab; and the rectangle abde, the uniform shearing stress due to a weight at the end. The whole depth dc at the abutment represents a total shearing stress equal to the sum of the dis- tributed and end loads; and the total stress at intermediate points is repre- sented by the corresponding ordinates. Fig. 177. Shearing Stress. Fig. 178. Shearing Stress. In a beam supported at both ends, and loaded at any point, the shearing stress in each segment is equal to the pressure on its abutment. The THE STRENGTH OF MATERIALS. pressures at a and b, Fig. 178, are as the segments m and n, and the shearing stress in the segments ad and db are equal to W x and W x ; repre- sented by the graduated rectangles on a d and db. When a concentrated load is moved over the beam, the shearing stress in each segment varies as the length of the other segment: from o to W, the weight; represented by the two graduated triangles, ab c, ab d, Fig. 179, in which the verticals a c and b d, at the ends, represent the weight. c Fig. 179. Shearing Stress. Fig. 180. Shearing Stress. When a number of weights are placed irregularly on a beam, the shearing stress of some is neutralized more or less by that of others; and, referring to any given section of the beam, the shearing stress is equal to the differ- ence of the sum of those portions of the weights placed on one side of the section which are conveyed to the abutment on the other side, and the sum of those portions of the weights on the other side which are conveyed to the abutment on the first side. When a beam, supported at both ends, is loaded uniformly, the shearing stress is o at the centre, as in Fig. 180, and increases uniformly towards the abutments, where it is equal to half the weight. When a load of uniform density, as a railway train, traverses a girder, the shearing stress at the front of the train increases as the square of the length of the loaded segment. Suppose that the train advances from b to a, Fig. 1 8 1, covering the whole length, the curve of increasing shearing stress, .b c, is parabolic, having its apex at b. When the girder is wholly covered, the shearing stress follows the triangular gradations shown by dot-lines. Fig. 181. Shearing Stress. Fig. 182. Shearing Stress. When a fixed uniform load and a rolling load are combined, the maximum shearing stress to which the girder is liable at different points of its length is shown by the combined ordinates in Fig. 182. Sectional area of a continuous web calculated from the shearing stress. "When the flanges are parallel," says Mr. Stoney, 1 "the theoretic area of a continuous web may be calculated from the shearing stress by the following rule : Sectional area of web = shearing stress unit-stress in which the unit-stress is the safe unit-stress for shearing. This gives the 1 The Theory of Strains in Girders and Similar Structures. DEFLECTION OF BEAMS AND GIRDERS. 527 minimum thickness, which, however, is often much less than a due regard for durability requires." " When a girder, with parallel flanges and a continuous web, is loaded in the manner described below, where / = the length, and / = the safe unit-strain for shearing force, the theoretic quantity of material in the web would be as follows:" Theoretic KIND OF LOAD. Quantity of Material in a Continuous Proportional numbers. Web. Fixed central load. . . \V W/ 12 2/ Concentrated rolling load.. W 3W/ 18 4/ Uniformly distributed load = W W/ 6 Distributed rolling load "W fwi 7 2 4 / DEFLECTION OF BEAMS AND GIRDERS. Compressive strain is taken as equal to tensile strain, per ton of direct stress on the fibres, and the strain is directly proportional to the stress, within the elastic limits. When a beam is deflected under a load, the lower side is lengthened and the upper side is shortened in propor- tion to the direct stress per unit of section of the fibres. In a beam of uniform strength, the fibres at the surface, on the up- per and lower sides, are, by the de- finition, equally stressed and equally strained throughout the length of the beam; and the form assumed by a straight parallel beam, when deflected under its proper load, is that of a cir- cular arc. Let a b c' c', Fig. 183, be a parallel beam, rectangular in section, having a constant depth, and of uniform strength, when loaded at the middle. Let its lower side assume, by deflec- tion, the form of the circular arc c' Fig. 183. Deflection of a Beam. ad' b, the ends ac', be', which were upright, in their normal position, are now convergent in the positions ad', be" ; and when produced, they meet at the centre of the arc, O, in the vertical radius d'O. The deflection at the centre, dd ' , is the versed sine of the arc. Let, 528 THE STRENGTH OF MATERIALS. R = the radius Od 1 , / = the length of the beam, or the chord, a b, b = the breadth of the beam at the middle, d = the depth of the beam at the middle, c d. a the sectional area of the beam, D = the deflection of the beam, dd' 9 I' = the difference of length of the upper and lower sides, E = the coefficient of elasticity, or the denominator of the fraction of the length, by which the beam is extended or compressed, per ton of direct stress per square inch of section, s' = the direct tensile stress on the extreme outer fibres, in tons per square inch, / = the direct compressive stress on the extreme outer fibres, in tons per square inch, W = the weight in tons. Note. The dimensions are to be all in inches, or all in feet. By the properties of the circle, the square of half the chord is equal to the product of the versed sine, or deflection, by the diameter minus the deflection; or ( - } = D x (2 R - D); or sensibly ( ) - & x 2 R; and (a) Again, by similar triangles, in Fig. 183, Qc" : Qa : : c' c" : ab\ or, in symbols, R : d ::/:/', substantially; whence Substituting this value of R in equation (a), ~ / 2 /' //' Now, /' = ' J + s ' . When the two stresses, s' and s", are equal to each Jt_j other, let them be represented by s. When they are not equal to each other, the deflection is nevertheless the same as if they were so, and that the direct tensile and compressive stress were each equal to the mean of / and /', or to Lf_. Putting iL = S) then /' = -^i-; and, substituting 22 h, 2 S/ 2 this value of /' in equation (<:), D = - -- ; or, o a\Li / x and, s = * 2 ............................ (2) DEFLECTION OF BEAMS OF RECTANGULAR SECTION. 529 DEFLECTION OF BEAMS OF RECTANGULAR SECTION. No. i. Rectangular beam, of constant depth, of uniform strength, loaded at the middle, Fig. 166, page 522. This beam is double-triangular in plan. The value of s, the direct stress on the fibres at the upper and lower surfaces, in terms of the weight, is, by formula 2, page 507, .(3) Equating this value of s and the above value (2), t _W/ whence w _ 4.62 ........................ <'> and if the circumferential arc of deflection be divided by the circumference, equal to 3.1416 d, the quotient is the angular deflection, or D=- (8) 3.i 4 i6<3 9 1482, or .661 4-5 2.289 6 X 12 12 15-5 2.0 2.222 9 x 12 12 17.0 2.5 I-635 12 X 12 12 - 27.5 3-25 1.992 Mr. Edwin Clark tested the transverse strength of red pine of large scantling selected from the scaffolding employed in constructing the Britannia Bridge : two whole balks 1 7 feet long, and a piece cut from the centre of a balk. Mr. C. Graham Smith gives the results of tests for transverse strength of 544 THE STRENGTH OF MATERIALS. pine timber of large scantling at Liverpool. The pieces were selected as average samples from cargoes. 1 The table No. 175 contains the leading results of the experiments of Mr. E. Clark and Mr. C. G. Smith. Table No. 175. TRANSVERSE STRENGTH OF PINE AND FIR. (Reduced and arranged from the experiments of Mr. Edwin Clark, and of Mr. C. Graham Smith. ) (Mr. Edwin Clark.) Breadth and Depth. Span. Application of the Load. Elastic Strength. Elastic Deflec- tion. Breaking Weight. Ultimate Deflec- tion. Ratio of Elastic to Breaking Weight. inches. feet. tons. inches. tons. inches. per cent. American Red Pine I. 12 X 12 15 Centre. 9.0 1. 00 14.82 4.OO 61 (Sp. gr., .509) 2. 12 X 12 15 Do. 9.0 1.25 13.24 3.10 68 (Sp. gr., .543) 3. 6x6 7-5 Do. 2.0 .62 3-29 1.68 61 (Mr. C. G. Smith.) Memel Fir. 4- 13-5 x 13-5 (from the butt) 10.5 Distributed. 38.0 37 61.00 62 5- 13-5 x 13-5 (from the top) 10.5 Do. 38.0 5i 61.00 62 Baltic Fir. 6. 6 x 12 12.25 Centre. 6.0 .66 8.50 1. 11 + 75 7. 6 x 12 12.25 Do. 6.0 .72 10.50 I-93 + 57 Pitch Pine. 8. 6 x 12 12.25 Do. 5.0 .28 10.2 1.31 5 9. 6 x 12 12.25 Do. 8.0 97 10.5 1.31 + 76 10. 14 x 15 10.5 Do. 40.0 49 60.0 1.14 67 ii. 14 x 15 10.5 Do. 35-o 49 59-2 59 Red Pine. 12. 6 X 12 12.25 Do. 5.0 .70 7-5 67 13. 6 x 12 12.25 Do. 5 >o .70 8-5 1.94 + 59 Quebec Yellow Pine. 14- 14 x 15 10.5 Distributed. 35- 39 61.0 58 15. 14 x 15 10.5 Do. 35- 39 61.0 58 16. 14 x 15 10.5 Centre. 30.0 .56 38.3 78 17. 14 x 15 10.5 Do. 34-0 Three beams of oak, mentioned by Mr. Baker, 2 appear to have been broken transversely by the following loads at the middle : 1. i inch square x 2 feet span 212 tons breaking- weight. 2. 8^ inches square x u feet 9 inches span 14.365 3. io 2 / 3 in. wide x 12% in. deep, 24 ft. 6 in. span.. 8.780 1 See Mr. Smith's paper on Pine Timber, read before the students of the Institution of Civil Engineers in 1875, and published in Engineering, vol. xix. page 392. 2 On the Strengths of Beams, Columns, and Arches. 1870. TRANSVERSE STRENGTH OF FIR AND OAK. 545 MM. Chevandier and Wertheim tested the transverse strength of rectan- gular beams of fir and oak from the Vosges. 1 Table No. 176. TRANSVERSE STRENGTH OF FIR AND OAK FROM THE VOSGES. (Reduced from MM. Chevandier and Wertheim's data.) VOSGES TIMBER. Specific Gravity. Breadth and Depth. Span. Breaking Weight at the middle. FIR. inches. inches. feet. pounds. tons. 530 11.4 x 12.8 42.64 14,120, or 6.30 .506 IO.O X 1 1.2 36.08 11,867, or 5.30 .548 8.8 x 9.6 29.52 7,584, or 3.38 .525 6.7 x 7.7 29.52 4,580, or 2.04 .481 3.65 x 4.85 29.52 1,137, or -508 493 9.7 x 2.16 9.91 2,017, or -9 00 479 9.5 x i. ii 9.91 581, or .260 OAK. 1.008 9.2 x 10.9 18.04 17,356, or 7.75 .958 8.6 x 9.3 18.04 15,816, or 7.06 .922 7.6 x 8.6 18.04 11,495, or 5.23 .928 6.3 x 74 18.04 12,155, or 543 .985 54 x 6.3 18.04 4,895, or 2.19 .636 3.26 x 3.20 9.84 1,188, or .530 759 3.07 x 3.16 8.20 1,617, or .722 .685 11.5 x 2.15 18.04 957, or .427 .824 5.64 x 1.66 9.84 825, or .368 .712 9.5 x i.i i 9.84 715, or .319 ELASTIC STRENGTH AND DEFLECTION OF TIMBER. Reverting to the conclusions of MM. Chevandier and Wertheim, on the strength and elasticity of timber, page 538, these experimentalists found that there was no limit of elasticity, properly so called, in wood; though there was a permanent set for every elastic extension. They, nevertheless, adopted empirically, as the limit of elasticity for tensile strength, the point at which a set of Va^oooth of the length is acquired. This is a fanciful distinction, for a set of i in 20,000 parts may be simply the effect of a straightening of the fibres. With this explanation, the following table, No. 177, of the tensile strength of timbers, condensed from their tables, is of some value; although the fractions of extension in the second last column are scarcely consistent with the results of the scanty experiments of others. 1 Morin's Resistance des Materiaux. 546 THE STRENGTH OF MATERIALS. Table No. 177. TENSILE STRENGTH OF TIMBER. (Reduced from the tables of MM. Chevandier and Wertheim.) ist SERIES. Comparative Elastic Strength per ton per square inch, taken when the set is i in 20,000. 2d SERIES. Elastic and Ultimate Strength. Specific Gravity. Elastic Strength, when set is i in 20,000. Ultimate Strength per square inch. Green Wood. Wood Dried. Total per square inch. Extension per ton per square inch in parts of the length. In closed premises. In the air and the sun. Acacia Fir tons. .814 .483 .627 1.046 1.096 .920 1.462 tons. 2.016 I.OI4 I.28I 1.229 .883 .762 tons. 2.O24 1.367 .027 .471 .491 037 .170 .462 .288 .149 957 .724 .942 .717 493 756 .812 .822 .808 .872 559 723 .692 .697 .601 .602 .674 477 tons. 2.O24 1.367 .814 .027 .471 .491 037 .170 723 .728 .712 657 .678 639 V8oi V 7 07 V6 9 o */632 */63 1/621 V 5 85 J /356 x /740 z /739 '/7I2 1/704 1/683 1/648 '/ 3 28 tons. 4.978 2.654 1.899 1.369 2.267 4-I2I 3-594 J-575 4-439 3.912 4.305 2.883 4-572 2.273 1.240 Hornbeam... Birch Beech Oak Do Pine Elm Sycamore. ... Ash . . Alder Aspen Maple Poplar The following are the results of experiments by Mr. Laslett on the elongation of hard woods under tensile stress. The specimens were 2 inches square; length, 36 inches. The elastic limit reached up to, or nearly to, the breaking point : Elastic Breaking ELASTIC EXTENSION. WOOD. per square inch. per square inch. Total. Per ton per square inch. Fraction of length. tons. tons. inch. inch. English oak ... 2.75 ... ... .25 ... ... .091 ... '/396 Dantzic oak .... ... 2.66 ... ... 2.66 ... ... .094 ... J /383 Indian teak ... 1.75 . ... 1.92 ... ... .18 ... ... .108 ... X /333 The following are the chief results of tests by Mr. Kirkaldy of the com- pressive resistance of two balks of fir White Riga and Red Dantzic, about 13 inches square, and 20 feet long, with square ends, in a horizontal position. They were "not very dry." The limits of the elastic strengths are taken at the points where the rate of compression for equal increments of pressure became accelerated. TRANSVERSE STRENGTH OF TIMBER. 547 COMPRESSION. WHITE RIGA. Elastic strength 133-9 tons. Do. per square inch . 792 Breaking strength *479 Do. per square inch -875,, Ratio of elastic to breaking strength 90 per cent. Elastic compression .523 inch. Do. per ton per square inch, in parts of the length r / 3 6 4 Final compression before rupture .642 inch. Set under total elastic stress ... .022 RED DANTZIC. 1 1 1. 6 tons. .627 138-0 775 8 1 per cent. .41 4 inch. '/ 3 64 . .548 inch. .02 MR. BARLOW'S EXPERIMENTS ON TRANSVERSE STRENGTH, I837. 1 Mr. Barlow made a number of experiments to test the transverse deflec- tion and strength of timber of average quality, taken as seasoned, from the stores in Woolwich dockyard. The specimens were 2 inches square, and tested on a span of 7 feet, except a few which were tested on a span of 6 feet. The average ratio of the elastic to the breaking strength is, from the table, 31 per cent.; but Mr. Barlow has not stated the conditions of the elastic limit prescribed by him. Table No. 178. ELASTIC TRANSVERSE STRENGTH OF TIMBER. (Condensed and adapted from Mr. Barlow's experiments.) Specimens 2 inches square, 7 feet span ; loaded at the middle. Name of Timber. Specific Gravity. Elastic Strength. Breaking Weight. Ratio of Elastic to Breaking Strength. Weight. Deflection. Weight. Deflection. Teak 745 579 969 934 .872 .756 993 .760 .696 553 .660 .657 553 753 738 .696 693 703 531 .522 .556 560 577 pounds. 300 150 150 2OO 225 200 150 225 150 125 150 150 150 125 150 125 150 150 125 125 ^o 150 2OO inches. I.I5I .822 .080 590 430 .266 .026 .685 134 755 931 .870 883 1.442 i. 006 1.006 1.885 .812 831 831 .800 pounds. 938 846 450 637 673 5 60 526 772 593 386 622 5ii 420 422 467 436 561 56i 325 370 5oi 5 10 655 inches. 4.32 5-92 5-92 8.10 6.00 4.86 5-73 8.92 5-73 6-93 6.00 5-83 4.66 6.00 6.00 6.00 6.42 6.42 8.58 5-00 5.00 5.00 4.00 per cent. 3 2 17.7 33 3i-4 33-4 36 28.5 29 25 32.4 24 29 36 30 32 29 27 27 38 34 30 30 30 Poon English oak . Do Canadian oak Dantzic oak Adriatic oak Ash Beech Elm Pitch pine . . Red pine New England fir Riga fir Do. (span 6 feet) Mar Forest fir Do. (span 6 feet) Do. Larch Do. (span 6 feet) Do. Do. Norway spar (span 6 feet) 1 On the Strength of Materials; edition of 1845. 548 THE STRENGTH OF MATERIALS. RULES FOR THE STRENGTH AND DEFLECTION OF TIMBER. The results of Mr. Laslett's experiments, tables Nos. 171 and 172, throw some light on the relations of the tensile, compressive, and transverse strength of timber. Employing formula ( 2 ), page 507, namely, s = W/ 1.155 bd 2 ' ' to calculate the direct tensile strength of the specimens, the results may be classified as follows: Calculated Tensile Strength WITHIN 5 PER CENT, of the Experimental Strength. Six HARD WOODS. Transverse Breaking Weight. Tensile Strength, calculated. Tensile Strength, experimental. Compressive Strength, experimental. English Oak (mean) Ibs. 68? tons. 2 3Q2 tons. 2 C4.6 tons. -3 -5 -2 7 Iron Wood 1.271 jy 4.4.28 A, "3 I I 5 208 Chow 07 c 3.^02 3.214. c 621 Iron Bark I 4O7 A OOO -j 740 > ^ 4 DO I Blue Gum .. . 712 2 4.77 2 7OO -3 078 Canadian Ash 638 2.2IQ 2.4.C3 2 ACT. Averages (hard woods) . . . 949 3-I5I 3.l6l 3.615 Calculated Tensile Strength MUCH GREATER than Experimental Strength. EIGHT HARD WOODS. Baltimore oak, African teak, Moulmein teak, greenheart, sabicu, average of American mahoganies, Eucalyptus mahogany, English ash : Averages, 967 3.354 2.120 3.493 NINE SOFT WOODS. Dantzic fir, Riga fir, spruce fir, larch, cedar, red pine, yellow pine, pitch pine, Kauri pine : 683 2.375 1.597 2.486 Calculated Tensile Strength MUCH LESS than Experimental Strength. Six HARD WOODS. French oak, Dantzic oak, American white oak, Eucalyptus Tewart, English elm, Rock elm : 750 2.607 3' 2 95 3-365 Averages of the Twenty Hard Woods preceding: 896 3.069 2.785 3.490 Averages of the Nine Soft Woods preceding : 683 2.375 1.597 2.486 Averages of Twenty-nine Woods, Hard and Soft: 830 2.853 2 -4!6 3.168 STRENGTH OF TIMBER OF LARGE SCANTLING. 549 This analysis shows that for only six out of twenty-nine woods does the formula ( i ) give the experimental tensile strength in terms of the trans- verse strength; and these are all hard woods. For the remainder of the woods, comprising the soft woods, the formula ( i ) shows a tensile strength varying extremely, both by excess and by deficiency, from the experimental strength ; and for all the soft woods the calculated tensile strength is far in excess of the experimental tensile strength. In every instance the experi- mental compressive strength is greater than the experimental tensile strength for the soft woods much greater; and the calculated tensile strength excepting for six hard woods, lies between these values. It is, therefore, to be inferred that the transverse strength is a function of the compressive strength as well as of the tensile strength and that it would be safe to calculate the transverse strength in terms of the mean of the tensile and compressive strengths, supposing that these values can be truly averaged for large scantlings. Calculating likewise the tensile strength of the pieces of soft woods tested for transverse strength by Mr. Fincham, table No. 173, page 543, they are as follows: Calculated Tensile Strength. Soft woods, six specimens, green, top 2.358 tons. Do. do. green, butt 2 -573 Do. do. dry, top 2.095 Do. do. dry, butt 2.024 Do. do very dry 2.403 Average, 2.290 Average from Mr. Laslett's experiments on soft woods, 2.375 showing a fair accord between the two calculated tensile strengths; though Mr. Fincham's 3-inch square specimens give a lower value than Mr. Laslett's 2-inch square specimens. CALCULATED TENSILE STRENGTH OF TIMBER OF LARGE SCANTLING. Selecting the experimental results for the transverse strength of beams of larger scantling, from six inches square upwards, the calculated tensile strengths, by formula ( i ), averaged for each set of specimens, are as follows : Calculated Tensile Strength. Maclure, last 3 pieces, table No. 174, page 543, Memel fir 1.950 tons. Smith, 2 175, 544) Do. 1.334 Smith > 2 175, 544, Baltic 1.400 Chevandier, 4 176, 545, Vosges 1.483 Average for Fir, 1.542 E. Clark, 3 pieces, table No. 175, page 544, Red pine 1.240 tons. Smith, 2 175, 544, Do 1.163 Average for Red Pine, 1.202 Smith, 4 pieces, table No. 175, page 544, Quebec yellow pine 1.200 tons. .Smith, 2 175, 544, Pitch pine 1.834 Baker, 2 544, English oak 1.416 Chevandier, 5 176, 545, Vosges oak 1.943 550 THE STRENGTH OF MATERIALS. FORMULAS FOR THE TRANSVERSE STRENGTH OF TIMBER OF LARGE SCANTLING. Adopting the foregoing data as the proper values of J, the tensile strength in tons per square inch, in the general formula ( i ), page 507, as applied to find the breaking weight of timber beams of considerable scantling, the numerical constant, 1.155 s > f r eacn is obtained: Ultimate Transverse Strength of Timber of Large Scantling, loaded at the middle. Fir ............................ W .................. ...... Red pine ..................... W^ 1 ' 39 ^ ........................ (3) Quebec yellow pine ........ w Pitch pine ................... W= 2 ' 12 ^ 2 ........................ (5) English oak w=i^ - (6) T- i i -ITT 2.2A.bd 2 i French oak W = (?) W = the breaking weight, in tons; b the breadth, d the depth, and / the span, all in inches. For other timbers, in the absence of direct experimental data, formulas may be deduced for transverse strength by substituting for s, in the general formula, the mean of the tensile and crushing resistances of a given wood, reduced in the proportion by which the strength of large scantlings is less than that of small scantlings ; which may be taken at two-thirds. Meantime, Mr. Laslett's data, table No. 171, may be utilized by fixing the value of the coefficient, 1.155 s t directly from the transverse breaking weights of the timbers, taken at two-thirds of the observed values. Invert- ing the general formula ( i ), page 507, W/ By means of this formula, the values of the numerical coefficients to be substituted for the coefficient in any of the formulas ( 2 ) to ( 7 ), for the ultimate transverse strength of other timbers, are found to be as follows in table No. 179: FORMULAS FOR THE TRANSVERSE DEFLECTION OF TIMBER BEAMS OF UNIFORM RECTANGULAR SECTION. The deflection of beams of small scantling may aid as a basis for calcu- lating the deflection of large beams, by means of the general formula ( 4 ), page 529, in which the value of E, the coefficient of elasticity, may be calculated from the various data already given for such timber by means of the inverted general formula ( 8 ), preceding. DEFLECTION OF TIMBER BEAMS. 551 Table No. 179. VALUES OF 1.155 J > NUMERICAL COEFFICIENT FOR THE TRANSVERSE STRENGTH OF TIMBER BEAMS; TO BE USED IN ANY OF THE FORMULAS ( 2 ) TO ( 7 ), page 550. (From Mr. Laslett's data.) Description of Timber. Values of 1.155*. Description of Timber. Values of 1.155 * Oak, English (average) 1.63 Iron Bark, Australia 3.87 French 2.4.1 Blue Gum, do >/ I. q6 Do 2 28 English fish 2 -37 Tuscan 2.08 Canadian ash 2. ^O Sardinian 2.08 Beech (estimated) 2.4.O Dantzic I.3O English elm I 08 Spanish . l.CA Rock elm Canada . 2. C7 American White 2.21 Hornbeam, England 2.C3 Baltimore . I QQ Dantzic fir 2 4.T African (or teak) 3.OC Riga fir . . 1.65 Moulmein teak 2. SI Spruce fir y Transverse Deflection of Rectangular Timber Beams of uniform section : W/ 3 4.62 (9) D = the deflection, / the span, b the breadth, and d the depth, all in inches; W the load at the middle in tons, E the coefficient of elasticity. The values of E and 4.62 E are given in the annexed table No. iSo. 1 SHEARING STRENGTH OF TIMBER. Oak treenails, firmly held, of from i inch to i^ inches -in diameter, were found by Mr. Parsons to have a shearing strength of about 2 tons per square inch of section. For the development of so much resistance, Pro- fessor Rankine deduces that the planks connected by the treenails should have a thickness of at least three times their diameter. Treenails of i^ inches, in 3-inch planks, bore only 1.43 tons per square inch; and in 6-inch planks, 1.73 tons. 1 It may here be stated, that, whilst the value of E possesses importance as an element in a scientific theory of deflection, it is not necessary, for the purposes of calculation for the deflection of beams, that the value of E should be exactly ascertained, since, in its employment in the formula for deflection, it is merged in the compound coefficient 4.62 E, the value of which can be determined, independently, from practical data. 552 THE STRENGTH OF MATERIALS. Table No. 180. VALUES OF E AND 4.62 E IN FORMULA ( 9 ), PAGE 551, FOR THE TRANSVERSE DEFLECTION OF TIMBER BEAMS. Description of Timber. By Laslett's Data. By Barlow's Data. Various Data. Averages. E. 4.62 E. E. 4.62 E. E. 4.62 E. E. 4.62 E. English oak 348 576 234 338 450 458 598 390 494 916 956 410 916 726 460 780 692 948 698 542 320 502 538 680 7H 578 3f 560 526 704 414 672 1611 2656 1080 1555 2080 2114 2761 1804 2276 4228 4412 1888 4228 336o 2118 3608 3196 4378 2559 2506 1476 450 746 376 934 654 2072 3445 1735 43" 3018 2939 2418 1227 2072 1615 3286 2187 2602 400 57 6 234 338 450 458 59 8 746 376 390 7'4 654 9l6 956 4 IO 916 726 460 7 80 692 948 698 5 88 320 524 266 5 02 776 538 616 7H 578 344 456 | S 28 622 J45> 534 624 1848 2656 1080 1555 2080 2114 2761 3445 1735 1804 3293 3018 4228 4412 1888 4228 336o 2118 3608 3196 4378 2559 2722 1476 2418 1227 2319 3630 2490 2920 3300 2669 1583 2100 2434 2968 2084 2465 2884 French do. Tuscan do Sardinian do Dantzic do American white do. Baltimore do Canadian do Adriatic do. African do. (or teak) Moulmein teak Poon Iron wood Chow Greenheart Sabica . Spanish mahogany.. Honduras do Mexican do Tewart Iron Bark Blue Gum . English ash 636 524 266 450 350 712 474 564 Canadian do. . Beech Elm Rock-elm 2319 2490 3H7 33o 2669 I5H 2585 2430 3257 1915 2920 i& 776 F. 756 S. 57 8 F^358 E.G. 460 S. 474 F. 464 8.690 F. 410 S-530 F. 504 F. 616 Memel fir Dantzic do . 3630 3491 2669 Spruce do. New England do. . . . Scotch do 1652 2124 2188 2141 346l 1891 2445 2328 2847 Larch Red pine Pitch do Yellow do Norway spar Kauri pine 1 E. C., E, Clark; S., G. G. Smith; F., Fincham. STRENGTH OF CAST IRON. 553 STRENGTH OF CAST IRON. TENSILE STRENGTH AND COMPRESSIVE STRENGTH. Mr. Hodgkinson's experiments and investigations form the basis of most of what is known on the strength of cast iron. To ascertain the relative strength of cast iron according to the form of the cross section, he tested specimens of cruciform, rectangular, and circular sections the first melting of the pigs. The area for each section, Figs. f *-- "V^S * INS 184, 185, and 186, was f -~ 23 ->, intended to be four square inches, but the castings were accurately measured, and the exact area of each was ascer- tained. The following were the average breaking weights or absolute tensile strengths per square inch for the different sections : Figs ' l84 ' l85 ' l86 Trial Sectlons for Cast Iron - Bowling iron, No. 2 Brymbo iron, No. 3 Blaenavon iron, No. 2 ... Section. Tensile strength per square inch. ( Cruciform 6.784 tons. (Rectangular 6.267 j Cruciform 6.661 ( Rectangular 6. 1 1 5 I Cruciform 6.253 I Circular 6.614 Total average 6.450 ,, From these results it appears, that, taking the strength of the cruciform section as i, the strengths of the other sections were relatively as follows: Cruciform. Bowling iron, No. 2, as i to .924 rectangular. Brymbo iron, No. 3, as i to .918 rectangular. Blaenavon iron, No. 2, ...as i to 1.054 circular. The section of the specimens tested by Mr. Hodgkinson for tensile strength was cruciform, and the specimens were V /111 /\ of the form Fig. 187; having a uniform section \ (III fffrs A for one foot of length. For compression, they were cylindrical, ^ inch in diameter, and were made to two heights, respectively equal to i diameter and 2 diameters, and they were placed for testing within a cylinder under a loaded plug, as shown in Fig. 188. He tested the strength of 16 denominations of cast iron, 51 specimens of which were tested for tension and 8 1 for compression. The results of the tests are condensed from the Commissioner's Figs. 187, 188. Specimens for Report on the Application of Iron to Railway Testing Tensile Strength and Compressive Strength. Structures, in table No. 181. 554 THE STRENGTH OF MATERIALS. Table No. 181. TENSILE AND COMPRESSIVE STRENGTHS OF CAST IRONS AND STIRLING'S IRON. (Mr. Hodgkinson.) Iron. Mean Specific Gravity. Mean Tensile Strength per Square Inch. Mean Compressive Strength per Square Inch. Ratio of Tensile to Compressive Strength. Height of Speci- men. Strength. Lowmoor, No. I ... Lowmoor, No. 2 ... Clyde, No. I 7.074 7.043 7.051 7.093 7.IOI 7.042 7.H3 7.051 7.025 7.024 7.071 7.037 6.989 7.II9 7.034 7.013 tons. 5.667 6.901 7.198 7-949 10.477 6.222 7466 6.380 6.131 6.820 6.440 6.923 6.032 6.478 6.228 5-959 inch. X 1/2 X 1/2 X I# X i# X 1% X 1/2 X I# X I'A X 1% X 1/2 X 1/2 X 1/2 *# X 1% X I* X 1/2 tons. 28.809 25.198 44430 41.219 41.459 39.6l6 49.103 45-549 47.855 46.821 40.562 35.964 52.502 45-7I7 30.606 30-594 32.229 33-921 44.723 45.460 33-390 33784 33.988 34.356 33.987 33.028 44.610 42.660 37.281 35.H5 34-43 33-646 : 5.084 : 4.446 : 6.438 : 5-973 ( : 5-759 : 5'53 : 6.177 I : 5.729 : 4-568 : 4.469 : 6.519 : 5.780 : 7.032 ) : 6.123 } : 4-797 : 4-795 : 5.256 : 5-532 : 6.557 : 6.665 .-5.186 : 5.246 : 4.909 : 4-963 : 5.635 I : 5.476 } : 6.886 : 6.585 : 5.985 : 5-638 : 5778 : 5.646 \ mean. i : 4.765 i : 6.205 i : 5.631 1 5-953 i 14.518 i : 6.149 i : 6.577 i : 4-796 i : 5-394 i : 6.611 i : 5.216 i : 4.936 i : 5-555 i : 6.735 i : 5.811 i : 5.712 Clyde No 2 ... . Clyde, No. 3 Blaenavon, No. i... Blaenavon, No. 2, ) ist sample ( Blaenavon, No. 2, ) 2d sample . ( Calder No i Coltness, No. 3 Brymbo, No. I Brymbo, No. 3 Bowling No 2 . Ystalifera anthra- cite, No 2 Yniscedwyn an- thracite, No. i.. Yniscedwyn an- thracite, No. 2.. Averages of cast ) irons ... . ) 7.055 6.830 38.525 i : 5.641 Stirling's iron, 2d aualitv 7.165 7.108 11.502 10.474 X 1/2 X 1/2 55-952 53.329 70.827 57.980 i : 4.865 i : 4-637 i : 6.762 i : 5-536 i 14.751 I : 6.149 Stirling's iron, 3d Quality Average of Stir- ) ling's iron ( 7.136 10.988 59.522 I : 5.417 STRENGTH AS AFFECTED BY THE MASS OF METAL. 555 It appears from the table that the tensile strength of cast iron varied from 5.667 to 10.477 tons, and averaged 6.830 tons per square inch. That the compressive strength varied from 25.198 tons to 52.502 tons, averaging 38.525 tons per square inch. That the compressive strength was from 4.518 to 6.735 times the tensile strength; average ratio of tensile to compressive strength, i to 5.641. That the specific gravity varied from 6.989 to 7.113, and averaged 7.055, and that, generally, the strength increased with the specific gravity, though there were many exceptions to such relation. That the tensile strength of Stirling's metal (a mixture of cast and wrought iron) averaged 10.988 tons per square inch, and the compressive strength 59.522 tons per square inch; ratio, i to 5.417. The average compressive resistances of the pieces one and two diameters high were respectively as 100 to 95.6. Dr. Anderson tested, at Woolwich Arsenal, 850 specimens of cast iron. The ultimate tensile strength of selected specimens varied from 4.90 tons to 14.5 tons per square inch, averaging 9.45 tons, and of all the 850 speci- mens, from 4.20 tons to 15.30 tons. He found that the average tensile strength of ordinary irons of commerce was 6 tons per square inch. It is probable that the higher strengths were those of bars of 20! or 3d meltings. STRENGTH AS AFFECTED BY THE MASS OF METAL. Mr. Hodgkinson, comparing the tensile strength of bars of cast iron, i inch, 2 inches, and 3 inches square, found that the relative strengths were approximately as 100, 80, 77. Captain James found that the tensile strengths of i-inch, 2-inch, and 3-inch bars were as 100, 66, 60; and that the tensile strength of 24 -inch bars cut out of 2 -inch and 3-inch bars had only half the strength of the bar cast i inch square. The ascertained inferiority in strength of massive castings as compared with thinner castings is attributable to the greater proportion of surface or "skin" on the thinner castings. It is known that the skin is harder and stronger than the interior of a casting. Besides, the interior of massive castings becomes more spongy in texture as the thickness is increased. STRENGTH OF CAST IRON AS AFFECTED BY COLD BLAST AND HOT BLAST. Mr. Hodgkinson tested several cast irons, made by cold blast and hot blast, with the following results, table No. 182; showing an average tensile strength, of all irons, 7.36 tons per square inch, and average compressive strength, 47.0 tons; ratio, i to 6.11. At the same time, it is shown that the hot-blast irons had 9.17 per cent, less tensile strength, but that they had 3.39 per cent, more compressive strength, than the cold-blast irons. Mr. Robert Stephenson concluded from experiments of more recent date, conducted by him, that the average strength of hot-blast iron was not much less than that of cold-blast iron; but that cold-blast irons, or mixtures of cold-blast irons, were more certain and regular, and that mixtures of cold- blast and hot-blast irons were better than either separately mixed. 556 THE STRENGTH OF MATERIALS. Table No. 182. STRENGTH OF COLD-BLAST AND HOT-BLAST IRON. (Mr. Hodgkinson.) Description of Iron. Tensile Strength per Square Inch. Compressive Strength per Square Inch. Cold Blast. Hot Blast. Cold Blast. Hot Blast. Carron iron No 2 tons. 7-45 6-43 7 .80 8.42 6.49 7.32 tons. 6.03 7.84 9.68 6.00 7-45 tons. 47-5 S'-S 41.65 36.50 tons. 48.50 59-50 64.9 38.50 36.90 Carron iron No 3 Devon iron No. 3 Buffery iron No i Coed-Talon iron, No. 3 Lowmoor iron, No. 3 Total averages .. . .... 7.40 44.3 49.70 Comparative averages of cold ) and hot blast f 7-. 7-52 36 6.83 47 44-3 .O 45.80 Sir William Fairbairn, writing in 1870, maintained that the quality of iron had been greatly improved since the introduction of the hot blast, and that nothing, at the time of writing, was said of the difference between hot-blast and cold-blast irons. Dr. Siemens, on the same occasion, stated that the ironmasters had seen the advantage of raising the temperature of the blast, and that, in using the Siemens-Cowper regenerative hot-blast stoves, the tem- perature had been raised as high as 1400 F., without any deterioration of the quality of the metal having been observed. 1 STRENGTH OF CAST IRON INCREASED BY REMELTING. The strength, as well as the density, of cast iron are increased by repeated remeltings. The increase of strength and density appears to be the conse- quence of the gradual abstraction of the constituent carbon of the iron, and the approximation of the metal in composition to wrought iron. Mr. Bramwell proved the increase of the tensile strength of Acadian cold-blast iron by remelting it. The tensile strengths of successive samples were as follows : ACADIAN IRON. Tensile strength per square inch. Sample bars. tons. i st samples 7.5 2d do. after 2 hours longer fusion 8.3 3d do. after i^ 10.8 4th do. remelted, with fresh pigs n.o 5th do. after 4 hours longer fusion 18.5 Maximum of 5th samples 19.6 1 Proceedings of the Institution of Civil Engineers, "Regenerative Hot Blast Stoves," by Mr. E. A. Cowper, vol. xxx. p. 321. STRENGTH OF CAST IRON INCREASED BY REMELTING. $$? Showing that the tensile strength was increased 150 per cent, by 8 hours of continued fusion, and by remelting. The compressive strength averaged 3^ times the tensile strength. 1 Sir William Fairbairn tested for compressive strength, samples of Eglinton No. 3 hot-blast iron of from i to 18 meltings the resistance was doubled by 1 8 meltings; but the maximum resistance was attained at the i4th melting, and amounted to 2.2 times the first resistance. The following are the results of these tests : EGLINTON No. 3 HOT-BLAST IRON. Melting. Compressive strength. tons. 44.0 43- 6 41.1 40.7 41.1 41.1 40.9 41.1 Melting. IO. II. 12. 14. 15- 16. 18. Compressive strength. tons. 57-7 .. 69.8 73- 1 .. 66.0 (defective) . . O C.O .. 76.7 .. 70.5 . 88.0 Remelting, or continued fusion, of cast iron is practised in the United States. The pig iron generally used has, in the state of pig, a tensile strength of from 5 to 6^ tons per square inch. When melted, it is kept for some time in a state of fusion, and the first castings have a tensile strength of about 9 tons per square inch. For guns, the metal is melted three or four times in an air-furnace, and at each melting is retained in fusion for from one to three hours before being poured; and, according to the experiments of Major Wade, the strength of iron so treated was succes- sively increased. The following are some of the results obtained by Major Wade: AMERICAN IRON. TENSILE STRENGTH. tons per square inch. Pigs-.... 5 to 6^ ist melting 9.3 2 2d do ii. 06 3d do 11.96 4th do 12.45 Maximum strength observed 20.5 Samples from 100 gun-heads 14.9 Proof bars (in other trials) 16.23 38 samples from a Rodman gun 15.3 to 19.8 Do. average 16.88 A lot of pig iron, in the crude state 5.66 27 guns cast from this pig iron, 3d melting 15. 75 The specific gravity of the metal was increased by successive meltings and protracted fusion, from 6.90, in some instances, to 7.40. The compressive strength of the irons tested by Major Wade, varied from 37.7 tons to 78 tons per square inch. The specimens were y 2 inch 1 The above particulars are reduced from the Proceedings of the Institution of Civil Engineers, vol. xxii. page 559. 558 THE STRENGTH OF MATERIALS. in diameter, and ij^ inches high. Some of the mean results were as follows : 2d Melting 3d Melting> No. i cast iron, 44.5 tons 62.5 tons. Mixtures of Nos. i, 2, 3, 69.4 74.6 It may be inferred that the ratio of tensile to compressive strength of the American irons above tested, was about i to 4. ELASTIC STRENGTH OF CAST IRON. Mr. Hodgkinson made experiments with round cast-iron bars of one square inch sectional area, and 50 feet long, suspended in a lofty building, to find the extension and permanent sets. These experiments were made C 2 3 4. 5 G G* 7 TONS LocuL per square inch. Fig. 189. Diagram to show Rate of Extension and Set of ic-feet bars of cast iron. Table No. 183. with the object of insuring exceptional accuracy of results. But, by much the greater number of Mr. Hodgkinson's experiments, both for extension and for compression, were made with bars limited to 10 feet in length, i inch square. The results of the observations on the extension and com- ELASTIC STRENGTH OF CAST IRON. 559 pression of 10 feet bars, are plotted in Figs. 189 and 190, in which the base- line A B represents the loads, and the verticals in light lines are the observed extensions, compressions, and sets, for the given loads. The curves A c and A D are traced through the ends of these verticals, and the vertical black lines show the extensions and sets, for integral tons of load. A 6 I 2 3 4 5 6 7 8 9 10 It 12 13 14 15 16 (7TONS Loads per square inch'. Fig. 190. Diagram to show Rate of Compression and Set of xo-feet bars of cast iron. Table No. 184. The following table, No. 183, is constructed from the diagram of exten- sion and set, Fig. 189, and it shows the mean extension and set for given stresses in integral tons up to 6^ tons on a i-inch square bar of average quality 10 feet long. The table No. 184 is likewise constructed from the diagram of compres- sion and set, Fig. 190, and it shows the mean compression and set for given stresses in integral tons up to 17 tons on a i-inch square bar of average quality 10 feet long. 560 THE STRENGTH OF MATERIALS. Table No. 183. MEAN REDUCED EXTENSION AND SET FOR GIVEN STRESSES, OF A BAR OF CAST IRON OF AVERAGE QUALITY, i INCH SQUARE, 10 FEET LONG. Deduced from diagram, Fig. 189, page 558. EXTENSION. SET. Total Extension. STRESS. Increment of Exten- Fraction of Length. Increment of Set for Total Set in Ratio of Total Set to sion for each Ton each Ton. In Inches. x ension. Total. Per Ton per Square Inch tons. inches. inches. ratio. length = i. inches. inches. ratio. ratio. I .0196 .0196 I I /6l22 .000163 .00058 .00058 I to 34 2 .O222 .0418 2.n V28 7 i .000174 .00139 .00197 3-4 to 21 3 .0238 .0656 3-35 Vi82 9 .000183 .00228 .00425 7-3 to 15.4 4 .0276 .0932 4-75 1/1287 .000194 .00325 .0075 13 to 12.4 5 .0300 .1232 6.29 V974 .OOO2O5 .00463 .01213 21 to 10.2 6 .0378 .l6lO 8.21 */745 .000223 .00720 01933 33 to 8.3 6.5 .0210 .1820 9.29 J /6S9 .000257 .00407 .0234 40 to 7.8 Table No. 184. MEAN REDUCED COMPRESSION AND SET FOR GIVEN STRESSES, OF A BAR OF CAST IRON OF AVERAGE QUALITY, i INCH SQUARE, 10 FEET LONG. Deduced from diagram, Fig. 190, page 559. COMPRESSION. SET. Total Compression. STRESS. Increment of Com- pression for each In Inches. Fraction of Length. Increment of Set for each Ton. Total Set. Ratio of Total Set to Com- pression. | Ion. Total. Per Ton per Square Inch tons. inches. inches. ratio. fraction. length = i. inches. inches. ratio. ratio. I .02 .02 I 1/6*00 .000167 .000567 .000567 I to 35-3 2 ,O2l6 .0416 2.08 1/2884 .000173 .00198 .00255 4-5 to 17.6 3 .O224 .064 3-2 X A*75 .000178 .OO2I2 .00467 8.2 to 13.7 4 .022 .086 4-3 Vi395 .OOOI79 .00230 .00697 12.2 to 12.3 5 .O222 .1082 /nog .000180 .OO26l .00958 17 to 11.3 6 .0228 .1310 55 J /gi6 .OOOl82 .OO3O2 .0126 22 to 10.4 7 .0234 1544 7-72 J /777 .000184 .00315 01575 28 to 9.8 8 .0236 .178 8.9 1/674 .OOOl85 00355 .0193 34 to 9.2 9 .0238 .2018 10.09 J /595 .000187 .0040 0233 41 to 8. 7 10 .023 .2248 11.24 z /534 .000188 .0043 .0276 48 to 8. i ii .0236 .2484 12.42 .000188 0043 .0319 56 to 7.8 12 .026 .2744 13.72 V437 .000191 .0056 0375 66 to 7-3 13 0258 .3002 15.01 1/400 .000193 .0077 .0442 78 to 6.8 14 .0306 3308 16.54 .000197 .0087 .0529 93 to 6.3 15 .0222 353 17-65 T /34<> .000196 .008I .061 1 08 to 5.8 16 .04 393 19-65 T /35 .000205 .0193 .0803 142 to 4- 9 17 033 .426 21.3 1/282 .000209 .0060 .0863 152 to 4-9 TRANSVERSE STRENGTH OF CAST IRON. 561 It is clear from the diagrams, Figs. 189 and 190, and the tables Nos. 183 and 184, that both the extension and the compression of cast iron, with the respective sets, begin at the beginning of the loading; and, strictly inter- preted according to the definition of elasticity, the evidence is to the effect that there is no such thing as perfect elasticity in ordinary cast iron. The progression of extension, compression, and set, moreover, is regular, and it is gradually accelerated whilst the stress is increased in arithmetical propor- tion. There is no sudden change in the rate of progression anywhere, no "yielding point" for cast iron, and no indication of a permanent elastic limit before rupture takes place. In this respect cast iron radically differs from wrought iron and steel, for in the behaviour of these metals the "yielding point" is a clearly defined characteristic. SHEARING STRENGTH OF CAST IRON. Professor Rankine states that the shearing strength of cast iron is 12.37 tons per square inch. But Mr. Stoney found by experiment that it was from 8 to 9 tons per square inch. Both of these data may be correct : it has been seen that cast iron varies very much in tensile strength, according to the character of the specimens operated upon. It is very probable that the shearing resistance of cast iron is, by reason of its comparative incompressibility, equal to its direct tensile resistance. MALLEABLE CAST IRON. The tensile strength of annealed malleable cast iron is guaranteed by manufacturers to 25 tons per square inch. It is capable of supporting 10 tons per square inch, tensile strength, without permanent distortion. TRANSVERSE STRENGTH OF CAST IRON. Cast- Iron Bars of Rectangular Section. Mr. Barlow found, by experi- ment, that for i -inch square bars of cast iron, the breaking weight in tons, applied at the middle, was expressed by the formula, in which b, the breadth, d the depth, and / the span, are in inches. Mr. Robert Stephenson arrived, by experiment, at exactly the same coefficient. If the coefficient be taken as only 12, the breaking weight of a i-inch square bar, at 12 inches span, is, by the formula, just i ton; and if the span, /, be expressed in feet, the formula ( i ), with a coefficient of 12, is re- solved into the form, If cast-iron bars were homogeneous, and of uniform density for all dimensions, the formulas i and 2 would give the breaking strengths correctly for all sizes of bars ; but so great is the diminution of strength in thicker castings, due to the comparatively open or spongy structure of cast iron in thick masses, that 3-inch square bars, relatively, have scarcely two-thirds of the transverse strength of i-inch bars, and the proper coefficient for 36 5 62 THE STRENGTH OF MATERIALS. formula (i), is only 8.6, as applicable to 3-inch bars. It is obvious that no constant coefficient can be employed, even in iron of the same denomina- tion, for the transverse strength of cast-iron bars, when the thickness is various. To apply the general formula (i), page 507, for the transverse strength of rectangular bars of cast iron, in terms of the tensile strength; namely, b d 2 s , x W=i.i55 -j-\ ( 3 ) in which <, d, and / are in inches, s, the tensile strength, in tons per square inch, and W, the breaking weight, in tons at the middle; the mean tensile strength of i-inch square bars of ten different irons was found by Mr. Hodgkinson to be 16,502 Ibs., or 7.36 tons, and their transverse strength, at 54 inches of span, was 464 Ibs. By the formula (3), taking the forces in pounds, the transverse strength is, 1.155* i x 16,502 = 353lb& . 54 or seven-ninths of the actual strength. The excess of actual strength, 3 1 per cent, results from the distribution of the stronger portion of the section of the bar at the outside, the skin, in fact, where its moment, or power of resistance, is, by reason of the leverage of the resistance, much more effective than that of the interior and weaker portions. By such tubular distribution, a greater total strength transversely is exerted than would have been exerted if the material had been of uniform tensile strength throughout the section, as was assumed in the construction of the formula. In bars of greater section, the influence of the skin on the strength is comparatively less. Accordingly, in the following examples selected for comparison, from data supplied by Mr. Edwin Clark, the excess of strength diminishes generally as the scantling of the specimen is increased. A tensile strength of 7 tons per square inch is assumed in the calculation of the strength, by the formula (3) BARS. TRANSVERSE STRENGTH. Width. Depth. Span. Calculated. Actual. Excess, Actual. (i.) i inch X i inch x 4.5 feet, .158 ton, .2^2 ton, 60 per cent. (2-) i X 3 x 1 8 .337 .429 27 (3-) 3 JJ x i x 2.25 I -376 53 (4-) 2 x 2 x 13-5 399 -475 19 (5-) 2 55 X 3 x 9 1-347 n I .800 35 (6.) 3 J? x 2 X 4-5 1.800 ?> 2 .410 34 (7-) 3 x 3 x 13-5 1-347 I .436 6.5 The i -inch square bar has 60 per cent, excess of strength. The 2d bar has only i inch of bottom skin for three times the depth of the ist, and so has only 27 per cent, excess. The 3d bar, of the same section as the 2d, was tried on its side, and has three times as much bottom skin as the 2d; and so has nearly double the excess of the 2d, but not so much as that of the ist, which has comparatively more side skin. The 4th bar, 2 inches square, has less skin in proportion to its bulk, and has only 19 per cent, excess of strength; whilst the 5th bar, of the same thickness, but deeper, has a greater excess, for its bottom skin has more leverage. The TRANSVERSE STRENGTH OF CAST IRON. 563 6th bar has the same section as the 5th, but is tested on its side, and has the same excess of strength; whilst the 7th bar, 3 inches square, has only 6^2 per cent, excess of strength above that calculated for it by formula (3). The strength of the yth bar is, on the contrary, 34 per cent, less than what would be calculated for it by the formula (i). Diminishing differences with increasing sections, are also exemplified by experimental observations of Mr. Hodgkinson, selected from one of his tables, 1 with bars of Carron No. 2, averaged for hot and cold blast, of three sizes, comprising two or more bars of each size. The sizes are here given in round numbers : BARS. TRANSVERSE STRENGTH. Width. Depth. Span. Calculated. Actual. Excess. (i.) I inch x i inch x 54 inches, .158 ton, .219 ton, 40 per cent. (2.) i x 3 x 54 1.381 1.736 26 (3.) i x 4 x 54 3.794 4.600 21 For the calculation of the transverse strength of cast-iron bars of rectan- gular sections, and of the larger scantlings, even of the commonest quality, formula (3) may, it appears, be safely employed, allowing a wide margin, with a minimum factor of 7 tons per square inch tensile strength. This gives a numerical coefficient of (1.155 x 7 = ) 8.08; say, 8, in formula ( 3 ). Transverse Strength of Rectangular Bars of ordinary Cast Iron, of the first melting. Tensile strength, 7 tons per square inch: Loaded at the middle, ...... W = ; ...................... ( 4 ) Loaded at one end, ......... W = -; ...................... ( 5 ) Round Cast-Iron Bars. The strength of round cast-iron bars, taking a tensile strength of 7 tons, is found from the general formula (15), page 510; in which .7854 x ^ = .7854 x 7 = 5.50. Transverse Strength of Round Bars of ordinary Cast Iron: Loaded at the middle, ...... W= 5 ' 5 ^ 3 ; .................... ( 6 ) Loaded at one end, ......... W^ 1 ' 375 ^; .................. ( 7 ) in which b, d, and / are in inches, and W is the breaking weight in tons. With tensile strengths greater than 7 tons, the constants to be used in these formulas are as follows: Tensile strength per Constant in Constant in Constant in Constant in square inch. formula (4). formula (5). formula (6). formula (7). 8 tons, 9-2 2.3 6-3 1.6 9 10.4 2.6 7-i 1.8 TO n-5 2.9 7-9 2.O ii 12.7 3-2 8.6 2.2 12 13.8 3-4 9-4 2.4 On the Strength and Properties of Cast Iron, 1846, pp. 398, 399. 564 THE STRENGTH OF MATERIALS. Test Bars. It is usual, in specifications for cast-iron work, to require that sample bars of cast iron, say i inch thick, 2 inches deep, and on bearings 36 inches apart, shall support a given weight applied at the centre. Ten years ago, a weight of 25 cwt. was considered sufficient as a test load; but the load has since been increased to 28 cwt. and 30 cwt. This is not very severe, for, by formula (i), based on the strength of i-inch square bars, the modern test-bar, of ordinary iron, should support 27.2 cwt. before breaking. TRANSVERSE DEFLECTION AND ELASTIC STRENGTH OF CAST IRON. Cast-Iron Rectangular Bars. Mr. Hodgkinson gives particulars of the deflection of rectangular bars under various loads. The deflections under medium loads are here averaged as follows; and the coefficients of elas- ticity E, are calculated by means of the formula ( 8 ), page 530: SPAN. LOAD. DEFLECTION. E. inches. tons. inches. Average of 8 bars, i inch square, 54 .100 .561 6076 i bar, i inch x 3 inches deep, 54 1.066 .216 6230 i bar, i inch x 5 inches deep, 54 3-348 .153 5966 Average value of coefficient of elasticity 6090 This value of E agrees with that which was found for direct tensile strength, table No. 183, page 560, and the resulting numerical coefficient, 4.62 E = 4.62 x 6090 = 28,136, say 28,000; which may be substituted for 4.62 E in the formula ( 8 ), page 530, to give the deflection of cast-iron bars, within ordinary elastic limits, thus : Deflection of Cast-Iron Rectangular Bars of Uniform Section. w 73 Loaded at the middle, D = .... (8) 28,000^ Loaded at one end, ... D = (9) 875 b a 3 D = the deflection, b - the breadth, d= the depth, /= the span, all in inches; W = the load in tons. Cast-Iron Round Bars. For round bars of uniform diameter, substitute the above-found average value of E, in the general formula ( 26 ), page 533. Then, 3.1416 x E = 3.1416 x 6090= 19,132, say 19,000. Deflection of Cast-Iron Round Bars. Loaded at the middle, D- W /3 .. ( 10 ) 19,000 d* Loaded at one end,... D = - ( n ) 594 ^ TORSIONAL STRENGTH OF CAST IRON. 565 TORSIONAL STRENGTH OF CAST IRON. The only direct experiments recorded, worth notice, on the torsional resistance of cast-iron, are those of Mr. Dunlop at Glasgow, in iSig. 1 They were made to ascertain the torsional strength of shafts as usually cast in Glasgow at the time. Two old bars of cast iron, about 5 feet long each, one of them 3 inches and the other 4 inches square, were turned down in the lathe at five different places, to ten different diameters, of from 2 to 4^ inches. The load was applied at the end of a lever 14 feet 2 inches long. Particulars of the experiments are given in table No. 185; the values of h, the shearing resistance, calculated by the general formula ( 3 ), page 535, are added. Table No. 185. TORSIONAL STRENGTH OF CAST IRON. 1819. (Reduced from Mr. Dunlop's data.) Diameters. Cubes of Diameters. Ratio of the Cubes. Breaking Weight. Ratio of the Breaking Weights. Shearing Stress per square inch. inches. tons. tons. 2 8 I .IIl6 I 8-530 2X 11.4 1.4 .1714 i-5 9-201 2^ 15.6 2.0 .182 1.6 7.123 2^ 20.8 2.6 .312 2.8 8.761 3 Failed 3X 34-3 4-3 .522 4-7 9.299 3X 42.9 54 554 5.0 7.902 3^ 52.7 6.6 .742 6.6 8.604 4 64 8.0 .865 7-7 8.265 4X 76.8 9.6 963 8.6 7.691 Average 8 T;; w o/ j It seems that the ultimate torsional strength increased very nearly as the cube of the diameter, and that the average torsional resistance per square inch of section was 8.375 tons. Assuming, as explained at page 561, that the shearing resistance of cast iron is equal to its direct tensile resist- ance, the general formulas for torsional strength ( i ), page 534, and ( 6 ), page 536, become, by substitution, For cast-iron round shafts, W = ' 2 ? s - R For cast-iron square shafts, W = '4 3.6 R R (12) (13) W = the force, in tons. R = the radius of the force, in inches. WR =the moment of the force, in statical inch-tons. d = the diameter of the round shaft, in inches. b the side of the square shaft, in inches. s = the ultimate tensile strength, in tons per square inch. Annals of Philosophy, vol. xiii. 1819. 565 THE STRENGTH OF MATERIALS. If the tensile strength, s, be taken at 7.2 tons, for iron of average quality, then, by substitution and reduction: Ultimate Torsional Strength and Sizes of Cast-Iron Shafts of average quality. Round shaft, W= 2 ( J 4) R d = Square shaft, W = ^ . ( 16 ) TORSIONAL DEFLECTION OF CAST-IRON BARS. In the absence of direct data for the torsional deflection of cast-iron bars, it is assumed that it is i ^ times that of wrought-iron shafts the same pro- portion as that of the transverse deflections of cast-iron and wrought-iron shafts, as indicated by a comparison of the formulas (8), page 564, and (5)j P a g e 59- Multiply, therefore, the second member of the formula (14), page 592, by 1^3; the coefficient 1070, or exactly 1073, becomes (1073x3/5 = ) 644: Torsional Deflection of Round Cast-Iron Bars. ' (.8) STRENGTH OF WROUGHT IRON. 567 STRENGTH OF WROUGHT IRON. TENSILE STRENGTH. Mr. Telford deduced from his experiments, an average tensile strength of 29.25 tons per square inch for wrought-iron bars. Mr. Barlow deduced from the results of eight bars of wrought iron Swedish, Russian, and Welsh, from i^ inches square to 2 inches in diameter an average tensile strength of 25 tons per square inch. Mr. Barlow also deduced from experiments on bars of from i inch in diameter to 2 inches square, that the elastic tensile strength of good medium wrought iron was 10 tons per square inch; and that the extension was at the rate of Vio.oooth part of the length per ton per square inch; and that, therefore, the elasticity was fully excited when the bar was stretched Vioooth part of its length. Sir W T illiam Fairbairn published, in 1861, results of experiments on the tensile strength of wrought iron, which are rendered, slightly adapted, in tables Nos. 186 and 187: Table No. 1 86. TENSILE STRENGTH OF WROUGHT IRON. 1861. (Sir William Fairbairn.) DESCRIPTION OF IRON. Mean breaking Weight per square inch. Ultimate Elongation. With Fibre. Across Fibre. Lowmoor iron (specific gravity 7.6885).... Lancashire boiler plates (9 specimens) Staffordshire iron (two ^-inch plates ") rivetted together) j tons. 28.66 21.82 21 28.40 2O. IO 22.30 26.71 27.36 22.69 26.80 26.56 26.65 37-96 21.25 22.29 tons. 23.43 20. IO 36 18.49 20.75 24.47 24.03 23.58 19.82 19.62 fraction of length. '/ 23 and V 3 6 Vs V, and /aa */ 2 oand'/26 'As and '/as Vis and '/a* V*, and Ya 3 ',: i 'As and '/ 35 Vas and Vas Charcoal bar iron . . Best best Staffordshire charcoal plate ) (4 experiments) J Best best Staffordshire plates (4 experiments) Best best Staffordshire plate Best Staffordshire Common Staffordshire Lowmoor rivet iron (2 experiments) Staffordshire rivet iron. . . . Staffordshire rivet iron Bar of the same, cold-rolled Staffordshire bridge iron Yorkshire bridge iron 568 THE STRENGTH OF MATERIALS. Table No. 187. TENSILE STRENGTH OF IRON AND STEEL PLATES THAT HAD BEEN SUBJECTED TO EXPERIMENT WITH ORDNANCE AT SHOE- BURYNESS. 1 86 1. (Sir William Fairbairn.) DESCRIPTION. THICKNESS OF PLATES, IN INCHES. Averages of plates of one make. X % # 1% 2 2^ 3 IRON. Lowmoor tons. 24-34 24.17 p tons. 25-75 23.22 tons. 29-43 26.47 tons. 24.16 22.3O 25.16 tons. 25.35 23.66 24.63 tons. 24.11 23.92 22.73 tons. 25.04 23-54 24.16 tons. 24.79 24.32 24.63 Thames Co.'s.... Beale & Co.'s.... Averages of j plates of the > same thickness ) 24.26 24.49 27-95 23.87 24-55 23-59 24.25 Average speci- ( fie gravity . . . j 7.70 7.72 7.72 7.72 STEEL. Howell& Co.'s | homogeneous > metal j Specific gravity... 30.70 33-69 30.91 26.2O 7 .8 9 27.04 7.91 27-5I 7.9T 27-39 7.91 29.06 Elongations before rupture of specimens, in part of the length (2^4 to 3 inches). IRON. Lowmoor Thames Co. per cent. 6.2 3-o per cent. 7-6 4.0 percent. IO.O 4.0 per cent. I 7 .6 14.6 19-3 per cent. 30-5 25-3 17.9 per cent. 28.8 32.0 16.0 percent. 32.0 26.5 23-3 percent. 20.5 I6. 5 16.1 Beale & Co Averages 4.6 5-8 7.0 17.2 24.6 25.6 27-3 Homogeneous ) metal J 25-6 IO.O 20.8 19-3 34-5 29-5 25-8 From the first table, No. 186, it appears that the strength of iron plates. in the direction of the fibre, varied from 28.66 tons for Lowmoor iron, to 20.10 tons for Staffordshire charcoal-iron; and that there is no tensile strength in the direction of the fibre so low as 20 tons per square inch. Also, that the averages of nine irons show, for the breaking weight,- With the fibre 23.68 tons per square inch. Across the fibre 2I -59 Difference 2.09 tons, or about 9 per cent. STRENGTH OF WROUGHT IRON. 569 From the second table, No. 187, it appears that the thicker plates have less tensile strength than the thinner plates; whilst the elongation before rupture is greater; thus: X to % inch thick. ij^ to 3 inches thick. -, ,. . tons. per cent. tons. per cent. Iron 25.57 elongation 5.8 24.06 elongation 23.7 Homogeneous metal... 3 1. 7 7 18.8 27.03 27.3 Sir William Fairbairn tested the resistance of iron plates to a bulging- stress. He stretched two ^-inch plates and two ^-inch plates over a cast-iron frame 12 inches square inside, as in Fig. 191, and subjected them to pressure from an iron bolt 3 inches in diameter, with a hemispher- ical end, which was applied to the plate. Fig. 191. Specimen Plate to resist Bulging Stress. Fig. 192. Effects of Bulging Stress. Sir W. Fairbairn. The ^-inch plates were indented ^ inch and ^ inch respectively, when they commenced to fail by cracking on the convex side, under a pressure of 4.7 tons applied at the centre. Under 7 tons of pressure they were cracked through. The j^-inch plates were indented .33 inch when they commenced to crack, under a pressure of 9 tons ; and they were cracked through under a pressure of 17 tons. See Fig. 192. The resistance to bulging in these experiments was in proportion to the thickness of the plates. To test the effect of cold-rolling on iron bars, Sir William Fairbairn tested three bars, and obtained the following results : Black bar from the rolls ........................ 26.0 tons. 20 per cent. turned to i inch in diameter ..... 27.1 22 cold-rolled to i inch in diameter 39.4 ,, 8 showing that the tensile strength was increased by one-half; but that the elongation was reduced to less than a half. With respect to the influence of temperature, Sir William Fairbairn, in 1857, found that the strength of ordinary Staffordshire iron plates, either with or across the grain, remained the same for temperatures varying from o F. to 400 F. At higher temperatures the strength declined, until, at a red heat, it fell from an ordinary average of 20 tons to 15^ tons per square inch. Mr. Thomas Lloyd tested the tensile strength of Staffordshire S. C. Crown bars, i^ inches in diameter, of one kind. The same bars were broken four times in succession, and the successive breaking weights were for the Tensile Strength. ist Breakage (10 trials) ............ 23.94 tons per square inch. 2d (io ) ............ 25.86 3d ,, (7 ) ............ 27.06 4th (6 ) ............ 29.20 5/O THE STRENGTH OF MATERIALS. showing a variation in the same bars of from 23.94 to 29.20 tons, or 18 per cent, of the maximum tensile strength. The tensile strength of i ^-inch round Staffordshire bars of various lengths was found by Mr. Lloyd as follows: Breaking Weight. 6 Bars, 10 feet long 32.21 tons. 6 Do. 3.5 3 2 - 12 6 Do. 3 32.35 6 Do. 2 3 2 - 00 6 Do. 10 inches long 3 2 - 2 9 showing that the strength was not affected by the length. Mr. Edwin Clark found the average tensile strength of iron per square inch as follows : Staffordshire boiler plates, y^ to II / I 6 inch thick, with fibre, 19.6 tons. i ,. i f with fibre, IQ.Q^ Do. special trials { across fibre! 16.82 ' Difference, i$j4 per cent, less 3.11 Best scrap rivet iron, ^ inch diameter, 24.0 He also found that the ultimate resistance to compression in a wrought- iron bar was 16 tons per square inch, at which pressure the metal began to ooze away. Under 1 2 tons per square inch, the set was so great that the form began to change; and this pressure was taken as the average ultimate resistance to compression that may be recognized in practice. The elastic limit of tensile strength was also taken as 12 tons per inch; and Mr. Edwin Clark concluded thus : " It is very nearly true, and very convenient in practice, to assume both the extension and compression to take place at the rate of one ten-thousandth Cro-Trw) f the length for every ton of direct strain per square inch of section " agreeing in this respect with Mr. Barlow. Mr. Edwin Clark found that the resistance of 7/6-inch rivet iron to shearing was as follows : Tons per Square Inch of Section. Resistance by single shearing, 24. 1 5 Do. by double shearing, 22.1 Do. in two ^-inch plates rivetted together (one section), 20.4 Do. in three ^/s-inch plates rivetted together (two sections), 22.3 Tensile strength, 24.0 When three ^-inch plates were rivetted together with a ^j-inch rivet, the frictional resistance to displacement of the middle plate, the hole through which was larger than the rivet, was from 6 to 8 tons. STRENGTH OF WROUGHT IRON. 571 EXPERIMENTS ON THE TENSILE STRENGTH OF WROUGHT IRON AND STEEL. By Mr. Kirkaldy, 1858-61. Mr. David Kirkaldy conducted, for Messrs. Robert Napier & Sons, an extensive series of trials of the tensile strength of iron and steel bars and plates, the results of which threw much light on the properties of these metals. 1 The specimens of bars were formed with a head at each end, united to the body of the bar by taper necks, to receive the shackles, as shown in Figs. 193 and 194. Screw bolts and nuts were shackled as in Fig. 195. o \ J \ o 9 9 O @ 9 L\ O d> Figs. 193, 194, 195. Mr. Kirkaldy's Test Speci- mens of Bars. Figs. 196 to 199. Mr. Kirkaldy's Test Specimens of Plates. The clear length of bar was about 7 inches. The plate specimens were formed as in Figs. 196 to 199, the ends of the thinner plates being forti- fied by flitches rivetted on both sides. The increments of load were applied slowly and gradually. The specimens of bar iron varied from ^ inch to i^ inch in diameter; but they were, for the most part, from 3^ inch to i inch in diameter. There did not appear to exist any material difference of strength that could be ascribed to difference of diameter. Tensile Strength and Elongation of Iron Bars. The average ultimate tensile strengths of iron bars, and their total elongations or stretching when fractured, were as follows : 1 The results of Mr. Kirkaldy's important investigations are published in his work, Experiments on Wrought Iron and Stee!, a mine of experiment and research. The data in the text are reduced from this work. It is right to explain that Mr. Kirkaldy expresses the resistance, in all cases, in pounds; and that they are, in the text, converted into tons. For, though the pound-unit commends itself as a basic unit of great simplicity, yet engineers are accustomed to think in terms of tons, and will continue to do so, until some universal decimal system is adopted, by which the ordinary ton may be superseded. It must be admitted that the ton of 2240 Ibs. is a barbarous unit, and that the New York ton of 2000 Ibs. is in every sense superior to the old British relic. 5/2 THE STRENGTH OF MATERIALS. Yorkshire rolled bars, ................... 27.39 tons. 25.2 per cent. Staffordshire do .................... 25.90 23.5 Lanarkshire do .................... 26 -55 T 9-4 Rivet iron, do .................... 26.00 20.5 Averages, 26.46 22.2 Hammered scrap, forged down, 23.85 ,, Bushelled iron (turnings), forged down, 24.95 l6 - 8 Crank-shaft, scrap iron, with fibre,... 20.37 21.8 ,, Do. do. across fibre, 18.55 I2 -5 Armour plate, across fibre, 16.92 9.0 ,, Averages, 20.93 I 7- The lowest tensile strengths of the better kinds were not more than i J/2 ton below the average for each brand. Contraction of the Sectional Area in Fracture. The reduction, in size, of a piece subjected to tensile strain is practically uniform throughout the portion that is strained, except near to the point of rupture, where the ^^Ci" piece is locally much K j more contracted in size, . ^--.rrf as illustrated by Fig. 200, -"- -y which shows the contrac- tion of a i -inch round bar of soft Bowling iron, with the line of fracture. The diameter, in this instance, was reduced 3/ 32 inch, and the sectional area was reduced at the fracture to 43.56 per cent, of the area of the original section. In the following selection of examples of fractured sectional areas, the percentage ranges from 29.5 per cent, to 87.8 per cent. Iron Bars Fractured Sectional Areas. Per cent, of original area. Swedish R. F. charcoal, 29.5 Staffordshire, charcoal, 38.4 Yorkshire, Lowmoor, 46.3 Staffordshire B. B. scrap, 47.6 Do. S. C. crown, 53.4 Scotch extra best best, 58.5 Do. best best, 68.9 Do. common, 71.6 Do. common, 85.2 Russian C. C. N. D. (for steel), 89.8 Average, 59.0 STRENGTH OF WROUGHT IRON. 5/3 Strength as the Diameter is Reduced by Rolling. Four pieces were cut off a i^-inch bar, reheated and rolled down to different sizes. They had the following tensile strengths: inch. tons. per cent. 1% ........................... 22.38 ......... 28.3 I ........................... 25.60 ......... 26.7 3 A ........................... 2 5-97 ......... 25.2 % ........................... 26.65 ......... 23.8 showing an increase of tensile strength, and less elongation. Strength as Affected by Turning, or Removing the Skin. Rolled bars i% inch diameter were turned down to i inch, and tested. The average results of four irons, so treated, were as follows : Tensile strength. Elongation. tons per square inch. per cent. Rough bars, .......................... 24.38 ............ 17.2 Turned bars, ....................... .. 25.00 ............ 19.3 showing that the turned bars were at least as strong as the rough bars. Strength as Affected by Forging. i^-inch round bars of four kinds of iron were reduced by forging to i inch and ^ inch in diameter. Tons per inch. Elongation. Rough bars, ................ 2 5- I 3 ......... 24.5 per cent. Forged bars, ................ 26.10 ......... 17.3 showing an increase of strength equal to i ton per square inch by the forging down, and a reduction of elongation. Strength as Affected by Reheating only. Five different irons, i inch in diameter, of which the collars had failed at the first trial, had the collars replaced. The effects of the reheating to which the five bolts were subjected in the operation are thus shown : Tons per inch. Elongation. 1 st Trial, ................. 25.86 ............ 10.1 per cent. 2 d Trial, ................... 24.88 ............ 32.6 showing a reduction of i ton per square inch of tensile strength, whilst the elongation was trebled. . On the contrary, two pieces of a 24 -inch bar of good iron were tested, one in its ordinary condition, and the other after it had been brought to a welding heat, and cooled slowly. Tons per inch. Elongation. In ordinary condition, ............. 25.27 ...... 22.3 per Cent. Heated, and cooled slowly, ....... 25.21 ...... 17.7 Strength as Affected by Intense Cold. Three pieces of a 24 -inch bar were tested, one at 64 K; the others, after having been exposed over night to intense frost, were broken at 23 F.: 574 THE STRENGTH OF MATERIALS. Tons per inch. Elongation. At 64 24.87 24.9 per cent. At 23 24.28 23.0 showing that at the lower temperature the strength was a little less by 0.59 ton. Strength as Affected by Notching the Bar. The two ends of i inch round specimen bars were screwed, and the screw at one end was divided by turning out a square notch or groove Y% inch wide, as at Fig. 201, leaving a diameter of .70 inch at the bottom of the groove. After having been broken at the notch, the body of the bar was turned down to the same diameter as the notch had originally, and the bar broken a second time, -Notched Specimen Bar. through the body. The following are selections from the results of such tests applied to bar irons; the strength of the rough bar being added for comparison. The elongations are not recorded; but the contracted sec- tional areas of fracture are here given : Lowmoor (hardest bar), . . . Bowling (softest bar), Govan Diamond, Tensile Strengths. Contracted Sectional Area. Notched. Turned Down. Rough Bar. Notched. Turned Down. Rough Bar. tons. 40-95 2 9 .87 30.96 29.87 tons. 31-57 25.01 25.70 28.17 tons. 29.10 26.20 25-7I 23-I5 per cent. 92.0 72.2 75-9 IOO.O per cent. 50.8 44.4 46.4 92.0 percent. 49.0 43- 6 50.8 96.1 Dundyvan, common, 32.91 27.61 26.04 85.0 58.4 59-9 showing a remarkable excess of resistance, more than 5 tons, at the notch, due apparently to the shortness of the notched portion, which was partially sustained by the thicker parts on each side, whilst the contraction of area was in a measure prevented. Strength of Bars or Bolts as Affected by Screwing. Screws with rounded threads were cut in three modes by means of old blunt dies, new sharp dies, and chasers; and tested for tensile strength. The following selection of results has been made for the purpose of fair comparison; premising that the diameter at the base of the thread was the same for all the screwed bolts of one diameter : Not Screwed. Screwed. Difference. tons. tons. bolts, screwed, 24.47 18.22 ,, chased, 24.45 17.20 i-inch bolts, screwed (old dies), 27.60 24.62 screwed (new dies), 27.10 19.04 chased, 27.95 20.02 bolts, screwed (old dies), 26.47 22.77 screwed (new dies), 26.47 : 947 chased, 26.47 18.70 6.25, or 25 per cent less. 7.25, or 29 2.98, or 1 1 8.06, or 30 7.93, or 28 3.70, or 14 7.00, or 26 8-77, or 33 STRENGTH OF WROUGHT IRON. 575 A decided variation of strength is due to the mode of cutting the screw. The blunt dies, compressing the metal, as Mr. Kirkaldy argues, harden it, and increase its tensile resistance. The sharp dies, cutting more readily, do not compress the metal so much as the blunt dies. The chaser does not compress it at all. Hence, the reduction of strength is greater in screwing with sharp dies than with blunt dies, and is greatest when the chaser is used. Strength of Bars as Affected by Welding. The pieces of bar iron to be tested were cut through the middle, and scarfed and welded in the ordinary manner. The tensile strength at the weld was in all cases less than that of the original bar, as well as the elonga- tion before fracture. The following are the averaged results for each class of irons: Tensile Strength. Elongation. per cent. less. per cent. inch diameter, ............ 17.6 ......... 6.7 ............ 30.2 ......... 7.3 ............ 14.5 ......... 5.9 ............ 15.1 ......... 10.5 Glasgow B. Best, i Farnley, i Govan B. Best, Govan Extra B. B. Average, 19.4 7.6 These averages conceal the excessive fluctuations of strength, which varied from 2.6 to 43.8 per cent, below the normal strength of the bars. Strength of Bar Iron in Resisting Stress Suddenly Applied. In a special series of trials, when the steelyard was duly loaded, and all taut, it was suddenly released by means of a trigger; so that the stress was delivered upon the specimen suddenly, but without any blow or jerk. Mr. Kirkaldy ascertained the value of the rupturing stress for each iron by taking the mean between the lowest stress that caused rupture and the highest that did not do so. The specimens consisted of i-inch round bars. The following are the comparative tensile strengths : Bradley charcoal, Bradley Crown S.C., Lowmoor . . LOAD APPLIED. DIFFERENCE. ELONGATION. Gradually. Suddenly. Gradually. Suddenly. tons. 25.45 27.82 26.48 26.81 26.70 26.70 24.87 19.54 tons. 22.05 22.10 21.37 20.91 20.45 21.07 22.50 15.82 tons, percent, less. 3.40, or 13.4 5.72, or 20.6 5.11, or 19.0 5.90, or 22.0 6.25, or 24.8 5.63, or 19.1 2.37, or 9.6 3.72, or 19.0 per cent. 30.2 25-3 27.O 24.9 21. 1 per cent. 40.1 22.5 25.0 23.6 25-3 20.1 17.3 16.9 Lowmoor, Glasgow B. Best, Glasgow B. Best, Glasgow B. Best ) (X-inch bars), ) Crank-shaft . 25.54 20.78 4.76, or 1 8.6 24.6 20.1 (5 specimens.) 5/6 THE STRENGTH OF MATERIALS. These results show that the tensile resistance to fracture by suddenly- applied stress is from 2 to 6 tons per square inch, or from 10 to 25 per cent, less than when it is gradually applied. The average elongation is also less, decidedly more so, if the exceptional specimen, Bradley charcoal bar, be omitted from the average. The Influence of Frost, at 23 F., on the tensile resistance of bar iron to strains suddenly applied was tried on seven specimens cut from a ^-inch bar of Glasgow B. Best iron. The average results showed 3.6 per cent, diminution of resistance; and a reduction of elongation from 25 per cent, to 20^ per cent. Influence of Additional Hammering on the Iron in a large Crank-shaft. Three pieces i^ inches square were cut out, and forged down to i^ inch, and turned to i inch in diameter. Compared with two pieces which were simply cut out, and turned down to i inch, the results were as follows : Tons per inch. Elongation. Cut out and turned down, 19.90 1 6. 8 per cent. Cut out, forged down, and turned,.... 23.70 11.7 showing 20 per cent, increase of strength with reduced elongation. Strength of Hammered Iron as Affected by Removing the Skin. Two i^-inch square bars of Govan hammered iron were turned down to i inch in diameter. Compared with i inch square Govan hammered bars, in their skins, they gave better results, thus : Tons per inch. Elongation. i -inch square bars, 28.60 20.6 per cent. i j^-inch square bars turned down,.... 30.35 23.5 Hardening Iron Bars. A i^-inch round bar of Bowling iron was cut into several pieces, which were turned and forged down, and hardened : Diameter. Tons per inch. Elongation. Turned to i inch, 2 7- I 5 28.3 per cent. Forged to .87 inch, hardened in water, 32.79 19.6 Do. .78 oil,... 28.85 J 9- 8 Do. .70 tar,... 28.06 22.4 showing that hardening in water increases the strength more than in oil or tar. The tensile strength of the second piece, above noted, namely, 32.79 tons per square inch, was the greatest strength of iron observed by Mr. Kirkaldy. Experiments on pieces cut from the large crank-shaft already mentioned, and from an armour-plate, and hardened, show that there was no increase of tensile strength by hardening, and that the elongation was reduced. Case-hardening Iron Bars. By case-hardening specimens of several irons, and cooling them in oil, or in water, or slowly, the loss of tensile strength averaged 2.21 tons per STRENGTH OF WROUGHT IRON. 577 square inch; whilst three-fourths of the elongation was gone. The averages may be placed together for easy comparison : CASE-HARDENED. Tensile Strength. Elonga- tion. In Ordinary Con- dition. A Forged and cooled in oil, .... tons. 2 5-39 23.70 2 5-36 22.11 per cent. 6.2 2.9 ii. 6 4-9 tons. 26.50 26.50 27.07 25-32 per cent. 26.6 26.6 23-8 20.7 B Forged and cooled in water, C Forsred and cooled slowly D. Turned down, and cooled slowly, Averages . 24.14 6.4 26.35 24.4 Cold-rolled Iron Bars. Five pieces of 24 -inch Blochairn bar-iron were treated as follows : Tons per inch. Elongation. Cold-rolled (2 pieces) 3 J -86 n. 8 per cent Cold-rolled and annealed (2 pieces) 26.50 25.6 In ordinary condition ( i piece) 27.06 22.8 showing that cold-rolling added nearly 5 tons to the strength, which was lost when the bars were subsequently annealed. Strength of Angle-iron, Ship-strap, and Beam-iron. The tensile strength of angle-irons, about ^-inch thick, is generally less by from i to 2 tons per square inch than that of bar-iron. The tensile strength of ship-strap and beam-irons, from ^ to i inch thick, is 2 tons less than that of angle-irons. The elongations, correspondingly, are also less. Tensile Strength of Iron Plates. Iron plates, of thicknesses varying for the most part from |^ inch to 24 inch thick, cut into specimens i y 2 and 2 inches wide, were tested : PLATES. TONS PER INCH. ELONGATION. With Fibre. Across Fibre. With Fibre. Across Fibre. Yorkshire tons. 24-75 23.01 22.89 23-37 21.96 tons. 22.64 21.40 21.39 19.22 19.56 per cent. 13-4 9-3 9-5 9.6 7.0 per cent. 8.0 5-3 5-2 2.8 3-2 Staffordshire Durham Shropshire Lanarkshire General averages 23.20 20.84 9.8 4-9 The greatest difference of the lowest tensile strength in any group was 3 tons per inch below the average of the group. In the Yorkshire plates it did not exceed 2 tons. The tensile strength across the fibre is from i ^ tons to 4 tons per inch less than that with the fibre. The average difference is 10 per cent. 37 578 THE STRENGTH OF MATERIALS. Fractured Sectional Area of Iron Plates. With Fibre. Across Fibre. Yorkshire 63.5 percent. 79.7 per cent, of original area. 76.5 83.7 Staffordshire crown S. C. 78.5 Bradley.... 84.3 Scotch best boiler 87.3 Staffordshire best best... 90.9 Scotch ship 95.4 Scotch common 94.4 89.9 92.0 93-6 94.6 97-5 98.5 Cold-rolled Iron Plates. Pieces of Blochairn plate .345 inch thick were reduced by cold-rolling to .238 inch thick, or to two-thirds: TONS PER INCH. ELONGATION. With Fibre. Across Fibre. With Fibre. Across Fibre. In ordinary condition 20.45 39-73 22.75 19.20 36.00 21.72 per cent. 4-4 O.I 8.0 per cent. 2.6 0.0 6.0 Cold-rolled Cold-rolled and annealed Cold-rolling nearly doubled the strength, but annihilated the elongation. By annealing, all but 2^ tons per inch of the extra strength was lost; but the original elongation was doubled. Strength of Iron Plates as Affected by Galvanizing. Fourteen specimens of Glasgow best boiler plate, from 3/ l6 to ^ inch thick, were prepared for trial, half the number having been galvanized. There was no perceptible difference in any respect between the galvanized and the ungalvanized plates. Specific Gravity of the Irons Tested. Armour-plate 7-6134 Angle-iron 7.6006 Iron plates 7.6287 Yorkshire rolled bars 7.7600 Staffordshire rolled bars... 7.6178 Lanarkshire rolled bars ... 7.6280 Crank-shaft 7.6307 The specific gravity was diminished by cold-rolling, though the tensile strength was increased ; as follows : Ordinary. Cold-rolled. Bar iron, specific gravity 7-636 7-582 Boiler-plate, 7.566 7.539 The specific gravity of iron was also diminished by stretching under tensile Stress : SPECIFIC GRAVITY. Before Stretching. After Stretching. Three i-inch Yorkshire bars, stretched to .90 inch... 7.752 7-674 Two .83-inch Blochairn bars, .76 ...7.636 7.569 Average for five bars 7.760 7- 6 3 2 showing an average reduction of . 1 28, or 1.65 per cent., in the specific gravity. STRENGTH OF WROUGHT IRON. 579 EXPERIMENTS OF THE STEEL COMMITTEE OF CIVIL ENGINEERS. 1870. The Steel Committee, who will be again noticed in treating of the strength of steel, tested the strength of a number of wrought-iron bars i y 2 inches diameter, consisting of twelve bars of Lowmoor iron, six bars of best Yorkshire iron, and six bars of usual S. C. Crown, or Stafford- shire iron. Table No. 189 gives condensed results of the experiments for the tensile strength of wrought-iron bars, in 10 feet of length; and table No. 188, the same for their compressive strengths. Note to Tables Nos. 188, 189. The lowest elastic strength in any group of bars did not exceed i ton per square inch less than the average elastic strength; say, not more than 10 per cent, less than the average for iron bars. A chemical analysis of these irons is given with that of the steels tested by the committee, in table No. 203, page 603. Table No. 188. COMPRESSIVE STRENGTH OF WROUGHT-!RON BARS. 1870. i% inches in diameter. Observations made on lo-feet lengths. (Reduced from results of experiments made by the Steel Committee.) Mark and Description. Elastic Strength (Compressive) in Tons per Square Inch. Elastic Compression. Elastic Compression per Ton per Square Inch, in parts of the Length. L S 3 Lowmoor tons. 13-5 13-5 13.0 per cent. .101 .106 .101 length = i. L S s L S 6 Averages 13-3 .102, or i in 977 .000077, or 1/12,987 L i Lowmoor 12.5 10.5 II.5 .090 L 2 L 3 Averages II.5 .089, or i in 1130 .000077, or 1/12,987 K C I Yorkshire. 13.0 13.0 .100 .103 K C 2 K C ^ Averages I 3 .0 .101, or i in 987 .000078, or Vi2,82i F R i, usual S. C. Crown F R 2, F R 3, Averages II.5 II.5 12.0 093 .095 .103 II.7 .097, or i in 1030 .OOOOSO, Or T /i2,5oo Yorkshire Summary 12.6 11.7 Averages. .097, or i in 1030 .097, or i in 1030 .000077, or 1/12,987 .000083, or 1/12,048 S C Crown Total averages 12. 1 .097, or i in 1030 .000080, or Via.sco THE STRENGTH OF MATERIALS. & s I I i j 6 fi O .2 rt n d 1 a D JS la 3 a ON oo - -g 1" c vO CO VO CO ^ *,n . ON t^. t^.toO\ t^ ri t>. to II 0, bflP, S .5 a'S OO to CO ON VO t^ M co vO M co ^ to to ON vo 00 VO N Ji>f **.* oo' M vN N CO rt CO tN CJ N c? M N N tN tN M to ON 1 I S^ g 35 S ON ON HH r^^ O^ >~* ) 1 C "" "c g rt rt rt .S.S rt W " U M M M M M W M o flj J-< j_, j^ j^ -. Si j_, PJ o O O O O O O JS rt W ft 2 o 2 vO CO vO OO ON ON O O O CO 8" J ON ON ON 000 1 covO 2 g; 8 . S aJ3 III 1! e'S i O O O O to O CO O O to M 10 to to CO O CO ^ C/3-^.rt ^f ^" 2" w N ^ M CO CO CO CO w w N CO 1-1 HI . 2; 1 i c N o u ^ ~ to J CO CO rj to d Q a f: = 1 vvmoor.. > g 1 i s cj C^O!C^ 3 uuu i-T rf co CO ^U C ii "o ^^ J Jl-l J ^^^ fe^fe >ti H STRENGTH OF WROUGHT IRON. 5 8l HAMMERED IRON BARS (SWEDISH). Table No. 190 contains a selection of results of trials made by Mr. Kirkaldy of the tensile and compressive strength of hammered bar-iron manufactured by Messrs. Gammelbo & Co., Nericia, Sweden. Table No. 190. STRENGTH OF SWEDISH HAMMERED IRON BARS. 1866. TENSILE STRENGTH. Size of Specimens (2, 3, or 4 of each scantling). Length for Elongation. Elastic Strength per Square Inch. Absolute Strength per Square Inch Elongation in parts of Length. Lowest. Highest. Average. tons. tons. tons. tons. per cent. l^-inch round, ) turned down from / 15 inches 10.75 12.00 11.05 18.80 24.6 bars 2 inches . j I -inch square 20.35 22.9 3-inch round 10 18.85 33-1 2-inch round 10 18.87 32.5 I -inch round IO 18.92 ^-inch round 10 23.90 5-9 Flat, 3 x y z inch 15 23.00 12. 1 Flat, 2 x Yz inch 15 20.62 21.6 Flat, lYit-Yt inch 15 20.55 16.4 I -inch square iron } converted into blis- > 12.77 1.2 tered steel ) 2 -inch round, case ) hardened j 22.50 I -inch round, case | hardened } 19-35 ^-inch round, case ) hardened \ 23-2 COMPRESSIVE STRENGTH. Size of Specimens (2, 3, or 4 of each scantling). Length for Compression. Elastic Strength per Square Inch. Absolute Strength per Square Inch. Compression in parts of Length. Lowest. Highest. Average. i^-inch round I y^, -inch round i^-inch round I -inch square 1% inches 15 3. i inch tons. 10.10 8-94 8.94 tons. 12.64 10-75 9.84 tons. 10.74 9-45 10.42 tons. 66.45 12-53 37-90 82.20 per cent. 45-4 3-7 33-i 53-3 I -inch square con-") verted into blistered > steel . ...) i inch 83.40 48-3 I ^-inch turned SHEARING S' II FRENGTH || .S. 5 82 THE STRENGTH OF MATERIALS. For transverse strength, four 2-inch square bars were tested, on a span of 25 inches, with the following results: Breadth. Depth. Elastic Stress. Deflection. Ultimate Stress. Deflection. inches. inches. pounds. inches. pounds. inches. I 2.04 X 2.02 7,5 .089 15,888 5-18 2 2. 02 X 2.O4 7,000 .088 14^75 4.98 3 1.95 X 2.02 6,000 .072 I3 5 9 6 5 5-85 4 2.00 X 2.00 6,000 .078 I'3338 5.38 Averages . 6,625 .082 ^S 16 5-35 or 2.96 tons or 6.48 tons The bars remained uncracked under the ultimate stress. For torsional strength, the averages for four bars turned to i ^ inches in diameter, with a length of 7 diameters, the stress being applied at the end of a i2-inch lever, were as follows: Elastic stress 1062 pounds, or .474 ton. Deflection in parts of a revolution = i... .on turn. Ultimate stress 2677 pounds, or 1.195 tons - Ultimate deflection 4.70 turns. One-inch square bars of the same manufacture were tested. For tensile strength, they broke with an average of 20.34 tons per square inch. Under bending stress, the average results of four bars, 1.04 inches wide by 1.05 inches deep, showed that they bore an elastic stress of 1250 pounds, with .216 inch deflection. The ultimate stress was 1978 pounds, with 6.60 inches of deflection. MR. J. TANGYE'S EXPERIMENTS ON THE COMPRESSIVE RESISTANCE OF WROUGHT IRON. A i-inch round bar of soft Lowmoor iron, 8 or 9 inches long, was planed on two opposite sides to a thickness of ^ inch, and was subjected to pressure on one side under a steel die ^ inch square, having an area of Y^ square inch. The following are the results of the tests; and they prove clearly that a unit of iron has a much greater power of resistance when it forms a portion of a larger mass, than when it is isolated in the manner customary in making experiments on resistance to compression: Load per square inch. 12 tons no impression. 16 20 24 28 32 36 40 slightest indentation, sensible to the finger-nail, distinctly visible, edge followed by finger-nail. indented about inch. STRENGTH OF WROUGHT IRON. 583 KRUPP AND YORKSHIRE IRON PLATES. 1875. Mr. Kirkaldy made an experimental inquiry into the relative properties of wrought-iron plates manufactured by Herr Krupp, Essen, and plates manu- factured in Yorkshire. The results are detailed in a valuable report by Mr. Kirkaldy, from which the following particulars have been extracted. Twenty-seven plates in all, not less than 4 feet by 3 feet, of three thicknesses, 2/3 inch, ^ inch, and ^ inch, were obtained from Mr. Krupp and from six Yorkshire manufac- turers; from which the specimens were cut out. The Yorkshire brands were: Low- moor, Bowling, Farnley, Taylor's, Cooper & Co., and Monkbridge. Tensile Strength. Table' No. 191 gives condensed results of the tests for tensile strength. Each entry for Krupp iron is an average for nine speci- mens; and for Yorkshire iron, for eighteen specimens. The results for the three thicknesses are nearly alike. The form of the specimens is shown in Fig. 202. Fig. 202. Test Specimens for Tensile Strength. Table No. 191. KRUPP AND YORKSHIRE IRON PLATES TENSILE STRENGTH. 1875. Thicknesses ^, ^, and % inch. Breadth, 2 inches; length for extension, 10 inches. (Reduced from Mr. Kirkaldy's Reports.) Description. Elastic Strength per square inch. Ultimate Strength per square inch. Ratio of Elastic to Ultimate Strength. Extension. Sectional Area of Fracture. At 30,000 Ibs. per square inch. Ultimate. LENGTHWAY. Unannealed KruDD tons. 1 1.6 12.4 tons. 22.7 21.3 per cent. 51-3 58.4 52.5 59.6 per cent. 1.30 .65 per cent. 25.4 16.7 per cent. 60.4 794 Yorkshire Annealed Krupn ... II.O 12.0 21.0 20.1 2.72 1.42 28.2 18.4 56.3 77.8 Yorkshire CROSSWAY. Unannealed Krupp . ... II.4 12.4 21.7 20.3 52.6 61.4 i-35 5i 174 II. 2 75.2 85.3 Yorkshire Annealed Krupp 10.8 12. 1 20.4 I 9 .2 52.7 62.8 2-37 .81 19.7 12.8 73-0 83.1 Yorkshire AVERAGES. Krupp Yorkshire II. 2 12.2 21.5 20.2 52.1 60.4 1.94 .85 22.6 14.8 66.2 81.4 THE STRENGTH OF MATERIALS. Effect of Drilled Holes and Punched Holes on the Tensile Strength. Specimens 8 inches wide were prepared according to Fig. 203, with two rows of rivet holes, .85 inch in diameter, in the central portion, 2^ inches apart, and the holes were at 2 inches pitch. The punched holes were conical, as usual. The reduction of width of solid metal by the holes Before Bulging. After Bulging. Fig. 203. Specimen Plate to test Effect of Drilling and Punching. Figs. 05. Test Specimen Plates for Bulging Stress. amounted to (.85 x 4 = ) 3.40 inches, and the net section was (8 3.4 = ) 4.6 inches wide, or 57.5 per cent, of the total width. Four Krupp specimens and nine Yorkshire specimens of varying thickness, were tested for each result. Table No. 192 gives some deductions from the elaborate results reported by Mr. Kirkaldy. Table No. 192. KRUPP AND YORKSHIRE IRON PLATES TENSILE STRENGTH OF DRILLED AND PUNCHED PLATES. 1875. Thickness Y%, ^, and ^ inch. Holes, .85 inch in diameter. (Deduced from Mr. Kirkaldy's Reports.) Description. Reduced Section, in parts of Total Section. Reduced Strength, in parts of Total Strength. Tensile Strength per square inch of Net Section, in parts of that of Entire Section. Total Elongation of Holes. DRILLED HOLES. Length way KruoD per cent. C7 C per cent. 6? 8 per cent. per cent. Yorkshire 5/O C7 C c6 o 3-7 T Q o ->/-> 3 D -y 99-O Crossway Krupp... C7 c 6r T 1 06 2 22 8 Yorkshire J/O C7 C 3/o 57.2 L 3-4 PUNCHED HOLES. Lengthway Krupp C7 r r r r 806 T 7 7 Yorkshire .)/;> C7 r 5 1 O CQ O 87 o 1 >7 8 i jt O *j Crossway Kruop C7 r CO O 87 o ill Yorkshire 5/O C7 c AH 6 82 8 7 O AVERAGES. Krupp C7 c c6 ^ 08 o 19 6 Yorkshire. ... ... C7.; C2 Q Q2 O 117 STRENGTH OF WROUGHT IRON. 585 Resistance to Bulging Stress. Discs 12 inches in diameter, cut in the lathe out of plates, were pressed into an aperture 10 inches in diameter, by a bulger-ram about 5 inches in diameter, of which the end was turned to a radius of 5 inches. The preparation for the trial, and the object after Figs. 206, 207, 208. Specimens for Resistance to Bending Stress. having been bulged, are shown in Figs. 204 and 205. A selection of the results is given in table No. 193. Each result for Krupp iron is an aver- age from four specimens; and for Yorkshire iron an average from six specimens. The stress was gradually increased until the specimen was pushed through the aperture, or until the specimen gave way either by cracking or bursting. Table No. 193. KRUPP AND YORKSHIRE IRON PLATES RESISTANCE TO BULGING STRESS. 1875. Discs 12 inches in diameter, pressed into lo-inch apertures. (Selected from Mr. Kirkaldy's Table.) Stress Bulging in Inches. Ultimate. J. nicRncss of Plate. Lbs. 25,000. Lbs., 100,000. Lbs., 200,000. Bulge. Stress. Effects. inches. inches. inches. inches. inches. tons. Unannealed. Krupp, 440 .82 2.14 3-27 62.10 uncracked 533 .64 1.86 340 73.20 uncracked 653 .50 1.59 2.82 3.36 97.06 i burst Yorkshire, .390 S3 ( 4 plates ) t 2.50 J 2.6 5 41.00 3 burst, i cracked .510 .61 f 5 plates ( I 1.85 ( 2.72 6l.03 5 burst .625 35 ) 4 plates) 1 1-57 J J 3 P^tes ) ( 2.83 J 2.52 73.83 burst Annealed. Krupp, 440 83 2.25 3-27 5540 uncracked 533 .72 1.96 3-39 71.28 uncracked 653 .56 i-75 { i plate ) ! 3-18 j 345 88.62 i burst Yorkshire, .390 93 2-73 3-19 47-35 i burst, i cracked .510 .72 ( 4 plates 1 ( 2.05 5 2.88 55-85 2 burst, i cracked .625 43 1.65 ( 3 Plates ) I 3-14 J 2.82 77.30 3 burst, i cracked 586 THE STRENGTH OF MATERIALS. Resistance to Bending Stress. Specimens, 2*1/2, inches wide, of plates of the three thicknesses, were bent double both hot and cold. First, by bend- ing them between supports 10 inches apart to a right angle, as in Fig. 206; then the cold specimens were doubled, as in Fig. 207, to a distance apart of four times the thickness; whilst the hot specimens were doubled flat, as in Fig. 208. Of the specimens of Krupp iron, thirty-six in number, all bore the test, except six which were more or less cracked. Of the speci- mens of Yorkshire iron, seventy- two in number, twenty-five only passed the tests uncracked. PRUSSIAN IRON PLATES. 1874. Two large iron plates, .64 inch thick, manufactured by Mr. Borsig, of Berlin, were tested for tensile strength by Mr. Kirkaldy in 1874. The following abstract contains the averages of four experiments to each result: Unannealed. Annealed. With Fibre. Across Fibre. With Fibre. Across Fibre. Elastic strength 13.00 tons 23.40 54-5 % 23-8 % 1 2. 60 tons 22.70 55-5 % U.6 % 12.73 tons 22.53 56.5 % 24-7 % 12.04 tons 21.70 55-4 % 15-1 % Ultimate strength Ratio Elongation The greatest deviation from the average elastic tensile strength was half a ton below it. The plates were tested for bulging strength. Discs, 12 inches in diameter two annealed and two unannealed were cut out of the plates and pressed through an aperture 10 inches in diameter, the same as shown in Figs. 204 and 205, page 584. Bulging. Pressure. 22.32 tons Unannealed. ; 90 inch 44.64 66.96 89.28 (ultimate) 104.18 ., 104.20 1.49 i-95 2.39 2.69 Annealed. .95 inch. 2.07 2.60 burst, one burst. TENSILE STRENGTH OF IRON WIRE. Mr. Barlow deduced from experiments by Mr. Telford on the strength of iron wire from T / IO inch to I / 20 inch in diameter, that the ultimate tensile strength was equivalent to 36 tons per square inch of section. The tensile strength of Warrington iron wire is given at page 247 : Unannealed, about 36 tons per square inch; annealed, about 24 tons per square inch. STRENGTH OF WROUGHT IRON. 587 American Wire. Mr. Roebling states that bar iron of from i inch to i y% inches square, fit to make the best quality of wire, should have a tensile strength of 60,000 Ibs., or 27 tons per square inch. The same iron, reduced to No. 9 wire, bears 100,000 Ibs., or 44^ tons per square inch; and, if drawn to No. 20 wire, it will bear from 20,000 Ibs. to 30,000 Ibs., or 9 to 13 tons, more. From these data it would appear that wire made of the best qualities of iron has about the same strength as some qualities of steel wire. The tensile strength of American iron wire, together with that of wire from the cables of a suspension bridge after having been 32 years in use, according to Professor Thurston, are as in table No. 194. Table No. 194. TENSILE STRENGTH OF AMERICAN IRON WIRE. 1875. Breaking Weight. . _ J A Actual. Per Square Inch. inch. pounds. tons. per cent. .029 75 56.90 IOO 0535 238 47.26 9 8. 9 .071 368 40.35 91.7 .08 474 42.10 98.7 .1205 963 37-70 96.7 134 1310 41.47 98.5 Wire from suspension \ bridge, .1236 diameter. > 1081 40.17 56 Average of 1 2 tests 1 SHEARING AND PUNCHING STRENGTH OF WROUGHT IRON. Swedish bar iron bore an average shearing stress of 15.20 tons per square inch; the ultimate tensile strength was 18.80 tons (page 581). The shearing resistance of bars 3 inches by j inch and i inch thick, flatwise, with parallel cutters, and to punching i-inch and 2-inch holes through bars % inch, i inch, and i ^ inches thick the power being applied through a hydraulic shearing press, was found by Mr. C. Little 1 to be : Per square inch of area cut. BARS. inch thick, shearing, and punching i-inch holes ......... 22.35 i i ......... 21.83 ^ punching 2-inch holes ........................... 19.00 i 2 ........................... 19.90 1/4 2 ........................... 19.50 The shearing resistance of " ordinary round bar iron of commerce," by direct pull, was ascertained by Chief Engineer W. H. Shock, of the United 1 Proceedings of the Institution of Mechanical Engineers, 1858; page 73. 588 THE STRENGTH OF MATERIALS. States Navy. The following are averages of the results from 12 specimens to each average. The diameters were exactly measured; the attachments were slightly rounded at the edges, and hardened : l DIAMETER OF BOLTS. Resistance in tons per square inch cut. inch. single shear. double shear. y 2 19.68 18.32 y% 17-41 17-23 2< 17-61 17-76 y% 18.50 16.88 i !7.9o 16.78 Averages ................. 18.22 17.40 Mean of the averages ................ 17.81 tons. The averaged results of experiments on the strength of rivetted joints, 2 showed that whilst the plates broke with a load of 19.44 tons per square inch, the rivets were sheared by a stress of 17.45 tons per square inch of section. The shearing strength of wrought iron, in view of the foregoing data, is taken at 80 per cent., or four-fifths of the ultimate tensile strength. TRANSVERSE STRENGTH OF WROUGHT IRON. Rectangular Bars of Wrought Iron. Wrought-iron bars are not readily ruptured by transverse stress. Their transverse elastic strength, therefore, naturally constitutes the chief matter of investigation. Actual data are extremely scarce. Mr. Barlow gives the approximate elastic tensile and transverse strengths of four bars of iron; of which the elastic tensile strength of the first bar was 9.5 tons per square inch; and of the others, 10 tons per square inch. The elastic transverse strengths of these bars are here given, as approximately observed, and as calculated from the observed tensile strength by formula ( i ), page 507. Elastic Transverse Strength. BARS - Calculated. Observed. 2 inches x 2 in. deep. 33 in. span. 2.66 tons. 2.50 + 10115. i-5 x 3 33 4-72 4-25+ i-5 x 3 33 4-72 4-25+ ,, !-5 x 2.5 33 3.28 3-+ The limit of elastic strength was not closely ascertained, but it was known to be greater than the observed strengths here noted. The calcul- ated strengths appear, therefore, to be substantially correct. The following is the form of the calculation, as exemplified for the first of these bars : ,. = 2.661 tons. 33 Mr. Edwin Clark tested three bars of wrought-iron, one i inch square, and two i^ inches square, for transverse strength, as follows: 1 Journal of the Franklin Institute, 1874. 2 Transactions of the North of England Mining Institute. STRENGTH OF WROUGHT IRON. 589 Elastic Transverse Strength. BARS> Calculated. Actual. i inch x i in. deep. 12 in. span. 1.155 tons. 1.117 tons. i-5 x i-5 3 6 I - 2 99 1-275 The Swedish bars, noticed at page 581, 2 inches square, had an ultimate tensile strength of 18.8 tons per square inch, and the i-inch square bars, 20.34 tons. By the formula ( i ), page 507, Ultimate Transverse Strength. SWEDISH BARS. Calculated. Observed. 2.04 in. x 2. 02 in. deep. 25 in. span. 7.230 tons. 7.093 tons, uncracked. 2.02 x 2.04 25 7.302 6.646 1.95 X 2.02 25 6.9II 6.234 2.00 X 2.00 25 6.948 5.955 Average for 2 -inch bars, ......... 7.098 6.482 Average for i-inch bars 1.04 in. x 1.05 in. deep. 25 in. span. 1.077 tons - 0.883 tons > uncracked. The calculated strength of the 2-inch bars averaged 9.5 per cent, in excess of the observed strength; and that of the i-inch bars, 22 per cent. But the bars were not broken, nor even cracked, and they would of course have borne a greater load before breaking. There is a regular and close correspondence between the calculated and the observed transverse strengths of wrought-iron bars above; contrasting with the diversity observed with cast-iron bars. The regularity results from the more nearly uniform texture and strength of wrought iron. Formula ( i ), page 507, in its general form, may be adapted for wrought iron by assuming an average tensile strength of 22.5 tons per square inch for the value of s. Then 1.155 s= 1.155 x 22.5 26. Transverse Strength of Rectangular Bars of Wrought Iron, average quality, Loaded at the middle,... W-^M 2 ............................ (i) Loaded at one end, ...... W = ^l ........................... (2) Round Iron. The strength of round wrought-iron bars, taking the same tensile strength, 22.5 tons, is found from the general formula ( 15 ), page 510; in which .7854x.y = .7854x 22.5 - 17.7. Transverse Strength of Round Wrought-iron Bars, average quality. Loaded at the middle,... W= 17 ' 7 ^ ........................... (3) Loaded at one end, ...... W = 4 ' 4 d *. ........................... (4) W = the breaking weight in tons; b the breadth, d the depth, and / the span, all in inches. 590 THE STRENGTH OF MATERIALS. TRANSVERSE DEFLECTION AND ELASTIC STRENGTH OF WROUGHT IRON. Wrought-Iron Rectangular Bars. The deflections of a few bars under given weights, applied at the middle, were observed by Mr. Barlow and Mr. Edwin Clark. To collate with these the observations of Mr. Kirkaldy on bars of Swedish iron, page 582, the results are here grouped together, and the value of E, the coefficient of elasticity, calculated by formula ( 6 ), page 529, are added: BARS. Span. Load. Deflection. 1 E. Barlow, 2 inches square, inches. 33 33 33 33 33 12 % 3 ^ 36 36 25 25 25 25 the fir tons. 2.50 2.00 4.00 4.00 4.OO .711 1.275 1. 1 45 1.695 1.695 3.348 3-125 2.679 2.679 st 10 bars (Ic inches. .100 .077 .088 .102 .104 .026 .320 .290 .400 350 .089 .088 .072 .072 mg span), I2,T54 16,616 3,730 7,532 12,764 10,228 7,948 8,220 8,454 9,638 7,566 7,004 7,830 7,260 10,228 2 * . " , 5x3 inches deep C X 2 1J E. Clark, inch square (3 bars),.... c c Swedish 2.04 x 2.02 deep, 2.O2 X 2. 02 I QC X 2 O^ 2.00X2.00 Average coefficient of elasticity, E, foi Adapting the general formula ( 8 ), page 530, the numerical constant becomes 4.62 x 10,228 = 47,253, say 47,000. Elastic Deflection of Uniform Rectangular Bars of Wrotight Iron, loaded at the middle. W/ 3 (5) 47, ooo bd* D = the deflection, b the breadth, d the depth, / the span, all in inches; W the weight in tons. Round Iron. For round bars, substitute the above-found value of E, in the general formula ( 24), page 533. Then, 3.1416 x = 3.1416 x 10,228 = 32,116, say 32,000. Elastic Deflection of Round Wrought-Iron Bars, loaded at the middle. W/ 3 D= 32,000 a 4 TORSIONAL STRENGTH OF WROUGHT IRON. (6) Data are extremely scarce. The Swedish iron, page 581, gave a shear- ing resistance of 15.2 tons per square inch. By the general formula STRENGTH OF WROUGHT IRON. 5QI ( i ), page 534, the breaking force at the end of a 1 2-inch lever, applied to a i i/o, -inch round bar, is 12 The actual force was found to be (page 582) = 1.195 tons - Taking the shearing strength of wrought iron at 80 per cent, of the tensile strength, as decided at page 588, put s = the ultimate tensile strength, then h = .So s, and, by substitution in the general formulas for torsional strength (i), page 534, and (6), page 536, For wrought iron round shafts,... W = - R 4.5 R For wrought iron square shafts,... W = l = -- ........... (8) W = the force in tons. R = the radius of the force in inches. W R = the moment of the force in statical inch-tons. tf^the diameter of the round shaft in inches. b = the side of the square shaft in inches. s = the ultimate tensile strength in tons per square inch. Take the tensile strength s equal to 22.5 tons per square inch as an average value j then, by substitution and reduction: Torsional Strength and Sizes of Wroiight-Iron Shafts of average quality. Round shafts : W = l^ .......................................... ( 9 ) R Square shafts :W=7i?A 3 ( n ) R ELASTIC TORSIONAL STRENGTH AND DEFLECTION OF WROUGHT-!RON BARS. The results of experiments with Swedish bars, page 582, show that the elastic torsional strength was 40 per cent, of the ultimate torsional strength. The elastic shearing stress is found by formula ( 3 ), page 535, , WR For Swedish hammered bars, page 582, WR = .474 ton x 12 inches - /TOO ^ ' 2 7 o X I 5.688, the moment of the force; and h- ' = 6.06 tons per square inch, the elastic limit of shearing stress. 59 2 THE STRENGTH OF MATERIALS. The value of E', the coefficient of torsional elasticity, as defined at page 536, is found for the Swedish bar, by the general formula (12), page 537 : TJ,, WR/ .474 X T2 X 10.5 JH, _ = __LLZ _ -=I22Q .873 x i.5*x.oii By inversion and reduction, the equation for torsional deflection is ob- tained : Elastic Torsional Deflection of Round Wrought-Iron Bars. n WR/ = 7^-^ ................................... < I4 > D = the total angular deflection in parts of a revolution. W the twisting force in tons. R = the radius of the force in inches. W R = the moment of the force. / = the length of the shaft in inches. d - the diameter of the shaft in inches. STRENGTH OF STEEL. 593 STRENGTH OF STEEL. MR. KIRKALDY'S EARLY EXPERIMENTS. In the course of the experiments already noticed, 1 page 571, Mr. Kirk-. aldy tested a great number of bars and plates of steel, the general results of which are given in a condensed form in tables Nos. 195 and 196. The bars were from ^ inch to i inch in diameter, and possessed an average tensile strength of from 60 tons per square inch for tool-steel, to 28 tons per square inch for puddled steel. The greatest observed strength was 66.2 tons. The steel plates were from s/ l6 to s/ l6 inch thick. Their tensile strength ranged from 45^ to 32 tons per square inch, with the fibre. The average tensile strength was 40 tons with the fibre; and across the fibre the tensile strength was 36^ tons, or 91 per cent, of the tensile strength in the direc- tion of the fibre. Table No. 195. TENSILE STRENGTH OF ROUND STEEL BARS. 1861. (Mr. David Kirkaldy.) NAME. Condition. Size. Breaking Weight per square inch. Elonga- tion in parts of length. Lowest. Highest. Average. Turton's cast steel, for tools \ Jowitt's do. do. I Do. do. chisels \ Do. double shear steel... / Do. cast steel for drifts... 1 Bessemer tool steel. . / Forged, reheated, and cooled gradu- ally. Rolled. J5 Forged. Rolled. inch. 53to.59 .5610.58 .56 to. 60 . 5 6&. 57 .6510.75 75 75 .5 7 to .60 57&-S9 .91 to .93 .56 75 55to.57 75 75 to.i.o 75 77 tons. 50.10 52.55 50.15 47.65 43-15 46.10 45.27 45.27 39-97 37.51 38.42 36.70 38.30 29.07 29.94 24.55 19.00 20.50 tons. 64.90 66.20 61.60 55-95 58.25 54-97 52.12 50.12 51.87 49.42 42.92 44-45 42.30 36.92 33-62 33.54 31-93 31.40 tons. 59.32 59.10 5575 52.87 51.76 49-75 47.90 47-60 46.56 45-15 41.08 40.47 40.05 32.37 3i.9i 31.32 28.24 28.05 per cent. 54 5.2 7-i 13-5 13-3 5-5 ' 12.4 8.7 9-7 10.8 15-3 137 11.9 1 8.0 '19.1 ii-3 12.0 9.1 Moss & Gamble, cast steel for rivets Naylor, Vickers, & Co., cast steel for rivets Wilkinson blister steel . Towitt's cast steel, taps Krupp's cast steel, bolts Homogeneous metal Do. do Forged. ?> Rolled. Forged. ?j Towitt's spring steel Mersey Co.'s puddled steel... Blochairn do. Do. do. Do. do. Averages 38.50 46.80 42.66 II. 2 1 Experiments on Wrought Iron and Steel. 38 594 THE STRENGTH OF MATERIALS. Table No. 196. TENSILE STRENGTH OF STEEL PLATES. 1861. (Mr. David Kirkaldy.) DESCRIPTION OF STEEL. Thickness of Plate. Breaking Weight per square inch. Elongation in parts of length. With Fibre. Across Fibre. With Fibre. Across Fibre. Turton & Sons, cast steel Shortridge & Co do inch. 3/i6 ! 3/ l6 & i/ 4 3/8 '/8 & 3/ l6 3/i6 5/i6 I tons. 42.10 42.97 36.48 33-75 45-28 45-80 45.64 43.00 32-32 3440 31-93 tons. 43-00 43-37 38-90 30.84 43-30 37-93 38.11 37.67 32.90 32.85 30.22 per cent. 5-7 8.6 17-5 19.8 2.8 4 i 3-6 8.2 5-9 6.2 3.6 per cent. 9.6 8.9 17-3 19.6 14.4 i-3 3-3 2.7 4.1 3-2 5-7 Naylor, Vickers, & Co., cast steel... Moss & Gamble, do. Shortridge & Co., do. ... Mersey Co puddled steel Mersey Co., " hard " puddled steel.. Blochairn puddled steel Blochairn, do. Shortridge & Co do Mersey Co., "mild" puddled steel- Mersey Co., do. do. Total averages 39-42 37-17 7.8 8.2 Averages for comparison of strengths ) lengthwise and across I 40.17 36.56 8.2 7.6 Mr. Kirkaldy discovered that the strength of steel was materially in- creased by hardening the metal in oil; and that it was materially reduced by hardening in water. Three pieces from a bar of chisel-steel were so treated, with the following results : Tensile Strength. Soft steel ............................................ 541^ tons. * cooled in water cooled in oil Coal-tar and tallow were used for cooling steel, and with good effect; but they were not so efficacious as oil. Steel plates, similarly treated in oil, acquired a gain of strength varying from 56.4 per cent, for the highest temperature at which they were cooled to 12.8 per cent, for the lowest. The shearing strength of steel rivet-iron was found, from seventeen tests, to average 74 per cent, of the ultimate tensile strength of the same bar. STRENGTH OF HEMATITE STEEL. 1866. Mr. Kirkaldy tested the strength of bar-steel manufactured by the Barrow Hematite Steel Company. Four samples were tested for each kind of stress: For tensile stress, cast steel, forged and turned to i^ inches diameter; length, 14 inches. For compressive stress, hammered cast steel, forged and turned to i^ inches diameter; length, 14 inches. For shearing stress, hammered cast steel, forged and turned to ij^ inches STRENGTH OF STEEL. 595 diameter. For transverse stress, hammered cast steel, i^ inches square; span, 25 inches. For torsional stress, Bessemer cast steel, forged and turned to i^ inches diameter; length, 8 diameters. ELASTIC, per square inch. Tensile strength 1 8. 63 tons. Compressive strength 23.21 Shearing strength ULTIMATE, per square inch. 32.27 tons. 71.24 25.21 Elongation. 19.2 per cent. {Elastic load 3.80 tons. Ultimate load ) (uncracked) j 7 ' 35 " Torsional strength ) F1ocfiV ^ ^ i-3 Deflection .122 inch. 6.64 ins. .008 turn. 1-54 STRENGTH OF KRUPP STEEL. 1867768. Blocks of Krupp's cast steel from the heads of broken crank-shafts of the " Jeddo " and the " Sultan," were cut up into numerous specimens by Fig. 209. Krupp Steel Crank-shaft, "Sultan." Mr. Kirkaldy, and tested for strength. The annexed Fig. 209 shows how the broken crank of the " Sultan " was divided and cut up. 1 Specimens. For tensile strength, ....... For compressive strength, For transverse strength,... < Depth .... ( Span For torsional strength, ..... < T 1 JEDDO. SULTAN. leter ... 1.25 inches. 1.128 inches. th 8.5 icter ... 10.0 1.128 th 1.128 1th 1.37 h 1.76 ... . 10 1.50 1.91 10 1.128 2 diamete rs. leter... 1.25 th 2 diameters. J The author is indebted to Mr. Longsdon for copies of the " Results of Experiments," from which the above particulars have been reduced. 59 6 THE STRENGTH OF MATERIALS. Average Results. For tensile strength: J EDDO . SULTAN. Elastic strength 18.53 tons. 19.10 tons. Do. extension 541 per cent .586 per cent. Do. do i in 185 i in 171. Elastic extension per ton per ) , , , i i ,1 f 7^28? or .000202. /o 2 66 or .000300. square inch ; length = !..../ ' 34 Breaking weight 41.18 tons. 42.07 tons. ^weight . elaStiC . t0 . bre . aking } Per cent. 4 5-4 per cent. Permanent extension 12.6 7.9 ,, Sectional area of fracture 77.4 76.9 For compressive strength: Elastic strength 21.13 tons. Do. compression .798 per cent. Do. do i in 125. Elastic strength, per ton per ) , square inch; length = i ... } '/** or 377- Breaking weight 89.30 tons. For transverse strength : Elastic stress 7.94 tons. 10.74 tons. Ultimate stress 21.31 27.14 Ratio 37.2 per cent. 39.6 per cent. Elastic deflection 055 inch. .082 inch. Ultimate deflection 1.49 1.19 For torsional strength : > .491 ton. .497 ton. Ultimate stress 1.280 1.068 Ratio 38.4 per cent. 47.3 per cent. Elastic torsion 005 turn. .on turn. Ultimate torsion 441 .339 The lowest ultimate tensile strength of the steel of the " Jeddo " was nearly 10 tons per square inch below the average; of that of the "Sultan" it was 2^ tons below the average. The strength of the specimens cut from the interior of the blocks averaged very little less than that of those from the exterior. The crank-shaft of the " Jeddo " was supplied to replace a broken shaft of wrought iron, noticed at page 576, of which the tensile strength averaged about 20 tons, as against 41.2 tons for the steel shaft. EXPERIMENTS OF THE STEEL COMMITTEE. A Committee of Civil Engineers 1 instituted and completed a series of experiments on the strength of steel bars, in 1868-70. They were con- 1 The Committee consisted of Messrs. W. H. Barlow, George Berkley, John Fowler, Douglas Galton, C.B., and J. Scott Russel. Mr. Berkley, Secretary; Mr. W. Parsey, Assistant- Secretary. STRENGTH OF STEEL. 597 ducted with every provision for insuring accuracy; and the results were printed in two reports, from which the following particulars are derived. First Series of Experiments. The first series of experiments, 203 in number, were conducted by Mr. Kirkaldy, under the instructions of the Committee, with his testing machine, in which the amounts of extension, compression, and deflection were read off a dial. The experiments were directed to test the resistance of steel bars to tension, compression, transverse strain, and torsion. Twenty-nine samples of steel bars, 2 inches square and 15 feet long, of the best marketable quality ordinarily made, were obtained from ten manufacturers; of these, 1 8 were of Bessemer steel, and 1 1 of crucible steel. Each bar was parted into lengths by a shaping machine, for bending, twisting, pulling, and thrusting, as shown by Fig. 210. For pulling, the specimen was prepared as in Fig. 211, and divided into inches of length; for bending, as in Fig. 212 ; for twisting, as in Fig. 213; and for thrusting, or compression, as in Fig. 214. For tensile, Bending. Twisting. Pulling. Thrusting. Do. Spare i> i i > j 11 Spare. Fig. 210. Specimen Bars how divided. SO INCHES- Fig. 2ii. Graduation of Bars for Pulling Stress. ~ =l 3= =3B ffl Fig. 212. For Bending Stress. Fig. 213. For Twisting Stress. Fig. 214. For Thrusting or Compressive Stress how divided. SPECIMEN BARS OF THE STEEL COMMITTEE, ist Series. compressive, and torsional tests, the bars were turned down to 1.382 inches in diameter, having a sectional area of 1.5 square inches, and highly polished. For bending, or transverse tests, they were planed to a section of 1.9 inches square. The final results have been condensed from the report, and are worked out in the following tables : Table No. 197 shows the tensile strength of the steel bars. Table No. 198 shows the elastic compressive strength of the steel bars. The ultimate compressive strength of short specimens, which is always an indefinite quantity, is not given here; but it may be stated that the short specimens required a great deal to crush them; and that the long specimens, 36 diameters in length, failed by buckling, when the elastic limit of stress was arrived at. Tables Nos. 199 and 200 show the transverse strength and the torsional strength of steel bars, distinguishing the elastic from the ultimate stress. 598 THE STRENGTH OF MATERIALS. I a co vo ! co M ^ C/5 ! 1 ^ "o w M t/3 U I g vU w B , c/5 w H STRENG diameter "S C/2 erf S w 'ENSILE inches fti 1 OH X w C/) 1 | * 1 ON o 1 8 T3 .2 E 1 3 rt ss c3 rO H 0) 1 o .s 6 H i'sjj | to Tf co to M CN ON o to III 10 M N 10 Jj to to tovo M N . M vo vo to C '55 g M M 00 10 M t^ ON tO Tt-00 vo S i. M M M t^ ONCO N O CO !> Q 4; X AH . S!! . rt ^ jn to ^J- co O W CO vo to "^ t^ to vo "3- ON r^ O J CO co O CO ^>% l> c . M co co ^ co > coco CO fl fl C fl fl g fl fl C fl < fl C fl I! s M M H J-H VH ^-( ID ? t y t ^-l II g, o o o o O O O CO O O O O ^00 O CO 00 O O to *> vo" ON tOVO 10 to ON tovO vo vo vo J a N M M N N N N W M N of-Sjjf 5 M-JT EL- s O ^** co O "j COOO ^ co N VO ^ to vo tovO ** CO Tf M CO ~ J>'2 q ^ 2 co IH M ON O to ONOO M M M W^ A 2| W N N M N (N M M N M N 1 : 'I a 3 V r/3 c/5 N ^ i JJ ^2 X S|i T - Tl ,H **"* Cu JH t/2 < ^ ^ - g> ^ .v ^ OJ 0^ a s g< 'O 'O 'O ^ O 'U 'O -^ OT 6 s > ^H VH >-H "*"" ^ vU O vl) *\ CJ O O CD (U g Ills HIS |* J$ ft c\j cj c^ o KWffll wu | I to to ^f ^F to ^F t-T M" co" M" ? STRENGTH OF STEEL. 599 Table No. 198. COMPRESSIVE STRENGTH OF STEEL BARS. 1868. Two-inch square bars turned to 1.382 inches in diameter (1.5 square inches of area). Lengths, various. (Reduced from the Experiments of the Steel Committee, ist Series.) BESSEMER STEEL. DESCRIPTION, with respective number of samples. Elastic Strength (compressive) in tons per square inch. Elastic Compression per ton per square inch, in parts of the length of 36 diameters. Length, i diam. ; 1.38 in. Length, 2 diams. ; 2.76 in. Length, 4 diams. ; 5-53 in. Length, 36 diams. ; 50 in. 5, Hammered, tyres,. 5, Hammered, axles,. 4, Hammered, rails,.. 4, Rolled, tons. 23.03 23-84 23.88 18.98 tons. 22.32 22.76 23.32 18.30 tons. 22.23 21.34 21.77 18.07 tons. 19.15 18.51 18.95 17.20 Length=i. .000065, or i/ 15>3 8 S .000062, or i/x6>fa .000065, or i/ IS , 3 8 S .000065, or 1/15,385 5, Hammered, tyres,. 4, Hammered, axles,. I, Hammered, rails,.. i, Rolled axles CR 24.02 26.90 26.78 21.87 UCIBLE 23.93 26.11 26.34 19.64 STEEL. 21.74 23.99 20.55 18.75 21. II 22.30 19.54 18.77 .000065, or 1/15,385 .000065, or 1/15,385 .000065, or 1/15,385 .000069, or 1/14,493 1 8, Bessemer steels,...! n, Crucible steels, 29, Steels, SUMI 22.43 24.89 dARY A\ 21.67 24.01 fERAGES. 20.85 22.26 18.45 20.43 .000064, or 1/15,625 .000066, Or x /i5,i52 2 3 .66 22.84 21.55 19.44 .000065, or Vi5,385 Table No. 199. TRANSVERSE STRENGTH OF STEEL BARS. 1868. Two-inch square bars planed to 1.9 inches square. Distance of supports, 20 inches. BESSEMER STEEL. DESCRIPTION, with respective numbers of samples. Elastic Stress. Ultimate Stress. Ratio of Elastic to Ultimate Stress. Ultimate Deflection. REMARKS. AVERAGES. 5, Hammered tyres, tons. t? 7-99 6.61 tons. I3-I7 13-20 12.85 n-75 per cent. 57-3 61.5 6l.2 56.3 inches. 3-82 4.08 3-94 4-03 Bent to 6 inches; uncracked. 5, Hammered, axles, 4 Hammered rails . 4, Rolled ; tyres, axles, rails, . . . CRUCIBLI 5, Hammered, tyres, II 8.36 4, Hammered, axles, 8.38 l f Hammered, rails, ... 1 i XT : STEEL 14.65 17.91 12.06 57-4 53-9 43-6 53-8 3-32 3-35 3-65 3-84 \ In most f cases bent f to 6 inches; ) uncracked. I, Rolled, axles, Su 1 8 Bessemer steels MMARY j IVERAGI 12.74 15.04 s. 59-5 52.2 3-97 3-54 1 1, Crucible steels, 29, Steels, 7-74 13.89 55-7 ' 3-76 6oo THE STRENGTH OF MATERIALS. Table No. 200. TORSIONAL STRENGTH OF STEEL BARS. 1868. Two-inch square bars, turned to 1.382 inches in diameter (1.5 square inch of section). Length for torsion, 8 diameters = 1 1 inches. (Reduced from the Experiments of the Steel Committee, 1st Series.) BESSEMER STEEL. DESCRIPTION, with respective number of samples. Elastic Stress at the end of a i2-inch lever. Elastic Torsion. Ultimate Stress at the end of a i2-inch lever. Ratio of Elastic to Breaking Stress. Ultimate Torsion, * uncracked, for 3.75 turns. Least. Greatest. Average. AVERAGES. tons. i turn = i. tons. per cent. i turn=i. i turn=i. i turn=i. 5, Hammered, tyres, .701 .014 i-54 45-4 I.SO 2-73 2.21 5, Hammered, axles, .667 .Oil 1.47 44.9 2-33 3-75* 3-07 4, Hammered, rails, .688 .012 1-45 46.8 2.10 3-75* 2-73 4. Rolled; tyres, ) axles, rails,... \ .569 .008 1.44 39.5 2.6l 3.75* 3-" CRUCIBLE STEEL. 5, Hammered, tyres, 736 .014 1.59 46.6 II 1.77 3-39 2.33 4, Hammered, axles, 731 .013 1.69 43-4 1-07 2.32 1.79 i, Hammered, rails, .714 .016 1.81 40.0 .86 .86 .86 i, Rolled, rails 554 .012 i-34 42.7 1-73 2.14 1.94 SUMMARY AVERAGES. 1 8, Bessemer steels, . . . .656 .Oil 1.47 44.6 2.78 1 1 , Crucible steels, .... .684 .014 1.61 42.5 i-73 29, Steels, .670 .013 1-54 43-6 2.26 The lowest tensile and compressive elastic strengths, ranged, for each group in the first series, about 5 tons per square inch below the averages given in the tables; say 20 per cent, below the averages. The same proportionate range is found in the elastic resistances to torsional and trans- verse stress. Second Series of Experiments (made at Woolwich Dockyard]. The object of the second series of experiments by the Steel Committee, was to make experiments on the tension and compression of long steel and iron bars, measuring the changes of length directly from the bars. For this purpose, 91 bars of steel and iron, each 14 feet long and i% inches in diameter, were obtained, consisting of 33 bars of Crucible steel, 34 bars of Bessemer steel, 1 2 bars of Lowmoor iron, 6 bars of best Yorkshire iron, and 6 bars of usual S. C. Crown, or Staffordshire iron. The extensions were measured on 10 feet length of each bar; and for compressive tests, the bars were cut to a length of 1 2 feet, and the measure- ments made on a length of 10 feet. The bars were tested in their natural skins. Before they were tested, they were thoroughly examined and straightened, and the diameters checked by means of vernier callipers, capable of showing a variation of a icooth part of an inch. The results for iron bars have been given at page 579. Table No. 201 gives the condensed results of the experiments for the tensile strength of steel bars, and table NH 00 ^-0000 vo tn vO HI rt-0 ^0 to tood | 00 t^ >* ONOO ON O ON ON ON ON ON to < n ^-oo' CO* 1 O c4 ft M M M CO ION M 00 O t^. 00 -^ M.S ^-g . vo w OO in T}- M in toco |2 *" O rt-oo to mo vo <* ^2-^S-S OO 0* 10 10 CO CO IPI ** vnvoS?3- Vcoco coco to rf co co CO co co CO *J CO rt ^ 1 -i IL C "^ hn 10 m O O M m 10 N rn en cT cT cT cT a OO W tr* S 'wo c II HHHK^H^^HH f"^ ^4 C3 CX3 g l' OOOOOOOOOO 000 "" o S 1 1 ' J 2 w _^j ^3 Ifffllfffl vo to O N . 0000 c/3 XN J>* 00 g jS' S JK ' ' ' ' ' > ^ o3 * ,, y -S UH W 2 i> o J ^ ^1 1 a si 2 c oo vn OO M O J>" O fO O vO ^ W M M ^ 10 t** 10 O g t^. vo >-< Tf 1 "s 1s"s > to Pi 10 ON Sec CO VO c 3 -S I u 4) 0) Pq IM H IH IN 0R Sal X"5 g -g x w<~ ^. oooooooooo 0000 W *\ ^-a S a O Q Q\ Q^ HH V.Q f/^ O ^-O O ^O 00 O O to 1O "5T CO OO rf CO vo M i ^ fi o, ^ n ||S.I w O 10 O O tooo O O I s * to m O tn to '++ ON ^ J H||!f _^*^ 0; J^vo CO 00 N n O f2 TT^ : : 4) F-H ' : ::::;: 1 j J M T3 " ^ C " *Q_. : : i'&i : : : c3 . g ^ - : 4) K $ ' : ' : 1 = = ' : $ g : co 1 1 i '" I ilf fl * ^ Description, g ^ Jf -d d 1 w P^jtn t/jWDiuai^ **a, ui.l s w ^- }!ff Crucible steels Bessemer steel 1 ^3 602 THE STRENGTH OF MATERIALS. Table No. 202. COMPRESSIVE STRENGTH OF STEEL BARS. 1870. \ l / 2 inches in diameter ; lo-feet lengths. (Reduced from the Experiments of the Steel Committee, 2d Series. ) CRUCIBLE STEEL. Description, and Reference Letter. Elastic Strength (compressive) in tons per square inch. Elastic Compression. Elastic Compression per ton per square inch, in parts of the length. a Chisel; 3 samples tons. 26.33 26.2 25.5 26.0 18.0 16.2 24.0 19.5 27.0 24.0 per cent. .198, or i in 506 .202, or I in 496 .186, or in 537 .198, or in 506 .137, or in 731 .126, or in 791 .180, or in 555 .138, or in 722 .204, or in 490 .185, or in 541 length=i. .000075, or 1/13,333 .000077, or Vxa^ty .000073, or '/i 3 ,699 .000076, or '/xs.xsl .000076, or Vis.iss .000080, or '/".soo 000075, or */ I3 ,33 3 .000071, or Vi^oSs .000075, Or '/X3.333 .000077, Or Viz.gSj b Tyre; 3 samples c 2 samples d Rods; 2 samples e f Gun-barrels; 3 samples f Hammered ; 2 samples 2 samples / Rods j Rolled k Fagotted, hammered ) and rolled ; 3 samp. ) 1 3 samples BESSEMEI 1 8.0 21.2 16.0 1 6.0 . STEEL. .133, or i in 751 .163, or i in 612 .125, or i in 801 .125, or i in 801 .000074, or x /i 3 ,si 4 .000077, or i/ I2)9 8 7 .000078, or x/12,821 .000078, or i/ I2>8a i m 2 samples " n Tyres, axles ; 3 samp. Crucible steels SUMMARY 23-3 17.8 AVERAGES. .175, or i in 570 .137, or i in 732 .000076, Or z /i3,25o .000077, Or Vi3,040 Bessemer steels All steels 20.5 .156, or i in 641 .000076, or VIS.HO BARS TESTED- o Crucible steel ; axles, ) rails, tyres (3 samp.) f p Bessemer steel ; axles, ) rails, tyres (3 samp.) j for Compres 23-0 24.0 sion, but not for Ext .172, or i in 581 .182, or i in 550 ension. .000073, Or '/I3.700 .000074, or 1/13,514 The lowest elastic strength in any group of bars, in the second series, did not exceed one ton per square inch less than the average elastic strength of the group say not more than 5 per cent, less than the average for steel bars. Table No. 203 shows the chemical composition and specific gravity of fourteen of the bars subjected to tests. Tensile Strength of Tempered Steel. The Steel Committee publish the results of experiments made at H. M. Gun Factory, Woolwich, on the comparative strength of untempered and tempered steel. A summary of the results is given in table No. 204. The specimens were .50 and .53 inch in diameter, and from i inch to two inches in length. STRENGTH OF STEEL. 60 3 Table No. 203. CHEMICAL ANALYSIS AND SPECIFIC GRAVITY OF STEEL AND IRON BARS. (Tested by the Steel Committee, 1870.) CRUCIBLE STEEL. CHEMICAL CONSTITUENTS. Ultimate Tensile Reference. Iron. Carbon. Silicon. Man- ganese. Sulphur. Phos- phorus. Gravity. Strength per square inch. p. cent. p. cent. p. cent. p. cent. p. cent. p. cent. tons. a 98.86 79 .115 .19 trace .01 7.839 52.76 b 98.67 67 .20 44 trace .02 7.831 51.01 c 98.87 57 .14 37 .01 .04 7.851 43-48 d 98.63 .90 39 .02 .06 7.844 41.85 e 98.87 .58 .22 30 .02 .01 7.825 40.50 f 98.88 47 .61 .02 .02 7.845 38.51 99.22 59 03 .14 trace .02 7.859 h 99.16 44 .14 23 .01 .02 7.850 35-47 i 99.16 52 .10 .19 trace 03 7.850 Averages, 98.89 .62 .114 34 .01 .026 7.842 42.15 BESSEMER STEEL. P 99.24 34 trace 35 .04 03 7-857 I 99-21 32 .01 .40 .04 .02 7.857 34.19 m 99.22 .05 .38 .02 .02 7.853 33-68 n 99-13 35 03 44 .04 .01 7.852 33-66 Averages, 99.20 33 .022 39 035 .02 7-855 33-84 YORKSHIRE IRON. II 99-49 23 .10 .08 .02 j .08 || 7.758 || 23.69 Table No. 204. TENSILE STRENGTH OF TEMPERED STEEL. Average Breaking Weight per Square Inch. Manufacturer, or MATERIAL. Number of After being Tempered in Oil at Contractor. Specimens. Tempered As Received. High Medium Low Water. . ^ Kelt. Heat. Heat. ? tons. tons. tons. tons. tons. tons. Krupp, Firth, Cast Steel,.... Steel, 9 3 32.1 34-4 "T 65.4 54-4 56.4 Firth, Homog. Steel, 217 31.6 47-7 48.0 47-0 44.4 Cammell, .. Steel, 2 26.6 Cammell, .. Homog. Steel, 61 29-5 54-6 45-7 51-5 49.0 51-1 Steel, 4 3!-7 36.6 Homog. Steel, 36 29.0 37-2 39-3 53-9 Styrian Steel,. 8 56.2 82.2 Moser Steel, 7 33-5 54-6 55-8 Whitworth, Steel, 2 38.2 48.2 6o 4 THE STRENGTH OF MATERIALS. STRENGTH OF FAGERSTA STEEL. 1873. Mr. Kirkaldy made a comprehensive set of experiments on the strength of steel manufactured at the Fagersta Works, Sweden. First Series of Experiments. Twelve hammered bars, 2 inches square, in four groups of different degrees of hardness, here distinguished as a, b, c, d, were tested for tensile, compressive, shearing, torsional, and transverse strength three samples for each test. For the transverse tests, the specimens were planed to 1.9 inches i 20' Fig. 215. Fagersta Steel Test Specimen for Bending or Transverse Stress. square; for the other tests, they were turned to a diameter of 1.128 inches, having i square inch of sectional area. The forms of the specimens are shown in Figs. 215, 216, 217. The condensed results are given in tables Nos. 205 to 207. Table No. 205. FAGERSTA STEEL BARS TRANSVERSE STRENGTH. 1873. 1.9 inches square; span, 20 inches. Load applied at the middle. BARS. Elastic Stress. Ultimate Stress. Ratio of Elastic to Ultimate Stress. Ultimate Deflection. Remarks. a tons. Q. 1-2 tons. 14. ^ per cent. 66.0 inches. 78 fractured b y--}--} Q.OQ IQ.C7 4Q.6 1.4-0 fractured c 8.l8 17. 0'? 4.8 3. 31 uncracked d 7.O4. 11.28 62.3 c.I I uncracked Averages 8.58 15.61 56.5 2.67 Table No. 206. FAGERSTA STEEL BARS TORSIONAL STRENGTH. 1873. Diameter 1.128 inches (i square inch section). Length for torsion, 8 diameters. Stress applied at the end of a 1 2-inch lever. BARS. Elastic Stress. Breaking Stress. Ratio of Elastic to Breaking Stress. Ultimate Angular Torsion. Least. Greatest. Average. a tons. .507 .502 .484 341 tons. .946 1.043 1.009 .679 per cent. 53-9 48.2 48.3 50.2 i turn=i. .207 .625 .897 3-053 i turn=i. 410 .938 1.255 3725 i turn=i. .2 9 I 793 1. 02 1 3.219 b . . c d Averages .458 .919 50.2 I.I95 1.528 I-33I STRENGTH OF STEEL. 605 Table No. 207. FAGERSTA STEEL HAMMERED BARS TENSILE, COMPRESSIVE, AND SHEARING STRENGTH. 1873. 2-inch square bars turned to 1.128 inches in diameter (i square inch of section). TENSILE STRENGTH. BARS. 10. 15 inches long. (9 diameters.) Elastic Strength in tons per square inch. Breaking Weight in tons per square inch. Ratio of Elastic to Breaking Strength. Permanent Extension. Sectional Area of Fractujer--- /, >''' a tons. 27.7O tons. ^S.OA per cent. 71.1 per cent. 1.8 petipent^y b . 28.Ii; S 47.6o CQ.4. c.i QW&7 c 2C..Q4 45.82 5 X 7 ;6.6 6.6 8c h d IQ.24 27.77 70 i i6c. 385 J'-'O Averages 25.25 39.16 64.8 7-5 78.9 COMPRESSIVE STRENGTH (per square inch). BARS. Length, i diameter, 1.128 ins. Length, 2 diameters, 2.25 inches. Length, 4 diameters, 4.51 inches. Length, 8 diameters, 9.02 inches. Elastic Strength. Elastic Strength. Destroying Weight. Elastic Strength. Destroying Weight. Elastic Strength. Destroying Weight. a tons. 28.57 27.98 26.78 1741 tons. 28.27 26.19 25-I3 18.75 tons. 75.85 77-37 69.64 54.15 tons. 28.27 26.19 23.81 18.30 tons. 59.92 52.50 47.01 36.50 tons. 27-53 25.89 23.51 18.16 tons. 45.62 42.50 37.87 21.05 b c d Averages 25.18 24.70 69.24 24.03 48.88 2377 36.76 SHEARING STRENGTH (per square inch). BARS. Ultimate Shearing Strength. Detrusion before Rupture, as a measure of hardness inversely. Per square inch. Per cent, of Ultimate Tensile Strength. Actual. In parts of the diameter. a tons. 27.42 35.60 31-99 20.28 per cent. 73.3 75-2 69.5 74.0 inch. 193 .249 .281 .323 per cent. 17 21 25 29 b c d Averages. . 28.82 73-5 .261 23 For Shearing Stress. For Pulling or Tensile Stress. or Figs. 216, 217. Fagersta Steel Test Specimens. 6o6 THE STRENGTH OF MATERIALS. Second Series of Experiments on Fagersta Steel. To test the influence of hammering, and of annealing steel bars. Four ingots 6 inches square, differing in hardness, -inch square bars. i k I Averages, ... 42.05 ... ... 34.96 ... ... 25.67 ... ... 60.59 ... 42.84 ... ... 32.21 ... ... 69.4 ... ... 51.6 ... ... 79.7 ... ... 5.7 ... 16.0 ... IO.I ... ... 34.23 ... ... 45.21 ... ... 66.9 ... 10.6 ... 3-inch square bars. i k I Averages, HAMMERE ... 20.85 ... ... 13.30 ... ... 12.68 ... D ANNEALED ... 31.66 ... ... 31.09 ... ... 25.28 ... ... 65.8 ... ... 42.8 ... ... 50.2 ... 1.7 ... ... 7.7 ... ... 25.2 ... ... 15.61 ... ... 29.34 ... ... 52.9 ... ... 11.5; ... >-inch square bars. i k I Averages, ... 31.12 ... ... 21.34 ... ... 14.24 ... ... 54.92 ... ... 36.66 ... ... 56.8 ... ... 8.3 ... ... 25.39 ... ... 56.1 ... ... 12.6 ... ... 22.23 ..- ... 38.99 ... ... 57.0 ... ... 9.5 ... 3-inch square bars. i k I Averages . ROLLED ... 2687 UNANNEALED. 59.71; ... 79-6 ... ... 48.7 -. ... 43.2 ... .6 ... ... 2.5 ... ... 31.1 ... ... 13.57 ... ... 10.22 ... ... 27.85 ... ... 23.64 ... 16 89 ... 28.41 ... ... 57.2 ... ... 11.4 ... >-inch square bars. i k I Averages, ... 35.09 ... ... 20.89 ". ... 15.09 ... ... 59.82 ... ... 40.50 ... ... 27.13 ... ... 56.2 ... ... 51.6 ... ... 55-6 ... 16.0 ... ... 22.2 ... ... 23.69 ... ... 42.48 ... ... 54.4 ... ... 15.2 ... 6o8 THE STRENGTH OF MATERIALS. Table No. 209 (continued}. ROLLED ANNEALED. Bars of Various Degrees of Hardness. Elastic Strength in tons per square inch. Breaking Weight in tons per square inch. Ratio of Elastic to Breaking Weight. Permanent Extension. 3-inch square bars. i k I tons. ... 19.78 ... ... 12.32 ... ... 10.22 ... tons. ... 32-55 ." ... 26.87 -.. ... 23.64 ... per cent. ... 60.7 ... ... 45-8 -. ... 47.7 ... per cent. 2.0 ... ... 3.8 ... ... 26.0 ... ... 14.11 ... 27.6Q . "U.4 10.6 j^-inch square bars. i k I ... 28.93 . ... I8. 39 - 12. AC ... 57.14 ... ... 35.81 ... 23.CO . ... 50.6 ... ... 51.4 ... C^ o ... 8.5 ... ... 9.8 ... 27 I Averages. ., ... 2O.OO ... 38.82 , ci.7 13.8 Third Series of Experiments on Fagersta Steel. To compare the tensile strength of steel bars, reduced by hammering and by rolling. Bars of three degrees of hardness were tested, say /, k, I. Of each degree, six 3-inch square hammered bars were tested, five of which were reduced by hammering to 2%, 2, ij^, i, and ^ inch square, and then turned to given diameters. In table No. 209 are given comparative results for the 3-inch and ^-inch square bars, hammered and rolled. The original table shows that the strength was proportionally increased as the bars were reduced in size. Fourth Series of Experiments on Fagersta Steel. To test the tensile and compressive strength of Fagersta steel plates, of /^> ~/i, Y%) YZI an d y% inch thickness, cut into strips 2^ inches wide. Table No. 210 gives the comparative results of the trials. Fifth Series of Experiments on Fagersta Steel. To show the variations in results for tensile strength, arising from differ- ences in the form and proportions of specimens. Two sets of specimens were prepared according to Figs. 218; one set was 10 inches wide and 10 inches long at the parallel middle portion; and the smaller set i*^ inches wide and 4}^ inches } ol[i S a * the middle. Condensed results are given in table No. 211; and the results of the i oo-inch bars, from table No. 210, are added, for comparison. STRENGTH OF STEEL. 609 Table No. 210. FAGERSTA STEEL PLATES TENSILE AND COMPRESSIVE STRENGTH. 1873. Specimens 2j^ inches wide, 100 inches long. TENSILE STRENGTH UNANNEALED. PLATES. Thickness. Elastic Strength (Tensile) in tons per square inch. Elastic Extension. Breaking Weight in tons per square inch. Permanent Extension. Ratio of Elastic to Breaking Strength. Sectional Area of Fracture. inch. tons. per cent. tons. per cent. per cent. per cent. H 17.37 .136 24.61 5.21 70.6 62.1 X 15.89 .124 24.17 10.17 65.7 46.3 3 /B H.34 .091 21.84 20.64 51.9 29.0 YZ 12.28 .082 22.39 16.30 54-8 38.8 H 11.65 .078 22.00 17-95 52.9 39-3 Averages 13.71 .102 or i in 980 23.00 14.03 59.2 43.i ANNEALED. 1 A 11.92 .096 20.30 10.98 58.7 364 X 13-30 .098 22.14 16.88 60. i H 11.56 .096 20.86 18.19 55-4 30.4 /4. 12.19 .088 22.09 19.15 55.2 357 * 11.25 .088 21. l8 1745 36.9 Averages 12.04 .093 or i in 1020 21.31 16.53 56.5 344 Elastic extension per ton ) Unannealed 0000744, or 1 /i 3 ,4 3 s. per square inch, J Annealed 0000772, or */ I8 a8, COMPRESSIVE STRENGTH UNANNEALED. PLATES. Thickness. Elastic Strength (Compressive) in tons per square inch. Elastic Compression. Elastic Compression per ton per square inch, in parts of the length. inch. tons. per cent. Length = i. X I7.8l .106 X 1 6.20 .115 X 11.83 .089 % I3.30 .099 H 11.38 .088 Averages 14.10 .100, or i in looo .000071, Or 1/14,100 ANNEALED. # 10.40 063 X 12.28 .088 H 11.25 .083 % 10.13 .074 H 8.98 .066 1 0.6 1 .075, or i in 1333 .0000707, or 1/14,143 6io THE STRENGTH OF MATERIALS. Table No. 211. FAGERSTA STEEL PLATES TENSILE STRENGTH AS AFFECTED BY THE FORM AND PROPORTIONS OF THE SPECIMENS. 1873. Averaged results of specimens from ^ to % inch thick. UNANNEALED. SPECIMENS. Elastic Strength ^ (Tensile) in tons per square inch. Breaking _ Weight in tons per square inch. Ratio of Elastic to Breaking Weight. Permanent Extension. Sectional Area of Fracture. Length = breadth.... Figs. 218 Length = 3 breadths Length = 44 breadths tons. 16.05 15.56 I3-7I tons. 26.39 25.56 23.00 per cent. 60.0 60.3 59.2 per cent. 29.7 35-o 14.0 per cent. 48.8 43-2 43-1 Length = breadth.... Figs. 218 Length = 3 breadths Length /|/| hffadths ANNEAI 13-53 12.98 12.04 ,ED. 23.65 23.16 21.31 57.0 56.0 56.5 33-i 39-3 16.5 39-1 36.5 34-4 Sixth Series of Experiments on Fagersta Steel. To test the influence of holes drilled and holes punched in steel plates. Specimens were formed 12^ inches wide, and otherwise like the broad specimen, Fig. 219, for comparison. There were three rows of rivet holes, 3 inches apart; and five holes in each row at 2^-inch centres. The holes were .77 inch in dia- meter, and made .77 x 5 = 3.85 inches of blank; the net section was (12.5-3.85 = ) 8.65 inches wide, or 69.2 per cent, of the total width. Table No. 212 gives some deductions from the reported results. Table No. 212. FAGERSTA STEEL PLATES TENSILE STRENGTH, WITH RIVET HOLES WHEN DRILLED AND WHEN PUNCHED. Specimens 12^ inches wide. Three rows of holes .77 inch in diameter. UNANNEALED. DRILLED HOLES. PUNCHED HOLES. PLATES. Thickness. Reduced Section in parts of the Total Section. Reduced Strength in parts of Total Strength. Tensile Strength per square inch of Net Section, in parts of that of Unreduced Reduced Strength in parts of Total Strength. Tensile Strength per square inch of Net Section, in parts of that of Unreduced Section. Section. inch. per cent. per cent. per cent. per cent. per cent. % 69.2 74.9 108.5 67.0 97.1 X 6 9 .2 76.6 III.O 68.5 99.2 % 6 9 .2 77.0 1 1 1.6 69.5 100.4 'A 69.2 78-3 H3.5 54.8 79-4 H 6 9 .2 77.2 1 1 2.0 51.0 74.0 Averages . . . 6 9 .2 76.8 III.3 62.2 90.0 STRENGTH OF STEEL. 611 Table No. 212 (continued}. ANNEALED. DRILLED HOL^S. PUNCHED HOLES. PLATES. Thickness. . Reduced Section in parts of the Total Section. Reduced Strength in parts of Total Strength. Tensile Strength per square inch of Net Section, in parts of that of Unreduced Reduced Strength in parts of Total Strength. Tensile Strength per square inch of Net Section, in parts of that of Unreduced Section. Section. inch. per cent. per cent. per cent. per cent. per cent. 1 A 6 9 .2 72.9 105.7 6 7 .I 97.1 X 6 9 .2 74-5 108.0 67.8 98.0 H 6 9 .2 75-i 108.8 66.4 9 6.1 % 6 9 .2 77.0 1 1 1.6 69.9 105.6 X 69.2 75-9 IIO.O 68.7 99.2 Averages... 6 9 .2 75-1 108.8 68.0 98.6 Unannealed. Annealed, per cent. per cent. Note. The average elongation with drilled holes, 14.9 18.0 Do. do. punched holes, 6.3 16.6 Do. do. solid plate, 29.7 33.1 Seventh Series of Experiments on Fagersta Steel. To test rolled steel plates under bulging stress. The specimens were discs, 12 inches in diameter, cut out in the lathe, and pressed through an aperture 10 inches in diameter. The bulger or ram was cylindrical, about 5 inches in diameter; and the preparation for the trial is shown in Fig. 204, page 584, and the finished article, after bulging, in Fig. 205. Table No. 213. FAGERSTA STEEL PLATES RESISTANCE TO BULGING STRESS. Discs 12 inches in diameter; aperture 10 inches in diameter. UNANNEALED. Stress Bulging in inches. Ultimate. Disc. Effect. Thickness. Lbs., 25,000. Lbs., 100,000. Lbs., 200,000. Bulge. Stress. inch. inches. inches. inches. inches. tons. M X 1.86 1.09 z 3.00 3-i i 14.50 31-94 buckled uncracked H .89 2.68 3.22 46.92 )) X .68 1-93 3-33 71.83 * 44 1.61 2.77 3-44 97.90 ANNEALED. X X 2.25 1.32 3-04 3.12 11-53 26.77 buckled uncracked X .94 3-23 43-67 a y* 73 2.06 , 3-34 67.28 H .52 1.72 3-14 3-45 90.09 6l2 THE STRENGTH OF MATERIALS. SIEMENS-STEEL PLATES AND TYRES. 1875. A number of steel plates manufactured to the specification of the Admiralty, by the Landore Siemens-Steel Company, were tested for the Company by Mr. Kirkaldy. By the terms of the specification, it was required that the ultimate tensile strength should be not less than 26 tons, nor more than 30 tons, per square inch, with an extension of 20 per cent, in a length of 8 inches. Strips cut lengthwise of the plate, ij^ inches wide, heated uni- formly to a low cherry-red heat, and cooled in water at 82 F., were to sustain bending double in a press, to a curve of which the inner radius was to be one-and-a-half times the thickness of the plate. Abstracts of the results of the tests for tensile strength are given in table No. 215, together with tests for the tensile strength of steel tyres. Twelve specimens of the plates of the 2d series in the table, were tested for bending, length- wise and crosswise, between supports at 10 inches apart. All the specimens bore the test uncracked. Plates of various thicknesses were tested for resistance to bulging stress, i2-inch discs having been forced through lo-inch apertures, in the manner before described, page 584. All the plates bore the test without cracking. Particulars are given in table No. 214. Two steel tyres, of which the tensile strengths were tested (3d series, table No. 215), were respectively 43 and 37 inches in diameter, and 2.32 and 2.10 inches in thickness. They were collapsed under transverse pressures of 42.22 and 52.16 tons; so that opposite sides of the hoop were pressed into contact with each other. The larger tyre burst at one of the bends ; the smaller remained unbroken. Table No. 214. SIEMENS-STEEL PLATES RESISTANCE TO BULGING STRESS. 1875. Discs 12 inches in diameter, pressed into lo-inch apertures. (Reduced from Mr. Kirkaldy's Reports.) Stress. Bulging in inches. Ultimate. Thickness of Plates. Ib. 25,000. ib. 100,000. Ib. 200,000. Bulge. Stress. EFFECTS. inch. inch. inches. inches. inches. tons. Unannealed. 37 .42 I.7I 3-15 63.750 uncracked. 7i .05 1.09 I. 9 6 3-48 145.500 do. Annealed. 37 .67 2.02 3-17 60.357 uncracked. .41 .56 1.84 3.22 68.191 do. .41 59 1.89 3-23 68.080 do. .50 .29 1.45 2.79 3.31 101.920 do. .62 15 1.40 2.51 3-38 II5.III do. .70 .10 1.26 2.18 3-42 123.260 do. STRENGTH OF STEEL. 6l 3 Table No. 215. SIEMENS-STEEL PLATES TENSILE STRENGTH. From .37 to .71 inch in thickness. (Reduced from Mr. Kirkaldy's Reports.) SERIES i. PLATES OF DIFFERENT THICKNESSES. 1875- Treatment, and Thickness of Plates. Elastic Strength per square inch. Ultimate Strength per square inch. Ratio of Elastic to Ultimate Strength. EXTENSION. Sectional Area of Fracture. At 60,000 Ibs. per sq. inch. Ultimate. LENGTHWAY. inch. Unannealed.... .37 Do 71 Means tons. 15.446 13.572 tons. 32.535 29.870 per cent. 47-4 45.4 per cent. 4.50 6.75 per cent. 22.3 24.5 per cent. 62.5 55-3 14.509 31.202 46.4 5.62 23-4 58.9 Annealed 37 14.062 13.929 13.303 13.125 11.741 10.937 30.143 29.647 29.491 29.388 27-595 26.821 46.6 46.9 45 i 44.6 42.5 40.7 8.00 8.08 8.50 8.66 13.80 17.72 24.8 21. 1 24.8 20.4 25.5 25.0 56.9 55.3 61.5 55-5 56.7 54-5 Do 4.0 Do. . .40 Do 50 Do . . 62 Do. . 70 Averages 12.848 28.848 444 10.79 24.6 56.8 CROSSWAY. Unannealed.... .37 Do 71 Means I5-3I4 13.571 32.442 30.062 47-2 45.1 4.52 7.07 22.4 247 62.5 56.4 14.442 31.250 46.1 5-79 23.5 59-5 Annealed 37 Do 40 13.928 13.840 13-393 13.303 11.741 10.937 29.856 29 ' 8 2! 29.366 29.705 27.040 26.885 46.6 46.3 45.6 44-8 43-4 40.6 9.39 9.07 7.81 8.50 16.61 17.30 26.4 26.3 20-4 20.2 22.7 26.0 53-4 50.4 61.0 53-3 647 49-3 Do. A2 Do. . C2 Do 62 Do 70 Averages 12.856 28.788 44-5 11.44 23.6 59.1 SERIES A nnealed 64 2. PLAI 'ES ANNE. 25-483 26.996 \LED, AN! > HARDEN ED. 24.1 20.2 47-5 5i-3 Do 62 Means 26.240 22.2 494 Hardened: Cherry-red, and \ cooled in water > .64 at 82 F ) 28.867 29.036 22.4 1 8.0 50.7 54-5 j Do 62 Means 28.951 20.2 52.1 614 THE STRENGTH OF MATERIALS. Table No. 215 (continued], SERIES 3. TYRES. EXTENSION. Diameter of Specimens. Elastic Strength per square inch. Ultimate Strength per square inch. Ratio of Elastic to Ultimate Strength. Sectional Area of Fracture. At 60,000 Ibs. per Ultimate. sq. inch. inches. tons. tons. per cent. per cent. per cent. per cent. istTyre, specimen. 1.511 17.098 29.853 57-2 6.58 1 8.8 55.8 1.511 17.321 30.800 5 6.2 6.20 23-6 51.9 2d Tyre, specimen. 1.511 18.482 30.083 61.4 6.48 17.7 58.8 1.511 18.840 3L075 60.6 5.59 16.9 70.4 Averages . . 17-935 30-453 58.8 6.21 19.2 59-2 WHITWORTH'S FLUID-COMPRESSED STEEL.* On Sir Joseph Whitworth's system of treatment, a pressure of 6 tons per square inch is applied as quickly as possible to melted steel, after it is taken from the furnace. A column 8 feet high is reduced i foot in height in the course of five minutes. Specimens for testing tensile resistance are cylindrical, formed as in Fig. 220; the central portion has a sectional area of % square inch, being Figs. 220, 22T. Whitworth's Fluid-Compressed Steel Test Specimens. .798 inch in diameter, and has length of 2 inches, or 2^ diameters. The upper and lower portions are screwed, and are seized by nuts. The usual appearance of broken specimens is shown at Fig. 221. Table No. 216 gives results of tests for the tensile resistance of fluid- compressed steel, and of the purest and best irons made in England. Sir Joseph Whitworth states that he can produce, with certainty, by com- pression, steel having 40 tons ultimate strength, with 30 per cent, ductility. In relation to this, Mr. F. W. Webb says that he has no difficulty in pro- ducing a mild cast steel having 30 to 32 tons ultimate strength, and 33 or 34 per cent, ductility. Sir Joseph Whitworth considers that there is no need for more than 30 per cent, of ductility; with this proportion, steel tears when ruptured, and does not fly to pieces. He expresses the value of steel by the sum of the tensile strength in tons per square inch, and the ductility in percentage of the length, found by fracturing specimens of the form, Fig. 220. Thus, for steel of 40 tons strength, and 30 per cent, ductility, the resultant value is (40 + 30 ) 70. 1 The materials for this notice are derived from the Proceedings of the Institution of Mechanical Engineers, 1875, page 268. STRENGTH OF STEEL AND IRON. 6i S Table No. 216. WHITWORTH'S FLUID-COMPRESSED STEEL, AND BEST IRONS TENSILE STRENGTH. i. FLUID-COMPRESSED STEEL. Arbitrary Distinguishing Colours for Groups. Ultimate Tensile Strength per square inch. Ductility, or Elongation. Uses to which the Steel is applicable. tons. per cent. ( Axles, boilers, connecting rods, Red, Nos. i, 2, 3.... 40 32 1 cross-heads, crank-pins, hydraulic j cylinders, cranks, propeller shafts, ( rivets, tyres, &c. C Cylinder linings, slide-bars for loco- motives, shafting, couplings, drill- Blue, Nos. i, 2,3... 4 8 24 -{ spindles, eccentric - shafts for punching machines, large swages, [ hammers, &c. ( Large planing and lathe tools, large Brown, Nos. i, 2, 3, 58 I? < shears, drills, smiths' punches, dies ( and taps, small swages, &c. Yellow, Nos. i, 2, 3, 68 IO ( Boring tools, finishing tools for plan- \ ing and turning. Special alloy with ) Tungsten ) 72 H For particular purposes. Note. In each group No. i is most ductile, No. 3 least ductile. 2. IRON. DESCRIPTION. Ultimate Tensile Strength. ELONGATION. Several Specimens. Averages. Several Specimens. Averages. WROUGHT IRON. Yorkshire tons per square inch. ( 31, 30, 29, 27, ) { 27, 26.8, 26.8 1 27, 24.8 27, 26.8, 26, 25 25, 24, 24, 24, 20 26, 24, 24 tons. 28.3 25. 9 26.2 234 24.7 per cent. ( 23, 22, 31, 41, ) ( 22, 43, 42 f 39,40 39, 40, 41, 38 35, 39, 34, 33, '5 30, 35, 28 per cent. 32 39-5 39-5 31.2 3i Lowmoor Northamptonshire . . . Staffordshire Do. (Dudley Ward). Averages 25.7 34-6 CAST IRON. ( 13, 12, II, II,) \ 10, 9-5, 7 } 10.5 ( .90, 1. 10, 1.00, ) 1 .6s,.75,.i2,.5o5 .72 6l6 THE STRENGTH OF MATERIALS. CHERNOFFS EXPERIMENTS ON STEEL. 1 Steel, when cast and allowed to cool quietly, assumes a crystalline struc- ture. The higher the temperature to which it is heated, the softer it be- comes, and the greater is the liberty its particles possess to group themselves into crystals. Steel, however hard it may be, will not harden if heated to a temperature lower than what may be distinguished as dark cherry-red (temperature a), however quickly it is cooled ; on the contrary, it will become sensibly softer, and more easily worked with the file. Steel heated to a temperature lower than, say, red but not sparkling (temperature &), does not change its structure whether cooled quickly or slowly. When the temperature, in rising, has reached b, the substance of steel quickly passes from the granular or crystalline condition, to the amor- phous, or wax-like structure, which it retains up to its melting point (tem- perature c}. The points #, b, and c, have no permanent place in the scale of temper- ature, but their positions vary with the quality of the steel; in pure steel, they depend directly on the quantity of constituent carbon. The harder the steel, the lower the temperatures. The tints above specified have reference only to hard and medium qualities of steel; in the very soft kinds of steel, nearly approaching to wrought iron, the points a and b range very high, and in wrought iron the point b rises to a white heat. The assumption of the crystalline structure takes place entirely in cooling between the temperatures c and b- } when the temperature sinks below b there is no change of structure. For successful forging, therefore, the heated ingot, after it is taken out of the furnace, must be forged as quickly as possible, so as not to leave any spot untouched by the hammer, where the steel might crystallize quietly, but that the formation of crystals should be hindered, and that the steel should be kept in the amorphous condition until the temperature sinks below the point b. Below this temperature, if the piece be left to cool in quiet, the mass will no longer have a disposition to crystallize, but will possess great tenacity and homogeneity of structure. When steel is forged at temperatures lower than b, its crystals or grains, being driven against each other, change their shapes, becoming elongated in one direction and contracted in another; whilst the density and the tensile strength are considerably increased. But the available hammer- power is only sufficient for the treatment of small steel forgings ; and the object of preventing the coarse crystalline structure in large forgings is more easily and more certainly effected, if, after having given the forging the desired shape, its structure be altered to the homogeneous amorphous condition by heating it to a temperature somewhat higher than b, and the condition be fixed by rapid cooling to a temperature lower than b. The piece should then be allowed to finish cooling gradually, so as to prevent, as far as possible, internal strains due to sudden and unequal contraction. 1 Abstracted from Remarks on the Manufacture of Steel, and the Mode of Working tt, by D. Chernoff, 1868; translated by Mr. William Anderson, C.E., 1876. Mr. Anderson has conferred a substantial favour upon the steel-manufacturing and steel-consuming com- munity by the translation and circulation of this valuable document. TRANSVERSE STRENGTH OF STEEL. 617 STRENGTH OF STEEL WIRE. Dr. Pole states that music-wire has a resistance equal to 90 tons per square inch. Mr. Roebling states that steel wire has been manufactured which would resist a tensile stress of 300,000 Ibs., or 134 tons, per square inch; but not in large quantity. Steel wire, No. 14 W.G., or .085 inch, about X / I3 inch, in diameter, made for purposes of steam-ploughing, has a tensile resistance of from 2000 Ibs. to 2240 Ibs., equivalent to from 160 to 175 tons per square inch. SHEARING STRENGTH OF STEEL. The ultimate resistance of steel to shearing stress varies from 69 to 78 per cent, of the ultimate tensile strength per square inch of section. Mr. Kirkaldy found, for 16 specimens from a bar of rivet steel, an average of 73-5 P er cent.; and the same for 12 specimens of Fagersta steel. Mr. J. T. Smith, in an article hereafter noticed, states that the force required to punch a hole ^ inch in diameter through the |^-inch webs of Bessemer steel rails varied from 46^ tons to 82^ tons, according to the hardness of the rail. When a taper of x /i6 inch was allowed in the hole, the shearing resistance to punching, per square inch of surface cut through, was such as to average 70.14 per cent, of the tensile strength for the softer steels, and 72.5 per cent, for the harder steels. Upon the whole, an average of 72 per cent, of the tensile strength may be accepted as the shearing resistance of steel. TRANSVERSE STRENGTH OF STEEL BARS. The instances of tests for the transverse strength of steel, detailed in previous pages, are resumed below, showing the dimensions of specimens, with their average ultimate tensile and transverse strengths. The transverse strengths, also, are calculated from the tensile strength by formula ( i ), page 507, and entered in the second last column of the table. The formula is, W = the breaking weight at the middle, in tons. #, d, /=the breadth, depth, and span, in inches. s = the ultimate tensile strength, in tons per square inch. Take the first example in the table: 1.75 inches square, 25-inch span, 32.27 tons tensile strength. W = "55 * '-75 3 * 32.27 = 7<99 tons 2 5 Actual weight applied = 7.35 tons (uncracked). These steels show a still closer correspondence of the calculated to the actual strengths than was shown by the wrought irons, page 589. Naturally, the transverse strength for uncracked specimens, as calculated above, is somewhat greater than the observed strengths, since the strength was not 6i8 THE STRENGTH OF MATERIALS. exhausted by actual fracture. The averages are practically identical, and the identity of the calculated with the experimental strengths, is a natural consequence of the homogeneity of the material. Number and Description of Specimens. SECTION. Ultimate Strength. Span. Tensile, Transverse. EFFECTS. Breadth. Depth per sq. inch. Calculated. Actual. inch. inch. inch. tons. tons. tons. 4, Hematite, page 591 8, Krupp, "Jeddo,"' page 596, i-75 1.50 i-75 1.91 25 10 32.27 42.07 7-99 26.59 7.35 27.14 uncracked. fractured. 4, Krupp, "Sultan," page 596, 1-37 1.76 IO 4I.I8 20.19 21.31 fractured. 18, Bessemer, p. 599,... 1.90 1.90 20 33-34 13.24 12.74 uncracked. uncracked in n, Crucible, p. 599,.. 1.90 1.90 20 36.30 14.38 15.04 most instances. 4, Fagersta, p. 604, . 1.90 1.90 20 39.16 I5-SI I5.6I half fractured, half uncracked. Averages, 16.32 16-53 The general formula ( i ) may be adapted for steels of a particular tensile strength, by substituting for (1.155 s ) i ts numerical value. Thus, for steel of 30 tons tensile strength, 1.155^=1. 155 x 30 = 34.6; and w _ 3 4.6 bd* ( 2 ) Ultimate Tensile Strength, tons. ?O . I Coefficient (1.155*)- . 34.6 Ultimate Tensile Strength, tons. 42 \ * ) Coefficient (i-i55*). 48 c; ?2 270 44 5 8 24. 7Q 1 A e C 2 O OT- ' ? r Oy'O 4O 3 T-J 4.6 5**" r -3 T GO ^6 . AI.6 48 DO' 1 r r A 38 . . 4-3.0 CQ JO-4 e 7 8 40 . . 46.2 O v J / <0 RULE. To find the Ultimate Transverse Strength of Rectangular Steel Bars. Multiply the breadth by the square of the depth, and by the coeffi- cient (1.155 s )> corresponding to the ultimate tensile strength, and divide by the span. The quotient is the breaking weight in tons, applied at the middle. Note. To find the coefficient for any other tensile strength, not given above, multiply the given tensile strength by 1.155. TRANSVERSE DEFLECTION OF STEEL BARS. For want of data, it is assumed that the deflection of steel bars is to that of iron bars of the same dimensions, in the ratio of their extensibilities, or inversely as their coefficients of elasticity. From the results of experi- ments on iron and on steel rails, it appears that the coefficients are practi- TORSIONAL STRENGTH OF STEEL. 619 cally as u to 13. Increasing, therefore, the numerical coefficients for wrought-iron bars, in formulas ( 5 ) and ( 6 ), page 590, in this ratio, the following formulas are deduced : Elastic Deflection of Uniform Bars of Steel, loaded at the middle. Square bars... . D = ( i } 56,000 bd 3 Round bars, D = , W/3 j7 ( 4 ) 38,000 d* D = the deflection, b the breadth, d the depth, / the span, all in inches; W = the weight, in tons. TORSIONAL STRENGTH OF STEEL BARS. The torsional resistances of steel, already recorded, are, with the ultimate tensile and shearing strengths, resumed below, and the calculated resistances, by formula ( i ), page 534, are added in the second last column. The shearing strength is taken at 72 per cent, of the tensile strength, as was settled, page 617. The torsional stress was applied, in the following experiments, at the end of a 1 2 -inch lever. The formula is, W = the breaking stress in tons. h = the shearing strength in tons per square inch. */=the diameter in inches. R = the radius of the force in inches. WR= the moment of the force, in statical inch-tons. Take the first example in the table below: i tons shearing strength, and a 1 2-inch radius : inches in diameter, 25.21 Breaking force, by formula^' 278 x I - 2 5 3x2 5- 2 i = I>I4I tons Description, and Number of Specimens. Diameter. Ultimate Strength. Ultimate Torsional Force. Tensile. Shearing. Calculated. Actual. 4, Hematite, page 595, 4, Kru P p,"Jeddo," 596, 4, Krupp, Sultan," 596, 1 8, Bessemer, 600, inches. 1.25 1.25 I.I28 1.382 1.382 I.I28 I.I28 I.I28 I.I28 I.I28 tons. 32.27 4I.l8 42.07 3343 36.30 38.04 47.60 45.82 27.37 39.16 tons. 25.21 say 72% jj j> )> V 27.42 35.60 31.99 20.28 28.82 tons. I.I4I 1.342 1.007 1.472 1.599 .912 I.I84 1.064 .674 .958 tons. 1.030 1.280 1.068 1.470 1.610 .946 1.043 1.009 .679 .919 ii Crucible, 600 ^ Fciffersta a 6oj. ^, Do. b* 604, 3 Do. c* .. , 604., 3, Do- d, 604, 12, Do. average, Averages of all I.I35 1.105 62O THE STRENGTH OF MATERIALS. The results of experiment and of calculation show a close correspondence. When the shearing strength is not known experimentally, substitute .72 s, or 72 per cent, of the tensile strength s, for h in the formula; and (6) (7) R Ultimate Torsional Strength of Steel Bars. When the tensile strength, s, is 30 tons, then = 6, and Generally, for tensile strengths of from 30 to 50 tons, the values of the numerical coefficients in formulas ( 7 ) and ( 8 ), are as follows : Tensile Strength. Coefficient. Tensile Strength. Coefficient. tons. (.20 s.) tons. (.20$). 30 6 32 6.4 34 6.8 35 7 36 7- 2 38 7-6 42 8.4 44 8.8 45 9 46 9-2 48 9.6 50 10 40 8 Elastic Torsional Strength of Steel Shafts. Hematite steel 41.5 per cent, of ultimate strength. Krupp, "Jeddo" 38.4 Krupp, " Sultan " 47.3 Bessemer 44.6 Crucible 42.5 Fagersta (average) 50. 2 Average 44.1 percent. ELASTIC TORSIONAL SHEARING STRESS AND DEFLECTION OF STEEL BARS. The elastic shearing stress /z, is found by formula ( 3 ), page 535. h WR / \ h = ^j*d* (9) For Hematite steel, for example, page 595, W R .428 ton x 12 inches = 5.136, the moment of the force, and h 5i_2 ^46 tons per square inch, the elastic limit of shearing stress. The coefficient of torsional elas- ticity, E', as defined at page 536, is found by formula ( 12 ), page 537 : E'- Q W */ - -48x i,x 10 I2)forHematite steel . .873^/ 4 D .873 X I.25 4 X .008 STRENGTH VARIES WITH CONSTITUENT CARBON. 621 For the several steels, the elastic shearing stress and coefficient of elas- ticity, calculated in the same way, are as follows : Steels. Specimens. Diameter and Length for Observation. Elastic Shearing Stress per square inch. Coefficient of Elasticity. Hematite, page 595 inches. inches. 1.25 x 10 tons. 9.46 E' 3012 Krupp, "Jeddo," 596 Krupp, "Sultan,".... 596 Bessemer, 600 1.25 X2.5 I.I28 X2.25O 1.382 x ii 10.85 13-95 10.73 1382 86 5 2472 Crucible, 600 I.382X ii 11.19 2025 Fasrersta a 604 .128 X Q 1C. 2^ Do b 604 .128 x o I S.IO Do. c, 604 .128x9 > .. 14.56 Do d 604. .128 X Q IO.2C, Do. average, 604 .128x9 13-77 Omitting the coefficients of elasticity for the "Jeddo" and the "Sultan," as the specimens were very short, the average of the remaining three co- efficients is 2503; and the value of .873 E' in formula ( 10), page 537, is (.873 x 2503 = ) 2185; say 2200. Whence, by substitution: Elastic Torsional Deflection of Steel Bars. WR/ D 2200 */ 4 (10) D = the total angular deflection in parts of a revolution. W = the twisting force in tons. R = the radius of the force in inches. W R = the moment of the force in statical inch-tons. /=the length of the shaft under torsional stress in inches. i, page 158). (Proceedings of the Institution of 622 THE STRENGTH OF MATERIALS. the last degree of carbonization, i% per cent., the steel becomes gradually weaker, until it reaches the form and strength of cast iron. Table No. 217. TENSILE STRENGTH OF STEEL CONTAINING DIFFERENT PROPORTIONS OF CARBON. Mr. T. Edward Vickers. Description of Steel. Proportion of Carbon! (approximate). Breaking Weight per square inch. ELONGATION. No. 2 No 4 per cent. ... -33 43 tons. ... 30.4 ... 34 O inches. ... .37, or 9.8 per cent. 37 or o 8 No. 5 No 6 ... .48 ... C -3 ... 37-5 ..- 42 ? ... .25, or 8.9 12 or 8 o No. 8 No. 10 No. 12 No 15 ... .63 ... ... .74 ... ... .84 ... I OO ... 45.0 ... ... 45-5 -.. ... 55.0 ... DO O ... .00, or 7.1 ... .69, or 5.0 ... .12, or 8.0 oo or 5 o No 20 I 25 60 o 62 or 4 4 A specimen bar was turned down to a diameter of ^ mcn at the middle, so as to form a circular notch. On being tested, it broke with 79^ tons per square inch, whilst the ordinary specimen bar of the same steel broke with 60 tons per square inch. Mr. Webb's datum above given is in harmony with Mr. Vickers' data. See also on this subject RAILWAY RAILS, at page 664. RESISTANCE OF STEEL AND IRON TO EXPLOSIVE FORCE. Sir Joseph Whitworth tested iron and steel by the explosive force of gunpowder. The specimens were cylinders having a bore of ^ inch, a diameter outside of i j inches, and a length of 4 inches. They were made open at the ends, and were closed for the purpose of the experiments. Table No. 218. RESISTANCE OF IRON AND STEEL TO EXPLOSIVE FORCE. METAL. Charge of Powder. Expansion in diameter at middle before Number of pieces when bursting. burst. grains. ratio. inch. pieces. Cast iron I c I OOOO "^6 Wrought iron, Staffordshire, coiled Fluid compressed steel, No. 3, red 95 275 6-3 18.3 .0997 .1659 5 2 Do. do. No. 3, brown .. 325 21.7 .0950 4 1 The intermediate percentages of carbon in column 2, from No. 4 to No. 15 inclusive, are merely approximate, having been interpolated in proportion to the Nos. of the steel. RECAPITULATION OF DATA. 623 RECAPITULATION OF DATA ON THE DIRECT STRENGTH OF IRON AND STEEL. Cast Iron, pp. 553 to 561. The ultimate tensile strength ranges from 5 to 7^ tons per square inch: first meltings, specimens under i inch in thickness. For thicker castings the strength diminishes. The com- pressive strength is from four and a half to about seven times the tensile strength. For general calculations, say, tensile strength 7 tons, compressive strength 49 tons. The ultimate tensile strength is increased by repeated remeltings to from 15 to 20 tons per square inch; and the compressive strength to from 70 to 80 tons. The elastic strength practically is equal to the ultimate tensile strength. Wrought Iron, pp. 567 to 591. The ultimate tensile strength of rolled bar iron varies from 22^ to 30 tons; rivet-iron from 24 to 27 tons. Plates from 20 to 23 tons; about i ton less crossway than length way of the fibre. The strength is reduced more than i ton by annealing. The resistance to compression is an indefinite quantity. The elastic tensile strength of iron bars averages not less than 50 per cent, of the ultimate strength; and that of iron plates is generally from 55 to 60 per cent, of the ultimate strength. The elastic strength of bars and plates, both tensile and compressive, may be taken at 1 2 tons. The elongation of wrought-iron bars, within the elastic limit, is at the rate of x / IO >ooo to VIS.QOO part of the length say, an average of x /ia,o part per ton per square inch; or a total of X / IOOO part of the length. The same fraction may be taken for compression within the elastic limit. Approximate Strength of Wrought-iron Bars in Terms of the Circular Inch (Mr. E. Clark). " A strength of 20 tons per square inch is nearly equivalent to one of 16 tons per circular inch. An ordinary i-inch round rod bears tensilely 16 tons, and weighs 8 Ibs. per yard. "For a round rod of any diameter, the square of the diameter, in quarter-inches, is the breaking weight in tons. " Half this quantity is the weight in pounds per yard. " A rod will be perceptibly damaged by half this stress, which can never be safely exceeded; one-third being sufficient in practice." Steel, pp. 593 to 615. By Mr. Kirkaldy's earliest experiments, it was found that the average tensile strength of bar steel varied from 60 tons for tool -steel, to 28 tons for puddled steel; and that of steel plates from 3/ l6 to s/i6 inch thick, from 32 to 45% tons. From subsequent experiments, it appears that the ultimate tensile strength of rolled bar steel varies from 30 to 50 tons per square inch. The average tensile strength may be taken at 35 tons, and the elastic strength, tensile and compressive, at 20 tons. The tensile and the elastic compressive strength of hammered steel bars is from 4 to 5 tons more than that of rolled bars. By annealing, the elastic strength of rolled steel bars is reduced 3 tons, and that of hammered bars 5 tons. Steel plates have elastic tensile and compressive strengths averaging about 14 tons; the ultimate tensile strength is from 22 to 32 tons, according to 624 THE STRENGTH OF MATERIALS. the proportion of constituent carbon. The strength is the same lengthwise and crosswise. Annealing reduces the tensile strength of steel plates, elastic and ultimate, by i ^ or 2 tons ; and the elastic compressive strength by twice as much. The elongation and the compression of steel bars within the elastic limit may be taken at Vi 3,000 part of the length per ton per square inch; or a total of Viooo part of the length. Inches Sfress per S Do. do. i 8.592 16.47 Do. do. 1.5 8.876 17-13 Do. do. i 19.20 Do. do. 2 8.614 20.25 Do. do. i 20.34 Do. do. 2 8.580 20.41 Do. do. 2 8.615 20.27 Do. do. 3 8.422 21.38 Do. do. 4 22.32 Gun met3l 12 copper i tin 12 Q/t i ^.y^. j> Do ii i I 3 71 Do 10 i 1 j- / 14. T\ Do Q i T" / J I7.OO jj )> 1 Useful Rules and Tables, page 205. STRENGTH OF COPPER AND OTHER METALS. Table No. 219 (continued). 627 DESCRIPTION OF METAL. Speci- fic Gravity Ultimate Tensile Strength per square inch. Experimentalist. Gun metal, average strength of good bronze Gun metal, average results of tests oH specimens from bronze guns elastic > strength, 6.56 tons ) Gun metal, American guns Gun-heads Breach-squares Small bars cast in same moulds with guns Small bars cast separately in iron moulds Do. do. in clay moulds Finished guns Alloys of copper and tin, unwrought Equivalents. By weight. iodi+ Sn, 84. 29 copper + 15.71 tin, gun metal... Sn, 82.81 +17.19 9Cu+ 8Cu+ 7Cu+ Cu+ Sn, 81.10 Sn, 78.97 Sn, 34.92 Sn, 15.17 Sn, o +1.90 +21.03 +65.08 +84.83 +ioo. brasses ...... small bells. . speculummet tin ........... 8.523 8.765 8.584 8-953 8.313 8.561 8.462 8-459 8.728 8.056 7-447 7.291 tons. 1473 12.19 13.24 20.76 18.76 16.82 11.51 10.3 to 23.3 16.1 15.2 17.7 13.6 1.4 3- 1 2-5 Aluminium bronze 90 copper, i aluminium 32.67 Do. maximum 43.00 Tin, cast . 2.11 Do. Banco 7.297 .95 Lead, cast .81 Do. sheet .86 Lead pipe i.oo Zinc, cast 1.336 Soft solder 2 tin, i lead Brass, fine or yellow 8.02 Brass, fine or yellow, 2 copper, i zinc 12.90 Brass tube, 62 copper, 38 zinc 46.00 Do- ,, 70 30 36-00 Do. wire Muntz's metal 3 copper, 2 zinc 22.00 Alloys of copper, zinc, iron, and tin" Sterro- metal " Copper 10, iron 10, zinc 80 7.000 3.17 Do. 60, 3, 39, tin i. 5 24.00 Do. 60, 4, 44, 2: Cast in sand 19.25 Cast in iron, annealed , 24.25 Cast in iron, forged red hot 31.00 Copper 60, iron 2, zinc 37, tin i 34.0 Do. 60, 2, 35, 2 38.0 Do. 55.0,,, 1.77,,, 42.36, ,,0.83: Cast 27.0 Forged red hot 34.0 Drawn cold 38.0 Dr. Anderson Wade )) Mallet Dr. Anderson Rennie Wade Rennie Navier Jardine Stoney Rankine Rennie Dr. Anderson Everitt Dufour Dr. Anderson Dr. Anderson 628 THE STRENGTH OF MATERIALS. Table No. 220. PHOSPHOR-BRONZE, BRONZE, AND BRASS. FROM LIEGE. TENSILE STRENGTH. Elastic Ultimate Ratio S< :t. Report.) Strength per Strength per of Elastic to At 20,000 Sectional Area of DESCRIPTION. square inch. square inch. Ultimate Strength. Ibs. per square inch. Ultimate. Fracture. PHOSPHOR-BRONZE. Lowest values tons. A 777 tons. Q 712 per cent. 31 t\ per cent. OQ per cent. 36 per cent. 0,6 I Highest values 4-/// 10.625 22.73O 685 .wy c I 3 33 4. 68 I Averages of 1 2 specimens 7482 15.386 48.6 i-59 II.4 87.4 ORDINARY BRONZE. Lowest values 7-221 Q 06 1 667 18 I 2 08 ; Highest values 8 7Q4. :7' I^.lSA S;.Q I IO 4. O QI 6 Averages of 6 specimens "/y*t 8.095 10.582 >V 7 6.5 .56 2.23 y*.v 95-6 BRASS A AIQ 12 284 367 q 80 16 i 81 7 TENSILE STRENGTH OF WIRE OF VARIOUS METALS. M. Baudrimont, in 1835, tested the strength of annealed metallic wires at various temperatures, from 32 F. to 392 F. 1 The wires of gold, platinum, copper, silver, and palladium were about Yeoth inch in diameter; the iron wires were Vi45 tn mc ^ m diameter. The results of the tests are Table No. 221. TENACITY OF METALLIC WIRES AT VARIOUS TEMPERATURES. 1835. Dia- Ultimate Tensile Strength. Tensile Strength per square inch. METAL. of wire Area. at 61 F. At 32 F. At 212 F. At 392 F. At 3 2F. At 212 F. 392 F. inch. sq. inch. Ibs. Ibs. Ibs. tons. tons. tons. Gold .0162 .000207 | S.6I i 5-42 4.64 4-49 3-86 3.80 12. 1 II.7 10.0 9-7 8-3 8.2 Maximum Minimum Platinum .Ol6l .000205 j 6.70 f 6.59 5-94 5.61 5-27 5-03 14.6 14.4 13.0 12.2 "5 II. Maximum Minimum Copper ... .0177 .OOO247 IO. II 10.02 8.80 8.73 7.92 7.27 18.3 18.1 16.0 I 5 .8 14.4 13 * Maximum Minimum Silver 0157 .OOOI93 7.86 7.78 6.74 6-39 5-13 5.10 18.1 18.0 15-6 I 4 .8 II-9 ii. 8 Maximum Minimum Palladium .0156 .000192 IO. 12 9.98 9.00 8.8q 7-99 7.41 23-5 23.1 2O.9 20.6 18.5 17.2 Maximum Minimum Iron .0069 .000373 II. 12 10.89 10.66 10. 16 11.31 11.15 133-2 I30-3 127.6 121.7 J 35-4 133-5 Maximum Minimum 1 Annales de Chimie, 1850. STRENGTH, OF STONE, BRICKS, ETC. 629 arranged in table No. 221, and it is shown that, ist, the tenacity varies with the temperature; 2d, it decreases as the temperature rises, except for iron; 3d, that iron presents a peculiar case. At 212 F. its tenacity is less than at 32 F., but at 392 F. it is greater. Table No. 222. TENSILE STRENGTH OF WIRE PHOSPHOR-BRONZE, COPPER, BRASS, STEEL, AND IRON. (Reduced from Mr. Kirkaldy's Report.) Unannealed. Annealed. Diameter. Ultimate Tensile Strength. Diameter. Ultimate Tensile Strength. Ulti- mate Exten- sion. DESCRIPTION OF WIRE. Total. Per square inch. Total. Per square inch. Phosphor-Bronze Lowest values ... inch. .0585 or 1/17.1 .0665 or 1/15 .063 or 1/15.1 Ibs. 394 tons. 43-59 71.21 56.28 inch. .1070 or J /9. 3 .1125 or 1/9 .1108 or 1/9 Ibs. 527 tons. 22.58 28.82 24.42 p. c'nt. 33-o 46.6 39- Highest values Averages of 20 ) specimens . . \ CoDoer... .0640 or 1/15.6 . .0605 or 1/16.5 .0600 or 1/16.7 .0580 or 1/17.2 j. 0580 or i/ I7 . 2 203 233 342 170 174 28.18 36.23 54.07 28.71 29.40 .0640 or i/ is .e .0605 or 1/16.5 .0600 or Vi6. 7 .0580 or 1/17.2 .0580 or i/ I7 .a 119 148 211 162 122 l6.52 23.01 33-32 27.36 20. 6 1 34-1 36.5 10.9 17.1 28.0 Brass Steel. . Iron, galvanized, BBC Do. do. BCE STRENGTH OF STONE, BRICKS, &c. Table No. 223. TENSILE STRENGTH OF STONE, BRICKS, AND CEMENT. DESCRIPTION OF MATERIAL. Weight per cubic foot. Ultimate Ten- sile Strength per sq. inch. Experimentalist. Sandstone Ibs. tons. KO Buchanan Whinstone , .i}^ 6iu Arbroath pavement .\j^ c6^ Caithness do O'-'J 4.71 White Marble 322 Do .$6* 24.6 Hodgkinson Flint-glass rod, annealed I O7 Fairbairn Green glass rod I.2Q White crown-glass rod I 14. Thin glass globes, cohesion 2 2"? Plaster of Paris Ibs. 71 Rondelet Mortar of quartzose sand and hydraulic ) lime \ 136 to 85 Vicat Mortar of quartzose sand and ordin- ) ary lime C 21 tO 51 630 THE STRENGTH OF MATERIALS. Table No. 223. (continued). DESCRIPTION OF MATERIAL. Weight per cubic foot. Ultimate Ten- sile Strength per sq. inch. Experimentalist. Adhesion of Plaster of Paris to brick ) or stone average . . ( Ibs. Ibs. 50 Rondelet Adhesion of bricks cemented with Port- } land cement, 12 months old, and i > cement to i sand . \ neat I tO I Gault-clay bricks, pressed; in air... Do. do. in water Gault-clay bricks, wire cut ; in air... Do. do. in water Gault-clay bricks, perforated; in air.. Do do. in water Stock bricks in air 4 1 46 68 47 108 84 78 44 46 43 39 83 g Grant 1 j> )) )) )) Do in water . / 06 W J 7O Staffordshire blue brick, pressed ) with frog ; in air J y 74 /y 56 )) Do in water 7fi 07 Do. rough, without frog; in air.... Do. do. do. in water Fareham red bricks; in air / u 48 40 126 Ol 47 29 8^ )) )) jj Do. do. in water 123 ?* 62 >> Portland cement : Seven-day tests l ^J 862 to 408 )> Average of do. per bush. 115.2 Ibs... Portland cement, 123 Ibs. per bushel, mixed with equal weight of Thames sand : Age in water, 7 days 90 neat cement. -,5-? 358.5 cement & sand TC7 )) 5? Do. i month ^ 3 AID *^/ 2OI Do. 6 do C2"? 284 Do. 12 do CA7 3IQ Do. 2 years t' OOO 3 C I Do. 4 do eg? *9 l6 1 Do. 7 do j'-'j COO l8A Portland cement, 112 Ibs. per bushel, mixed with various proportions of sand; 12 months old: 3 sand, i cement J7^ 241 5 do. i do 2T/I 7 do. i do 161 55 Roman cement averages : Age in water, 7 days yj QO )) Do. i month T T C Do. 6 do 1 L y 2io JJ Do. 12 do 286 Do. 2 years 2/1 "2 Do. 4 do 281 J5 Do. 7 do 5 T C J l J )> 1 Proceedings of the Institution of Civil Engineers, vols. xxv. and xxxii. STRENGTH OF STONE, BRICKS, ETC. 63! Table No. 224. CRUSHING STRENGTH OF STONES AND BRICKS. DESCRIPTION OF MATERIAL. Specific Gravity. Tons per square inch. Experimentalist. Granite : Aberdeen blue. 2 625 d.87 Rennie Peterhead 3 7O Cornish 2662 >/ u 2 8^ >j Dublin . 466 \Vilkinson Wicklow I C2 N e wry * 86 j> Mount Sorrel . 2 67 1; 3-uw 57A Fairbairn Whinstone Scotch . 3 7O Buchanan Greenstone Irish Q 3O "Wilkinson Sandstones and Grits : Arbroath pavement yo w 3 C2 Buchanan Craigleith freestone 2 AC2 J'j* 2 6l Rennie Derby grit, friable sandstone *^!H 2 3l6 I 4.O Yorkshire pavin " 2 ?O7 2 c c Red sandstone Runcorn *OD Q7 L Clark Quartz rock, Holyhead, across lami- ) nation C 7/ 11.40 Mallet Do parallel to lamination . 62C Marble : Statuary vf.-O T AA Rennie Italian white veined A. "32 Irish 4-J^ 675 to Q oo Wilkinson Limestone : 2.S84. 2 AA Rennie Purbeck . . 2 CQQ A. OQ Magnesian **yryy i ^6 Fairbairn Anglesea 2 72O ^ ^8 L. Clark Irish >j 5 06 to 7 56 "Wilkinson Chalk 22/1 Rennie Slates : Irish, on bed of strata \ 10 60 Wilkinson Do on edge of strata... 623 Bricks : Red .7C8 Rennie Yellow-faced, baked. . J3V AAO Do. burnt .qq.\.j 64.^ Gault-clay pressed I 1 1 1 Grant Do wire-cut 884 Do. perforated l.lSo Stock I OA.A Fareham red 2 SOO Staffordshire blue, pressed with frogs Do. rough, without frogs Stourbridge fire-clay 3.100 ^8 J5 L. Clark Do. do / .670 J. R. Walker Tividale blue 620 Brickwork in cement not hard 2^2 E Clark THE STRENGTH OF MATERIALS. Table No. 224 (continued}. DESCRIPTION OF MATERIAL. Specific Gravity. Tons per square inch. Experimentalist. Portland cement 3 months old I 7O Grant i do to i sand I. II i do to % sand A? Portland cement Q months old T-J 267 }> j (Jo to i sand 2 O4. )> i do to 5 sand .71 Portland cement, concrete blocks 12- inch cubes compressed, 12 months old: i cement to i sand and srravel Total Crushing Weight, tons. I7O i to 3 1 1 C C >j i to 6 QI O ?> Mortar : Lime and river sand per square inch. IQA Rondelet Do do beaten 266 Lime and pit sand 2;8 Do do beaten 3C7 Glass jj/ 12.31 to 14.23 >) Fairbairn STRENGTH OF ELEMENTARY CONSTRUCTIONS. RIVET-JOINTS IN IRON PLATES. There are two elements by which the strength of rivetted joints is deter- mined: the tensile strength of the perforated plate, and the grip and shearing strength of the rivets. Strength of Perforated Iron Plates. The usual effect of perforation by punching, is a weakening of the metal about the holes ; so that the tensile strength per unit of section between the holes is less than that of the unpierced plate. Yorkshire iron (table No. 192, page 584) loses from 13 to 17 per cent, of its tensile strength, and Krupp iron from 10 to 13 per cent., by punching. It is generally assumed, according to Mr. Wilson, that hard plates of fair quality lose from 20 to 24 per cent, of their tensile strength by punching for steam-joints, but that many soft plates do not lose more than 8 per cent. When plates are drilled, on the contrary, it is considered that the tensile strength remains unimpaired. But, Mr. J. Cochrane found from experiments with bar iron that there was no loss of tensile strength by punching holes in the bars. 1 Low- moor and Staffordshire bars were planed down to a nominal thickness of YZ inch, and shaped to a width of 2 inches. One-inch holes were made through the bars in three ways: ist, by drilling; 2d, by punching ^ inch too small and rimering out to the size; and 30!, by punching at once to the full size. The tensile resistances per square inch were as follows : Formation of Holes. Lowmoor Iron. Staffordshire Iron. [/ Drilling 24.72 tons. 23.15 tons. Punching and rimering 25.51 23.15 Punching 24.53 23.69 No doubt the holes were punched with a wide clearance in the die a provision which very much facilitates the separation of the metal, eases the punch, and eases the stress on the metal. Strength of Rivetted Joints. Sir William Fairbairn, in 1838, deduced from experiment with small specimens of i^-inch iron plate, that double- ri vetted lap-joints were stronger than single-rivetted joints, and that their relative values were as follows : Tensile strength of the solid plate, as 100 Do. double-rivetted lap-joint, as 70 Do. single-rivetted lap-j oint, as 56 ^Proceedings of the Institution of Civil Engineers, vol. xxx. 1869-70, p. 265. 634 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Mr. Bertram's Experiments. At Woolwich Dockyard, Mr. W. Bertram tested various plate-joints. The results were reported, and they were inves- tigated by the author, in i860, 1 in connection with Mr. Bertram's method of welding joints the scarf-weld and lap-weld, Figs. 224 and 225, in which the lap is i ^ inches. Staffordshire plates of good quality were selected for Fig. 223. Entire plate. Fig. 224. Scarf- welded joint. Fig. 225. Lap-welded joint. Fig. 226. Single-rivetted joint, by hand. O Fig. 227. Single-rivetted joint, by hand, snap-headed. Fig. 228. Single-rivetted joint, by machine. G m Fig. 229. Single-rivetted joint, with countersunk head. Fig. 230. Double-rivetted joint, snap-headed. ! Fig. 231. Double-rivetted joint, countersunk and snap-headed. Fig. 232. Double-rivetted joint, with single welt, countersunk and snap-headed. Boiler- Plate Joints, tested by Mr. Bertram. the trials: of three thicknesses, ^-inch, 7/ l6 -inch, and ^-inch; and made up into ten varieties of specimens, 4 inches broad and 24 inches long, in which the rivet-joints were made with 24 -inch rivets at a pitch of 2 inches. Three specimens of each variety of joint, for each thickness of plate, were tested, and the results averaged for each set of three specimens. These joints are illustrated and described at Figs. 223 to 232. The net sectional area of 1 Recent Practice in the Locomotive Engine, 1858-59; also Railway Locomotives, 1860, by D. K. Clark. Blackie & Son. See these works for an extended notice of plate-joints. RIVET-JOINTS. 635 plate, in the line of the rivets, was 62.5 per cent, of the solid section. The sectional area of a ^-inch rivet is .4417 square inch, giving for two rivets a shearing section of .8834 square inch. Shearing Section of Rivets. Net Sectional Area. plate, .94 sq. ins. .8834 sq. in., or 94 per cent, of net section. 7/ l6 1.094 .8834 or 80.8 % 1-25 -8834 or 70.7 The fractures took place, in nearly all cases, in one of the plates, in the line of the rivet-holes; but, in a few cases, the rivets were shorn across. The normal strength for the solid plates was nearly uniform, and averaged, for all thicknesses, 20 tons per square inch. Table No. 225. ULTIMATE TENSILE STRENGTH OF WELDED AND RIVETTED JOINTS OF BOILER-PLATE. Tensile strength of the entire plate, 20 tons per square inch. (Reduced, in 1860, from Mr. Bertram's experiments.) Description of Joint. Form of Joint. Net Ultimate Tensile Strength of Joint ; that of the entire plate=ioo. M-inch Plate. 7/i6-inch Plate. ^-inch Plate. Average for three thicknesses. i Entire plate Fig. 223 Fig. 224 Fig. 225 per cent. 100 faulty So per cent. 100 106 6 9 per cent. 100 102 66 per cent. IOO 104 62 2. Scarf- welded joint 3 Lap-welded joint 4. Single - ri vetted joint, ) by hand ( Fig. 226 Fig. 227 Fig. 228 Fig. 229 40 50 40 44 50 52 54 5o 60 56 52 52 50 53 49 49 5. Single-rivetted joint, by J hand, snap-headed.. ( 6. Single - ri vetted joint, i by machine.. . \ 7. Single - rivetted joint, } with countersunk J- head ) 8. Double-ri vetted joint, ) snap-headed \ Fig. 230 Fig. 231 Fig. 232 59 53 52 70 72 60 72 69 65 67 65 59 9. Double-ri vetted joint, } countersunk and > snap-headed ) 10. Double-rivetted joint, } with single welt, ( countersunk and I snap-headed . . j From these data, it appears that the scarf-welded joint is as strong as the entire plate, and that the strength of the lap-welded joint averages only 636 STRENGTH OF ELEMENTARY CONSTRUCTIONS. five-eighths, or 62 per cent, of that of the entire plate. The varieties of single-rivetted joints average nearly equally strong for each variety; and they have only half the strength of the entire plate, excepting the snap-headed, which has rather more than half the strength. Of the double-rivetted joints the ordinary lap is the strongest, having two-thirds of the entire strength; the welt-joint is weakest. Comparing the different thicknesses of plate, the averages of all the lap-joints, at the foot of the table, show that the ^-inch is the strongest, that the 7/ l6 -inch is nearly as strong, and that they are about one-fourth stronger than ^-inch lap-joints, relatively to the thickness of plate. Leaving the averages, the drift of the evidence is, that the thinner the plate the more efficient the joint. The single-rivetted joints, No. 4, have successively 40, 50, and 60 per cent of the strength of the entire plates, and the double-rivetted joints, 59, 70, and 72 per cent; insomuch that the 2/6-inch single-rivetted joint is absolutely stronger than the thicker joints the actual breaking weights being successively 16, 17%, and 18 tons for the j^-inch, 7/ l6 -inch, and ^-inch joints. For the double-rivetted joints, the actual breaking weights are about 23.5, 24.5, and 21.5 tons; showing that the 7/ l6 -joint is absolutely stronger than the ^-inch, and that the 3/^-inch joint has only one-twelfth less absolute strength than the ^-inch joint The double-rivetted welt -joint, similarly, is more efficient for the thinner plates, and its absolute strength is practically the same for them all. It appears, then, that ^4 -inch rivetted plates are practically stronger than 7/ l6 -inch and ^-inch rivetted plates; and that, of the ^-inch joints, the order of strength is as follows : Tensile Strength. Entire plate, ^-inch thick 100 Double-rivetted lap-joint, average 71 Double-rivetted single welt-joint 65 Single-rivetted lap-joint, average 55 These proportions do not differ widely from those that were given by Sir William Fairbairn. It appears that countersunk rivetting does not impair the strength of the joint, as compared with external heads. To bring out the comparative weakness of the joints of the thicker plates, the fourth line of the following tablet, which is obtained by dividing the third by the second line, shows that the tensile strength per square inch of net section of the ^-inch single-rivetted joint, was nearly nine-tenths of that of the entire plate; whilst that of the 7/ l6 -mch joint was just over eight- tenths; and of the ^ -inch joint, seven-tenths. **"* ^--h ^nch. Net section, 62.5 % 62.5% 62.5 % of that of entire plate. Net tensile strength, av.,... 43.5 51.5,, 55 Do. per square inch of ) net section, in parts of > 70 82 88 that of entire plate,... j 50 6 9 66 of that of entire p.ate. RIVET-JOINTS. 637 The lap-weld joint is strikingly weaker than the body of the plate, though there is no reduction of section. The weakness arises from the indirectness of the lap, for the joint, though solid, is not straight. The experiment proves that the lap is essentially an element of weakness, irre- spective of the loss of strength by rivet-holes: the thicker the plate, the greater is the distorting leverage, insomuch that the absolute strength of the ^-inch lap-welded joint was not greater than that of the 2/6-inch joint. The annexed Figs. 233 and 234, show the ultimate distortion by the oblique stress on lap-joints. Scale, One-half. Figs. 233, 234. Ultimate effects of Oblique Stress on Lap-joints. On the principle here noticed, one may account for the practically equal strength of the joints made with countersunk rivets, compared with those having external rivet-heads, notwithstanding the greater reduction of solid section by countersinking: the leverage is shortened, and it may be measured from the centre of the cylindrical part of the rivet in the line a a, Fig. 235, or thereabouts, towards the inner side of the plate. On the same principle, the conical form of punched holes reduces the leverage and the obliquity of the pulling stress. 1 Scale, One-half. Fig. 235. Diagram to show Stress on Countersunk Rivets. As the double-ri vetted joints, No. 8 of the series, exhibited respectively 59, 70, and 72 per cent, of the tensile strength of the entire plate, it appears that its resistance per square inch of net section, was 94, 112, and 115 per cent, of that of the entire plate. There is an apparent anomaly here : it may be supposed that the normal strength of the particular plates exceeded 20 tons per square inch, aided, perhaps, by the frictional grip of rivets, first pointed out by Mr. Edwin Clark (see page 5 70). Mr. J. G. Wright's Experiments.' 2 ' Mr. Wright gives the strength of two 1 The author believes he was the first to publish the rationale of the strength and the weakness of ri vetted joints, as the cause of the grooving of plates at such joints. 2 Discussion upon Mr. W. R. Browne's paper, " On the Strength and Proportions of Rivetted Joints," in the Proceedings of the Institution of Mechanical Engineers, 1872. 638 STRENGTH OF ELEMENTARY CONSTRUCTIONS. specimens of single-rivetted square lap-joints, and two of diagonal joints, at angle of 45, which were tested by Mr. Kirkaldy. They were made with ^-inch Staffordshire plate, exactly .38 inch thick, 12 inches wide, with 2^-inch lap, punched holes, and six if-inch rivets in the square joint, at 2 inches pitch. The diagonal joint was made with eight rivets of the same size and pitch. The ultimate tensile strength of the solid plate was 19.69 tons per square inch with the fibre, and 16.80 tons across. The section of the entire plate was (i2x.38 = )4.56 square inches, and the total ulti- mate strength with fibre was (4.56 x 19.69 = ) 89.8 tons. Ultimate Tensile Strength. Entire plate 89.8 tons. Square joint 43.0 or 48 per cent. Diagonal joint 58.0 or 64 Square Joint. Diagonal Joint. Net sectional area, 59.4 per cent. 91.7 per cent, of entire plate. Net tensile strength, 48.0 64.0 Do. per square inch 1 of net section, in I R parts of that of | entire section... j The diagonal joint was one-third stronger than the square joint; although, per square inch of net section, it opposed less resistance, because its resist- ance, which was necessarily exerted in an oblique direction, was a resultant compound of shearing resistance with the lengthwise resistance of the plates. The net sectional area of the square joint was 2.71 square inches; and the shearing section of the rivets, 3.11 square inches, or 115 per cent, of the net section. Mr. L. E. Fletcher's Experiments. In these experiments, to be after- wards noticed, a double-rivetted lap-joint, made with punched holes, zigzag, of 7/ l6 -inch Staffordshire plate, in a 7-feet Lancashire boiler, was burst with a force of 20.01 tons per square inch of net section between the rivets in line, the fracture taking place in the plate. With so high a tensile resistance, it is probable that the strength of the plate was very little, if at all, impaired by punching. The rivets were placed at 2.44 inches pitch in line, and had an average diameter of T 3/ l6 inch. The ultimate strength of the joint may be taken as two-thirds of that of the solid plate being in the ratio of the net sectional area to the section of the entire plate. Messrs. John Elder 6 Go's Experiments. A double-rivetted lap-joint of 2^ -inch iron plate failed with a force of 15.06 tons per square inch of the net section between the rivets, the strength of the solid plate being 20.5 tons; also, a similiar joint of 9/ l6 -inch plate failed with a force of 14.28 tons per square inch of net section, whilst the strength of the solid plate was 20.2 tons. Here, it was found that the net tensile resistance of the plates between the holes was less by one-fourth than the direct strength: confirmatory of the deductions on the comparative weakness of the rivet-joints of the thicker plates. Mr. Brunei's Experiments. Mr. Brunei made experiments on double- rivetted double-welted plate-joints, of which the author published an RIVET-JOINTS. 639 analysis in 1 85 8-5 9. l The specimens were of ^-inch best Staffordshire plates, 20 inches wide, butt-jointed, with a covering or fishing plate on each side, 10 inches deep, put to- gether With punched holes Scale, One-twelfth. and rivets, as in Figs. 236-238, showing chain rivetting and zigzag rivetting. See tablet, p. 640. The ist specimen failed with 153 tons, shearing 10 rivets; and the 2d specimen failed with 164 tons, breaking a plate through the rivets; mean strength, 158.5 tons, = 15.85 tons per square inch of the entire section. Fig. 236. The 3d and 4th specimens failed with 167 and 147 tons respectively, through a line of rivet-holes ; mean strength, 157 tons, = 15.7 tons per square inch of the entire sec- tion. Fig. 236. The 5th, 6th, 7th, and 8th specimens broke with 158, 1 60, 1 6 1, and 168 tons re- spectively; mean strength, 162 tons, = 16.2 tons per square inch of the entire section. The fractures took place in the plates, following, in one case, the zigzag course of the rivets. In two cases, the rivets partly failed. Fig. 237. The Qth and roth speci- mens broke through the plate with 171 and 176 tons re- spectively; mean strength, 173^ tons, = 17.35 tons per square inch. Fig. 238. Five solid %-inch plates, from 1 2 to 1 6 inches in width, of the same quality as the speci- mens, were broken by from 1 9. 4 to 2 2 tons per square inch; mean strength, 20.6 tons. O o O o c o o o o Figs. 236-238. Rivetted Plate-Joints. Tested by Mr. Brunei. The third line following the tablet, p. 640, shows that the strength of the plate per square inch was impaired by from i to 7 per cent, by punching. The average efficiency, or actual strength, of the double-welt double- Recent Practice in the Locomotive Engine, 1858-59; also Railway Locomotives, 1860, page 2 640 STRENGTH OF ELEMENTARY CONSTRUCTIONS. rivetted joint may be taken as 80 per cent, of that of the entire plate, when the net sectional area is 83 ^ per cent. BRUNEL'S EXPERI- MENTS. Rivets, Pitch, Sectional Area of Plate. Total Shearing Section SPECIMEN. dia- meter. versely. Entire. Net, transversely. of Rivets. Nos. inch. inches. sq. in. sq. inch. per cent. square inches. I and 2, Fig. 236, 4 10 8.28 82.8 7.42, or 90 % of net sect. 3 and 4, Fig. 236, 3 / 4 10 8.125 81.25 8.84, or 1 10 5 to 8, Fig. 237,.. gandio, Fig. 238, X X 4 5 10 10 8.125 8. 5 81.25 85 8.84, or 1 10 ,, ,, io.6i,ori25 ,, ,, Chain. Chain. Nos. i and 2. Nos. 3 and 4. Net sectional area 82.8 % 81.25 % Net tensile strength 77 76 Strength per square inch of j net section, in terms of > 93 93.5 that of entire section... . I Zigzag. Nos. 5 to 8. 85% 78.6 92.5 Chain. Nos. 9 and ic 85% 8 4 99 Mr. R. B. Longridge reported the results of tests for the strength of rivetted joints in iron boiler plates, made for him by Mr. Kirkaldy. RIVET-JOINTS IN STEEL PLATES. The results of experiments on the strength of rivet-joints in steel plates, conducted by Messrs. David Greig and Max Eyth, Professor A. B. W. Kennedy, and Mr. C. H. Moberley, have, with those on rivet-joints in iron, been exhaustively analysed in the Strength of Materials, in preparation by the author; and noticed in summary in his work on the Steam Engine. The conclusions arrived at on the proportions and strength of rivetted joints in boiler plates of iron and of steel, ^-inch thick, are collected in the table No. 227. The relations of thickness of plate, diameter of rivets, and pitch of rivets, here shown for ^-inch plate-joints, are applicable to other thicknesses of plates, and are generalized as follows. The "spacing" denotes the distance apart of the two rows of rivets in double-rivetting. Table No. 226. STANDARD PROPORTIONS OF RIVETTED JOINTS IN IRON AND STEEL. Thickness of plates unity or i Diameter of rivets thickness of plate, Pitch of rivets (single-r i vetting). ... { <; hickness of plate, ( diameter of rivets, Pitch of rivets (double-rivetting)... { t hickness f P late > ( diameter of rivets, Diagona, pitch (double-rivetting).. { SSf^St Spacing (double-rivetting) longitudinal pitch, Lap (single-rivetting) { thickness of plate, I diameter of rivets, Lap (double-rivetting) I thi ^ness of plate, I diameter of rivets, 5/3 .56, or 6 3 10.48, or 5.24,01 RIVET-JOINTS. 641 i O cfl ffl a rt * g g S g H O <*> 5 *> & ^ w 5 i I 1 g M r^ to to .5 1 8 NOO VO W ON M || 9 O ON 10 O ON vo ! . d c co M M M 00 ON *5S OT IT) VO M O% w" vo 111 o "' Jj ONOO VO 00 00 ^J- t-I ^ o oo o t^ O *^ IP l' 13 i c/2 4> o \O N rf s, ^^s vO* ^t" vo 000 M M M c g "85 1-1 "c vO O O to r>. O 2 s - u o^ O* vO Jj 00 CO ON tovo vo ONOO ON |}1: u D 1 Cfl O 00 vo vo vovO I O 00 O OO t^OO iff ^ J v vo O O O VO N *I 43 " vo vovo i: *; vo vo O g t^- !>. VO 000 vo vo O 1 * 3 5 " oo" oo' i> VOVOM 00 OO t^ r- r vo f 2* *o a O *iV* 1 jj* a 5 w -^ ^-00* Jj ON ONOO ^" ^t" O^ 000 1-1 w a M M M d ill ||| i c vo vo vo 4) <-> N W W IO VO VO ^o Sl o- X, vo vo vc r-t-r- 3 ui 8' fe rg N N d CO fO fO 1 3 (5 . rt s i. c cnscnScns ws ens ens I i,l |^X sap soo soo ens ens ens J fi 5 I -u ; *j til ! p2 W3 'O 1 1 H Z > . 'Cod O ^ (O ^ 8 8. * Oi en M V % If! 11 1 s-s? ^ SP8 i ""3-- S, 5 S - w H $ "^ . "S I ||| || I S 5 H-5 H o bo ^g 41 6 4 2 STRENGTH OF ELEMENTARY CONSTRUCTIONS. On these proportions, the diameter and pitch of rivets suitable for plates of from y% inch to JI /i6 inch in thickness, are as given in the following table, No. 227^. The calculation is not extended for thicknesses greater than II / I 6 inch, for which the corresponding rivet has a diameter of i ^ inches, as this size of rivet is assumed to be the maximum properly available in practical operations. Table No. 2270. STANDARD RIVETTED JOINTS OF MAXIMUM STRENGTH, IN IRON AND STEEL PLATES, OF VARIOUS THICKNESSES. (Net section for single-riveting, 62.5 per cent, of whole plate-section; for double-riveting, 75 per cent. ) PITCH OF RIVETS. LAP, Thickness of Plate. Diameter of Rivets. Single Double Rivetting. Single Double Rivetting Lonei- T^- tudinal. D^gonal. Spacing. Rivetting. Rivetting. inch. inches. inches. inches. inches. inches. inches. inches # X 2 /3 I X 9 /i6 X I 5 /i6 3 /i6 ?/8 I I */z I/^ 27 /32 1/^5 2 X X I x / 3 2 ll A 1/8 iX 2>6 /i6 ft I 2 / 3 2> i 7 /& jI3 /32 1^ 3X H X 2 3 2X I XI /i6 2X 7 /i6 ^ 2'/ 3 3^ 2^ 2 4M y* I 2 2 / 3 4 3 2X 3 sX 9 fi6 \y% 3 4^ 3M 2/^ 3 3/ 6 r 7^ H J X 3'/, 5 3X 2 I 3/ I6 3X 6^ " /I6 * 3 2 / 3 4^ 3V.6 4^ 7X Other proportions than those given in table No. 2270 may, of course, be adopted; and, in fact, must be adopted for plates of greater thickness than IZ /i6 inch. By means of the following general formulas the pitches of rivets, and their diameters, producing joints of equal resistance, may be found for plates thicker than / l6 inch. Pitch of Rivets for Equal Resistance. _-7354 P=^* + d ZLV.ZL I-57/ 1-57 7854" / = thickness of plates, in inches. d= diameter of rivets, in inches. p = pitch of rivets, in inches. r = ratio of shearing section of rivets to net section of plates. CO PILLARS OR COLUMNS. 643 STRENGTH OF PILLARS OR COLUMNS. Mr. Stoney lucidly develops the leading principles of the resistance of columns supporting incumbent loads, 1 which are, no doubt, strictly appli- cable to columns of perfectly homogeneous material. He shows that the strength of very long square or round pillars varies directly as the 4th power of the diameter, and inversely as the square of the length; that it depends not on the direct strength of the material, but on the coefficient of elasticity, which represents the stiffness and capability of resisting flexure; that the strengths of similar long columns are as the squares of any linear dimension, or as the sectional areas, whilst their weights are as the cubes of the dimension; and that, if the strengths of long pillars of similar section are the same, while the length varies, the sectional areas vary as the lengths, and the weights vary as the squares of the lengths. Finally, that the weight which produces moderate flexure in a very long pillar is also very near the breaking weight, as a trifling additional load bends the pillar very much more, and strains the fibres beyond what they can bear a conclusion of great practical importance, which has been corroborated by experience. MR. HODGKINSON'S INVESTIGATIONS. The following is an abstract of Mr. Hodgkinson's conclusions on the resistance of cast-iron columns, under loads. The mode of fracture of cast- iron struts or columns under compression, is the same when the height is greater than the diameter of the specimen, and not greater than four or five times the diameter. When the height is greater, the specimen bends. Fracture usually takes place by the two ends of the specimen forming cones or pyramids, splitting the sides, and throwing them out. Sometimes the end slides off as a wedge, the height of which is somewhat less than 1.5 diameters. The tensile and compressive resistances average as i to 6.6; or, as the specimens were of unequal quality, the ratio should be i to 7 or 8, giving 49 tons per square inch for the ultimate compressive resistance. Long Columns. Experiments were made on castings of Lowmoor No. 3 iron. Of three cylindrical columns having the same diameter and length, the first had the ends rounded; the second, one flat end and one round end; the third, both ends flat. The strengths were as i, 2, 3 nearly. A long flat-end column has the same strength as a round-end pillar of half the length. The same properties apply to pillars of steel, wrought-iron, and wood. Swelling a pillar at the middle adds not more than one-seventh to the strength. The power of resistance is as the 3.6 power of the dia- meter, and as the 1.7 power of the length. These remarks apply to all pillars not less than 30 diameters in length, up to 120 diameters; and the following are the formulas for the breaking load of flat-ended columns : Solid columns, W = 44 '. ( 4 ) Hollow columns, W = 44 _ Z , ( 5 ) L in which D is the external diameter in inches, d the internal diameter in inches, L the length in feet, W the breaking load in tons. 1 The Theory of Strains, 1873, P a g e 2 44- 644 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Short Flexible Columns. The resistance is compounded of compressive resistance and transverse resistance. Let W = the breaking load, and c - the direct compressive resistance of the column (say 49 tons x sectional area in square inches), then, having first calculated the breaking weight W by the above formula, ( 4 ) or ( 5 ), the strength is found by the formula, This formula is inferred from the nature of the compound strain the results given by it are nearly correct, but rather excessive. The strength of long similar columns is nearly as the sectional area, or nearly as the square of the diameter; it is as the 1.865 power. The strength of taper columns is to that of cylindrical columns as D 2 D' 2 to D 4 , the extreme diameters of the taper column being D and D'. If the two columns be of the same length and solid content, the cylindrical one is the stronger. The strength of a column of a double-flanged section is only three-fourths of that of a uniform hollow cylinder of equal weight; and that of a column of cruciform section is less than half. The strength of a solid square cast-iron column is 50 per cent, more than that of a round column of the same diameter. A column irregularly fixed, so that the pressure is taken diagonally, has only a third of the strength when squarely fixed. Cast-iron pillars, with discs on the ends, are somewhat stronger than those with simply flat ends. Solid square cast-iron pillars bend or break in the direction of a diagonal. A slight inequality in the thickness of hollow cast-iron pillars does not reduce the strength materially. Square is the strongest section for timber rectangular in form. COMPAKATIVE STRENGTH OF LONG COLUMNS. Cast iron ............................................. 1000 Wrought iron ..... .................................... 1 745 Cast steel ............................................ 2518 Dantzic oak .......................................... 109 Red deal ...................................... . ...... 78.5 3.6 POWERS OF DIAMETERS. Diameter. Power. Diameter. Power. Diameter. Power. Diameter. Power. I I 3 52.196 d 632.91 10 3981.07 i-5 4-3 3-5 90.917 7 II02.4 II 5610.7 2 12.125 4 H7.03 8 1782.9 12 76745 2.5 27.076 5 328. 3 2 9 2724.4 1.7 POWERS OF LENGTHS. i i 7 27-33 T 3 78.3 21 176.92 2 3-25 8 34.29 H 88.8 22 191.48 3 6.47 9 41.9 15 99.85 24 222.0 4 10.55 10 50.12 16 111.43 26 254-3 5 15.42 ii 58.93 18 136-13 28 288.5 6 21.03 12 68. 3 3 20 162.84 30 324.4 PILLARS OR COLUMNS. 6 4S Mr. F. W. Shields gives the safe load on hollow cast-iron columns of good construction, with flat ends, and with base plates. 1 THICKNESS. inches. and upwards. Load per square inch of Sectional Area. Length 20 to 24 Diameters. tons. 2 25 to 30 Diameters. The reduction of the load per square inch with the thickness, is devised to allow for liability to weakness from inequalities of the casting. Mr. GORDON'S RULES. The first and second formulas were deduced by Mr. Lewis D. B. Gordon from the results of Mr. Hodgkinson's experiments. As here given, they show the total breaking weight of a cast-iron column with flat ends. The succeeding formulas for the strength of columns of wrought- iron and steel have been constructed on the basis of Mr. Gordon's formulas. 1. For solid or hollow round cast-iron columns: W-- ( 7 ) I+ ^ 2. For solid or hollow rectangular cast-iron columns: W = -^ ( 8 ) 1+ 500 3. For solid rectangular wrought-iron columns (Mr. Stoney): W = -*- ( 9 ) l+ - 3000 4. For columns of angle, tee, channel, or cruciform iron (Mr. Unwin): - w=-^ do) I+- 900 5. Solid round columns, of mild steel (Mr. Baker): W=-_ (ii) 1+ 1400 1 Transactions of the British Association, 1861. 646 STRENGTH OF ELEMENTARY CONSTRUCTIONS. 6. Solid round columns, strong steel (Mr. Baker): J _J_ _ 900 7. Solid rectangular columns, mild steel (Mr. Baker): W = -^- ................................. (13) I + 24 8o 8. Solid rectangular columns, strong steel (Mr. Baker): W = -^ ................................... (14) I6OO W = the breaking weight in tons. a = the sectional area of the material in inches. r=the ratio of the length to the diameter. The diameter for calculation is the least diameter of the section, or that in the direction of which the piece is most flexible. MR. HODGKINSON'S RULES FOR TIMBER COLUMNS. When both ends are flat and well-bedded, and the length exceeds 30 diameters : Long square columns of Dantzic oak (dry): W=io. 95 ^l ................................. ( 15 ) Long square columns of red deal (dry): W= 7 .8o^- .................................. ( 16 ) Long square columns of French oak (dry): 17 W = the breaking weight in tons. d= the breadth in inches. /= the length in feet. When timber columns are less than 30 diameters in length, their strength is calculated by formula ( 6 ), page 644, for which the value of W is to be calculated by one of the above formulas. When the column is oblong in section, multiply the result as found for the shorter dimension of the section by the ratio of the longer to the shorter dimension. MR. BRERETON'S EXPERIMENTS ON TIMBER PILES. Mr. R. P. Brereton gives the loads that could be borne, as found from experiments, by large fir or pine timber 1 2 inches square, of various lengths : 10 feet long bore 120 tons; 20 feet long, 115 tons; 30 feet long, 90 tons; 40 feet long, 80 tons. Mr. Stoney plotted these results, and constructed the following table from the curve : CAST-IRON FLANGED BEAMS. 647 i" Ratio of length to least breadth 10 15 20 25 30 35 40 45 50 Weight that can be borne in tons per ) square foot of section ) 120 118 "5 100 90 84 80 77 75 Checking this table by Mr. Kirkaldy's experiments (page 547) on balks of Riga and Dantzic timber, having a length of 20 feet, which was 18% times the width, the actual breaking weights were Total. Per square foot of Section. By Mr. Stoney's Curve. Riga 148 tons, or 126 tons, 116 tons. Dantzic 138 or in 116 116 Mean 119 showing a close correspondence between the results of experiments con- ducted independently. Mr. Hodgkinson's rule gives results which are rather less than those that are given in Mr. Stoney's table. MR. LASLETT'S EXPERIMENTS ON COLUMNS OF WOOD. A notice of Mr. Laslett's experiments is given at page 541. He deduces from his experiments, repeated for many kinds of wood : that the maximum resistance of square pieces to compression is exerted when the sectional area in square inches is to the length in inches approximately as 4 to 5, for equal seasoning and equal specific gravities. According to this deduc- tion, the maximum resistance of 1 2-inch square balks on end, would be exerted when they are 1 5 feet in length. CAST-IRON FLANGED BEAMS. Mr. Hodgkinson tested, for transverse strength and deflection, a number of cast-iron model beams of various proportions, and he discovered that the maximum strength of double-flanged beams, for a given sectional area, was realized when the area of the upper flange was one-sixth that of the lower flange. This conclusion harmonizes with the fact that the resistance of cast iron to compression is from 5 to 6 times the tensile resistance; and, as a scientific fact, it has its value. The general formula (19), page 511, for flanged beams, is as follows : in which a is the sectional area of the lower flange ; a!' that of the web, d" the reputed depth of the beam and of the web, taken as the total depth minus the thickness of the lower flange, / the span, all in inches, and W the breaking weight at the middle, in tons. Sections of cast-iron beams which have been tested, are given in Figs. 239-258, comprising 17 model beams tested by Mr. Hodgkinson, and 6 model beams by Mr. Berkley; like- wise 1 1 large beams. 1 Mr. Berkley's model beams are in pairs, and have the 1 See Hodgkinson on the Strength and other Properties of Cast Iron, 1846; and the Proceedings of the Institution of Civil Engineers, vol. xxx. page 252 (Mr. Berkley's paper on the " Strength of Iron and Steel"). 6 4 8 STRENGTH OF ELEMENTARY CONSTRUCTIONS. same dimensions as Nos. 6, 7, and 12 respectively. Table No. 228 con- tains, in part i, the needful particulars of those beams, and of the ultimate breaking weight, both actual and as calculated by formula (i). The deflection and the elastic strength of the beams are given in part 2 of the table. A tensile strength of 7 tons per square inch has been adopted in the calculation of the ultimate strength of Mr. Hodgkinson's model beams, 6^2 tons for the large beams, and 10^/2 tons for Mr. Berkley's model beams; for, though his test-castings bore a greater tensile load than 10^ tons, they were too short in the tested portion, which was i inch square and only i y 2 inches long, for the action of simple tensile resistance. No. i. No. 2. Nos. 3, 4. No. 5. Nos. 6, 18, 19. Nos. 7, 20, 21. No. 8. No. 9. No. TO. No. it. NOS. 12, 22, 23. No. 13. No 14. No. 15. Scale of Nos. i to 17 One-tenth full size. No. 16. No. No. 24. No. 25. No. 26. Scale of Nos. 24 to 27 One-twentieth full size. No. 27. Figs. 239-258. Sections of Cast-Iron Beams tested for Transverse Strength by Mr. Hodgkinson, Mr. Berkley, and others. Table No. 228. CAST-IRON FLANGED BEAMS. 649 Table No. 228. STRENGTH OF CAST-IRON FLANGED BEAMS. PART i. ULTIMATE TRANSVERSE STRENGTH. i. MODEL BEAMS. Thickness. Sectional Area of Flanges. Breaking Weight at the middle. Refer- Depth ence Number. Span. at middle. d Web. Lower flange. Lower flange. a Upper flange, ratio to lower. Calculated. W Actual. W feet. inches. inches. inches. sq. inches. ratio. tons. tons. Mr. HODGKINSON'S BEAMS. Assumed tensile strength, 7 tons per square inch. i 4-5 5/^ .29 39 69 I tO I 2.47 2.98 2 M .30 55 98 I tO 2 3-27 3.29 3 }> .32 57 1.20 I to 4 3.83 3.69 4 it 33 56 1. 2O i to 4 3-87 3.64 5 J5 35 .51 ?-57 i to 4^ 4.68 4.79 6 ,, 38 53 2.20 i to 4 6.45 6.46 7 J} 34 56 2.8 9 i to 5^ 7.85 7-47 8 ,, 33 57 2.31 i to 3.2 6.49 6.71 9 |f 35 537 2.92 I to 4.3 8.04 7-54 10 ,, 34 54 3-57 I to 5.6 9.56 8.68 ii 5> .266 .66 4.40 i to 6 10.98 11.65 12 335 65 4-31 i to 7 11.00 10.40 13 M 3-32 i to 6. 7 9.02 9.40 H 7-'o 6-93 38 75 4-54 i to 6 10.26 9.90 15 4.10 .40 74 4-44 i to 6 5-41 6.05 16 9.0 IO X 25 77 4.72 i to 8.3 13-28 12.80 17 4.5 5/? .40 .46 1.04 none 3-83 3-93 Averages for 17 beams i to A.. 6 7.07 7.01 Mr. BERKLEY'S BEAMS. Assumed tensile strength, 10.5 tons per square inch. 18 4-5 $yi .38 53 2.20 I to 4 9.67 10.00 19 J} 9.67 10.00 20 ,, 34 .56 2.'8 9 i to 3/2 11.85 ".75 21 ,, 11.85 ".85 22 ,, 335 .65 4.31 i to 7 16.47 14-25 23 tt it M 17.08 18.00 Averages for 6 beams i to <; " 12.76 12 64 L ^ j. j wt y vy **. V4 t- 2. LARGE BEAMS. Assumed tensile strength, 6.5 tons per square inch. (Reported by Mr. Hodgkinson.) 24 3 beams j 18.0 17 .625 1-25 10.31 i to 4. 6 24-93 25-0 25 11.67 9 I -5 I r 12. OO i to 1.33 21.24 2O. O 26 1 27-4 30.5 2.08 2.07 25.05 I tO 2. 1 94.64 76.6 + 27 2 23.1 36.1 3-36 3-12 74.60 none 330.0 153.0 + Mr. CUBITT'S BEAMS. 28 15.0 7.15 1.04 59 7.98 i to 3.6 7-75 7.00 29 , 7.17 1. 10 59 8.ii i to 3.6 7.96 7-13 30 , 10.75 93 .04 5.25 i to 2.3 11.02 II.5O 31 5 10.75 1-05 05 5.42 i to 2.3 11.71 12.00 32 s , 12.75 73 .88 4-47 I tO 2. 7 "95 10.25 33 , 12.8 95 .09 5-59 I tO 2. 25 14.89 15-75 34 4 } 14.0 .91 .00 6.50 none i8.39 12.38 35 5 , 17.25 .68 .84 4.96 i to 2. 2 19-39 16.00 36 fi 7-5 7.15 1.06 59 8.03 i to 3.4 15.63 15-63 37 6 M 10-75 .92 .02 5.16 i to 2.25 21.76 23.87 Averages for n of these large beams i to 3. 17 17.00 17.08 Total averages for 34 beams i to 4. 30 11.31 11.26 650 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Table No. 228 (continued}. PART 2. DEFLECTION AND ELASTIC STRENGTH. i. MODEL BEAMS. Limits of Elastic Strength. Ratio of Reference Number. Form of Beams. Elastic Strength to Breaking Weight. Coefficient of Elasticity. Deflection at middle. Load. Inches per ton of load. tons. inches. tons. inches. per cent. E. Mr. HODGKINSON'S BEAMS. 2 3 4 6 Elliptical. 45 5.670 .079 88 5 2 32 I Do. 49 7.244 .068 9i 4980 9 Uniform depth. 33 6.872 .048 9i 4372 10 Do. 36 7-454 .048 86 3420 ii 12 Do. .48 10.254 .047 99 3324 13 14 Elliptical. Uniform depth. .46 .60 8.503 9.204 11 90 93 5438 4284 15 Do. Do. .70 55 3.787 11.450 ^048 63 89 5520 5208 17 Segmental. .42 3.700 .114 94 5600 Averages for 10 beams... . .... 88 47<;8 Mr. BERKLEY'S BEAMS. H"/ 18 19 Uniform depth. Do. 245 .271 7.00 8.00 035 034 lo 7400 7634 20 Do. 387 IO.OO 039 85 5486 21 22 Do. Do. .264 9.00 8.00 037 033 76 56 5760 5274 23 Do. 303 10.00 .030 55 5i54 Averages for 6 beams . . . 70 6118 / 2. LARGE BEAMS. (Reported by Mr. Hodgkinson.) 24 3 beams Segmental. 1. 00 20.0 .050 80 4632 8 Segmental. 33 1.29 10.0 76.6 033 .017 5o P 4236 27 .68 153-0 .0044 ? ? Mr. CUBITT'S BEAMS. 28 Uniform section. 1-54 6.0 257 86 4760 29 Do. 1.215 5- 243 70 4706 30 Do. 645 7-o .092 6l 5340 3i Do. 1.04 II. 095 92 4900 32 Do. .60 9.0 .067 88 5566 33 Do. .90 15.0 .060 95 5 32 34 Uniform depth. .41 6.0 .070 49 4840 35 Do. .76 16.0 .048 100 5226 36 Uniform section. 3i 10. .031 64 4884 37 Do. .261 20. o .013 84 4762 Averages for 12 beams 77 4906 Total averages for 28 bean IS 79 5"3 CAST-IRON FLANGED BEAMS. 651 1 No. 26 z No. 27 as the appara 3 No. 32 4 No. 34 5 No. 35- 6 No. 37 Notes to Table No. 228. 'broke apparently in consequence of an accidental shake." 'with this load, 153 tons in the middle, the experiment was discontinued, :us was overstrained." 'bottom flange unsound." 'bottom flange unsound." 'bottom flange unsound." ' nearly but not quite sound." The contents of the table exhibit a surprisingly close conformity throughout between the calculated and the actual ultimate strengths of such beams as were sound, and were fairly tested and broken. The total averages for 34 cast-iron beams show that, with a ratio of upper to lower flange of i to 4.30, the breaking weight, as calculated, was 11.31 tons, and as tested, 11.26 tons. It further appears that the ultimate strength of a cast-iron beam is scarcely affected by the proportionate size of the upper flange; and that the formula ( i ), page 647, may be adopted for the calculation of the strength of flanged cast-iron beams of any ordinary section. It is sufficient to employ the factor, 7 tons, for castings of less than ^ inch in thickness, and 6.5 tons for those which have a thickness of i inch and upwards. Taking the span in feet : Ultimate Transverse Strength of Cast-iron Flanged Beams. For the thinner castings ...... W = d " ^ a + 2 a//) ............. ( 2 ) O For the thicker castings ...... w = T(6.5'+i.9'O ............ ( 3 ) VV = the breaking weight in tons at the middle; a the sectional area of the lower flange, and a" the sectional area of the web, taken at the reputed depth, both in square inches; / the span in feet. The reputed depth is the total depth minus the thickness of the lower flange. The following are the constants for other factors of tensile strength : Tensile strength per square inch. Constants in formula 2 or 3. for a. for a". 6 tons ........ ......................... 6 ............ 1.7 &A ................................. 6 -75 ............ 2.0 lYz ,, ................................. 7-5 ............ 2 - 2 8 ................................. 8 ............ 2.3 9 ................................. 9 ............ 2 - 6 10 ................................. 10 ............ 2.9 Approximate Rule for the Strength of Cast-iron Flanged Beams. Referring to formula ( 2 ) for a tensile strength of 7 tons : To the sectional area of the lower flange add a fourth of the sectional area of the web, calculated on the total depth, both in inches; multiply the sum by the total depth in inches, and by 2^ ; and divide the product by the span in feet. The quotient is the breaking weight at the middle, in tons. For any other tensile strength, use it as the multiplier instead of 2^, and divide the product by 3, and by the span. The quotient is the breaking weight. 652 STRENGTH OF ELEMENTARY CONSTRUCTIONS. ELASTIC STRENGTH AND DEFLECTION OF CAST-IRON FLANGED BEAMS. From the observations of the experimentalists on the deflection of the beams noted in the second part of table No. 228, it is shown that the elastic strength that is, the limit of load for uniform increments of deflection, irrespective of set as given in the table, averages about 80 per cent, of the ultimate strengths. The value of the coefficients of elasticity, E, were cal- culated tentatively by means of the formula ( 13 ), page 531, for beams of constant depth and uniform strength loaded at middle; but, since all the beams excepting six of Cubitt's beams were proportioned for carrying uniform loads, the tentative values found for these beams were modified according to their special forms, on the principles already adopted in the general section on the deflection of beams, pages 529, 530, to give the proper values of E. The general average value for E is 5113, and the numerical coefficient of the formula ( 13 ), page 531, being multiplied into this value, gives the resultant coefficient for beams of cast iron. The general formula is as follows ; c being the coefficient : Deflection of Cast-iron Flanged Beams. D _ W/3 (cE) P a e 6 5 6 ; an d the safe distributed load in the last column is taken as one-third of the breaking weight at the middle, according to a factor of 6. To find, from table No. 230, the ultimate strength of, or the safe per- manent load for, a joist, for any other span than 10 feet, multiply the tabular weight for the beam of the given section by 10, and divide the product by the given span. Inversely, to find the span for a joist of a given section, with a given weight, multiply the tabular weight by 10, and divide the product by the given weight 654 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Table No 230. SOLID-ROLLED WROUGHT-IRON JOISTS : DIMENSIONS, WEIGHT, AND STRENGTH. SPAN, 10 FEET. Depth of Beam. Breadth of Flanges. THICKNESS. Reputed Weight per lineal foot. Ultimate Strength, Loaded at the Middle. Safe Perma- nent Load, Uniformly Distributed. Of Web. Of Flanges. inches. 16 inches. ctf inch. ...3/4 - 9/16 ...9/16... 9/16 ...7/16... 3/4 ...7/16... 7/16 ...3/8 -. 7/16 ...7/16... 3/8 ...3/8 ... 3/8 "J/i6 .. inch. ... I3/I6 ... I3/I6 . 7/8 . Ibs. ... 62 ... tons. .. 84 cwts. ;6o H IA. ... 5 78 6 cU 60 ... 60 ... 68 ... 67 453 44.7 12 12 ' 6 c 15/16 n/i6 ... 56 42 61 .. 4 1 ? 407 qoo 10 10 10 9>A ... 9# 4^ ... 4 # ... 4^ ... 4/2 ... 3# cv 5/8 Q/l6 36 T.2 34 ... 26 ... % s 227 ... 173 167 177 9/16 ... 11/16... 1/2 0/16 32 7Q , 24 2Q 18.6 21 124 I/LO 8 8 ... ^ /8 5 4 9/16 1/2 . 29 ... 21 20 .. i c.4. 133 IO7 8 8 ... 7 7 2X 2^ 3/8 7/16 ... 15 I C 9-5 ... Q. 1 , ... IV^J ^ 62 3^ ^M 5/16 ...5/16... 5/16 ...9/32... 5/16 ...5/16... 5/16 ...7/16... 3/8 ...3/8 ... 5/16 ... 1/4 ... i/4 ... 1/4 .- 3/i6 ... 1/4 ... 3/i6 1/2 7/16 . 19 IQ , 1 1.6 . 1 0.4. 77 6q 7 7 ... j/8 ... 2^ 2i 7/16 3/8 . 14 14 ... 7-6 ... 6.6 ... 7-8 ... ,8 ... ... 15.1 ... .. 86 . 51 ... 44 52 -. 39 40 ... 101 35 cy 6X 6X ... 6X 6 ... 5/2 5 ... 4# 4 4 ... 3^ 3 3 3X ... 2X .- 2 ... 5 ... 2 ... 4^ .- 3 ... 2 3 2 I# ... 3 2 *. 13/32 .. 3/8 .- 7/16 .. 9/16., 7/16 .. 1/2 ... 7/16 .. 5/16... 3/8 .. 5/16... 7/32 .. 5/16... 7/32 ... 18 ... ii ... 30 ... 10 27 13 ... 8 ... 12 ... 8 ... $ 1 A ... 10 ... ^ 6 3^8 ... 2.45 ... i. ii ... 2.5 ... 1.22 40 21 25 ... 16 7-4 ... 17 8.1 TRANSVERSE STRENGTH OF WROUGHT-IRON JOISTS. A number of solid-rolled joists of uniform symmetrical section were Figs. 259-262. Sections of Solid Wrought-iron Flanged Beams, or Joists. tested by Mr. Kirkaldy for Mr. Moser, the sections of four of which have WROUGHT-IRON FLANGED BEAMS OR JOISTS. 655 been ascertained approximately, and are here annexed, Figs. 259-262, with the following particulars : JOISTS. Weight per foot. Depth. Breadth. Web. Thick- ness. Flange. Mean thickness- Sectional Area. Web at reputed depth. One Flange. Total. A Ibs. 43-56 37.54 26.22 19.16 inches. 11.75 9.85 7.90 7.07 inches. 5.70 4.60 376 3.00 inch. .60 .50 .50 .50 inch. .643 .804 .619 . 4 85 sq. ins. 6-537 4.523 3.640 3.293 sq. ins. 3.665 3.700 2.329 1-455 sq. ins. 13.34 11.50 8.033 5.870 B D . E The elastic and ultimate transverse strengths of these beams are reduced from the observations of Mr. Kirkaldy, and are given in the table No. 231. In the last column, the probable real ultimate strengths of the beams are added; they are computed at twice the elastic strength, in correspondence with the well established ratio of the tensile elastic and ultimate strengths of wrought iron. The ultimate strength, column 4, was calculated by formula (19), page 511, in which the ultimate tensile strength, s, is taken as 20 tons : Table No. 231. TRANSVERSE STRENGTH OF SOLID-ROLLED WROUGHT- IRON JOISTS. (Results of Experiment.) JOISTS. Span. Elastic Strength. Ultimate, or Breaking Weight. Cause of Failure. Probable Real Breaking Weight. Observed. Calculated. Observed. A : feet. 20 10 20 10 20 10 10 5 tons. 10.714 21.428 8.482 17.857 4.0l8 8.705 5402 11.607 tons. 20.390 40.780 15.060 30.120 8.203 16.406 8.150 21.124 tons. 14.310 32.450 11.445 26.530 6.371 15.112 8.150 19.520 Buckling )) j> )> ?> 5J JJ tons. 21.428 42.856 16.964 35-7I4 8.036 17.410 10.804 23.214 D E 11.027 20.331 16.736 22.053 As the cause of failure was buckling, it is clear that the beams would have supported greater weights if they had been supported laterally. That the want of such support was the cause of the weakness, is evidenced by the fact that the observed breaking weights more nearly approach the calculated weights for the shorter than for the longer spans. The probable real break- ing weights average more than the weights, as calculated from the experi- mental data. Adapting the formula (19), page 511, by assuming the value of s 20 tons, then 656 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Ultimate Transverse Strength of Solid Wrought-iron Joists of Uniform Symmetrical Section. In tons, 55*') .......................... (s) >... .(6) ... 0.03 / W = the breaking weight at the middle. a - the sectional area of the lower flange, in square inches. a" the sectional area of the web, taken at the reputed depth, in square inches. ^ = the reputed depth in inches; that is, the total depth minus the average thickness of one flange. /=the span in feet. Approximate Rules for the Strength of Solid Wrought-iron Joists of Ordinary Proportions. Reduce the second coefficient in the numerator of the above formulas, to i, and increase the depth to the total depth, d, of the beam. ist Approximate Rule. In Tons. To the sectional area of one flange add one-fourth of the sectional area of the web, calculated on the total depth, both in inches; multiply the sum by the depth in inches and by 7, and divide by the span in feet. The quotient is the breaking weight at the middle. In Hundredweights. Substitute 133 for the multiplier 7 in the preceding calculation. The second last column in table No. 230 was calculated by this rule. The formulas are In tons, W = ............................... (7) 2d Approximate Rule. In Tons. Multiply the breadth of the joist by the square of the depth in inches, and by 0.6; and divide the product by the span in feet. The quotient, plus i, is the breaking weight at the middle. In Hundredweights. Substitute 12 for the multiplier 0.6 in the preceding calculation. The quotient, plus 20, is the breaking weight. The formulas are : In tons, W = +i ................................... (9) ,,r 12 bd 2 / x In cwts., W = -,- + 20 ................................. (10) W = the breaking weight at the middle, b the breadth, and d = the depth in inches; /=the span in feet. Note. The ist approximate rule is better than the 2d rule. WROUGHT-IRON FLANGED BEAMS OR JOISTS. 6 S7 DEFLECTION AND ELASTIC STRENGTH OF SOLID WROUGHT-IRON FLANGED BEAMS OR JOISTS OF UNIFORM SYMMETRICAL SECTION. With the particulars already given for the beams A, B, D, and E, page 655, and the subjoined deflections under given loads at the middle, within the elastic limits, the values of the coefficient of elasticity, E, calculated from the general formula (13), page 531, by inversion, are as follows: BEAM. Span. Load at the Middle. Deflection. E. feet. tons. inches. 20 8.929 .848 13,196 10 8.929 .132 10,588 B. ,- 20 7-143 I.I50 13,100 10 14.286 330 11,414 20 3-571 1.420 12,120 . 10 8.929 .440 12,232 IO 5-357 .405 13,692 5 8.929 .158 7,314 Average coefficient of elasticity, E, excluding the last, as exceptio for 7 beams, nal 12,334 To adapt the general formula just named, the constant becomes 4x 12,334 = 49,336; or (49336-^123=) 28.5, when the span is in feet: Deflection of Solid Rolled Wrought-iron Flanged Beams of Uniform Symmetrical Section, loaded at the Middle. D = W/3 28.5 # +1.1550") D = the deflection in inches, W = the load in tons, / the span in feet, d" the reputed depth in inches, being the whole depth minus the thickness of the lower flange; a the sectional area of the lower flange, and a" the sectional area of the web reckoned on the reputed depth, both in square inches. Note. If the same weight be uniformly distributed, the divisor 45.7 is to be used. Approximate Rule. Load at the Middle. To the sectional area of one flange add one-fourth of the sectional area of the web, calculated on the total depth, both in inches; multiply the sum by the square of the depth in inches, and by 114, making a product A. Multiply the load in tons by the cube of the span in feet; and divide this product by the product A. The quotient is the deflection in inches. Load Uniformly Distributed. Use the multiplier 183 in the calculation, instead of 114. STRENGTH OF RIVETTED WROUGHT-IRON JOISTS. Compared with solid-rolled joists, the strength of rivetted joists is less, and the deflection is greater. A series of rivetted plate-joists of uniform 42 658 STRENGTH OF ELEMENTARY CONSTRUCTIONS. section were constructed and tested for deflection by the late Mr. Thomas Davies, in I856. 1 The sections of these beams are shown by Figs. 263-268; No. 7. Figs. 263-268. Sections of Rivetted Wrought-iron Joists. they consist of plate-webs and flanges, united by angle-irons, of which the scantlings are given on the figures, No. of joist, i. 2. 3. 4. 5. 7. Weight of joist,... 4.25 ... 6.5 ... 20.5 ... 14.75 J 3-62 ... 13.62 cwts. Span for trial, n.66 ... 16.5 ... 28.5 ... 28.5 ... 22.5 ... 22.5 feet. The loads rested on spaces at the middle of the beams, 2 1 inches wide. The elastic strengths and deflections of the joists, as deduced from the record of the results, were as follows : Elastic strength,.... n% ... n% ... 10^ ... 7 ... 13 ... 13 tons. Deflection, 437 ... .625 ... 2.000 ... 1.620 ... .875 ... .875 inches. All the beams except No. 2 were unsymmetrical, and an approximate rule for strength and deflection may be constructed, by making a calculation for each beam in the position in which it was tested, and in an inverted position, in terms of the flange and angle-irons which are undermost, in each position respectively; and finding the mean results. In this way the 1 See a paper read at a meeting of the Edinburgh Architectural Institute, February 18, 1856. WROUGHT-IRON FLANGED BEAMS OR JOISTS. 059 breaking weights, columns 2 and 3 in the following tablet, were arrived at for e'ach joist, applying the formula ( 5 ), page 656, for solid-rolled joists. Two-thirds of these mean calculated breaking weights are given in the 4th column ; and they are probably the real breaking weights, since they average exactly twice the observed elastic strength given in the last column. JOISTS. Calculated Breaking Weight. Mean Breaking Weight. Two-thirds of the Mean. Observed Elastic Strength. No i, tons. 34. 5 2 tons. tons. tons. Do. inverted, 38.65 36.^9 24.4 flj? No 2, 31.80 21.2 nK No 3, 37.36 28.Q2 33.14 22.1 I0|/- No 4 . 13.74 Do. inverted, 27.03 20.40 13-6 7 No 5, .. 33.47 ) ( 13 No. 7 (No. 5 inverted), 5M9 ( 42.53 28.4 < j ) 13 Averages. 22 O no By an appropriate alteration of the coefficient in the rule for solid-rolled joists, therefore, the following rule is obtained: Approximate Rules for the Strength of Rivetted Wrought-iron Joists. In tons. Find the sectional areas of the upper and the lower flanges with their angle-irons respectively;- to half the sum of these areas add one- fourth of the sectional area of the web, calculated on the total depth, all in inches; multiply this last sum by the depth in inches, and by 4^3; and divide by the span in feet. The quotient is the breaking weight at the middle. In hundredweights. Substitute 94 for the multiplier 4^ in the preceding operation. The formulas are: In tons, W = Incwts, (12) (13) in which a' is half the sum of the upper and lower sectional areas, a" the sectional area of the web, d the depth, / the span, and W the load at the middle. Note to the rule. If the beam is symmetrical in section, the section for one flange only is calculated. Similarly, let the deflections for the elastic strengths, for each beam in its first position, and as inverted, be calculated by the formula (u), page 657, for solid-rolled joists. They are given in the 2d column of the 66o STRENGTH OF ELEMENTARY CONSTRUCTIONS. following tablet, and the mean for each is given in the 3d column. The actual deflections, in the 4th column, are greater than those in the 3d column, in the ratios indicated in the last column. JOISTS. Calculated Deflections. Mean Calculated Deflections. Actual Deflection. Ratio of Actual to Calculated Deflections. No i inches. .227 .203 I.I48 1.483 inches. .215 .418 I.3I6 1. 606 .654 inches. 437 .625 2.000 1.620 | -875 ( .875 ratio. i to 2.033 i to 1.495 i to 1.520 i to 1.009 i to 1.338 i to 1.338 i to 1.479 Do. inverted, No. 2, No. i,., Do inverted No 4. 2.130 1.083 Do inverted No. 5, . 730) 578J g No. 7, . No 7 (No 5 inverted) Average ratio, not includin There is considerable variation in the ratio of the calculated to the actual deflections; the average is i to i^. Modify accordingly the coefficient in the approximate rule for solid-rolled joists, page 657 : Approximate Rule for the Deflection of Rivetted Wr ought-Iron Joists. Load at the middle. Find the sectional areas of the upper and the lower flanges with their angle-irons respectively; to half the sum of these areas add one-fourth of the sectional area of the web, calculated on the total depth, all in inches ; multiply this last sum by the square of the depth in inches, and by 75, making a product A. Multiply the load in tons by the cube of the span in feet; and divide this product by the product A. The quotient is the deflection, in inches. Load uniformly distributed. Use the multiplier 120 in the calculation, instead of 75. Note to the Rule. If there be no flanges, the angle-irons alone are to be taken as representing flange-area. Remarks. i. The experimental elastic strengths, as well as the deflec- tions, of Nos. 5 and 7 beams, which were in fact the same beam in reverse positions, are identical. The identity here observed is confirmatory of the general principle of the elasticity of beams, enunciated at page 517. 2. It follows, from experimental tests, that the strength of solid-rolled joists is to that of rivetted joists, of equal weights, as i^ to i; and that their deflections, under equal loads, are as i to i^. BUCKLED IRON PLATES. Buckled plates, so named by Mr. Mallett, the inventor, are bulged plates, which are curved or arched, with a very small rise or curvature, springing from the edges of the plate, a narrow strip of which, all round, RAILWAY RAILS. 66 1 is retained in the original plane of the plate. Buckled plates are very rigid, and are capable of sustaining heavy loads. When bolted down, or rivetted all round the edges, they offer twice the resistance that they do if simply supported; and if two opposite sides be wholly unsupported, the resistance is only s/ 8 ths. Less than 2 inches of rise is sufficient for a ^-inch plate, 4 feet square. A ^-inch Staffordshire plate, 3 feet square, with a 2-inch flat border, buckled with a rise of i ^ inches, is crippled with a load of 9 tons distributed over half the surface ; if rivetted down, 1 8 tons are required to cripple it. Plates of soft puddled steel bear twice these loads before being crippled. The strength appears to increase as the square of the thickness. The factor of safety adopted by Mr. Mallet is 4 for steady loads, and 6 for moving loads. RAILWAY RAILS. RAILS OF SYMMETRICAL SECTION. These are beams of limited depth and considerable flange-area, and the strength should be calculated by formula (22), page 512, repeated below; for the application of which the section of a double-headed rail is to be divided for the calculation, according to the annexed diagram, Fig. 269. a, a, a, a, are the flange portions, c d the web, d the depth of the rail, and d" the vertical distance apart of the centres of the flanges. That the sectional area of the flange may be correctly ascertained, Nl 17 Fig. 269. For Transverse Strength of Rail of Symmetrical Section. Fie. 270. Squaring the Section of a Double- headed Rail. the surface should be divided into thin strips parallel to the neutral axis, as in the diagram, the area of each of which should be calculated. If the outer contour of the flange is circular, as is usually the case, the resultant centre of the flange a a may be taken as passing through the centre of the circle. If the flanges are otherwise formed, the position of their centre of gravity, ascertained by the rule, page 514, may be taken. Approximate results may be obtained by squaring the section of the flanges, by the eye, and calculating from the centres of the rectangles, as in Fig. 270. 662 STRENGTH OF ELEMENTARY CONSTRUCTIONS. W/ (2) (4 a' - +1.155 /'73 ' = 131.56 tons, total tensile resistance. 1.99 Fig. 272. Section of Steel Flange-Rail for the Metropolitan Railway. RAILWAY RAILS. 66 7 The breaking weight at the middle, on a span of 60 inches, is, then, , 17 131.56 x _:L_D DO = 29.59 This, it may be noted, is an application of formula ( 25 ), page 516. Let the Metropolitan rail be calculated for its transverse strength when upside down: the product 4.324 x (35 x 1.73) is divided by 2.51 inches, the distance of the upper surface, now downwards, of the head of the rail from the neutral axis : 4-3 2 4x(35 x I> 73)= I04t30 tons> tota i tensile resistance upside d and the breaking weight at the middle, on a span of 60 inches, is W = *4.3 x 3.374 x 4 = 6 tons . DO To compare these calculated results with the results of Mr. Kirkaldy's experimental tests, these are, with Mr. Fowler's permission, here ab- stracted : Table No. 234. TRANSVERSE STRENGTH OF STEEL FLANGE-RAILS FOR THE METROPOLITAN RAILWAY. 1867. Span 60 inches. Load applied at the middle- (Reduced from Mr. Kirkaldy's Reports.) Ultimate Strength. Elastic Strength. Elastic Deflec- tion per ton. Load. Deflec- tion. Load. Deflec- tion. Set. S olid rail, normal position ist specimen tons. 30-393 29.632 29.043 28.457 28.733 inches. 942 8.69 10.54 9.18 14.78 tons. I2.50O II.607 11.607 12.500 11.607 per cnt. inches. .255 .232 .238 .250 .232 inches. .022 .028 .030 .021 .030 inches. 2d do ^d do 4th do 5th do Averages 29.270 10.52 11.964 41 .2414 .026 .0202 Solid rail inverted. 22.OI4 542 I5.I80 6 9 .290 235 .040 .0191 Normal position, holes y punched in the flanges, f Average of six speci- ( I5.6l8 .669 11.904 76 .023 .0197 Normal position, holes ") drilled. Mean of two \ specimens ) 23.934 4.51 12.500 52 .267 .025 .0214 Note. The holes, punched or drilled, were i.io inches in diameter, .75 inch from the edge of the flange. 668 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Fig. 273. Section of Iron Flange-Rail. Scale, %th. The breaking weight of the rail varied, in six specimens, from 28.46 tons to 30.39 tons; average, 29.27 tons. The calculated breaking weight is 29.59 tons, or about y$ ton in excess of the average. Inverted, one specimen broke with 22.01 tons; the calculated breaking weight is 23.46 tons, or nearly i^ tons more. The elastic strength in this position was greater than in the normal position. Influence of Holes in the Flange. When punched, the effect was to reduce the breaking weight nearly a half. When drilled, the ultimate strength was only reduced about a sixth. But the elastic strength remained, in both cases, unimpaired; and the elastic deflection per ton was practically identical in all cases averaging about .20 inch per ton. Wrought-Iron Flange- Rails. The annexed sec- tion, Fig. 273, shows a wrought-iron flange-rail, 5 inches high, weighing 75 Ibs. per yard. Ten specimens of rail of this section, of Cleveland manufacture, were tested for transverse, tensile, and compressive strength by Mr. Kirkaldy. The samples for the tensile and compressive tests were cut from the middle of the head and of the flange. Tensile. Compressive. HEAD: Elastic strength per square inch ...... 13.21 tons. 18.13 tons- Ultimate strength ...... 20.93 67.00 FLANGE: Elastic strength per square inch.... 13.62 Ultimate strength ,, ....21.83 Ultimate transverse strength, span 3 feet ...... 33.60 tons. The centre of gravity of the reduced section, that is, the neutral axis of the entire section, shown in the Fig. 273, is 2^ inches above the base of the section, or the height is one-half the total height of the rail. The resultant centre of tensile stress is 1.974 inches below the neutral axis, and that of compressive stress is 1.683 inches above it, as indicated. The vertical distance apart of these resultant centres is (1.683 + I< 974 =: ) 3-^57 inches, and by Rule 4, *%. "2 ( - 6.92x3.657x5.058 =23.62 tons per square inch, the ultimate tensile strength of the wrought-iron flange-rail, in its lower or flange portion. DEFLECTION OF RAILS. Double-headed Rails. Formulas for the deflection of double-headed rails are deduced by equating the values of s, the tensile strength per square inch, given by formula ( 2 ), page 662, and by formula ( 2 ), page 528; thus : = - ; whence, <*-7 RAILWAY RAILS. 669 From this equation, the following values of D and E are deduced : W/3 _ 4E ( 4 tfV /2 + W/3 4D The values of E, by formula ( 6 ), for the rails tested by Mr. Price Williams, as detailed in table No. 232, page 663, are as follows: Iron Rails, double-headed. Steel Rails, double-headed. E. E. L ............ 11,146 ............ A ............ 13,038 N ............ 10,571 ............ B ............ 13,588 M ............ 9,683 ............ C ............ 12,982 P ............ 12,457 ............ D ............ i3>7 Averages... 10,964 ................................ I 3 I 54 That is, the iron rails were extended, say, '/".ooo of their length per ton of tensile stress per square inch of section ; and the steel rails were extended, say, Vis.ooo of their length per ton per square inch. It has already been deduced from direct experiments on the elongation of bars (see pages 623, 624), that the extension of iron was from I / IO>0 oo to I / I3>00 o part of the length, and that of steel was I / I3>00 o part of the length, per ton per square inch. Thus, the deductions from experiment on transverse resistance, are strongly corroborated by the results of experiment on direct tensile resistance. Substituting these values of E, in round numbers, in formula ( 5 ), the following formulas for the deflection of iron and steel rails, like those tested by Mr. Price Williams, are derived : Deflection of Double-headed Rails, within the Elastic Limit, Loaded at the Middle. IRON... . D = - _ __ . ( 7 ) 44,000 (4 a d 2 + 1.155 STEEL .......... D = D = the deflection at the middle, in inches. W=the load at the middle, in tons. tf' = the net sectional area of one flange, in inches (excluding the central portion pertaining to the web). d= the total depth of the rail, in inches. dT = the vertical distance apart of the centres of the flanges, in inches. /' = the thickness of the web, in inches. /=the length of span between the supports, in inches. Flange-Rails. By a similar process, equating the values of s, given by formula ( 4 ), page 665, and by formula ( 2 ), page 528, formulas are deduc- ible for the deflection of flange-rails within elastic limits : 670 STRENGTH OF ELEMENTARY CONSTRUCTIONS. W// ^ = 4 ^ ED ; and W/3 ^, = 6.92x4x^^3 ED A; 6.92 a 3 A / 2 whence the following values of D and E : To find the value of E by the formula ( 10 ), for Mr. Fowler's steel rail, investigated at page 666, for which the value of the quantity A is 6.983: 1 1.964 tons x 60 inches 3 x 1.99 inches : 4.5 inches x 3.374 inches x .2414 inch: : . . . = 7264. 27.68 x 4.5 inches x 3.374 inches x .2414 inchx 6.983 That is to say, the flange of Mr. Fowler's steel rail is extended I / 72 64 part of its length per ton of tensile stress per square inch. This fraction is considerably greater than the fraction that was found for the double-headed steel rails tested by Mr. Price Williams, averaging I / I3>000 part. The greater extensibility, nearly twice as much, is in accordance with the relative tensile strengths of the steels of which the different rails were made 35 tons per inch for the flange-rail, and 45 tons for the double-headed rails. Substituting in formula ( 9 ), the value of E, just found, the formula is reduced to the following form for the deflection of steel flange-rails like Mr. Fowler's: Deflection of Steel Flange- Rails, within the Elastic Limit, Loaded at the Middle. D= i / \ 200,000 dd 3 A To find, in the absence of data, the probable numerical constant for the deflection of iron flange-rails, let the constant in this formula be reduced in the ratio of 52,000 to 44,000, the correlative constants for steel and iron, in formulas ( 7 ) and ( 8 ); or to 2oo,ooox = 170,000: Deflection of Iron Flange-Rails, within the Elastic Limit, Loaded at the Middle. D- ..................... 170,000 aa 3 A D = the deflection at the middle, in inches. W = the load at the middle, in tons. ^ x = the height of the neutral axis of the reduced section, above the base of the section, in inches. d= the total height of the rail, in inches. E = the elasticity, or deflection, in sixteenths of an inch per ton of load. S = the working strength, or load, in tons. /=the span when loaded, in inches. b = the breadth of plates in inches, supposed uniform. / = the thickness of plates in sixteenths of an inch. n - the number of plates. RULES FOR THE ELASTICITY OF LAMINATED SPRINGS. RULE i. To find the elasticity of a laminated spring. Multiply the breadth in inches by the cube of the thickness of each plate in sixteenths of an inch, and by the number of plates ; multiply the cube of the span in inches by 1.66. Divide the second product by the first. The quotient is the elasticity in sixteenths of an inch per ton of load. RULE 2. To find the span due to a given elasticity, and number and size of plates. Multiply the elasticity by the breadth in inches, and by the cube of the thickness in sixteenths, and by the number of plates; and divide by 1.66. Find the cube root of the quotient. The result is the span in inches. RULE 3. To find the number of plates due to a given elasticity, span, and size of plate. Multiply the cube of the span in inches by 1.66. Multiply the elasticity by the breadth of plate in inches, and by the cube of the thickness in inches. Divide the first product by the second. The quotient is the number of plates. Note i. The span and the elasticity are those due to the spj/Jng when weighted. Note 2. When extra thick back and short plates are used, they must be replaced by an equivalent number of plates of the ruling thickness, prior to the application of Rules i and 2. This is found by multiplying the number of extra-thick plates by the cube of their thickness, and dividing by the cube of the ruling thickness. Conversely, the number of plates of the ruling thickness given by Rule 3, required to be removed and replaced by a given number of extra-thick plates, are found by the same calculation. Note 3. It is assumed that the plates are similarly and regularly formed, and that they are of uniform width, and but slightly tapered at the ends. RULES FOR THE WORKING STRENGTH OF SPRINGS. RULE 4. To find the working strength of a laminated spring. Multiply the breadth of plates in inches by the square of the thickness in sixteenths, and 1 Railway Machinery, 1855, page 242. Also, Railway Locomotives, 1860. 672 STRENGTH OF ELEMENTARY CONSTRUCTIONS. by the number of plates; multiply the working span in inches by 11.3. Divide the first product by the second. The quotient is the working strength in tons of load. RULE 5. To find the working span due to a given working strength, and number and size of plates. Multiply the breadth of plate in inches by the square of the thickness in sixteenths, and by the number of plates ; multi- ply the working strength in tons by 11.3. Divide the first product by the second. The quotient is the working span in inches. RULE 6. To find the number of plates due to a given working strength, span, and size of plate. Multiply the working strength in tons by the span in inches, and by 11.3; multiply the breadth of plate in inches by the square of the thickness in sixteenths. Divide the first product by the second. The quotient is the number of plates. RULE 7. To find the required original compass of the spring. Multiply the elasticity in sixteenths per ton by the working strength in tons, and add the product to the desired working compass. The sum is the whole original compass, to which an allowance of from ^ to 3/ 8 inch should be added, for the permanent setting of the spring. Note i. The span is that due to the form of the spring when weighted. Note 2. Extra thick back or short plates must be replaced by an equivalent number of plates of the ruling thickness, before applying the Rules 4 and 5. To find this, multiply the number of extra-thick plates by the square of their thickness, and divide by the square of the ruling thick- ness. Conversely, the number of plates of the ruling thickness given by Rule 6, required to be removed and replaced by a given number of extra- thick plates, are found by the same calculation. Helical Springs. Most of the data on the strength of helical springs are indefinite and contradictory. It may be assumed that the elasticity, or deflection per unit of load, is as the fourth power of the diameter or of the side of the bar, if round or square, of which the spring is constructed ; as the cube of the mean diameter of the coil or helix, as the number of free coils of the springs, and as the load applied. In the " Report on Safety Valves," 1 the following formula is propounded : E = Compression or Extension of one coil, in inches. d= diameter from centre to centre of steel bar composing the spring, in inches. w - the weight applied, in pounds. D = the diameter, or the side of square, of the steel bar of which the spring is made, in i6ths of an inch. C = a constant which, from experiments made, may be taken as 2 2 for round steel, and 30 for square steel. The deflection for one coil is to be multiplied by the number of free coils, to obtain the total deflection for a given spring. 1 Transactions of the Institution of Engineers and Ship-builders in Scotland, 1874-75, page 39. ROPES. 6/3 It is also stated in the Report that the relation between the safe load, size of steel, and diameter of coil, has been deduced from the works of Professor Rankine; and that it may be taken for practical purposes as follows : D= 4.29 , for round steel, ................. (2) ROPES HEMP AND WIRE. HEMPEN ROPES. By the old ropemakers' rule the breaking strength in hundredweights was equal to four times the square of the girth of the rope in inches. This is equivalent to Gregory's rule, by which the breaking strength in tons was equal to one-fifth of the square of the girth in inches. The square of half the girth represented the weight in pounds per fathom. The following table of the strength of cordage, is reduced from Mr. Glynn's 1 data. The ropes recorded in the second part of the table are machine-made ropes. Made by the warm register, the rope is stronger and more durable than by the cold register, as it is more thoroughly penetrated by the tar. But it is less pliable, and cold-register rope is now generally used for cranes, and block and tackle. Table No. 235. BREAKING STRENGTH OF TARRED HEMP ROPES. (Reduced from Mr. Glynn's data.) Size of Rope. Made by the Old Method. Made by the Register. Girth. Diameter. Common Staple Hemp. Best Petersburg Hemp. Cold Register. Warm Register. inches. inches. tons. tons. tons. tons. 3 95 2.22 2.70 3-30 3-85 3X 4 .11 .27 3-33 3-92 3-87 4.67 5.00 5.85 1:1! 4X 43 4.60 5-55 7.29 8.68 5 59 5-95 7.08 9.15 10.71 $y* -75 6.90 8. 3 I 11.07 13.00 6 .91 8.10 9.65 10.94 14.80 6% 2.07 9.16 10.54 15.46 18.10 7 2.24 10.24 12.26 18.00 21.00 7/2 2-39 11.15 1373 20.60 24.10 8 2.54 12.00 14.30 23-43 2742 Specimens of white 2-inch rope, exhibited at Kew Gardens, bore the following breaking weights : 1 On the Construction of Cranes and Machinery, page 94. 43 6/4 STRENGTH OF ELEMENTARY CONSTRUCTIONS. 2-inch Neapolitan, 2.75 tons, breaking weight. Konigsberg, 1.97 French, 2.17 St. Petersburg, 2.17 Italian, 2.32 Specimens of rope supplied by the National Association of Rope and Twine Spinners, were tested by Mr. Kirkaldy. Rope. Circumfer- ence. Weight per Fathom. Ultimate Strength. Total. 3 erLb.-w'ght per Fathom. Russian rope, 48 t Machine yarn, 50 Hand-spun yarn, 51 breads inches. 5.26 5.07 5-39 pounds. 5-74 5-35 6.04 tons. 4-95 5.14 8.16 tons. .863 .961 1.350 Extension in 50 inches Length. Stress per Pound-weight per Fathom. 500 Ibs. 1000 Ibs. 1500 Ibs. 2000 Ibs. 2500 Ibs. 3000 Ibs. Russian rope, inches. 3.38 3-25 3.26 inches. 5.29 4-53 4.46 inches. 6.59 5.56 5.29 inches. 6.56 5. 9 I inches. 6^5 inches. 6.63 Machine yarn Hand-spun yarn, The bearing capacity of a hemp rope is proportional to its thickness, the number of its strands, the slackness with which they are twisted, and the quality of the hemp. Karl von Ott states that ropes 0.866 inch in diameter, whose threads had been shortened by twisting l / 5 tn > X tn > and ^d of their original length, broke respectively with loads of 6834 Ibs., 5335 Ibs., and 4519 Ibs. He adds that the ultimate strength of ropes, according as they are wetted, or tarred, or dry, usually varies between 7000 Ibs. and 11,400 Ibs. per square inch, and that a working strength of one-sixth, say 1422 Ibs., or 0.63 ton per square inch may be adopted. WIRE ROPES. The strength of wire ropes of iron and steel, manufactured by Messrs. R. S. Newall & Co., is given in table No. 236, together with that of hemp rope, for comparison. From the table, the following data are derived : 1. The Breaking Strength is About i ton per Ib. weight per fathom for round hemp rope. 2 iron 3 to 3^ steel 2. The Working Load is 3 cwts. per Ib. weight per fathom for round hemp rope. 6 iron 10 steel 3. The Working Load in Cwts. is s/eths of the square of the circumference in inches, for round hemp rope. About 5 times iron 9 times steel ROPES. 6 75 Table No. 236. STRENGTH OF ROPES HEMP, IRON, STEEL. (Messrs. R. S. Newall Co.) i. ROUND ROPES for Inclined Planes, Mines, Collieries, Ships' Standing Rigging, &c. HEMP. IRON. STEEL. TENSILE STRENGTH. Circum- ference. Weight per Fathom. Circum- ference. Weight per Fathom. Circum- ference. Weight per Fathom. Working Load. Ultimate Strength. inches. pounds. inches. pounds. inches. pounds. cwts. tons. .. 2V .. 2 ... I ...I 6 2 . I I 9 3 2 \/ 4 ... T fyh. ... 2 ... 12 ... ... 4 ... I* i l A i l A 15 5 4 l /2 C . -, 18 ... ... 6 ... 2 3% 1/8 2 21 7 j- T/ 7 2 ^At A 2 */ 2/1 8 2% 27 9 ... 6 ... Q 2^/9. ,. C J i 7 A . ^ ,. ^O .. ... IO ... 2^ 33 ii .. 6y 2 .. ... IO ... ... 6 . . . 2 - I/ ^6 ... 12 . 2X 4 39 13 ... 7 ... 12 ... ... 27A .. ... 7 ... 2X . A y^7 ... 42 ... 3 7 1 A 1 45 15 7 5^ . 14 . ... 8 ... 23/( ..5 - & ... 16 . . 3% 5i 17 ... 8 ... ... 16 ... ... ^ft ... ... 9 ... ... 2#... ... 5X ... 54 ... ... 18 ... 3/^2 IO 2/8 6 60 20 8 y^ ... 18 ... .3 C/C ...ii 6// 66 ... 22 ... 3X 12 4 72 24 ... 9> ... ... 22 ... ... 3JA ... ...13 ... ... 3X - ... 8 ... ... 78 ... ... 26 ... IO ... 26 4 14 84 28 ... 4X ... ...15 ... ... 3ft ... ... 9 ... ... 90 ... ... 3 ... II ... 3 4ft 16 96 32 ... ... 4^ . ...18 ... ... 3^ ... ...10 ...108 ... ... 3 6 ... 12 34 4/s 20 3% 12 120 40 2. FLAT ROPES. For Pits, Hoists, &c. &c. 4 x i J-^ ... 20 ... 2/ X l /z - - T T AA 2O 5 x iX 24 2/2 X 13 23 5> x i^ ... 26 ... 23/ -^ yf ... I I . 60 .. 27 .. 28 3 x 16 2 x y 2 10 6 4 28 6 x i^ ... 30 ... 3/4 x ... 18 ... 2% X ^ ... II ... ... 72... ... 32 ... 7 x i^ 36 3^ x 20 x 12 80 36 8X x 2*^ ... 40 ... ... 22 ... 2>^ x X ... 13 ... ... 88... ... 40 ... 8/"2 x 2X 1 45 4 x 2 5 2 X x H 15 100 45 9 x 2^ ... 50 ... 4X x */ ... 28 ... 3 x ... 16 ... ... 112 ... ... 50 ... 9^ x 2 ft 55 4/2 x 32 3/4 x ,, 18 128 56 10 X 2>^ ... 60 ... 4^ x ... 34 ... 3/2 x ,, ... 20 ... ... 60 ... 6;6 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Table No. 237. STRENGTH OF CABLE FENCING STRANDS AND SOLID FENCING WIRE. (Reduced from Messrs. Francis Morton & Co.'s "Standard Quality" Table.) Size of Fencing Strand. Solid Wire of Equal Diameter. Length of One Ton. Ultimate Strength per Square Inch. Strand. Wire. Strand. Wire (annealed). 4fc No.*/, No. O inch. .326 yards. 3,000 yards. 2,700 tons. 2.419 tons. 2.000 * No. 5/0 I j .300 3,800 3,200 1.828 1.683 4t N - 4 / 2 .274 5,600 3,850 1.562 1.402 fNo. 3/o 3 .250 6,000 4,650 1.340 1.169 No. oo 4 .229 6,200 5,500 1.160 .988 wjjxh No. o 5 .209 7,800 6,600 .893 .817 % No. i 6 .191 9,800 7,900 .714 .682 H No. 2 7 .174 1 1 ,000 9,550 .627 .566 Jb Na 2A 8 .159 15,000 11,400 49 1 473 Note. The number, size, and strength of the Iron Wire quoted in this Table are the same as those of Ryland's Warrington Wires, table No. 82, page 247. Table No. 238. TENSILE STRENGTH OF AMERICAN IRON WIRE ROPE AND HEMP ROPE. (Mr. ]. A. Roebling.) Trade Number. Circumfer- ence of Wire Rope. Circumfer- ence of Hemp Rope of equal Strength. Ultimate Strength. Trade Number. Circumfer- ence of Wire Rope. Circumfer- ence of Hemp Rope of equal Strength. Ultimate Strength. No. inches. inches. tons. (English.) No. inches. inches. tons. (English.) FINE WIRE. H 3.26 8X 18.2 I 6.62 i5# 67.3 15 2. 9 8 7X 14.5 2 6.20 I4# 59 16 2.68 6X "-3 3 5-44 13 49 17 2.40 S/2 8 4 4.90 12 39-6 18 2.12 5 x 6.9 5 4.50 I0# 32 9 9 4^ 5-3 6 3-91 9X 24.7 20 63 4 3-72 7 3.36 8 18.4 21 53 3-3 2.57 8 2.98 7 14.5 22 3i 2.8 i-93 9 2.56 6 10.4 23 23 2.46 i-5 10 2.45 5 7.8 24 .11 2.2 1.16 COARSE WIRE. 25 94 2.04 94 II 4-45 I0# 33 26 .88 i-75 74 12 4.00 10 27-3 27 .78 1.50 5i 13 3-63 9/2 22.7 CHAINS. 6/7 French Wire Rope. For mining purposes, each strand consists of a core of hemp and 12 wires; and the rope has 5 or 6 strands on a central hemp core. Flat ropes are formed by laying 3 or 4 ropes side by side, and binding or lacing them with annealed wire; but flat ropes are seldom employed. Table No. 239. FRENCH IRON WIRE ROPES FOR MINING SERVICE. (Manufactured by MM. Harmegnies, Dumont, & Co., Anzin.) Working depth, 400 metres, or 440 yards. FLAT ROPES. ROUND ROPES. Number of Strands. Width. Thickness. Weight Yard. Working Load. Number. Diameter. Weight Y?rd. Working Load. strands. inches. inch. Ibs. tons. No. inches. Ibs. tons. 8 5-1 .87 16 5 10 1.30 6. 5 3 8 4-7 79 13 4-5 II 1. 10 5 2-5 6 3-9 83 12 4 12 . 9 8 3-8 2 8 4-3 .67 II 3-5 13 83 3 i-5 6 3-5 79 10 3 H 71 2.6 i 6 3-2 .67 9 2-5 15 .63 2 75 6 3-2 63 8 2 16 59 i-5 5 6 2.8 59 7 1.8 17 Si i .25 9 2.4 55 6 i-5 Note to Table. i. Steel wire ropes may be a third less in weight than iron wire rope for the working load. 2. Hemp ropes should be a third heavier than iron wire rope for the same working load. Steel Wire Ropes. Ropes consisting of 26 steel wires, No. 14 W. G., or .085 inch in diameter, are made for steam ploughing purposes. The weight of the rope is about 2 Ibs. per yard, less than i ton per 1000 yards. Each wire, it is said, bears a tensile stress of from 2000 Ibs. to i ton; and, at this rate, the rope should have a tensile resistance equal to 24 or 26 tons. CHAINS. Chains are constructed either with open links, Figs. 274 and 275, or with stud-links, Figs. 276, 277, 278, and 279. The standard proportions of the links of chains, in terms of the diameter of the bar iron from which they are made, are as follows : Extreme Length. Extreme Width. Stud-link 6 diameters 3.6 diameters. Close-link 5 3.5 Open-link 6 3.5 Middle-link 5.5 3.5 End-links 6.5 4.1 End-links are the links which terminate each i5-fathom length of chain; they are made of thicker iron, generally 1.2 diameters of the common links. 6/8 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Ordinary Stud-link Chain-cable. The admiralty test for the tensile strength of ordinary stud-link chain-cables, is at the rate of 630 Ibs. per circular ^6 -inch section of one side of a link: equivalent to 22.92 tons per square inch of one side, or to 1 1.46 tons per square inch of both sides taken together, just within the elastic limit. No. 4. No. 6. Figs. 274-279. Links of Chain-Cables. No. i, Circular Link. No. 2, Oval Link. No. 3, Oval Stud-Link, with pointed stud. No. 4, Oval Stud-Link, with broad-headed stud. No. 5, Obtuse-angled Stud-Link. No. 6, Parallel-sided Stud-Link. The weight of a link in similar cables, increases as the cube of any lineal dimension, say the thickness; and the weight per yard increases as the square of the thickness of chain. Whence the formula Weight per yard of Common Stud-link Chain-cable. W = 26.9 */ 2 ; or, in round numbers, 27 d* ( i ) W = the weight per yard in pounds; d= the thickness of the chain, or the bar from which it is made, in inches. The weight of a bar of iron a yard long is 10 Ibs. per square inch of sec- tion, or 7.854 Ibs. per circular inch; that is, a i-inch round bar weighs 7.85 Ibs. per yard, whilst a stud-chain cable of i-inch iron, weighs 26.9 Ibs. per yard, or 3.42 times the weight of a i-inch bar. Generally, therefore, a stud chain-cable weighs 3.42 times as much as a bar of the same size and length. The table No. 240 contains the dimensions, weights, and strengths of ordinary stud-link chain-cables. Column 4 shows the weight of 100 fathoms of cable in 8 lengths; including 4 swivels and 8 joining shackles. The sixth column gives the ultimate strength by actual tests made at Woolwich, in 1842-43, averaging, as shown in the last column, 16 tons per square inch, or two-thirds of the strength of the original bar, assumed at 24 tons per square inch. The safe working-stress is 5.73 tons per square inch of both sides together, or half of the proof-stress. Open-link Chains, Figs. 274 and 275. The chain, Fig. 275, is sometimes called a close-link chain, to distinguish it from the circular-link chain, Fig. 274. The ultimate strength, generally, is the same as that of stud-link chains; but the elastic limit is less than that of the others, and the proof- stress for close-link chains is just two-thirds of that for stud-link chains, or LEATHER BELTING. 679 Table No. 240. ORDINARY STUD-LINK CHAIN-CABLE WEIGHT AND STRENGTH. Dimensions of Link. Weight of 100 Fathoms. Average Ultimate Strength. Admiralty Proof-stress adopted by Lloyds'. Ultimate Strength per square inch of Both Sides of Link. Diameter of each Side. Length of One Link. Width of One Link. Total. Per Fathom (6 Feet). inches. inches. inches. cwts. Ibs. tons. tons. per cent. tons. 7/ l6 2^j i-575 9.20 "3 3% # 3 1.8 12 13-4 4K 9 /i6 3^ 2.025 15.2 17.2 5/^ ft 3^ 2.25 18.75 21 9-58 7 73 I 5 .6 "/* 4/^ 2.475 22.7 25-4 %/4 4/^ 2.7 27 30.2 J3-5 1 io/^ 75 15-3 7 /% sX 3-i5 36.75 41.2 20.38 13* 67 16.9 I 6 3-6 48 53-8 24-25 18 74 15-4 \y% 6^ 4-05 60.75 29.54 22^ 77 14.9 ify 7^ 4-5 75 8 4 28>^ i^ 8/^ 4-95 90.75 101.6 34 !/ 9 5-4 108 121 59-58 40^ 68 16.9 l $i 5.85 126.75 142 47^ tl ioK 6-3 6-75 i47 168.75 164.6 I8 9 74.12 92.88 55^ 6 3X H lil 2 12 7.2 192 215 99-54 72 1 72 i 5 .8 2j^ 12^ 216.75 242.8 2X 1-3^2 8. i 243 276.2 9 1 /^ 14* 8.55 270.75 303.2 lOl l /2 2^2 15 9.0 300 336 \\2. l /2 2$ \&y z 9.9 363 406.6 136^ Average s . . 72 15-9 i. The Safe Working-stress is taken at half the Proof-stress. 2. The Proof-stress and Safe Working-stress for close-link chains are respectively two- thirds of those of stud-link chains. 7.64 tons per square inch of section of both sides, or 410 Ibs. per circular j^-inch of section of one side. The safe working-stress is half the proof- stress, or 3.82 tons per square inch of section. The weight of close-link chain is about three times the weight of the bar from which it is made, for equal lengths. Karl von Ott, comparing the weight, cost, and strength of the three materials, hemp, iron wire, and chain iron, concludes that the proportion between the cost of hemp rope, wire rope, and chain, is as 2:1:3; and that, therefore, for equal resistances, wire rope is only of half the cost of hemp rope, and a third of the cost of chains. LEATHER BELTING. According to the experiments of Messrs. Briggs and Towne, the tensile strength of single leather belts, .219 inch thick, was, Per Per square inch wide. inch of Section. Through the lace-holes, 210 Ibs 960 Ibs. Through the rivet-holes, 382 1740 Through the solid parts, 675 3080 68o STRENGTH OF ELEMENTARY CONSTRUCTIONS. Messrs. Norris & Co.'s beltings, as tested by Mr. Kirkaldy, gave the following results for ultimate tensile strength : Table No. 241. TENSILE STRENGTH OF LEATHER BELTING. SIZE. English Belting. Helvetia Belting. DOUBLE. i ^ inches Ibs. I4,86l Ibs. 17 622 7 6,IQ^ 1 1, 080 6 5,60^ IO,4.?6 4., 36 5 6 2O7 2 2,942 4,237 SINGLE. jo 8846 ii 888 4.,O6O 5.426 A ^,24.8 2,04.8 i l A -J QO7 3 -277 Spill's machinery belting is manufactured from flax-yarn, saturated with a compound substance said to be incapable of decomposition. According to the annexed results of tests it is stronger than leather belts : * No. i, No. 2, 5 No. 3, 10 Leather belt, 4 Tensile Strength, per inch Wide. 5 inches wide, I2 54 Ibs. 1489 525 Untanned leather belts are said to be half as strong again as tanned leather belts. Mr. John Mason, of Bulkley, Barbadoes, uses belts cut from raw cowhide, simply dried in the sun. They last longer, he says, than leather belts, and are made at a fourth of the cost of the latter. 2 India-rubber belts, made of American cotton canvas, cemented in layers by vulcanized india-rubber, and covered by a compound of rubber, have been proved to possess considerably greater frictional adhesion than leather belts. STRENGTH OF BOLTS AND NUTS. Mr. Brunei's Experiments? Mr. Brunei tested the tensile strength of screwed bolts and nuts of Shropshire iron, from ^ inch to ij^ inches in 8 Fig. 280. Screw Bolt and Nut. diameter, applying the stress between the head and the nut, when 16 inches 1 Exhibited Machinery of 1862, page 423. 2 Engineering, June 19, 1874. 3 The particulars of these experiments are derived from the Author's work on Railway Locomotives, 1860. BOLTS AND NUTS. 68 1 apart, and placed as in Fig. 280. The length of the screwed part was 3^ inches. In most instances, the bolt snapped at the base of the screwed part. Diameter Total Breaking Weight. Breaking Weight per sq. inch. inches. tons. tons. 10 .................. 3 2 I2 ................. 27 15% .................. 25 20 .................. 25 21 .................. 21 2 9 ........ .......... 2 3 To find to what extent the screwing of a bolt diminishes its tensile strength, Mr. Brunei tested four i^-inch bolts and nuts to the annexed form, Fig. 281, on which the screwed part was enlarged to i^ inches in diameter. The bolts were broken in the shank, and the average breaking weight was equal to 25.2 tons per square inch, showing an addition of 2.2 tons per inch, as compared with the screwed shank, Fig. 280. Inversely, it may be inferred that the strength of i^-inch bolts was reduced 2.2 tons, or 8 per cent, by screwing. Fig. aSi.- Screwed Enlarged Bolt and Nut. The heads of the i ^-inch bolts were i % inches thick, and they stood fast during all the trials. The depth of the nuts of these bolts varied from i % inch to 24 mcn - Nuts i inch deep, or 8 / I0 ths of the diameter, stood well. Do. y% or 7/ I0 ths thread strained. Do. 24 or 6 /ioths thread stripped. The thread, it appears, was stripped when the depth of the nut was only 3/ 5 ths of the diameter. Nevertheless, in ordinary good practice, a depth of half the diameter has been found sufficient for both the head and the nut. But it may well be better to make them deeper, to allow for contingencies. Working Stress for Screwed Bolts. A working stress of i ^ tons per square inch has been assigned for screwed bolts. In France, it has been taken as high as 324 tons P er square inch. Whitworttts System of Standard Sizes of Bolts and Nuts. The thickness of the bolt head is ^6 ths of the diameter, and that of the nut is equal to the diameter. The angle of the triangular thread is, in this system, 55. The top and the bottom of the thread are rounded off, and the reduction so made of the exact height of the triangle is one-third ; that is, one-sixth from the top, and one-sixth from the bottom. The actual height of the thread becomes rather more than 3/ s ths, and less than 2 / 3 ds, or about 63 per cent., of the pitch. See table No. 242, next page. For screws with square threads, the number of threads per inch is one- half of the number for triangular threads. 2/8 3X 4 4X 2.509 ... 2.634 ' 3/2 3X 3X- 3 3 BOLTS AND NUTS. 683 The American standard pitches are nearly identical with the Whitworth standards. American Standard Sizes of Bolts and Nuts. (United American Railway Master Car-builders' Association, in Convention at Richmond, I Va., June 15, 1871.) ROUGH BOLTS. The breadth across the flats of the bolt-head and the nut = i YZ diameters + Y% inch. The thickness of the head = ^ diameter 4- J /i6 inch. The thickness of the nut = i diameter. FINISHED BOLTS. The breadth across the flats of the bolt-head and the nut = i y 2 diameters + x /i6 inch. The thickness of the head and of the nut = i diameter - X / X 6 inch. Diameter. Number of Threads per Inch. Diameter. Number of Threads per Inch. Diameter. Number of Threads per Inch. inches. threads. inches. threads. inches. threads. X 20 IM 6 3^ 3 5/i6 18 1/2 6 4 3 % 16 Itt S/2 4X 27/ & 7/i6 14 Iff s 4X V/4 /2 13 tg s 4X 2% 9/i 6 12 2 4/2 5 ". 2/2 H II 2X 4/2 5X Z/2 ti 10 2/2 4 S/2 2*/S % Q 2^ 4 Sti 2 3 /S i 8 3 3/2 6 2X !# 7 3X 3/2 W 7 3/2 3/4 Table No. 243. WHITWORTH'S STANDARD PITCHES FOR SCREWED IRON PIPING. Diameter of Piping. Number of Threads per inch. Diameter of Piping. Number of Threads per inch. Diameter of Piping. Number of Threads per inch. inches. threads. inches. threads. inches. threads. % 28 H 14 1/2 II X 19 X H 1% II 3 A 19 i II 2 II /2 H iX II above 2 8 M. Armengaud gives a table of the dimensions of bolts and nuts, based on the average practice in France. It is here translated into English measures, for threads of triangular and of square section. The thickness of the nut for triangular threads is equal to the diameter of the bolt, as in Whitworth's system. The depth of the square thread is nearly equal to half the pitch, or to the thickness of the thread. 684 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Table No. 244. ARMENGAUD'S FRENCH STANDARD BOLTS AND NUTS. With Hexagonal Heads and Nuts. i. TRIANGULAR THREAD (Equilateral Triangle). SCREW. HEAD AND NUT. Diameter of Bolt and Screw. Diameter at Bottom of Thread. Number of Threads per inch. Thickness of Head. Thickness of Nut. Breadth across the Flats. Working Tensile Stress. millimetres. inches. inches. threads. inches. inches. inches. Ibs. 5 .20 13 18.1 .24 .20 55 44 7.5 30 .22 16 30 30 .68 99 10 39 31 14.1 38 39 .88 178 12.5 .49 39 12.7 44 49 1.04 277 15 59 .48 11.5 52 59 1.20 400 17-5 .69 .58 10.6 .58 .69 1.40 545 20 79 .66 9.8 .66 79 1.50 7i3 22.5 .89 .76 9.1 72 .89 1.68 902 tons. 25 .98 .84 8.5 .80 .98 1.84 .50 30 1.18 1.02 7-5 94 i.iS 2.16 73 35 1.38 1.20 6-7 i. 08 1.38 2.48 99 40 1.58 1.40 6.0 1.22 1.58 2.80 1.30 45 1.77 1.56 5-5 1.36 1.77 3-20 1.64 50 1.97 1.74 5-i 1.50 1.97 3-44 2.03 55 2.17 1.92 4-7 1.64 2.17 3-76 2-45 60 2.36 2.08 44 1-74 2.36 4.08 2.92 65 2.56 2.26 4.1 1.92 2.56 4.40 3-42 70 2.76 2.44 3-8 2.06 2.76 4.70 3-97 75 2-95 2.60 3-5 2.20 2.95 5.00 4.56 80 3-15 2.78 3-4 2.34 3-15 5-35 5.12 2. SQUARE THREAD. Depth of Thread. LUllb. 20 79 .072 6.57 1.82 32 25 .98 .O8l 5.97 2.01 5 1 30 1.18 093 5.40 2.22 73 35 1.38 .10 4-93 2.41 99 40 i-57 .106 4-53 2.63 1.30 45 1-77 .114 4.20 2.85 1.64 50 1.97 .128 3-91 3-07 2.03 55 2.17 13 3.65 3-30 2-45 60 2.36 .14 3-43 3-50 2.92 65 2.56 15 3-23 3-70 342 70 2.76 .158 3-o6 3-92 3-97 75 2-95 .166 2.92 4.13 4.56 80 3-15 .174 2.76 4.36 5.18 , 8 5 3-35 .183 2.63 4.58 5.85 90 3-54 .192 2.51 478 6.56 95 3-74 .200 2.41 5.00 7oO 100 3-94 .209 2.31 5.22 8.10 105 4-i3 .220 2.22 5-43 8-93 1 10 4-33 .226 2.13 5.66 9.80 115 4-53 .230 2.06 5.87 10.71 120 4.72 2.24 2.00 6.08 11.66 SCREWED STAY-BOLTS AND STAYED SURFACES. 685 SCREWED STAY-BOLTS AND STAYED SURFACES. Screwed Stay-Bolts. Sir William Fairbairn tested the strength of 24 -inch stay-bolts, with enlarged ends, screwed into ^5-inch plates of copper and Figs. 282, 283. Flat Stayed Plates. of iron, some of them being rivetted or headed in addition, as in the Figs. 282 and 283. BOLTS. PLATES. Breaking Weight. 1 . Copper into copper, screwed and rivetted, 7.2 tons. 2. Iron into copper, do. do 10.7 3. Iron into copper, screwed only 8.1 4. Iron into iron, screwed and rivetted 12.5 Notes. ist Test. The bolt broke through the shank. 2d Test. The rivet-head was broken off, and the bolt was drawn out of the plate, strip- ping the thread. $d Test. The bolt stripped the thread of the plate. 4th Test. The bolt broke through the shank; screw and plate uninjured. Flat Stayed Plates. Sir Wil- liam Fairbairn tested two flat boxes, Fig. 284, 22 inches square, having top and bottom plates of ^-inch copper and 3/fr- inch iron respectively, inclosing a 2*4 -inch water-space; stayed with ^/jg-inch iron stays, having enlarged ends screwed and rivetted into the plates, to re- present the Conditions Of the Fig. 284. -Flat Stayed Plates. firebox of a locomotive. The stays were placed at intervals of 5 inches p 11 ll EJ O( 81 Oi 81 01 Oi 01 0! 0! Oi <3:l OC 30OOOOOOOOOC )0 o o o o o o o o o o o o o o )OOOOOOOOOOC 686 STRENGTH OF ELEMENTARY CONSTRUCTIONS. in the first box, and 4 inches in the second. Under the pressure of water, the sides of the first box commenced to bulge or swell between the stays at a pressure of 455 Ibs. per square inch, and the box was burst at 815 Ibs., by the drawing of the head of a stay bolt through the copper plate, as shown. In the second box, the bulging commenced at a pressure of 5 1 5 Ibs. per square inch; it amounted to Ji inch at 1600 Ibs; and at 1625 Ibs. one of the stays was drawn through the iron plate, stripping the thread. At the 5-inch intervals, rupture took place when the stress on each stay- bolt was 9 tons; at the 4-inch intervals, the ultimate stress was n^ tons. Flat Plates of Marine Boilers. Two experimental flat boxes were tested at Plymouth Dockyard, by Mr. Phillips. 1 They were constructed respec- tively of 7/i6-inch and of ^-inch iron plates, stayed at intervals of 15^ inches by 15^ inches, with i^-inch screwed stay-bolts rivetted over at the ends, giving a superficies of 240 square inches for each bolt. Tested by hydrostatic pressure, the plates were bulged between the stay- bolts, and were finally pushed off, or drawn away from the bolts, under the following pressures : PLATES. Sectional Area of Stay-bolts. Bursting Pressure per Square Inch of Surface. Total Pressure for each Stay-bolt. In Body. At Thread. inch. 7/i6 y* square inch. I. 4 8 1.48 square inch. (say) 1.2 (say) 1.2 Ibs. 105 140 tons. 11.25 H73 When nuts were applied to the ends of the stay-bolts through the 7/ l6 -inch plate, they bore a pressure amounting to 165 Ibs. per square inch, on the plate; when the box gave way at a rivetted joint. Rules for Flat Stayed Surfaces. Mr. Wm. Bury 2 propounds the following rules for the staying of the flat surfaces of marine boilers: ist. The diameter of the screwed stays over the threads, should never exceed three times the thickness of the plates. 2d. The working steam-pressure allowed per square inch of section of the stay-bolts, at the threads, is 5000 Ibs. 3d. The formula for the working pressure in pounds per square inch, with the above-named proportions, is, 112 (thickness of plate in sixteenths inch) _ ,, area of stayed surface for each stay, in square inches = ' which appears to agree with safe practice. Mr. Bury gives the following data, by this rule : 1 See Engineering, September I, 1876, page 185. '''Engineering, September 15, 1876, page 236. HOLLOW CYLINDERS. 687 Table No. 245. PROPORTIONS OF FLAT STAYED SURFACES OF BOILERS. For a working pressure of $o Ibs. per square inch. Diameter of Sectional Area Area of Surface for Distance of Centres Stay-bolts. at Threads. each Stay-bolt. of Stay-bolts. inch. square inch. square inches. inches. i/i 0.8 133 1 1 *A. I X I.O 166 13 I5/ I6 I.I 183 \y% 1.2 200 14/4 11/2 i-5 250 15^ Mr. Bury recommends that nuts should be applied on the uptake ends of the bolts outside the plates, where they are above the water-line. He reckons on a bursting pressure six times the working pressure. HOLLOW CYLINDERS: TUBES, PIPES, BOILERS, &c. RESISTANCE TO INTERNAL OR BURSTING PRESSURE. TRANSVERSE RESISTANCE. The action of a centrifugal pressure within a cylinder is illustrated by Fig. 94, page 2 74. The resistance offered by the sides of the cylinder to internal pressure transversely, is not uniformly exerted throughout the thick- ness of the sides. On the contrary, the resistance varies, and is a maximum at the inner surface of the cylinder, and when the stress on the inner .sur- face does not exceed the limit of elastic resistance, the tensional stress diminishes uniformly through the thickness of the sides, and is a minimum at the outer surface. For cast-iron, in which the strain increases approximately in proportion to the stress, this simple ratio of decrease holds approximately up to the bursting strength, which is measured by the total resistance opposed to breakage when the internal surface is strained to the ultimate limit of its tensile strength. But in the stretching of wrought iron and steel, there is a break in the uniformity of the stretching, at the yielding point, as is shown very clearly by Fig. 222, page 624; for, beyond the yielding point, the extension proceeds in a greatly accelerated ratio with the stress. Take, for instance, the cast-iron cylinder of a hydraulic press, 10 inches in diameter internally and 20 inches externally, shown in cross section in Fig. 285. Divide the thickness of it into an indefinite number of concentric rings of equal thicknesses, a, b, c, d, e-, and suppose, only for the sake of argument, that the first or innermost ring is stretched by internal pressure to 1 1 inches in diameter inside. All the other rings will be stretched to larger diameters, in such proportions that, whilst the circumferential extension is the same for all the rings, the increase of diameter will be inversely as the original diameter of each ring, so that the outermost, or 20-inch ring, will be stretched only ^ inch in diameter, or half the diamet- 688 STRENGTH OF ELEMENTARY CONSTRUCTIONS. rical stretch of the innermost ring. The comparative stretches of the suc- cessive rings are shown by shadings on the right side of the figure; and the shadings at the same time show by their thicknesses the relative stresses on the successive rings. Fig. 285. Diagram to show the Stretching of Hollow Cylinders by Internal Pressure. 56 78 910 as e Fig. 286. Diagram to show the Hyperbolic Ratio of the Stress throughout the Thickness, by In- ternal Pressure. Since the stress is inversely as the radial distance from the centre, if its values be represented by ordinates to the radius treated as a base-line, cae, in the longitudinal section, Fig. 286, they will, if connected at the ends, form a hyperbolic curve a' e'; and the area comprised between the curve and the base-line, is a measure of the total resistance of the section. Let r = the inside radius c a, r' = the outside radius c e, s = the maximum tensile stress, in tons per square inch, <^=the inside diameter = 2 r, d' the outside diameter = 2 r', R = the ratio of the outside diameter to the inside diameter = = , d r p the internal pressure in tons per square inch. Then, the rectangular area c c' a' a is a measure of, or is equal to, rxs, for a length of i inch parallel to the axis ; and the area of resistance under the hyperbolic curve a a' e' e, is equal to, for both sides (r x s) x hyp log R x 2. The internal pressure to be resisted, for i inch of length, is equal to / d, the product of the inside diameter and the hydrostatic pressure per square inch; and it is equal to the resistance; that is, p d= (r x s x hyp log R x 2). Or, p d 2 r s x hyp log R = ds x hyp log R ; and / = j-xhyp log R ( i ) s = hyp log R hyp log R= (3) These formulas express the relations of the internal pressure, and the maximum tensile stress on the metal at the inner surface, within the limits of elastic strength. They are given as rules, below. HOLLOW CYLINDERS. 689 Bursting Strength. For cast-iron cylinders, the foregoing formulas may also be employed in calculations for the bursting strength, and the corre- sponding ultimate breaking strength. To calculate the bursting strength of wrought iron and of steel cylinders, let the base-line cae, Fig. 286, represent, as before, the inside radius ca, and the outside radius c e. Draw the verticals c c' and a a', to measure the ultimate tensile strength of the metal per square inch; conceive the verti- cals to be bisected at points which may be indicated as c" and a", and draw c" a" e" parallel to the base. The rectangle a a" e" e would represent the resistance of the section due to the elastic strength of the material, which is uniform throughout the thickness and is taken as half the ultimate tensile strength. Draw intermediate vertical ordinates through the radial intervals of the thickness, between a and c; and set off the lengths of the upper segments, above the middle level a" e" , to represent the values of the uni- formly varying tensions in excess of the elastic limit, forming a hyperbolic curve, say, a 1 e'. The resistance of the section thus treated, consists of two parts: the uniform resistance aa" e" e, equal to (r'-r)^y 2 s; and the varying resistance a" a' e e", equal to (rx ) x hyp log R. Twice the sum of these resistances is equal to the internal pressure per inch of length of the cylinder; whence, 1 (R + hyplogR-i) 2 /,\ (R + hyplogR-i)'" ) = +i ............................. (6) RULES FOR THE STRENGTH OF HOLLOW CYLINDERS, WITHIN THE LIMITS OF ELASTIC STRENGTH. RULE i. To find the Internal Pressure for a given maximum tensile stress on the material. Multiply the hyperbolic logarithm of the ratio of the external to the internal diameter, by the maximum tensile stress in tons per square inch of the metal. The product is the internal pressure in tons per square inch. RULE 2. To find the maximum Tensile Stress on the sides for a given inter- nal pressure. Divide the pressure in tons per square inch by the hyperbolic logarithm of the ratio of the external to the internal diameter. The quo- tient is the maximum tensile stress on the metal in tons per square inch. 1 The formula (4) is thus deduced : / d=( (r' - r) x x 2 ) + ( 2 ( r x L) x hyp log R); or, p d- ( d -^-x s) + (-- x s x hyp log R). Dividing both sides and, substituting R for , the formula for the pressure becomes, p = , (R + hypiogR-D (4) 44 6Q3 STRENGTH OF ELEMENTARY CONSTRUCTIONS. RULE 3. To find the Ratio of the Outside Diameter to the Inside Diameter, for a given maximum tensile stress on the sides, and a given internal pressure. Divide the pressure by the stress, both in tons per square inch. The quotient is the hyperbolic logarithm of the ratio of the diameters, for which the ratio may be found in a table of hyperbolic logarithms. RULES FOR THE BURSTING STRENGTH OF HOLLOW CYLINDERS. Cast Iron. The rules and formulas ( i ), ( 2 ), and ( 3 ), may be employed for calculat- ing the bursting strength, and the corresponding ultimate tensile strength, of cast-iron hollow cylinders. Wrought Iron and Steel. RULE 4. To find the Bursting Pressure for a given ultimate tensile strength of the material. To the ratio of the outside to the inside diameter, add the hyperbolic logarithm of this ratio, and from the sum deduct i. Multi- ply half the remainder by the ultimate tensile strength in tons per square inch. The product is the bursting pressure in tons per square inch. RULE 5. To find the Ultimate Tensile Strength of the material for a given bursting pressure. To the ratio of the outside to the inside diameter, add the hyperbolic logarithm of this ratio, and from the sum deduct i. Divide twice the bursting pressure in tons per square inch by the remainder just found. The quotient is the ultimate tensile strength in tons per square inch. RULE 6. To find the Ratio of the Outside to the Inside Diameter, for a given bursting pressure and ultimate tensile strength. Divide twice the bursting pressure by the tensile strength, and add i to the quotient. The sum is equal to the ratio plus the hyperbolic logarithm of the ratio. The value of the ratio is found by trial and error, in a table of hyperbolic logarithms. Notes to Rules i to 6. i. The hyperbolic logarithm of a number is equal to the product of its common logarithm by 2.3026. 2. The pressure and the tensile stress may be expressed in pounds or in hundredweights, instead of tons. ist Example. Let the inside diameter of the cast-iron cylinder of a hydraulic press be 10 inches, the outside diameter 30 inches, and the ultimate strength of the metal 7 tons per square inch; to find the bursting pressure. The ratio of the diameters is ( =) 3, of which the hyper- 10 bolic logarithm is 1.0986 (see table No. 2, page 61). By rule i, the burst- ing pressure is (1.0986 x 7 =) 7.69 tons per square inch. Average Stress on the Metal. As the total transverse resistance per inch of length of cylinder is equal to p d, which is the product of the inside diameter by the bursting pressure per square inch, the average stress on the metal is equal to rf . ; that is to say, it is equal, in tons per square inch, d-d to the product of the inside diameter by the bursting pressure in tons per square inch, divided by the difference of the inside and outside diameters. HOLLOW CYLINDERS. 691 In the foregoing example, the average stress, which bursts the cylinder, is equal to / I0 x 7- 9 = ^ 3.845 tons per square inch of section little more 30- 10 than half the direct tensile resistance of the metal. 2d Example. A steam-boiler, 7 feet in diameter inside, of 7/ l6 -inch wrought-iron plates, was burst at a longitudinal double-rivetted joint by a pressure of 310 Ibs. per square inch. The outside and inside diameters, d and d', were 84.875 inches and 84 inches respectively, the ratio of which is 1.0104. By formula ( 5 ) the ultimate tensile strength was _ 3'*' 62 62 =29,886 Ibs., i. 0104 + hyp log 1.0104 i i. 0104 + . 010345 i .020745 or 13.34 tons per square inch of the section of the solid plate. $d Example. A cast-iron pipe, 10 inches in diameter inside, is ^ inch in thickness. What is the bursting strength when the ultimate tensile strength of the material is equal to 7 tons per square inch? The ratio of the outside to the inside diameter is as 11.5 to 10, or as 1.15 to i; and, by formula ( i ), 7 x hyp log 1.15 = 7 x .1398 = .9786 ton, or 2192 Ibs. per square inch, is the bursting pressure. APPROXIMATE RULES FOR TRANSVERSE RESISTANCE TO BURSTING PRESSURE. When the diameter is very considerable, compared to the thickness, the transverse resistance to bursting pressure may be taken approximately as directly proportional to the thickness of the metal, and inversely propor- tional to the diameter. The total pressure on a i-inch length of section of both sides together, is equal to the product of the diameter by the pressure per square inch. Let d\h^ diameter, in inches; / = the thickness of metal at each side, in inches ; s = the ultimate tensile strength of the metal, in tons per square inch ; and p - the pressure in pounds per square inch. Then dp is the total pressure on a i-inch length of both sides together; 2 / is the sectional area of both sides; and 2/.$-x 2240 = dp, or, RULE 7. The bursting pressure in pounds per square inch of surface is equal to 4480 times the product of the thickness by the ultimate tensile strength per square inch, divided by the diameter. RULE 8. The thickness of metal required at each side is equal to the product of the diameter and the pressure per square inch, divided by the ultimate tensile strength in tons per square inch, and by 4480. RULE 9. The ultimate tensile strength in tons per square inch of section of metal is equal to the product of the diameter by the bursting pressure in pounds per square inch; divided by the thickness of metal and by 4480. Note. When the material is made of jointed plates, the tensile stioigth of the whole plate is to be multiplied by the coefficient of strength of the joint, to give the reduced strength to be employed as the value of s in the calculation. 692 STRENGTH OF ELEMENTARY CONSTRUCTIONS. LONGITUDINAL RESISTANCE TO BURSTING PRESSURE. When the ends of a cylinder are closed, and make one piece with the cylindrical portion, the total longitudinal resistance to internal pressure is b b d d Fig. 287. Diagram for the Resistance of a Flat-headed Cylinder to Bursting Pressure. directly proportional to the thickness of the metal, and to the diameter; whilst the total bursting force, acting on the ends, is proportional to the Table No. 246. EXPERIMENTAL RESISTANCE OF SOLID-DRAWN TUBES TO BURSTING PRESSURE AND COLLAPSING PRESSURE. (Deduced from Messrs. Russell's data. ) WROUGHT-!RON TUBES. External Diameter. Thickness. ' Internal Diameter. Bursting Pressure. Collapsing Pressure. Difference of Burst- ing and Collapsing Pressures. Per square inch of Surface. Per square inch of Section of Metal. Per square inch of Surface. Per square inch of Section of Metal. inches. 3 1 X 3 l /s i* 2^ 2X 2 I* Avera B.W.G. 10 10 II II II 1 1 12 12 ges, omi inch. 134 134 .120 .120 .120 .120 .109 .109 tting the inches. 2.982 2.857 2.760 2.510 2.260 2.OIO 1.782 1-532 last tube Ibs. 4800 4500 4500 5200 5000 5900 5900 5600 tons. 23.84 21.42 23.10 24.28 21.02 22.06 21-53 17-57 Ibs. 3300 3150 3500 3500 3600 4500 4900 4OOO tons. 17.86 16.40 19.53 17.89 16.74 18.82 20.07 H.33 tons. 5.98 5.02 3-57 6-39 4.28 3-24 1.46 3-24 22.40 18.20 I'A 2 iH 13 13 15 17 .095 .095 .072 .058 HOMOGEN 2.810 2.060 1.856 1.409 EOUS ME" 3600 7600 4000 4600 PAL TUBE 23-77 36.78 23.02 24.94 3150 22.20 4600 24.32 3500 21.70 4000 25.02 i# 13 .095 BESSEMER STEEL TUBES. 1.56 j) 7800 28.92 4600 i 18.91 2^ I* l/B # 5/i6 5/i6 5/i6 HYI r# i % X )RAULIC T proved to 1 11,000 6,OOO 4,000 12,000 'UBES. 14-73 4.29 2.23 2.15 HOLLOW CYLINDERS. 693 square of the diameter. It results that the longitudinal resistance per square inch to bursting force is inversely proportional to the diameter. Let the circle and the rectangle, Fig. 287, be a cross section and a longitudinal section of a cylindrical boiler; the area of the circle is a measure of the longitudinal pressure of the steam on the ends of the boiler. Set off the interval a a on the circular section, and the interval ccon. the longitudinal section; and draw the diameters ab,ab, and the parallels cd,cd. The areas of pressure to be resisted are respectively the two triangular spaces ab, and the rectangle cd; and, since the former have only half the surface of the latter, it follows that the longitudinal stress per square inch on the shell is only half the transverse stress, and that the amount of the longitudinal resistance is proportionally twice as much as that of the trans- verse resistance. On the same showing, a hollow sphere resists twice the pressure per square inch, that a tube of equal diameter and equal thickness can do. Wrought-iron Tubes. Messrs. J. Russell & Sons tested the resistance of solid-drawn wrought-iron tubes to bursting pressure, and to collapsing pres- sure, on the results of which table No. 246 is based. The bursting pressure of the wrought-iron tubes in tons per square inch of section of metal, appears to be practically constant; and it may be taken for practical purposes that the ultimate strength is measured by the tensile strength of the material. Resistance of a Lancashire Boiler to Bursting Pressure. A boiler 7 feet in diameter, made of 7/i6-hich plates, was tested by Mr. L. E. Fletcher, and bore a pressure of 310 Ibs. per square inch, when it failed at one of the longitudinal seams, which were double-rivetted. Applying rule 9, page 691, the ultimate tensile strength was equal to ( 8 4 mches x 310 Ibs. = ) 4375 x 440 13.29 tons per square inch of section of the entire plate. This instance formed the subject of the 2d example, page 638. In this instance, the tensile strength, as calculated by the exact rule 5, page 690, is 13.34 tons per square inch. This is .05 ton, or about 2 / 5 ths of i per cent, more than is given by the approximate rule. Take the correctly calculated strength, 13.34 tons, with the net section of plate between the rivets, which was two-thirds of the section of the con- tinuous plate. Then 13.34x^-20.01 tons per square inch, the tensile strength of the plate between the rivet-holes. Resistance of a Cylindrical Marine Boiler and a Superheater to Bursting Pressure}- A cylindrical boiler, 1 1 feet 3 inches in diameter inside, of ^-inch plates, double-rivetted, was burst by a hydraulic pressure of 230 Ibs. per square inch, equivalent, by rule 9, to J 35 x 2 3 s. _ 2 ^ tQns p er S q uare j nc jj 75 *448o of section of the solid plate. The rivet-holes were i x /i6 inch in diameter, at 2^4 inches pitch, leaving 61.36 per cent, of solid metal between; and the ultimate tensile strength of metal left between the rivet-holes was 9.241 x 100 . , -? -= 15.06 tons per square inch. * \) A superheater, 99.915 inches in diameter inside, of 9/ l6 -inch plates, 1 The data are derived from Engineering, July 21, 1876, page 47. 694 STRENGTH OF ELEMENTARY CONSTRUCTIONS. double-rivetted, with ^/ l6 -mch rivet-holes at 2^ inches pitch, was burst at the same time by a hydraulic pressure of 245 Ibs. per square inch, equiva- lent to 99-9*5 x 2 ^5 = 9.655 tons per square inch of the solid plate. The . .562" x 4480 metal left between the rivet-holes was 67.6 per cent, of the entire section, and the resistance of that metal was 9.655 x - - = 14.28 tons per square inch of its net section. '' Now the tensile strength of the plates of the boiler and the superheater, tested by Mr. Kirkaldy, averaged 20.5 tons per square inch for the boiler, and 20.2 tons for the superheater. These instances have already been noticed at page 638, in the discussion of rivet-joints, and they forcibly demonstrate the essential weakness of rivetted lap-joints in very thick plates. The net tensile resistance of the plates between the holes was reduced a fourth. Cast-Iron Pipe. For a lo-inch pipe, ^ inch thick, having an ultimate tensile strength of 7 tons per square inch, the bursting pressure is, by formula ( 7 ), page 691, 4.4.80 x .7? x 7 ,, , '-2 ' = 2352 Ibs. per square inch. 10 By a previous calculation, page 691, with the exact formula ( i ), the burst- ing pressure was found to be 2192 Ibs. per square inch, showing that the ordinary approximate formula ( 7 ) gives 160 Ibs., or 7^ per cent., more than the correct formula. RESISTANCE OF HOLLOW CYLINDERS TO EXTERNAL COLLAPSING PRESSURE. Solid-drawn Tubes, By the action of a centripetal force on the outside of a hollow cylinder, compressive stress is produced, tending to collapse the cylinder. In table No. 246, the resistance of wrought-iron tubes to col- lapse is given. It is less than the resistance to bursting, and the difference between the bursting and the collapsing pressures increases with the dia- meter, as shown in the last column. When plotted and arranged into a curve, or, as in this case, a straight line, the value of the difference, in terms of the diameter, is 2^3 (d- i), which probably holds for diameters up to 6 inches. When the diameter d is only one inch, d- i =o, and the difference vanishes. The average bursting pressure being 22.40 tons per square inch of section of metal, the collapsing pressure is 22.40-2^ ( Fig. 288. Fig. 289. Fig. 290. Illustrations of Stress in Framed Work. ji . j, '!, OU CL CL- Fig 291. tutes the fundamental feature of framed work, as distinguished from solid work or web-work. When a load W is applied direct to a vertical pillar ab, Fig. 288, the resistance is in the line of the stress, and no framework is employed. But, if the load be applied at c, Fig. 289, hori- zontally apart from ~> T. 2 i Let the girder, Fig. 298, be doubled in length, to comprise six lower bays, and five upper bays, as in Fig. 300; and loaded at the middle. The O w Fig. 300. Warren-Girder. horizontal stresses in the flanges are accumulated from each end towards the middle, where they are a maximum, as in tablet c. Tablet c. . Bays in compression, .. Bays in tension, ad *r a' a" r'V 1 c' c" a" d\ db" c'c, yff cc<> Vb Stresses as I 2 T. 4 5 1 6 e 4" 3 2 I Valuation of the horizontal stress in terms of the load. The unit-stress i, in the tablet c, is measured by a d' 9 Fig. 300, the horizontal component of the oblique thrust c s a, of which c s d' is the vertical component, or half the weight. The value of the unit-stress relatively to that of c~d' t or half FRAMED WORK. 70 1 the weight, may be measured by means of a scale of parts. Or, trigono- metrically, let the angle at a be signified by a, then, c^d' : ad 1 : : sine a : cosine a : : ^ W : unit-stress; and T/ iir cosine a / x unit-stress at a = y z W : ( i ) sine a In the Warren-girder, the angle a is 60; and unit-stress a.ta = j4 W'-^=^A W x .577; or .060 unit-stress at a = . 2885 W (2) showing that the horizonal unit-stress in each of the end bays is equal to the weight at the centre multiplied by .2885. The stresses in the other bays, above and below, are in simple proportion to their numerical order from the support at each end towards the centre : stress on any bay = unit-stress x N, ( 3 ) in which N is the order-number of the bay. The stress on the central bay is also expressed by the equation, stress on the central bay = unit-stress x , ( 4 ) in which n is the total number of bays. Also, stress on the middle pair of bays = unit-stress x (5) 2 In the example, Fig. 300, the stress on the central bay c' c", by formula (3) or (4), is, (.2885 Wx6), or (.2885 W X I1-I)= 1.731 W; (a) and the stress in the central pair of bays is, (.2885 Wx 5), or (.2885 WxH-^-V 1.443 W (b) It appears that the stress at the middle of the longer boom is greater than the stress at the middle of the shorter boom by one unit-stress. Valuation by momerits, The tension in the central bay is given by the expression (A), page 699, namely -, in which W is the weight, / the span, 4f = 6 bays, and d the depth = .866 (sine a) proportionally, the length of a bay being i. The tension is, then, ^ - = 1.732 W, as already found (a). 4 x .866 Stress in the braces. The stress in the braces is to half the weight, as thie length of a brace is to the depth of the girder, or as radius to sine a, therefore, Stress in each brace = y z W x L_ = ^ . . ( 6 ) sine a 2 sine a In the Warren-girder, sine a -.866, and the stress in the brace (Warren-girder) is = .577 W,...( 7 ) 2 x .866 which is twice the unit-stress in the flange. 702 STRENGTH OF ELEMENTARY CONSTRUCTIONS. THE WARREN-GIRDER LOADED AT AN INTERMEDIATE POINT OTHER THAN THE CENTRE. In a Warren-girder, Fig. 301, loaded at d, as in an ordinary loaded beam, the weight on the supports at a and b are respectively - W andyW, (8) in which / is the total number of bays in the longest flange, and m and n the number of bays to the left and to the right of the weight. Ow Fig. 301. Warren-Girder, loaded at any intermediate point. Stress in the Braces. The stresses in the braces d c' and d c", which immediately support the weight, are as n and m, or inversely as the lengths of the two segments; and, adapting formula (6), n W Stress in the braces of the longer side = x (o) / sine a j , i m W , \ Do. do. shorter do. = x _ ( 10 ) / sine a Sine a^ .866, and in this example the stresses are, 2 W In the longer side = x = .385 W, ,. (c) 6 .866 In the shorter side = -^- x-:= .770 W, (d} 6 .866 transmitted to the supports a and b, and there resolved into vertical and horizontal components. Second Process for the stress in the braces Unit-coefficient of diagonal stress. The sum of the stresses (c] and (d) is 1.155 W, which bears to the weight W the ratio of the length of a brace to the depth of the girder; since i. : .866 : : 1.155 : i. Divide the coefficient 1.155 by the number of diagonals, 2 / or 12, and the quotient .09625 is a unit-coefficient per diagonal. Multiply this unit-coefficient by 2 m and 2 , or the number of diagonals in the longer and the shorter sides : .09625 x 4 diagonals = .385 .09625 x 8 do. = .770 The products are the coefficients (c) and (d). This process for arriving at the stresses in the diagonals is the simplest where a number of calculations are to be made for one girder. FRAMED WORK. 703 Horizontal Stress in the Booms. The unit-stress at each end of the girder, adapting formula (i), is as follows: n' cosine a Unit-stress at support a = -j- W g . ne q . (12) Do. support sine a In Fig. 301 the number of bays m' and n' are respectively four and two bays, to the left and to the right; together, six bays = /. Then, Unit-stress at support a - -^-W x . 577 = . 192 W; Do. support = 6 W; and the unit-stress at b is equal to twice the unit-stress at a. The stress in the intermediate bays, between each support and the weight, is as before (formula 3), For the long end a d, (unit-stress at a) x N ; ............ ( 14 ) For the short end b d, ( do. at b] x N; ............ (15) in which N is the order-number of the bay, on either side of the weight, reckoned from the point of support at the same side. The successive stresses thus calculated are given in the following tablet/, in which the unit-stress at a is taken as i, and that at b is proportionally as 2. Tablet /(Fig. 301). t In compression C T C $ r s r 1 " r"V c'r" c"c In tension . ci (i d d 1 a" a'" d" d db tib The horizontal stresses ) are as .. \ 2 3 4 5 6 7 8 6 4 2 The maximum stress is in the bay c' /', over the weight, in compression. The tensile stress in the two bays a'" d and d b ', contiguous to the weight, are as 7 and 6 respectively, and they do not balance each other. But, as a matter of fact, there is a balance of stress, and it is completed by the difference of the horizontal components of the stresses in the two braces dc, dc", from which the weight is directly suspended, being respectively equal to the unit-stresses for the long and short ends. The difference of these is 2 - i = i, or one unit-stress in the direction db\ and (6+ i =)y is the total stress in the bay db ', which balances the opposite stress, also 7, in the bay da"} THE WARREN-GIRDER UNIFORMLY LOADED. A uniform load on a Warren-girder is, in fact, a load equally divided and applied to the apices of the web, as, for example, in Fig. 302, in which the 1 With this explanation, it may be said with propriety that the sum of the increments of stress on the one side of the weight is equal to the sum of the increments of stress on the other side. But, abstractly, it is an erroneous assumption. 704 STRENGTH OF ELEMENTARY CONSTRUCTIONS. shorter flange, which is uppermost, comprises six apices, on which the weights W, W", &c., are placed. Stress in the Braces. The stress in the braces caused by each weight may be calculated separately in the manner already explained. The unit- W O w" O w* O AAAAA7\ O Fig. 302. Warren-Girder, uniformly loaded Longer Flange Undermost. coefficient of stress for one diagonal in 12, as in Fig. 302, is 1.155 + 12 = .09625. The stresses caused by the weight W, which is i diagonal from a, and 1 1 diagonals from b, are, In the brace i 09625 x n diagonals = 1.058 W,; In the braces 2 to 12 09625 x i do. = .096 W,. Calculating, in the same way, the stresses caused by the other weights, the constituents of stress on each diagonal are obtained, the coefficients of which are given in the following table, No. 249, in which compressive stress is distinguished as + , and tensile stress as - . The resulting coefficient of stress in each brace is given in the second last column : Table No. 249. COEFFICIENTS OF STRESS IN THE BRACES OF A WARREN- GIRDER UNIFORMLY LOADED, WITH THE LONGER FLANGE UNDER- MOST. Fig. 302. Ratio of Segments of Girder. W I tO II. W" 3 to 9. W" 5 to 7. W 4 7 to 5. w s 9 to 3. W 6 II tO I. Resultant Stress in each Diagonal. Units of Result- ant Stress. braces. I 2 3 4 1 + 1.058 ) + .096] - .096 + .096 - .096 + .096 + .866 -.866 + .866 ) + .289 j -.289 + .289 + .674 -.674 + .674 -.674 + .674 ) + 48i( + .481 -.481 + .481 -.481 + .481 -.481 ON ON ON ON ON ON OO OO OO OO OO OO r) N M 10 8 6 4 2 zontal stress in bays, units,., j 6. Accumulated increments of) Q Q stress in bays, units, } 24 28 ^ 6 l6 2 4 3 3 4 3 6 FRAMED WORK. 707 Longer Flange Uppermost, Fig. 303 Half of Girder. 1. No. of braces, and No. of bays, 12345 6 2. Units of resultant stress in braces, -5 +5 -3 +3 -i +i 3. 4. Resultant stress of braces ) 5 + 5 5 + 3 3 + 3 3+1 i + i at apices, ....................... J or 10 8 6 4 2 5. Horizontal components of \ these, or increments of hori- > 10 8 6 4 2 zontal stress in bays, units,.. ) 6. Accumulated increments of I _ jg ^ stress in bays, units, .......... j 7. Total horizontal stress in bays, ) units, ............................ j -* -^ The resultant stress at the central apex, between braces 6 and 7, is, in both cases, equal to o; and therefore there is no increment of horizontal stress at the centre. The increment of horizontal stress in the central bay, No. 6, is, in both cases, equal to 2 units, and the increments increase by 2 units, from bay to bay, up to bay No. 2, where the increment amounts to 10 units. So that, inversely, the increments of horizontal stress in the flange diminish in arithmetical progression as they approach the centre. Let n = the number of braces, or the total number of bays, in half the girder; N = the order-number of a given bay, counted from the end of the girder, above and below; W' = the weight on one apex, for which the unit of horizontal stress, transmitted through one of a pair of diagonals, is .2885 W; The equations, for the horizontal stress in the given bay, based upon the foregoing analysis, are as follows : i st. When the longer flange is undermost: Horizontal stress in a given bay = .2885 W'( + (N - i) (2-N)) ...... (19) 2d. When the longer flange is uppermost: Horizontal stress in a given bay = .2885 W'((;/ i)-f (N - i) (2*2 N)) (20) These formulas are very easy of application. The reasoning by which they have been constructed by the author is given in the foot-note. 1 1 The increments of horizontal stress, at the several bays, line 5, in the "Analysis of stress," are in arithmetical progression, having the common difference, 2, originating at the centre. The order-number of the terms of the progression, counting from the centre, is expressed by (n - (N - i) ) ; and (n - (N - i) ) x 2, is the value of the increment in units. For the first increment, for example, in bay No. 2, N = 2, and ;/ 6 ; and the value of the increment is (6 -(2- i) ) x 2=10, as given in the analysis. If N', N", N'", &c., represent for the moment the successive order-numbers of the bays following No. 2 bay, the values of the successive accumulated increments of stress, line 6, are as follows : In No. 2 bay, (n - (N - i) ) x 2 No. 3 ( - (N - i) ) -t- ( - (N' - i) ) x 2 No. 4 (_(N and so on. The value for each bay, putting N for the order-number of the bay, and condensing the expression, is 708 STRENGTH OF ELEMENTARY CONSTRUCTIONS. ROLLING LOAD ON THE WARREN-GIRDER. Concentrated Rolling Load on the Warren-Girder. The stress caused in successive diagonals by a passing weight is alternately tensile and com- pressive; and the stress is a maximum when the load is on the apex. Distributed Rolling Load on the Warren-Girder. Suppose that the rolling load is practically of uniform distribution, as a railway train, the stresses in the diagonals and flanges may be tabulated and analyzed, as exemplified at pages 704 and 705. If the train extend over the whole of the girder, the case becomes one of a girder uniformly loaded. The stresses in partially- covered girders may be analyzed in like manner, and the changes in direction and intensity of stress determined. But it is essential, at the same time, that the stresses caused by the permanent weight of the bridge should be determined; since the actual ultimate stress in any member is the resultant of the action of the whole of the load, both permanent and passing. The maximum stress in the flanges takes place when the passing load covers the whole of the girder. PARALLEL LATTICE-GIRDER. Latticing is the combination of two or more systems of triangulation in the web of a girder, in which the diagonals cross each other. The number of apices is proportionally multiplied, and the length of the bays is propor- tionally shortened. The effect is that the weight is distributed over a greater number of points in the flange, the graduations of stress in the flange are reduced, whilst also the stress in the diagonals is proportionally reduced. There is a special advantage in lattice- work, in affording the means of stiffen- ing the braces by simple connections at the intersections. If the diagonal stresses be calcul- Fi g . 3 o 4 .-Paraiiei Lattice-Girder. ated, in the first instance, as for a single triangular system, let them be divided by the number of systems in the lattice ; the quotient is the aliquot part of the stress, as distributed to each diagonal. The fundamental triangulation is shown by thick lines, Fig. 304; and in this instance, where only one additional system is interpolated, the stress in the fundamental diagonals is reduced to a half of what they would sustain if they stood alone. THE PARALLEL STRUT-GIRDER. In the parallel strut-girder, Fig. 305, having vertical and diagonal bracing, supporting a single weight, W, on the upper flange at the centre, the vertical (KX2x(N-i))-(Nx2x(N-i))+ I+(N " l) x2x(N-i) = (N-i)x(2(tt-N) + N) To this is to be prefixed the initial horizontal stress at bay No. I, which is n units, or 6 units, for Fig. 302, and (w-i) units, or 5 units, for Fig. 303. Thence the formulas (19) and (20) : Horizontal stress in a given bay = . 2885 Wi( + (N- i) (2-N)), (19) when the longer flange is undermost ; and Horizontal stress in a given bay = .2885 Wi(- l) + (N- i) (2 -N) ),.... (20) when the longer flange is uppermost. xxxxxxx FRAMED WORK. 709 brace or strut cd receives and supports the whole of the weight; and the compressive stress is divided and transmitted by tension through the diagonals dc and dc", according to the parallelogram of forces, of which Figs. 305 and 306. Parallel Strut-Girders. the diagonal de, equal to twice dc, the depth of the girder, represents the weight. The depth dc represents half the weight, and the tensional stress in each of the diagonals d c' and d c" is represented in magnitude and direction by the diagonals. The tensions of these diagonals balance each other horizontally at their intersection at the lower flange at d, and thus they do not throw any horizontal stress on the lower flange. STRUT-GIRDER WITH A CONCENTRATED MOVING WEIGHT. When the load moves over the girder, each strut requires to be braced by a pair of diagonals intersecting at the foot of the strut, in the same way as the strut cd, under the fixed weight in Figs. 305 and 306, is braced by the diagonals dc and dc". The result is a system of cross-bracing, or counter- bracing, by crossed ties, as in Fig. 306. The extra braces at the outer struts ac', be" (Fig. 305), are not necessary, but they are introduced to complete the design. The maximum stress is imposed on each strut when the weight passes over its summit. If the weight move on the lower flange, the maximum stress on a given strut is imposed when the weight passes the lower end of the next strut on the side of the more distant support; and the maximum stress on any strut, by the lower flange, never exceeds half the weight. (Fig. 305.) The tension in the diagonal d c is resolved into compressive stress in the upper bay c c and the strut c a. The compressive stress in the strut c a is resolved into tensile stress in the lower bay a d and the outer diagonal a c" \ and lastly, the stress in the outer diagonal a! c" is resolved into compressive stress in the outer bay c" c and in the strut c" a, of which the former is transmitted to the middle bay c c. A similar action takes place in the other half of the girder, and the horizontal stresses in one half balance those in the other. The vertical stress in the lateral struts is obviously, by transmission, equal to YZ W. That is, the stress in the lateral struts is only half the stress on the central strut, which supports the whole of the weight. The compressive stress in the outermost struts, or half of the weight, is received and resisted by the supports at a and b. The outer bays of the lower flange, a a' and b' b, are not subjected to any transmitted stress. The horizontal stresses in the bays of the upper and lower flanges are, then, in the following ratios : STRENGTH OF ELEMENTARY CONSTRUCTIONS. No. of bay i, 2, Compressive stress in upper flange, as i, 2, Tensile stress in lower flange, as o, i, 3> 4 2, I I, O Trigonometrically, the weight is supposed to be divided into two halves, each of which is represented by the depth cd, and causes the diagonal stress on either side. Then, Stress in strut cd \ stress in diagonal dc : : sine a : radius or i; a being the angle a d c' formed by the diagonal with the lower flange. Therefore, Stress in each diagonal R Stress in Stmt T/ W or Stress in each diagonal = sine a W sine a 2 sine a (21) When the distance apart of the struts is equal to the depth of the girder, the angle a = 45, and sine a =.707; then, W Stress in every diagonal = = .707 W. In the upper flange, the stress caused by each diagonal being as the half weight dc to the length of a bay cc" , or as sine a to cosine a; then Stress caused in each upper bay = cosine a (22) sine a When the angle a = 45, cosine a - sine a; and Stress caused in each upper bay = y 2 W, and Stress accumulated in middle upper bay = W. In the lower flange, the middle lower bays are in tension = ^ W, due to the thrust of the struts at a and V . If the extreme bays of the lower flange be removed, and the girder be supported direct at the ends of the tipper flange, as in Fig. 307, the stresses Fig. 307. Parallel Strut-Girder. Lower end bays removed. Fig. 308. Parallel Strut-Girder, loaded at an intermediate point. in the structure remain unaltered, since there is no horizontal stress in the extreme bays of the parallel girder, Fig. 305. It was seen that the func- tion of the end struts was only to support the girder. STRUT-GIRDER LOADED AT AN INTERMEDIATE POINT, OFF THE CENTRE. The strut-girder, Fig. 308, four bays in length, is loaded at cd, one bay from the support b, and three bays from the support a. Let /=the total FRAMED WORK. number of bays, and m and n = the numbers of bays to the left and to the right of the weight. The loads on the supports at a and b, as well as the compressive stress on the struts to the right and left, are respectively (23) being in the inverse ratio of the distances of the weight from the supports; or they are, in the present example Stress in the struts of the longer side ................ j^ W, Do. do. shorter side ............... % W. The stresses in the diagonals are in the same proportion, thus : Stress in the diagonals of the longer side = -^- -. - ................ (24) / sine a Do. do. shorter side = ^- -. - ................. (25) ?* I sine a If the angle a = 45, then sine a = .707, and the stresses are, In the diagonals of the longer side ......... }i x - = -3535 W, .707 Do. do. shorter side... . 3^ x - = 1.060=; W. .707 The unit or increment of horizontal stress in the bays of the upper and lower flanges is as follows : Unit of stress in the bays of the longer side, -r W : ...... ( 2 6 ) Do. do. shorter side, ^W cosine a ..... ( 27 ) / sine a When the angle a = 45, cosme q = I? an d the unit of stress is, sine a In the bays of the longer side .............................. % W, Do. do. shorter side ............................ &:Wfr the accumulated stress in the several bays is, by formula (3), page 701, For the longer side ............ W cosmea xN ............... (28) / sine a For the shorter side ...... W cosme q xN... .. (29) / sine a in which N is the order-number of the bay, in the upper flange, on either side of the weight, reckoned from the point of support. For the lower flange, N is the order-number less i, seeing that, as before explained, there is no transmitted horizontal stress in the lower bays situated next the points of support. 712 STRENGTH OF ELEMENTARY CONSTRUCTIONS. The resultant stresses in the several bays are, in the example, Fig. 308, relatively, as follows: No. of bay i, 2, 3, 4 Compressive stress in upper flange as i, 2, 3, 3 Tensile stress in lower flange as o, i, 2, o Here it is apparent that, whilst the resultant stresses in the bays 3 and 4 of the upper flange balance each other, there is no tensile stress in No. 4 bay of the lower flange to balance the stress in No. 3. But the balance is supplied by the difference of the horizontal components of the diagonal stresses which meet below the weight at d. STRUT-GIRDER UNIFORMLY LOADED. The general conditions of stress in the strut-girder uniformly loaded, as in Fig. 309, are similar to those in the Warren-girder, as elucidated, page 704. Fig. 309. Strut-Girder uniformly loaded. Stress in the Struts. The stress is calculated by an adaptation of formula (17), page 706, in which sine a becomes = i, seeing that the angle of the strut with the flange is a right angle. The formula becomes, Stress in a given strut -n" W, ( 3 ) in which n" the number of weights between the strut and the centre of the girder. Stress in the Diagonals. This stress is found by formula (17) Stress in a given diagonal = -? sme a (so For illustration, the diagonals i and 8 carry the seven weights W to W 7 suspended between them; and each sustains half the number, or 3^ W, and the stress in each is 3% L, Similarly, the diagonals 2 and 7 carry sine a the five weights W" to W 6 between them, each sustaining 2^ weights, or 2^2 W'; and the stress in each is 2*4 The diagonals 3 and 6 carry the sine a weights W"' W 4 W 5 between them, each supporting i^ W', and the stress is Lastly, the diagonals 4 and 5 carry the weight W 4 between them, sine a each supporting W,, and the stress is W sine a No diagonal stress is transmitted across the centre of the girder: in this respect the strut-girder differs from the Warren-girder. FRAMED WORK. 713 Stress in the Flanges. The stress in the flanges increases by diminishing increments at each apex towards the centre, where it is a maximum. The increments consist of the horizontal components of the diagonal stresses developed at each apex. The accumulated horizontal stress in each bay is expressed by the following formulas, which have been constructed in a manner similar to that by which formulas (19) and (20) were constructed: Let n = the number of bays in the length of half the girder ; N = the order-number of a given bay, in the upper or the lower flange, a - the angle between the diagonal and the flange, W = the weight on one strut, for which the unit of horizontal stress in the flange, transmitted through the next diagonal, is W cosme a > sine a For the horizontal stress in a given bay : i st. In the Upper Flange: Stress in a given bay = W'-, smea N ( - -) ............... (32) 2d. In the Lower Flange: Stress in a given bay = W COsme a (N - i) x (n - *Lzl)... ( 33 ) When the distance apart of the struts is equal to the depth of the girder, w , cosine <*_ W ' sine o The gradation of stress in the flanges may be given for Fig. 309, con- taining 8 bays. No. of diagonal and bays, .................. Increments of stress in diagonals, ] sine a Increments of stress in bays of upper j flange (unit = W' cosine ) (^ ^ '# ^ units ' sine a Accumulated stress in do. do. , 3 y z 6 7 y 2 8 units. Increments of stress in bays of lower ( T/ / l7 - f a V O ^ 7z 2 72 I 72 UnilS. flange, ; j / Accumulated stress in do. do. , o 3^ 6 7^ units. STRUT-GIRDER TRAVERSED BY A LOAD UNIFORMLY DISTRIBUTED. The struts require to be counter-braced, and the stresses are calculated as in the immediately preceding case. ROOFS. i. The weight of and load upon a roof are taken as uniformly distri- buted over the surface of the roof; and the total weight on each pair of rafters, couple, or truss, is equal to the sum of the weight of the truss itself, and of so much of the roof as is carried between two trusses. STRENGTH OF ELEMENTARY CONSTRUCTIONS. In the triangular roof-truss, a be, Fig. 310, the total weight, W, may be considered as localized at the supports a and b, and the ridge c\ a fourth Figs. 310 and 311. Triangular Roof-Trusses. , and a half at c. The tension in a b is that due to the each at a and weight, YZ W, at feet - 5 tons; and it is O X O ICCl resolved into 4^ tons in C, and .875 tons in F. This tension in F is resolved into 1.54 tons in C and in D. The total stresses in the three tension-rods C, D, and E are, then, as follows : Totals. In C, stress through F by direct weight of roof,.. 3.18 tons. stress by tensile force of E, 4.75 stress from F by do. do., 1.54 9.47 tons. In D, stress through F by direct weight of roof,.. 3.18 stress from F by tensile force of E, 1.54 4.72 In E, stress, 5.00 So much by way of analysis. But Mr. Stoney shows a method of deducing the stresses, by starting from the stress on the abutment and working thence towards the centre. Referring to Fig. 312, and adopting the same data as above, the reaction of the left abutment is 4 tons, of which i ton is directly balanced by the weight W concentrated there, leaving 3 tons to be resolved in the directions of A and C, into 10.35 tons and 9.47 tons respectively. The pressure of W", 2 tons, is resolved into 1.8 tons on F and .875 tons on A; and (10.35 - -$75 = ) 9-475 tons is tne thrust in B. At a, the stresses in C and F, which are known, are resolved by the intermediate substitution of their resultant into the stress 4.72 tons in D, and 5 tons in E. 4. In Fig. 313, the middles of the rafters are strutted by struts c' d and c" d, meeting at d in the horizontal tie-bar, and tied to the ridge by the ver- tical rod c d. The weight of the roof is localized at #, c ', c, c", and b, in the proportions ^ W, % W, ^ W, ^ W, and y% W. In the truss ac'd, the weight on c' is equally sustained by the limbs c' a and c' d, y% W being borne by the abut- ment, and J$ W being transmitted through the tie-rod cd to the ridge c. As ^ W is also transmitted from c", in the right hand Fig 3I3 ._ A . Truss Roof . rafter, the total weight at the ridge is (% + ^4+^)W=^W. This is just what the ridge would have borne, without the intervention of the struts; and the function of the struts is chiefly to assist the rafters in resisting transverse stress. The pull in the vertical tie-rod c d is (^ + y% = ) ^ W. To find the stresses in the rafters and the horizontal member a b, Dr. Rankine distinguishes the main truss a cb and the secondary trusses ac' d and dc" b. The tension in ab is the sum of the tensions due to the first and second trusses ; the thrust in a c', likewise, is the sum of the thrusts, and that in c' c is the thrust of the first truss only. Suppose the span /= 20 STRENGTH OF ELEMENTARY CONSTRUCTIONS. feet ; the pull in a d by the first truss is by formula by the second truss, ^ * '* = */ 6 W. The sum of these pulls is (j^ + x / 6 = ) y z W, the resultant tension in a b or a d. The thrust in a c by the first truss, is to the relative tension in a d, as a c to a d. The length of a c = \/(a d) 2 + (cd)* =12.5 feet, and the thrust is, y W x 5 =.4! 7 W. In the second truss, the thrust in a c' is the same 10 fraction of the tension in ad, due to the truss; or it is T /e Wx 12.5 = 10 .242 W. The sum of the first and second thrusts, or .658 W, is the resultant thrust in a c ' . The value of the thrust in a c, by the first truss, may be found in terms of the relative tensions in a d and c d; for it is equal to ^/tension in (0) 2 + tension in (cd)*= \J (YzY + (Y%Y = -4 J 7; as nas already been found. 5. In Fig. 314, each rafter is divided into three equal parts, which are supported by two struts, c' d and c'" d' for the left-hand rafter, and c 4 d" and c" dim the right-hand rafter; suspended by vertical rods c' d\ c d, and c" d" '; united by the main tie-rod a b. The total weight W, uniformly distri- buted, is localized at a c'" c' c c" c b in the proportions, c c c c '/ 6 W, '/6 W, / 6 W, /e / 6 W, '/ I2 W. Fig. 314. Trussed Roof. Three trusses are recognized here: the first truss a c b, the second a c d, and the third a c'" d'. Half the stress at c'" 9 the summit of the isosceles or third truss, is transmitted by the vertical rod c' d' to c ', where the stress is increased to ( J /6 + Yi2 = ) }i W. This load is transmitted to a and d' at the base of the truss, in the inverse ratio of the segments a d' and d' d; that is, two thirds, or (^ x 2^ = ) i/ 6 W, is transmitted through the strut cd and rod cd, to c. An equal quantity, x / 6 W, is transmitted from the right-hand rafter, and the sum x / 3 W added to T / 6 W, makes ^ W, the resultant load at the ridge. Suppose the span /= 60 feet and the rise d = 15 feet, the pull in a b or a d due to the first truss, is by formula (A),/^ = ^ W; by the second truss, 4" the pull is 8 / 9 ths of what it would be if the truss were isosceles, or it is * w x % = w/ the third 'A w x y 3 / 24 FRAMED WORK. 717 is the resultant stress in a d' and d" b; and the sum of the first and second pulls is the resultant stress in d' d", or fe W. The pull in the main tie-rod a fr, due to the second strut, was said to be 8 / 9 ths of what it would have been for an isosceles truss. In general terms, the fraction of what it would be for an isosceles truss of the same height and length is the ratio of the product of the segments into which the tie-rod or base of the truss is divided by the vertical rod from its apex, to the square of half the base. In this instance the base is divided into */$ and I / 3 , and 2 / 3 x I / 3 - 2 / 9 ; also ^ x y 2 = % or 2 / 8 j and the ratio of 2 / 9 to 2 /s is 8 to 9, or 8 / 9 ths. 1 The thrusts in the rafters may be found by the method already applied, in the previous case; and the same general process is applicable to roofs of more extensive construction. Professor Rankine gives general equations, for the stresses in roofs of the strut-and-rod class, Fig. 314; and Mr. Stoney shows how the stresses may be found in employing the parallelogram of forces, from the pressure on an abutment. 2 By the application of the principle of the parallelogram of forces, the stresses in crescent and other forms of girders and roofs may be determined. 1 See page 508, at top. 2 Civil Engineering, page 472. The Theory of Strains, page 159. WORK, OR LABOUR. UNITS OF WORK OR LABOUR. The fundamental units of work the foot-pound and the kilogrammetre have been denned at page 312; and their relations with those of horse- power have been stated at page 158. Horse-power. Horse-power is a measure of the rate at which work is performed. One horse-power is the expression of 33,000 foot-pounds of work done per minute, or 550 foot-pounds of work done per second. It is nearly identical with the French horse-power (cheval-vapeur, or cheval\ which is equal to 75 kilogrammetres of work done per second. As a kilo- grammetre is equal to 7.233 foot-pounds, the " c/iei>a/" is equal to (75 x 7.233 = ) 542.5 foot-pounds of work per second, which is 1,37 per cent, less than the English measure of a horse-power. Mechanical equivalent of heat. The values of the mechanical equivalent of heat in English and in French measures are denned at page 332, and their relations are stated at page 159. An English unit of heat is the quantity which is required to raise the temperature of water at or near 39.1 F., the temperature of its maximum density, through i F. ; and its mechanical value or equivalent is equal to 772 foot-pounds. One horse- power is therefore equivalent to (33,000^772 = ) 42^ heat-units per minute. A French unit of heat is equal to that which is required to raise the temperature of i kilogramme of water through iC. ; and its mechanical equivalent is 424 kilogrammetres = 3063. 5 foot-pounds. LABOUR OF MEN. Mr. Smeaton concluded that the power of an ordinary labourer at ordi- nary work was equivalent generally to work done at the rate of 3762 foot- pounds per minute. But, according to a particular estimate made by him for pumping up water 4 feet high, by good English labourers, their power was equivalent to 3904 foot-pounds of work per minute; and this he estimated as twice that of ordinary persons " promiscuously picked up." Mr. John Walker found that the force exerted by an ordinary labourer in raising weights for driving piles, average daily work, was 12 Ibs. In working daily at a winch or a crane-handle, the average force was 14 Ibs. moving at the rate of 220 feet per minute, equivalent to (14 x 220 = ) 3080 foot-pounds per minute. Mr. Glynn says that a man may exert a force of 25 Ibs. at the handle of a crane for short periods; but that, for continuous work, a force of 15 Ibs. is all that should be assumed, moving through 220 feet per minute. The power of a man would thence be (15 x 220 = ) 3300 foot-pounds per minute. LABOUR OF MEN. 719 Mr. G. B. Bruce states that, in average work at a pile-driver, a labourer exerts a force of 16 Ibs., plus the resistance of the gearing, at a velocity of 270 feet per minute, for 10 hours a day, making one blow every four minutes. The power is (16 x 270 = ) 4320 foot-pounds per minute. Mr. Joshua Field, in 1826, tested the performances of men at a crane of rough construction, in ordinary use. The barrel was n^ inches in diameter to the centre of the chain; and the driving -gearing consisted of two pinions and two wheels, geared successively, with an 1 8-inch handle. The ratio of the power to the weight was i to 105. The loads were thus so proportioned as to be reduced successively to from 10 to 35 Ibs. at the handle ; frictional resistance being additional. The load was raised through a height of 16^ feet in each experiment. The results were as follows: Table No. 251. POWER OF MEN AT A CRANE. Statical Equiva- No. of Resist- T s\r*A Time lent Power in Ex- peri- ance of the Load at the -LiOao. Raised. in Raising. Foot- pounds REMARKS. ment. Handle. per Minute. Ibs. Ibs. minutes. ft. -Ibs. I 10 1050 i-5 11,550 Easily done by a stout Englishman. 2 15 1575 2.25 11,505 Tolerably easily by the same man. 3 20 2100 2.0 17,325 Not easily by a sturdy Irishman. 4 25 2625 2-5 17,329 With difficulty by a stout Englishman. 5 30 315 2.5 20,790 Do. by a iTondon man. 6 35 3675 2.2 27,562 With the utmost difficulty by a tall Irishman. 7 55 55 2-5 24,255 Do. do. by a London man. 8 55 )) 2.8 3 21,427 With extreme labour by a tall Irishman. 9 10 55 ,5 5, J> 3-0 4.05 20,212 15,134 Withvery great exertion by a sturdy I rishman. With the utmost exertion by a Welshman. ii J5 Given up at this time by an Irishman. Mr. Field states that No. 4 gave a near approximation to the maximum power of a man for 2 ^ minutes. In all the succeeding trials, the men were so much exhausted as to be unable to let down the load. It would appear from this table that the maximum net pressure at the handle for constant working would not exceed 15 Ibs., exclusive of frictional resistance. The loads were only from y 2 to i ^ tons, much below the capacity of the crane; and the frictional resistance was disproportionally great for the work of one man. The author has found that when cranes were worked up to their capacity, the men could without difficulty exert a net pressure of 30 Ibs. at the handle, exclusive of frictional resistance, for a short time. In one instance, he observed that a very strong man raised 23 cwt. by a 30 cwt. crane, when he exerted a net force of 100 Ibs. at the handle. An ordinary man at the same crane, raised 14 cwt. with difficulty, with a net force of 56 Ibs. at the handle; and the same man raised without difficulty 10 cwt., with a force of 40 Ibs. at the handle. A man can exert on the handle of a screw-jack, of say 1 1 inches radius, a net force of 20 Ibs., without difficulty. From the foregoing data, it appears that the average net daily work of an ordinary labourer at a pump, a winch, or a crane, may be taken at 720 WORK, OR LABOUR. 3300 foot-pounds per minute; and, allowing one-third more for the frictional resistance of the machine or apparatus, the total work done would be at the rate of 4400 foot-pounds per minute. It may be added that, taken generally, well-fed English labourers can turn a crank by hand, at the rate of from 25 to 30 turns per minute, for a continuance, exerting a pressure of 20 Ibs. at the handle; or they can apply a pressure of 28 or 30 Ibs. for a short time, or from 50 to 56 Ibs. at an emergency. M. Cornet's estimate of the work of a labourer in France, turning a crank, amounts to 6 kilogrammetres per second, equivalent to 2604 foot- pounds per minute, for 8 hours a day above ground, or 6 hours a day in a mine. This, it is presumed, is the net work. LABOUR OF HORSES. According to Messrs. Boulton & Watt's estimate of the power of a dray- horse, it could do work equivalent to 33,000 foot-pounds per minute, for 8 hours a day. Tredgold estimated the work of a horse at 27,000 foot-pounds per minute, for 8 hours a day. Simms tested the labour of horses in raising water : 23,412 foot-pounds per minute, for 8 hours a day. 24,360 6 27,056 4% 32,943 . 3 He preferred the performances for 6 hours and 3 hours a day, as they were unobjectionable to the health and durance of the horses. Rennie found that a horse weighing 1 1 cwts. could draw a canal boat at a speed of 2^ miles per hour, with a pull of 108 Ibs., over a distance of 20 miles per day. This performance is equivalent to a work of 23,760 foot- pounds per minute. He estimated that the average work of horses, strong and weak, is at the rate of 22,000 foot-pounds per minute. Mr. Beardmore found that a horse eight years old, weighing 10^ cwts., performed 39,320 foot-pounds of work per minute, for 8 hours a day. It is inferred from the foregoing data, that the maximum work done by an average horse, per day of 8 hours, is at the rate of 25,000 foot-pounds per minute. At the same time, it appears from the results of trials at Bedford, to be noticed subsequently, that the average work of a horse is 20,000 foot- pounds per minute. See page 963. Good horses can draw a load of i ton at the rate of 2 ^ miles per hour, during from 10 to 12 hours. Mr. A. Wilson found that, in India, a pair of well-fed bullocks raised 82 bags of water 22 feet high in i hour, for a morning's work of 4^ hours. Each bag contained 4^ cubic feet of water, and the work was equivalent to 8000 foot-pounds per minute. WORK OF ANIMALS IN CARRYING LOADS. 1 Men Carrying by Hand. Labourers wheeling millstone in barrows, on the quays of Paris, a distance of 22 to 27 yards, making 25 to 30 trips per 1 Data derived from Les Moyens de Transport, by M. Alfred Evrard, 1872, vol. i. LABOUR OF ANIMALS. 72 1 hour; the daily work performed is equivalent to the carriage of from 330 to 400 Ibs. i mile. The following are other cases of daily labour, showing the useful weight carried i mile: In Belgium, working in couples, one man carries 560 Ibs. i mile. At Port Royal, loading pig-iron, one man carries 160 Ibs. i mile. At Paris, loading sugar-loafs, 86 Ibs. Men Carrying on the Back. Carrying tiles or bricks, net load 106 Ibs.; day's work 600 Ibs. carried i mile. Carrying coal in mines, net load 90 to 100 Ibs.; day's work 344 Ibs. i mile. Another case, net load 100 to 130 Ibs.; day's work 340 Ibs. i mile. Loading coke into waggons, net load 100 Ibs. ; day's work 270 Ibs. i mile. Discharging coke on the ground, net load 100 Ibs.; day's work 330 Ibs. i mile. Discharging coal on the ground, Port Royal; net load 106 Ibs.; day's work 370 Ibs. i mile. Discharging coal on the ground, Paris; net load no Ibs.; day's work 560 to 960 Ibs. T mile. Charging small coal into boats, Rive-de-Gier; 190 Ibs. net load; day's work 1230 Ibs. i mile. On the back of a Horse. The load carried by a horse on its back varies generally from 220 to 390 Ibs., about 27^ per cent, of the weight of the animal. Carrying a man of 176 Ibs. weight, at about 3^ miles per hour; day's work 4400 Ibs. i mile. Carrying a load of 260 Ibs. for 10 hours at 2^ miles per hour, 6540 Ibs. i mile. Trotting with a man of 176 Ibs. weight, at 5 miles per hour, for 7 hours; day's work, 6100 Ibs. i mile. On the back of a Mule. At Buenos Ayres; net load, 170 to 220 Ibs.; day's work 6400 Ibs. i mile. In Spain, net load 400 Ibs. at 2.9 miles per hour; day's work 5300 Ibs. i mile. In France, net load 330 Ibs. at 2 miles per hour; day's work 5000 Ibs. i mile. On the back of an Ass. Load 176 Ibs., carried 19 miles; day's work 3300 Ibs. i mile. The asses in Syria can carry from 450 to 550 Ibs. of grain. - On the back of a Camel. Load 550 Ibs. carried 30 miles per day for 4 days, resting on the 5th day. For 4 days, day's work 16,500 Ibs. i mile. For 5 days, 13,000 Ibs. i mile. The ordinary load for a dromedary is 770 Ibs. On the back of a Lama. Load no Ibs.; day's work 2000 to 3000 Ibs. i mile. 46 FRICTION OF SOLID BODIES. The friction between surfaces pressed together, whether flat or round, of which one is moved on the other, is said to be in the direct ratio of the pressure, and to be independent of the velocity, and of the area of the surfaces pressed together, up to what may be called the elastic limit, or the Table No. 252. FRICTION OF JOURNALS IN THEIR BEARINGS. Diameters from 2 to 4 inches. Speeds varied as I to 4. Pressures up to 2 tons nearly. (Reduced from M. Morin's data. ) Description of Surfaces in Contact. LUBRICANT. Coefficient of Friction. Ordinary Lubrication. Continuous Lubrication. JOURNALS. BEARINGS. Cast iron on cast iron Cast iron on gun metal Cast iron on lignum vitae Wrought iron on cast iron Wrought iron on gun metal Wrought iron on lignum ) vitae ( ( Lard, olive oil, or tallow The same lubricants, ) and wetted \ pressure=i .07 to .08 .08 .054 .14 .14 .07 to .08 .16 .16 .10 .14 .07 to .08 .07 to .08 .09 .19 .25 .11 .19 .10 .09 .12 IS pressure^ i. .03 to .054 ' Asphalte. Surfaces unctuous (^ Unctuous and wetted . . . ( Lard, olive oil, or tallow i Surfaces unctuous .03 to .054 \ Unctuous and wetted . . . ( Slightly unctuous .09 .03 to .054 .03 to .054 f Wood slightly unctuous | Oil, or lard { Unctuous Unctuous, with mixture ) [ of lard and plumbago ) Olive oil, tallow, or lard... ( Olive oil, tallow, or lard I Grease y Unctuous and wetted.... ( Slightly unctuous Oil, or lard Unctuous Gun metal on gun metal... Lignum vitae on cast iron Lignum vitae on lignum ) fOil \ Lard ( Lard .07 ( Unctuous Lard vitae \ FRICTION: M. MORIN'S EXPERIMENTS. 723 limit beyond which abrasion takes place. Friction is of two kinds : sliding or rubbing, and rolling; and the frictional resistance to the commencement of motion, after two bodies have rested sometime in contact, is greater than the friction between bodies of which one is already in motion upon the other. M. Morin's experiments afford the principal available data for use. Though the constancy of friction holds good for velocities not exceeding 15 or 1 6 feet per second; yet, for greater velocities, the resistance of friction appears, from the experiments of M. Poiree, in 1851, to be diminished in some proportion as the velocity is increased. Table No. 253. FRICTION: RESULTS OF EXPERIMENTS. (Reduced from M. Morin's data.) Description of Surfaces in Contact. Disposition of Fibres. State of the Surfaces. Coefficient of Friction. Oak on oak parallel dry pressure=i. 48 Do soaped .16 Do perpendicular dry .14- Do on end, on side .19 Different woods on oak parallel .36 to .40 \Vrou cr ht iron on oak .62 Do do ... wetted .26 Do. do soaped .21 Cast iron on oak dry .40 Do do y wetted .22 Do do soaped IQ Brass on oak drv .62 \Vrought iron on elm .2; Cast iron on elm .20 Leather on oak .30 to .3C Do do wetted .20 Leather belt on oak (flat) dry .27 Do. on oak pulley Leather on cast iron & on bronze Do. do. do. Do. do. do. Do. do. do. Wrought iron on cast iron perpendicular wetted unctuous and wet oiled slightly unctuous 47 (after resting in contact.) .56 .36 23 15 .18 Do on bronze .18 Cast iron on cast iron .15 Do. on bronze -15 Bronze on bronze drv .20 Do on cast iron 22 Do on wrought iron slightly unctuous .16 Oak, elm, yoke elm, cast iron, ) wrought iron, steel, and ( bronze, sliding on each ( other, or on themselves j 'lubricated in^j the ordinary ! way ,with lard, f <{ tallow, oil, &c.J continuously ) lubricated } [_ slightly unct'ous .07 to .08 .05 15 724 FRICTION OF SOLID BODIES. It appears from the table No. 252 that the frictional resistance of metal journals revolving in metal bearings is uniform for all metals, with one exception; and that the resistance, with continuous lubrication, is only 56 per cent, of the resistance with ordinary lubrication: Lubrication. Coefficient. JOURNAL. BEARING. Cast iron in cast iron N ordinanr / -7 to .08, or */ 14 to '/ I2 . Cast iron in gun metal / I mean .075, or I /i 3 . 3 Wrought iron in cast iron V continuous f -03 to .054, or r / 33 to Wrought iron in gun metal ) \ mean .042, or I / a4 Gun metal in gun metal; ordinary, .10, or I / I0 ih. 2.5 Additional data derived from the friction of mill-shafting, are given under SHAFTING. FRICTION ON RAILS. M. Poire'e's experiments on the Paris and Lyons railway were made with a waggon, presumably having four wheels, of which the brake was screwed up, so that the wheels were skidded. The resistance to traction, or the friction on the rails, at various velocities, was as follows : Table No. 254. SLEDGING FRICTION OF A WAGGON ON RAILS. M. Poiree's Coefficients. Empty Waggon, 3.40 tons. STATE OF THE RAILS. Dry. Very Dry. Damp. Dry and Rusty. Dry. Springs gagged. Velocity of Waggon. feet per second. 13 tO 20 20 tO 26 26 to 33 33 to 46 46 to 60 60 to 72 miles per hour. 9 to 14 14 to 18 l8 tO 22 22 tO 30 30 to 40 40 to 50 weight=i. .208 .179 .167 .144 weight=i. .246 .222 .202 .I8 7 weight=i. .110 .083 weight=i. .201 .182 175 .162 ~36 weight=i. .200 .172 .154 .132 Comparative Coefficie Speed, 30 feet per second ; or 20 miles per hour. nts of Frictioi 175 .169 i for Diffe r ent Weigh ts ; Rails I )ry. Empty waggon, 3.40 tons Loaded waggon, 6.45 At speeds under 20 miles per hour, it appears from the table that, when the rails are dry, the coefficient of friction, or the adhesion, is one-fifth of the weight, and that on very dry rails it is one-fourth. As the speed is increased, the adhesion is reduced. These data are corroborative of the results of the author's experiments on the ultimate tractive force of locomotives on dry rails, from which he obtained a coefficient of friction equal to one-fifth of the weight, at speeds of about 10 miles per hour. WORK ABSORBED BY FRICTION. 725 They are corroborated by the experience of train-brakes on the District Railway, London. The gripping force of the brake is greatest just before the train is brought to a state of rest. It is seen from the table that, on damp rails, the coefficient of friction, at 20 miles per hour, was reduced to one-ninth. In the second part of the table, it is seen that the coefficient of friction increases in a ratio rather less than that of the weight. WORK ABSORBED BY FRICTION. The product of the total pressure between the rubbing surfaces by the coefficient of friction, is the total frictional resistance; and the product of this resistance by the space through which it acts, is the work done, or absorbed, by friction. Let W = the load or pressure, in pounds, /=the coefficient of friction between the two surfaces, s = the space passed through by one surface on the other, in feet, t the time in minutes, v = the velocity at the surface, in feet per minute, S = the speed of revolution, or number of turns per minute, U = the work absorbed, in foot-pounds, H = the horse-power absorbed, d=\he diameter of the axle-journal, or of the pivot, in inches, r = ihe radius of the axle-journal, or of the pivot, in inches, a = half the angle at the apex of a conical journal or a conical pivot, /=the axial length of the rubbing surface of a conical pivot, in inches. The space described by a cylindrical journal for one turn, is equal to 3. 14 d, in inches, or to .26 d, in feet; and by the flat end of a cylindrical pivot, the space described is equal to two-thirds of this quantity, in the ratio of the mean diameter to the extreme diameter, or .175 */, in feet. For a conical pivot, the mean diameter is, as for a flat pivot, two-thirds of the ' extreme diameter of the rubbing surface of the pivot, and the space described for one turn is expressed by .175 d, in feet. But the pressure on the surface of the conical pivot, compared with the pressure on a flat pivot, is greater in the ratio of the slant length of the pivot to the extreme radius of the rubbing surface, or as radius to sine a; or as A/ r 2 + / 2 to r. The pressure on the surface of a conical journal, is to that on a cylin- drical journal of the same length and extreme diameter, as radius to cosine a. Work Absorbed by Friction. 1. On a flat surface U=/Wx s ( i ) 2. On a cylindrical journal, for one turn U =/"W x .26 d..... ( 2 ) 3. On the square end of a cylindrical pivot, I TT = / W x 1 7 <: *._/> J Fig. 318. Formation of Teeth of Wheels. Performing the same operation, conversely, for the teeth of the pinion, the wheel and the pinion so constructed work truly together. In the wheel and pinion, Fig. 316, are shown the generating circles, having diameters equal to the radii of the respective wheels. 1 Whilst a wheel and pinion, or two wheels, having their teeth so con- structed, work truly together, they do not work truly with wheels or pinions of other diameters. For bevil-wheels, this peculiarity is of no moment, since they can only work by pairs ; but, for spur-wheels, it is convenient that any two wheels of the same pitch should be capable of working truly together. This property of interchangeableness is secured by the employment of the same generating circle for both flanks and faces, and for all diameters, as well as for racks. The minimum number of teeth may be taken as 12, and the diameter of the common generating circle may be taken as equal to the radius of the pinion of 12 teeth. Teeth of wheels so formed have the flanks rather excessively tapered; but, by reducing the taper, or thinning the tooth, for a distance of half the height of the flank measured from the root of the tooth, the working durability of the tooth is much increased, and steadiness of action is promoted even when the tooth has become considerably worn. If the full form of the flank of the tooth be retained, a shoulder is gradually worn into it, by which a tendency is induced to force the wheel and pinion out of gear; but in cutting away the flanks near the base, whilst the working portion of the flank-surface remains, shoulders can only be formed after very excessive wear. 2 Involute Teeth. The teeth of two wheels will work truly together, when their acting surfaces are involutes. The involute curve may be described 1 There is an error in the form of the flanks of the teeth as shown in Fig. 316; they should have been straight and radial. 2 Mr. Robert Wilson, of Patricroft, employs this method of forming the teeth of spur- wheels. The author is indebted to him for the above particulars about it. FORM OF THE TEETH OF WHEELS. 733 mechanically: Let A, Fig. 319, be the centre of a wheel, and mna a thread lapped round its circumference. The curve a b h, described by a pin at the end of the thread when unwound from the circle, is an involute. Fig. 319. Formation of Teeth of Wheels. The curve is also formed by causing a straight ruler R to roll on the circle, Fig. 319, with a pin at the end, tracing the involute qp. That two wheels with involute teeth should work truly, the circles from Fig. 320. Formation of Teeth of Wheels. which the involute forms for each wheel are generated, must be concentric with the wheels, with diameters in the same ratio as those of the wheels. Let AT, BT, Fig. 320, be the pitch-radii of two wheels to work together; 734 MILL-GEARING. through T draw any straight line D E, and with the perpendiculars A D and B E, describe the circles D H and E F. The involutes K T H and G T F give the forms of the teeth. To describe the teeth of a pair of wheels, of which A c and B c, Fig. 321, are the pitch radii, draw c d and c d perpendicular to the radials B d and A d; these radials are the radii of the involute circles from which the acting faces of the teeth are formed. Involute teeth have the disadvantage of being, when in contact, too much inclined to the radial line, by which an undue pressure is excited on the bearings. But they have the advantage of working truly, even at varying distances apart of the centres, and any two wheels of a pitch will work together in sets, however different the diameters. PROPORTIONS OF THE TEETH OF WHEELS. Referring to the annexed Fig. 322, the leading dimensions are indicated by literal references, and are thus distinguished in the table No. 257, in Fig. 322. Proportions of Teeth of Wheels. whj^Tftie first and second scales of proportions are, both of them, used by engineers in good practice; 1 and the third is the scale of Sir Wm. Fairbairn. 2 TABLE No. 257. TEETH OF WHEELS: PROPORTIONAL DIMENSIONS. (Fig. 322.) ELEMENTS. ist Scale. 2 d Scale. 3 d Scale. A+C Pitch of teeth, I 15 or I 1. 00 C& Thickness of teeth, S/TT or .4.S 7 or .47 .41 Ad Width of space 61 or 1 s 8 or $1 re Ad C^ Play i or 07 .IO a c Length above pitch-line, s/ IO or .30 5X r -37 W C c Length below pitch-line 4/10 or 40 6 */ or 4.3 4.O A a -\-fic Working length of tooth, 6/ IO or 60 ii or 73 7O Ca Whole length of tooth, 7/10 or .70 12 or .80 .7C C c A a Clearance at root, I / IO or .10 i or .07 .05 C b Thickness of rim, 7 or .4.7 Note to table. The proportion of clearance at the root of the tooth is usually varied from, say, x / I0 th of the pitch for the smaller wheels, to x /aoth for the larger wheels. 1 Engineer and Machinist's Assistant, page 89. 2 Mills and Millwork, Part 2, page 33. PROPORTIONS AND STRENGTH OF THE TEETH OF WHEELS. 735 STRENGTH OF THE TEETH OF WHEELS. The tooth of a wheel in action, is a beam fixed at one end and loaded at the other; and, as the available strength of a wheel-tooth is that of its weakest exposure, the strength should be calculated for the con- tingency that the whole of the force may be applied to the tooth at one corner, D, Fig. 323. The tooth, it is conceivable, may be broken across at any of the lines B b, B c, or B d; but, under uniform con- ditions, the actual line of fracture would be B ^, which is the base of a right-angle triangle having the equal sides D B and D b, and of which the height D e is equal to half the base B b. Now, the height of the tooth D B, is the slope of the right-angled triangle D B e, and is 1.414 times D e\ and the values of the elements for the application of the formula, according to the 3d scale of proportions in table No. 257, are: P = the pitch = i. /=the length of the "beam" = D e = .75 P 4- 1.414 = .53 P. b - the breadth of the beam = B = .53Px2 = 1.06 P. d= the depth of the beam, or the thickness of the tooth = .45 P. s = the tensile strength of the material in tons per square inch. W = the breaking weight at the corner of the tooth, in pounds. Adapting the general formula (4), page 507, the coefficient .289^ becomes (.289 x 2240 s = ) 647, and, in terms of the above symbols, Fig. 323. Strength of Teeth of Wheels. w _ (647 ........................... To express the breaking strength of the tooth in terms of the pitch, sub- stitute the equivalent values of b, d, and /, in formula ( 4 ): - W = 647^i.o6^x(. 4S P)- = 647fx . 40S ^. or W = ( 2 6 2 s) P 2 ................................................ (5) VALUES OF THE NUMERICAL COEFFICIENT IN FORMULA ( 5 ). Ultimate Tensile Strength Coefficient per Square Inch. (262 s). tons. 7. Cast iron, ........................ 262 x 7= 1834, say 1800 8. Do., ........................ 262 x 8 = 2096, 2100 9. Do., ........................ 262 x 9 = 2358, 2400 10. Do., ........................ 262x10 = 2620, 2600 11. Do., ........................ 262x11 = 2882, 2900 12. Do., ........................ 262x12 = 3144, 3100 12. Gun-metal, ....................... 262x12 = 3144, 3100 20. Wrought iron, .................. 262x20 = 5240, 5200 30. Steel, .............................. 262x30 = 7860, 8000 736 MILL-GEARING. For wheels of ordinary cast iron, when the tensile strength is not given, assume 7 tons per square inch as the tensile strength, and adopt the co- efficient 1800. Then, The Ultimate Transverse Strength of the Teeth of ordinary Cast-Iron Wheels, in terms of the pitch, is, W-i8oo P 2 (6) Inversely, P= -- (7) To express the breaking strength of the tooth in terms of the thickness of the tooth, d, which is equal to .45 P. The pitch P = , and by sub- 45 stitution in formula ( 5 ), W= 262 sx ( ) 2 ; or, ^ 2 ......................... (8) VALUES OF THE NUMERICAL COEFFICIENT IN FORMULA ( 8 ). Ultimate Tensile Strength Coefficient per Square Inch. (1294 s). tons. 7. Cast iron, .................. i2Q4x 7= 9058, say 9000 8. Do., .................. I294X 8=10,352, 10,000 9- Do., .................. I294X 9=11,646, 12,000 10. Do., .................. 1294x10=12,940, 13,000 11. Do., .................. 1294x11 = 14,234, 14,000 12. Do., .................. 1294x12 = 15,528, 16,000 12. Gun-metal, ................. 1300x12=15,528, 16,000 20. Wrought iron, ............ 1294x20 = 25,880, 26,000 30. Steel, .............. .......... 1294x30=38,820, 39,000 Again assuming a tensile strength of 7 tons per square inch for ordinary castings, The Ultimate Transverse Strength of the Teeth of ordinary Cast-Iron Wheels in terms of the thickness, is, /~~W Inversely, = d A/- - ........................... ( 10 ) v 9000 The excess of transverse strength of cast iron, in thicknesses of from i to 3 inches, above that which is calculated from the tensile strength, as detailed at page 555, affords a margin for weak forms of teeth, and for the loss of strength by trimming or by wear, and especially by the removal of the skin. The excess, it is true, diminishes as the thickness increases ; but, on the contrary, the diminution of strength is less by the wear of the thicker teeth than by that of the thinner teeth. Thus, there is a natural adjustment of the supply of strength in excess to the requirement. WORKING STRENGTH OF WHEEL-TEETH. It is usual to act on a factor of safety of 10, for wheel-teeth: Formulas ( 6 ), ( 7 ), ( 9 ), and ( 10 ), thus adapted, become, HORSE-POWER OF TOOTHED WHEELS. 737 For the Working Strength of Wheel-teeth of ordinary Cast Iron : In terms of the pitch, ......... W = 180 1^. ...................... ( n ) Do " " P =V^- - (I2 > In terms of the thickness,. . . . W = 900 d 2 ........................ ( 13 ) ~ For wheels of cast iron of greater strength, or of gun-metal, wrought iron, or steel, the coefficients for a factor of safety of 10, are one-tenth of those which are given at pages 735 and 736, and they may be substituted in the above formulas, thus, FOR WORKING STRENGTH. In formulas (n) and (12). In formulas (13) and (14). Coefficient for gun-metal, .............. 310 ............ 1600 Do. wrought iron, .......... 520 ............ 2600 Do. steel, ..................... 800 ............ 3900 Sir William Fairbairn states that, for wooden teeth, a thickness i^ times that of cast-iron teeth is sufficient; 3/ s ths of the pitch goes to the thick- ness of the wooden cogs, and 2 / 5 ths to that of the iron teeth of the wheel geared with the wooden teeth. There is no clearance, as the teeth are accurately trimmed. BREADTH OF THE TEETH OF WHEELS. When the breadth of a tooth is just twice its whole length, its ultimate resistance to transverse stress is approximately equal to its resistance to diagonal stress applied at one corner. A greater breadth than twice the length, therefore, is not reckoned to add to the transverse resistance of the tooth; but it is necessary for durability. Breadth in relation to Working Stress. Tredgold fixed the maximum average working stress at the pitch-line at 400 pounds per inch of breadth of teeth. Sir William Fairbairn adopted this datum. HORSE-POWER TRANSMITTED BY TOOTHED WHEELS. A horse-power is work done at the rate of 33,000 pounds through i foot, per minute; or 550 foot-pounds per second. Let, H = the horse-power transmitted. W = the stress in pounds, at the pitch-line. v = the velocity at the pitch-line, in feet per second. S = the speed in turns per minute. D = the diameter in feet. That is to say; ist. The horse-power transmitted is equal to the product of the stress by the velocity at the pitch-line, divided by 550. 2d. The stress at the pitch-line is equal to 550 times the horse-power divided by the velocity at the pitch-line. 3d. The velocity at the pitch-line is equal to 550 times the horse-power, divided by the stress. 738 MILL-GEARING. The speed of the wheel, in turns per minute, is equal to v x ; or That is to say; ist. The speed of the wheel is equal to 19.1 times the velocity at the pitch-line, divided by the diameter. 2d. The velocity at the pitch-line is equal to the product of the diameter by the speed, divided by 19.1. To find the horse-power in terms of the stress, the diameter and the speed ; W v or W, D, and S. By formula (15), H = - ; and, substituting the value of 10,500 (20) That is to say, the horse-power transmitted is equal to the product of the stress at the pitch-line, by the diameter and by the speed; divided by 10,500. For the horse-power in terms of the pitch. Substitute in formula (20), the value of W in terms of the pitch, that is, by formula (n), for cast iron, 10,500 (for cast iron) H = - -- ( 22 ) That is to say, the horse-power that may be transmitted by a cast-iron wheel is equal to the product of the square of the pitch by the diameter and by the speed; divided by 58.3. The horse-power per foot of diameter and per turn per minute, is P 2 (for cast iron) H = ; ( 23 ) being equal to the square of the pitch divided by 58.3. The formulas ( 22 ) and ( 23 ) are available for the calculation of the horse-power of wheels made of other metals, by using the proper constants, as follows: FOR HORSE-POWER. In Formulas (22) and (23). Coefficient for gun-metal 10,500 ^310 = 33.9 Do. wrought-iron 10,500-^-520 = 20.2 Do. steel 10,500^-800 = 13.1 Table No. 258 gives particulars of dimensions, stress, and horse-power of toothed spur-wheels of ordinary cast iron, for various pitches. The thickness, column 2, is given as 45 per cent of the pitch; the working WEIGHT OF TOOTHED WHEELS. 739 stress at the pitch-line, column 3, is calculated from the pitch by formula (n); the horse-power transmitted at i foot per second, column 6, is calculated from the stress by formula (15), and the horse-power per foot of diameter and per turn per minute, in the last column, is calculated by formula (23). This table affords data for performing all the usual calcula- tions for the horse-power of toothed wheels. 1 Table No. 258. STRENGTH AND HORSE-POWER OF SPUR-WHEELS, Made of ordinary cast iron. BREADTH OF TEETH. HORSE-POWER TRANSMITTED. Pitch of Teeth. Thickness of Teeth. Working Stress at Pitch-line. Least Usual At i Foot per Second Per Foot of Diameter Breadth. Breadth. at the and per Turn Pitch-line. per Minute. inches. inches. inches. pounds. inches. H. P. H. P. thickness X 2. I 45 1 80 .90 2 X 327 .0172 & 1 28l 405 1. 12 1-34 2. l / 2 and 3 3X and 3^ .511 .736 .0268 .0386 l$ 79 551 1.58 4X 1. 000 .0525 2 .90 720 1. 80 6 1.310 .0686 for 400 Ibs. per inch. 2 .90 720 1. 80 6 1.310 .0686 2X .01 911 2.28 6 1.656 .0868 2> .12 II2 5 2.8 1 6 2.045 .1007 2 /4^ 24 1361 3-40 7 2.474 .1297 3 35 l620 4.05 9 2.945 .1544 3X .46 1901 4-75 9 3.456 3/^s 57 2205 5.51 10 4.010 .2100 4 .80 2880 7.20 14 5.236 .2744 4X 2.02 3645 9.11 6.627 3472 5 2.25 4500 11.25 15 to 16 8.182 4028 6 2-73 6480 16.20 16 to 18 11.782 .6176 Note. For mitre and bevil wheels, the mean diameter, breadth, and thickness of teeth are to be used in calculation. WEIGHT OF TOOTHED WHEELS. Spur-wheels. The weight of spur-wheels of usual proportions, per inch of breadth, increases in a ratio greater than the diameter, but less than the square of the diameter. As ascertained by plotting the particulars of a great number of wheels, the relation of the diameter and the weight per inch wide, is represented by the general expression a d+ b d 2 , in which d is the diameter, and a and b are constants for each pitch. Within the ordinary practical limits of breadth for each pitch, the weight varies as the breadth of the wheel, and may be taken at a constant per inch of breadth. 1 The author is aware that Tredgold, Fairbairn, and others, give higher values for the strength and power of the teeth of wheels than he gives in the text. 740 MILL-GEARING. Table No. 259. WEIGHT OF CAST-IRON SPUR-WHEELS, Per Inch of Breadth. T}-4._T_ DIAMETERS IN FEET. .rltCn. 50 75 i ,.5 | , 2-5 3 4 Usual Breadth. inches. cwt. cwt. cwt. cwt. cwt. cwt. cwt. cwt. inches. 6 1.27 1.66 2.07 2.97 i6to 18 5 1. 08 1.40 75 2.52 15 to 16 4^ .707 .984 1.28 .60 2.30 H 4 .638 .888 1.16 44 2.07 14 3^ .569 792 1.03 .29 1.85 10 3X 535 744 .968 .21 74 9 3 .319 .500 .696 .905 13 .62 9 1% .218 .297 .466 .648 .844 .05 5i 7 2^ .202 .275 431 .600 .781 .98 .40 6 2X .185 253 397 .552 .719 .90 .29 6 2 .110 .I6 9 .231 .362 .504 .656 .82 .18 6 i# .100 153 .209 .328 .456 594 74 .06 4X i# .089 137 .187 293 .408 531 .66 95 ( 3X not exc'ding 5 ft. I $% above 5 feet. i* .077 .121 .165 .259 360 .469 59 .84 \2j4 not exc'ding 4 ft. ( 3 above 4 feet. i .068 ,105 .143 .224 .312 .406 5i 73 2j^ Ib. Ib. Ib. ib. ib. Ib. ib. ib. # 6 9 13 22 32 43 55 84 2 < 6 13 22 32 43 55 84 2 % 5 8 12 20 3i 44 IX x 6 10 15 27 42 I# DIAMETERS IN FEET. Pitch. 5 6 7 8 9 10 II 12 inches. cwt. cwt. cwt. cwt. cwt. cwt. cwt. cwt. 6 3.98 5.09 6.30 7.63 9.06 1 0.60 12.24 14.00 5 3.38 4.32 5.36 6.48 7.70 9.00 10.40 11.88 4/2 3.08 3-94 4.88 5.90 7.01 8.20 9-47 10.82 4 2. 7 8 3-55 4.40 5-33 6-33 740 8.55 9-77 3^ 2.48 3-i7 3-93 4-75 5.64 6.60 7.62 8.71 3X 2 -33 2.98 3-69 4.46 5-30 6.20 7.16 8.18 3 2.18 2.78 3-45 4.18 4.96 5.80 6.70 7.66 2X 2.03 2.59 3.21 3-89 4.62 5.40 2K 1.88 2.40 2.98 3.60 4.28 5.00 2X 1-73 2.21 2-74 3-31 3-93 4.60 2 1.58 2.02 2.50 3.02 3-59 I* 1-43 1.82 2.26 2.74 IX 1.28 1.6 3 2.02 iX i-i3 1.44 I .98 For pitches of i inch and upwards, the weight per inch of breadth, for a given diameter, increases directly as the pitch. FRICTIONAL WHEEL-GEARING. 741 The following formulas are deduced from the actual weights of spur- wheels of from fa inch to 4 inches pitch, and from 6 inches to 1 2 feet in diameter. Weight of cast-iron spur-wheels of from i inch to 6 inches pitch, per inch of breadth : x (i+.io//) ................... (24) Weight of cast-iron spur-wheels of pitches less than i inch : For y% inch pitch, W = . 0935 ^+.0235 d* ......... ( 25 ) For ^ W = .0935 3 pulleys, 5 ft. 228 speed,,. ..120 ; 2 28x5 SO/2 pulley, 22; ) 17 tons ) 2 7 159 3 16x4 6S 14 T r 182 ( i 6 2o8 4 30x5 52 24 ( 2 I 8 V 156 5 18x4 65 12 i ,,3 ft. 6 in 223 Table continued. Velocity of Belt. Tension on Belt. Belt Thickness and Width of Belt. Horse- power Trans- mitted. Surface per Min- ute per H.P. Feet per Minute. Feet per Second. Total. Per inch Wide. feet. feet. H. P. Ibs. Ibs. sq. feet. , f double, 23 y z ins. 3582 59-7 125 1152 49 54-9 1 / 29 3582 59-7 175 1612 56 49-5 1 I4K HX 3498 3498 58.3 58.3 125 125 1180 1180 81 33-6 33-6 3 12 2859 47-7 9 1038 86.5 31-8 f 17 *J 26^ . total, 3920 65.3 457 3849 59 46.5 65 5 single, 22 2453 2555 116 136 23.6 20.2 Concluding Table of the Driving Power of Leather Belts. On the whole, it may be concluded that Messrs. Briggs and Towne's data afford a satisfactory basis for the application of general practical rules. Table No. 261 gives particulars of the practical driving-power of leather belts .22 inch thick, per inch wide, based on their data for arcs of contact, of from 90 to 270, assuming 66^5 Ibs. per inch wide, as the maximum working strength. The maximum transmitted working stress, calculated by them by means of formula (12), is given in column 2 ; the 3d and 4th columns were calculated by means of formulas ( i ) and ( 6 ) ; and the horse-power in column 4, multiplied by 33,000, gives column 5. The sum of the tensions, in column 6, is calculated by adding to the transmitted stress in column 2, twice the difference between it and 66^. Thus, for the first case, 66 2 / 3 -32.33 = 34.33, which is the slack tension, and 32.33 + 750 MILL-GEARING. (34.33 x 2)= 101.00, in column 6. The last column gives the resultant stress, by the parallelogram of forces, caused by the tensions of the belt, on the bearings of the shaft. Messrs. Briggs and Towne give many instances in practice, in corrobora- tion of their deductions. Table No. 261. DRIVING-POWER OF LEATHER BELTS. Maximum Working Strength 66% Ibs. per inch wide, single thickness, .22 inch. (Based on Messrs. Briggs and Towne's data. ) A _.. Maximum "\\7rtvlrtMrr Power transmitted, per inch wide. Sum of the Resultant Arcs of Contact. VvorKing Stress trans- mitted, per inch wide. At i foot per second, Velocity of Belt. Per foot of Diameter of Pulley, and per Turn per Minute. on both sides of a Belt, per inch wide. x ressure on the Journals, per inch width of Belt. degrees. Ibs. horse-power. horse-power. foot-lbs. Ibs. Ibs. 90 32.33 .059 .00308 102 101.00 71.42 100 34.80 .06 3 .00331 109 98.53 7547 no 37-07 .067 00353 116 96.26 78.85 120 39.18 .071 .00373 123 94.15 81-53 135 42.06 .076 .00400 132 91.27 84.32 150 44.64 .O8l .00425 140 88.69 85.67 180 49.01 .089 .00467 154 84.32 84.32 210 52.52 .095 .00500 165 80.8 1 78.05 240 55-33 .100 .00527 174 78.00 67.59 270 57-58 .105 .00548 181 75-75 53-56 Note. The thickness of belt is .22 inch for a maximum working strength of 66% Ibs. per inch wide. For any other thickness the data in the table are to be altered in the ratio of . 22 to the thickness. INDIA- RUBBER BELTING. Driving Belts 1 manufactured from American cotton canvas, cemented in layers by vulcanized india-rubber, and coated with the same material, have been tested for strength and adhesion. It is stated that a strip i inch wide bears a tensile stress of 200 Ibs., and that the india-rubber belt possesses about three times the surplus or effective adhesion of leather belts. WEIGHT OF BELT-PULLEYS. The weight of pulleys of the same diameter, varies within much wider limits than that of spur-wheels, not merely because the breadth varies very much, but also that there is greater variation per inch of breadth. The following formulas for the weight of drum-pulleys are, therefore, the expres- sion of average weights for pulleys of medium proportions. For pulleys designedly strong and heavy, up to 30 inches in diameter, the weight per inch of breadth may be as much as 25 per cent, more than the average, or, for particularly light pulleys, as much lighter. As the diameter increases, the percentage of variation diminishes; and for 6-feet or y-feet pulleys, it may never exceed 10 per cent, either way. 1 Manufactured by the North British Rubber Company. BELT-PULLEYS AND BELTS. 751 The author is indebted to Mr. R. Heber Radford for the examples of the actual weights of pulleys as used in the Sheffield, Manchester, and Bradford districts, given in tables No. 262, 263, and 264. The averaged weights per inch wide of the Sheffield and the Manchester finished pulleys, in terms of the diameter ranging from i foot to 4 feet, are found, by plotting, to be expressed by the same formula. For the pulleys of the Bradford district, the formula is slightly different from that; but the chief interest of the Bradford ex- amples, consists in the data they afford of the reduction of the weight of the rough castings, by the operations of turning, boring, and slotting. From these data, the following formulas have been deduced, showing that the increase of weight per inch wide, is simply as the increase of diameter : Table No. 262. WEIGHT OF FINISHED CAST-IRON PULLEYS, SHEFFIELD DISTRICT. (Examples contributed by Mr. R. Heber Radford.) WEIGHT, WEIGHT, tir- j. r turned, bored, and slotted. TIT; J*.L turned, bored, and slotted. Diameter. Width. Total. Per inch wide. Diameter. VYiutn. Total. Per inch wide. feet. ins. inches. Ibs. Ibs. feet. ins. inches. Ibs. Ibs. O 6 28 4-7 2 6 10 224 22.4 3 6 63 10.5 2 6 12 232 19-3 3 10 86 8.6 2 8 9 110 12.2 3 12 102 8.5 2 \\y 2 6 140 23-3 5 7 66 94 3 o 8 120 15.0 6 9 75 8.3 3 o 12 1 08 9.0 1 9 9 80 9.0 3 o H 136 97 1 9 10 84 8.4 3 5 8 160 20.0 2 6/ 2 114 17.5 3 6 6 160 26.7 2 O 10 120 12.0 3 7 12 175 14.6 2 10 I 5 8 I 5 .8 4 o 8 212 26.5 2 16 170 10.6 4 o 8 225 28.1 2 5 6 110 18.3 4 i# 12 338 28.2 Table No. 263. WEIGHT OF FINISHED CAST-IRON PULLEYS, MANCHESTER DISTRICT. (Examples contributed by Mr. R. Heber Radford.) Dia- WEIGHT turned, bored, and slotted. Dia- WEIGHT turned, bored, and slotted. Diameter. Hole. Total Per inch wide. Diameter. Width. meter of Hole. Total. Per inch wide. feet. ins. inches. inches. Ibs. Ibs. feet. ins. inches. inches. Ibs. Ibs. i 4 3 2^ 28 9-3 2 6 6 2^ 105 17-5 2 6 2^ 74 12.3 2 6 8 2 116 14.5 2 6 2 * 76 12.7 3 o 6 2^ 129 21.5 2 8 2^ 84 10.5 3 o 6 2 134 22.3 2 2 6 2 87 14.5 3 6 6 2^ 137 22.8 2 3 6 2^ 93 15-5 3 6 6 2 144 24.0 752 MILL-GEARING. Table No. 264. WEIGHT OF ROUGH CASTINGS, AND FINISHED CAST- IRON PULLEYS, BRADFORD DISTRICT. (Examples contributed by Mr. R. Heber Radford.) WEIGHT, WEIGHT, Diameter. Width. Diameter of Hole. Rough Castings. turned, bored, & slotted, Reduction of Weight. Total. Per inch wide. Total. Per inch wide. feet, inches. inches. inches. Ibs. Ibs. Ibs. Ibs. per cent. 10 3 I# 16 5-3 13 4-3 19 4X 2 21 5.0 18 4.0 14 2 4 2 31 775 27 6.75 13 4 4^ 2X 44 9.26 38 8.0 13-6 6 5X 2^ 63 11.4 53 9.6 16 8 9 2^ 104 1 1.6 92 10.2 11.5 10 10 2^ 132 13.2 118 ii. 8 10.6 Weight of Pulleys per inch wide, in the Lancashire and Yorkshire Districts, from i foot to 4 feet in Diameter. Rough Castings, W = 7.625^- 1.5 (13) Turned and Finished Pulleys, W= 7^-1.75 ( J 4) W=the weight of the pulley in pounds per inch wide. d= the diameter in feet. Note. These formulas are probably applicable for pulleys of from 10 inches to 10 feet in diameter. From the weights of a very large number of rough castings of pulleys, ranging from i foot to 7 feet in diameter, as used in the London district, supplied by Mr. Charles Mackintosh, the following formulas have been deduced. For rough castings above 2 feet in diameter, the weight in- creases simply as the diameter increases. For diameters less than 2 feet, the weight increases with the square of the diameter. The same propor- tional reduction for the finished weight may be applied to the London pulleys as was done to the Lancashire pulleys : Weight of Pulleys per inch wide, in the London District, from i foot to 7 feet in diameter. -o , ( not exceeding 2 feet in diameter, W = 3*/ 2 + 3 ngs \ 2 feet in diameter and upwards,.... W= 12^^-9.5 (15) (16) Turned and ( not exceeding 2 feet in diameter, W = 3 d z - .625 d+ 2.75 ( 17 ; finished pulleys ( 2 feet in diameter and upwards, W = 1 1.625 d- 9.25 ( 18 ) The weights of pulleys, rough as cast, and turned and finished, have been calculated by means of the foregoing formulas, for diameters increasing from 10 inches to 8 feet; given in table No. 265. It is apparent that the London pulleys are much heavier than the country pulleys. The reduc- tion of the weight of the rough castings by turning and finishing, varies from 13 per cent, for 1 2-inch pulleys, to 10^ per cent for 2-feet pulleys, and 9 per cent, for 8-feet pulleys, for the country pulleys ; and from 1 5 to 7 per cent, for the London pulleys. BELT-PULLEYS AND BELTS. 753 Table No. 265. BELT PULLEYS CALCULATED WEIGHTS. Lancashire and Yorkshire. London. Lancashire and Yorkshire. London. Diameter. Weight per inch wide. Weight per inch wide. Diameter. Weight per inch wide. Weight per inch wide. Rough Castings. Turned and Finished. Rough Castings. Turned and Finished. Rough Castings. Turned and Finished. Rough Castings. Turned and Finished. inches. Ibs. Ibs. Ibs. Ibs. feet. Ibs. Ibs. Ibs. Ibs. 10 4.8 4.08 5-i 4-33 2.75 19.5 17.50 24.2 22.2 II 5-5 4.67 5-5 4.68 3 21.4 19.25 27.2 25.1 12 6.1 5-25 6.0 5-13 3-25 23-3 21.00 30.2 27-9 !3 6-7 5-83 6.6 5-67 3-5 25.2 22.75 33-2 30.1 H 7-5 6.42 7-i 6.12 4 29.0 26.25 39-5 36.8 15 8.0 7.00 7-7 6.67 4-5 32-8 29.75 45-5 42.4 16 8.7 7-58 8.3 7.22 5 36.6 33-25 51-5 4 8.1 18 9-9 8.75 9-7 8.51 5-5 40.4 36.75 57-5 53-8 20 II.2 9.91 n-3 10.00 6 44.2 40.25 64.0 60.0 21 11.6 10.50 12.2 10.9 6.5 48.1 43-75 70.0 65.7 feet. 2 13-7 12.25 15.0 13-50 7 51.9 47-25 76-3 71.7 2.25 15-7 14.00 18.1 16.4 7-5 55-7 50-75 82.5 77-3 2-5 17.6 15-75 21. 1 19-3 8 59-5 54.25 90.5 84-3 ROPE-GEARING. 1 Round hemp-ropes working in grooved wheels are occasionally employed instead of belts or toothed wheels for transmitting power from the engine. The fly-wheel is made considerably wider than a spur fly-wheel would be, but rather less than a belt-wheel would be, and V grooves are turned out of the circumference, the sides of which are at an angle of 40, and the number and size of which are regulated by the quantity of power to be taken off. The ropes are usually 5^ and 6^ inches in circumference for larger powers, and 4^ inches for smaller powers. To prevent wear and tear of rope, the circumference or the diameter of a pulley should be at least 30 times that of the rope, and the shafts should be at a distance apart of from 20 to 60 feet. The number of ropes required for the transmission of a given power is determined from the circumferential velocity of the fly-wheel, which is gener- ally between 3000 and 6000 feet per minute. Mr. Durie instances the rope-gearing of Messrs. Nicoll's factory at Dundee, which was erected in 1870. The power of the engine varies from 400 to 425 indicator horse- power. The fly-wheel, 22 feet in diameter, makes 43 turns per minute, with a surface velocity of 2967 feet per minute; it is 4 feet 10 inches wide, and has 18 grooves, each of which is occupied by a 6}^-inch rope, trans- mitting the power of say 23 indicator horse-power. Five ropes are employed to transmit the power to the ground floor, over a 7^-feet pulley; 1 See a paper read by Mr. James Durie, at the Institution of Mechanical Engineers, published in Engineering, November 3, 1876, page 394. 48 754 MILL-GEARING. four ropes drive a 5^-feet pulley on the first floor; and six ropes drive a 5 ^-feet pulley on the second floor. Lastly, on the other side of the engine- shaft, the power is transmitted by three ropes to a weaving-shed, on a 7%- feet pulley. For 23 horse-power, the stress on each rope is ( 33 x 2 3 _ ^ 256 Ibs., less the resistance of the engine. When the ropes become too slack, they are cut and re-spliced, and the work of a rope under such treatment is temporarily performed by the other ropes driving the same pulley. In another example 40 indicator horse-power is disposed of, for each 6^2 -inch rope, at a velocity of 3784 feet per minute; and the equivalent stress is (33 x 4 _ ^ ^49 Ibs. for each rope. 37 8 4 Taking the ultimate strength of a 6^-inch rope at 10 tons, or 22,400 Ibs., it would appear that the working stress is only about i y z per cent, of the ultimate strength; giving a factor of safety, 67. It is believed that an economy of power is effected by the substitution of rope-gearing for toothed-gearing. A 6^-inch rope is equivalent, accord- ing to Mr. Durie, to a leather belt 4 inches wide, for the transmission of work, at say 3000 feet per minute. From some comparative experiments made by Mr. W. A. Pearce, Dundee, it appears that a 6-inch rope in a grooved pulley possessed four times the adhesive resistance to slipping exhibited by a half-worn ungreased 4-inch single belt. The ropes used for gearing are made of carefully selected hemp : the fibres very long, well twisted and laid, yet soft and elastic. The splice should be uniform, of the same diameter as the rope, and 9 or 10 feet long. TRANSMISSION OF POWER BY ROPE TO GREAT DISTANCES. Wire-Ropes. M. Him, in 1850, made many trials with endless bands of steely iron passing over pulleys for the conveyance of power to great distances; but he finally adopted iron wire-ropes, unannealed, working over grooved pulleys of large diameter. M. Umber, 1 in 1859, described the apparatus. The pulleys may be of hard wood; they are formed with a groove slightly rounded, about 2 inches deep and i y z inches wide, lined at the bottom with leather or gutta-percha. They should be at least i metre, or 3.28 feet, in diameter, and should be driven at the greatest practicable speed. A diameter equal to 200 times that of the cable, is the most suitable proportion. The distance apart of the driving and the driven pulleys should be not less than from 130 to 160 feet, and the pulleys may be placed at any greater distance apart. The greater the distance apart, the steadier the movement. The velocity of the rope is about 50 feet per second, or 3000 feet per minute. At this rate, a force of n Ibs. would be equi- valent to i horse-power. The most common sizes of wire-rope employed are as follows : Diameter. Weight per Metre. Weight per Yard. 4 millimetres, or .16 inch. .10 kilogramme. .20 pound. 6 or .24 .17 .34 9 or .35 .31 .62 12 or .47 .45 .90 At Colmar, a force of 47 horse-power is transmitted a distance of 250 1 Annales des Fonts et Chaussees, 1859. TRANSMISSION OF POWER BY ROPE. 755 yards by a ^-inch wire-rope, over two pulleys of 3 metres, or about 10 feet in diameter, making 95 turns per minute. The rope is supported at the middle of the span by pulleys of i metre in diameter. The frictional resistance is less than 3 per cent. The ropes receive a coat of a mixture of oil and tar twice per month, and they wear well. The ropes, in all cases, consist of 36 wires, in six strands of 6 wires each, on a core of hemp. Each strand likewise is formed on a hempen core. The hempen cores are favourable for flexibility. From another account, it appears that 100 horse-power can be transmitted 1 20 yards without any intermediate support, by an endless wire-rope of .40 inch in diameter, over pulleys from 13 to 14 feet in diameter, making 100 turns per minute, equivalent to a velocity of rope of upwards of 4000 feet per minute. For longer distances, the rope is supported at intervals of 1 60 yards by y-feet pulleys. The calculated loss of power in transmitting 120 horse-power is 17^ per cent, or 21 horse-power. The sources of loss are: i st. The resistance of the air to the arms of the wheels. 2d. The resistance of rigidity of the rope in passing over the wheels. 3d. Axle-friction: fixed loss 2^ per cent, for the large pulleys, and i per cent, for every 1000 yards. In an excellent illustrated account of M. Hirn's rope-transmitter, by Mr. H. M. Morrison, 1 he states that soft willow wood succeeds best as lining for the large pulleys. The pulleys were constructed successively of copper, hardwood, and polished cast iron, and were also faced with leather, horn, india-rubber, lignum-vitse, and boxwood ; but all these materials failed, as the facings were soon worn out, and when the groove was of metal or hardwood, and did not itself wear, it destroyed the rope. The tension in the upper rope, he says, is just double that in the lower rope. The best method of changing the direc- tion of transmission of the power, at any point in its course, has been found by experience to be by the use of bevil-wheels. Directing pulleys are not so good for the purpose. For high speeds, the pulleys should be of best cast steel, as iron pulleys may fly to pieces by centrifugal force. Mr. Morrison states that the fine makes of ropes are constructed of 6 strands of 1 2 wires each 72 wires in all; and that in America, the wires are still finer and closer, and as many as 135 in number. Cotton Ropes. Mr. Ramsbottom, in 1863, applied cotton ropes or cords, for driving the traversing cranes at Crewe workshops. 2 The cords are made of soft white cotton, y% inch in diameter when new, and weighing i y z oz. per foot; they soon become reduced to 9/ l6 inch thick by stretching, and they last about eight months. They are, when new, rubbed over with a little tallow and wax. The total lengths of each of the two cords, in three different shops, are respectively 800, 320, and 560 feet. The pulleys over which they are passed are not less than 18 inches in diameter, or 32 diameters of the cord; and in the first of the above shops, alone, the cord makes from 12 to 20 bends according to the machinery in action. The groove of the driving-pulleys is V-shaped, at an angle of 30, and the cord is gripped between the inclined sides. The cord is supported at intervals of 12 or 14 feet by flat slippers of chilled cast iron. The velocity of the cord is 5000 feet per minute; and as some of the pulleys make 1000 turns per minute, they require to be perfectly self-balanced, 1 Proceedings of the Institution of Mechanical Engineers, 1874. 2 See his paper on the subject, in the Proceedings of the Institution of Mechanical Engineers, 1864. 756 MILL-GEARING. in order to run with steadiness and ease. In the overhead traversers, the total leverage is slightly over 3000 to i ; and in lifting a load of 9 tons, the actual pull on the rope is 17 Ibs. A tightening stress on the cord of 109 Ibs., applied by means of a weighted pulley, is found to keep it steady, and to give the required degree of hold on the main driving-pulley. In the wheel-shop, two dozen pairs of blocks and ropes had previously been employed, requiring a large number of labourers to work them; whilst now the two traversing cranes, with two men to work them, do the whole work of the shop, and it is done much more quickly than before. SHAFTING. Shafting is subject to two kinds of stress: transverse and torsional. The dimensions of shafts are settled by conditions of stiffness, or resistance to deflection, under the action of either kind of stress. TRANSVERSE DEFLECTION OF SHAFTS. The deflections of cast-iron, wrought-iron, and steel bars or shafts, loaded at the middle, are given by formulas (8) and (10), page 564; (5) and (6), page 590; and (3) and (4), page 6 19. The deflection under the same weight uniformly distributed, is $ths of that under the weight when placed at the middle. Altering some of the measures, let D = the deflection, in inches. W the weight, in pounds. /= the length or distance between centres of bearings, in feet. d= the diameter of the round shaft, in inches. b the side of the square shaft, in inches. The modified formulas, adapted for uniformly distributed weight, are given in two series; first, for shafts simply supported at the ends; second, for shafts fixed at both ends, as are continuous shafts. In settling the divisors for the second series, it is assumed that the deflections are one-half of those of the first series. The deflections, actually, are not so great as one-half; and margin is thus left for deflection arising from the mode of application of the torsional force, and for the excess of deflection at the loose end of the line of shafting. Transverse Deflection of Shafts, under Uniformly Distributed Weight. Supported at the ends. Fixed at the ends. W /3 W /3 Cast-iron shafts : Round, D = __ : D = _ _ .... ( i ) 39,400 d^ 79,000 d 4 W /3 W /3 Square, D = W / ; D = W / L ... (2) 58,000 b^ 116,000 4 Wrought-iron shafts:-Round,D = i; D=_ 4 ... (3) Square, D = ; D = _ ... (4) 97,500 b^ 195,000 b 4 Steel shafts: Round, D = -^-; D = ^ l " ...... (5) 78,800 158,000 T^ W / 3 ~ W / 3 , , \ Square, D = -- ; D = ...... (6) 116,000 232,000 TRANSVERSE DEFLECTION OF SHAFTS. 757 The working limit of deflection is taken as */ IOO inch per foot of length, or of the distance of bearings; and the limiting value of D in inches is 100 ' By substitution of this value for D ; and by reduction and inversion : _ The Diameter and the Side for the Limiting Transverse Deflection. Supported at the ends. Fixed at the ends. W / 2 IV / 2 Cast-iron shafts : Round, d* = : d* = ........ ( 7 ) 394 790 W / 2 W 7 2 W / 2 W 7 2 Wrought-iron shafts : Round, d* = L : d 4 = . . . ( o ) 664 1330 Square, t< = ; 6*, ......... ( IO ) 975 195 W 7 2 W / 2 Steel shafts: Round, A torsional deflection of i, in a length equal to 20 diameters of the shaft, is a good working limit of deflection ; that is, J /36o th part of a turn, or .00278 turn, for 20 diameters. Now, for cast-iron, W R = - ; wrought iron, W R = l6 ' 6o * D ' ; steel, W R .34.30* P. and> substi . / / tuting .00278 for D', and ^ for /, in these equations, and reducing: The Working Moment of the Force, and the Diameter, for the Limiting Torsional Deflection. For cast iron, ....... W'R=i8.5 o ? W 03 O ill 6 S bo'^3 bJD^ HH o O "^ ^ TJ- \JT\ vr> u-i to S - L-O 9 TJ" ~ ySLl &-S 2 w>i |3' OO OO m ON ON "^ CJ co \O ON m ON OO \O P-, ,-, co \o ^O Is ON J^ i-. 00 CO i-. co OO ON CO ON CO u~> M u"i oo' <* rfoo NvOOOOOnoON .S S __. X 2" U V9 C ^H 80 feet. 74 feet. from the centre of the grate, J Total distance from centre of grate ) to base of chimney, ............ j 7 " Height of chimney above level of ) r . floor, .............................. J 96 feet 9 inches ' A Green's fuel-economizer was placed in the main flue; it had 12 rows of 4^-inch cast-iron pipes, 8 feet 9 inches long, placed vertically 84 tubes in all having a collective heating surface of 850 square feet, exclusive of the connecting pipes at top and bottom. The feed-water was passed through the economizer on its way to the boiler, and absorbed a portion of the waste heat. The fire-grates were tried at 2 lengths, 6 feet and 4 feet. The shorter grate gave the more economical result, but it generated steam less rapidly. Three modes of firing were tried; spreading, coking, and alternate firing. With round coal, on the whole, the greatest duty was obtained by coking firing, with the least smoke. With slack, alternate-side firing had the advantage. Fires of different thicknesses were tried : 6 inches, 9 inches, and 1 2 inches. It was found that 9 inches was better than 6 inches, and 12 inches better than 9 inches. Air admitted at the bridge gave a slightly better result than by the door; and the admission of air in small quantity on the coking system, prevented smoke. The doors were double, slotted on the outside, and pierced with holes on the inner side. The maximum area of opening was 31^ square inches for each door, being at the rate of 2 square inches per square foot for the 6-feet grates, and 3 square inches for the 4-feet grates. The amount of opening was regulated by a slide. The standard fire adopted for trial was 1 2 inches thick, of round coal, treated on the coking system, with a little air admitted above the grate, for a minute or so after charging. The water was evaporated under atmospheric pressure. The quantity of refuse from the Hindley Yard coal, averaged in the trials with the marine boiler, to be afterwards described, 2.8 per cent, of clinker, 2.8 per cent, of ash, and .8 per cent, of soot; in all, 6.4 per cent. Making allowance for the difference of soot, the total refuse may be taken, in the trials of the stationary boilers, at 6 per cent. General Deductions. The advantage of the 4-feet grate over the 6-feet grate, was manifested by comparative trials with round coal 12 inches thick, LANCASHIRE STATIONARY BOILERS AT WIGAN. 773 and slack 9 inches thick. With the 4-feet grate, the evaporative efficiency, taking averages, was 9 per cent, greater than with the 6-feet grate; though the rapidity of evaporation was 15 per cent, less, at the same time that 19^ per cent, more coal was burned per square foot per hour. When equal quantities of coal were burned per hour, the fires being 12 inches thick, 8 per cent, more efficiency and 12 per cent, greater rapidity of evaporation were obtained from the shorter grate. Thus : Coking Firing. Length of grate, 6 feet. 4 feet. State of damper, two-thirds closed, fully open. Coal per hour, 4.0 cwts. 4.14 cwts. Coal per square foot of grate per hour, 14 Ibs. 23 Ibs. Water at 100 evaporated per hour, 65 cubic feet. 7 2. 6 cubic feet. Water at 212 per pound of coal, 10.10 Ibs. 10.91 Ibs. Smoke per hour: Very light, 4.3 minutes. 4. i minutes. Brown, 0.4 0.3 Black, o.o ,, o.o To compare the performances with coking and spreading firing, having i2-inch fires for round coal, and 9-inch fires with slack: Whilst, with round coal, the rapidity of evaporation was the same with both modes of firing, the efficiency was from 3 to 4 per cent, greater with coking. With slack, on the contrary, the spreading fire evaporated a fourth more water per hour than the coking fire, though with 4^ per cent, less efficiency. With thicknesses of coking fire, 6 inches, 9 inches, and 12 inches, for round coal; and 6 inches and 9 inches for slack; the results were in all respects decidedly in favour of the thicker fires rather than the thinner fires. Comparing the thinnest and the thickest fires, from 5^ to 20 per cent, more water was evaporated per hour by the thickest fires, and from 1 1 to 1 8 per cent, more per pound of fuel. The effect of the admission of air above the grate, continuously or intermittently, for the prevention of smoke, as compared with that of its non-admission, was ascertained with round coal, and with slack. The averaged results showed that by admitting the air above, the evaporative efficiency was increased 7 per cent. ; but that the rapidity of evaporation was diminished 3}^ per cent. Comparing the admission of air above the fuel at the door, and at the bridge through a perforated cast-iron plate; it was found that the admission at the bridge made a better performance, by about 2^ per cent., than at the door. To try the effect of increasing the supply of air above the fuel, the door- frame was perforated to give an additional square inch of air-way per foot of grate, making up 3 square inches; an allowance of i square inch was also provided at the bridge. Round coal was burned on the coking system, 1 2 inches thick, on 6-feet grates, with a constant admission of air above the fuel. When the supply. by the door was increased from 2 inches to 3 inches per square foot of grate, the evaporative efficiency fell oft 8^ per cent., and the rapidity 3 per cent. When an extra inch was supplied at the bridge, making up 4 square inches per foot of grate, the evaporative efficiency only fell off 0.65 per cent., and the rapidity i^ per 774 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. cent. The effect of this evidence is, that the bridge is the better place for the admission of air, and that if the air be admitted by the bridge alone, the area of supply may be beneficially raised to 4 square inches per square foot of grate. Comparing the effect of the admission of air in a body, undivided, with that of its admission in streams, on a 6-feet grate, with coking fires 12 inches thick of round coal; there was 6j^ per cent, of loss of efficiency by the admission in a body, though the smoke was equally well prevented. Mr. Fletcher concludes that the greatest rapidity of evaporation was obtained, when the passages for the admission of air above the fuel were constantly closed; that the next degree of rapidity was obtained when they were open only for a short time after charging, and the lowest when they were kept open continuously. He also concludes that, whilst, in realizing the highest power of a free-burning and gaseous coal, smoke is prevented ; yet, in realizing the highest power of the boiler, smoke is made. In burning slack, smoke was prevented as successfully as in burning round coal, though its evaporative efficiency was from i to i y% Ibs. of water per pound of fuel, less than with round coal. To work out the problem of firing slack without smoke, and without loss of rapidity of evaporation; trials were made at the boilers of 16 mills, when the slack was fired on the alternate-side system. No alterations were made in the furnaces in preparation for these trials; in many instances, the fire-doors had no air-passages through them. The grates were from 3 feet 7 inches to 7 feet long; they averaged 6 feet in length. Number of boilers fired, 65 boilers. Slack burned per boiler per week of 60 hours, 17-35 tons. Slack per square foot of grate per hour, 1 9. 2 5 Ibs. Smoke per hour : Very light, 11.5 minutes. Brown, 2.3 Black, 0.3 14.1 In 1 2 instances, no black smoke whatever was made. It is said that the steam was as well kept up, and the speed of the engines as well maintained, as before the trials were made. COMPARATIVE PERFORMANCE OF THE STATIONARY BOILERS AT WIGAN. There were made altogether about two hundred and ninety trials with the three boilers, of which sixty may be regarded as comparative trials of the boilers. The results of these sixty trials are embodied in the table No. 270, page 776. The second part gives the best results that had been obtained from each boiler, supplied with round coal, on the coking system; and with air admitted through the doors for a few minutes after charging. Suffice it, meantime, to remark that the performance of the double-flue boilers amounted practically to the same as that of the water-tube boiler. Thus, LANCASHIRE STATIONARY BOILERS AT WIGAN. 775 the means of the double-flue boilers compare as follows with the results of the conical water-tube boiler : AVERAGES OF SIXTY TRIALS, WITHOUT ECONOMIZER Water at 100 consumed Water at 212 per hour. per Ib. of coal. Double-flue boilers, 79.65 cubic feet 10.31 Ibs. Conical water-tube boiler, 78.95 10.34,, BEST RESULTS OBTAINED : WITHOUT ECONOMIZER Double-flue boilers, 81.92 10.86,, Conical water-tube boiler, 79. 1 7 10.58,, WITH ECONOMIZER Double-flue boilers, 90.72 11.56,, Conical water-tube boiler, 86.31 11.82 In doing the same work, it is to be noted that the water-tube boiler was 2 feet shorter, and 6 inches less in diameter, than the double-flue; and that it had 48 square feet, or 6 per cent, less area of heating surface. A trial was made with the object of testing the comparative merits of the plain double-flue and the water-tube flue, by shutting off the draught from the external flues, and leading it direct from the internal flues to the chimney, with the following results (grates 6 feet long, coking firing, 12 inches thick): WITHOUT ECONOMIZER Water at 100 consumed Water at 212 per hour. per Ib. of coal. Iron double-flues, 82.97 cubic feet 8.23 Ibs. Water-tube flue, 80.00 8.50,, WITH ECONOMIZER Iron double-flues, 98.85 10.08,, Water-tube flue, 89.08 10.16 Showing that the double flues, having 33 square feet, or nearly 8 per cent, more heating surface than the water-tube flue, evaporated more water per hour, but with rather less efficiency than the water-tube flue. The evaporative power of the boilers was rather increased than diminished by the closing of the external flues, though there was a sacrifice of evaporative efficiency. Water-tubes. Four water-tubes were inserted in each flue of the iron-flue boiler, 5 ^ inches in diameter inside, and 2 feet 7 ^ inches long, making an addition of 30 square feet, or 6^ per cent., to the flue-heating surface, or 4 per cent, of the total heating surface. The result of the insertion showed equal rapidity of evaporation, and a gain of 3 per cent, in efficiency ; as follows : Water at 100 Water at 212 F. consumed per hour. per pound of coal. Without water-tubes, 9 I - I 5 cubic feet i o. 43 Ibs. With water-tubes, 91.12 io-77 776 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. w RMANCE O] H < c^ M H h4 I (2 O U Q 3 O X-N .^ W p/^ in py^ rH "c Q i >* PH & fa $ 1 ] SULTS 1 i 1 W P4 w w ^ g CK ffi ^ | H 1 GH U CJ T d (M 3 rt H t^O ui O O O <5 o o odd a co . oo 10 ooo CO ci N M HH' vo cocOTf h- 1 H- 1 (-H HH B b 2 cf N O O w o o d o 1 1 ON i HH vn ' "tf- co T}- TJ- \O Tj- CO 1-1 xovO vo vO* jr S.I S,l-a HN t^ O\ CX5 t-^r^ ^ CO tH t-l OO t>xt^ W O *O to I r-. vo vo vo ThOO rt- N co C< co co gS'S'sIS >< do o" d d d cf\ 1 cfv odd O O O O ^ws M s a'ss.c itiiuS * *# JW -"' CO u vr> os t^ O O vQ -" fe t>. 1-^vO to 10 CO ^ M" ci ci coroco O C/3 is) O O >-" OO !>. OO Tf vr> rf C^' N CO CO CO vo r-00 r}- O N 9 - CO co co co kijij TO? tJ^co^-tH WON jW < t^t^N VO^J-VT) rj M . cr> "t t^ w VO !> >-> CO CO oo oo t^i>. t-. ON M \f) o 00 rj- O\ ^t- co' o' oo o\ t-^OO t^ t" JU*lL 3IIJ* . M 'co t/i VO 1-1 Tt- HH t>. N N Tj- w co Os O\ i OO I-H Os t^> c3 vo co -5t UJ o ioo vr> N* ri ci VO* 1 vo" c4 CJ N n- TJ- T^- Tj- o^Jj hil t^sl * vn Ji ^ - - o. co N ~ XO CO N ^ 1 1 1 '^ ... . . . . 1 i i i *rt *3 : : ; # :& : : : . i5 2 i h O U Q O Q Z < oT 5- PQ ROUND COAL, COKING FIRING, 12 ins. WITHOUT ECONOMIZER. Double-flue boiler, with iron tubes.... Double-flue boiler, with steel tubes.... Conical water-tube boiler Double-flue boiler, with iron tubes.... Double-flue boiler, with steel tubes.... Conical water-tube boiler SPREADING FIRING, 12 inches thi< Double-flue boiler, with iron tubes.... Double- flue boiler, with steel tubes (no Conical water-tube boiler Double-flue boiler, with iron tubes.... Double-flue boiler, with steel tubes.... Conical water-tube boiler If! J W OJ X . o ^ ^^^ s, fi*5S 2 ^X) o S3 ^S|< C q3 > i i'rt 33.H ggg QQu LANCASHIRE STATIONARY BOILERS AT WIGAN. 777 Smoke per Hour. OOO OOO OOOO d d d o" d d do o" d ooo odd o o oooo d d o" d d d 3 ONOO OOO Tt-O O w d d d d d o" do" o" d 1 1 CO O vo o" d O vo r^ O vo ^t- o" d d d d d I 1 tCvoO ON N co OOCO COOO N d d O Tj-vd w c4 -I -! 1 CJ vO VO vO t^ ON 1-1 vo N co n vo do" - CON r $0$ co N vo N N N N H- oo r- ^ Tj-vO ON N N N OO ON vo t^ d o" ON >-< M d d d d vo "-i OO VO^OO vo voOO vo J41tl N ON >-i OO "-I OO OOOON v ONVO r^Tht^^l- N N 1-1 O N ON ~ co N CO OO VO vo ON CO O ONVO VO CO CO Tt- CO CO CO CO CO CO 3 u Illtl D t^ vo N N **3" c$ ON ^J- t"> O ^ O vo N i- OO HH vo N i-< O ONOO oo t^ t^ r-* oo oo t^oo o ^S VO t^ CO N co T}- vo vovd ON OO t- ONOO OO 00 IHjl vovO ON vo ONVO co t^*. O O ON t-^OO HH ci i-I O ON O* O _, N N IN N W N N rj-NN ONI vOMvoON OO ON Tj~ i t^. ON O ON "^~ co . .... O vO OO 1 co O W ON ON O fX) |_| M NN M-ll-lM "jii Tj" ON CO O CO O t^VQ \O t*^ vovO 00 OO t^ >-i VO rf VO vO N vO w vo t^ TJ- ON HH vo vot- rf CO voj- 4 TJ- iis.ll i^OQ ^ | - 1 ^' ww^O t^. TJ- VO ONVO OO CO Tj- N I CO t- Tj-OQ VO CO 1 CO >-< vo CO CO TT o vo' rt-vd N N N rj- co TJ- TJ- H! id xo J2 K S S*"M '' III vo CO " ~ \ i i M o K M A u Q O % Q 2 jj 1 3 : : i : i i i i i WITH ECONOMIZER. Double-flue boiler, with iron tubes Double-flue boiler, with steel tubes Conical water-tube boiler Double-flue boiler, with iron tubes (no trials) Double-flue boiler, with steel tubes Conical water-tube boiler Means of the Two Sizes of Grate. Double-flue boiler, with iron tubes Double-flue boiler, with steel tubes Conical water-tube boiler Average of the three boilers "^ - % i ; i ; ; ; ; % ROUND COAL, COKING FIRING, 12 WITHOUT ECONOMIZER. Double-flue boiler, with iron tubes Double-flue boiler, with steel tubes Conical water-tube boiler Double-flue boiler, with iron tubes Double-flue boiler, with steel tubes Conical water-tube boiler Means of the Two Sizes of Gr Double-flue boiler, with iron tubes Double-flue boiler, with steel tubes Conical water-tube boiler Average of the three boile 7/8 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Green's Fuel-Economizer. From the average results of various compara- tive trials, burning round coal and slack, and with coking firing, on 6-feet and 4-feet grates, it appeared that, burning equal quantities of coal per hour, the rapidity of evaporation was increased 9.3 per cent., and the efficiency 10 per cent, by the addition of the economizer. Temperature of the Products of Combustion, and of the Feed-water, when the water is passed through the Economizer. Average results : With 6-feet Grate. With 4 -feet Grate. Before. After. Before. After. Temperature of gases in the flues before ) , o o and after traversing the economizer, j 49 34 3 Temperature of the feed-water, 47 157 41 137 Whence it follows that, to raise the temperature of the feed-water through 100 F., the gases were cooled down through an average of 250 F. Temperature of the Products of Combustion, without the Economizer. The variations to which the temperature of the escaping gases is subject, are illustrated in the annexed statement, showing the temperature with different thicknesses of fire, burning round coal with coking firing, without the economizer. ROUND COAL. With 6-feet Grate. Thickness of fires, inches 12 9 6 Coal per foot of grate per hour,... pounds 19 20 20 Water at 1 00 evaporated per hour, cu. feet 85.7 85.5 81.2 Water at 212 per pound of coal, pounds 10.12 9.79 9.16 Temperature in chimney-flue, 630 556 539 Smoke per hour Very light, minutes 2.0 o.o 0.5 Brown, minutes 0.4 o.o o.i Black,.. ...minutes o.o o.o o.o With 4-feet Grate. 12 9 6 23 24 24 72.8 70.7 6l.7 10.90 9.95 9.21 505 451 445 2.8 0.4 o.o O.I 0.0 0.0 O.O 0.0 O.O It is shown that the temperature in the chimney flue is lower with the 4-feet grate, than with the 6-feet grate; it averages 107 lower, and correspond- ingly, the evaporative efficiency averages higher. But, with the same grate, both the evaporative efficiency and the temperature become less with the thinner fire, due, no doubt, as Mr. Fletcher points out, to the passage of a greater surplus of air through the thinner fire. Volume of Air Supply and Products of Combustion. The volume of air entering the ash-pit and passing through the grate, when the doors were closed, was found, by means of Biram's anemometer, to be, for grates 4 feet long, with fires 9 inches thick, from 245 to 250 cubic feet per pound of coal burned ; the average velocity of entrance into the ash-pit, which was 2 feet square, having been observed to be 9.3 feet per second. As the composition of the coals has not been given, it may only be assumed roughly, that the coal chemically consumed 140 cubic feet of air for the combustion of one pound ; and, if the above-noted quantities of air supplied be exact, it would follow that a surplus of air amounting to from 75 to 80 per cent, was present. This is questionable, and it is probable, in the scarcity of data, that the observations for velocity were made at the centre of the draught- way, where the velocity was a maximum, and that no correction was made for the inferior velocities at other parts of the section. LANCASHIRE STATIONARY BOILERS AT WIGAN. 779 From an analysis of the products of combustion in the chimney, it appeared that there was no appreciable quantity of carbonic oxide present. Trials under Steam of more than one Atmosphere of Pressure. As the experiments at Wigan were made under one atmosphere of pressure, a few trials were made under an effective pressure of 40 Ibs. per square inch, with the following comparative results : At atmospheric At 40 Ibs. pressure. per square inch. Water at 100 evaporated per hour, cubic feet, 83.6 80.4 Water at 212 per pound of coal, pounds, 10.76 9.53 showing a reduction of i ^ pounds of water in evaporative efficiency, at the higher pressure, which is more or less accounted for, first, by the greater total heat of steam at the higher pressure, requiring more fuel-heat for its formation; secondly, by the higher temperature of the water in the boiler at the higher pressure, which would to some extent check the absorption of the last portions of heat from the gases before they escaped into the chimney- flue. Still, the difference is excessive. Trials with D. K. ClarKs Steam-induction Apparatus for the Prevention of Smoke. In Clark's smoke-preventer, the air was admitted through the door, regulated in quantity by a flap-valve, and deflected upwards upon an air-plate placed across the furnace above the dead-plate, and against the furnace-front. Steam from an auxiliary boiler was conducted by a pipe above the air-plate, and was discharged in four jets over the fire, towards the bridge. In passing over and beyond the edge of the air-plate, the steam induced the air which passed forward from the door under the air- plate, and carried it onward above the fire thus forcibly mingling it with the combustible gases, and at the same time increasing the draught. The trials were made in three ways ist, with the jets and the air-valves constantly open; 2d, with the jets and the air-valves open for a minute or so only, after each charge; 3d, with the jets constantly open, while the air- valves were closed. It was found that, when the jets were constantly open, the quantity of steam consumed from the auxiliary boiler to supply them amounted to one-thirtieth of the quantity of water evaporated. The following are the comparative results of performance on 6-feet grates, with the steam-inductor, and with the ordinary fire-door and the split bridge. The jets and air-valves of the steam-inductor were open for a minute or so only after each charge; and, taking the interval between the charges at fifteen minutes, it is evident that the quantity of steam consumed by the nozzles was insignificant : Without Economizer. ROUND COAL, 6-feet Grate ; Firing, 12 inches thick. Coking. Spreading. Coal per sq. foot of grate per hour, steam inductor, pounds, 18.77 23.86 Water at .oo'perhour, { Smoke per hour, ordinary door Very light, .......................................... minutes, 3.1 5.3 Brown, ................................................ 0.8 4.9 Black, ................................................ o.o 3.3 With Economizer. Coking. 18.20 91.77 ;8o EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Without Economizer. SLACK, 6 feet Grate; Firing, 9 inches thick. Coking. Spreading. Coalpersq. foot of grate per hour, steam-inductor, pounds, 18.7 With Economizer. Coking. 20.7 IOI OQ Waterat 100 per hour, j g^^^ CUblC feet ' 7 6 *f s 71 C8 Wfltprat2T2nernoundroal \ steam-inductor, pounds, 9.17 Water at 212 per pound coal, | gplit bridge 3 gg 10.65 9-7 "1 Smoke per hour, ordinary door Very light minutes o "> '"J O O Brown . ... ... oo o o Black. . o.o o.o It is shown that, with round coal and coking firing, there was no advantage by the steam-inductor, except in reducing the smoke, whilst the evaporation was rather less rapid than with the ordinary door; but that, with spreading firing, the evaporation was more rapid by 17 percent. With slack, the evaporation was decidedly superior, both in rapidity and efficiency, with the steam-inductor. Trials with Self-feeding Fire-grates ( Vicar? System). The fire-bars are impressed with a slow reciprocating movement, the effect of which is to cause the fuel to travel gradually and steadily from the front to the back. The comparative performances of Vicars' grate and the ordinary grate, are shown by the subjoined results: Waterat .oo'perhour, ROUND COAL, 4-feet Grate. *"' Waterat 3I2 - per pound coal, 9! 6 feet. 77-94 67.56 9.52 With Economizer. 78.97 71.58 10.56 9-23 SLACK, length of Grate, 4 feet. S Vicars' grate,.... cubic feet, 64.95 coking firing,.... 60.72 spreading ... 77.72 ( Vicars' grate, pounds, 9.82 Waterat 212 per pound coal, < coking firing, 9.58 ( spreading 8.94 It is seen that, with slack, Vicars' grate had the advantage both in rapidity and efficiency of evaporation over hand-firing coking, and that it also evaporated more rapidly with round coal, but less efficiently; though, if the rapidity had been the same, the efficiency would probably also have been the same. Compared with spreading firing, Vicars' grate was superior in evaporative efficiency as well as in the prevention of smoke, though it did not evaporate so rapidly. In burning large quantities of coal continuously on Vicars' grate, the rapidity of evaporation fell off in the longer trials, and to some extent also the efficiency. The 6-feet grates were very little behind the 4-feet grates in efficiency. Comparative Performance in Calm and Windy Weather. A high wind invariably increased the performance. The average results under all con- ditions showed that 10 per cent, more coal and 12 per cent, more water were consumed, and that the evaporative efficiency was increased 4.4 per cent. MARINE BOILER AT WIGAN. 78 1 Comparative Performance when the Natural Draught was increased by the aid of an Auxiliary Furnace. An auxiliary furnace was put in action at the bottom of the chimney, so as to increase the draught. The effect, taking the mean of a number of trials, was to raise the rapidity of evaporation from 72.96 to 84.09 cubic feet of water at 100, per hour, whilst the water evaporated per pound of fuel was raised from 10.77 to 10.81 pounds. The mean efficiency, thus slightly raised, was in fact an average of two opposite effects ; for, with round coal, the efficiency was reduced, whilst with slack it was increased, by the additional draught. Mr. Fletcher's Conclusions. Mr. Fletcher draws the following conclusions from the experiments on stationary boilers at Wigan: ist. That the coals of the South Lancashire and Cheshire district, though of a bituminous and free-burning character, can be economically burned in the ordinary class of mill-boiler, without smoke. 2d. That the double-flue Lancashire boiler, whether with steel or.- iron flues, and the Galloway, or water-tube boiler, are practically equal in performance; and that both of them develop, when suitably set and fired, high economic results. 3d. That external brickwork flues, though adding but little to the yield of steam, save fuel. 4th. That the addition of a feed-water heater or economizer is a decided advantage, not only in increasing the yield of steam, but also in diminishing the annual cost of boiler repairs and coal. EVAPORATIVE PERFORMANCE OF SOUTH LANCASHIRE AND CHESHIRE COALS, IN A MARINE BOILER, AT WIGAN. I866-68. 1 The marine-boiler was a copy of the test-boiler at Keyham Dockyard. The shell was rectangular, 5 feet wide, for two furnaces, 7 feet 8 inches long, and 8 feet 10 inches high. The furnaces were i foot 8^5 inches wide, 2 feet 83/8 inches high at the front, rising to 3 feet high at the back, and 6 feet deep from front to back or tube plate. There were 124 flue-tubes, 2% inches in diameter inside, and 5 feet long, placed at a pitch of 3^3 inches from centre to centre. The chimney was 18 inches in diameter, and 52 feet 8 inches high above the boiler, or 59 feet 8 inches above the level of the grates. The proportions of furnaces which were finally adopted, after many preliminary trials, were as follows : Dead-plate, i o inches long, 1 6 inches below the crown of the furnace; grates, 3 feet long, inclined 24 inch to a foot; bars, ^ inch thick, air-spaces y z inch; bridge built up to a level 9 inches below the crown, and 9^ inches above the grate. The fire-doors were fitted with a sliding grid for the admission of air into a perforated box inside the door. In the first instance, there were 730 perfora- tions, giving an area of 33 square inches, or 3.2 inches per square foot of grate. They were afterwards reduced to 342 in number, 16^ square inches in area, or 1.6 inch per foot of grate. During the preliminary experiments, it was found of advantage to reduce the length of grate from 4 feet to 3 feet, to adopt a blind dead-plate in preference to a perforated one, and to slightly lower the grate. Fires of 6 inches, 9 inches, 12 inches, and 14 inches in thickness were tried; the greater the thickness the better was the performance. The firing was tried on the spreading and on the coking systems. 1 The author is indebted for the particulars of these trials to Mr. Lavington E. Fletcher's Report. See note, page 771. 782 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Coking-firing was adopted as the standard method, with fires of 14 inches and 12 inches thickness. The furnaces were charged alternately, and the entrance for air through the door was allowed to remain open for a few minutes after each charge was delivered, for the prevention of smoke. For each trial, 1000 Ibs. of round coal was consumed, lasting 3 hours 27 minutes as an average; average rate of consumption, 290 Ibs. per hour, or 28 Ibs. per square foot of grate per hour. The feed-water was supplied at ordinary temperatures. The steam was generated under one atmosphere of pressure, and escaped direct into the air. Total grate-area, 10.3 square feet. Total heating surface : Plate, above the grate,.... 95 square feet. Tubes, outside surface, ... 413 508 Ratio of grate-area to heating surface, say i to 50. For some trials, an inverted bridge was added at the back of the furnace, 9 inches clear of the first bridge. By thwarting the current, it was instru- mental in preventing smoke, and in slightly increasing the evaporative efficiency, by 1.7 per cent. ; though at a loss of 7^ per cent, of evaporative rapidity. The i4-inch fire excelled the 9-inch fire, burning coal at the rate of 27 Ibs. per square foot, by 7 ^ per cent, for rapidity and efficiency of evaporation. Comparing the coking and the spreading systems, there was 6% per cent, gain by the coking system in efficiency, with a loss of 10 per cent, in rapidity. When air was shut off at the doorway, the smoke-making was accompanied by 4 per cent, loss of efficiency, with a small advance of i ^ per cent, of rapidity. When the trials were prolonged, to burn 1500 Ibs. of coal, as against the standard of 1000 Ibs., the rate of consumption of fuel was i Ib. more per square foot per hour, and of water 2^ per cent, less; the average efficiency was reduced 5 per cent. Table No. 271 shows the general results arrived at by Messrs. Richardson and Fletcher, compiled from the Report of Mr. Fletcher. The vacuum in the chimney [at the base, probably] was observed to vary from ^ inch to fully 7/ j6 inch of water; and in the flame-box from % inch to fully s / I6 inch. The fires were maintained at 14 inches thick, and the coal was stoked, on the coking plan, in charges of from 29 to 38 Ibs., at intervals of from ii to 17 minutes. The perforations in the fire-doors were opened intermittently, and the doors were opened a little, occasionally, after firing. Each trial lasted for from 3 to 4 hours. The quantity of ash varied from 1^/2, to 7 per cent., and of clinker from 0.6 to 3 per cent. From the table, it appears that the quantity of water evaporated varied from 44.12 to 51.63 cubic feet per hour, at the rate of from 10.37 to 12.54 Ibs., at 2 1 2, per pound of coal, averaging 11.54 Ibs.; and that the coal was burned at the rate of from 25^ to 31^ Ibs. per square foot of grate per hour. The duration of the smoke, which was very light, varied from 0.2 to 6 minutes per hour; the mean duration was 2.4 minutes in the hour. A mixture of Hindley Yard coal and Welsh coal-dust, in the proportion MARINE BOILER AT WIGAN. of 2 to i, was tried, and the results are given in the table, showing an evaporation of 11.83 Ibs. of water per pound of the fuel, and at the rate of 41.38 cubic feet per hour. Table No. 271. SOUTH LANCASHIRE AND CHESHIRE COALS RESULTS OF TRIALS IN A MARINE BOILER AT WIGAN. 1866-68. (Compiled from the Report of Mr. Lavington E. Fletcher to the Association for the Prevention of Steam-Boiler Explosions.) Total area of fire-grates, 10.3 square feet. COAL. Coal Con- sumed iCur. Coal per Square Foot of Grate per Hour. Water Con- sumed from 100 per Hour. Water per Square Foot of Grate per Hour. Water Evapor- ated from 212 per Pound of Coal. Smoke Hour. FIRST SERIES OF TRIALS. Hindley Yard. cwt. 2 ?2 Ibs. 2t 24 cub. feet. 4.6 17 cub. feet. 4. 4.8 Ibs. 12 7Q minutes, very light O 2 Worsley Top Four Feet 2*88 2 I 36 4.8 SO 2 O4. ^jy IO 77 4. O Upper Crumbouke 2 64 28 74. 48 17 6-vq A 57 II 71 1 7 Lower Crumbouke 2 47 26 4.1 q.0. 1^ 48 60 4. 72 11.^1 12 4.^ i'8 Upper Three Yards 2 48 27 OO 46 26 4. 4Q 1 ^-4i 1 1 60 a 1 ? Six Feet Rams 2 44. 26 50 44. 7S 4-71 1 1 74. o 2 O Great Seven Feet 2 77 2Q 71 "\1.74 *H J x 4 08 II 71 C.Q Blackrod Yard . . 2 71 2C 14. AC 77 4. 4.O 12 l8 2 4. Pemberton Four Feet 2 64 m y m ie \- JQ 87 ej 67 SOI .we II 71 2 Q Haigh Yard. 2 7S 2C C-J 4.7 78 4 oo 12 C4. O 1J Furnace Mine 2 66 28 07 44 4Q 4. 72 10 40 I 4. Bickerstaffe Four Feet. . 2 ^4. ^u.yj 27 67 4S 28 II 08 O O Rushy Park and Little Delf, mixed Ince mixed . . 2 . 2 6l 30.29 28 64 50.67 4.6 ?2 4-92 A r T II.2Q IO QQ 4-3 i 6 Arley Mine 2 26 24. 46 44. 12 H-O 1 4. 28 12 l8 O 4 Average results of 15 samples ) of coal . . ) 2-55 27.63 47-25 4-59 H-54 2.4 Mixture of 2 Hindley Yard coal ) and i Welsh coal-dust \ 2.21 24.OO 41.38 4.02 11.83 0.0 I 2 3 4 5 6 7 NOTE. The quantities in column 6 have been recalculated. D. K. C. The effect of reducing the flue-surface w^s tried by plugging up one- half of the number of flue-tubes, in alternate diagonal rows, so that the tube- surface was reduced by 206.5 square feet. The comparative results obtained with 1 2-inch fires were as follows : Flue-tubes Half the Tubes all open. plugged up. Coal per square foot of grate per hour, Ibs., 25 24 Water at 100 evaporated per hour, cubic feet, 45.78 43oi Water per pound of coal, as supplied at 212, Ibs., 12.41 12.23 Smoke per hour very light, minutes, 2.8 8.0 Showing that with half the tubes, the performance was nearly as good as with them all open. EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Messrs. Nicoll & Lynn, for the Board of Admiralty, made two inde- pendent series of trials of the South Lancashire and Cheshire coals ; in the second of which the draught was increased by a steam-jet from a neighbouring boiler. The average results of these trials are placed beside those of Mr. Fletcher's trials in the annexed table, No. 272. Table No. 272. SOUTH LANCASHIRE AND CHESHIRE COALS SUMMARY RESULTS OF PERFORMANCE IN THE WIGAN MARINE BOILER, In three series of trials, by Messrs. Richardson & Fletcher, and by Messrs. Nicoll & Lynn. __ Particulars. Trials by Messrs. Richardson and Fletcher. Trials of Messrs. Nicoll and Lynn. Without Jet. With Jet. Area of fire-grate square feet 10.3 10.3 10.3 Coal per hour, cwt. 2 1 5 27.63 2-53 27.50 370 41.25 Coal per sq. foot of grate per hour, Ibs. \Vater from 100 per hour cubic feet 47.25 4-59 11.54 48.30 4.69 11.92 69.13 6. 7 I 11.36 Water per sq. foot of grate per hour, Water from 212, per pound of coal, Ibs. Duration of smoke, in the hour, ) ,,- i i . / IIHIlUlCo verv lisriit \ 2.4 I.I 0.0 TRIALS OF NEWCASTLE AND WELSH COALS IN THE WIGAN MARINE BOILER. A mixture of Davidson's Hartley and Hasting's Hartley (Newcastle coals), and a mixture of Powell's Duffryn, Nixon's Navigation, and Davis's Abercwomboy (Welsh), were tried in the Wigan boiler being the same as some coals that had been tried at Keyham Dockyard in 1863. The general results of the trials are, for comparison, placed together with those of the South Lancashire and Cheshire coals, thus : Table No. 273. NEWCASTLE AND OTHER COALS: COMPARATIVE RESULTS OF EVAPORATIVE PERFORMANCE. COALS. Coal per Square Foot of Grate per Hour. Water at 100 per Hour. Water at 112 per Ib. of Coal. Newcastle, Ibs. 28.83 cubic feet. Ci.4? Ibs. II. (K Welsh, ' J 26.2O 4860 12.4.4. South Lancashire and Cheshire : Average, 27 6^ 4.7 2S I I CA Highest evaporative efficiency ) (Haigh Yard) \ 25-53 47.38 12.54 Lowest evaporative efficiency ) (Worsley Top Four Feet), ) 31.36 48.50 10.37 MARINE BOILER AT NEWCASTLE-ON-TYNE. 785 Showing that the average of the South Lancashire and Cheshire coals is inferior in rapidity and in efficiency of evaporation to both of the other coals, and that though the best of the South Lancashire coals has a greater evaporative efficiency than the others, the rapidity of evaporation was less. This comparison is corroborative of the deductions made from Delabeche and Playfair's analysis and trials of coals from the several districts (see page 413). EVAPORATIVE PERFORMANCE OF NEWCASTLE COALS IN A MARINE BOILER, AT NEWCASTLE-ON-TYNE, 185 7.* v \ ^ ' These experiments were made to test the evaporative power of the steam-coal of the Hartley district of Northumberland. The experimental boiler was of the marine type, 10 feet 3 inches long, 7 feet 6 inches wide, and 10 feet high; with 2 internal furnaces, 3 feet by 3 feet 3 inches high, and 135 flue-tubes above the furnaces, in 9 rows of 15 each, 3 inches in diameter inside, 5^ feet long. The dead-plates were 16 inches long, and 2 1 inches below the crown of the furnace. As the result of many preliminary trials, two standard lengths of fire-grates were fixed upon 4 feet 9 inches, and 3 feet 2^ inches, with a fall of y 2 inch to a foot; and the fire-bars were cast y?. inch thick, with air-spaces from ^ to ^ inch wide. The fire-doors were made with slits ^ inch wide and 14 inches long, for the admission of air. The chimney was 2 feet 6 inches in diameter. A water-heater was applied at the base of the chimney, in the thoroughfare; it contained 76 vertical tubes, 4 inches in diameter, surrounded by the feed-water. Total area of fire-grates, 4 feet 9 inches long, 28^ square feet. PO- do. 3 2% 191^ Heating surface of boiler (outside), 749 square feet. Do. water heater, 320 Ratio of larger grate-area to heating surface of boiler, i to 26.28 Do. smaller do. do. i to 38.91 Two systems of firing were adopted, as "standards of practice:" First, ordinary or spreading firing, in which the fuel was charged over the grate, and the whole of the supply of air was admitted through the grate. Second, coking-firing, in which the fuel was charged, i cwt. at a time, upon the dead-plate, and subsequently pushed on to the grate, making room for the next charge; and air was admitted by the doorway as well as by the grate. Four systems of furnace were tried, of which Mr. C. W. Williams' was adjudged by the experimentalists to have rendered the best performance. According to this system, air was admitted above the fire at the front of the furnace, by means of cast-iron casings, having apertures on the outside, with slides, and perforated through the inner face, next the fire, with numerous ^-inch and ^-inch holes, having a total area of 80 square inches, or 5.33 square inches per square foot of grate. Alternate firing was adopted by Mr. Williams. The general results of the experiments are given in table No. 274. 1 The author has derived the particulars of those trials from the Report of Messrs. Long- ridge, Armstrong, 6 Richardson to the Steam Collieries Association of Newcastle-on-Tyne. 1857- 50 786 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. > _g d a-ll 8 i . ** OJ kt Al, L 21 % 06 O- r! o 2 i o3 t> f yi y RESULTS OF EVAPORAI t^. W ition of Newcastle-on-Tyne. Remarks on the Prevention of Smoke, &c. admitted entirely through te. Much smoke, often veryde admitted through both the g .d the door. No smoke. 3 cJ C 'rt -2 g 1 'K 3 . feet per pound of coal. Tem ure in uptake, 448. Much sm< through the grate and the d< ed 7o cubic feet through ate and 88 cubic feet through ors, per pound of coal. Tem ure in uptake, 480. No sm< ition of smoke, practically pel Do. do. Do. do. ention of smoke, practically :t; temperature at base of cl y above 600. 00 i !x c3 C "^ 3 ^_* oo H rr^ '*"' W KJ? S JL H <5 ^<< ^ CJ 0) CJ t*-H H ^Q te o w < - ^_ -^ v^ V -v ' v Y ' tBERLA w ^ H ollieries rt T3 S W| o*| g -<' 8 to J>.00 CO N i-i d >-< to J3 *-^ ;> U ^ ^ rt 1 1 ^ 1 _o . ON g ajB, > vO LH CO ^ il ^ O *-" OJ w w ^" ^* to fc 5 a; ^ a fa 3 fe Ptf *o i,' 3 s o a ^ 5ss | 00 2 O OO ONVO N t^ to ON ON & to H u 2 o PP o y o ^ fo * H C/3 3 w ^ 2 ^3 _0 s|| 10 O I 2 9 8 to t^OO vO O "if 5 & M .5 -4.J .5 4^ ^ . i . J> D ^ Sfi M w M s s * g ^ ^ -XJ T3 'd ^ . oo H I 1 o MARINE BOILER AT NEWCASTLE-ON-TYNE. 787 The experimentalists reported that Mr. Williams' plan gave the best results, and they concluded: " ist. That by an easy method of firing, com- bined with a due admission of air in front of the furnace, and a proper arrangement of fire-grate, the emission of smoke may be effectually prevented in ordinary marine multitubular boilers whilst using the steam-coals of the Hartley district of Northumberland. 2d. That the prevention of smoke increases the economic value of the fuel and the evaporative power of the boiler. 3d. That the coals from the Hartley district have an evaporative power fully equal to that of the best Welsh steam-coals, and that, practically, as regards steam navigation, they are decidedly superior." These gentlemen made a trial of Aberaman Welsh coal, and they found that its practical evaporative power, when it was hand-picked, and the small coal rejected, was at the rate of 12.35 H>s. of water per pound of coal, evaporated from 212; this may be compared with the best result from Hartleys coal, large and small together, in table No. 274, which was 12.53 Ibs. water from 212 per pound of coal, or with another result of experiment, with Hartley coal, not given in the table, showing 12.91 Ibs. water per pound of coal. As a check on these results, they ascertained the total heat of combustion of the two coals here compared, by means of an apparatus constructed by Mr. Wright, of Westminster, so contrived that a portion of coal is burned under water, and the products of combustion actually passed through the water, which absorbs the whole heat of combus- tion. The following are the comparative values : Water practically Evaporated Total Heat of Combustion per Pound of Coal. in Evaporative Efficiency. Welsh coal, hand-picked, 12.35 Ibs 14.30 Ibs. Hartley coal, large and small, 12.91 14.63 The experimentalists also point out the " elasticity of action " of the Hartley coals: they burned them at rates varying from 9 to 37^ Ibs. per square foot of grate per hour without difficulty, and without smoke. The Welsh coal, burned at the rate of 34^ Ibs. per foot per hour, melted, it is said, the fire-bars after an hour and a half s work. TRIALS OF NEWCASTLE AND WELSH COALS IN THE MARINE BOILER AT NEWCASTLE, FOR THE BOARD OF ADMIRALTY. By Messrs. Miller & Taplin. 1858. Messrs. Miller & Taplin, representing the Board of Admiralty, conducted, in 1858, a series of trials at Newcastle, with the same marine boiler as was employed by Messrs. Longridge, Armstrong, & Richardson, the object of which was to investigate the comparative evaporative power and other pro- perties of Hartley coal and Welsh steam-coal, and the merits of Mr. Williams' plan of smoke-prevention. The fire-bars were i^f inch in thickness, and had ^j-inch air-spaces. The feed-water was passed through the heater, except when otherwise stated. Mr. Williams' apparatus was constantly in action when Hartley coal was burned without smoke; and it was closed when this coal was tried for smoke making, also when Welsh coal was burned. 788 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. ARI . oo 10 oo co H - u ^ i H C/3 S fa O * ^g I ssi fa W 5 Pi Pi < (S* * s ** > H O g a Pi e 2^2 3 S S | | s .'* rS^' I fl . W O i-3 I 1 H S ig. M r.cooo co i>. d MI d d ON d VO to ON ON O ON q oo^o* o d d d M M M M VO ^ cOOl^ w CN>-O OO N M M* C* CO rororo ^j ^" ^^vO ^ 1*** ^" cooO O co O t^ ON .o 00 r^ Tj* M t^ t*^ Tj" ON O vO M ON O ,-; ON co *^-VQ' vd T"-OO" ** M" vd 10 * oo' 3 OO ON ONOO t- t^-00 00 ON 00 iri 10 vO O O covO 00 ri- 1>. to HH co yjO^OMCO O OONQtOvo vd t~"od d M" co tovd oo" o^ t^. ON ON M MO ON \o oo -^ TJ- Tj- q M oo' rt- t^. co co TJ- vo* HH MM M M M i-i gO'OOOO ovovdvo'vo O O * co TJ-vn I IS 'do d dodo' do* d d^ ' ~J n3 T3 "d .. .2 : JS g ^ . .^S^Q w ^ ciOOO O OOOO O O O O-.8-_OO"rH"t1 ffiQQQ p PPPP p p p P^^^PPog ON O *-< M co -< N COTJ- tnvo MM M MMM MARINE BOILER AT NEWCASTLE-ON-TYNE. 789 I **" ft i 3-6 \"0 &a Is 8 . c> dS d> co ONO N ro O N ^.1 12 ^5 rJ3 1 3|| ; W S JT iiife^ gS2| a ^ ft rtOO N N \O O wvo c^i o\ N r-co O vO t^ t~~ r^oo co co i-o r^. c\ O vo COVOCOOO "* _. CO" 6 O NH f4 cf\ -9 c/3 JS ^ = = | 1 1 ^ CM PL, 00 CO ^3 S WELSH COAL, when was passed directly without the heater. Blaengwarn Merthy Woolwich Docky * 790 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. During the trials of Hartley coal, the fires were maintained at from 1 2 to 14 inches in thickness on the grates, the coal was stoked on the coking system, the fresh charges of coal having been delivered at the front, on each side of each grate alternately; and the incandescent fuel pushed forward towards the bridge before charging. During the trials of Welsh coal, the fires were maintained at from 8 to 10 inches in thickness; and, in charging, the fresh coal was thrown where it was required, all over the fire; the burning fuel never being touched by any firing tool. The cinders that fell through the grates were constantly raked together and thrown upon the fires. The results of the trials made by Messrs. Miller & Taplin have been analyzed and compiled into the table No. 275, in which the results of the performances of the West Hartley coals are grouped, to which is added the results obtained from Lambton's Wallsend house coal, as a bituminous or highly smoky coal. The results of the trials of the South Welsh coal are likewise grouped in the table. Separate trials of each coal were made, in which the feed-water was delivered direct into the boiler, the heater having been for this purpose disconnected. Messrs. Miller & Taplin concluded from the results of their experiments : ist, that when the smoke from Hartleys coal is consumed, the evaporative value of this coal is nearly equal to that of Welsh coal, whilst its rapidity of combustion and evaporation of water is greater; 2d, that the Hartleys coal is less liable to be broken up by movement than Welsh coal ; and that it is less disintegrated by long exposure to the atmosphere than Welsh coal; 3d, that Hartleys coal may be burned without making smoke, by the use of Mr. C. W. Williams' apparatus. TRIALS OF WELSH AND NEWCASTLE COALS IN A MARINE BOILER AT KEYHAM FACTORY. 1863. Trials of Welsh and Newcastle coals, singly and in combination, were con- ducted with the coal-testing boiler at Keyham Factory, by Mr. T. W. Miller. The boiler is the pattern from which the Wigan boiler, described at page 781, was made, and of which it was a copy, except that the flue-tubes are 2 inches. The dead-plate was 10 inches below the crown of the furnace, and 6 inches in length. The grate was made of two different lengths, 4 feet and 3 feet, and inclined 2 inches per foot. The bridge was 8 inches below the crown. Two different doors to each furnace were employed during the trials : one, a common door, with a few small perforations for air; the other was made double, and air entered from the bottom, and passed through numerous ^-inch holes into the furnace. The air-way in the second door amounted to 60 square inches, equal to 8.6 inches per square foot of the longer grates, and 11.4 inches for the shorter grates. The charges of coal were from 16 to 19 Ibs. Total grate-area, 4 feet long, 13. 75 square feet. ,, 3 io-3 Total heating surface, plate, 72.5 tubes (outside), 324.5 397.0 Ratios of grate-areas to heating surface : larger grates,. . . i to 29 smaller ... i to 38.5 STATIONARY BOILER IN AMERICA. 791 Mr. Miller reported that "the combinations, in equal proportions, of Welsh and Newcastle coals, while they produced, on the average, nearly equal economical results, measured by the quantity of water evaporated by i Ib. of fuel, they produced on the average greater rapidity in evaporation y and that they on the average produced the least amount of smoke." He also found that the small Welsh coal could be burned beneficially in mix- ture with Newcastle coal. The general results of his experiments are given in table No. 276. EVAPORATIVE PERFORMANCE OF AMERICAN COALS IN A STATIONARY BOILER. 1843. The American coals, of which the composition was given page 418, were tried for their evaporative performance by Professor Johnson, with a flat- ended cylindrical boiler, 3*4 feet in diameter and 30 feet long, with two thorough internal flues, 10 inches in diameter. The grate was placed below the boiler at one end, and was 5 feet long by 3 feet wide ; it was 9 inches below the boiler at the front, and 10 inches at the back. The bars were ^ inch thick, with ^-inch air-spaces. The grate could be shortened 8 inches, by inserting a perforated air-plate at the bridge; and n^ inches at the front, by inserting a coking-plate. The air for com- bustion was heated in a chamber under the ash-pit, before passing through the grate. The gases passed under the boiler to the back, returned through the inside flues, and made another circuit of the boiler by a wheel-draught through side flues. The chimney was 18 inches square, and 61 feet high above the grate. Area of grate : Full length, 16.25 square feet. With air-plate, i4-7 With air-plate and dead-plate, 1 1.375 Heating surface : Lower flue, 130 Two flue-tubes, 157 Two side-flues, 90.5 Total, 377.5 :ea heating surface: Ratio of grate-area to J Full length) , tQ With air-plate, i to 26.8 With air-plate and dead-plate, i to 33.2 The water was evaporated into steam of from 6 Ibs. to 7 Ibs. per square inch above the atmosphere. The coal was delivered in charges of from 100 Ibs. to 1 10 Ibs. The condensed results of performance are given in table No. 277. The average results were that in burning 7 Ibs. per square foot of grate per hour, 9^ Ibs. of water was evaporated from 2 1 2 F. per pound of coal. The inferior evaporation for the bituminous caking coals is accounted for by its imperfect combustion, evidenced by the smoke which escaped in considerable quantity. 792 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. fe H OLD 3 si ll H T3 1/5 g | ; ; 55 ; | t-.-i-li'a'fe '3 3 y 4 s-afc* ^ .S ;? tt> 1 1 Q 00 t^OO *** c-T CO o 8 5*2 8* CO pa 1 1 00 2 | BiH-j^l - | fottflfcflMl a I S M n gftfr. | 5 5^t ^h C^ CO ro fO C) CO CO co co 1 'eS g -d g M S ~ o T3 t, i 2 c " N N - N N O vo OS VO w covo ;> > ^ O ON O\ C) ON ON I-H ON O O ON ** ^H JJ^S " " " ~ IgJJgj JNN t^VOO co ^w o 1 S K CO S&8'& CO CO CO CO CO CO CO vo U*a 3 a J! 5 'ft^R a 8 ^^^ 3 ^g, vO VO OO vo vo vO t^ IW ON to ON t"^ O^ 5 W N ' ' M 1 ON ON co covO ON q w q rh q ci ci N t?i N N CO u lit 1 a- ."iTt s _ ^ r ^ ^ ^ fT "w : ! : : "^ : : : : I | ||;]] 1? '7~ A T~ "I t? r; tj || ^ Jll'l'l' 1 * 2 I ^ 2 iS S rt liiiij^^ fc S.>S tJte oj 2 lplllll With perforated door Hartley Main, ^ 9J O ^ bJD^ 1 rt t^ _(-; S ^ ^Jllliliiil ^ ^ IM MARINE BOILER AT KEYHAM FACTORY. 793 i 1 c3 I & .9 .S.S 1 8 i5 i w i/T uT 1 t ' i \N S^SNXN y Q 13 .2 TH T*- T+- O 00 iOO\ .2 g J3 I S w SN ^ ^i ^ CO S S tf tf V J3 H 2 a 8 < O ^ rt S r It w 2 2" M 10 CO > & | " = s s 1 1 E N CON N N *0 _;^_; ! = = - 1 " " ^.2 .2 1 ^ l w ji S s 1 S -r M OT CO sr , < "w'S . ? I j_, )-, C ^-< vO HI -^- r+- O\ M O uj 00 VO mvO CO COCO 5 OC\OOOQ>O CO O vO HH M O T*- vo t^ i ^- M 1, |j,|ls,| Th Tj- in to 1-1 vO N t2i CO fOvO tOvO O N 10 CO O VO Tf IO 8 S vo 1 VO ^ * crtf a ffi 3 fi ci ci N ei co CO CO CO * CO ^} !>. W 1 C3 15 o ^ y .a Z |1|| 5J r>oo NH t* 10 o\ o ^a co co co coVo T+- CO CO O t^ CO vo' vo' co CO CO q co ^ of ^H rj < ^ 9 S y-^ Q .2 s *> I-' H a j.| g rf 0^ S" ^vo" ^-00 i- ^- v CO* i- ct fl . i>.vo* t> N CO pi ~ pi pt s ^h 3 0SY| d d 1 1 ' |a 4//& Series. WITH SMALLER GR With common doors- Welsh Coal, Half Welsh sma Davidson's Hartl Half Welsh beans, ] ting's Hartley, . . . ^O ! ' _ri ? w w Q. rt H .8 ^ H H M c5 a S z 794 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Table No. 277. AMERICAN COALS: RESULTS OF THEIR EVAPORATIVE PERFORMANCE WITH A CYLINDRICAL STATIONARY BOILER AT THE NAVY YARD, WASHINGTON. 1843. (Reduced from the Report of Professor W. R. Johnson.) COAL. Area of Fire- grate. Coal Consumed per Hour. Coal per Square Foot of Grate per Hour. Water Evapor- ated from Ordinary Tempera- ture per Hour. Water per Square Foot of Grate per hour. Water Evapor- ated from 212 F. per Ib. of Coal. sq. feet. Ibs. Ibs. cub. feet. cub. feet. Ibs. Anthracites (7 samples),... 14.30 94.94 6.64 12.37 0.87 9-63 Free-burning bituminous coals (n samples), 14.14 99.16 7.01 13-73 0.97 9.68 Bituminous caking coals (Virginian, I o samples) , 14.15 105.02 742 12.16 0.86 8.48 Averages, 14.20 9971 7.02 12.75 0.90 9.26 CONDITIONS OF SMOKE. Anthracites, No smoke. Free-burning bituminous coals, Little smoke, and mostly when charging. Bituminous caking coals, Smoke considerable, in one instance constant. The air-plate was tried both open and closed for each coal, with various effect, good and bad. The average results for the open air-plate proved a gain in efficiency and a loss in rapidity, thus : Gain of Efficiency. Loss of Rapidity. COALS. per cent. per cent. Anthracites, 0.43 14.9 Free-burning coals, 2. 13 2.68 Caking coals, 1.96 1.48 Foreign and western coals, 3.38 5.37 Showing that the open air-plate was most beneficial for efficiency and least injurious for rapidity with the smoke-making coals. Surplus air in the products of combustion. By analysis, it was found that twice the quantity of atmospheric air that was chemically necessary, passed through the furnace. Temperature of the air and smoke. The average temperatures were as follows : External air, 73 F. Air on arriving at the grate, 250. Heated 177. Ga^s on arriving at the chimney, 292. Excess above steam 65. Draught-gauge, 307 inch of water. Influence of soot in the flues. It was observed that whilst, in the perform- ance of the anthracites, day after day, the temperature at the chimney and the evaporative efficiency were practically constant, with the smoky coals the temperature rose and the efficiency fell off. In three instances of caking coal, the temperature rose 75 from an average of 298 F., on the MARINE BOILER IN AMERICA. 795 first day, to 373 F. on the last day; and the efficiency fell off i lb., from 8.66 to 7.68 Ibs. These effects are due, of course, to the accumulation of soot on the surface of the boiler, and the impediment thus caused to the passage of heat. Level of the grate. The grate was tried at 7 inches and 12 inches below the crown, the standard level having been 9 inches. The trials showed that the 7-inch level was 5^ per cent, better than the 9-inch, and the 9-inch level 8 per cent, better than the 1 2-inch. Effect of cutting off the two side-flues. Reducing thus the heating surface by 90.5 square feet, the comparative performance with and without the side-flues was as follows : ANTHRACITE : Water per lb. of Coal. Water per hour. Without side-flues, 9.96 Ibs. ... 13.33 cubic feet. With do 10. 1 1 ... 14.03 CAKING COAL: Without side-flues, 7.80 ... 11.72 With do 8.52 ... 11.30 Showing that, whilst the average rapidity was not affected, the efficiency was diminished by the closing of the side-flues. EVAPORATIVE PERFORMANCE OF AN EXPERIMENTAL MARINE BOILER, NAVY YARD, NEW YORK, U.S. 1 Mr. Isherwood made trials of an experimental multitubular marine boiler under cover, on land. The boiler was covered with felt stitched on canvas. Steam of 20 Ibs. effective pressure per square inch was generated and blown off. The shell was 7 feet 7 inches deep, 3 feet i ^ inch wide, and 6 feet 5 inches high, with a single fire-flue 26 inches in diameter, and 24 flue-tubes above the fire-flue, 3 inches in diameter outside, 5 feet 10^ inches long. With a 4-inch dead-plate, the grate was 5 feet long, inclined, being 12 inches below the crown at the front, and 15^2 inches at the bridge. Area of fire-grate, 10.8 square feet. Heating surface : furnace, 1 9. 7 square feet. smokebox, 2 5-75 tubes, 100.78 uptake, 4.02 150.30 Ratio of grate-area to heating surface, i to 14. The fires were 5 inches in thickness, using Pennsylvanian anthracite of medium quality. The refuse averaged about 20 per cent, of the fuel. In the following selection of the results of the performance, the equiva- lent quantities of water as evaporated from and at 212 R, are substituted for the original quantities: 1 Experimental Researches in Steam Engineering, vol. ii. 1865. EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Table No. 278. EVAPORATIVE PERFORMANCE OF EXPERIMENTAL MARINE BOILER, AT THE NAVY YARD, NEW YORK, U.S. ist Area of Grate. Series of Heating Surface. Trials: Varyi Ratio of Heating Surface to Grate. ng Rate of Comt Coal per Square Foot of Grate. *)ustion. Water per Pound of Coal from and at 212. square feet. square feet. ratio. Ibs. Ibs. A 10.8 ^o-S 14 5-57 9.27 B 10.8 150.3 14 10.99 8. 95 C 10.8 150-3 14 16.57 7-94 D 10.8 I S-3 14 22.10 7.80 E 10.8 150.3 14 27.76 7.40 ^d Series of Trials: Varying Area of Grate. I 8.64 149.0 17.24 15 8.58 J 6.48 148.0 22.84 15 8.28 K 4.32 147.0 34.03 15 8.93 Ajh Series of Trials : Constant Total Quantity of Fuel Consumed per Hour, with Varying Grates. L 8.64 149,0 17.24 20.73 8.03 M 6.48 148.0 22.84 27.42 7-43 N 4-32 147.0 34.03 27.58 7.24 %th Series of Trials : Heating Surface of Tubes cut off. V 10.8 45-5 4.21 16.57 '.; 5.91 w 10.8 45-5 4.21 16.58 6.10 X 10.8 45-5 4.21 11.77 6.64 EXPERIMENTS ON THE COMPARATIVE EVAPORATIVE PERFORMANCE OF STATIONARY BOILERS IN FRANCE. 1874. A commission was appointed by the Sotiete Industridle de Mulhouse to test, under identical conditions, the comparative performance of a French or elephant boiler, a double-flue Lancashire boiler, and a so-called " Fairbairn" boiler. 1 Lancashire boiler. 6.56 feet in diameter, 25.75 feet long; flues 27.5 inches in diameter, with internal fire-place 28.5 inches in diameter. Shell-plates ..64 inch, flue-plates ^ inch thick. Grate inclined; mean level below crown, 1 6 inches. Fire-bars .6 inch thick, air-spaces j^ inch. "Fairbairn" boiler. Two cylinders 4.1 feet in diameter, 25.75 feet long; central fire-tube 27.5 inches in diameter, enlarged at the end to form an internal fire-place 28.5 inches in -diameter. The two cylinders were united to a third above them, 3.75 feet in diameter, 23 feet long, by three neck- ings or pipes, 14 inches in diameter, from each lower cylinder. Plates 1 Bidletin de la Societe Industrielle de Mulhouse, June, 1875. See also Proceedings of the Institution of Civil Engineers, vol. xliii. p. 377, where an abstract of the Report is pub- lished. STATIONARY BOILERS IN FRANCE. 797 y 2 inch. Grate inclined; mean level below crown, 16 inches. Fire-bars .6 inch, air-spaces % inch. French boiler. Body 3.74 feet in diameter, 29.5 feet long. Three heaters, 1.64 feet by 32.8 feet long, united to the body by three neckings to each. Plates of body % inch; of heaters .4 inch thick. Grate hori- zontal; level below middle heater, 18 inches, and below side heaters, 16 inches. " Fairbairn." Lancashire. French. Length of boiler, feet 23 & 2;. 75 25 75 29 5 & 32 8 Total heating surface, . . . square feet I OI7 ^:>-/5 612 607 Length of grates, feet A. cq At ? UJ/ A 2 T Combined width of grates, A C -3 ^O J 4C-? A 76 Total grate-area, square feet 2O.5 20 5 2O I Ratio of heating surface to grate-area,.... Total capacity cubic feet i to 49.5 6A2 C i to 29.8 6"?7 5 i to 30.3 C or T \Vater-space, CAA 7 U J/O AI 2 5 5J 1 - 1 4O8 I Steam-space, Q7 8 225 o 1 2"? O Heating surface per cubic ) ( f foot of water,.... } sc l uare feet 1.8 7 1.48 1.49 Total weight, with accessories, tons IQ 6 166 IA C Weight per square foot of heat- ) ,, ing surface, \ 42.4 59-7 *4O 52.5 The gases in the Fairbairn boiler passed from the flues by the sides of the lower cylinders, and returned by the sides of the upper cylinder, towards the chimney. In the Lancashire boiler, they passed from the inside flues on each side to the front, and thence under the boiler to the chimney. In the French boiler, the current was not divided, but after heating the three heaters it wound round the boiler. The flues delivered into the same chim- ney. The temperature in the flues, just at the chimney, about 4 inches above the bottom, was taken every five minutes. The steam was maintained at from 4.6 to 5 atmospheres. The feed-water was supplied at from 79 to 84 F. The regular daily work lasted from 6 a.m. to 6 p.m., with i ^ hour interval; working time, 10^ hours. The coal consumed in getting up steam was included in the consumption. Two days before the trial, each boiler was emptied and was thoroughly cleaned inside and outside. Each trial lasted several days consecutively. The coals consumed were Ronchamp and Saarbrucken, the general composition of which is indicated by the following analysis t 1 Gaseous Elements onfy, or "Pure Fuel" Carbon. Hydrogen. Oxygen and Nitrogen. Actual Heat of Combustion of One Pound of Pure Fuel. Ronchamp per cent. 88 5o per cent. A 60 per cent. 672 English units. 16 416 Saarbrucken, ypy 81 10 A 75 HTC TC -32O 1 " Calorimetric Trials and Analysis of Coals and Lignites," by A. Scheurer-Kestner and C. Meunier-Dollfus. See Proceedings of the Institution of Civil Engineers, vol. xliii. p. 396. EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Table No. 279 FRENCH AND ENGLISH BOILERS. RESULTS OF EVAPORATIVE PERFORMANCE. (Reduced from the Report of the Mulhouse Commission.) FUEL. RONCHAMP AND SAARBRUCKEN COALS. BOILER, AND FUEL. Coal Consumed per Hour. Water Evapor- ated 3$ Hour from and at 212 F. Water per Ib. of Entire Coal. Tem- pera- ture of Gases. Air drawn in per Ib. of Coal. Total. Per Sq. Foot of Grate. Ash. Total. PerSq. Foot of Grate. RONCHAMP COAL. Heavy Firing. "Fairbairn," Lancashire cwt. 3.39 3.50 3-69 Ibs. 18.53 I9-I5 20-57 pr. c't. 13-8 I4.I I4.I cub. ft. 56.06 53-45 54-73 cub. ft. 2.73 2.61 2.72 Ibs. 9.21 8.50 8.26 Fahr. 421 572 562 cu. ft. 226 183 I 94 French, RONCHAMP COAL. Light Firing. ' ' Fairbairn, " 1.96 1.91 2.04 10.70 10.41 11.36 13-5 I 4 .6 13.6 3i-i4 30-52 31-38 1.52 1.49 1.56 8.86 8.92 8.58 337 406 425 26l 194 193 Lancashire. French, SAARBRUCKEN COAL. " Fairbairn " 3.04 3.02 3-n 16.59 16.50 17.32 10.6 9-7 9-4 43-20 40.69 41.89 2. II i-99 2.08 7-93 7-51 7-51 402 554 544 195 1 80 179 Lancashire, French GENERAL AVERAGES OF THE FOREGOING PER- FORMANCES. ' ' Fairbairn, " 2.80 2.81 2.95 I5-27 15-35 16.42 12.6 12.8 12.4 43-74 41-55 42.67 2.12 2.03 2.12 8.67 8-33 8.12 387 5" 5io 227 1 86 189 Lancashire French, AVERAGES OF 3 DAYS' PER- FORMANCE, when equal rates of evaporation were effected. Lancashire. . 3.57 3-57 iq.50 19.87 54.10 54-32 2.64 2.70 8.44 8-49 587 572 165 197 French, EVAPORATIVE PERFORMANCE OF LOCOMOTIVE BOILERS. The author collected, from various trustworthy sources, the results of the performance of locomotive boilers, of the earliest as well as the most recent designs, and has reduced them and placed them together with the results of his own observations, in table No. 2 So. 1 Boilers of nearly every size and variety that have been used in England, are represented in the table; the areas of grate vary from 6 to 24 square feet, the heating surfaces from 40 to 2000 square feet, and the ratios of sur- face to grate, or the surface-ratios, from 40 to i to 100 to i. The fuel used was coke, except in a few specified instances of boilers designed for burning coal, in which coal was used. 1 These data are derived from the author's work on Railway Machinery, 1855, page 156; and Railway Locomotives, page 33*. Reference is made to these works for information on the details of the boilers. LOCOMOTIVE BOILERS. 799 Table No. 280. LOCOMOTIVE-BOILERS: PROPORTIONS AND RESULTS OF EVAPORATIVE PERFORMANCE. The Fuel used was Coke, except when Coal is specifically stated. No. Name of Locomotive. Area of Fire- grate. Heating Surface (Tubes measured on the Outside). rlatio of Heating Surface to Grate. Coke consumed >er Square Foot of Grate per hour. Water Consum- ed per Square Foot of Grate per hour. Water Evapor- ated per Pound of Coke, from and at 212 F. j EARLIEST LOCOMOTIVES. Killingworth, . sq. ft. 7.O sq. ft. 41.25 ratio. 6 Ibs. 44 (coal) cu. ft. 2.3 Ibs. 4.O2 2 3 Do. improved, . . . Rocket, . . ..... 10.9 6 124 138 11,4 21 57 (coal) 35.5 4 3 5.32 6.27 4 Phoenix, 6 / 326 55 54 5-7 7.86 Atlas . Q.2O 275 30 60 5- *4 6.35 Star, 7.76 359 2 92 8.22 ~-jj 6.53 8 Average of 4 locomotives, Soho 6.5 8.44. 348 412 53-5 35 90 100 130 9.8 10 13 O3 8.04 7.42 7.38 10 ii Hecla, 8.34 418 49 92 125 II II. 3 6^65 12 13 Bury's goods locomotives, Bury's passenger ,, GT. WESTERN RAILWAY. Ixion 9.2 9.2 13.4. 461 387 699 50 42 52 in 112 138 9.24 8.1 5 1C 6.15 4-93 8-33 IiJ Hercules, 13.6 vyy 699 51.4 IO5 15 10.70 16 17 Etna, Capricornus, Giraffe, 11.4 12.5 467 608 41 48.6 97 76 10.7 8.8 8.21 8.61 18 IQ Mentor, Cyclops, Royal Star, . . 136 ii. 7 699 822 51-4 7O 69 91 8 10.8 8.67 8.85 20 21 Pyracmon Class, Aiax. 18.44 13.67 1363 IO67 74 78 69 84 8-4 II. 2 9-09 0.90 22 23 24 25 26 27 28 2 9 30 21 Great Britain, Iron Duke, Great Britain Variety, .... Courier Variety, LONDON AND NORTH- WESTERN RAILWAY, &c. A, York & North-Mid- ) land Railway, j Hercules, York&North- 1 Midland Railway, . .. ) Sphynx, Man., Sheffield, [ ( & Lincoln Railway, j (Later engines.) Heron, L. & N.-W. Ry., No. 291, No. 300, ,, ,, SOUTH-EASTERN RAILWAY No 142, 21 21 23.62 9 .6 9.6 10.56 10.5 19 22 14 7 1938 1938 1866 903 828 1056 7 82 1449 1263 1158 2 92 92 79 94 86 100 74-5 76.26 57-41 788 8* 90 75 132 105 i57 90 56-5 5o-7 62.25 (coal) II II 8.6 17 15 22.1 II. I 6.2 6.6 8 77 9-95 9.17 8.60 10.52 10.70 10.41 9.29 9^28 10.15 32 No. 118, 26.2^ 063.5 /" 36.7 30. 86 (coal) 4. ^4 10.60 2-2 No 58 12.25 7O5 7 j i 57 6 61 22 (coal) 8.60 10 13 34 No. 44.49 (coal) 7.35 11.91 2C No 142 . . . 14.7 II58.2 78.8 55.71 7.73 q 77 3* No. 105, IO.5 623.1 59.3 55.91 9.43 11.68 ^7 No Q IO. 5 - J 623. 1 cq. 3 66.19 IO.OO 10 06 8oo EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Table No. 280 (continued). No. Name of Locomotive. Area of Fire- grate. Heating Surface (Tubes measured on the Outside). Ratio of Heating Surface to Grate. Coke Consumed Per Square Foot of Grate. Water Consum- ed per Square Foot of Grate. Water Evapor- ated per Pound of Coke, from and at 212 F. sq. ft. sq. ft. ratio. Ibs. cu. ft. Ibs. LONDON AND SOUTH- WESTERN RAILWAY. 38 Snake, 12.4 '" 985 79 87 12.26 10 no ,, 57 (coal) 9.65 10.57 41 Canute, cold feed- water, 1 tiles, f ,, 49 (coal) 7-77 9.90 42 Canute, feed- water heat- > ed, no tiles, . ) ,, If ,, 42 (coal) 6.42 9-54 43 Canute, cold water, no ) ', tiles } }) M 58 (coal) 8.89 9-56 44 Canute, feed- water heat- ) ed tiles \ I 46 (coke) 6.46 8.76 45 Canute, feed- water heat- ) ed, no tiles, j" ,, ,, 49 (coke) 7.17 9-13 46 Canute, cold water, no ) tiles, f J J 54 (coke) 8.69 10.04 CALEDONIAN RAILWAY,&C. 47 48 49 No. 33, Caledonian Ry., No. 42, No. 43, 10.5 10.5 10.5 i 79 75 75 42 11 9-2 12.46 IO. II 11.31 50 No. 51, 10.5 788 75 45 6-7 11.04 51 52 }No. 13, | 10.5 10.5 788 75 75 57 11.6 8.2 8.09 10.71 53 No. 13, 9.0 788 87.6 102 14.7 9-52 54 Nos. 125, 127, n-37 1050 t2 66 8.66 9.72 55 No. 102, ii. 8 974 2-5 94 10.3 8.15 56 Orion, Sirius, E. &G.Ry., 12.23 758 62 44 6.29 10.71 57 America, Nile, ,, 1 1. 10 736 66.3 70 8.8 9-3 1 58 Pallas, 16.04 818 51 38 6 10.47 59 Brindley, ,, 9-15 802 87.65 54 7.2 60 Orion, G. & S.-W. Ry., 9.24 495 53.6 84 9-4 8 28 61 Queen, 10.5 65-5 87 10 8^7 PORTABLE STEAM-ENGINE BOILERS. 80 1 EVAPORATIVE PERFORMANCE OF PORTABLE STEAM- ENGINE BOILERS WITH COAL. 1872. The results of the excellently conducted trials of portable steam-engines exhibited at the show of the Royal Agricultural Society, at Cardiff, in 1872, were fully reported by the judges, Mr. F. J. Bramwell and Mr. W. Mene- laus. 1 To this report, with the valuable tables appended to it, prepared by the consulting engineers, Messrs. Eastons & Anderson, the author is indebted for the data with which he has formed the table No. 281. The fuel used was Llangennech (Welsh) coal; an analysis of it, by Mr. G. J. Snelus is given at page 415, ante. The quantity of ash and clinker averaged, so far as it was observed, about 6 per cent, of the fuel. The boiler was of the ordinary pattern, having a firebox and multitubular flues ; but Messrs. Davey, Paxman, & Co.'s boiler contained, in addition, ten circulating wrought- iron bent water-tubes, 2^ inches in diameter, in the firebox, rising from the sides to the top. Table No. 281. PORTABLE STEAM-ENGINE BOILERS. PROPORTIONS AND RESULTS OF EVAPORATIVE PERFORMANCE. 1872. (Compiled and reduced from the Report of the Judges, Royal Agricultural Society's Show, Cardiff.) Fuel : Llangennech (Welsh) Coal. Area of Fire- Coal No. Constructors. grate. Heating Surface (Tubes measured Ratio of Heating Surface to Trial Con- sumed per Sq. Foot of Equivalent Water Evaporated from and at Equiva- lent Water Evapor- As Re- Nor- mal. duced for Trial. on the Outside). Fire- grate. Trial- grate H^ur. 212 F. per Square Foot of Grate, per Hour. ated per Pound of Coal. sq. ft. sq. ft. sq. ft. ratio. Ibs. Ibs. cu. ft. Ibs. I J Marshall, ) ( Sons, & Co. \ 4-4 3-o 283.5 94-5 15-7 161 2.58 10.23 2 \ Clayton & \ \ Shuttleworth 5-3 3-2 220.0 69 12.8 151 2.42 11.83 1 Clayton & \ Shuttleworth j it ,, ,, J 12-5 148 2.36 ii. 81 3 Hayes 5-i 5-i 170.6 33 14.8 66.5 1.06 4-59 4 ( Davey, Par- ) \ man, & Co. \ 3-75 3-75 168.4 45 10.3 114 1.83 11.02 5 Tuxford & Sons 6.13 193.0 6 Brown & May... 3-2 3-2 I59-I 5 9-53 104 1.66 10.89 7 Tasker & Sons... 4-7 4-7 158.0 34 13.0 119 1.91 9-33 8 j Reading Iron- ) | Works } 7.2 2-37 211. 89 20.4 214 3-43 10.49 9 Lewin 4-3 1.6 I5I.6 10 j E. R. & F. ( Turner 3-5 3-5 187.8 54 20.7 204 3-26 9-93 ii 1 Barrows & j Stewart 5-0 5-o 129.8 26 13-6 1 20 1-93 8-97 12 j Ashbey, Jef- \ fery, & Luke 5-5 2.0 204.5 1 02 3i-i 319 5.10 9.27 1 The Trials of Portable Steam- Engines at Cardiff; Report by the Judges. 1872. 51 8O2 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. RELATIONS OF GRATE-AREA AND HEATING SURFACE TO EVAPORATIVE PERFORMANCE. SPECIAL EXPERIMENTS ON THE RELATIVE VALUE OF THE DIFFERENT PARTS OF HEATING SURFACE. Mr. Graham's Experiments , 1858. Mr. John Graham published, in , 1 an account of his experiments on the proportional evaporative value of the different parts of the heating surfaces of boilers. ist Series. Four open tin pans, 12 inches square, in a row, set in brick- work. A grate 1 2 inches square was set directly under the first pan, 29 inches below it, from which a flash-flue 3 inches deep conducted the . gaseous products under the other pans towards a chimney. The first pan showed " the direct heating effect of fire;" the second, the effect of an "equal surface of blaze," the third and fourth, the effect of heated air only. With a " moderately strong draught " the quantities of water evaporated per hour were proportionally as follows : Percentage of Evaporative Duty. ist pan, ........ . ............... as 100 ......... 67.6 2d ........................ 27 ......... 18.2 3<* ........................ 13 ......... 8.8 4th ........................ 8 ......... 5.4 IOO.O Showing that two-thirds of the whole evaporation was effected from the first pan, and only a twentieth from the last pan. 2d Series. Three cylinders of ^-inch plate, 3 feet in diameter, and 3 feet long, open to the atmosphere, in a row end to end, were set in brickwork. A grate was placed under the first cylinder, 3 feet long and 2 feet wide, and 9^ inches below the cylinder; with a flash-flue under the second and third cylinders, concentric with them, of 4 inches radial width, and carried up on each side to the level of the centre of the cylinders. The average results of eleven trials for evaporation, with the calculated heating surfaces, were as follows : Area of grate, ........................................ 6 square feet. Heating surface of ist cylinder, ............ !-53 Do. 2d do ............. 14.13 ,, Do. 3d do ............. 14.13 38.79 Worsley coal consumed, 7 2 Ibs. per hour, or 1 2 Ibs. per square foot of grate per hour. Water evaporated from 60 F., 4.55 Ibs. per pound of coal ; the duty was proportionally as follows : Percentage of Duty. For Whole Surface. Per Square Foot. i st boiler, as 100 66.4, or 73 percent. 2d 34.7 23.0, 18.5 3d 16 ^0.6, 8.5 IOO.O IOO.O 1 Transactions of the Literary and Philosophical Society of Mana/iester, vol. xv., 1858. COMPARATIVE EFFICIENCY OF HEATING SURFACE. 803 Showing that about three-fourths of the evaporative work per square foot of surface was done by the first cylinder, and only a twelfth by the third cylinder. Experiments of Messrs. Woods and Dewrance, I842. 1 Mr. Edward Woods and Mr. John Dewrance, in 1842, tested the evaporative duty of successive portions of the flue-tubes of a locomotive boiler, 5 feet 6 inches long, divided into six compartments by vertical diaphragms. The first compart- ment was 6 inches long, and each of the others 12 inches. It was found that the evaporative duty of the first compartment was about the same per square foot as that of the fire-box; that of the second compartment about a third of that value; that of the remaining compartments very small; and that the first 6 inches did more work than the remaining 60 inches of tube. Experimental Dedttdions of M. Paul Havrez, 1874. The important deduction, that the evaporative performance of similar boilers per unit of grate-area, increases with the square of the surface-ratio, is confirmed by the deduction made by M. Paul Havrez of the following law, from the perfor- mances of locomotive boilers: 2 That the quantities of water evaporated by consecutive equal lengths of flue-tubes decrease in geometrical progres- sion, whilst the distances from the commencement of the series increase in arithmetical progression. The point, he adds, at which the law begins to prevail, is that at which the radiation of heat from the fuel ceases, and heat is communicated by conduction alone. One of the experiments of which the results were investigated by M. Havrez, was made by M. Petiet, of the Northern Railway of France, who repeated the experiment of Mr. Woods and Mr. Dewrance, and tested the evaporative value of the different parts of a locomotive boiler having tubes of a length of 1 2 feet 3 inches divided into five compartments. The first compartment consisted of the fire-box, with 3 inches of length of the tubes; and the four tube-sections were 3.02 feet long. Using coke and briquettes as fuel, the average results were as follows : Fire-box ist Tube ad Tube sd Tube 4th Tube Section. Section. Section. Section. Section. {60.28 box 16.15 tubes. 76.43 179 179 179 1 79 sq.ft. Water evaporated per \ square foot per hour, > 24.5 8.72 4.42 2.52 1.68 Ibs. with coke ) Water evaporated per \ square foot per hour, > 36.9 11.44 5.72 3.52 2.31 Ibs. with briquettes j M. Havrez's law of progression is traceable here, and whether it be exact, or only approximately true, the rapidly diminishing evaporations are corro- borative of the results of previous experiments. If the successive evapora- 1 The Engineer, March, 1858. 2 " Evaporation in Steam-boilers decreasing in Geometrical Progression," by M. Paul Havrez, Annales du Genie Civil, August and September, 1874; abstracted in the Proceed- ings of the Institution of Civil Engineers, vol. xxxix., page 398, 1874-75. 804 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. tions be set off as ordinates to a base line representing the advance of the heating surface, and contoured, the area of the figure is a measure of the total evaporation. The area would bulk largely at the first part, and taper down quickly towards the end; and it is easily comprehended that such areas of evaporation for boilers of different total lengths or quantities of surface, may increase practically as the squares of the total surfaces, supposing that the final temperatures of the gases in leaving the boilers were the same. FORMULAS FOR THE RELATIONS OF GRATE-AREA, HEATING SURFACE, WATER, AND FUEL. It is well known that, in a given boiler, in which the grate and the heat- ing surface are constant and, of course, also the ratio of the surface to the grate-area, the greater the quantity of fuel consumed per hour the greater also is the quantity of water evaporated; but that the production of steam increases at a less rate than the combustion, in other words, that the quan- tity of water evaporated per pound of fuel is diminished. But it has remained a question: At what rate does this diminution of efficiency take place? The answer is supplied by the fact, generalized from the experimental obser- vations on stationary, portable, marine, and locomotive boilers, detailed or noticed in preceding pages, that the total quantity of water evaporated per square foot of grate is expressed by a constant quantity, A, plus a constant multiple, B c, of the fuel consumed per square foot of grate, or by the general formula The sense of this equation is, that though the water evaporated per square foot of grate does not keep pace with the fuel consumed, yet that the quantity of water increases by equal increments for equal increments of fuel per square foot of grate. Again, on the inverse supposition, that the efficiency of the fuel remains constant, how is the performance of a boiler affected by the proportions of the grate-area and the heating surface? The author, in 1852, investigated this question by the aid of the observations already noticed, of the evapora- tive performance of locomotive-boilers, using coke; and he deduced from them, that, assuming throughout a constant efficiency of the fuel, or pro- portion of water evaporated to the fuel, the evaporative performance of a locomotive boiler, or the quantity of water which it was capable of evapo- rating per hour, decreases directly as the grate-area is increased; that is to say, the larger the grate the smaller is the evaporation of water, at the same rate of efficiency of fuel, even with the same heating surface. 2d. That the evaporative performance increases directly as the square of the heating surface, with the same area of grate and efficiency of fuel. 3d. The necessary heating surface increases directly as the square root of the performance ; that is to say, for example, for four times the performance, with the same efficiency, twice the heating surface only is required. 4th. The necessary heating surface increases directly as the square root of the grate, with the same efficiency; that is to say, for instance, if the grate be enlarged to four times its first area, twice the heating surface would be COMPARATIVE EFFICIENCY OF HEATING SURFACE. 80 5 required, and would be sufficient, to evaporate the same quantity of water per hour with the same efficiency of fuel. Let W be the quantity of water evaporated per hour, and C the weight of coke consumed per hour, W and C varying so as to preserve a constant ratio to each other; let h= the heating surface, and^= the area of grate, in square feet; then in which m is a constant. When the water, W, is expressed in cubic feet, and 9 Ibs. of water is evaporated per pound of fuel, the value of m, deduced from the results of forty experiments, was found to be .00222, and .OO222 g (3) Reduced to the standard of one square foot of grate, let w and c be the weights of water and fuel respectively, per square foot of grate, in constant ratio to each other; then, dividing the above formulas respectively (4) and w (cubic feet) = .00222 ( ) 2 o ( 5 ) Showing that, when the ratio of water to fuel is constant, the performance of the boiler, per square foot of grate, increases as the square of the ratio of the heating surface to the grate-area. The following table of examples, extracted from Railway Machinery?- shows how closely the evaporation proceeded according to the square of the surface-ratio, when 9 Ibs. of water, at the ordinary temperatures and pressures, was evaporated per pound of coke. Table No. 282. OF RELATIVE HEATING SURFACES AND RATES OF CONSUMPTION OF WATER IN LOCOMOTIVE BOILERS. (Railway Machinery.) Consump- Classified Groups of Locomotives. Surface- ratio. tion of Water per Hour per Sq. Foot of Water per Pound of Coke. Number of Ex- periments. Grate. ratio. cubic feet. Ibs. Orion, Sinus, Pallas, E. & G. Ry, 52 6.15 9 13 C. R. Passenger Engines, 66 8 " Q.I 17 Snake L & S W Ry. 72 12 8q 2 Sphynx A Hercules, QO 18 **y 8.Q2 8 The quantities of water are thrown into the parabolic curve, AC, Fig. 328, next page, being ordinates to the base-line, AB, on which the relative surface-ratios are measured. It was thus found, that, practically, there can never be too much heating surface, as regards economical evaporation, but there maybe too little; and 1 Railway Machinery, page 158. 8o6 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. that, on the contrary, there may be too much grate-area for economical evaporation, but there cannot be too little, so long as the required rate of combustion per square foot, does not exceed the limits imposed by physical conditions. ~ A o Heating Surface-ratios. Fig. 328. Diagram to show Rate of Economical Consumption of Water per hour per foot of Grate, for given Surface-ratios. To co-relate the formula ( i ), in which the surface-ratio is constant, with the formula ( 4 ), in which the evaporative efficiency of the fuel is constant, it may suffice for the present to observe that the quantity B c is constant for all surface-ratios, and that the quantity A varies as the square of the surface-ratio. Let the surface-ratio = r, then A = ar 2 , in which a is. <5 a constant which is specific for each kind of boiler; and (6) w = the water evaporated in pounds per square foot of grate per hour. c the fuel consumed in pounds per foot per hour. E = = the efficiency of the fuel, or the weight of water evaporated c per pound of fuel. A = ar 2 = a constant, which is specific for each kind of boiler. B = a constant multiplier, specific for each kind of boiler. r - = the ratio of the heating surface to the grate-area. o a = a constant, specific for each kind of boiler. When the water and fuel per foot of grate per hour are given, the value of the required surface-ratio is found from the above formula, for ar 2 = w ~Bf, and I - =FT- .......................... (7) When the water per foot of grate per hour, and the surface-ratio, are given, to find the fuel per foot of grate per hour required to evaporate the water: ^c-w ar 2 , and w - ar* B (8) IV When the efficiency E = , of the fuel is given, that is, the weight of water evaporated per pound of fuel; also, the surface-ratio; to find the fuel that COMPARATIVE EFFICIENCY OF HEATING SURFACE. 807 may be consumed per square foot of grate per hour corresponding to that efficiency. As - =E = c f =B + ; then ar* = c (E - B) ; and c c c- ar* (9) When the efficiency E or , and the fuel consumed per foot of grate per hour, are given, to find the surface-ratio required to effect that evaporation. As already found, ar* = c (E - B), and r* = c ~ ; whence, Newcastle Marine Boiler, page 785. Select for comparison, from tables Nos. 274 and 275, pages 786 and 788, the performance of this boiler with a grate-area of 22 square feet, and 749 square feet of heating surface, 34.05 times the grate, with increasing rates of combustion of coal per square foot per hour. Find the corresponding weights of water evaporated per square foot, and plot them to a vertical scale, ace -ISC .too so ccal- Fig. 329. Newcastle Marine Boiler. Diagram to show Relation of Water and Coal per square foot of Grate-area, 22 square feet. Surface-ratio, 34.05. upon a base-line AB, Fig. 329, measuring the weights of coal consumed. They are found to lie in, or close to, a straight line, D C, drawn obliquely upwards from a point, D, in the ordinate of zero, at a level which is 25 Ibs. above the base-line, and the general formula ( 6 ) becomes w =25 + 971^; ( IJ ) in which ar* = 25, and 6 = 9.71. The annexed table, No. 283, shows the correspondence of the actual quantities of water evaporated, with those which are calculated from the coal consumed, by this formula ( 1 1 ). 1 1 The diagonal line C D, in Fig. 329, does not exactly strike the average of the results for the grate of 22 square feet alone ; but it is the average for the results obtained from the various sizes of grate taken together. For reference to the line A E, see page 817. 8o8 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Table No. 283. NEWCASTLE MARINE BOILER RELATIONS OF COAL AND WATER. Grate 22 square feet. Surface-ratio 34.05. Nos. of Experi- ments in Tables No. 274 and 275. Coal per Foot of Grate per Hour. Water per Pound of Coal, from and at 212 F. Total Water per Foot of Grate per Hour. Water per Pound of Coal, according to Formula. Observed. By Formula (n). Difference by Formula. Ibs. Ibs. Ibs. Ibs. per cent. Ibs. 5 17.27 11.70 202.0 192.7 -4.6 ii. 16 H 22.08 11.41 251.9 2394 -5.0 10.84 15 23.04 10.62 244.7 248.7 + 1.6 10.79 16 25.97 11.17 290.1 277.2 -4.4 10.67 17 26.05 10-33 269.1 277.9 + 3-3 10.67 6 26.98 10.80 291.4 287.0 -i-5 10.64 18 28.51 10.58 301.6 301.8 0.0 10.58 Since A = ar* = 25, in the present instance; a -| = 5i_i = .02156, and ar* = .02 1 56 ?- 2 . By substitution, the following formula is obtained, which applies to all surface-ratios in the Newcastle boiler: 12 Table No. 284. NEWCASTLE MARINE BOILER RELATIONS OF COAL AND WATER. Varying grate-area and surface-ratio. Calculations for normal surface-ratio 34.05 by formula ( 1 1 ). Coal per Square Foot of Grate per Hour. Water per Sq. Foot of Grate per Hour, for Normal Surface-ratio, 34.05. No. of Experi- ment. Grate-area. Surface- ratio. Actual. Reduced in the Ratio of the Squares of the Sur- face-ratios, for Normal Reduced in the same Ratio as for the Coal. Calculated from Column 5 by Formula Difference by Formula. Ratio, 34.05. ( " ) (i) Ujj sq. feet. (3) ratio. Ibs. Ibj (6) Ibs. (7) Ibs. (8) per cent. 10 42 17.83 1 6.0 58.35 563.1 591.6 + 5.1 II 17.6 63.82 583.3 644.7 + 10.5 12 13 33 22.7 18.13 20.36 66.12 45.8i 5924 4237 667.0 469.8 + 12.6 + 10.9 2 28.5 26.28 19.0 31.90 355-0 334-8 - 5-7 5 22 34.05 17.27 17.27 202.0 192.7 - 4.6 14 22.08 22.08 251.9 239-4 - 5-0 15 23.04 23.04 244.7 248.7 + 1.6 16 }) 25.97 25.97 290.1 277.2 - 4-4 17 ?J }) 26.05 26.05 269.1 277.9 + 3-3 6 26.98 26.98 291.4 287.0 18 28.51 28.51 301.6 301.8 o.o 4 19.25 38.91 17.25 13-21 165.5 153.3 - 74 21 18 4I.6l 18.67 12.50 139.7 146.4 + 4.8 22 24.89 16.67 182.7 186.9 + 2.3 7 27.36 18.32 208.3 202.9 - 2.6 8 15-5 48.32 3740 18.57 1974 205.3 + 4.0 COMPARATIVE EFFICIENCY OF HEATING SURFACE. 809 The results of the other experiments with the Newcastle boiler, made with different areas of grate, may be reduced for direct comparison with those made with the 22-feet grate, by reducing both the coal and the water per square foot per hour, in the ratio of the squares of the respective surface- ratios, whilst the ratio of the coal and water, or the efficiency, remains constant. The table No. 284 shows the reduced water (column 6) corresponding to the reduced coal (column 5), for the normal surface-ratio 34.05. In column 7, the reduced waters are given as calculated by the formula ( 1 1 ) ; and the differences by the formula, which are, upon the whole, inconsiderable, are given in the last column. To show the suitability of the formula ( 1 2 ) for the calculation of water evaporated, from the given surface-ratios, as they are, the annexed table, No. 285, shows, by comparison (columns 5 and 6), the actual and calculated quantities of water evaporated by the coals (column 4), with the ratios in column 3. The percentages of differences are identical with those already exhibited in the previous table. Table No. 285. NEWCASTLE MARINE BOILER RELATIONS OF COAL AND WATER. Varying grate-areas and surface-ratios. Calculations for the actual ratios, by formula ( 12 ). Number of Experi- ment. Grate- area. Surface- ratio. Coal per Square Foot of Grate per Hour. Water per Square Foot of Grate per Hour, for the given Surface-ratios. Actual, as from and at 212 Fahr. Calculated by Formula (12) Difference by Formula. W (2) (3.) (4) (5} (6) (7) square feet. ratio. Ibs. Ibs. Ibs. per cent. 10 42 17.83 16.0 1544 162.2 + 5-1 II 17.6 160.9 177-7 + 10.5 12 w JL 18.13 162.5 182.8 + 12.6 13 33 22.7 20.36 188.3 231.5 + IO.9 2 28.5 26.28 19.0 2II.5 1994 - 5-7 5 22 34.05 17.27 202.0 192.7 - 4.6 15 || 22.08 23.04 251.9 244.7 2394 248.7 + 1.6 16 25.97 290.1 277.2 - 4-4 17 w B 26.05 269.1 277.9 + 3-3 6 26.98 291.4 287.0 18 M 28.51 301.6 301.8 0.0 4 19.25 38.91 17.25 216.1 200.1 - 74 21 1 8 4I.6l 18.67 208.5 218.6 + 4.8 22 24.89 272.8 279.0 + 2.3 8 15-5 4^32 27.36 37-40 311.1 397.6 303.0 413.5 + 4.0 The consistency of the results of the application of the formula under widely varying proportions of boiler, and varying rates of combustion, affords evidence of the correctness of the principles on which it is based. Wigan Marine B oiler , page 781. The trials of this boiler were made with a constant grate of 10.3 square feet area, and a constant surface of 508 square feet, giving a surface-ratio 8io EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. of 50. The average results of the trials selected for the present purpose, are placed in the following table, together with the quantities of water evaporated, as calculated by the following formula deduced from the plotting of the results: 10-75' (13) Showing a smaller constant and a greater multiple than the formula of the Newcastle boiler. Substituting for 25 the general expression a r*, and reducing for the value of w, the general formula is, 10.75 which may be employed for different surface-ratios. Table No. 286. WIGAN MARINE BOILER RELATIONS OF COAL AND WATER. Grate 10.3 square feet; surface-ratio 50. DESCRIPTION OF COALS. Coal per Foot of Grate Hour. Water per Pound of Coal, from and at 212 Fahr. Total Water per Square Foot of Grate per Hour. Observed. By Formula (IS]- Difference by Formula. South Lancashire and Cheshire ) coals Mr Fletcher's trials \ Ibs. 27.63 27.50 41.25 28.8 3 26.2O Ibs. 11.54 11.92 11.36 11.95 12.44 Ibs. 318.8 327.8 468.6 344-5 325-9 Ibs. 322.1 320.6 468.6 334-9 306.6 per cent. + 1.0 -2.2 O.O -2.8 -6.0 South Lancashire and Cheshire ( coals Messrs. Nicol & Lynn... \ Hartley's (Newcastle) coals Welsh coals It appears from this table that the South Lancashire and Cheshire coals, and the Newcastle coals, were equally efficient; and that the Welsh coals had a slightly greater evaporative action than the others. Experimental Marine Boiler, Navy Yard, New York, U.S., page 795. This boiler affords examples of very low surface-ratios. With its normal proportions, 10.8 square feet of grate and 150.3 square feet of surface, the surface-ratio is 14. When the flue-tubes were stopped off, the surface-ratio was only 4.21. By the plotting of the experimental results, reduced for a uniform surface-ratio of 14, the following formula was derived: w = . 0204 r*+ 7.624 c ( 15 ) It is seen in the following table, that the calculated evaporation is con- siderably in excess of the actual reduced evaporation, in the extreme instances of the flash-flue and the small surface-ratio, 4.21. It is obvious that such dissimilar cases as those of a flash-flue and a multitubular boiler, are not directly comparable. COMPARATIVE EFFICIENCY OF HEATING SURFACE. 8ll Table No. 287. EXPERIMENTAL MARINE BOILER, NAVY YARD, NEW YORK RELATIONS OF COAL AND WATER. Varying grate-area and surface-ratio. Calculations for normal surface- ratio 14. Coal per Square Foot of Grate per Hour. Water per Square Foot of Grate per Hour, for Normal Ratio 14. Index to Grate- Surface- Reduced in the Ratio of Calculated r_.^__ Experi- ment. area. ratio. Actual. the Squares of the Sur- Reduced in the irom Column 5 Difference by face-ratios, for Normal Same Ratio. by Formula Formula. Ratio 14. (15)- square feet. ratio. Ibs. Ibs. Ibs. Ibs. per cent. X 10.8 4.21 11.77 130.2 865 996.1 + 15 V ?? j) 16.57 183.2 1082 1400 + 29.4 w ?? )) 16.58 183.3 III9 1401 + 25.2 A ?) 14 5-57 5-57 51.63 46.4 - IO.I B jj )) 10.99 10.99 98.39 87.7 - 10.9 C j? 16.57 16.57 I3L7 130.3 - I.O D >> D 22.10 22.10 172.5 172.4 0.0 E j) j) 27.76 27.76 205.4 215.5 + 4.9 I 8.64 17.24 15 9.88 84.80 79-3 - 6.5 L JJ jj 20.73 13-66 109.60 I08.I - 14 J 6.48 22.84 15 5.64 46.67 47.0 + 0.7 M JJ >j 22.84 10.30 76.54 82.5 + 7-1 K 4.32 34-03 15 2.54 22.67 23-3 + 2.8 N >> 27.58 4.67 33-81 39-6 + 17 Wigan Stationary Boilers, page 771. The data afforded by these typical boilers are specially useful, as they represent classes of boilers in general use in England. The several experimental results, required for the present purpose, are collected in the annexed table. The first two are the results for flash-draughts, for which the side and bottom flues were cut off, and the gases were conducted direct to the chimney after having passed through the tubes. By plotting the coal and water reduced according to the squares of the surface-ratios, for a uniform ratio of 30, this formula was obtained, w = 20 + 9.56*: And in the general form, for various ratios, w = .0222 r 2 + 9.56^ ( 16 ) ( By the formula ( 16 ), the quantities of water in column 6 of the table No. 288 were calculated from the reduced coals in column 5. The agreement of the reduced and the calculated quantities of water (columns 6 and 7) is very close, excepting for the flash-draught. 812 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Table No. 288. WIGAN STATIONARY BOILERS RELATION OF COAL AND WATER. Varying grate-area and surface-ratio. Calculations for ratio 30. BOILERS (Without Economizer). Grate- area. Surface- ratio. Coal per Square Foot of Grate per Hour. Water per Square Foot of Grate per Hour for Ratio 30. Actual. Reduced in the Ratio of the Squares of the Surface- ratios, for Ratio 30. Re- duced in the same Ratio. Calcu- lated from Column 5 by Formula (16). Difference by For- mula. (') Galloway, flue-tubes ) only ( (2) sq. feet. 31.5 )) )) )) )) )) )) 21 )) )) (3) ratio. 13.70 14.74 22.8 23-5 24.4 5) )) 254 34-3 35-5 36.5 j> (4) Ibs. 18.58 19.91 I8. 3 I4.O 17.26 1 8.6 19.1 16.71 21.8 23.0 21.5 22.7 (5) Ibs. 89.10 82.47 31.68 22.82 26.03 28.12 28.87 23.31 16.68 16.43 14.52 15-33 (6) Ibs. 757-3 678.8 322.9 230.4 271.5 290.2 293-7 251.0 179.6 179.2 158.0 165.1 (7) Ibs. 871.8 808.4 322.9 238.2 268.8 288.8 296.0 242.8 179.5 I77.I 158.8 166.6 (8) per cent. + 15.0 + ig.O 0.0 + 3-4 - I.O - 0-5 + 0.8 - 3-3 o.o - 1.2 + 0.5 + 0.9 Lancashire, flue- \ tubes only ... C Galloway, complete... Lancashire and Gal- ) lowav C Lancashire Do Do Do. with water ) tubes ] Galloway Lancashire and Gal- } lowav ... . . ( T -y ' Lancashire Do Stationary Boilers in France, page 796. The proportions and the results of performance- are treated in the follow- ing table. The following special formulas have been deduced for the three boilers respectively, and for the three collectively : "Fairbairn" Lancashire French All the boilers w = .oii26r* w = .omr 2 + 7.7 c ............ ( 18 ) + 8.0 c ............ ( 19) + 8.0 c ............ ( 20 ) + 7.82 c ........... ( 21 ) It is seen that the same formula applies to the Lancashire and the French boilers; and that, therefore, the reporters of the trials were justified in asserting that these boilers were equally efficient. The comparatively inferior quantity evaporated in the first trial in the table, resulted probably from an excessively large surplus of air admitted into the furnace: the total quantity of air in that instance, amounted to 261 cubic feet per pound of coal. COMPARATIVE EFFICIENCY OF HEATING SURFACE. Table No. 289. STATIONARY BOILERS IN FRANCE RELATIONS OF COAL AND WATER. Calculations of evaporative performance for surface-ratio 30. Ronchamp coal. BOILERS. Grate- area. Surface- ratio. Coal per Square Foot of Grate per Hour. Water per Square Foot of Grate . per Hour, for Surface-ratio 30. Actual. Reduced in the Ratio of the Squares of the Surface- ratios for Ratio 30. Reduced in the same Ratio. Calculated from Col- umn 5, by Formulas ( 18 ), ( 19 ), (20). I Difference by For- mulas. "Fairbairn"... jj Lancashire jj 55 French . sq. feet. 20.5 55 55 55 55 20.1 55 55 ratio. 49-5 55 29.8 55 jj 30.3 55 5) Ibs. 10.70 18.53 10.41 19.15 19.50 11.36 19.87 20.57 Ibs. 3-93 6.8 1 10.55 19.41 19.76 11.14 19.48 20.16 Ibs. 34.8 62.7 94.1 165.0 166.8 95-5 165.4 1 66.6 Ibs. 40.7 63.3 92.5 161.8 164.5 97.1 162.3 167.6 per cent. + 17 + 0.9 *7 - -9 - 4 + -7 - -9 + 0.6 55 5J Locomotive-Boilers, page 798. The experimental trials from which the evaporative performances of locomotives have been tabulated, have, of course, been conducted under various conditions. There is, nevertheless, a remarkable degree of har- mony amongst them, for, when plotted, they are seen, with a very few exceptions of early date, to follow the laws of evaporative performance already enunciated. Even the performance of the boiler of the primitive Killingworth engine, when the evaporative efficiency is increased by one- half to represent the value of coke compared with coal as imperfectly burned in that boiler, range as well as should have been expected, with those of other locomotives. In fact, the improved Killingworth boiler exhibits a performance above the general average. Using good coke as fuel, the evaporative performance of locomotive- boilers in which the flue-tubes are spaced sufficiently apart to admit of a free circulation of water around them, is substantially embraced by the following formula when the surface-ratio is 75, which is a good practical ratio : w= 100 + 7. 94 500 Ibs., or 8 cubic feet. pound of coal, ........................ j Do. per horse-power, ...... 62.4 or i Do. per square foot of) grate, 9. Ibs.,.. ..... .......... say \ 9 "-45 8 16 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Table No. 291. PORTABLE-ENGINE BOILERS: CALCULATED EVAPORATIVE PERFORMANCE. From and at 212 F., at the rate of 10 Ibs. of water per pound of coal. Coal Consumed per Hour. No. of Boiler. Surface- ratio. Grate-area. Per Square Total Water Evaporated per Hour. Foot of Total. Grate. ratio. square feet. Ibs. Ibs. Ibs. cubic feet. I 6 4 4-4 2340 102.96 1029.6 16.50 10 54 3-5 16.66 58.31 583.1 9-34 6 5o 3-2 14.30 45.76 457.6 7.33 4 45 3-75 11.57 43-24 4324 6-93 2 4i 5-3 9.60 50.88 508.8 8.15 12 37 5-5 7.82 43.01 430.1 6.89 9 35 4-3 7.00 30.10 301.0 4.82 7 34 4-7 6.60 31.02 310.2 4-97 3 33 5-i 6.22 31.72 317.2 5.08 5 3i 6.13 5.50 33-71 337-1 5.40 8 29 7.2 4-23 30.45 304.5 4.88 ii 26 5.0 3-86 19.30 193.0 3-09 Averages, 40 4.84 9.14 44.24 442.4 7.09 To evaporO ate 8 cubic feet ofl 40 5.46 9.14 49.92 499-2 8.0 water per hour, J GENERAL FORMULAS FOR PRACTICAL USE. By the French experiments with stationary boilers, the Lancashire and French boilers were, by the formulas, page 812, identical in performance; and the so-called "Fairbairn" boiler was nearly as effective as these, within 3}^ per cent. The three forms of boiler may, therefore, be accepted as equally efficient; and they may be classed with the Wigan boiler, as of equal efficiency, with the same coal, and with the same management. The performance of the Howard boiler, likewise, is conformable to the formula for the Wigan boiler; and the Howard boiler is a type of the " sectional " kind of boilers. The formula for the Wigan boiler is, therefore, applicable to all stationary boilers, other than multitubular, with good coal and good management. The performances of the Newcastle and the Wigan marine boilers, are nearly alike. Thus, for a surface-ratio 30, the corresponding quantities of water, w, for different rates of coal, c, per square foot of grate per hour, are as follows : Coal, c= 10 20 30 Newcastle boiler, w= 116.5 2I 3-6 310.7 Wigan boiler, w= 116.5 224.0 331.5 Differences, w= o.o 10.4 20.8 Less than Wigan, ... o.o 4.6 6.3 40 Ibs. 407.8 439-0 3 1 - 2 7.1 per cent. GENERAL FORMULAS FOR PRACTICAL USE. 817 Halve the difference, and take a mean of the formulas ; the mean will be a satisfactory general formula for marine boilers : Newcastle, ...... w = . 02156 T^ + Q.yi c Wigan, ........... w = . 01 r*+ 10.75 c Mean, ......... o/ = .oi6 r* + 10.25 c ............... (28) For coal-burning locomotive boilers a mean of the two formulas adduced, page 813, which are nearly identical, will be a satisfactory formula: S. E. Railway, ........ w = .oog f 2 -f 9.6 c L. & S. W. Railway, w = .009 r* + 9.82 c Mean, ............... w = .oog r 2 + 9.y c .............. ( 29 ) The general formulas which have been deduced are here collected to- gether : Formulas for the Relation of Coal and Water consumed in Steam-boilers per square foot of grate-area per hour, and the ratio of the heating-surface to the area of the fire-grate. Water taken as evaporated from and at 2 12 F. Stationary Boilers, ......... w = .o222 r 2 + 9.56^: ............ (3) Marine Boilers, ... .......... w= .016 r 2 + 10.25 c ............ ( 3 1 ) Portable-engine Boilers,.. w= .008 r 2 + 8.6 c ............ ( 32 ) Locomotive Boilers ) / x (coal-burning), / "" w = 9 r+ *"l < ........ (33) Locomotive Boilers ) / x (coke-burning),}-" ^=-'78>"+ 7-94^- . ( 34 ) Limits to the Application of the Formulas ( 30 ) to ( 34 ). There are minimum rates of consumption of fuel below which these formulas are not applicable. The limit varies for each kind of boiler, and it varies with the surface-ratio. It is imposed by the fact that the maximum evaporative power of fuel is a fixed quantity, and is naturally at that point of the scale, say E in Fig. 329, page 807, where the reduction of the rate of combustion for a given ratio, procures the absorption into the boiler of the whole of the proportion of the heat which is available for evaporation. In the combustion of good coal the limit of evaporative efficiency may be taken as measured by 1 2 ^ Ibs. of water from and at 212 F. ; and in that of good coke by 12 Ibs. of water from and at 212 F. The dotted line EA, Fig. 329, represents the correct course of the diagram towards the zero point, indicating a constant proportion of w 12.5 c, for coal; or w = 1 2 c, for coke. To ascertain the minimum rates of combustion of coal for stationary boilers, to which the formula ( 30 ) applies : The limit is reached when w becomes equal to 12.5^; or when I2.5^ 9. 5 6) c = 2. 94.*:. By reduction, c = ' 222 r* = . oo 7 5 5 r 2 . For a given surface- 2.94 52 8i8 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. ratio r, the limiting value of c is found by multiplying the square of the ratio by .00755. For the other kinds of boiler, the limiting values of c are found in the same way. They are here placed all together : Stationary boilers, limiting value of c - .007 5 5 r*. Marine boilers, =.oojr 2 . Portable-engine boilers, . =.oo2r*. Locomotive boilers (coal-burning), =. 00325 r 2 . (coke-burning), =.0044^. For lower values of c, or consumptions of fuel per square foot of grate per hour, the values of w, the corresponding quantities of water, are simply 12.5 c for coal, and 12 c for coke. The annexed table, No. 292, contains the limiting values of c for given surface-ratios r. Table No. 292. MINIMUM VALUES OF c, OR MINIMUM QUANTITIES OF FUEL CONSUMED PER SQUARE FOOT OF GRATE PER HOUR, FOR GIVEN SURFACE-RATIOS, TO WHICH THE FORMULAS ( 30 ) TO ( 34 ) ARE APPLICABLE. 5 10 SUR 15 FACE-RAl 20 IDS. 30 40 50 Minimum Consumption of Fuel per Square Foot of Grate per Hour. Stationary . Ib. .2 17 .05 .1 .1 lb. 7 7 .2 3 4 lb. i-7 1.6 4 7 I.O Ibs. 3-0 2.8 .8 1:1 Ibs. 6.8 6-3 2.9 4.0 Ibs. I2.I II. 2 3-2 5.2 7.0 Ibs. 18.9 17.5 K II.O Marine, Portable Locomotive (coal-burning),. Do. (coke-burning), Locomotive (coal-burning),. Do. (coke-burning), 60 Surface- 70 ; 75 ratios (con 80 tinued}. 9 100 II.7 16 15.9 21 18.3 25 20.8 28 26.3 36 32.5 44 The only limit to the application of the formulas ( 30 ) to ( 34 ), to ascend- ing values of c, or quantities of fuel per square foot per hour, is the limit of endurance of the fuel itself under the action of the draught: from 100 Ibs. to 1 20 Ibs. per square foot per hour, for ordinary hard coal or coke. Beyond this limit, the fuel is liable to be shaken and partly dispersed, unconsumed, by the force of the draught; although coke has been known to withstand the draught of a locomotive when consumed at the rate of 130 Ibs. per square foot per hour. EVAPORATIVE EFFICIENCY OF STEAM-BOILERS. 819 Table No. 293. EVAPORATIVE PERFORMANCE OF STEAM-BOILERS, FOR INCREASING RATES OF COMBUSTION AND DIFFERENT SURFACE-RATIOS. For best coal and best coke ; surface-ratio 30. Kind of Boiler, Water from and at and Fuel. 212 F. per Hour. Fue 5 per Squ IO are Foot 15 of Grate 2O per Hou 30 r, in pouj 40 ids. 50 Stationary, coal, formula ( 30 ). Per square foot Per Ib. of coal Ibs. 62.5* 12-5 Ibs. 116 11.56 Ibs. I6 3 10.89 Ibs. 211 10.56 Ibs. 307 IO.23 Ibs. 402 1 0.06 Ibs. 49 8 9. 9 6 Marine, coal, for- mula (31). Per square foot Per Ib. of coal 62.5* 12.5 "7 11.69 168 11.25 219 JO-PS 3 22 10.69 424 I0.6I 527 10.54 Portable, coal, formula ( 32 ). Per square foot Per Ib. of coal 50 IO 93 9-3 136 9.01 179 8.95 265 8.83 35 1 8.77 437 8.74 Locomotive (coal- burning), formula ( 33 ). Per square foot Per Ib. of coal 57 11.4 i5 10.5 '54 10.26 202 IO. IO 299 9-97 396 9.90 493 ! 9-86 j Locomotive (coke- burning), formula (34). Per square foot Per Ib. of coke 56 11.14 95 9-54 135 9.02 175 8-75 254 8-47 334 8-35 413 8.03 Surface-ratio 50. Kind of Boiler, and Fuel. Water from and at 212 per Hour. Fue 5 i per Sqt 10 are Foot '5 of Grate 2O per Hoi 30 ir, in pou 40 nds. 50 Stationary, coal, formula (30). Per square foot Per Ib. of coal Ibs. 62.5* 12-5 Ibs. 125* 12.5 Ibs. 187.5* 12.5 Ibs. 247 12-33 Ibs. 342 II. 4 I Ibs. 438 10.95 Ibs. 534 10.67 Marine, coal, for- mula (31). Per square foot Per Ib. of coal 62.5* 12-5 125* 12-5 187.5* 12.5 245 12.25 348 11.58 450 11.25 552 11.05 Portable, coal, formula ( 32 ). Per square foot Per Ib. of coal 62.5* 12-5 106 10.6 149 9-93 I 9 2 9 .6 2 7 8 9.27 364 9.10 450 9.00 Locomotive (coal- burning), formula ( 33 ). Per square foot Per Ib. of coal 62.5* 12.5 1 20 ii-95 168 ii. 20 217 10.85 3H 10.45 4" 10.26 508 10.15 Locomotive (coke- burning), formula (34). Per square foot Per Ib. of coke 60* I2.O 120* I2.O 164 10.91 20 3 io. 16 283 9.42 362 9-05 8*83 Surface-ratios 75. Kind of Boiler, and Fuel. Water from and at 212 per Hour. Fue 30 1 per Sqi 40 iare Foot 5 of Grate 60 per Hoi 75 ir, in pou 90 nds. 100 Locomotive (coal- burning), formula (33). Per square foot Per Ib. of coal Ibs. 342 "39 Ibs. 439 10.97 Ibs. 536 10.71 Ibs. 63 I 10.65 Ibs. 778 10.37 Ibs. 927 10.26 Ibs. IO2O 10.20 Locomotive (coke- burning), formula (34). Per square foot Per Ib. of coke 338 11.27 418 10.44 497 9-94 576 9.61 695 9.26 8l 5 9-05 894 8-94 1 These quantities fall below the scope of the formulas for the water, as explained in the text. 820 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. APPLICATIONS OF THE GENERAL FORMULAS FOR THE EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. The table No. 293, preceding page, contains the relative quantities of fuel consumed and water evaporated, for surface-ratios and rates of combustion per square foot of grate per hour, within the range of ordinary practice. It is seen that, with the surface-ratios 30 and 50, the boilers are in the order of evaporative efficiency as follows : SURFACE-RATIO 30. Marine. Stationary. Locomotive (coal-burning). Portable. Locomotive (coke-burning). SURFACE-RATIO 50. Marine. Stationary. Locomotive (coal-burning). Do. (coke-burning). Portable. Table No. 294. EQUIVALENT WEIGHTS OF BEST COAL AND INFERIOR FUELS. To be used with formulas ( 30 ) to ( 34 ), page 817. Relative Heating Power. Equivalenl Weight of Best Coal. Equivalen Weight of Inferior Fuel. Relative Heating Power. Equivalent Weight of Best Coal. Equivalen Weight of Inferior Fuel. Relative Heating Power. Equivalen Weight of Best Coal. Equivalen Weight of Inferior Fuel. best coal=i best coal=i best coal=i. best coal=;i best coal = i best coal=i. 100 I I. 70 .70 143 40 .40 2.50 99 99 I.OI - -6 9 1.45 39 39 2.56 98 .98 1.02 68 .68 1.47 38 .38 2.6 3 97 97 1.03 67 .67 1.49 37 37 2.70 96 .96 1.04 66 .66 1.52 36 .36 2.78 95 95 1.05 65 .65 1.54 35 35 2.86 94 94 1.06 64 .64 1.56 34 34 2-94 93 93 i. 08 63 63 1.59 33 33 3-03 92 .92 1.09 62 .62 1.61 32 32 3-i3 9i .91 1. 10 61 .61 1.64 3i 3i 3-23 90 .90 i. ii 60 .60 1.67 30 30 3-33 89 . 88 !88 1. 12 I.I4 59 58 59 58 1.69 1.72 3 .29 .28 3-45 3-57 87 .87 I.I5 57 57 i-75 27 27 3-7o 86 .86 1.16 56 56 1.79 26 .26 3-85 85 .85 1.18 55 55 1.82 25 25 4.00 84 84 1.19 54 54 1.85 24 .24 4.17 83 -83 1.20 53 53 1.89 23 23 4-35 82 .82 1.22 52 .52 1.92 22 .22 4-55 81 .81 1.23 5i 5i 1.96 21 .21 476 80 .80 1.25 5o .50 2.OO 20 .20 5.00 79 79 1.27 49 49 2.04 JQ .19 5.27 78 78 1.28 48 48 2.08 18 .18 '5.56 77 77 1.30 47 47 2.13 17 17 5.88 76 .76 1.32 46 .46 2.17 16 .16 6.25 75 75 i-33 45 45 2.22 15 15 6.67 74 74 1-35 44 44 2.27 14 .14 7.14 73 73 1-37 43 43 2-33 13 13 7.69 72 .72 i-39 42 .42 2.38 12 .12 8-33 7i 7i 1.41 4i .41 2-44 II .11 9.09 10 .10 IO.O EMPLOYMENT OF FORMULAS FOR INFERIOR FUELS. 821 Portable-engine boilers are clearly inferior in efficiency to coal-burning locomotive boilers, and they may be constructed like these with sensible advantage. Employment of the Formulas (30) #7(34) for Fuels of Inferior Heating Power. i st. To find the evaporative performance of a given weight of inferior fuel, per square foot of grate per hour. Substitute, for the given weight of inferior fuel, the equivalent weight of best coal, and find by the formula the water evaporated. The equivalent weight of best coal is found by multiplying the weight of inferior fuel by the number in column 2 of the table No. 294, opposite the relative heating power of the inferior fuel. 2d. To find the weight of an inferior fuel required for a given evaporative performance. Find, by the formula, in its inverted form, on the model of the equation ( 9 ), page 807, the weight of best coal required, and substitute for this weight the equivalent weight of the inferior fuel. The equivalent weight of inferior fuel is found by multiplying the weight of best coal by the number in column 3 of the table, opposite the relative heating power of the inferior fuel. A table of relative heating powers of fuels is given at page 769. STEAM ENGINE. ACTION OF STEAM IN A SINGLE CYLINDER. PRESSURE OF STEAM DURING EXPANSION IN A CYLINDER. When steam is admitted to a cylinder during a portion of the stroke, then cut off, and expanded in the cylinder, upon the piston, for the remainder of the stroke, the pressure on the piston, during the period of admission, is or ought to be uniform, whilst the pressure during the period of expansion falls as the piston advances and the steam expands. In engines in good working order, the expansion follows substantially the law of Boyle, or Mariotte, according to which the pressure falls in the inverse ratio of the expansion. Substantially, it is said, for the actual changes of pressure seldom follow the law exactly. The pressure usually falls more rapidly in the first portion of the expansion, and less rapidly in the last portion, than is indi- cated by the law of the inverse ratio; and thus, the final pressure may be, and it usually is, greater than that which would be deduced from the ratio of expansion. But the fulness of the expansion-curve depicted on the indicator-diagram, near the end, compensates for the hollowness near the beginning; and, sinking details, it is found that, practically, the area bounded by the curve is equal to that which would be bounded by a hyperbolic curve formed according to Mariotte's law. 1 It is, therefore, assumed, for purposes of illustration and the calculation of power, that the expansion of steam in the cylinder takes place according to Mariotte's law: the curve representing the diminishing pressures due to the increasing volume being a portion of a hyperbola. To formulate the method of describing a hyperbolic curve of expansion over a given base-line : Let L = the length of the stroke, in feet, supposing that there is not any clearance. /=the period of admission, or the cut-off, in feet. s = any greater part of the stroke measured from the commencement, in feet. P = the total initial pressure, in pounds per square inch. P' = the total pressure in pounds per square inch, at the end of a given part of the stroke s. P" = the total final pressure, or the pressure at the end of the stroke, in pounds per square inch. 1 Mr. David Thomson arrived at the same conclusion in his excellent paper "On Compound Engines," in the Proceedings of the London Association of Foremen Engineersy. September, 1873. PRESSURE OF STEAM DURING EXPANSION. 823 ThenF.i ; .. ( z ) expressed by the following rule : RULE i. To find the pressure at any point of the period of expansion when the initial pressure is given. Multiply the initial pressure in pounds per square inch by the period of admission in feet, and divide the product by the distance of the given point from the beginning of the stroke. The quotient is the pressure in pounds per square inch. The pressure P' may also be found from the final pressure P" by the formula giving the rule : RULE 2. To find the pressure at any point of the period of expansion when the final pressure is given. Multiply the final pressure in pounds per square inch by the length of the stroke, and divide the product by the distance of the given point from the beginning of the stroke. The quotient is the pressure in pounds per square inch. Note. When there is clearance, it is to be reckoned in parts of the stroke, and added to the values of L, /, and s, before using these for calculation. Let the base-line mn, be the length of the stroke, say 6 feet; me the initial pressure, say 63 Ibs. ; cd the period of admission, say one-third of the stroke. Suppose, for simpli- city, that there is not any clear- ance. Draw the perpendicular^', from the point of cut-off, and divide the period of expansion d'n into any suitable number of parts, say 10 parts, at the points i, 2, 3, &c. Calculate by the rule the Several pressures at the points Fig. 330. Construction of a Hyberbolic Curve. i, 2, 3, &c., and set them off by the scale of pressure on vertical ordinates from the points ; the curve dg traced through the ends of the ordinates is the hyperbolic curve of expansion. At the successive points of the base of the expansion-line, which are, d'> J > 2 > 3> 4, 5> 6, 7> 8 > 9> . the values of the ordinates, or pressures, are 6 3> 5 2 -5> 45. 39-4, 35. S^S* 28 - 6 > 26.1, 2 4- 2 , 22.5, 2 1 Ibs. per sq. in. ; or, putting the initial pressure = i, they are relatively as i, .833, .714, .625, .555, .500, .455, .417, .385, .357, .333. The extreme ordinate ng is thus found to be a third of dd', or 21 Ibs., and the ordinate No. 5 is a half, or 31.5 Ibs. As an example of the cal- culations for an ordinate, take No. 2. The period of admission is 2 feet, the divisions of the base of expansion are - foot, and the length m 2 is 2.8 feet; 10 then, by the rule, the pressure measured by No. 2 ordinate, is, 824 STEAM ENGINE SINGLE-CYLINDER. The calculation may generally be simplified by taking, as a datum, the length of stroke = i. In this instance, the period of admission would be = .333, the period of expansion = .666, and each tenth division = .0666. By the rule, 6 3 lbs. x.333 _ 21 _ 333 + (.0666x2) -.467" as before. To illustrate generally the application of the hyperbolic law of expansion, showing that the product of the pressure and the volume at any point of the expansion-curve is constant, let the base-line A B represent the course of a piston in a cylinder, and the volume described by it. Sup- posing that there is no clearance, let steam of 10 Ibs. total pressure A c, be admitted for a space i foot in length, A D. The rectangle A E is the product of the pressure and volume of the steam admitted. If expanded to the double volume A d, and to half the pressure de, the area of the elongated rectangle AC is equal to that of the initial rectangle A E. Expanding further, to four volumes A d', and to the fourth part of the initial pressure, d' e, the new rectangle Ae' is equal to each of the others Ae and A E. Similarly, the Fig. 331 To Illustrate the Hyperbolic Law of the Expansion of Steam. rectangles Ae" and Ae'", for a fifth and a sixth of the initial pressure, and five times and six times the initial volume, are each equal to the initial rectangle AE. The hyperbolic curve containing these rectangles may be indefinitely extended at either end, to embrace, on the one part, intense pres- sures and small volumes, and, on the other part, very low pressures and large volumes. THE WORK OF STEAM BY EXPANSION. Proceeding, now, to a consideration of the area of the diagram, Fig. 331; as the area of the rectangle A E, is the product of the pressure and volume, and expresses the work done upon the piston by the steam in entering and occupying the cylinder, so, likewise, the hyperbolic area, D E d'" e'", expresses the work done by the steam by expansion within the cylinder after it is shut in. 'This area, and consequently the quantity of work done, may be computed by means of the known relations of hyperbolic superficies with their base-lines: according to which, if the base-lines AD, Ad, Ad', &c., extend in a geometrical ratio, or as i, 2, 4, 8, 16, &c., the successive areas De, ne', &c., increase in an arithmetical ratio, or as i, 2, 3, 4, &c. On the principles of logarithms, which represent, in arithmetical ratio, natural numbers in geometrical ratio, special tables of so-called hyperbolic logarithms are compiled, to facilitate the calculation of the areas of work due to various degrees of expansion. The hyperbolic numbers consist, in fact, of the multiples of common logarithms by 2.302585, which, thus modified, become direct expressions of the proportions borne by the work by expansion THE WORK OF STEAM BY EXPANSION. 825 pertaining to different degrees of expansion, to the initial work done by the steam during its admission into the cylinder; but they are not employed as logarithms. For example, the initial volume being expressed by i, and the total volumes by expansion, by the following numbers in geometrical ratio, i, 2, 4, 8, 16, the hyperbolic logarithms of these numbers are, in arithmetical ratio, .000, .693, 1.386, 2.079, 2 -772, being as o, i, 2, 3, 4, and these logarithms express the actual ratio of the whole work by expansion, for different degrees of expansion, to the initial work of the steam, expressed as i. The total work done by a quantity of steam expanded successively from the initial volume, i to 2, 4, 8, 1 6 volumes, will therefore be in the proportions of i, i +.693, 1 + 1.386, 1 + 2.079, 1 + 2.772, or i, 1.693, 2.386, 3.079, 3.772, showing that, for an expansion of 16 times, the initial work done by the steam during its admission is nearly quadrupled. But it is necessary to make a deduction for the back pressure from the condenser, to find the effec- tive work of the steam. Suppose a cylinder of 5 feet stroke, represented by A B, Fig. 331, with a piston having an area of i square inch, into which steam of 10 Ibs. pressure per square inch is admitted for i foot of the stroke, A D, against a uniform back pressure of, say, 2 Ibs. per square inch, for the whole stroke. Let the steam be expanded through the re- maining four-fifths of the stroke, and construct the diagram of work, Fig. 332, in which the 2-lb. zone of resistance or back pressure is shaded. Then, At the end of the ist, 2d, 3d, 4th, 5th foot of stroke, The total pressures are.... 10 5 3 */ 3 2)4 2 Ibs. per sq. inch; The back pressures are... 222 2 2 Ibs. do. do.; The effective pressures.... 8 3 i i/ 3 y z o Ibs. do. do. The total work done by expansion up to the end of each foot of stroke, is represented by the hyperbolic logarithm of the ratio of expansion, the initial work being = i. Thus, At the end of the ist, 2d, 3d, 4th, $th foot of stroke, The steam is expanded into 234 5 volumes, Of which the hyperbolic ) *l : - c -', , < T logarithms are [ ~ ^ *' 10 *'& ll The initial duty being as... i i i i i (unity), And the total duty as i 1.69 2.10 2.39 2.61 826 STEAM ENGINE SINGLE-CYLINDER. As the initial work, represented by i, is 10 foot-pounds, being 10 Ibs. exerted through i foot, and the resistance is 2 foot-pounds for each foot of the stroke, At the end of the ist, 2d, 3d, 4th, 5th foot of the stroke, The work by expansion is o.o 6.9 n.o 13.9 16.1 foot-pounds; The total work done is 10 16.9 21. o The total resistance is 246 The total effective work is 8 12.9 And the gain by expansion is o 61 15.0 23.9 8 15.9 26.1 10 16.1 do. do. do. 87 99 101 per cent. From the foregoing particulars, it appears that the total work of the steam, by expanding it to five times the initial volume, is fully 2^ times the initial work done without expansion. When the back pressure is allowed for, the effective work, 16.1 foot-pounds, is only twice the initial work, 8 foot- pounds; making a gain of 101 per cent, when the expansion is extended to the extreme limit, where the positive pressure becomes equal to the back pressure. It further appears that the effective work of the steam expanded down to the back pressure from the condenser, is just equal to the work developed by expansion alone. The initial work is balanced in amount by the resist- ance, each of them being 10 foot-pounds. The same conclusions apply to a non-condensing cylinder discharging the steam into the atmosphere. Let the total initial pressure, A C, Fig. 332, be 75 Ibs. per square inch, and suppose the steam to be expanded five times, as before, down to a pressure of 1 5 Ibs. per square inch, and then exhausted into the atmosphere, maintaining a back pressure of 15 Ibs. per square inch throughout the stroke, represented by the shaded zone. On a piston of i square inch area, the proportions of work will be as follows : At the end of the ist, The total work done is as The total work done is ) actually ] 75 The total resistance is The total effective work is The gain by expansion is In this case, where steam of five atmospheres is expanded five times, and exhausted into the atmosphere at a pressure of one atmosphere, the pro- portions of work done are the same as when steam of 10 Ibs. pressure per square inch is expanded five times and exhausted at a pressure of one-fifth, or 2 Ibs. per inch; and they indicate equal degrees of efficiency of the steam in the way it is applied. ist, 2d, 3d, 4th, 5th foot of stroke, I 1.6 9 2.10 2-39 2.6 1 75 126.7 157.5 179.2 195.7 foot-pounds. 15 30 45 60 75 do. 60 96.7 112.5 119.2 120.7 do. o 61 87 99 101 per cent. CLEARANCE IN STEAM-CYLINDERS. 827 It may be concluded, generally, that when the steam is expanded down to the back pressure in the cylinder, whether from the condenser or from the atmosphere, the effective work done in the cylinder is just equal to the total work done by expansion, the total initial work being just balanced and neutralized in amount by the resistance of back pressure. And the utmost useful ratio of expansion, looking to the operations within the cylinder, is measured by the number of times which the total back pressure is contained in the total initial pressure of the steam in the cylinder. Indeed, it may be affirmed that four-fifths of this measure of expansion is sufficient as a limit, for it has been shown that whilst the gain by expansion to four times is 99 per cent., that of a fivefold expansion is 101 per cent, which is only 2 per cent. more. Another reason usually advanced for arresting the fall of pressure, in expanding, at a higher limit than the back pressure, is based on the frictional or passive resistance of the engine. This resistance is to be opposed by the steam in the cylinder; and the total pressure, it is said, should not fall below that which is equivalent to the back pressure, plus the frictional resistance, since, it, is argued, if the pressure at any part of the stroke do fall below the sum of these resistances, the excess of these above the positive pressure is so much dead resistance, and is so much in reduc- tion of the useful efficiency of the steam. This argument is plausible, but fallacious ; and it would be valid only on the supposition that the engine could move without, at the same time, doing its proper duty in driving shafting and machinery. The supposition is, of course, impossible. But, why draw the line of so-called useless resistance at the fly-wheel shaft? The shafting for driving the machinery also opposes dead resistance, and before the engine can move at all, the resistance of the shafting must be overcome. The resistance of all the machinery must likewise be overcome. The useful work to be done must likewise be overcome; in fact, the whole of the work, dead and alive, must be overcome. So the argument leads to the absurd conclusion that the pressure in the cylinder should not fall below the total mean pressure exerted; and as it is not to fall below, neither can it reach above the mean pressure, for that would imply an additional initial force, which would render a greater mean pressure, which is absurd. If the argument had any truth in it, it would lead necessarily to the abandonment of all expansive working, and to the employment of a uniform pressure, with the admission of steam throughout the whole of the stroke. CLEARANCE IN STEAM-CYLINDERS. The clearance, or free space, between the piston when at the beginning of a stroke, and the slide-valve, is filled with steam of the initial pressure at the commencement of each stroke ; and this padding, as it may be called, does no work directly, and is entirely non-effective in non-expansive engines. But in expansive-working cylinders, the clearance-steam does its proper quota of work, in conjunction with the other steam, during the period of expansion. The volume of the clearance may be measured in parts of the stroke supposed to be multiplied into the area of the piston ; and it is here taken, for purposes of discussion, at 7 per cent, of the stroke. 828 STEAM ENGINE SINGLE-CYLINDER. FORMULAS FOR THE WORK OF STEAM IN THE CYLINDER. Now, let L = the length of stroke, in feet, / = the period of admission, or the cut-off, in feet, excluding clearance, c = the total clearance at one end of the cylinder, the volume being measured in feet of the stroke, L' = the length of the stroke, plus the clearance, or L + c, I' = the period of admission, plus the clearance, or /+ c, R = the nominal ratio of expansion, or L -r- /, R' = the actual ratio of expansion, or L' + /', a = the area of the piston in square inches, P = the total initial pressure in Ibs. per square inch, supposed to be uniform during admission, p = the average total pressure, in Ibs. per square inch, for the whole stroke, p' = the average back pressure, in Ibs. per square inch, for the whole stroke, w = the whole work done in one stroke, in foot-pounds, w' = the work of back pressure for one stroke, in foot-pounds, W = the net work done in foot-pounds. The actual ratio of expansion is . l+c I' The work done during admission is equal to the total pressure on the piston, a x P, multiplied by the period of admission, or a P /, which is the work in foot-pounds, and this work is done by a volume of steam measured by the period of admission, plus the clearance, or by / + c - l'\ and as 1=1' -c, then whole work done during admission = a P /= a P (/' - c) ..... ( 3 ) To find the work done by expansion to the end of the stroke, the total pressure on the piston, a P, is to be multiplied by /', the period of admis- sion plus the clearance, and by the hyperbolic logarithm of R', the actual ratio of expansion, or whole work done during expansion = a P /' x hyp log R', ... ( 4 ) which is the work done by expansion, in foot-pounds. Add together these two quantities of work, ( 3 ) and ( 4 ), and reduce; then, for the total work, w, done by the steam in one stroke of the piston, a/ = 0P[/'(i+hyplogR')-f] ...................... ( 5 ) The work of back pressure for one stroke is w' = ap'*L; ............................... ( 6 ) and the net work, such as may be measured by an indicator-diagram, is w w'\ or, W = *[P(/'(i+hyplogR')-')-/L] ............ ( 7 ) INITIAL PRESSURE IN THE CYLINDER. 829 RULE 3. To find the net work done by steam in the cylinder for one stroke of the piston, with a given cut-off. i. To the hyperbolic logarithm of the actual ratio of expansion, allowing for clearance, add i; multiply the sum by the period of admission, plus the clearance, in feet; from the product subtract the clearance, and multiply the remainder by the total initial pressure in Ibs. per square inch. The product is the total work done in foot-pounds per square inch on the piston. 2. Multiply the average back pressure in Ibs. per square inch by the length of the stroke ; the product is the negative work of back pressure in foot-pounds per square inch. 3. Subtract the second product from the first product; the remainder is the net work in foot-pounds per square inch on the piston. 4. Multiply the area of the piston by the net work per square inch; the product is the net work in foot-pounds done in the cylinder for one stroke. Note. When the period of admission and the clearance are expressed as percentages of the stroke, the percentages are to be converted into feet of the stroke. The actual ratio of expansion is found by dividing TOO plus the percentage of clearance, by the sum of the percentages of admission and clearance. To exemplify the application of the rule, take a non-condensing steam- cylinder 3 feet in diameter with a stroke of 5 feet, and initial steam of a total pressure of 70 Ibs. per square inch on the piston, cut off at one-fourth of the stroke, and expanded during the remaining three-fourths. The aver- age back pressure is 17 Ibs. per square inch, and the clearance is 5 per cent, of the stroke. What is the whole work done in one stroke? The steam is cut off at 15 inches, to which the clearance, which is 5 per cent, of the stroke, or 3 inches, is to be added. The sum is 18 inches, or 1.5 feet, and the actual ratio of expansion is 5 L_3 = 3.5, of which the hyperbolic logarithm is 1.204; to this add i, making 2.204, to be multiplied by 1.5, making 3.306. From this product subtract the clearance .25 feet, leaving 3.056. Then 3.056 x 70 Ibs. = 213.92 foot-pounds of total work per square inch of piston; and 213.92x1017.87 square inches area of piston = 217,750 foot-pounds, the total work done in one stroke. The back pressure 1 7 Ibs. per square inch x 5 = 85 foot-pounds per square inch for the whole stroke; and 85 x 1017.87 = 8653 foot-pounds, the negative work of back- pressure. Finally foot-pounds. Total work done on the piston, for one stroke, ....... 2 1 7,750 Negative work of back pressure, for one stroke, ...... 8,653 Difference, or net work for one stroke, ........ 209,097 INITIAL PRESSURE IN THE CYLINDER. Inverting formula ( 7 ), the required initial pressure for a given net quan- tity of work in one stroke, is as follows : P- fl[/;(H.hyplogR')-'] ' f The initial pressure required to produce a given average total pressure per square inch for a given actual ratio of expansion, is found by sub- 830 STEAM ENGINE SINGLE-CYLINDER. stituting, for W, its equivalent a L (p-p')> in formula ( 8 ); and reducing. Then p- ^ L _ . ..(9) /'(i+hyplogR')-' AVERAGE TOTAL PRESSURE IN THE CYLINDER. The average total pressure, /, in the cylinder, in terms of the initial pres- sure^ for a given actual ratio of expansion, is found by dividing the second member of the equation ( 5 ), by the area of the piston and by the length of the stroke; or by a simple inversion of equation ( 9 ) : The average total pressure, /, in terms of the total work done for one stroke, is also, AVERAGE EFFECTIVE PRESSURE IN THE CYLINDER. The average effective pressure is found by subtracting the average back pressure from either of the above values of p, formula ( 10 ) or ( 1 1 ), or it is found by dividing the second member of equation ( 7 ) by the area of the piston and by the length of the stroke : giving, by reduction, (p -/) = P [/'(i+ hyp log R')-j _ f _ (I2} THE PERIOD OF ADMISSION AND THE ACTUAL RATIO OF EXPANSION. The actual rate of expansion required for the production of a given average total pressure from a given initial total pressure may be found ten- tatively by inverting the formula ( 10 ), for initial pressure, and reducing, by which the following formula is obtained : gR'^-f-^- -i ....................... (13) Here, there are two unknown quantities, namely, hyp log R' and /'. RULE. Multiply the length of stroke by the mean pressure, and divide by the initial pressure; and to the quotient add the clearance, making a sum A. Assume a period of admission, and add to it the clearance, to make a value for the divisor /', and find the corresponding value for hyp log R', the hyper- bolic logarithm of a ratio of expansion. Find the ratio in a table of hyper- bolic logarithms, and by it divide the sum of the stroke and the clearance. If the quotient be equal to the assumed period of admission plus the clear- ance, it follows that the assumed period is the required period of admission, and the ratio of expansion is the required actual ratio. But if the quotient be greater than the sum of the assumed period and the clearance, then the . assumed period of admission is too long. If the quotient, on the contrary, ACTUAL RATIO OF EXPANSION. 831 be less, the assumed period is too short. Try again, and assume a shorter or a longer period of admission, as the case may require, until the required period of admission and ratio of expansion have been arrived at. This is a long rule, but the operation of it is less tedious than may be imagined. For example, reverting to previous data, take the stroke = 5 feet; clearance .25 feet, total initial pressure = 70 Ibs., and average total pressure = 42.78 Ibs. per square inch; to find the required period of admission. Then 42.78 x q + 25 = 3. 306 (Sum A) 70 Assume a period of admission, 1.75 feet; then i.75+-25 = 2.oo (Sum B) And, 3.306-^2 = 1.653, from which deduct i; the remainder .653 is the hyperbolic logarithm of the ratio of expansion, 1.92. Now, the stroke plus the clearance is 5.25, and = 2.73 feet, as a period of admission plus clearance; and 2. 73-. 25 = 2. 48 feet. But this is greater than the assumed period namely, 1.75 feet. Try, therefore, a smaller period to begin with, say i foot then i+. 25 = 1. 25 (Sum B) 3.306^1.25 = 2.61; and 2.61-1 = 1.61, which is the hyperbolic logarithm of the ratio 5 ; then ^-^ = 1.05 feet; and 1.05 -.25 = . 80 foot. But .80 foot is less than the assumed period, namely, i foot; and i foot is too short. The required period must be less than 1.75, and more than i foot; and nearer to i foot than to 1.75 feet. Try 1.25 feet, then 1.25 + .25 = 1.50 (Sum B) 3.306^-1.50=2.2040; and 2.2040-1 = 1.2040, which is the hyperbolic logarithm of the ratio 3.5; then ^^ = 1.5 feet; and 1.5 -.25 = 1.25 feet, which is equal to the period last assumed. The required period of admis- sion is, therefore, 1.25 feet; and the ratio of expansion is 3.5. Note. Calculation for this rule may be shortened by using the following table (No. 295), page 836, particularly when the clearance is 7 per cent, of the stroke, as is assumed in the composition of that table. When the clearance deviates by i or 2 per cent, from the standard of the table, suitable allowances may be made on the results drawn from the table, by which near approximations may be made. Take the last example, in which the clearance is 5 per cent, of the stroke. Reduce the given mean pressure to the expression .611, which is its relative value when the initial 832 STEAM ENGINE SINGLE-CYLINDER. pressure is taken as i, thus 42.78-^-7o = .6n. Looking down the fourth column of the table, the nearest values are .619 and .608, corresponding to the ratios of expansion 3.5 and 3.6, the exact ratio being 3.5. The corresponding periods of admission in column 3 are 23.6 and 22.7 per cent, of the stroke, and adding to these 2 per cent., to compensate for the difference of clearance 5 per cent, in the example, as against 7 per cent, in the table the sums average about 25 per cent., which is the correct admission. RULE 4. To find the Period of Admission required for a given Actual Ratio of Expansion. Divide the length of stroke plus the clearance by the actual ratio of expansion; and deduct the clearance from the quotient. The remainder is the period of admission. 2. When the Quantities are given as Percentages of the Stroke. Add the percentage of clearance to 100, and divide the sum by the actual ratio of expansion; and deduct the percentage of clearance from the quotient. The remainder is the period of admission as a percentage of the stroke. The Period of Admission required for a given Actual Ratio of Expansion is RULE 5. The Pressure of Steam expanded in the Cylinder, at the end of the Stroke, or at any other point of the Expansion, is found by dividing the initial pressure by the ratio of actual expansion calculated to the given point of the stroke. The quotient is the pressure at that point. Or, multiply the initial pressure by the period of admission plus the clear- ance, and divide the product by the length of the part of the stroke described up to the given point, plus the clearance. The quotient is the pressure at that point. THE RELATIVE PERFORMANCE OF EQUAL WEIGHTS OF STEAM WORKED EXPANSIVELY. The steam may be said to be measured off for each stroke of the piston, a cylinder-full at a time, of expanded steam; whilst the final pressure is a measure of the density, and therefore of the weight, of this steam. The mean pressures, again, are measures of the total performance of the same body of steam. It follows, that the relative total performance is directly as the mean pressure, and inversely as the weight of steam condensed or as the final pressure, and that, if the former be divided by the latter, the quotients will show the relative total performance of a given weight of the steam, as admitted and cut off at different points, and expanded to the end of the stroke, with a clearance of 7 per cent, of the stroke, as follows : When the steam is cut off at i, #, fa Y^ Vs. fa Vio, VisO fstroke > the relative total performance per unit of steam is directly as the average pressures, i.ooo, .969, .860, .637, .567, .457, .413, .348,... (A) and is inversely as the final pressures, i.ooo, .769, .532, .298, .250, .182, .159, .128. WORK DONE BY ADMISSION AND EXPANSION. 833 The relative, or proportional, total performance of given equal weights of steam are therefore in the ratio of the second last row of figures divided by the last row of figures; the total performance for steam admitted for the whole stroke, without any expansion, being taken as i. Thus, _i .969 .860 .637 .567 .457 -413 -348 i' -769' .532' .298' .250' .182' .159' .128 or the quotients, i.oo, 1.26, 1.62, 2.13, 2.27, 2.51, 2.60, 2.72.... (B) These quotients may be found, otherwise, from the actual ratios of expansion, which are inversely as the final pressures, by multiplying the average pressures by the respective ratios. For example, when the steam is cut off at ^, the actual ratio of expansion is 1.3, and the mean pressure .969 x 1.3 = 1.26, which is the relative efficiency, as already found above. It is seen that the total work or performance of a given weight of steam is fully doubled by cutting off and expanding at a fourth of the stroke, as compared with the admission of steam for the whole of the stroke. In these comparisons of the relative performance of steam worked expan^ sively, the opposition of back pressure has, for simplicity, been omitted from the calculations. Taking the back pressure as constant with all ratios of expansion, it would constitute a uniform quantity to be deducted from each of the total mean pressures, of which the ratios are given in line A ; and as the remainders would thus decrease more rapidly than the total pressures, it would follow that the quotients, line B, would increase less rapidly than as they are there shown to increase. PROPORTIONAL WORK DONE BY ADMISSION AND BY EXPANSION. To ascertain in what proportions the whole work for the stroke is done by admission and by expansion, leaving unconsidered the back pressure : the work by admission is in proportion to the period of admission, and if this be subtracted from the proportional mean pressure, the remainder is the proportional work by expansion. Thus, when the steam is cut off at x Ao, these fractions are the periods of admission, and are proportional to the work by admission, and are decimally as follows : i.ooo, .750, .500, .333, .250, .200, .125, .100, .066, which being subtracted from the relative total average pressures, the re- mainders are the relative work by expansion : .000, .219, .360, .393, .387, .367, .332, .313, .282; the sum of the last two rows, or the total average pressures, being as i.ooo, .969, .860, .726, .637, .567, .457, .413, .348; which are the same as the values in line A, page 832. Here it appears that the quantity of work done by expansion, arrives at a maximum when the period of admission is about one-third of the stroke. 53 ^34 STEAM ENGINE SINGLE-CYLINDER. With a greater or a less admission it is reduced. But the proportion of work by expansion, relative to the work by admission, increases regularly as the admission is reduced. Thus, taking the work for the periods of admis- sion successively, as i, i, i, i, i, i, i, i, i, the corresponding proportions of work done by expansion, are successively as o, .29, .72, 1.31, 1.55, 1.83, 2.66, 3.13 4.27. The loss by clearance-space neutralizes a considerable proportion of the gain by expansion, as appears from the following examples. THE INFLUENCE OF CLEARANCE IN REDUCING THE PERFORMANCE OF STEAM IN THE CYLINDER. To note the effect of clearance in reducing the efficiency of steam in the cylinder, let the steam be admitted for one-fourth of the stroke, and let cdgnm be the indicator-diagram described, with a perfect vacuum, of which the base m n is the length of the stroke = 100, and the extension of the base, m m', is the length of the clearance = 7. The average pressure, //, is, by the for- mula ( 10 ), .637, when the initial pres- sure is i. The loss of pressure by clearance, is represented by the initial area mm'c'c, the pressure being = i, and the volume = 7 per cent, of that of the stroke. Averaged for the whole stroke, that is, multiplying i by 7 and dividing by 100, the average loss of pressure for the whole stroke is i x _? .070; and 100 Fig. -Dmgmm h t^how influence of Clear- jf th j s aver age loss be added to the aver- age pressure, the sum, .637 + .070 = .707, expresses the relative efficiency with which a given weight of steam would be worked if there were no loss by clearance. It shows that there would be a gain of 1 1 per cent. This greater relative efficiency is represented on the diagram by the upper line/'/'. The relative efficiency may be otherwise found by means of formula ( 10), for the average total pressure, the item of clearance being eliminated from it. Suppose the clearance in the diagram, Fig. 333, to be included as part of the stroke, then the period of admission becomes 32 per cent, and the length of stroke 107 per cent.; and, when the initial pressure is i, 32 (i+hyp log ^ or 3.35) 107 107 = .66i, the average pressure, as against .637 the average pressure with clearance. But, as the strokes are different, the average pressures are to be multiplied INFLUENCE OF CLEARANCE. 835 by their respective strokes, to give the proportion of the efficiencies; thus, .637 x 100 = 63.7 relative efficiency, for 25 % admission, with 7 % clearance; .661x107 = 70.7 do. 32 do. without clearance; being in the same ratio to each other, as the values .637 and .707 already found. The comparison is extended for other periods of admission by simply adding the average loss .070, to the corresponding average pressures in the 4th column of the table, No. 295. Thus, When the steam is cut off at full stroke, #, ^, */ 3 > #i Y*> 'Ao, Vis of stroke, the average pressures representing the relative work, when the pressure during admission = i, are i. ooo, .969, .860, .726, .637, .457, .413, .348, and, adding the loss by 7 per cent, of clearance, .070, the increased relative work done by a given weight of steam, if there were no clearance, would be 1.070, 1.039, -93> -796, .707, -5 2 7> -483, showing that the gain would be 7, 7.2, 8.1, 9.6, ii.o, 15.3, 17, 20 percent, which is lost by clearance. Table No. 295. RATIOS OF EXPANSION OF STEAM, WITH RELATIVE PERIODS OF ADMISSION, PRESSURES, AND TOTAL PERFORMANCE. To facilitate calculations about steam expanded in cylinders, the table No. 295 has been composed. The actual ratios of expansion, column i, range from i.o to 8.0, for which the hyperbolic logarithms are given, for ready reference, in column 2. The 3d column contains the periods of admission relative to the actual ratios of expansion, as percentages of the stroke, calculated by Rule 4. The 4th column gives the values of the mean pressures relative to the initial pressures, the latter being taken as i, calculated by formula ( 10 ). The 5th column gives the values of the initial pressures relative to the mean pressures, when the latter are taken as i. These values are the reciprocals of those of the 4th column; at the same time they may be calculated by formula ( 9 ). In the calculation of these last three columns, 3, 4, and 5, clearance is taken into account, and its amount is assumed at 7 per cent, of the stroke. In the 6th column, of final pressures, they are such as would be arrived at by the continued expansion of the whole of the steam to the end of the stroke, the initial pressure being equal to i. They are the reciprocals of the ratios of expansion, column i, as indicated by Rule 5. The 7th column contains the relative total performance of equal weights of steam worked with the various actual ratios of expansion: the total performance when steam is admitted for the whole of the stroke without expansion being equal to i. They are calculated on the principle exem- plified at page 832. 836 STEAM ENGINE SINGLE-CYLINDER. Table No. 295. EXPANSIVE WORKING OF STEAM: ACTUAL RATIOS OF EXPANSION; WITH THE RELATIVE PERIODS OF ADMISSION, PRES- SURES. AND PERFORMANCE. Clearance at each end of the Cylinder, 7 per cent, of the stroke. (SINGLE CYLINDER.) I ACTUAL RATIO OF EXPANSION ; Or Number of Volumes to which the Initial Volume is Expanded. 2 HYPERBOLIC LOGARITHM of Actual Ratio of Expansion. 3 CORRESPONDING PERIOD OF ADMISSION, or CUT-OFF. Clearance, 7 per cent, of the Stroke. 4 AVERAGE TOTAL PRESSURE. 5 TOTAL INITIAL PRESSURE. 6 TOTAL FINAL PRESSURE. 7 RATIO OF TOTAL PER- FORMANCE of Equal Weights of Steam. (Col. 4 -=-Col. 6.) initial volume stroke = 100. initial pres- mean pres- initial pres- with 100 % of admission SSI. sure=i. sure;^. sure=i. = 1.000. 1.0 .0000 100 1. 000 .000 1. 000 1. 000 1.05 .0488 95.0 9997 .003 .952 1.050 .1 0953 90.3 or 9/10 .996 .004 .909 1.096 .15 .1398 86.0 .990 .010 .870 1.138 .18 .1698 83.3 or s/6 .986 .014 .847 1.164 .2 .1823 82.1 983 .017 .833 1.180 23 .2070 80.0 or 4/5 .980 .020 .813 1.206 .25 .2231 78.6 977 .024 .800 1. 221 3 .2624 75-3 or # .969 .032 .769 I.26l 35 .3000 72-3 .961 .041 .741 1.297 39 3293 70.0 or 7/xo 953 .049 .719 I-325 4 3365 69.4 .951 .052 .714 1-332 45 .3716 66.8 or 2/3 942 .062 .690 1.365 5 4055 64-3 932 073 .666 1-399 54 4317 62.5 or % 925 .O8l .649 1.425 55 .4382 62.0 .922 .085 .645 1.429 .6 .4700 59.9 or 3/ s 913 .095 .625 1.461 .65 .5008 57-9 .107 .606 1.490 7 .5306 56.0 .894 .II 9 .588 1.520 75 559* 54.1 883 .132 571 1.546 .8 .5878 52.4 873 .145 555 1-573 .85 6i53 50.8 .864 157 .541 1-597 .88 .6314 50.0 or y z .860 .163 532 1.616 9 .6419 49-3 .854 .171 .526 1.624 95 .6678 47-9 .846 .182 5 J 3 1.649 2.0 .6931 46.5 .836 .196 .500 1.672 2.1 .7419 44.0 .818 .222 .476 1.718 2.2 7885 41.6 799 .251 455 1.756 2.28 .8241 40.0 or 2/ 5 .787 .271 439 1-793 2-3 .8329 39-5 .782 .279 435 1.798 2.4 8755 37.6 or y* .766 305 .417 1-837 2.5 .9163 35-8 .750 333 .400 1.875 2.6 9555 34-2 .736 359 .385 1.912 2.6 5 9745 33-3 or i/ 3 .726 377 377 1.925 2.7 9933 32.6 .719 391 370 1-943 2.8 1.030 31.2 .706 .416 357 1.978 2.9 1.065 29.9 or 3/10 .692 445 345 2.006 3-o 1.099 28.7 .679 473 333 2.039 3-i 1.131 27.5 .665 .504 323 2.059 3-2 1.163 26.4 .652 534 313 2.083 3-3 1.194 25.4 .641 .560 303 2.115 EXPANSIVE WORKING AND PERFORMANCE OF STEAM. 837 Table No. 295 (continued}. I ACTUAL RATIO OF EXPANSION ; Or Number of Volumes to which the Initial Volume is expanded. 2 IYPERBOLIC LOGARITHM of Actual Ratio of Expansion. 3 CORRESPONDING PERIOD OF ADMISSION, or CUT-OFF. Clearance, 7 per cent, of the stroke. 4 AVERAGE TOTAL PRESSURE. 5 TOTAL INITIAL PRESSURE. 6 TOTAL FINAL PRESSURE. 7 RATIO OF TOTAL PER- FORMANCE of equal Weights of Steam. (CoL 4-5-CoL 6.) initial volume stroke = 100. initial pres- mean pres- initial pres- with 100 % of admission SKI. sure=i. sures^. sure=i. = 1.000. 3-35 1.209 25.0 or ]i .637 1.570 .298 2.129 3-4 1.224 24-5 .631 1.585 .294 2.146 3-5 1.253 23.6 .619 1.615 .286 2.104 3-6 I.28l 22.7 .608 1.645 .278 2.187 3-7 1.308 21.9 597 1.675 .270 2.2II 3-8 1-335 21.2 .589 1.698 .263 2.240 3-9 1.361 20.4 579 1.727 .256 2.202 4.0 1.386 19.7 or Vs .567 1.764 .250 2.278 4.1 1.411 19.1 559 1.789 .244 2.291 4.2 1-435 18.5 551 I.8I5 .238 2.315 4-3 1.459 17.9 .542 1.845 233 2.326 44 1.482 17-3 533 1.876 .227 2.348 4-5 1.504 1 6.8 or i/6 .526 I.90I .222 2.370 4.6 1.526 16.3 .518 1.930 .217 2.387 4-7 1.548 15.8 .511 1-957 .213 2.399 4.8 1.569 15-3 .503 1.988 .208 2.4l8 4.9 1.589 14.8 494 2.024 .204 2.422 5.0 1.609 14.4 or */7 .488 2.049 .200 2.440 5.2 1.649 13.6 .476 2.IOI 193 2.466 5-4 1.686 12.8 .462 2.164 .185 2.497 5-5 1.705 12.5 or x /8 457 2.188 .182 2.5II 5.6 1.723 I2.I .450 2.222 .178 2.528 5.8 1.758 11.4 .438 2.283 .172 2-547 5-9 1-775 1 1. 1 or i/g 432 2.315 .I6 9 2.556 6.0 1.792 10.8 .427 2.342 .167 2.567 6.2 1.825 10.3 .419 2.387 .161 2.585 6.3 1.841 10.0 or i/zq 413 2.421 .159 2.597 6.4 1.856 9-7 .407 2.457 .156 2.609 6.6 1.887 9.2 or i/n .398 2.513 .152 2.6l9 6.8 1.917 8.7 .388 2-577 .147 2.639 7.0 1.946 8.3 or 1/12 .381 2.625 143 2.664 7.2 1.974 7-9 373 2.681 139 2.683 7-3 1.988 7-7 or 1/13 .369 2.710 137 2.693 7-4 2.001 7-5 365 2.740 135 2.703 7.6 2.028 7.1 or i/ I4 357 2.801 .132 2.7II 7.8 2.054 6.7 or '/is .348 2.874 .128 2.719 8.0 2.079 6.4 or 1/16 342 2.924 .125 2.736 The pressures have been calculated on the supposition that the pressure of steam, during its admission into the cylinder, is uniform up to the point of cutting off, and that the expansion is continued regularly to the end of the stroke. In practice, of course, there are deviations from these ideal conditions. Wiredrawing action occasionally causes a fall of pressure during admission, and the opening of the exhaust before the piston arrives at the end of the stroke causes the expansion-line to fall away towards the end. 838 STEAM ENGINE SINGLE-CYLINDER. The allowances necessary to be made for these deviations, as well as for the back pressure of the air in non-condensing engines, and that from the condenser in condensing engines, and for compression of exhaust steam towards the end of the return stroke, will be considered at a subsequent stage. The calculations have been made for periods of admission ranging from 100 per cent, or the whole of the stroke, to 6.4 per cent, or x /i6th of the stroke. And though, nominally, the expansion is 16 times in the last instance, it is actually only 8 times, as given in the first column. The great difference between the nominal and the actual ratios of expansion is caused by the clearance, which is equal to 7 per cent, of the stroke, and causes the nominal volume of steam admitted, namely, 6.4 per cent., to be augmented to 6.4+7 = 13.4 per cent, of the stroke, or more than double, for expansion. When the steam is cut off at I /g th, the actual expansion is only 6 times ; when cut off at J / 5 th, the expansion is 4 times ; when cut off at yd, the expansion is 2^3 times; and to effect an actual expansion to twice the initial volume, the steam is cut off at 46 *4 per cent, of the stroke, not at half stroke. Though a uniform clearance of 7 per cent, at each end of the stroke has been assumed as a fair average proportion for the purpose of compiling the table, the clearance of cylinders with ordinary slides varies considerably say, from 5 to 8 or 9 per cent. With the mean clearance, 7 per cent., that has been assumed, the table gives approximate results sufficient for most practical purposes; they will economize calculation, and they are certainly more trustworthy than such as can be deduced by calculations based on simple tables of hyperbolic logarithms, where clearance is neglected. It has already been exemplified at page 831, how the table may serve in making approximate calculations when the clearance is other than 7 per cent. TOTAL WORK DONE BY ONE POUND OF STEAM EXPANDED IN A CYLINDER. If i Ib. of water be converted into steam of atmospheric pressure 14.7 Ibs. per square inch, or 2116.8 Ibs. per square foot it gradually occupies a volume equal to 26.36 cubic feet; and the work done in acquir- ing this volume under one atmosphere is equal to 2116.8 Ibs. x 26.36 feet = 55,799 foot-pounds. The equivalent quantity of heat expended is i unit per 772 foot-pounds, or, altogether, 55,799-^-772 = 72.3 units. This is precisely the work of i Ib. of steam of one atmosphere, acting on a piston without expansion. The gross work thus done on a piston by i Ib. of steam, generated at total pressures varying from 15 Ibs. to 100 Ibs. per square inch, varies, in round numbers, from 56,000 to 62,000 foot-pounds, equivalent to from 72 to 80 units of heat. The simple work of a pound of steam, without expansion, thus exempli- fied, is reduced by clearance according to the proportion it bears to the net capacity of the cylinder. If the clearance be 7 per cent, of the stroke, then 107 parts of steam are consumed in doing the work of a stroke, which is represented by 100 parts, and the work of a given weight of steam without expansion, admitted for the whole of the stroke, is reduced in the ratio of 107 to 100. Having determined, by this ratio, the quantity of work by i Ib. of steam without expansion, as reduced by clearance, the work for various ratios of expansion may be deduced from that, in terms of the WORK DONE BY ONE POUND OF STEAM. 839 relative performance of equal weights of steam, as exemplified, page 835, and given in the 7th column of table No. 295. To find the total actual work of i Ib. of steam, for any ratio of expan- sion, it is only necessary to multiply the simple work, without expansion, as reduced by clearance, by the ratio or relative performance just referred to. The simple work of a pound of steam does not greatly vary with the pres- sure; and, for present purposes, the work of steam of a total pressure of 100 Ibs. per square inch will be calculated and tabulated. This pressure corresponds to a net pressure, above the atmosphere, of 85 Ibs. per square inch a convenient average standard of pressure. The volume of i Ib. of saturated steam of 100 Ibs. per square inch is 4.33 cubic feet, and the pressure per square foot is i44x 100=14,400 Ibs.; then, the total simple work or total initial work, as it may be called is, 14,400x4.33 = 62,352 foot-pounds. This amount is to be reduced for a clearance of, say, 7 per cent, thus: 62,352 x = 58,273 foot-pounds, 107 which is the total simple work of i Ib. of steam of 100 Ibs. total pressure per square inch, after the loss by clearance is deducted ; and, divided by Joule's equivalent, 772, it is equal to 75.5 units of heat. Now, the total or constituent heat of i Ib. of loo-lb. steam, reckoned from a temperature of 212 F., is 1001.4 units; reckoned from 102 F., the temperature of water from the condenser under a pressure of i Ib. per square inch, the con- stituent heat is 1111.4 units. The equivalent of the net simple work, 75.5 units, is, then, 7.5 per cent, of the total heat reckoned from 2i2F., or 6.7 per cent, if reckoned from 102 F. For shorter admissions, with com- plementary expansion, the work is increased as in the following examples : When the steam is cut off at i, k, *A, %, Vs, */*, Vio, Vis o f stroke, the actual ratios of expansion are, i, 1.3, 1.88, 3.35, 4.0, 5.5, 6.3, 7.8 times; the comparative performances of i Ib. of steam are as i, 1.261, 1.616, 2.129, 2.278, 2.511, 2.597, 2.719, and the total actual work of i Ib. of xoo-lb. steam is in the same proportion, 5 8 ,273, 73,513, 94,2oo, 124,066, 132,770, 146,325, I S l ,3?o, 158,414 foot-pounds. The equivalents, as heat, of the actual work done, are 75-5, 95-2, 122.0, 160.7, 171-9, 189.5, 196.1, 205.2 units, which are, in parts of the constituent heat reckoned from 102 F., equal to 6.7, 8.5, 1 1.0, 14.5, 15-5, i7-o, T 7-6, 18.5 per cent. From these examples, it appears that the total work done by i Ib. of steam, without making any allowance for back pressure or other contin- gencies, varies from about 60,000 foot-pounds when applied without expan- sion, to about double that, or about 120,000 foot-pounds, when expanded three times, cutting off at about 27 per cent, of the stroke; and to about 840 STEAM ENGINE SINGLE-CYLINDER. 150,000 foot-pounds, or 2^ times the first performance, when expanded about six times, cutting off at about 10 per cent, of the stroke. Also, that, of the heat consumed in the formation of steam, not 7 per cent, is converted into total work when there is no expansive action; that substantially with an expansion of six times there is only 17% per cent converted ; and that even with an expansion of eight times, when the steam is cut off at Vis** 1 ? I GSS than 20 P er cent., or one-fifth of the heat consumed, is converted into work. The remainder of the heat is lost, as for the pur- pose of the steam-engine. CONSUMPTION OF STEAM WORKED EXPANSIVELY PER HORSE-POWER OF TOTAL WORK PER HOUR. The measure of a horse-power is the performance of 33,000 foot-pounds per minute, or of 33,000 x 60 = 1,980,000 foot-pounds per hour. This work is to be divided by the work of i Ib. of steam, and the quotient is the weight of steam or water required per horse-power per hour. For example, the total actual work done in the cylinder by i Ib. of zoo-lb. steam, without expansion, and with 7 per cent, of clearance, is 58,273 foot-pounds; and 1,9 0,000 _ _^ ^ O f s team, is the weight of steam consumed for the total 5 8 > 2 73 work done in the cylinder per horse-power per hour. For any shorter period of admission, with expansion, the weight of steam per horse-power is less, as the total work by i Ib. of steam is more, and may be found by dividing 1,980,000 foot-pounds by the respective total work done; or by dividing 34 Ibs. by the ratio of performance, column 7, table No. 295. In this way it is found that, when the steam is cut off at i, Y^ fa fa x /s fa x /.o, Vis of stroke, the quantities of steam, or water as steam, consumed per horse-power of total work per hour, are 34.0, 26.9, 21.0, 1 6.0, 14.9, 13.5, 13.1, 12. 5 Ibs. Further, allowing that 10 Ibs. of steam are generated by the combustion of i Ib. of coal, the fuel consumed per horse-power of total work per hour is, 3.40, 2.69, 2.10, i. 60, 1.49, 1.35, 1.31, 1.25 Ibs. TABLE (No. 296) OF THE TOTAL WORK DONE BY i POUND OF STEAM OF 100 LBS. TOTAL PRESSURE PER SQUARE INCH. The table No. 296, which follows, is compiled on the basis of the con- ditions above laid down, which are repeated under the heading of the table, for ready reference. The ist, 2d, and 3d columns are repeated from table No. 295. The 4th column, of total actual work done by i Ib. of steam of 100 Ibs. total pressure, is calculated by multiplying the work without expansion, namely, 58,273 foot-pounds, by the ratios in column 3, for the proportional work when expanded. The 5th column contains the equivalent of heat converted into work, which is found by dividing the work in foot-pounds by Joule's equivalent, 772; and the 6th and 7th columns give these values as percentages of the total heat of steam raised from 212 and 102 F. respectively. The 8th column contains the quantity of steam consumed for the total work done per horse-power per hour. WORK DONE BY ONE POUND OF STEAM. 841 Table No. 296. TOTAL WORK DONE BY ONE POUND OF STEAM OF 100 LBS. TOTAL PRESSURE PER SQUARE INCH. ASSUMPTIONS. That the initial pressure is uniform; that the expansion is complete to the end of the stroke; that substantially the pressure in expansion varies inversely as the volume; that there is no back pressure; and that there is no compression. Volume of I Ib. of steam of 100 Ibs. pressure per square inch, or 14,400 Ibs. per square foot, 4.33 cubic feet. Product of initial pressure and volume, 62,352 foot-pounds. Constituent heat of I Ib. of this steam Reckoned from 212 F., 1001.4 units. Reckoned from 102 F., 1111.4 units. Clearance at each end of the cylinder, 7 per cent, of the stroke. ACTUAL RATIO OF EXPAN- SION. CORRESPONDING PERIOD OF ADMISSION or CUT-OFF, in percentage of Stroke. TOTAL ACTUAL WORK DONE by i Ib. of loo-lb. Steam. EQUIVALENT OF HEAT converted into Work. Quantity of Steam con- sumed per Horse-power of actual Work done per Hour. Ratio of Work done (col. 7, table No. 295). Actual Work done. Heat con- verted. Percentage of Consti- tuent Heat converted, as calculated from 212 F. and 102 F. (i) (2) (3) (4) (5) (6) (?) (8) initial vol. = i. per cent. foot- pounds. units. %from 2I2;F. %from 102 F. Ibs. 1.0 100 .000 58,273 75.5 7-5 6.7 34-0 1.05 95 .050 6l,I93 79-3 7-9 7-1 32-4 I.I 90.3 or 9/ IO .096 63,850 82.7 8-3 7-5 31.0 I.I5 86.0 .138 66,310 85.9 8.6 7.8 29-9 1.18 83-3 or s/6 .164 67,836 87.9 8.8 7-9 29.2 1.2 82.1 .180 68,766 89.1 8.9 8.0 28.8 1.23 80.0 or 4/5 1. 206 70,246 91.0 9.1 8.2 28.2 1.25 78.6 1. 221 71,151 92.2 9.2 8-3 2 7 .8 1-3 75-3 or # I.26l 73,513 95.2 9-5 8.5 26.9 i-35 72.3 1.297 75,575 97-9 9-8 8.8 26.2 1-39 70.0 or 7/ IO I-325 77,242 100. 1 IO.O 9.0 25.6 1.4 69.4 1.332 77,6i6 100.6 10. 1 9.1 25.5 1.45 66.8 or 2/ 3 1.365 79,555 102.9 10.3 9-3 24.9 i-5 64-3 1-399 81,546 105.6 10.6 9-5 24-3 1.54 62.5 or % 1.425 83,055 107.6 10.8 9-7 23.8 i-55 62.0 1.429 83,299 107.9 10.8 9-7 23-7 1.6 59.9 or 3/ 5 1.461 85,125 110.3 II.O 9-9 23-3 1.65 57-9 1.490 86,828 112.5 n-3 10.2 22.8 i-7 56.0 1.520 88,598 114.8 11.5 10.4 22.4 1-75 54.1 1.546 90,115 116.7 11.7 10.5 22.0 1.8 52.4 1-573 91,680 118.7 11.9 10.7 21.6 1.85 50.8 1-597 93,o65 120.5 12.0 10.8 21.3 1.88 50.0 or y z 1.616 94,200 122.0 12.2 II.O 21.0 1.9 49-3 1.624 94,610 122.5 12.3 ii. i 20.9 1.95 47-9 1.649 96,100 124.5 12.4 1 1.2 20.6 2.0 46.5 1.672 97,432 126.2 12.6 1 1-3 20.3 2.1 44.0 1.718 100,266 129.7 13.0 ii -7 I 9 .8 2.2 41.6 1.756 102,366 132.5 13.2 11.9 194 2.28 40.0 or 2/5 1-793 104,466 135-3 13-5 12.2 19.0 2-3 39-5 1.798 104,855 135-7 13.6 12.3 18.9 2.4 37.6 or 3/ s 1.837 107,050 138.6 13.9 12.5 18.5 2.5 35-8 1.875 109,266 I4I.5 I4.I 12.7 18.1 2.6 34-2 1.912 1 1 1,400 144.3 144 13.0 17.8 842 STEAM ENGINE SINGLE-CYLINDER. Table No. 296 (continued}. ACTUAL RATIO OF EXPAN- SION. CORRESPONDING PERIOD OF ADMISSION or CUT-OFF, in percentage of Stroke. TOTAL ACTUAL WORK DONE by i Ib. of loo-lb. Steam. EQUIVALENT OF HEAT converted into Work. Quantity of Steam con- sumed per Horse-power of actual Work done per Hour. Ratio of Work done (col. 7, table No. 295). Actual Work done. Heat con- verted. Percentage of Consti- tuent Heat converted, as calculated from 212 F. and 102 F. (*) (2) (3) (4) (5} (6) (?) (8) initial vol. = i. per cent. foot- pounds. units. % from 212 F. %from 102 F. Ibs. 2.65 33-3 or 1/3 1.925 112,220 1454 14.5 I3- 1 177 2.7 32.6 1-943 113,244 146.7 14.7 13.2 I 7 .6 2.8 30.2 1.978 115,244 149.2 14.9 13.4 17.2 2.9 29.9 or 3/ IO 2.006 116,885 I5I.4 I5.I 13.6 16.9 3-o 28.7 2.039 118,820 153-9 154 13-9 I6. 7 3.1 27.5 2.059 119,970 155-4 15-5 13-9 I6. 5 3-2 26.4 2.083 121,386 157.2 15-7 14.1 I6. 3 3-3 25.4 2.115 123,278 159.6 1 6.0 14.4 16.1 3-35 25.0 or ]i 2.129 124,066 160.7 16.1 14.5 1 6.0 3-4 24.5 2.146 125,066 162.0 16.2 14.6 15.8 3-5 23-6 2.164 126,125 163.4 16.3 14.7 15-7 3-6 3-7 22.7 21.9 2.187 2.2II 127,450 128,860 165.1 166.9 16.5 16.7 14.9 15.0 15-5 15.4 3-8 21.2 2.240 130,533 169.1 16.9 15.2 15.2 3-9 20.4 2.262 131,800 170.7 17.1 15.4 15.0 4.0 19.7 or 1/5 2.278 132,770 I7I.9 17.2 15-5 14.9 4.1 l % 1 2.291 133,500 172.9 17.3 15.6 14.8 4.2 18.5 2.315 134,900 174.8 17-5 15.8 14.7 4.3 17.9 2.326 135,555 175.6 17.6 15.8 14.6 4.4 17-3 2.348 136,825 177.2 17.7 15.9 14.5 4-5 1 6.8 or Ve 2.370 138,130 178.8 17.9 16.1 14-34 4.6 16.3 2.387 139,100 180.2 1 8.0 16.2 14.23 4-7 15.8 2-399 139,800 181.1 18.1 16.3 14.16 4-8 15-3 2.418 140,920 182.5 1 8.2 16.4 14.05 4-9 14.8 2.422 141,210 182.8 18.3 16.5 14.03 5.0 14.4 or 1/7 2.440 I42,l8o 184.2 18.4 16.6 13.92 5.2 13-6 2.466 143,720 186.2 18.6 .16.9 I3-78 5-4 12.8 2.497 145,525 188.5 18.8 16.9 13.60 5-5 12.5 or y 8 2.5II 146,325 189.5 18.9 17.0 13-53 5-6 12. 1 2.528 147,320 190.8 19.1 17.2 13-44 5.8 1 1.4 2.547 148,390 192.2 19.2 17.3 13-34 5-9 1 1. 1 or 1/9 2.556 148,940 192.9 19-3 17.4 13.29 6.0 10.8 2.567 149,586 193.7 19.4 17-5 13-23 6.2 10.3 2.585 150,630 19.5 17.6 I3-I4 6-3 i o.o or VIQ 2-597 151,370 196.1 19.6 17.6 13.08 6.4 9-7 2.609 152,033 196.9 19.7 17.7 13.02 6.6 9.2 or Vn 2.619 152,595 197.7 19.8 17.8 12.98 6.8 8-7 2.629 153,810 199.2 19.9 17.9 12.87 7.0 8.3 or 1/12 2.664 155,200 201. 1 20.1 18.1 12.75 7-2 7-9 2.683 156,330 202.6 20.3 18.3 12.66 7-3 7.7 or i/,3 2.693 156,960 203.3 20.3 18.3 12.61 7.4 7-5 2.703 157,560 204.1 20-4 18.4 12.57 7-6 7-i or i/ I4 2.7II 157,975 204.6 20.4 18.4 12.53 7-8 6.7 or Vis 2.719 158,414 205.2 20.5 18.5 12.50 8.0 6.4 or Vie 2.736 159,433 2O6.5 20.7 1 8.6 11.83 CYLINDER-CAPACITY AND WORK DONE. 843 APPENDIX TO TABLE No. 296. TABLET OF MULTIPLIERS for the total Work done by i Ib. of Steam of other Pressures than 100 Ibs. per Square Inch, to be applied to the total actual Work as given in the table. See explanatory notice of the table, page 840. Total Pressures below 100 Ibs. per Square Inch. Total Pressures above 100 Ibs. per Square Inch. Ibs. per square inch. multiplier. Ibs. per square inch. multiplier. 65 975 IOO .000 70 .981 no .009 75 .986 120 .Oil 80 .988 130 .015 85 .991 140 .022 90 95 995 .998 If? .025 .031 An initial total pressure of 100 Ibs. per square inch has been adopted for the table, as an average pressure in ordinary good practice, and the contents of the table are good as approximate values for other pressures considerably different from 100 Ibs., more or less. A tablet is, however, appended to the table No. 296, containing multipliers for various other total pressures, which may be applied to the total actual work given in the table for the purpose of determining the correct total quantities of work for steam of the respec- tive pressures. These multipliers are arrived at by multiplying the total pressure of any other given steam per square foot, by the volume in cubic feet of i Ib. of such steam, and dividing the product by 62,352, which is the product in foot-pounds for steam of 100 Ibs. pressure. The quotient is the multiplier for the given pressure. From the tablet it appears that, between the extremes of 65 Ibs. and 160 Ibs. per square inch, the deviation from the work done as given for 100 Ibs. pressure does not exceed 2^ and 3 per cent. NET CYLINDER-CAPACITY RELATIVE TO THE STEAM EXPENDED AND WORK DONE IN ONE STROKE. The quantity of cylinder-capacity required for the performance of a given weight of steam admitted for one stroke depends on the volume of the steam, and on the ratio of expansion. If the given weight admitted be multiplied by the volume of i Ib. of the steam and by the actual ratio of expansion, the product is the gross cylinder-capacity, including clearance. For example, if i Ib. of steam of 100 Ibs. pressure per square inch be admitted for the whole stroke, without expansion, the gross capacity is the volume of i Ib. of such steam, namely, 4.33 cubic feet; and the net capacity, supposing the clearance to be 7 per cent, of the stroke, is 4-33 IOO 100 + 7 = 4.047 cubic feet. 844 STEAM ENGINE SINGLE-CYLINDER. If, again, 2 Ibs. of steam of 100 Ibs. pressure be admitted and expanded into three times its initial volume, the gross capacity is, 4.33 x 2 x 3 = 25.98 cubic feet; and the net capacity is 25.98 x 1^? = 24.28 cubic feet. 107 From this is derived following rule for net capacity : RULE 6. To find the net Capacity of Cylinder for a given Weight of Steam admitted for one stroke, and a given actual ratio of expansion. Multiply the volume of i Ib. of the steam by the given weight in pounds, and by the actual ratio of expansion. Multiply the product by 100, and divide by 100 plus the percentage of clearance. The quotient is the net capacity of cylinder. Again, the quantity of cylinder-capacity required for the performance of a given amount of total actual work, in one stroke, depends on the initial pressure, and the actual ratio of expansion, according to the follow- ing rule : RULE 7. To find the net Capacity of Cylinder for the performance of a given amount of total Actual Work, in one stroke, with a given initial pressure, and actual ratio of expansion. Divide the given work by the total actual work done by i Ib. of steam of the same pressure, and with the same actual ratio of expansion; the quotient is the weight of steam necessary to do the given work, for which the net capacity is found by Rule 6, preceding. Conversely, the weight of steam admitted per cubic foot of net capacity, for one stroke, is the reciprocal of the cylinder-capacity per pound of steam, as obtained by Rule 6. Likewise, the total actual work done per cubic foot of net capacity, for one stroke, is the reciprocal of the cylinder-capacity per foot-pound of work done, as obtained by Rule 7. Finally, the total actual work done per square inch of piston, per foot of the stroke, is Vi44 tn P art f tne wor k done per cubic foot; a prism i inch square and i foot long being I / I44 th P art f a cubic foot. The work, in either measure, is in direct proportion to the mean total pressure per square inch. TABLE OF RELATIONS OF NET CAPACITY OF CYLINDER TO STEAM ADMITTED AND WORK DONE. The table No. 297 gives the net capacity of cylinder required in relation to the quantity of steam of 100 Ibs. total pressure per square inch consumed, and of work done, in one stroke. Columns i, 2, and 3, are the ratios of expansion, periods of admission, and total actual work done by i Ib. of steam of 100 Ibs. pressure, repeated from columns i, 2, and 4, in the previous table, No. 296. In the 4th column are the net capacities of cylinder required for each pound of steam admitted for one stroke, found by Rule 6; and in the 5th column are the net capacities of cylinder for 100,000 foot-pounds of total actual work done in one stroke, found by Rule 7. The 6th column contains the weights of steam of 100 Ibs. pressure admitted to the cylinder for one stroke, per cubic foot of CYLINDER-CAPACITY AND WORK DONE. 845 net capacity. These values are simply the reciprocals of those in column 4. The yth column contains the total actual work done by steam of 100 Ibs. initial pressure, for one stroke, per cubic foot of net capacity. These values are the products of the reciprocals of those in column 5, by 100,000, since this is the number of foot-pounds for which the values in column 5 are calculated. The 8th column gives the total actual work per square inch of piston-area, per foot of stroke, in foot-pounds; the initial pressure is 100 Ibs., and the following pressures, for the different ratios of expansion, may also be read as percentages of the initial pressure. The total actual works are directly proportional to the mean total pressures per square inch, as given in column 4, table No. 295, page 836, where the initial pressure is taken as i; and the former have been found by multiplying the latter respectively by 100. The contents of the table No. 297 are calculated for steam of 100 Ibs. per square inch, total initial pressure ; but they are available, with the aid of a set of multipliers, for other pressures. For column 3, the total actual work done, the multipliers have already been given in the tablet appended to the table at page 848, for pressures of from 65 Ibs. to 160 Ibs. per square inch. For column 4, of the net capacity of cylinder per pound of steam expended in one stroke, the multipliers are simply the ratios of the volume of i pound of zoo-lb. steam to the respective volumes of i pound of steam of other initial pressures. Thus, for steam of 65 Ibs. total pressure, of which the volume of j pound is 6.49 cubic feet, to be compared with 4.33 cubic feet, which is the volume of a pound of loo-lb. steam, the multiplier is and the net capacity of cylinder per pound of 65-00. steam, when the steam is admitted for the whole of the stroke, is 4.05 cubic feet (for loo-lb. steam) x 1.5 = 6.07 cubic feet. For column 5, of the net capacity of cylinder per 100,000 foot-pounds of total actual work done in one stroke, the capacity for other pressures is modified, in the first place, in the inverse ratio of the multipliers, as found for tablet, page 848, for loo-lb. steam, and steam of the given pressure; secondly, in the ratio of the volume of i pound of loo-lb. steam, to that of a pound of the given steam. For example, for steam of 65 Ibs. total pressure per square inch, the weight of loo-lb. steam to do a given total work, as compared with the weight of 65-lb. steam, is as .975 to i.ooo, and the volumes of a pound each of the two steams are respectively 4.33 and 6.49 cubic feet, or as i to 1.5 as already found. The values in column 5 are therefore to be increased in the compound ratio of .975 to i.ooo and i to 1.5 or, combined, as .975 to 1.5, or as i to 1.54. The multiplier for 65-lb. steam is thus 1.54. 846 STEAM ENGINE SINGLE-CYLINDER. Table No. 297. NET CYLINDER-CAPACITY, WITH RELATION TO STEAM ADMITTED, AND TOTAL ACTUAL WORK DONE. For Steam of 100 Ibs. total pressure per square inch. Clearance at each end of the Cylinder, 7 per cent, of the stroke. Net Capacity of Per Cubic Foot of Net Cylinder. Capacity of Cylinder. Perioojooo WEIGHT TOTAL TOTAL ACTUAL ACTUAL RATIO OF EXPAN- SION. PERIOD OF ADMISSION, or CUT-OFF, as a Percentage of Stroke. TOTAL ACTUAL WORK DONE by i Ib. of loo-lb. Steam. Per pound of loo-lb. Steam, admitted in one Stroke. Foot- pounds of Total Ac- tual Work done by Steam of 100 Ibs. OF STEAM of 100 Ibs. Total Pressure admitted for one Stroke, ACTUAL WORK DONE by Steam of 100 Ibs., Total Initia Pressure, in one WORK DONE per Sq. Inch of Piston, per Foot o Stroke, by 100 Ibs. pressure, in one per Cubic Stroke, per Steam. Stroke. Foot. Cubic Foot. (i) (2) (3) (4) (5) (6) (7) (8) initial vol- ume = i. per cent. foot-pounds. cubic feet. cu. feet. pound. foot-pounds. foot- pounds. 1.0 100 58,273 4.05 6.94 .247 14,400 100 1.05 95.0 6l,I93 4-25 6.95 235 14,388 99-97 I.I 90.3 or 9/ IO 63,850 445 6.97 .225 14,347 99.6 1.1* 86.0 66,310 4.65 7.02 .215 14,245 99.0 1.18 83.3 or s/6 67,836 4.78 7.04 .209 14,204 98.6 1.2 82.1 68,766 4.86 7.06 .206 14,164 98.3 1.23 80.0 or 4/5 70,246 4.98 7.09 .201 14,104 98.0 1.25 78.6 71,151 5.06 7.II .198 14,065 977 i-3 75-3 or % 73,513 5.26 7 .l6 .190 13,966 96.9 1-35 72.3 75,575 5.46 7.23 .183 13,831 96.1 1-39 70.0 or 7/ IO 77,242 5.63 7.28 .178 13,736 95-3 1.4 69.4 77,616 5.67 7.30 .176 13,699 95.1 145 66.8 or 2/3 79,555 5.87 7.38 .170 !3,55 94.2 i-5 64-3 81,546 6.07 7-45 .165 13,423 93-2 1.54 62.5 or }i 83,055 6.23 7.50 .161 13,333 92.5 i-55 62.0 83,299 6.27 7-53 .159 13,280 92.2 1.6 59.9 or 3/ 5 85,125 6.47 J 55 I3,!4i 9 J -3 1.65 57-9 86,828 6.68 7-69 .150 13,004 i.7 56.0 88,598 6.88 7-77 .145 12,870 89.4 1-75 54.1 90,115 7.08 7.92 .141 12,626 88.3 1.8 52,4 91,680 7.30 7-95 .137 12,579 87.3 1.85 50.8 93,065 7-49 8.04 134 12,438 86.4 1.88 50.0 or l / 2 94,200 7.61 8.08 !3i 12,376 86.0 1.9 49-3 94,610 7-69 8.13 .130 12,300 85.4 1.95 47-9 96,100 7.89 8.22 .127 12,165 84.6 2.0 46.5 97,432 8.09 8. 3 I .124 12,034 83-6 2.1 44.0 100,266 8.50 8.49 .118 11,778 81.8 2.2 41.6 102,366 8.90 8.70 .112 11,494 79-9 2.28 40.0 or 2/5 104,466 9.23 8.83 .108 n,325 78.7 2-3 39-5 104,855 9-3i 8.89 .107 11,249 78.2 2.4 37-6 or 3/ 8 107,050 9.71 9.07 .103 11,025 76.6 2.5 35-8 109,266 10.12 9.26 .099 10,799 75.0 2.6 34-2 111,400 10.52 945 .095 10,582 73-6 2.65 33-3 or 1/3 112,220 10.72 9.56 093 10,460 72.6 2.7 32.6 113,244 10.93 9.65 .091 10,363 71.9 2.8 30.2 115,244 11-33 9.83 .088 10,173 70.6 2.9 29.9 or 3/ 10 116,855 11.74 10.04 .085 9,960 69.2 CYLINDER-CAPACITY AND WORK DONE. 847 Table No. 297 (continued). Net Capacity of Per Cubic Foot of Net Cylinder. Capacity of Cylinder. . Perioo.ooo TTt_ _. WEIGHT TOTAL TOTAL ACTUAL ACTUAL RATIO OF EXPAN- SION. PERIOD OF ADMISSION, or CUT-OFF, as a Percentage of Stroke. TOTAL ACTUAL WORK DONE by i Ib. of loo-lb. Steam Per pounc of loo-lb. Steam, admitted in one Stroke. root- pounds of Total Ac- tual Work done by Steam of loo Ibs. OF STEAM of 100 Ibs. Total Pressure admitted for one Stroke, ACTUAL WORK DONE by Steam o 100 Ibs., Total Initia Pressure, in one WORK DONE per Sq. Inch of Piston, per Foot o Stroke, by zoo Ibs. pressure, per Cubic Stroke, per Steam. in one Stroke. Foot. Cubic Foot. (x) (2) (3) (4) (s) (6) (7) (8) ini ticiJ vol ume=i. per cent. foot-pounds. cubic feet cu. feet. pound. foot-pounds. foot- pounds. 3-0 28.7 118,820 12.14 10.22 .082 9,785 67.9 3-i 27-5 109,970 12.55 10.46 .080 9,560 66.5 3-2 26.4 121,386 12.95 10.67 .077 9,372 65.2 3-3 25.4 123,278 13.35 10.83 .075 9,234 64.1 3-35 25.0 or % 124,066 13.56 10.93 .074 9,H9 637 34 24-5 125,066 13.76 11.00 073 9,091 6 3 .I 3-5 23.6 126,125 14.16 II. 21 .071 8,961 61.9 3-6 22.7 127,450 14-57 H.43 .069 8,749 60.8 37 21.9 128,860 14.97 1 1. 60 .067 8,621 597 3-8 21.2 1 30,533 15.38 11.78 .065 8,489 58. 9 3-9 20.4 131,800 15.78 11.98 .063 8,347 57-9 4.0 19.7 or x/- 5 132,770 16.19 12.19 .062 8,203 56.7 4.1 19.1 133,500 16.59 12.39 .060 8,071 55.9 4.2 18.5 134,900 17.00 1 2.6o .059 7,936 55-i 4-3 17.9 135,555 17.40 12.80 .058 7,8 1 2 54-2 4-4 17.3 136,825 I7.8l 13.01 .056 7,686 53-3 4-5 1 6.8 or x/ 6 138,130 18.21 13.19 .055 7,58i 52.6 4-6 10.3 139,100 18.62 13.38 .054 7,474 51.8 4-7 15.8 139,800 19.02 13.58 053 7,364 51.1 4-8 15.3 140,920 19-43 13.79 .051 7,252 50.3 4.9 14.8 141,210 19.83 14.01 .050 7,138 49.4 5.0 14.4 or i/ 7 I42,l8o 20.23 14.23 .049 7,027 48.8 5.2 13.6 143,720 21.04 14.64 .047 6,831 47-6 5-4 12.8 H5,525 21.85 15.02 .046 6,658 46.2 5.5 12.5 or y 8 146,325 22.25 15.20 .045 6,579 45-7 5-6 12.1 147,320 22.66 15-38 .044 6,502 45-0 5.8 1 1.4 148,390 23.47 15.80 043 6,329 43.8 5-9 1 1. 1 or I / 9 148,940 23.87 1 6.0 1 .042 6,246 43-2 6.0 10.8 149,586 24.28 16.23 .041 6,161 42.7 6.2 10.3 150,630 25.09 16.63 .040 6,013 41.9 6.3 10.0 or Vio 151,370 25.49 16.83 .039 5,942 4L3 6.4 9-7 152,033 25.90 17.04 038 5,868 40.7 6.6 9.2 or i/n 152,595 26.71 1747 037 5,724 39-8 6.8 8-7 153,810 27.52 17.89 .036 5,590 38.8 7.0 8.3 or x/ 12 155,200 28.33 18.27 035 5,473 38.1 7-2 7-9 156,330 29.14 18.64 0343 5,365 37-3 7-3 7-7 or x/ I3 156,960 29.54 18.83 0339 5,3H 36.9 7-4 7-5 157,560 29.95 19.01 0334 5,260 36.5 7.6 7.1 or '/i 4 157,975 30.76 19.47 .0325 5,136 35-7 7-8 6.7 or Vis 158,414 31.57 19-93 .0317 5,o 1 8 34-8 8.0 6.4 or Vie 159,433 32.38 20.31 0309 4,923 34.2 848 STEAM ENGINE SINGLE-CYLINDER. APPENDIX TO TABLE No. 297. TABLET OF MULTIPLIERS FOR NET CYLINDER-CAPACITY, STEAM ADMITTED, AND TOTAL WORK DONE. For Steam of other pressures than 100 Ibs. per square inch. (See explanatory notice of the table, page 844. ) MULTIPLIERS. Total Pres- sures per Square Inch. For Column 3. Total Work. For Column 4. Capacity. For Column 5. Capacity. For Column 6. Weight of Steam. For Columns 7 and 8. Work. Ibs. 65 975 1.50 1.54 .666 .65 70 .981 1.40 1-43 .714 .70 75 .986 31 i-33 .763 75 80 .988 .24 1.25 .806 .80 85 .991 17 1.18 .855 .85 90 995 .11 i. ii .901 .90 95 .998 .05 1.05 .952 95 IOO 1. 000 .00 1. 00 .00 .00 no 1.009 .917 .909 .09 .10 120 I.OII 843 .833 17 .20 130 1.015 .781 .769 .28 30 140 1.022 730 .714 37 .40 150 1.025 683 .667 46 5 1 60 I.03I .644 .625 55 .60 For column 6, of the weights of steam per cubic foot of net capacity for one stroke, the weights given for loo-lb. steam are to be multiplied by the reciprocals of the multipliers for column 4, corresponding to other pres- sures. Thus, for 65-lb. steam, the multiplier is the reciprocal of 1.5, or .666. For column 7, of the total actual work done, per cubic foot of net capacity, for one stroke, the work given for zoo-lb. steam is to be multiplied by the reciprocal of the multiplier for column 5, as determined for the given other pressure. Thus, for 65-^. steam, the multiplier is the reciprocal of 1.54, or .65. For the total actual work done per square inch of piston per foot of stroke, by steam of any other pressure, the values given in column 8, for loo-lb. steam, are to be multiplied by the given pressure and divided by 100. For 65-10. steam, for example, the multiplier is, in fact, 5- .65, the 100 same as for column 7. Otherwise regarded, the values in column 8 may be taken as percentages of the work for steam as admitted for the whole of the stroke, whatever the initial pressure may be. It is apparent that the multipliers for columns 7 and 8, are simply equal to the respective total pressures divided by 100, since the multiplier for loo-lb. pressure is taken as i. These multipliers must obviously be in the direct ratio of the pressures; and, of course, the multipliers for column 5 must be in the inverse ratio of the pressures. COMPOUND STEAM ENGINE. 8 49 COMPOUND STEAM-ENGINE. The compound steam-engine, consisting of two cylinders, is reducible to two forms, in which, first, the steam from the first cylinder is exhausted direct into the second cylinder, as in the Woolf engine; and, second, the steam from the first cylinder is exhausted into an intermediate reservoir, whence the steam is supplied to, and expanded in, the second cylinder, as in the " receiver-engine." In the Woolf engine, according to the original type, the pistons of the two cylinders move together, and make the same strokes simultaneously, the steam from the top and the bottom of the first cylinder, being exhausted into the bottom and the top, respectively, of the second cylinder. In the receiver-engine, the pistons are connected to cranks on one shaft, at right angles to each other. It is, in the first place, assumed that there is no clearance at either end of either cylinder; that there is no frictional resistance nor other hindrance in the engine to the flow of steam; that the pressure of expanding steam is inversely as the volume; that the expansion of the steam is continued in both cylinders to the end of the stroke; and that in the second cylinder the steam acts against a perfect vacuum on the other face of the piston. It is assumed, further, that no intermediate fall, or " drop " of pressure takes place between the first and second cylinders ; that is, that the initial pressure in the second cylinder is equal to the final pressure in the first cylinder; also, that there is no loss of steam by condensation within the cylinders. Taken generally, the work done by expansion into the second cylinder, is that due to the increase of volume of the steam in this, the second stage. WOOLF ENGINE IDEAL DIAGRAMS. The diagrams of pressure of the Woolf engine are produced in Fig. 334, as they would be described by the indicator, according to the arrows. In these ideal diagrams, pq is the atmo- spheric line, mn the straight line of perfect vacuum, cd the straight line of admission; the terminal lines, me and ng, straight and perpendicular to the atmospheric line, dg the hyper- bolic curve of expansion in the first cylinder, and^ the consecutive expan- sion-line of back pressure for the re- turn stroke of the first piston, and of positive pressure for the steam-stroke of the second piston. At the point h, at the end of the stroke of the second piston, the steam is exhausted into the condenser, and the pressure falls to the level of perfect vacuum, mn. The diagram pertaining to the second cylinder, below the curve gh, is char- Fig. 334. Woolf Engine : Ideal Indicator Diagrams. acterized by the absence of any specific period of admission; the whole of the steam-line gh being expansive, and generated by the expansion of the initial body of steam contained in the first cylinder into the second. The 850 STEAM ENGINE COMPOUND CYLINDERS. initial volume the volume of the first cylinder is, however, gradually re- duced by the advancing movement of the first piston, by which the initial steam is gradually driven into the second cylinder; until, when the stroke is completed, the whole of the steam is transferred from the first, and is shut into the second cylinder. The first cylinder, then, acts as a collaps- ible head to the second cylinder for each steam-stroke of the latter; the second cylinder being, at the beginning of the stroke, augmented by the capacity of the first; and at the end of the stroke, reduced to its normal dimensions. The final pressure and volume of the steam in the second cylinder are, consequently, the same as if the whole of the initial steam had been admitted at once into the second cylinder, and then expanded to the end of the stroke; the hypothetical period of admission being such a frac- tion of the stroke of the second cylinder as would represent the ratio of the volume of the first cylinder, which is the volume of the steam admitted, to that of the second cylinder. The net work of the steam, according to the supposed distribution in one cylinder only, would be the same as in the two cylinders compounded. Construct a combined diagram, with a continuous expansion-line, by piecing the first upon the second of the two diagrams, Fig. 334, page 849, representing the work as if it were done in one cylinder equal in capacity to the second of the compound cylinders. For this purpose, the first diagram is to be contracted to a length bearing the same proportion to the length of the second diagram, as the volume of the first cylinder to that of c d Fig- 335- Woolf Engine : Ideal Diagrams, reduced. Fig- 336. Woolf Engine : Ideal Diagrams, combined. the second cylinder in this example, one-third. It is necessary to do so in order to reduce the two elements of the combined diagram to the same hori- zontal scale for measurement. When the strokes of the cylinders are equal to each other, the capacities are in the simple ratio of their areas. In this example, therefore, the length of the first diagram is to be reduced to one- third of the length of the second. For this purpose, l&ghmn, Fig. 335, annexed, be the diagram from the second cylinder, over which the original diagram from the second cylinder is shown in dotting. Draw gg"' parallel to the base, and mhc perpendicular to the base, equal to the height of the first diagram; set off g'"g" equal to one-third of the base, and upon this WOOLF ENGINE IDEAL DIAGRAMS. 851 reduced base g'"g" complete the first diagram cdg", by drawing cd parallel to the base, and equal to one-third of it, and the expansion-curve dg". For the lower part of the figure, the line of back pressure g"h may be described by the method of ordinates, repeated from the curve gh. Thus the contracted diagram cdg"h is completed. To combine the first diagram, thus reduced to uniformity of scale, with the second diagram, let again ghmn, Fig. 336, annexed, be the second diagram, and describe the first diagram as reduced, in a reversed position cdg"h', at the head of the second diagram, the same letters of reference being used. Finally, continue the hyperbolic expansion- line dg" to the end of the stroke at h. Then the area gg*h t is equal to the axeB.gg ff ft', and, when substituted for it, completes the regular indicator diagram cdhmn, with a continuous expansion-line dg"h. The substitution is, in fact, necessary, since the lower part of the first diagram partly overlaps the second diagram. According to this combination, the upper part of the diagram, Fig. 336, above the curve gh, namely, cdhg, represents the diagram, as contracted, for the first cylinder, modified in form, but unaltered in area; and the lower part, below the curve gh, remains unaltered, both in form and in area, as the diagram for the second cylinder. The combined diagram, as a whole, exactly measures the whole net work done in both cylinders, and is such as would be formed by admitting and expanding the same quantity of steam in one cylinder having the dimensions of the second cylinder, with the period of admission, cd, equal to one-third of the capacity of the first cylinder, or one-ninth of the capacity or the stroke of the second cylinder. It follows further, that the work effected by expansion into the second cylinder of the Woolf engine, that is, the total work arising from expansion against the second piston, plus the gain of work in the first cylinder by the gradual reduction of back pressure in accordance with the expansion, is equal to that which would be effected by delivering the whole of the steam into the second cylinder before expansion is commenced, as in the receiver- engine. By this distribution, the upper part of the combined diagram, Fig. 336, cut off by the horizontal line gg", would measure the net work of the first cylinder, as there would be a uniform back pressure equal to ng on the piston; and the lower part of the diagram, below gg", would mea- sure the work of the second cylinder, with a period of admission equal to gg", the capacity of the first cylinder at the pressure ng, and with expansion to the end of the stroke. To exemplify the foregoing conclusions, under the conditions originally stated, suppose that the steam is admitted to the first cylinder at a total initial pressure of 63 Ibs. per square inch; that the areas of the first and second cylinders are respectively i and 3 square inches, and that the common length of stroke is 6 feet. The steam being cut off in the first cylinder at one-third of the stroke for the period of admission, cd, Fig. 335, page 850, it is expanded to three times its initial volume, and to one-third of the initial pressure, namely, ng, equal to 21 Ibs., at the end of the stroke. The steam is admitted to the second cylinder at the same pressure, ng, and is expanded there to three times the volume it acquired in the first cylinder, or to 3 x 3 = 9 times the initial volume in the first cylinder. At the same time, the pressure is reduced in the second cylinder to one-third of the final pressure in the first cylinder, or to one-ninth of the initial pres- sure there, namely, to 7 Ibs. per square inch, measured by mh, Fig. 335. 852 STEAM ENGINE COMPOUND CYLINDERS. The work of the compound engine may be calculated from the combined diagram, Fig. 336, page 850; regarding the upper part of the figure, above the line gg ff , as the net work of the first cylinder, according to the equivalent distribution mentioned at page 851, where the action is compared to that of a receiver-engine; and the lower part, below gg", as the work of the second cylinder, with a period of admission equal to gg", and an expansion to the end of the stroke. For the first section, the total work, over the base ", is calculated, and the work of the pressure, ng, as back pressure, on the same base, is deducted from it to give the net work. Now, the total pressure, nc, calculated on the area of the second cylinder, is 63 Ibs. * 3 square inches = 189 Ibs.; and the period of admission, cd, is one-ninth of the stroke, or 2 / 3 foot. The total initial work is, then, 189 x 2/3 = 126 foot- pounds, and the total work, with an expansion of three times for the stroke, gg", 2 feet, is 126 x (i +hyp log 3) = 264.4236 foot-pounds. The work of the back pressure, ng, or 21 x 3 = 63 Ibs., into the stroke, gg", or 2 feet, is 63 Ibs. x 2 feet= 126 foot-pounds, and the net work, above the line gg", is 264.4236 - 126 = 138.4236 foot-pounds. For the second section, according to the equivalent distribution, the initial work is equal to that of the back pressure on the first piston, which has just been calculated, namely, 126 foot-pounds, and the work for an expansion of three times, through the stroke, nm, is found by what is only a repetition of the calculation for the upper section, to be 264.4236 foot-pounds. The sum of the two sections is the total net work of the two cylinders; thus: Upper section, 138.4236 foot-pounds. Lower section, 264.4236 Total net work, 402.8472 Otherwise, the combined diagram, Fig. 336, represents the whole of the work as if it were done in one cylinder equal in capacity to the second cylinder assumed, in this instance, to have the same diameter and stroke. The period of admission, cd, is one-ninth of the stroke, or 6 -=- 9 = ^ foot ; the initial pressure being 63 Ibs. x 31 = 89 Ibs., and the initial work 189 x YZ = 126 foot-pounds. The whole work of the stroke is, therefore, 126 x (i 4- hyp log 9) = 126 x 3.1972 - 402.8472 foot-pounds; as was calculated before. i RECEIVER-ENGINE IDEAL DIAGRAMS. The hypothetical distribution which has been described for the Woolf engine, according to which all the steam with which the second cylinder is charged, is supposed to be admitted into the second cylinder before expan- sion begins, is that which actually takes place in the receiver-engine, the second general combination of the compound engine, in which the pistons of the two cylinders are connected to cranks at right angles to each other, on the same shaft, with an intermediate receiver. The receiver is occupied by steam exhausted from the first cylinder, and it supplies steam to the RECEIVER-ENGINE IDEAL DIAGRAMS. 853 second cylinder, in which it is cut off and then expanded to the end of the stroke. On the assumption that the initial pressure in the second cylinder is equal to the final pressure in the first cylinder, and, of course, equal to the pressure in the receiver, the volume cut off in the second cylinder must be equal to the volume of the first cylinder, for the second cylinder must admit as much at each stroke as is discharged from the first cylinder. For illustration, suppose again that the areas and capacities of the first and second cylinders, with the same length of stroke, are as i to 3, and that the steam is cut off iat one-third of the stroke, and equally expanded in both cylinders, the ratio of expansion in each cylinder being thus equal to the ratio of the capacities of the cylinders. With this distribution, the volume admitted to the second cylinder is equal to the volume discharged from the first cylinder, and there is no intermediate fall of pressure. The ideal diagrams of pressure which would thus be formed are shown in juxtaposi- tion in Fig. 337. Here, pq is the atmospheric line, cd is the line of admission, and hg the exhaust-line for the first cylinder, both of them being -XO Fig. 337. Receiver-Engine : Ideal Indicator Diagrams. Fig. 338. Receiver-Engine: Ideal Diagrams, reduced and combined. parallel to the atmospheric line; and dg is the expansion-curve. In the region below the exhaust-line of the first cylinder, between it and the line of perfect vacuum, the diagram of the second cylinder is formed; hi, the second line of admission, coincides with the exhaust-line hg of the first cylinder, and thus shows that, in the ideal diagrams, there is no intermediate fall of pressure. The line of perfect vacuum, ol, is parallel to the atmos- pheric line, and ik is the expansion-curve. The arrows indicate the order in which the diagrams are formed. The expansive working of the steam, though clearly divided into two consecutive stages, is, as in the Woolf engine, essentially continuous from the point of cut-off in the first cylinder to the end of the stroke of the second cylinder, where it is delivered to the condenser; and the first and second diagrams may be placed together and combined to form a continu- ous diagram. For this purpose, take, as was done for the Woolf engine, the second diagram as the basis of the combined diagram, namely, hiklo, Fig. 338, adding the atmospheric line,/^. The period of admission, hi, is one-third of the stroke, and as the ratios of the cylinders are as i to 3, hi 854 STEAM ENGINE COMPOUND CYLINDERS. is also the proportional length of the first diagram as applied to the second. Produce oh upwards, and set off oc equal to the total height of the first diagram above the vacuum line; and, upon the shortened base hi, and the height he, complete the first diagram with the steam-line cd, and the expansion-line, di. By this construction, the regular indicator diagram cdklo is formed, as applied to the second cylinder, and it measures the whole net work done in both cylinders : the upper section, cdth, being the measure of the net work done in the first cylinder, and the lower section, hiklo, being the measure of the work of the second cylinder. Resuming the data supplied for exemplifying the Woolf engine, with reference to the ideal diagrams for the receiver-engine, Fig. 337, let the areas of the first and second cylinders be respectively i and 3 square inches, the stroke 6 feet, and the initial pressure in the first cylinder 63 Ibs. per square inch. The steam being cut off at one-third of the stroke of the first cylinder, the final pressure is 21 Ibs., and the total initial work therein is equal to ocKcd = b$ Ibs. x 2 feet= 126 foot-pounds; for the whole stroke the total work is 126 x (i +hyp log 3) = 126 x 2.0986 = 264.4236 foot-pounds, the same as was found for the Woolf engine. For the second cylinder, the initial pressure is 21 Ibs. x 3 square inches = 63 Ibs., and the initial work is represented by oh x hi, which is equal to 63 Ibs. x 2 feet = 126 foot-pounds. For the whole stroke, the total work is i26x(i +hyp log 3)=i26x 2.0986 = 264.4236 foot-pounds. The work of the back pressure, oh, on the first piston, which is continued for the whole of the stroke, is to be deducted from this total work; it is represented by the rectangle ohxhg, equal to 21 Ibs. x 6 feet =126 foot- pounds. Then (264.4236-126=) 138.4236 foot-pounds is the net or effec- tive work for the second cylinder for one stroke. This, the work for the second cylinder, is to be added to the work for the first cylinder, and the sum, 402.8472 foot-pounds, is the united work for one stroke of the two cylinders. This is the same quantity of work as was calculated from the Woolf diagrams. It is obvious that the two combined diagrams, Figs. 336 and 338, pages 850 and 853, are identical in form and development. WORK OF STEAM AS AFFECTED BY INTERMEDIATE EXPANSION. That the work of expanding steam is to be calculated from the expansion upon a moving piston only, is obvious enough when it is considered that the steam may expand into an intermediate receiver, and into intermediate passages, without doing any work on a piston, whilst at the same time the pressure falls or " drops " as the volume is enlarged. Under these circum- stances, the second cylinder receives the steam at a lower pressure and in larger volume than it has when there is no intermediate expansion and fall of pressure; and there is less work done, whilst the ratio of active expan- INTERMEDIATE EXPANSION IN THE WOOLF ENGINE. 855 sion is necessarily reduced. If the second cylinder, however, be enlarged in capacity, in proportion to the enlargement of the volume of the steam and the fall of pressure, by intermediate expansion, the ratio of expansion, and the work done in it, would remain the same. Whilst, then, there is no reduction of work consequent on intermediate expansion of the steam, provided that the ratio of expansion originally designed be maintained by means of a second cylinder of suitably large capacity; there is actually a reduction of work, or loss of effect, by such intermediate expansion, when the capacity of the second cylinder remains the same. INTERMEDIATE EXPANSION IN THE WOOLF ENGINE. To proceed with the investigation of such loss by intermediate expansion as is suffered in the Woolf engine, take the example of Woolf engine already treated, with the same proportions and dimensions, and suppose that the total capacity of the passages from the first to the second cylinder is one- third, or 33 T / 3 per cent, of the capacity of the first cylinder. The ideal diagrams Fig. 335, and the combined diagram Fig. 336, page 850, are reproduced, partly in dot-lining, in Figs. 339 and 340 annexed, the same letters of reference being applied. To these are added the modifications introduced by the intermediate fall of pressure. The admis- sion and expansion of the steam in the first cylinder are indicated, as Fig. 339. Woolf Engine : Diagrams showing intermediate fall of pressure. nt. Fig. 340. Woolf Engine : The same diagrams reduced and combined. before, by the straight line cd, representing an initial pressure of 63 Ibs. per square inch, and the curve dg, or d g* ', with a terminal pressure, ng, or n'g", of 2 1 Ibs. But when the exhaust is opened, at the end of the stroke,^ or g" t the steam expands into the intermediate space, and occupies a total volume equal to i J /s or Vs times the capacity of the first cylinder, before the second piston commences its stroke. The final pressure is, at the same time, reduced, in the inverse ratio, to ^ths of 21 Ibs., or 15.75 Ibs. from ng to ng\ or from ifg* to n"g 5 , Fig. 340. With this lower pressure, and the aug- mented initial volume, i T / 3 times the capacity of the first cylinder, the 856 STEAM ENGINE COMPOUND CYLINDERS. steam expands into the second cylinder, and acquires a final volume by expansion equal to the capacity of the second cylinder plus the intermediate space, or 3 I / 3 times the capacity of the first cylinder. Hence the ratio of expansion into the second cylinder is ^-^-=2.5; and the final pressure 1 /3 nth", is (15.75 Ibs. -=-2.5=) 6.3 Ibs. per square inch. The actual curve of expansion, g 4 /i", is, like the normal curve gh, an elongated hyperbolic curve an elongation of the hypothetical expansion-curve, g 5 ti", which flows from the augmented initial volume, as illustrated in the annexed Fig. 341, in which those curves are reproduced, and in which the augmented initial volume is measured by the extension, nn'"ri, of the base-line; this extension comprises n' ri", the capacity of the first cylin- der, and n'"n, the capacity of the intermediate space, one- third of riri" ; making together i */3 times the capacity of the first cylinder. At the end Fig. 34i.-Woolf Engine ^Expansion Curves for first and f * e Stl ? ke <>f the SCCOnd second cylinders. cylinder, when the piston has arrived at m, the first piston has arrived at n", leaving the clear interval of intermediate space, ri"n, open to the second cylinder, which, added to the capacity nm of this cylinder, makes ri"nm, the final volume of the steam expanded into the second cylinder, equal to 3 J / 3 times the capacity of the first cylinder. Of this total volume," the section n'"n", i T / 3 times the capacity of the first cylinder, is the hypothetical period of admission, composed of the inter- mediate space ri"n, and the capacity of the first cylinder nn" ' ; the remain- ing section ri'm being the hypothetical period of expansion. The curve of back-pressure, g' h'", on the first piston, has the same initial and final pres- sures as the expansion-curve g 4 h". In piecing the first and second diagrams, to form the combined diagram, Fig. 340, the triangular area of positive pressure on the first piston, g*g s h'", is replaced by the equal triangular area g 4 g s h", and thus the united work of the two cylinders is indicated by the seven-sided diagram cdg" g 5 h"m n. The effect of the intermediate fall of pressure in reducing the performance of the expanding steam, is clearly shown by the combined diagram, Fig. 340, in which the section g" h of the normal expansion-curve dg" /i, is replaced by the lower expansion-curve g 5 h", with the vertical line g"g s denoting the intermediate fall of pressure. The four-sided area g"g*h"h expresses the net loss of useful expansive work caused by the intermediate fall : being the balance of loss after deducting the gain by the reduction of back-pressure on the first piston, measured by the area g"g 5 ti" h', from the loss of pres- sure on the second piston, measured by the area, gg 4 h" h, shown in both the Figs. 339 and 340. The loss of work by expansion of steam in the Woolf engine, into an intermediate space between the first and second cylinders, may be shown and calculated in the same way for other volumes of intermediate space say, one-half more, and as much more as the capacity of the first cylinder. Take" all four cases, as follows: INTERMEDIATE EXPANSION IN THE RECEIVER-ENGINE. 857 Intermediate Space. Ratios of Expansion. Combined Ratio. istcaser-Nil ( ist cylinder r to 3 (ad i to 3 i to 9 ad case:- V 3 capacity of IS t cylinder i " tc y linder ' ' 3 ) 2d i to 2.5 i to 7.5 ist cylinder i to 3 2d i to 2 V 3 i to 7 i to 6 4 thcase:-i | ist cylinder i to 3 \2d I tO 2 Applying these ratios to the initial steam admitted to the first cylinder, the total initial work is, as before, 126 foot-pounds, and the total net work for one stroke of the two cylinders is as follows : l Foot-pounds. ist case: 126 x (i +hyp log 9) or 3.1972 = 402.8472 2d case: 126 x (i + hyp log 7.5) or 3.0149 = 379.8774 3d case: 126 x (i +hyp log 7) or 2.9459 = 371.1834 4th case: 126 x (i +hyp log 6) or 2.7918 = 351.7668 INTERMEDIATE EXPANSION IN THE RECEIVER-ENGINE. With respect to the loss by intermediate expansion and fall of pressure in the receiver-engine, take examples based on the same data as have been applied to the discussion of the Woolf engine; and suppose, in the first instance, that the steam is expanded in the receiver into i J / 3 times, or four-thirds of, its volume when exhausted from the first cylinder, the pres- sure being proportionally reduced to three -fourths of the final pressure in the first cylinder, prior to its being admitted into the second cylinder. With this modification, the action of the steam is represented diagram- matically in Figs. 342 and 343 annexed; the first for the two cylinders separately, the second being the combined diagram. These diagrams are fundamentally the same as the ideal or normal diagrams, Figs. 337 and 338, page 853, constructed for the receiver-engine without any intermediate fall of pressure, and the same letters of reference apply to the same parts. The admission-line cd t one-third of the stroke, and the expansion-curve dg, for the first cylinder, Fig. 337, are the same as in the normal diagram, showing an expansion of three times; the initial pressure, oc, being 63 Ibs. per square inch, and the final pressure, Ig, being equal to 21 Ibs. on the square inch area of piston. At the end of the stroke, the pressure of the steam, as it is exhausted into the receiver, falls one-fourth, to 15.75 Ibs., measured by lg' t which gives the level of the back pressure on the first piston, g'h', parallel to gh. The steam exhausted from the first cylinder is at the same time expanded to i*/ 3 times the capacity of the cylinder. A volume of steam i x / 3 times the first cylinder must, therefore, be admitted 1 These calculations, as well as others which are quoted in this section on compound engines, are detailed at length in a work on the steam engine, by the author, now in course of preparation h 858 STEAM ENGINE COMPOUND CYLINDERS. to the second cylinder, of the reduced pressure 15.75 Ibs. per square inch. The length h' /', set off on the line h'g', the measure of the enlarged volume, is the period of admission into the second cylinder, and it is i x / 3 times hi, or i I / 3 thirds of the length of the stroke h'g', or 2^ feet. The point /', in fact, obviously lies in the normal curve of expansion ik; and from this point to the end of the stroke, at k, the two curves of expansion, namely, the normal curve ik, and the new curve i'k, so far as it extends, are iden- tical, with a terminal pressure, Ik, of 7 Ibs. per square inch. The outline of the combined diagram is, then, cdini'klo. -SD Fig. 342. Receiver-engine .-Diagrams showing Intermediate Fall of Pressure. Fig. 343. The same diagrams reduced and combined. The rate of expansion in the second cylinder, according to this distribu- tion, is not so high as in the first cylinder the initial volume being i r / 3 times the first cylinder, and the final volume being the capacity of the second cylinder, or 3 times the first cylinder. The ratio of expansion is, therefore, =t- =2.25. On the first diagram, Fig. 342, it appears that, whilst there is a gain of net work to the first cylinder when compared with the normal performance, measured by the drop of pressure, gg', for the whole stroke gh; there is, on the contrary, a loss of work to the second cylinder, measured by the same drop of pressure, hh', for the area hii'h'. The net loss is directly indicated on the combined diagram, Fig. 343, in which the gain in the first cylinder is measured by the rectangle hinh', and the loss in the second cylinder by the trapezoid hii'h'. The difference of these, the small tri- angular area ii'n, is the measure of the net loss. Taking four cases for comparison, corresponding to those calculated for the Woolf engine, the results of calculation are as follows z 1 The augmented initial volumes for expansion in the second cylinder, and the actual ratios of expansion in the two cylinders, are, 1 These calculations, as well as others which are quoted in this section on compound engines, are detailed at length in a work on the steam engine, by the author, now in course of preparation. WORK OF THE WOOLF ENGINE. 859 ist case Augmented Initial Volume in Parts of the Ratios of Expansion. Combined Ratio. First Cylinder. ( i st cylinder,... i to 3 ' ( 2d do. ...i to 3 i to 9 2d case... 3d case... 4th case . 1 1/ 3 . fist cylinder,... i ( 2d do. ...i to 3 to 2.25 i to 6.75 i to 6 i to 4.5 i# /istcylinder,...i to 3 tO 2 to 3 to 1.5 ' ( 2d do. ...i ( i st cylinder,... i '(2d do. ...i The net works are calculated in terms of the initial work, 126 foot-pounds, and the combined ratios of expansion, with an allowance for the net work acquired by the first cylinder due to the intermediate fall of pressure: foot-pounds. ist case: 126 x(i + hyp log 9) or 3.1972 =402.8472 2d case: 126 x (i^+hyp log 6.75) or 3.1595 =398.0970 3d case: 126 x (i'/ 3 + hyp log 6) or 3.1251 - 393.7542 4th case: 126 log 4.5) or 3.0041 = 378.5166 By comparison, it appears that the loss of work by intermediate fall of pressure, is less in the receiver-engine than in the Woolf engine; being only 6 per cent, of loss in the former, as against 12.7 per cent, in the latter, when the pressure falls to half the final pressure in the first cylinder. WORK OF THE WOOLF ENGINE, WITH CLEARANCE. Let Figs. 344 and 345 represent the diagrams of pressure from the first and second cylinders of a Woolf engine having the same dimensions, pro- Fig. 344. Woolf Engine : Diagrams with Clearance. Fig. 345. Woolf Engine : The same diagrams reduced and combined. portions, and letters of reference, as in the preceding examples, with the addition of a clearance at each end of the first cylinder, measured by cc' 86O STEAM ENGINE COMPOUND CYLINDERS. or mm', equal to 7 per cent, of the stroke, or .42 foot. The rectangular clearance space cc'm'm, measures the passive work of the clearance, or the product of the pressure m c by the period of the clearance cc'. As the steam is cut off at a third, or 33^3 per cent, of the stroke, the actual ratio of expansion is (see page 828) I0 + t. = 2.653. The initial o5 /3 ' i pressure being 63 Ibs., as before, the final pressure, ng, is ^- = 23.75 Iks.; " Do and the final volume, taking the working capacity of the first cylinder as i, is i + (7 per cent.), or 1.07. If there be no more clearance, or no inter- mediate space, between the first and second cylinders, the initial volume for expansion into the second cylinder is 1.07; and the final volume is 3. The ratio of expansion in it is, therefore, 3. =2.804; and the final pressure, 23.75 I>07 m h, is g =8.47 Ibs. per square inch. In view of these pressures, it is apparent that, whilst the work of admis- sion during the period c d is the same as it was when there was no clearance, namely, 126 foot-pounds, the work by expansion is greater, for the final pressures are respectively as follows : With no Clearance. With Clearance. In the first cylinder, 21 Ibs. per sq. inch. 23.75 Ibs. per sq. inch. In the second cylinder,... 7 8.47 If it were practicable to construct the compound cylinder so that there should not be any intermediate clearance, the employment of a smaller cylinder, as a prefix to a given cylinder, for receiving the charge of steam to be expanded through both cylinders, would have the economical effect of reducing the percentage of end-clearance, measured in parts of the larger cylinder. But it is not practicable to do so; and it remains to trace the influence of intermediate space combined with the initial clearance of the first cylinder, on the action and work of the steam in the second cylinder. Take, as before, four cases, and suppose that the volume of the inter- mediate space, including what is technically the clearance of the second cylinder, is a simple fraction of the capacity of the first cylinder plus its clearance, 7 per cent, or of 1.07 times the capacity of the first cylinder, as follows : for the ist, 2d, 3d, 4th case, the intermediate spaces are, o, x / 3 , YZ, i, part of the capacity of the first cylinder plus its clearance; or they are, o, -357? -535? I -7 f tne capacity of the first cylinder. Add to these 1.07, the capacity of the first cylinder plus its clearance; and the sums are the total initial volumes for expansion in the second cylinder, 1.07, 1-427, 1.605, 2 - I 4> times the capacity of the first cylinder. Again, to the same values of the intermediate space, add 3, the capacity of the second cylinder; and the sums are the final volumes by expansion in the second cylinder, 3'> 3-357> 3-535? 4-7 times the capacity of the first WORK OF THE WOOLF ENGINE. 86l cylinder. The ratios of expansion in the second cylinder are the quotients of the final by the initial volumes : 2.804, 2.352, 2. 202, 1.902, ratios of expansion. The intermediate falls of pressure are, in parts of the final pressure in the first cylinder, o, i^, J / 3 , y 2 of the final pressure; or, putting the final pressure equal to 23.75 Ibs., as was found, they are 0, 5'94> 7-9 2 , 11.87 I DS - P er square inch. The initial pressures for expansion in the second cylinder are, 1, 24, 2 /3J y* of tne nna -l pressure in the first cylinder; or 2 3-75? I 7-8i, I 5-83, 11.87 Ibs. per square inch; and the final pressures in the second cylinder are, 8.47, 7.57, 7.19, 6.24 Ibs. per square inch. The combined ratios in the four cases are as follows : ist case: ist ratio of expansion, i to 2.653 COMBINED RATIO. 2d do. i to 2.804 i to 7.438 2d case: ist ratio of expansion, i to 2.653 2d do. i to 2.352 i to 6.241 3d case: ist ratio of expansion, i to 2.653 2d do. i to 2.202 i to 5.843 4th case: ist ratio of expansion, i to 2.653 2d do. i to 1.902 i to 5.046 The initial work of the steam of 63 Ibs. total pressure, admitted into the first cylinder, for 2 feet of the stroke, and with a clearance of 7 per cent., or .42 feet, is as follows: Work done on the piston, 63 Ibs. x 2 feet =126 foot-pounds. Work done in the clearance,... 63 Ibs. x .42 foot = 26.46 Total initial work of the steam, 63 Ibs. x 2.42 feet= 152.46 This sum is the initial work on which the work by expansion is calculated ; whilst it is 26.46 foot-pounds in excess of the initial work done on the piston. The total work is, then, calculated as follows : NET WORK IN ist case: 152.46 x (i +hyp log 7.44) or 3.0069 = 458.27 FOOT-POUNDS. less, work in initial clearance, 2 6. 46 431.81 ad case: 152.46 x (i -i-hyp log 6.24) or 2.8310 = 431.47 less 26.36 405.11 3d case: 152.46 x(i+hyp log 5.84) or 2.7647 = 421.35 less 26.36 394-99 4th case: 152.46 x (i +hyp log 5.05) or 2.6194 = 399.29 less 26.36 372.93 862 STEAM ENGINE COMPOUND CYLINDERS. The calculations have been made in this form for the sake of comparison with those that were made for the work when there was no clearance (page 857). They can be made more directly by means of the formula ( 7 ) at page 828. The reductions of net work, in the 2d, 3d, and 4th cases, are successively 6.2, 8.6, and 13.7 per cent, of the work in the ist case. WORK OF THE RECEIVER-ENGINE, WITH CLEARANCE. For the work of receiver-engines, with clearance, taken at 7 per cent., at each end of the stroke of each cylinder, the annexed Fig. 346 shows the diagrams of pressure, using the same data and letters of reference as in Fig. 342, p. 858, with the clearance measured by cc', hh', or oo, equal to 7 per cent, of oL The steam being cut off at x / 3 d, the actual ratio of expansion in the first cylinder is, as for the Woolf engine, p. 860, 1- = the final pressure, Ig, is 5, * 33 7s +7 = 2.653; = 23.75 Ibs., which is also the pressure in the receiver, when there is no intermediate fall of pressure. The same is the initial pressure oh, in the second cylinder, with the clearance oo' . The volume admitted into the second cylinder is equal to the capacity of the Fig. 346. Receiver-engine: Diagrams with Clearance. Fig. 347. Receiver-engine : The same diagrams reduced and combined. first cylinder plus its clearance, or to one-third of the capacity of the second cylinder plus its clearance; that is, to one-third of 107 per cent., or 35 2 / 3 per cent., which consists of the clearance, 7 per cent., and (35 2 /s -7 = ) 28 2 / 3 per cent, of the stroke of the second cylinder. The steam admitted into the second cylinder thus occupies less than one-third of the stroke, by 4 2 / 3 per cent., as indicated by the length of the period of admis- sion, hi, in the diagram. As the steam is expanded from the capacity of the first cylinder plus its clearance, to that of the second cylinder plus its clear- ance, the ratio of expansion in the second cylinder, is necessarily equal to the ratio of the capacities of the two cylinders, which is 3 ; and T ' = 3 ; and the final pressure, Ik, is 23^5 = 7.92 Ibs. per square inch. WORK OF THE RECEIVER-ENGINE. 863 The combined diagram, Fig. 347, shows a dislocated expansion-line, in two parts : dg for the first cylinder, and ik for the second cylinder. The first part, dg, is extended continuously to the end of the stroke at k', and shows the loss of work caused by the excess of the volume of clearance of the second cylinder over that for the first, as measured by the area of the strip igk'k. For the other three cases, of intermediate falls of pressure, respectively }^ th, J / 3 d, and J^ of the final pressure in the first cylinder, the relations are as follows: For the ist, 2d, 3d, 4th case the augmented initial volumes for expansion in the second cylinder are, i, i J / 3 , i^, 2 times the capacity of the first cylinder plus the clearance; or they are 1.07, 1.427, 1.605, 2 x 4-95 foot-pounds. The works in the second cylinder are calculated from these data, with the ratios of expansion, as follows : Foot-pounds. ist case: 152.46 x (i+hyp log 3)= 3 J 9-93 less the work in clearance, 29.92 290.01 2d case: 152.46 x (i +hyp log 2.25) = 276.10 less the work in clearance, 22.44 2 53-66 3d case: 152.46 x (i +hyp log 2) = 258.13 less the work in clearance, !9-95 238.18 4th case: 152.46 x (i +hyp log 1.5)= 214.28 less the work in clearance, I 4-95 I 99'33 The total net work in both cylinders for one stroke, is found by adding together the three portions of work for each case : NET WORK RATIO in Foot-pounds, of Net Work. i st case : first cylinder above final pressure, 132.25 intermediate, o.oo 132.25 second cylinder, 290.01 422.26, as roo 2d case: first cylinder above final pressure, 132.25 intermediate, 35- 6 4 167.89 second cylinder, 253.66 421.55, as 99.8 3d case: first cylinder above final pressure, 132.25 intermediate, 47-5 2 179.77 second cylinder, 238.18 417.95, as 99.0 4th case: first cylinder above final pressure, 132.25 intermediate, 71.22 203.47 second cylinder, J 99-33 402.80, as 95.4 WORK OF THE RECEIVER-ENGINE. 865 Here it is seen that the reduction of the quantity of work performed for one stroke of the pistons, by intermediate falls of pressure, does not exceed i per cent, when the fall amounts to one-third of the final pressure in the first cylinder; and it is less than 5 per cent, even when the fall amounts to half the final pressure. The proportional reduction of work is something less with clearance, as in this instance, than without clear- ance, as exemplified at page 859. The work for one stroke may be calculated in terms of the combined ratios of expansion for the two cylinders ; making allowances for the loss by clearance, and the gain to the first cylinder by the intermediate fall of pressure. The total initial work for expansion is, as was found (page 863), 152.46 foot-pounds; and the ratios of expansion are as follows: ist case: first ratio of expansion, i to 2.653 COMPOUND RATia second do. i to 3.000 i to 7.959 2d case: first ratio of expansion, i to 2.653 second do. i to 2.250 i to 5.969 3d case: first ratio of expansion, , i to 2.653 second do. i to 2.000 i to 5.306 4th case: first ratio of expansion, i to 2.653 second do. i to 1.500 i to 3.979 With respect to the first case, it is obvious from an inspection of the combined diagram, Fig. 347, page 862, that the calculation of the work in terms of the initial work for expansion, and the total ratio of expansion, covers the whole area of the diagram, including the clearance-areas, thus : 152.46 x (i +hyp log 7.959) or 3.0743 = 468.71 foot-pounds. From this is to be deducted the work of the initial clearance in the first cylinder, 26.46 foot-pounds, represented by the area cc ' o" o; and also the work of the excess of clearance in the second cylinder, over and above that of the clearance in the first cylinder, calculated on che pressure in the receiver. As the clearance of the second cylinder is, like that of the first, 7 per cent, of the stroke, or .42 feet, the volumes of the respective clear- ances are in the ratio of the capacities of the cylinders, or as 3 to i, and are measured by the spaces oo' and oo" on the diagram. The clearance steam of the first cylinder, therefore, when transferred to the second cylinder, fills only one-third of its clearance space, measured by oo", and of the pressure oh; and additional steam from the receiver, of the same pressure, is required to fill the remaining two-thirds of the clearance of the second cylinder, or .42 x 2/$ = .28 foot, measured by o" o' . The work of the two clearances, to be deducted, is, then, as follows : Work of clearance of first cylinder, cc' o" o* 63 Ibs. x ) , , r , 3 in. x .14 foot, or 63 Ibs. x ,' in. x .42 foot } = 26 '^ 6 foot-pounds. Excess of clearance of second cylinder, h' o' o" , \ , 23.75 Ibs. x 3 in.x.28 foot ( = Work of clearances, 46.41 do. 55 STEAM ENGINE COMPOUND CYLINDERS. The gross work of the diagram being, as above, 468.71 foot-pounds. The work of clearances to be deducted is 46.41 do. Net work for one stroke, 422.30 do. For the 2d case, the gross work is 152.46 x (i +hyp log 5.969) or 2.7866 = 424.84 foot-pounds. Deduct the work of the clearances, as above,. . . 49.41 do. 378.43 do. To this is to be added the compensatory gain by the fall of the pressure in the reservoir, which is equal to 5.94 Ibs. per square inch. It is to be multiplied into the length of the stroke of the first cylinder, for the reduction of back-pressure, and the clearance of the second cylinder for the saving of passive work in the clearance. The stroke is, as reduced, 2 feet; the clearance is .42 foot, and the sum of these is 2.42 feet. Then the work of the gain is, 5.94 Ibs. x 3 in. x 2.42 feet = 43.12 foot-pounds, which is to be added to the remainder, above,. . .378.43 do. making the net work for one stroke 421.55 do. The calculations of net work are similarly performed for the 3d and 4th cases, and they are all brought together for the four cases for comparison, as follows : Foot-pounds. ist case: 152.46 x (i +hyp log 7.959) or 3.0743 = 468.71 deduct for clearances, 46.41 422.30 2d case: 152.46 x (i + hyp log 5.969) or 2.7866 =424.84 deduct for clearances, 46.41 378.43 add for fall of receiver-pressure 5.94 Ibs. x 3 in. x 2.42 feet, 43.12 421.55 3d case: 152. 46 x (i + hyp log 5.306) or 2.6688 =406.87 deduct for clearances, 46.41 360.46 add 7.92 Ibs. x 3 in. x 2.42 feet 57-5 4 J 7.96 4th case: 152.46 x (i + hyp log 3.979) or 2.3810 =363.01 deduct for clearances, , 46.41 316.60 add 11.87 Ibs. x 3 in. x 2.42 feet, 86.18 402.78 The net works thus obtained are the same as those that were deduced from the diagrams treated separately (page 864). WORK IN WOOLF AND RECEIVER ENGINES COMPARED. 867 COMPARATIVE WORK OF STEAM IN THE WOOLF ENGINE AND THE RECEIVER-ENGINE. It has been shown that the work of steam in the compound engine, when there is no clearance and no intermediate fall of pressure, is the same in amount, whether performed on the Woolf system or the receiver-system; but that, when there is an intermediate fall of pressure, with the enlarge- ment of volume by which it is accompanied, the work done on the receiver- system is greater than that on the Woolf system; that is to say, the reduction of work by fall of pressure is less rapid with the receiver than on the Woolf system. This is apparent in the following comparative note of the perfor- mances, from which it also appears that, whilst the receiver-engine does more work, it expands the steam to a less number of times than the Woolf engine : WOOLF ENGINE (no clearance). RECEIVER-ENGINE (no clearance). Ratio of Expansion. Net Work. Ratio of Expansion. Net Work. ist case: 9.0 402.85 ft.-pds 9.0 402.85 ft.-pds. 2d case: 7.5 379-88 6.75 398.10 3d case: 7.0 37*-i8 6.0 393-75 4thcase: 6.0 35 x -77 4-5 378.52 In fact, it was found that the reduction of work in the 4th case, when the pressure fell to one-half intermediately, was about 13 per cent, in the Woolf engine, and only 6 per cent, in the receiver-engine. The apparent anomaly that the engine in which the greater expansion of steam takes place, performs a less net work, is explained by the fact that in the former, the Woolf engine, much of the initial work of the steam for the second cylinder is lost in the intermediate space; whilst, in the latter, the receiver- engine, there is no loss of this kind. By the addition of clearance to each cylinder, equal to 7 per cent, of the stroke at each end, the actual ratios of expansion are sensibly reduced as compared with the ratios without clearance, in the Woolf engine, from 9 to 7.4 when there was no intermediate fall of pressure, and from 6 to 5 when there was a fall of one-half. In the receiver-engine, the reduction of ratio is less than in the Woolf engine: it is from 9 to 8 when there is no fall, and from 4.5 to 4 when there is a fall of one-half. Thus, the effect of the addition of clearance is clearly to reduce the net expansion. At the same time, it increases the net work done, as appears from the following statement: WOOLF ENGINE 7 % clearance. RECEIVER-ENGINE 7 % clearance. Ratio of Expansion. Net Work. Ratio of Expansion. Net Work. istcase: 7.44 431.71 ft.-pds 7.96 422.3oft.-pds. 2d case: 6.24 405.11 5-97 421.55 3d case: 5.84 394-99 5-3* 417.96 4thcase: 5.05 372-93 3-9 8 402.78 Taking only the 4th case : in the Woolf engine, the net work is raised, by the addition of clearance, from 352 to 373 foot-pounds; and, in the receiver- engine, from 378.5 to 403 foot-pounds. With clearance, as without clearance, it is found that the reduction of net work, by intermediate fall of pressure, is less in the receiver-engine, 868 STEAM ENGINE COMPOUND CYLINDERS. where it is only 4^ per cent., with clearance, than in the Woolf engine, where it amounts to about 14 per cent, when the pressure falls interme- diately one-half. As the combined ratios of expansion in the receiver-engine are less, for each case, than in the Woolf engine; so the terminal pressures of the expanded steam in the second cylinder, on passing to the condenser, are greater in the receiver-engine than in the Woolf engine : For the ist, 2d, 3d, 4th case, with 7 per cent, clearance, the terminal pressures in the second cylinder are, for the W T oolf engine, 8.47, 7.57, 7.19, 6.24 Ibs. per square inch, and for the receiver-engine, 7.92, 7.92, 7.92, 7.92. In the first case, the terminal pressure in the Woolf engine is greater than in the receiver-engine; for there was no intermediate space assumed in the former, whilst clearance-space for the second cylinder was assumed in the latter; but, in the other cases, the terminal pressures in the former fall consecutively below that of the first case. They also fall below those of the latter, which remain constant for all the cases. This constancy of terminal pressure in the second cylinder of the receiver-engine, simply follows from the fact that the terminal volume of the expanded steam is always the same, that of the second cylinder plus the clearance, what- ever be the intermediate fall of pressure; whilst in the Woolf engine, on the contrary, the terminal volume is equal to that of the second cylinder, increased by the volume of the intermediate space, and the terminal pressure must be less as the terminal volume is increased. As the terminal pressure in the receiver-engine is thus shown to be, in all practical cases, greater than in the Woolf engine, other conditions being the same, it directly follows that the work performed in expanding from a given initial pressure to the several terminal pressures, must be greater in the receiver-engine than in the Woolf engine. It may be gathered from these arithmetical deductions that the receiver- engine is an elastic system of compound engine, in which considerable latitude is afforded for adapting the pressure in the receiver to the demands of the second cylinder, without considerably diminishing the effective work of the engine. In the Woolf engine, on the contrary, it is clearly of much importance that the intermediate volume of space between the first and second cylinders, which is the cause of an intermediate fall of pressure, should be reduced to the lowest practicable amount. Supposing that there is no loss of steam in passing though the engine, by cooling and condensation, it is obvious that whatever steam passes through the first cylinder, must also find its way through the second cylinder, neither more nor less. By varying, therefore, in the receiver-engine, the period of admission in the second cylinder, and thus also the volume of steam admitted for each stroke, the steam will be measured into it at a higher pressure and of a less bulk, or at a lower pressure and of a greater bulk: the pressure and density naturally adjusting themselves to the volume permitted to escape from the receiver into the cylinder. With a sufficiently restricted admission, the pressure in the receiver may be main- tained at the pressure of the steam as exhausted from the first cylinder. On the contrary, with a wider admission, the pressure in the receiver may FORMULAS AND RULES FOR COMPOUND ENGINES. 869 fall or " drop " to three-fourths, or even one-half of the pressure of the exhaust steam from the first cylinder. There is a means of counterbalancing the loss of performance by inter- mediate fall of pressure, by so enlarging the second cylinder as to effect the same ultimate ratio of expansion behind the pistons, as would be effected in the originally designed engine if there were no intermediate fall. For example, when the capacities of the first and second cylinders are as i to 3, and the steam is cut off in each at one-third of the stroke, without any intermediate fall, the steam, if there be no clearance, is expanded into nine times its initial volume. But, when there is an intermediate fall of pressure, of, say, one-fourth of the final pressure in the first cylinder, involving an increase of volume of steam in the ratio of 3 to 4, the second cylinder must be correspondingly enlarged in the ratio of 3 to 4, in order to contain the charge of steam for expansion, when cut off, as before, at one-third of the stroke. By such enlargement of the second cylinder, in the ratio of the intermediate enlargement of the steam, the same ultimate ratio of expansion is secured, and an equivalent performance is effected. Such a remedy, when specially applied for the purpose of counterbalancing ineffective expansion of steam, involves the employment of enlarged cylinders, and entails the objections of increased weight, bulk, and cost of machinery. It would be more useful as a remedy, when applied to the Woolf engine, than to the receiver-engine. FORMULAS AND RULES FOR CALCULATING THE EXPANSION AND THE WORK OF STEAM IN COMPOUND ENGINES. In view of the preceding discussions of the expansive working of steam in compound cylinders, the following algebraic symbols are used : a = the area of the first cylinder in square inches. a' = ihe area of the second cylinder in square inches. r = the ratio of the area of the second cylinder to that of the first cylinder. L = the length of the stroke in feet, supposed to be the same for both cylinders. /=the period of admission to the first cylinder, in feet, excluding clearance. c=ihe clearance at each end of the cylinders, in parts of the stroke, in feet. L' = the length of the stroke plus the clearance, in feet. /' = the period of admission plus the clearance, in feet. s = the length of a given part of the stroke of the second cylinder, in feet. P = the total initial pressure in the first cylinder, in pounds per square inch, supposed to be uniform during admission. P' = the total pressure at the end of the given part of the stroke, s. p = the average total pressure for the whole stroke. R = the nominal ratio of expansion in the first cylinder, or L-?-/. R' = the actual ratio of expansion in the first cylinder, or L' -=-/'. R" = the actual combined ratio of expansion behind the pistons, in the first and second cylinders together. R'" = the actual ratio of expansion, or number of volumes into which the steam occupying the first cylinder at the end of the stroke, is expanded in the second cylinder at the end of any part of the return stroke, s : the special initial volume, or the capacity of the first cylinder, being = i. n = the ratio of the final pressure in the first cylinder to any intermediate fall of pressure between the first and second cylinders. N = the ratio of the volume of the intermediate space in the Woolf engine, reckoned up to, and including the clearance of, the second piston, to the capacity of the first cylinder plus its clearance. z/ = the whole net work in one stroke, in foot-pounds. 8/0 STEAM ENGINE COMPOUND CYLINDERS. Formulas and rules may be constructed on the basis of the combined ratios of expansion behind the two pistons : the combined ratio being the product of the actual ratios of expansion in the first and the second cylinders. When, as usually happens in practice, intermediate expansion takes place between the cylinders, if the ratio of this expansion be multiplied into the combined ratio of expansion behind the pistons ; or, if the three individual ratios of expansion in the first and second cylinders, and in the intermediate space, be multiplied together, the product is the ratio of total expansion of the steam within the engine, to the end of the stroke of the second cylinder, when it is discharged into the condenser. For example, if the steam be expanded to three times its volume in the first cylinder, twice in the second cylinder, and one-and-a-half times in the intermediate space ; the combined ratio of expansion behind the pistons is the product of the first and second of these ; that is, In first cylinder, i to 3, or 3 In second cylinder, i to 2, or 2 Combined ratio of expansion behind ) , ,. pistons, } Ito6 > Or6 Intermediate expansion, i to 1.5, or 1.5 Ratio of total expansion, i to 9, or 9 Or, the individual ratios may be placed consecutively thus : In first cylinder, i to 3, or 3 Intermediate expansion, i to 1.5, or 1.5 In second cylinder, i to 2, or 2 Ratio of total expansion, i to 9, or 9 Conversely, when the total expansion is given, the expansion behind the pistons may be calculated by dividing the total ratio by the ratio of inter- mediate expansion. Thus, if the total ratio be 9, and the intermediate ratio be 1.5, the combined ratio behind the pistons is -2- = 6; or i to 6. Generally, if the total ratio be divided by any one of the individual ratios, the quotient is the product or combination of the two others. Further, if two ratios be equal to each other, the combined ratio is equal to the square of one of them; and, conversely, the square root of a given ratio is the value of two elementary ratios, which when combined yield the given ratio. Thus, if there be two ratios, each equal to 3, then 3 x 3, or 3 2 = 9, the combined ratio formed by those two. Conversely, (^9 = ) 3 is the value of two elementary ratios which, if combined, form the ratio 9. Similarly, the two equal ratios which, when combined, form the ratio 7, are each equal to *J 7 = 2.65; the square of 2.65, or 2.65 x 2.65, being equal to 7. FORMULAS AND RULES FOR COMPOUND ENGINES. 871 To find the actual ratio of expansion in the first cylinder. This is found by the formula, page 828, when the stroke, the period of admission, and the clearance are given. It is equal to 7 = R' ('5) That is to say, the actual ratio of expansion in the first cylinder is equal to the quotient of the length of stroke plus the clearance divided by the period of admission plus the clearance. For example, if the steam be cut off at one-third of the stroke, and the clearance be 7 per cent., the length of stroke being equal to i; then the stroke plus the clearance is equal to 1.07, and the period of admission is equal to .3333 + .07 =.4033; and I '' = 2.653, the actual ratio of expansion. -433 To find the ratio of Intermediate Expansion. According to the assumption that the volume of a given weight of steam is inversely as its elasticity, or its pressure per square inch, the enlargement of volume, or expansion, may be deduced from the pressures before and after expansion. Thus, if the pressure be reduced from 20 Ibs. to 15 Ibs. per square inch, the volume, inversely, is enlarged in the ratio of 15 to 20; and, if the initial volume be 20 taken as i, then, by proportion, 15 : 20 : : i : 1.33; or i x = 1.33. Thus, in reducing the pressure ^th, the volume is enlarged ^d, and the ratio of expansion is 1.33. By the notation, n is the ratio of the final pressure in the first cylinder to the intermediate fall of pressure between the first and second cylinders ; or, it is the denominator of the fractional part of the final pressure, expressing the fall of pressure. When the fall is ^th, therefore, n = ^; and the remaining pressure is ^ths, and is as 3, or n-i. The pressures, then, before and after the fall, are as ;/ : n i ; and, inversely, the volumes are as n - i : n. Taking the capacity of the first cylinder with its clearance, as i, the expanded volume is found by the proportion n - i : n \ : i : n ; and the ratio of intermediate expansion is equal to n ~ I (16) n- i Substituting 4 for n in this expression, the ratio of expansion in the preceding example, is -i = A= 1.33, as already found. It is necessary, in the receiver-engine, thus to reckon backwards, from the observed pressures to the volumes, in order to find the intermediate ratio of expansion, since the volume of the receiver affords no evidence whatever of the amount of expansion between the first and second cylinders. The same process may, of course, be applied in the Woolf engine, to find the intermediate expansion; but the ratio of this expansion is, otherwise, exactly and directly determinable by the volume of the intermediate space. The ratio of the capacity of the first cylinder plus the clearance, to the intermediate space, is, by the notation, as i to N ; and the sum of these, 8/2 STEAM ENGINE COMPOUND CYLINDERS. or the enlarged volume, is as (i + N). The ratio of intermediate expansion is, therefore, as i to (i +N); or it is (i+N). (17) To find the value of N in terms of the intermediate fall of pressure: The intermediate ratio was found, in terms of the ratio of pressure n, to be 11 ; and i+N = -^-; so that n- i n i That is, the volume of the intermediate space relative to that of the first cylinder plus its clearance, is equal to the quotient of the final pressure in the first cylinder, divided by the reduced pressure after the fall, minus i. For example, the final and reduced pressures being 20 Ibs. and 15 Ibs. respectively, 20^-15 = 1.33; and 1.33-1 =.33, which is the value of N, the intermediate space, relative to the capacity of the first cylinder plus its clearance, taken as i. In this calculation, the actual values of the pressures have been used, instead of their relative values as indicated in the above expression (18); but the result is the same, for, putting for n the ratio 4, then, -* = -=1.33; and I -33- I =-33> as before. 4~ J 3 The capacity of the intermediate space in the Woolf engine, is found by multiplying that of the first cylinder plus its clearance by the ratio N. To find the Ratio of Expansion in the second cylinder. In the Woolf engine. This would be expressed by the ratio of the capacity of the first cylinder to that of the second cylinder, if there were no clearances nor other intermediate space. With clearances and intermediate space, the ratio of expansion in the second cylinder is less than that, and is equal to the ratio of the capacity of the first cylinder plus its clearance plus the intermediate space, to the capacity of the second cylinder plus the intermediate space, this last being taken to include the clearance of the first and second cylinders. Taking the capacity of the first cylinder plus its clearance as i, that of the intermediate space is N. The capacity of the second cylinder, with its clearance, is expressed by the ratio r; without clearance, it is less than r by as much in proportion as the capacity of the cylinder is less than the cylinder plus the clearance, or as L is less than (L + c\ or L'. The reduced ratio is, then, r x ; and the ratio of expansion JLj in the second cylinder is as (i + N) is to (r x ) + N; or JL< (rx y Ratio of expansion in second cylinder = - ................ ( J 9 ) That is, the ratio of the first to the second cylinder is multiplied by the length of stroke, and divided by this length plus the clearance; and the ratio of the intermediate space is added to the quotient, making a sum, say, A. FORMULAS AND RULES FOR COMPOUND ENGINES. 8/3 To the ratio of the intermediate space is added i, making a sum, say, B. Sum A is divided by sum B, and the quotient is the ratio of expansion in the second cylinder. For example, let r=3, ^" = .333, L = 6 feet, and L' = 6 plus 7 per cent, of 6, or 6.42. Then (3* ^-) + -333 6 -42 _ - i +.333 ' 353> the ratio of expansion in the second cylinder. In the receiver-engine. The actual ratio of expansion, in the second cylinder, is not affected by clearance, assuming, of course, that the per- centage of clearance is the same as in the first cylinder. When there is no intermediate fall of pressure, the ratio of expansion is simply that of the first and second cylinders, or r. But, with an intermediate fall, this ratio is reduced as the ratio of intermediate expansion is increased, namely - and it is as this ratio inversely, or, Ratio of expansion in second cylinder = r x - = ' - ' ...... ( 20 ) n n For example, putting the ratio of the cylinders, r = 3, and the ratio, , of the intermediate fall to the final pressure in the first cylinder, = 4, as before; then, \4~ I ) 3 = 3^ij = 2.25, the actual ratio of expansion in the second 4 4 cylinder. To find the total actual Ratio of Expansion as well as the combined actual Ratio of Expansion behind the two pistons. The total actual ratio of expan- sion is, as was stated (page 870), the product of the ratios of the three consecutive expansions: in the first cylinder, in the intermediate space, and in the second cylinder. For the Woolf engine. The expressions of these expansions are numbered (15), (17), and (19), and their product is as follows: L > ;or, Total actual ratio of expansion = x (r + N), or R'( + N) . . . (21) / -II JL/ That is to say, the ratio, r, of the first to the second cylinder is to be multiplied by the length of stroke, and divided by this length plus the clearance ; and the ratio-value of the intermediate space, N, is added to the quotient. The sum is then multiplied by the actual ratio of expansion in the first cylinder, and the product is the total actual ratio of expansion. For example, let the steam be cut off at a third of the stroke of the first cylinder, with a clearance of 7 per cent. \ let the ratio, r, of the cylinders be 3, and the ratio-value, N, of the intermediate space, .333 or ^d. Then, the stroke of the first cylinder being = i, the actual ratio of expansion in 8/4 STEAM ENGINE COMPOUND CYLINDERS. it, R', as was exemplified at page 871, is 1.07-=- .4033 = 2.653. The modified ratio of the cylinders is 3 x = 2.804; and 2.804 + -333 = 3- 13 7- Finally^ 2.653x3.137 = 8.322, the total actual ratio of expansion. It may be observed, that the fraction, -i-, above employed, is equivalent to the 6 I -7 fraction, ^ -, employed for the same purpose, in the example, page 873. The combined actual ratio of expansion behind the pistons, in the Woolf engine, is the product of the first and third of the above-cited expressions, namely (15) and (19), or, / \ (22) That is to say, the product, as above found, for the total expansion, is to be divided by the ratio-value of the intermediate space plus i ; the quotient is the combined actual ratio of expansion behind the pistons. For example, resuming the data of the preceding example, the final product expressing the total actual ratio of expansion, was found to be 8.322; and the divisor to be applied to it, is i + .333 = 1.333. Then, '^ 22 = 6.242, the combined o55 actual ratio of expansion behind the pistons. For the Receiver-engine. The total actual ratio of expansion is the product of the expressions of the three consecutive expansions, numbered (15), (i 6), and (20); their product is as follows: L' n (n i ) r L' -n , , \ xxL - 1 ~*'*f ................... (23) That is to say, the ratio, r, of the first and second cylinders is to be multi- plied by the actual ratio of expansion, R', in the first cylinder. The product is the total actual ratio of expansion. For example, making, as before, r= 3, and R' = 2.653, the product (3 x 2.653 = ) 7-959? * s the total actual ratio^ of expansion. The product of the first and third of the above three expressions, namely (15) and (20), gives the value of the combined actual ratios of expansion behind the pistons; thus, That is to say, the ratio of the first and second cylinders is multiplied by the actual ratio of expansion in the first cylinder, and by the ratio of the intermediate fall of pressure to the final pressure in the first cylinder minus i; and the final product is divided by this ratio simply. The quotient is the combined actual ratio of expansion behind the pistons. For example, resuming the product in the last preceding example, and taking ;/, the ratio of the intermediate fall of pressure =4; then 3 x 2.653 x ini = 7.959 x 3 4 4 = 5-9^9, the required ratio. FORMULAS AND RULES FOR COMPOUND ENGINES. 8/5 To find the Work done in the two cylinders of compound engines The Woolf engine. It has already been stated that the formula ( 5 ), page 828, for the work of steam expanded in one cylinder, applies also to the work of steam in the Woolf engine, when the combined actual ratio of expansion behind the pistons in the two cylinders, is given. Thus, the net total work for one stroke of the two pistons, quoting that formula, is, ( 2 5) RULE i. To find the net work done by steam in the two cylinders of a Woolf engine, for one stroke, with a given combined actual ratio of expansion. To the hyperbolic logarithm of the combined actual ratio of expansion behind the two pistons, add i ; multiply the sum by the period of admission to the first cylinder plus the clearance, in feet; and from the product subtract the clearance. Multiply the area of the first piston, in square inches, by the initial pressure in pounds per square inch, and by this remainder. The product is the net work in foot-pounds. For example, let the 2d case, pages 860, 86 1, be calculated by this rule: a P = i x 63 = 63 Ibs., /' = 2.42 feet, c = .42 foot, and R"=6.24. Then, .42 (i+ hyp log 6.24) -.42] .42 x 2.8310) - .42] = 63 (6.85 1 - .42) = 405.20 foot-pounds, = 6$ [2.4 = 63 [ (2. as was before calculated, allowing for small errors of approximation. The Receiver-engine. A complete formula for the work of the receiver- engine necessarily comprises three elements : First, the expression of the gross work, including the work of the clearances; second, the deduction for the passive work of the clearances; third, the addition for the gain of work by the reduction of the back pressure on the first piston when there is an intermediate fall of pressure. Beginning with the first case, pages 863, 864, in which there is no intermediate fall of pressure, the total initial work of the steam admitted to the first cylinder is expressed by a P /'; whence the total work with expansion is gR") .................... (26) This measures the total area of the diagram, Fig. 347, page 862, including the clearances. The work of the clearance of the first cylinder, cc' o"o, is The work of the clearance of the second cylinder is the rectangle hh'o'o, which includes the section hoo" of the first clearance; and, deducting this, the remainder, which is the rectangle h' o' o" , is to be added to the first clearance. To express this remainder algebraically, the volumes of the first and second clearances, oo" and od ', are in the ratio of the areas of the cylinders, or as i to r, and the volume of the difference, o' ' o ', is as c (r i). The height, d ' h' ', is the final pressure in the first cylinder, and is equal to the 876 STEAM ENGINE COMPOUND CYLINDERS. initial pressure divided by R', the actual ratio of expansion in the first cylinder; or, P TT/* Therefore the work of the excess, o"o r , of the second clearance is, and the two works of the clearances are together, to be deducted from the gross work by expansion (26). Whence the equation for the net work, in the first case : or K. )], ......... (27) when there is no intermediate fall of pressure. Before reducing this formula to a rule, it may be remarked that it gives values which approximate closely to the true values, for cases in which there are intermediate falls of pressure such cases as usually occur in practice; and, for ordinary practical purposes, the results of the application of this formula will be sufficiently near to exactness. It was found, in fact (page 864), that the reductions of work by intermediate falls, as compared with the work done when there was no fall, were as follows: When the pressure falls to 24, 2^, y?, of the final pressure in the first cylinder, the reduction of work is, 0.2, i.o, 4.6 per cent, of that in the first case. The intermediate fall of pressure is rarely so much as two-thirds; and even with this fall the reduction of work, it is seen, only amounts to i per cent. The slightness of the reduction results from the fact, as was before explained, that though the actual ratio of expansion, with intermediate falls, is less than when there is no intermediate fall, yet the loss of work by such reduc- tion of expansion is practically compensated by the gain of net work on the first piston by the fall of back pressure against it. Adopting, then, the formula (27) as applicable for all cases of receiver- engines arising in practice, it is required only to give the actual ratio of expansion in the first cylinder, and to multiply this ratio by the ratio of the capacities of the two cylinders, to arrive at the ratio of expansion to be employed in the formula. This is literally the actual combined ratio of expansion for the first case, without intermediate fall of pressure, as was found (page 865), represented by R" in the formula (27). FORMULAS FOR COMPOUND ENGINES. 8/7 RULE 2. To find the net work done by steam in the two cylinders of a receiver-engine for one stroke, with a given actual ratio of expansion in the first cylinder. Multiply the actual ratio of expansion in the first cylinder by the ratio of the two cylinders, and to the hyperbolic logarithm of the compound ratio add i; multiply the sum by the initial period of admission to the first cylinder, plus the clearance, in feet (product A). Divide the ratio of the two cylinders, minus i, by the actual ratio of expansion in the first cylinder; add i to the quotient, and multiply the sum by the initial clearance in feet (product B). Subtract product B from product A, giving the remainder C. Multiply the area of the first cylinder, in square inches, by the total initial pressure in pounds per square inch, and by the remainder C. The product is the net work in foot-pounds for one stroke. This rule is applicable to any of the four cases, page 865 : a= i square inch, P = 63 Ibs. per square inch, ^ = .42 foot, /'=2.42 feet, R' = 2.653, R" = 7.959, r=3, and hyp log ^' = 2.0743. Then, on the model of the given formula (27), w = 63 [2.42 (i + 2.0743) - .42 (i + = 63 (7.440 - .737) - 422.29 foot-pounds, as was before calculated for the first case. Or, following the wording of the rule: The combined actual ratio of expansion is 7.959, of which the hyperbolic logarithm is 2.0743; adding i to this, the sum, 3.0743, is multi- plied by 2.42, the initial period of admission plus the clearance, and 3.0743 x 2.42 = 7.440 (product A). Again, the ratio of the cylinders is 3, and 3-1 = 2; the actual ratio of expansion in the first cylinder is 2.653, and 24-2. 653 = . 75 4. Adding i to this quotient, the sum is multiplied by the initial clearance .42, or 1.754 x .42 = .737 (product B). The difference of products A and B is (7.440 - .737 = ) 6.703, and this, multiplied by 63 Ibs., the initial pressure per square inch, and by i, the area of the piston in square inches, gives 6.703 x 63 x i = 422.29 foot-pounds, the work of one stroke. 8;8 STEAM ENGINE COMPOUND. COMPRESSION OF STEAM IN THE CYLINDER. The work expended in compressing such exhaust steam as is not per- mitted to escape during the return-stroke of the piston, and is shut into the cylinder against the retiring piston, is to be reckoned against the quantity of steam thus reclaimed. For every phase of the distribution there is a par- ticular period of compression, by the adoption of which the resulting effi- ciency of the steam, for a given distribution, is raised to a maximum. The method of determining the best period of compression will be given in the author's work on The Steam Engine. The following table, No. 298, contains the best periods of compression for several periods of admission, with 7 per cent, clearance, and for several back exhaust-pressures. It is seen, by the table, that, the more expansively the steam is worked, the greater should be the period of compression that is, the exhaust port should be closed the earlier in the course of the return-stroke; and that the greater the proportion of back-pressure to initial-pressure, the less should be the period of compression. Table No. 298. COMPRESSION OF STEAM IN THE CYLINDER. BEST PERIODS OF COMPRESSION : Clearance 7 per cent. Total Back-pressure, in Percentages of the Total Initial Pressures. Cut-off in Percentages 2/^j 5 IO 15 2O 2 5 30 35 of the Stroke. Periods of Compression, in parts of the Stroke. per cent. per cent. per cent. per cent. per cent. per cent. per cent. per cent. per cent. 10 65 57 44 32 15 58 5 2 40 2 9 23 2O 52 47 37 27 22 25 47 42 34 26 21 17 30 42 39 32 25 2O 16 14 12 35 39 35 29 23 19 15 13 II 40 36 32 27 21 18 14 13 II 45 33 30 25 20 17 14 12 10 5 30 27 23 18 16 13 12 IO 55 27 24 21 17 15 13 II 9 60 24 22 19 15 14 12 II 9 65 22 2O 17 15 14 12 IO 8 70 19 17 16 14 14 12 10 8 75 17 16 14 13 12 II 9 8 NOTES TO TABLE. i. For periods of admission, or percentages of back-pressure, other than those given, the periods of compression may be readily found by interpolation. 2. For other clearances, the values of the tabulated periods of compression are to be altered in the ratio of 7 to the given percentage of clearance. PRACTICE OF EXPANSIVE WORKING OF STEAM. ACTUAL PERFORMANCE OF STEAM IN THE STEAM-ENGINE. In working steam expansively, the practical performance is affected by several circumstances. There is the influence of the wire-drawing of the steam during its admission into the cylinders; of the needful opening of the exhaust passages before the end of the stroke, for the escape of the steam from the cylinder; of the back exhaust pressure on the piston, and the closing of the exhaust passage before the end of the return-stroke, with the consequent shutting in and compression by the piston of a portion of the exhausting steam. These influences have been analyzed and measured by the author. He concluded that, when the cylinders were liberally pro- portioned, first, the possible loss by early exhaust was of no importance, and that the early release was, on the contrary, beneficial, in facilitating a complete exhaust during the return-stroke; second, that the loss by wire-drawing was of little or no moment, and that, as wire-drawing was, to some degree, equiva- lent to an earlier cut-off, it might even prove advantageous in point of economy; third, that the loss by back exhaust pressure in excess of the atmospheric resistance in non-condensing engines, in good practice, is of little or no importance. These conclusions were based upon the performance of locomotives, fitted with the link-motion, and worked with steam of 100 Ibs. effective pressure per square inch in the boiler; but they are applicable to all classes of steam-engine. 1 The only obstacle to the working of steam advantageously to a high degree of expansion in one cylinder, in general practice, is the condensation to which it is subjected, when it is admitted into the cylinder at the begin- ning of the stroke, by the less hot surfaces of the cylinder and the piston ; the proportion of which is increased with the ratio of expansion, so that the economy of steam by expansive working ceases to increase when the period of admission is reduced down to a certain fraction of the stroke, and that, on the contrary, the efficiency of the steam- is diminished as the period of admission is reduced below that fraction. The initial condensation here pointed out, is succeeded by the re-evaporation of a portion of the condensed steam during the later portion of the period of expansion ; because, as the pressure falls, the temperature of the steam, and of the water which it con- tains, also falls, until it ultimately descends below the actual temperature of the cylinder, when the heat of the cylinder is absorbed by the water, and 1 See Railway Machinery, 1855, pp. 69-99; a l so a paper on " The Expansive Working of Steam in Locomotives," in the Proceedings of the Institution of Mechanical Engineers, 1852, pp. 60-82, and 109-128. 880 PRACTICE OF EXPANSIVE WORKING OF STEAM. evaporation takes place. The author, in 1851, experimentally demonstrated the existence of this condensation in the cylinders of locomotives. Its reality and importance are now thoroughly understood and admitted. 1 He deduced from his experiments that, in jacketless cylinders, imperfectly pro- tected, the quantity of steam condensed amounted to from n to 42 per cent, of the whole of the steam admitted to the cylinders, according to the period of admission, ranging from 75 to 12 per cent, of the stroke. 2 The author also deduced that, on the contrary, when the cylinders of locomotives were thoroughly protected and heated in the smoke-box, there was no evidence to prove that initial condensation took place in the cylinders, to any important extent, within the limits of the expansive-working that was practised. By the application of a jacket of steam from the boiler, to the cylinder, a material increase in the efficiency of the steam has, in most cir- cumstances, been effected. But, it is incontestable that the jacket, though it diminishes, does not wholly prevent initial condensation of the steam admitted. By the compounding of cylinders, steam may be worked more expansively, and with a greater degree of efficiency, than in a single cylinder; for, obviously, the fluctuations of temperature which give rise to the condensa- tion that interferes with the action of steam worked expansively, are divided and reduced to one-half, in each cylinder, of what they amount to when the whole of the expansive action is confined to one cylinder. DATA OF THE PRACTICAL PERFORMANCE OF STEAM. Single- Cylinder Condensing Engines: Steam-jacketted and covered. The following data are reduced from the recorded performances of the engines : 3 1 The author was the first, so far as he is aware, to discover and demonstrate the exist- ence of initial condensation in steam-cylinders, and to prove that it increases rapidly and to a formidable extent as the ratio of expansion is increased. See his paper on "Ex- pansive Working of Steam in Locomotives," in the Proceedings of the Institution of Mechanical Engineers, 1852, page 109. See also Railway Machinery, 1855: "When steam is admitted to the cylinder while the latter is comparatively cold, a very sensible condensation of the steam takes place during admission, which continues to a certain extent during expansion. The heat thereby separated is absorbed by the material of the cylinder, and raises its temperature. A portion of this heat passes off, and is irrecoverably lost ; the remainder is re-absorbed by the precipitated steam during the expansion of the existing steam, if the expansion be long enough continued that is, until the temperature of the latter has fallen below that of the cylinder. This is clearly proved by indicator- diagrams taken at very slow speeds, on which occasions, the cylinder is cold enough to exhibit these operations in high relief." page 84. 2 In condensing engines, the loss by initial condensation may be much greater than 40 per cent., for which examples will here be given. Mr. Sutcliffe has followed the same method of analysis in stationary engines, and in the seventh edition of Hopkinson on the Indicator, published in 1875, he appears to have precisely adopted the conclusions and even the language of the author. " The initial condensation," he says, page 298, "relatively to the initial measure of steam used, and the pressure of steam found at the end of the stroke, is greater as the cut-off is earlier; by the diagrams referred to, and others from the same engines [referring to the Corliss engines at Saltaire], we find the initial con- densation, relatively to the terminal vario-thermal line, to be as follows : " At 7.4 expansions =27.0 per cent. 9.04 ,, =36.67 per cent. 11.4 =46.67 per cent." 8 For Nos. I, 2, 3, 4, Proceedings of the Institution of Mechanical Engineers, 1862, 1867, 1868. For No. 5, Report of the American Commission on the Vienna Exhibition, vol. iii., page 23. SINGLE-CYLINDER CONDENSING ENGINES. 881 Table No. 299. WORK OF EXPANDED STEAM: SINGLE-CYLINDER CONDENSING ENGINES. 1 Actual Ratio of Expan- sion. Weight of Steam per I.H.P. as per Indicator. Coal Con- sumed I.H. r p. Total Initial Pressure at Cut-off. As cut off. As expanded I 2 3 4 5 Corliss, Saltaire, 5.20 6.62 4.08 3.31 6.60 6.30 6.50 4.90 Ibs. 14.51 15-43 17.28 17.83 13.12 14.66 14.00 14.28 Ibs. 16.03 20.78 20.97 19.05 17.20 16.73 18.28 15.58 Ibs. 2.5 Ibs. 34^ 55 51 50 44 43 50 47 46~ 15 19.75 23.25 27.25 30.5 17-5 23 29 50 Allen Engine Do Do Crossness Pumping ) Engine .. I Do. Do. Crossness averages, East London Pumping Engines : 72-inch cylinder 6.07 i-93 2.80 3-62 4.38 5.10 2.81 3-66 3.65 IO.O 14.27 16.24 14.25 12.92 12.25 1 1. 60 13-95 11-33 I4.8I 16.95 15-74 15.22 14.91 13.58 1377 16.93 14.44 14.83 2.2 Do Do Do. Do 8o-inch cylinder QO do *r 100 do Sulzer Engine (Corliss / srear), . . \ 3-3 With regard to No. i, the Corliss engine at Saltaire, it is stated by Mr. Hopkinson that 6.96 Ibs. of water were evaporated per pound of coal. For No. 3, with good boilers, allow, say, 8^ Ibs. of steam per pound of coal. For Nos. 5 and 6, the quantities consumed were observed. Then, the following are the actual quantities of steam consumed, compared with the quantities indicated : Steam Consumed per I.H.P. per Hour. I. Saltaire, 17.4 Ibs. 3. Crossness (estimated) 18.7 4. East London : 72-inch cylinder (ratio 3.62), 20.72 80 21.38 90 , 18.82 loo 20.08 5. Sulzer, 19.6 More than Sensible Steam cut off. 20 per cent. 60 35 Compound Condensing Engines, with and without steam-jackets. The following data are deduced from particulars supplied by the constructors : 56 882 PRACTICE OF EXPANSIVE WORKING OF STEAM. Table No. 300. WORK OF EXPANDED STEAM : COMPOUND CONDENSING ENGINES. Actual Ratio of Expansion. Weight of Steam per I. H. P. as per Indicator. Coal Con- sumed I.H. r p. Total Initial Pres- sure at Cut-off. As cut off. As ex- panded. 40. Day, Summers, & Co., Receiver, ) Marine, steam -jacketted, ) ( 1st cyl. 1.715 ( 2d cyl. 2. 220 Ibs. 14.74 10-75 Ibs. I7.OO 11.99 Ibs. 2.IO Ibs. 49K Both 3.807 5#. John Elder & Co., Receiver, ) Marine, steam-jacketted, j j 1st cyl. 1.850 ( 2dcyl. 1.852 H-45 13.21 14.85 14.85 1.61 56 Both 3.426 6. J. &E.Wood, Receiver, Station- ) ary no jackets . \ j istcyl. 4.010 j 2d cyl. 1.857 10.94 13-34 10.77 12.03 2.I4 1 85K Both 7.446 7. Donkin, Woolf, Stationary, 2d ) cylinder only steam-jacketted, ... } \ istcyl. 5.269 1 2d cyl. 2. 590 10.09 II. 12 19.16 1.9 5t# Both 13.650 8. Donkin, Woolf, Stationary, ) steam-jacketted, . . ... . . . . j ( istcyl. 2.486 j 2d cyl. 3. 22 1 I3.I8 13.87 17.85 50^ Both 8.007 9. Donkin, Woolf, Stationary, with- ) out steam in jackets, \ ( istcyl. 3.165 | 2d cyl. 3.221 15-59 18.73 I9.OI 48^ Both 10.200 10. Thomson, Woolf, Stationary, ) steam-jacketted \ ( istcyl. 2.985 J2d cyl. 3.384 10.84 12.71 15.27 36^ Both 10. 100 1 This quantity is the result of an estimate. The actual quantity of coal consumed per indicator horse-power was 2.67 Ibs., from which one-fifth, estimated as for general heating purposes, was deducted, leaving 2.14 Ibs. per indicator horse-power, as consumed by the engine. The quantities of steam consumed from the boiler were observed in each trial, except for the first three, and were as follows : WOOLF ENGINES. Steam consumed per I.H.P. 7. Donkin, 2d cylinder, steam-jacketted,... 20.55 Ibs. 8. steam-jacketted, 22.51 9. no steam in jacket, 32.72 10. Thomson, steam-jacketted, 20.93 More than Sensible Steam cut off. 103 per cent. more. I 10 93 Single- Cylinder Engines, steam-jacketted, non-condensing. The trials of SINGLE-CYLINDER NON-CONDENSING ENGINES. 883 portable engines at Cardiff, in I872, 1 afford various examples, from which the following deductions are made : Table No. 301. WORK OF EXPANDED STEAM: PORTABLE ENGINES. CONSTRUCTORS. Actual Ratio of Expansion. Weight of Steam per I.H.P., as per Indicator. Total Initial Pres- sure at Cut-off. As cut off. As expanded II 12 13 14 II 17 Marshall, Sons, & Co., Davey, Paxman, & Co...... Brown & May, 4.8 5.0 R6J3.S 2.4 3.8 2.7 5.0 Ibs. 16.87 H-93 20.52 25.32 18.54 20.08 16.28 Ibs. 29.82 26.45 28.87 30.27 23-93 22.13 29.98 Ibs. 741080 73 73 52 72.5 77-2 63.0 Tasker, Reading Engine Works, . . . Turner Ashby & Co., The quantities of water, as steam, actually consumed per indicator horse- power, are subjoined; together with the effective mean pressures in the cylinders. And, to make a comparison with what the performance would amount to if the atmospheric pressure were removed, as if the steam were condensed, one atmosphere, or 1 5 Ibs. per square inch, is added to the effec- tive mean pressure, as given in the second last column, with the weight of steam per total indicator horse-power accruing, in the last column : Effective Mean Pressure. Steam Consumed per I.H.P. More than Sensible Steam cut off. Effective Mean Pressure plus 15 Ibs. Steam Con- sumed per Total I.H.P. Ibs. Ibs. per cent. Ibs. Ibs. II 3L25 25.9 54 46.25 174 12 33-9 29.6 99 48.9 20.7 13 29.2 3L8 55 44-2 21.2 H 297 37-9 5o 44-7 25.2 15 37-0 24.1 30 52.0 17.2 16 36.24 27-6" 37 51.2 19-3 17 20.4 43-2 165 35-4 24.7 Single- Cylinder Engines, completely covered and heated; non-condensing. Average results of the trials of the " Great Britain " locomotive made in 1850, analyzed by the author in 1852, and published in 1856,2 are given in the following table, No. 302. The cylinders are "inside," being placed 1 The Trials of Portable Steam-engines at Cardiff. Report by the Judges, 1872. 2 Railway Machinery, 1856, page 80. See also a paper "On the Behaviour of Steam in the Cylinders of Locomotives during Expansion," by D. K. Clark, in the Minutes of Proceedings of the Institution of Civil Engineers, vol. Ixxii. 1882-83; page 275. 884 PRACTICE OF EXPANSIVE WORKING OF STEAM. within the smoke-box, and totally surrounded by the atmosphere of hot burnt gases. The cylinders are 18 inches in diameter, with a stroke of 24 inches; and 8 feet driving wheels. The ratios of expansion are here reckoned in terms of the whole of the stroke, though they were not so reckoned in the original investigation. Table No. 302. WORK OF EXPANDED STEAM " GREAT BRITAIN " LOCOMOTIVE. Weight of Steam per Total Initial Notch. Cut-off. Actual Ratio of Expansion. Indicator Horse-power, as per Indi- Pressure per Square Inch at Cut-off. cator. No. per cent. ratio. Ibs. Ibs. ISt 6 7 1.45 28.97 79-4 3d 50 1.90 24.52 72.6 5th 29 2.94 19.74 67.4 The general effect of the observations was that there was no material degree of initial condensation of the steam in the cylinders of the " Great Britain," confirmatory of the results of the author's experiments with the well-protected and heated inside cylinders of locomotives on the Edinburgh and Glasgow Railway. 1 Here follows a statement of steam consumed per indicator horse-power, taking the initial quantities cut off for the quantities actually consumed; showing the relative quantities that would have been due if the atmospheric pressure had been removed, as in a condensing engine: Notch. Effective Mean Pressure per Square Inch. Steam Consumed per I. H. P. Effective Mean Pressure plus 15 Ibs. per Square Inch. Steam Consumed per Total I.H.P. Ibs. Ibs. Ibs. Ibs. ISt 68.1 28.97 83.1 23-7 3 d 53-5 24.52 68.5 19.2 5th 32.1 19.74 47.1 13-5 American Marine Engines? Mr. Emery tested the condensing engines of the U.S. steamers Bache and Rush, having compound cylinders fully steam-jacketted and lagged, and the Dexter and Dallas, having single cylinders felted and lagged only. The Bache was tried in four ways : with and without steam in the jackets, using both cylinders, and using only the second cylinder. The following are the best results of performance for each series of trials : 1 Railway Machinery, page 8.2, &c. ; and the table in the same book, page 151. 2 Journal of the Franklin Institute, February and March, 1875. See also notices of the trials in the Proceedings of the Institution of Civil Engineers, vol. xl. page 292, and vol. xli. page 296. AMERICAN MARINE ENGINES. 885 Table No. 303. WORK OF EXPANDED STEAM : AMERICAN MARINE ENGINES CONDENSING. STEAMER. Actual Ratio of Expan- sion. Indicator Horse- power. Weight of Steam per I. H. P. per Hour. Total Initial Pressure per Square Inch. r ? y ndicator, First Cylinder. Actually Con- sumed. 18 19 20 21 22 23 24 Bache (Woolf). Without steam in jackets : First cylinder, 3-75 9.15 2-73 6.66 ^\ 5.63) 6.91 16.85 3-77 9.19 2.86 6.98 2-35 5-73 2-34 5-7M 2.09 5.10 1.74 4.24 11.82 7.62 5-32 12.62 8.57 5-ii 2.18 2.46 6.22 1.60 4-03 4.46 349 2.08 5.07 3.13 2.32 55-93 77.06 46.40 46.40 7745 99.18 110.50 104.03 102.26 134.53 47.24 71.75 89.14 54.84 74.62 116.01 66.74 266.5 168.7 185.9 292.4 196.2 138.0 221.4 234.3 Ibs. 1541 15.67 15.37 14.55 I4.IO 14.97 15.85 15.85 14.85 17.27 21.03 17.76 17-35 16.42 15.58 16.25 24.05 I7.I 19.7 16.2 I6. 3 20.3 19.2 20.1 22.8 Ibs. 23.76 23.04 23.21 25.11 20.71 20.33 20.37 21.97 22.38 21.17 35.08 29.62 26.25 27.11 24.09 23.15 34.03 18.4 22.1 23.9 23.9 3L8 26.7 26.9 3LO Ibs. about 90 90 90 90 90 90 9 90 90 90 90 90 90 90 90 90 27 82.3 50.2 80.4 53-6 47 394 Both cylinders,.. First cylinder, Both cylinders First cylinder . Both cylinders, With steam in jackets : First cylinder Both cylinders,.. First cylinder, Both cylinders, First cylinder Both cylinders .. First cylinder, Both cylinders First cylinder Both cylinders,. . .... First cylinder, Both cylinders First cylinder, Both cylinders, Without steam in jacket : Second cylinder only, Do. do Do. do. . . With steam in jacket : Second cylinder only Do. do Do. do Do. do. Rush (Receiver). Steam-jacketted : First cylinder, Both cylinders, First cylinder, Both cylinders, Dexter (single cylinder). Felted and lagged, Do. do Do. do Dallas (single cylinder). Felted and lagged, Do do Do do 886 PRACTICE OF EXPANSIVE WORKING OF STEAM. French Stationary Engines^ M. Hallauer reports the results of experi- ments on a 24-inch single-cylinder engine, worked by steam superheated 150 degrees, and lagged and felted yielding 135 indicator horse-power; and his own experiments on a Woolf engine, with steam-jacketted cylin- ders : FRENCH STATIONARY ENGINES CONDENSING. Actual Ratio of Expansion. Steam Consumed per I. H. P. per Hour. Total Initial Pres- sure in Cylinder. 25 Hirn (superheated steam in ) single cylinder) ( 4 Ibs. 15-5 Ibs. 60 26 Leloutre ( Woolf engine, ) steam-jacketted) ( 12 24.83 GENERAL DEDUCTIONS FROM THE DATA OF THE ACTUAL PERFORMANCE OF STEAM. Single Cylinders, with steam-jackets; condensing. The analysis of diagrams from the Allen engine, No. 2, page 88 1, indicates that the expansion-ratio 6.62 was better than the ratios 4.08 and 3.31. Mr. C. T. Porter maintains that the ratio 8 is best for the Allen engine. For the Crossness engines, No. 3, the ratio 6 appears to be the best; though perhaps the Corliss, No. i, with the ratio 5.2, is fully as good as the Crossness. The Sulzer (Corliss gear), No. 5, with the ratio 10, is not so efficient as these others. Of the East London engines, No. 4, the 72-inch engine appears to have greater efficiency when expanding 5.10 times than for less ratios, and of the expan- sion-ratios for the four observed consumptions of water per indicator horse- power, the highest ratio, 3.66, gives the greatest efficiency. Again, the Bache marine engine, No. 21, page 885, yielded the highest observed efficiency with a ratio 5.11, but, by plotting, it is easily shown that the efficiency is practically the same for a ratio of 6. Non- Condensing. The portable engines, page 883, supply examples: For initial pressures between 70 Ibs. and 80 Ibs., -omitting No. 13 as unequal. No. Total Maximum Pressure. Expansion- ratio. Water per Normal I. H. P. per Hour. Water per Total I.H.P., if Atmosphere were removed. Ibs. Ibs. Ibs. 16 77.2 2.7 2 7 .6 19-3 15 72.5 3-8 24.1 17.2 ii 74 to 80 4.8 25.9 17.4 12 73 5.0 29.6 20.7 For lower initial pressures. 14 f 2.4 37-9 25.2 17 63 5.0 43-2 24.7 1 "Recherches Experimentales Sur les Machines a Vapeur." By MM. Hallauer and Dwelshauver-Dery; Bidletin de la Societe Industrielle de Mulhouse, 1877, page 190. GENERAL DEDUCTIONS. 887 It is seen that the highest efficiency is attained with an expansion-ratio of 3.8, whether against or without atmospheric resistance. Single Cylinders without steam-jackets; condensing. The most favourable results of Nos. 20^ 23, 24, and 25, are here abstracted: No. 2O Expansion-ratio. . ^.32 ., Water Consumed per I.H.P. per Hour. 26.25 Ibs 23 .. > ^~. 4..4O 23 Q ,. -3.4.0 . *yy .. 23.Q 24. . e o? 26 7 ,. -\.\-\ . 26 Q 2? .; .. 4.n .. . 18.62 steam It appears that, using ordinary steam, expansion-ratios within the limits of 3^ and 4^ are practically of equal efficiency, and that they give the highest efficiency. The same ratios probably apply to the use of super- heated steam, of which there is only one result, No. 25. Non- Condensing. The results from the cylinder of the "Great Britain" locomotive, page 884, show that the highest ratio of expansion that was tried, namely 2.94, gave the highest efficiency. A greater ratio of expansion would probably have given a still greater efficiency. Compound Cylinders with steam-jackets; condensing. Receiver-engines. Comparing the marine engines, Nos. 40 and 50, it appears that the ratio of expansion, 3.426, gave more efficiency than 3.807. But in the Bache and the Rush, Nos. 19 and 22, it appears that a ratio of from 6 to 7 was most efficient. With Nos. 40 and 5 a, the total initial pressure was 49% Ibs. and 56 Ibs. absolute per square inch; but in Nos. 19 and 22, it was 90 Ibs. and 82 Ibs. Woolf Engines. The ratio 13.65 for No. 7 was better than the ratio 8.007 f r No. 8, requiring 20.55 Iks. an( ^ 22 -5 I Ibs. of water per indicator horse-power respectively. No. 10, with a ratio 10.1, required 20.93 Ibs. of water; whilst No. 26, with a ratio of 12, required 24.83 Ibs. The most efficient ratio of expansion is most probably about 10. Proportional Ratios of Expansion in the first and second cylinders Receiver- engines. Comparing Nos. 40, 5 a, and 22, the best action is seen to be obtained with equal ranges of expansion ; for No. 5 a is better than No. 4 a, and No. 22, in which the ratios are equal, being each 2}^, is the best. Woolf Engines. Nos. 7, 8, and 10 form a curious series: XT. Expansion-ratio, Total Expansion- Steam consumed in First Cylinder. ratio. per I.H.P. 8 2.486 8.007 22.51 Ibs. or7i per cent, more than per diagram. 10 2.985 10.100 20.93 or 93 7 5-269 13-650 20.55 or I0 3 It appears that, as the initial proportion of steam condensed in the first cylinder increases, as shown by the percentages in the last column, the efficiency increases. The best performances of the Bache, No. 19, are made with a total expansion of from 6 to 9, having from 2^ to 3^ expan- 888 PRACTICE OF EXPANSIVE WORKING OF STEAM. sion-ratios in the first cylinders, and 2^ in the second. No. 26, with a total expansion-ratio 12, consumed 24.83 Ibs. of steam per indicator horse-power per hour; and the Bache, No. 19, for an expansion-ratio of 16.85, consumed 25.11 Ibs., the highest rate of consumption recorded for Woolf engines. Compound Engines without steam in jackets; condensing. The only example of receiver-engine is No. 6, which is not provided with steam- jackets. Nos. 9 and 18 are examples of Woolf engines. The results are here brought together : No. Total Initial Pressure per Square Inch. Actual Ratios of Expansion. Steam Con- sumed per I.H.P. per Hour. Coal per I.H.P. per Hour. 6. Receiver Ibs. Q C i/ ( 4.0IO J 3 1-857 ( _ 2. 14 Ibs. (estimated, 9. Woolf. 5/z 48^ ' 7446 3 ( 3-165 ) 3 3.221 f 32.72 see note to table, page 882). 1 8. QO ( 10.200 ) f 375 "^ 3 2.44 f 21 ?6 QO te) ( 2.73 ) 3 2.44 1 23 OA QO (. 6.66 J (-2.31 } 5 2.44 1 23.21 te) In the Woolf engines, it appears from the results, that the highest effi- ciency is obtained by an expansion-ratio of about 3 in the first cylinder, with a total ratio of 7. In the receiver-engine No. 6, the ratio in the first cylinder was carried to 4, with a total ratio of 7^, with an apparently excellent result. CONCLUSIONS ON THE ACTUAL PERFORMANCE OF STEAM. For the development of the highest efficiencies of steam, as used in the steam-engine, the steam-jacket or other means for protecting the steam from the cooling and condensing action of the cylinder, must be employed. The superheating of steam prior to its introduction into the cylinder is probably the most efficient means that may be employed for this pur- pose. The application, to the cylinder, of hot gases hotter than the steam is probably the next best means; and next comes the steam- iacket. CONCLUSIONS ON THE PERFORMANCE OF STEAM. 889 The importance of sustaining the temperature of steam expanded in a cylinder, preventing its falling low and leading to the cooling of the cylinder, is strikingly proved by the foregoing hypothetical calculations of the consumption of steam per indicator horse-power in non-condensing cylinders, on the assumption that the resistance of the atmosphere is removed, likening the conditions to those of condensing engines. In the cases of the portable engines and the locomotive, the consumptions, on this supposi- tion, amounted only to 17.2 Ibs. per indicator horse-power per hour, for the portable engine (No. 15), with an expansion-ratio of 3.8; and to 14.8 Ibs. for the locomotive (No. 18), with an expansion-ratio of 2.94. These results are below anything that has been recorded of single-cylinder condensing engines for the same ratios of expansion; and their superiority is due doubtless to the fact that the temperature of non-condensing cylinders never falls below 212. It is deducible from the results, that the compound steam-engine is more efficient than the single-cylinder engine, and that, of the two kinds of compound engines, the receiver-engine is more efficient than the Woolf engine. The reasons for the superiority of the receiver-engine have been partly pointed out in the comparative analysis, page 867. There is another reason in the fact that whilst the temperature of the first cylinder of the receiver-engine never falls below, nor even down to, that of the receiver, which stands at a constant pressure and temperature; in the Woolf engine, on the contrary, the average temperature in the first cylinder must be that of the steam expanding into the second cylinder, which falls continuously with the expansion. The most efficient ratios of expansion, together with the quantities of steam, or water, from the boiler, consumed per indicator horse-power per hour, deduced from the foregoing analyses, are placed for comparison in the table No. 304. It is scarcely necessary to observe, that the evidence of the initial condensation of steam during the period of its admission into the cylinder, is of great importance, and that, clearly, there is a wide margin for economy in the employment of steam for the production of power. Mr. Bramwell, in an excellent and interesting paper on marine engines, in I872, 1 showed that the average consumption of coal per indicator horse-power per hour, by steam-ships with compound engines in long sea voyages, varied from 2.8 Ibs. to 1.7 Ib. in nineteen steamers, for which the average consumption amounted to 2.11 Ibs. The foregoing deductions are consistent with, and are corroborated by, these facts. In the same paper, 2 Mr. Bramwell states that in H.M.S. Briton, fitted with compound engines on the system of Mr. E. A. Cowper, the steam was heated within a steam-jacket, on its passage from the first to the second cylinder, and that the consumption of coal, at nearly maximum power, was 1.98 Ibs. per indicator horse-power per hour, and that, at a third of the power, the consumption of coal was as low as 1.30 Ibs. This evidence is confirmatory of the conclusion that the work of steam is most efficiently developed when it is previously superheated. 1 " On the Progress effected in the Economy of Fuel in Steam Navigation, considered in Relation to Compound-Cylinder Engines and High-Pressure Steam," in the Proceedings of the Institution of Mechanical Engineers, 1872. 2 Page 153. 890 PRACTICE OF EXPANSIVE WORKING OF STEAM. From the foregoing and other evidence discussed in the author's work on the Steam- Engine, the following summary table has been prepared, showing the most economical results of performance of single-cylinder and compound-cylinder steam-engines under various conditions. These results are not put forward as final; but simply to indicate the directions in which the best action of the steam-engine may be obtained. Table No. 304. PRACTICAL PERFORMANCE OF STEAM-ENGINES: THE MOST EFFICIENT RATIOS OF EXPANSION, AND THE QUANTITIES OF WATER CONSUMED FROM THE BOILER PER INDICATOR HORSE-POWER. DESCRIPTION OF CYLINDERS. Most Efficient Ratio of Expansion. Steam, or Water from the Boiler, con- sumed per I.H.P. per Hour. initial volume pounds. Single cylinder, with steam-jacket, condensing : Thoroughly steam-jacketted j short ^tordS"*' 4 6 21 20.6 Only side-iacketted \ c ng stro ^ e "" 3-2 21.7 ' J ( Short stroke.... 5 23 Single cylinder, with steam-jacket, non-condensing, Single cylinder, without jacket, condensing : 4 25 Long stroke 4-5 20 Short stroke.... 4.25 25 Single cylinder, without jacket, condensing, steam ) superheated j 1 J /' 2 Single cylinder, without jacket, non-condensing, ) i8'/ cylinder well protected . . j /*> Compound cylinder, steam-jacketted, condensing: Receiver . . 10 15 to 16 Woolf 12 14 to 18 Compound cylinder, no jackets, condensing : Receiver 6% 2"^ Woolf 21 FRICTIONAL RESISTANCE OF STEAM ENGINES. See page 951. FLOW OF AIR AND OTHER GASES. DISCHARGE OF GASES THROUGH ORIFICES. Am. Gases and vapours act like liquids in flowing through orifices and tubes, in virtue of the difference of the inside and outside pressures; and the velocity of flow is regulated with respect to the fundamental formula for gravity, page 279, v = 8 fro m the last of which the particulars given in the text are derived; also Eli W. Blake, in The Engineer, December, 1869,. page 418; Wilson on Elastic Fluids, in Engineering, vol. xiii, page 35, &c., 1872. 8 9 4 FLOW OF AIR AND OTHER GASES. more than 58 per cent., the velocity of efflux, at constant density, that is, supposing the initial density to be maintained, is given by the formula, ^ = 3-5953 J h (6) -z/=the velocity of outflow in feet per minute, as for steam of the initial density. ^=the height in feet of a column of steam of the given absolute initial pressure, of uniform density, the weight of which is equal to the pressure on the unit of base. The lowest initial pressure to which the formula applies, when the steam is discharged into the atmosphere at 14.7 Ibs. per square inch, is (14.7 x I^ = ) 25.37 Ibs. per square inch. A number of examples of the 5 8 application of the formula are given in table No. 306, for initial absolute pressures of from 25.37 Ibs. to 100 Ibs. per square inch. The truth of the formulas is confirmed with a surprising degree of exact- ness by the experiments of Mr. Brownlee. Table No. 306. VELOCITY OF EFFLUX OF STEAM INTO THE ATMOSPHERE. Absolute Initial Pressure per Square Inch. Outside Pressure per Square Inch. Ratio of Expansion in Nozzle. Velocity of Efflux, as at Constant Density. Actual Velocity of Efflux, Expanded. Weight of Steam dis- charged per Minute, per Square Inch. Ibs. Ibs. ratio. feet per second. feet per second. pounds. 25-37 14.7 1.624 863 1401 22.81 3 14.7 1.624 867 1408 26.84 40 14.7 1.624 8 74 1419 35.18 45 14.7 1.624 8 77 1424 39-78 5o 14.7 1.624 880 1429 44.06 60 14.7 1.624 88 5 1437 52.59 70 14.7 1.624 889 1444 6l.07 75 14.7 1.624 891 1447 65.30 90 14.7 1.624 8 95 1454 77-94 IOO 14.7 1.624 8 9 8 1459 86.34 FLOW OF AlR THROUGH PIPES AND OTHER CONDUITS. Mr. Hawksley 1 states, as the result of varied experience, that the formula put forward by him for the flow of water in pipes, given at page 933, may be employed also for the flow of air in pipes. It is, (7) in which v is the velocity in feet per second, h is the head in feet of air, d is the diameter in feet, and / is the length in feet. But, it is convenient to express the head in inches of water. Taking the density of water as 1 Proceedings of the Institution of Civil Engineers, vol. xxxiii., page 55- FLOW OF AIR THROUGH CONDUITS. 895 815 times that of air, the multiplier (?^ = ) 68 is to be placed under the sign, when ^ = 48 ^/ -_, and by reduction, Flow of Air through pipes. (8) 156,800 z/ = velocity, in feet per second. h = the head, in inches of water. ) hyp log Yjt^_ (p -P')z/... (17) When the back-pressure P" is less than P', the lower limit of the positive pressure, there is a sudden fall of pressure at the end of the stroke, from P' to P", and the expression for the total net work is : PV (18) WORK OF DRY AIR IN A NON-CONDUCTING CYLINDER, ADIABATICALLY. The specific heat of i Ib. of air in foot-pounds of work, is Unit. Foot-pounds. At constant pressure, 2377x772 = 1 83.45 = K. At constant volume, 1688x772=130.3 =K'. Difference, 53.15 =a= K-K'. The difference, 53.15 foot-pounds, is equal to the value of the constant a, formula ( i ), for air, as given in formula ( 3 ), page 899. The ratio of the specific heats at constant pressure and constant volume is as 1.408 to i, or 1.408, whether they are expressed in heat-units or in foot-pounds. ADIABATIC COMPRESSION OF A GAS. Suppose that the gas, having the initial pressure P, volume V, and temperature T, is compressed adiabatically, and attains the pressure P', volume V, and temperature T'; the relations of the pressure, volume, and temperature are as follows: Adiabatic Compression. Showing that, for air, the pressure according to the adiabatic curve, varies inversely as the 1.408 power of the volume: that, in fact, the product P V y is a constant quantity ; and that the absolute temperature varies as 902 WORK OF GASES COMPRESSED OR EXPANDED. the .29 power of the pressure, and inversely as the .408 power of the volume. For instance, when the pressure is doubled, or as i to 2 the volumes are inversely as i to 1.636, or directly as, i to .611 the absolute temperatures are as i to 1.222 Table No. 308 contains the relative values of the ratios of the initial T" V P' and final temperatures , and volumes , for given ratios , of initial and final pressures, 1.2 to 10 times, calculated by means of formulas ( 21 ) and ( 19 ), with columns of differences to facilitate the calculation of interpolations when required. 1 Table No. 308. COMPRESSION OR EXPANSION OF AIR WITHOUT RECEIVING OR GIVING OUT HEAT. Corresponding ratios of pressure, temperature, and volume, according to equations ( 19 ) and ( 21 ). Ratio of Greater to Less Pressures. Ratio of Greater to Less Absolute Temperatures. Inverse of these Ratios. Ratio of Greater to Less Volume. Inverse of these Ratios. Ratio of Less to Greater Absolute Temperatures. Ratio of Less to Greater Volumes. numbers. differ. numbers. differ. numbers. differ. numbers. differ. 1.2 1.054 48 -948 41 1.138 132 .879 91 i-4 I.IO2 44 .907 34 I.27O 126 .788 72 1.6 I.I46 40 873 30 1.396 122 .716 57 1.8 I.I86 36 -843 25 1.518 118 .659 48 2 1.222 35 .8l8 22 1.636 114 .611 40 2.2 1.257 32 .796 20 1.750 112 -571 34 2.4 1.289 30 .776 18 1.862 109 537 30 2.6 2.8 I-3I9 1.348 29 27 -758 -742 16 15 I.97I 2.077 106 105 .^07 .481 26 23 3 1-375 26 .727 13 2.182 102 -458 20 3-2 1.401 25 .714 2.284 100 .438 19 3-4 1.426 24 .701 ii 2.384 99 .419 16 3-6 1.450 23 .690 ii 2.483 97 -403 15 3-8 1-473 22 .679 10 2.580 96 .388 14 4 4.2 1.495 1.516 21 21 .669 .660 9 9 2.676 2.770 94 93 -374 .361 13 12 4-4 1-537 2O .651 9 2.863 93 349 II 4.6 1-557 ig .642 7 2.955 338 10 4.8 1.576 19 -635 8 3.046 89 .328 9 5 1-595 86 .627 32 3.135 434 319 39 6 1.681 77 595 26 412 .280 29 7 1.758 70 -569 22 3^981 396 .251 2 3 8 1.828 63 -547 18 4-377 382 .228 18 9 1.891 59 .529 16 4-759 370 .210 15 10 1.950 .513 5.129 .195 i 2 3 4 5 1 This table is abstracted from a masterly paper, " Etude Theorique sur les Machines a Air Comprime;" by M. Mallard (Bulletin de la Societe de V Industrie Minerals, 1866-67). ADIABATIC COMPRESSION. 903 Let mn be the length of a cylinder filled with i Ib. of gas, having the initial pressure P, volume V, and temperature T. Let the gas be com- pressed adiabatically by a piston into the volume V, to the pressure P', and the tem- perature T'. The work of compression is measured by the area dgnd', and as shown by Mr. J. H. Cotterill, 1 it is expressed by for air, 130.3 (T'-T) . 22 > Add the work of driving the compressed air out of the cylinder into the reservoir, mea- sured by the rectangle dm, equal to V'P'; and subtract the work contributed by the air from the initial source, the atmosphere, for instance, which presses on the other face of the piston, measured by the rectangle gm, equal to V P. The net work expended is, Fig. 349. Compression of Air adiabatically. W = K' (T'-T) + V'P'-VP. ForairW=i 3 o.3(T'-T) + V'P'-V (23) By formula ( i ), V P' - (K - K') T', and V P - (K - K') T, (a being = K V'P'-VP = (K-K') (T'-T);) for air, V'P'- VP = 53.15 (T'-T) / " Substituting the value of V'P'-VP, in equation (23), and reducing, it becomes, Work expended in Compressing i pound of Dry Gas, in terms of the temperatures, W=K(T'-T); for air, W= 183.45 (T'-T) (24) That is, the net work expended in compressing i pound of gas, is equal to the increase of temperature, or the difference of the initial and final temper- atures, in Fahrenheit degrees, multiplied by the specific heat in foot-pounds at constant pressure. f When the initial temperature only is given, T'-T = T ( -i), and by substitution in formula ( 24 ), the final temperature may be found when the pressures are given : Work of Compressing i pound of Dry Gas {formula to aid in finding the final temperature). for air, W= 183.45 (25) P T' Corresponding to the ratio of the pressures r , the value of is found for air, in the table No. 308; thence the work, and also the final temperature. 1 Notes on the Theory of the Steam- Engine. 1871. 904 WORK OF GASES COMPRESSED OR EXPANDED. To express the work in terms of the pressures, P V = (K - K') T, by for- P V mula ( i ), and T = 7 . Substitute this value of T in formula (24) - 9 and Jv Js. also ( ) ' 29 for -; then, by reduction, Work of Compressing i pound of Dry Gas, in terms of pressures and initial volume. forair,W = 3 .45PV((.^r-i) The value of V, the volume of a pound of air, may be found for various pressures and temperatures by the formulas ( i ), ( 2 ), page 898. P V P' V Again, substitute the value of T = and T' = , in equation / \ j j "- &- &- ~~ " (24); and reduce: Work of Compressing i pound of Dry Gas, in terms of pressures and volumes. W=-A-(P'V'-PV); for air, W = 3 -45 (P'V'-PV) (27) JV JV To exemplify the rise of temperature by adiabatic compression, take atmospheric air at 62 F., or (461+62 = ) 523 F. absolute temperature. P' In doubling the pressure, the ratio = 2, and by the table No. 308, the corresponding ratio of the absolute temperatures is 1.222; whence, 523 x 1.222 = 639, the increased absolute temperature, and 639-461 = 178 F., the final temperature. For ratios of pressure, 2, 3, 4, 5, 10, the ratios of the initial and final absolute temperatures are, 1.222, 1.375, 1.495, i-595> i-95 and when the initial temperature is 62 F., the final temperatures are, 178 258 321 373 559. It may be noted that, in this example, for the ratios of pressure, 2, 3, 4, 5, and 10, the final temperatures are, very roughly, 3, 4, 5, 6, and 9 times the initial temperature 62. ADIABATIC EXPANSION OF GASES. Adiabatic expansion is a duplicate, in reverse, of the adiabatic compres- sion of a gas against a piston, and the primary formula ( i ), page 898, (K-K')T, with its derivatives (19) to (27), are applicable, by reversing the order of the symbols of initial and final pressures, volumes, and temperatures, defined at page 898. ADIABATIC EXPANSION. 905 The compressed-air engine differs from the compressing engine, in being controlled by a valve by which the supply of air to the cylinder is cut off at any point of the stroke, and any degree of expansion is effected. The air may thus be worked in three ways: ist, when it is completely expanded down to atmospheric pressure before it is exhausted; 2d, when it is admitted for the whole of the stroke, and exhausted at full pressure; $d, when it is only partially expanded, and exhausted at a pressure above atmospheric pressure. Referring for explanations to the discussion of adiabatic compression, it is sufficient now to repeat the formulas for compression, as adapted for adiabatic expansive-working. When a gas is completely expanded behind a piston from the pressure P, volume V, and temperature T, to P', V, and T', the relations are as follows : Adiabatic Expansion of a Gas. c P /V'V 4 8 / x '> for air > ^ = ............ (29) The table, No. 308, contains corresponding values of ratios of pressures, volumes, and temperatures, to save calculation. IST. WHEN THE GAS is COMPLETELY EXPANDED DOWN TO AN EQUALITY WITH THE BACK-PRESSURE. In the diagram, Fig. 349, let m n be the length of the stroke of a cylinder, into which i pound of a gas of the pressure P is admitted, occupying the portion of the stroke cd, or the volume V; and let the gas be expanded to the end of the stroke, and the volume V, and the pressure P', equal to the pressure of the surrounding medium, constituting back-pressure. The initial work, during admission, is measured by the rectangle dm, equal to VP, and the back-pressure by the rectangle gm, equal to V P'. The work of ex- pansion between the initial and final temperatures T and T', is measured by the area dgnd', and is expressed by, J>4'(T-T') = K'(T-T')j for air, 130.3 (T-T) ...... (32) That is, the work by simple expansion is equal to the fall of temperature, or the difference of the initial and final temperatures in Fahrenheit degrees, multiplied by the specific heat in foot-pounds at constant volume. Add the initial work, and deduct the work of back-pressure, and the net total work expended is, W = K'(T'-T) + VP-V'P', ) y-\ for air, W= 130.3 (T -T) + VP - V F j V By substitution and reduction, as was done for compression, page 903: 906 WORK OF GASES COMPRESSED OR EXPANDED. Work performed by One Pound of Dry Gas expanded down to the back- pressure, in terms of the temperatures. W = K(T-T); for air, W= 183.45 (T-T) (34) That is, the net work performed is equal to the fall of temperature, or the difference of the initial and final temperatures, in Fahrenheit degrees, mul- tiplied by the specific heat in foot-pounds at constant pressure. T , When the initial temperature only is given, T-T = T ( i - J_ )- } and by substitution in formula ( 34 ): Work performed by One Pound of Dry Gas expanded down to the back- pressure {formula to aid in finding the final temperature). W = KT(i-^); for air, W= 183.45 T(i- -II) (35) T' The value of corresponds in table No. 308, column 3, to the ratio of p the initial and final pressures, . Thence the work may be found; also the final temperature. To express the work in terms of the pressures, P V = (K - K') T, by P V formula ( 28 ), and T = ,. Substitute this value for T in formula K K (35 ); and also (-p-)' 29 for ; then, by reduction, Work performed by One Pound of Dry Gas, in terms of pressure and initial volume. . _ _ . (36) for air, W = 3.45 P V P' V Again, substitute the value of T = 7 , and T' = in equation (34)- and reduce, Work performed by One Pound of Dry Gas, in terms of pressures and volumes. .(PV-FV); for air, W = 3.45 (PV-FV).... (38) Jv Js. To exemplify the fall of temperature by adiabatic expansion, take atmo- spheric air at 62 R, or (461 + 62=) 523 F. absolute temperature. In p reducing the pressure to a half, the inverse ratio = 2, and the correspond- ing ratio of temperatures, column 3, table No. 308, is .818; whence 523 x ADIABATIC EXPANSION. 9O/ .818 = 428, the final absolute temperature, and 461-428= -33 F., the final temperature. Similarly, for inverse ratios of pressure, 2, 3, 4, 5, 10, the ratios of the initial and final absolute temperatures are, .818, .727, .669, .627, .513, and when the initial temperature is 62 F., the final temperatures are, -33, -81, -in , -133, -i 93 F. These instances illustrate the limitless possibilities of producing cold by the expansion of air. It is clearly as impracticable to work a compressed- air engine in such low temperatures, when every particle of moisture and lubricant would be frozen, as amongst the high temperatures previously noticed. 2D. WHEN THE GAS is ADMITTED TO THE CYLINDER FOR THE WHOLE OF THE STROKE. In this case, there is no expansive working, and the gas is exhausted at full pressure. The work done by i pound of dry gas is W = V(P-P'), ........................ (39) in which P and P' are the initial and the exhaust pressures. P V = (K - K') T, by formula ( 28 ), page 904, and, by inversion, and, by substitution and reduction, W = (K-K')T(i-^); for air, W = 53.15 T(i-J'). ... (41) A.gain, the general equation for the work done by i pound of dry gas is (formula ( 34 ), page 906), W = K(T-T); andW = KT(i-), ....... (42) in which T and T' are the initial and the final temperatures. Equating these expressions for W, ( 41 ) and ( 42 ), and, reducing, 1 for air, = . 71 + . 29 ; ..... ( 43 ) By either of these formulas, (43, 44), the final temperature T' is found, P' when the initial temperature T is given. For a ratio, for air, -= ^, or p -, = 2, for instance, with the initial temperature 62 F., or absolute tempera- 1 This method of finding the final temperature, by equating the two expressions for W, is borrowed from M. Mallard. See the preceding note, page 902. 908 WORK OF GASES COMPRESSED OR EXPANDED. ture 523, the final temperature T' = 5 23 (.7i+.29x %) = 523 x. 855 = 447; and 461 - 447 = - 14 F. To facilitate calculation, by means of formula ( 43 ), the values of the ratios of the absolute temperatures corresponding to given ratios of the pressures, are given in table No. 309. Table No. 309. COMPRESSED-AIR ENGINE: AIR ADMITTED FOR THE WHOLE OF THE STROKE. CORRESPONDING RATIOS OF PRESSURES AND TEMPERATURES. Ratio of the Final to the Initial Pressure. Ratio of the Initial to the Final Pressure. Ratio of the Final to the Initial Absolute Temperatures. Ratio of the Final to the Initial Pressure. Ratio of the Initial to the Final Pressure. Ratio of the Final to the Initial Absolute Temperatures. I I I '/6 6 .758 X 2 .855 // 7 751 '/s 3 .806 # 8 .746 1 A 4 .782 '/9 9 .742 '/s 5 .768 x /xo 10 739 The final temperatures of air under adiabatic expansion, and also when exhausted at full pressure, without expansion, due to given ratios of pres- sure, are detailed, for comparison, in table No. 310, in the second and third columns. The reduced efficiency by adiabatic expansion, supposing the initial temperature to fall to 62 F., given at page 910, is here given in column 4. The same, for full pressure, without expansion, is given in column 5. It is calculated thus, in the first instance, for example : The final temperature, column 3, is - 14 F., and is (62 + 14 = ) 76 below 62, being the range of the temperature in doing work. But the range of temperature in compress- ing the air adiabatically to twice the initial pressure is (178 (as at page 904) -62 = ) 116; and (T~?X 100 = ) 66 per cent, is the reduced efficiency without expansion, as in column 5. The ratios of these reduced efficiencies, Table No. 310. COMPRESSED-AIR ENGINE: AIR EXPANDED ADIABATI- CALLY, AND AIR ADMITTED FOR THE WHOLE STROKE. COMPARA- TIVE FINAL TEMPERATURES, AND REDUCED EFFICIENCIES. Initial temperature = 62 F. Ratio of the Final Temperature. Reduced Efficiency. Ratio of Reduced Efficiencies : Initial to the Final Pressure. With Adiabatic Expansion. Without Expansion. With Adiabatic Expansion. Without Expansion. sion and with Complete Expansion. Fahr. Fahr. per cent. per cent. per cent. 2 -33 -14 82 66 80 3 -81 -40 73 52 71 4 - in -52 67 44 66 5 -133 -60 63 39 62 10 -193 -75 5i 27.5 54 EFFICIENCY OF COMPRESSED-AIR ENGINES. 909 in columns 4 and 5, are given in the last column; found thus, in the first example, for instance: ( x 100 = ) 80 per cent. These ratios may also 82 be calculated as the ratios of the ranges of temperature in the two cases. In the first instance, for example, ( - 33 + 62 = ) 95, and ( - 14 + 62 = ) 76, are the ranges for adiabatic expansion, and without expansion; and (L- x 100 = ) 95 80 per cent, is the ratio of the reduced efficiencies. The table indicates, generally, the economical disadvantage of working compressed air without expansion. 30. WHEN THE GAS is BUT PARTIALLY EXPANDED. The absolute temperature of the gas, when expanded, falls from T to T'' at the end of the stroke. Here, it is suddenly exhausted into the surround- ing medium, and the temperature falls still further, to T". The work done by i pound of gas, in terms of the extreme temperatures, is, by the general formula ( 34 ), page 906, W = K(T-T"); for air, W= 183.45 (T-T) ............ (45) whence, as in ( 35 ), W=KT (!-); for air, W= 183.45 T(i-l^) ........ (46) (47) When the successive pressures, P, P', P", are known, the ratios of the temperatures in these last two formulas are easily found in the table No. 308, p" p" p' page 902, from the ratios of the pressures , or 7, and ; when the calculation for the work may be completed. The temperatures T' and T' may be found from T; first, for T', by inverting equation (31 ), page 905, for air,T = T - Thence, the value of T', the ultimate temperature, is found according to formula ( 44 ), page 907, to be, ''=T'(. 7 i + .29 ..... (49) EFFICIENCY OF COMPRESSED-AIR ENGINES. The work by expansion would be an exact duplicate, in reverse, of the work expended for compression, and the two works would be equal to each other, if the reverse actions took place between the same temperatures, pressures, and volumes. The efficiency of the combined compressor and motor would be equal to 100 per cent., irrespective of losses by friction and clearance. But, under practical conditions, the initial temperature for 9IO WORK OF GASES COMPRESSED OR EXPANDED. expansion is not more than that of the surrounding atmosphere; and, in working, by expansion, back to atmospheric pressure, even between the same extremes of pressure, the volumes are smaller, since the temperatures are lower; and the efficiency must, of course, be less than 100 per cent. In working air, under these conditions, between two given pressures, first compressively, and, second, expansively, let the ratios of the pressures, P which are the same in both actions, be , P being the higher pressure, and P' the lower, or atmospheric pressure. Put T" for the higher temperature by adiabatic compression, whilst T is, as before, the atmospheric tempera- ture, and T' the final temperature by expansion. Then, according to for- T" T /PY 2 9 mula(3i), = - = ( - ) ; that is to say, the ratios of the absolute tempera- /P\.2 9 tures are equal to each other, since they are each equal to \p,J . It follows that, T" :T : :T : T'; and that T"-T : T-T' : : T" : T; that is to say, the range or difference of the temperatures for compression, to 6^ per cent, of the weight of the air. Particulars of the compression of air, dry and moist, are given in table No. 311: COMPRESSION AND EXPANSION OF MOIST AIR. 913 Table No. 311. COMPRESSION OF AIR, DRY AND MOIST. TEMPERATURE AND WORK. (Deduced from M. Mallard's data.) Final Pressures. Final Temperatures for Compression. Work Expended in Com- pressing i pound of Air. Moisture Required to Produce Saturation in Initial Temperature=68 Fahr. Imtia.1 Pres- sure = i Air with Air with parts of the Weight of Atmosphere. Dry Air. Sufficient Moisture. Dry Air. Sufficient Moisture. the Air Com- pressed. atmospheres. Fahr. Fahr. foot-pounds. foot-pounds. per cent. i# 133 94 13,300 13,200 2.4 2 I8 5 III 23,500 22,500 3-0 2^ 22 9 124.5 30,500 29,OOO 3-6 3 266 135-5 37,000 35,000 4.0 ^ 300 145-4 43,200 40,500 44 4 330 153-5 48,500 45,000 4-8 4/2 357 161.6 53,600 49,000 5-1 5 383 167 58,500 52,500 54 s l A 407 173 63,200 56,500 5-7 6 428 179 67,000 60,000 6.0 6/ 2 440 184 71,000 63,000 6.2 7 470 190 75,000 66,000 6.4 7/2 49 194 78,300 68,300 6.6 Work in Expansion. There is a slight gain in work done, by the presence of vapour in the air, in a state of saturation; but it may be neglected in ordinary calculations. Table No. 312. EXPANSION OF AIR, DRY AND MOIST. TEMPERATURES. (Reduced from M. Mallard's data.) Temperatures. Ratio of Expansion. Final. Initial. Dry Air. Air with Suffi- cient Moisture. Fahr. Fahr. ratio. ratio. 32 40 1.05 1. 10 32 50 1.13 1.24 32 60 1.22 1.38 3 2 62 1.23 I.4I 32 68 I..28 1.50 32 70 1.30 1.56 32 80 1-37 1-75 32 90 1.47 2.00 32 100 2.28 32 no 1.67 2.6 3 32 120 1.76 3.00 32 130 1.88 345 32 140 2.0O 4.00 914 WORK OF GASES COMPRESSED OR EXPANDED. Temperature in Expansion. When moisture is present in air in the condition of saturation, the fall of temperature during expansion, is greatly less than what takes place when dry air is expanded. That a compressed- air engine may work without the freezing of any moisture or vapour in the air, it should not exhaust at a temperature lower than the freezing-point. Table No. 312, page 913, shows a few examples of the maximum ratio of expansion that may be practised, with given initial temperatures, when the final temperature is to be 32 F. : The table shows that air at 120 F. may be introduced into the cylinder at a pressure of 3 atmospheres, and expanded to atmospheric pressure, without risk of interference from the freezing of moisture; whilst with dry air, the maximum pressure, under the same condition, is only 1.76 atmo- spheres. AIR MACHINERY. MACHINERY FOR COMPRESSING AIR, AND FOR WORKING BY COMPRESSED AIR. COMPRESSION OF AIR BY WATER AT MONT CENIS TUNNEL WORKS (COMPRESSEURS A COLONNE D'EAU). 1 The motive power was derived from the fall of a column of water, having a head of 85^ feet, acting on the principle of a hydraulic ram, the water, by the power of its fall, compressing a given quantity of air at each stroke. There were 1 1 rams, to each of which the water was conducted from the reservoir by a 24-inch pipe. Each ram made from 2^ to 3 strokes per minute, and the air was compressed to 6 atmospheres of total pressure. The volume of air at atmospheric pressure, shut in and compressed for service at each stroke of the ram, was measured by a column in the air-limb of the pipe, 2.04 feet in diameter and 14.1 feet high, making a volume of 46.1 cubic feet of atmospheric air, or (46.1^6 = ) 7.68 cubic feet of com- pressed air. The volume of compressed air for 2 ^ strokes per minute was 19.2 cubic feet per minute. The net horse-power is [ (6 x 15) x 144 x 19.2 xhyp log 6]-^33ooo = 13.51 horse-power. The total expenditure of power in the water for generating compressed air was, 2.o4 2 x.7854x 14.1 x62^ Ibs. x85^ feet x 2^ strokes _ jg H p 33000 The efficiency was, thus, equal to 73 per cent. COMPRESSION OF AIR BY DIRECT-ACTION STEAM-PUMPS. In the temporary machines used at the works for the St. Gothard tunnel, the steam-piston, 19.7 inches in diameter, was fixed to the same rod with the air-piston of 17.73 inches, with a stroke of 4 feet. The air-pumps worked in water. The minimum number of double strokes per minute was 5, but the machine could make 20 per minute. In compressing air to 3 atmospheres, the efficiency, according to the indicator-diagrams, was 84 per cent. These pumps have been replaced by others on Colladon's system, in which the air-cylinder is kept cool by exposing every piece that is in contact with the air when undergoing compression, to currents of cold water. The pump makes 90 revolutions per minute, and is maintained sufficiently cool in compressing air to 8 atmospheres of pressure. 1 Simms 1 Practical Tunnelling, 3d edition, 1877, page 261. 916 AIR MACHINERY. COMPRESSED-AlR MACHINERY AT POWELL DUFFRYN COLLIERIES. 1 This machinery was constructed by Messrs. J. Fowler & Co., for Sir George Elliott. There is a pair of horizontal air-compressing engines connected to one shaft, the steam-cylinder and the air-cylinder being in one line, on the same rod. The steam-cylinders are 34 inches, and the air-cylinders 40 inches in diameter, with a stroke of 6 feet. The engine is worked with steam of 70 Ibs. effective pressure, cut off at one-fourth, and is fitted with Cornish steam- and exhaust-valves, 8 inches and 9 inches in diameter. The engines make 20 turns per minute, giving 240 feet of piston per minute, to indicate 482 horse-power, against a pressure of air of 40 Ibs. per square inch above the atmosphere. The air-cylinders are immersed each in a cold-water bath, open at the upper side. Experiments were made with a double-cylinder air-compressing engine, similar in arrangement to the above, having 1 6-inch cylinders for steam and for air, of 30 inches stroke, with an air-receiver 5 feet in diameter and 24 feet long. The steam was cut off at 80 per cent. The air-engine was an ordinary semi-portable engine, having two lo-inch cylinders of 12 inches stroke, cutting off at three-fourths. The air from the receiver was led into and passed through the boiler of the portable engine, and was thereby cooled down to within 5 of the atmospheric temperature before it passed into the cylinder. The principal results of the trials are quoted from the paper and given in table No. 313; in which the two lines, 7 and 12, have been calculated and added by the author. Table No. 313. AIR -COMPRESSING ENGINES, AND COMPRESSED-AIR ENGINES, AT POWELL DUFFRYN COLLIERY RESULTS OF TRIALS. Pressure of Air in Receiver, Effective, Ibs. 40.0 34-0 28.5 24 19 i. Effective mean pressure in steam-cylinders, Ibs. 2. Do. do. air-cylinders, Ibs. 3. Speed of piston, per minute, feet 4. Effective mean pressure in air-engine, Ibs. 5 Speed of piston per minute feet 26.3 24.0 190 35-6 1 08 25.1 22.7 155 29.8 IO4 21.5 19-5 140 24.7 104 19.7 16.5 no 21.0 1 08 16.6 '4-5 60 17.0 88 Air-compressing engine 6 In steam-cylinder (A), . I.H.P. Co. 4 46.2 35.8 25.8 11. 8 7 In air-cylinder (B), I.H.P. 52.6 4O.7 32.2 21.7 10. 1 8 Air-engine cylinder (C), . I.H.P. 18.3 14.7 12.2 10.8 7.1 9 Do brake (D), H.P. 1C. 7 12. C, IO.2 9.O 5.4 10. Efficiency of D in parts of A, per cent. u. Do. C ,, A, percent. 12. Do. B ,, A, percent. 25.8 30.8 87.7 27.1 31-8 88.0 28.5 34-1 9.8 34-9 41.9 84-3 45-8 60.2 85-4 13. Total Pressure in receiver, atmospheres 14. Actual final volume in air cylinder of com- pressing engine initial vol. I 3-72 .380 3-31 425 2.94 .470 2.63 .518 2.29 575 15. Final volume according to the adiabatic curve . . initial vol. ~~ I 393 .427 465 503 555 1 6. Final volume according to the hyperbolic curve initial vol. I .269 .302 340 .380 437 Actual mean pressure : 17 From indicator-diagram Ibs. 24.0 22.7 19.5 16.5 14.5 18. By the adiabatic- curve, Ibs. IQ By the hyperbolic-curve . . Ibs. 23-5 10.7 21. 1 17.6 18.6 ic.. 9 164 14.2 13-8 12.2 Proceedings of the Institution of Mechanical Engineers, 1874. HOT-AIR ENGINES. QI/ HOT-AIR ENGINES. Engines worked by heated air are of two classes: ist. Those in which the air is heated and cooled alternately by contact with hot and cold surfaces; and, 2d, those in which the air is mixed with the hot products of combustion when heating surface is not used. IST CLASS. RIDER'S Hox-Am ENGINE. In this engine, which is called a compression-engine, two single-acting cylinders are placed vertically, a little apart, connected at the upper part by a regenerator composed of thin plates. One of these is the working or hot cylinder, under which a fire is maintained, the other is the air-pump, or cold cylinder, surrounded by water to cool the air which is drawn into it, and which is pumped back into the hot cylinder. The plungers of these cylinders are worked by cranks placed at an angle of 95 on a shaft overhead. The working plunger of the i horse-power engine has a dia- meter of 6^ inches, with a stroke of 9^ inches; the pump-plunger is 6^ inches in diameter, with a stroke of 8.6 inches. " The compression (pump) piston first compresses the cold air in the lower part of the compression-cylinder into about one-third of its normal volume, when, by the advancing or upward motion of the power (working) piston, and the completion of the down-stroke of the compression-piston, the air is transferred from the compression-cylinder, through the regenerator, and into the heater, without any appreciable change of volume. The result is a greater increase of pressure, corresponding to the increase of temperature, and this impels the power-piston up to the end of its stroke. The pressure still remaining in the power-cylinder, and reacting on the compression- piston, forces the latter upward till it reaches nearly to the top of its stroke, when, by the cooling of the charge of air, the pressure falls to its minimum [about atmospheric pressure], the power-piston descends, and the compres- sion again begins. In the meantime the heated air, in passing through the regenerator, has left the greater portion of its heat in the regenerator-plates, to be picked up and utilized on the return of the air towards the heater." From indicator diagrams, taken at 120 turns per minute, it appears that the effective mean pressure in the working cylinder was 16.8 Ibs., and that in the pump was 7.15 Ibs. per square inch. Reducing the pump-pressure o { in the ratio of the strokes, it becomes 7.15 x = 6.47 Ibs.; then (i 6. 8 9-5 - 6.47 = ) 10.33 Ibs. per square inch is the net effective pressure on the working plunger, from which the power is to be calculated. The area of the plunger is 35.78 square inches, and the net indicator horse-power is 35.78 Ibs. x I0 # Ibs. x .80 foot x 120 = 6 hors er 33,000 It is stated that the quantity of coal consumed is from 2 to 3 Ibs. per net indicator horse-power. An engine of ^ horse-power was tested to deliver from 650 to 700 gallons 91 8 AIR MACHINERY. of water per hour, 90 feet high, with a consumption of 4 Ibs. of coal per hour. 1 Taking a mean of 675 gallons, the performance is equivalent to (675x10 Ibs. x 90 feet -- 60=) 10,125 foot-pounds per minute, or to (10,125 * 33> ooo = ) -37 horse-power of net duty, for which (4 Ibs. -^.307 = ) 13 Ibs. of coal was consumed per horse-power. 2D CLASS. BELOU'S Hox-AiR ENGINE AT CUSSET. The air is supplied by a feeding cylinder, i metre in diameter, with i y% metres of stroke, in which it is compressed, and from which it is discharged into a close furnace, where it is heated by the combustion produced by it. Thence, it is passed to the working cylinder, 1.4 metres in diameter, with i^ metres of stroke, where it acts with full pressure and expansively, after which it is exhausted into the atmosphere. These cylinders are double- acting. The feeding cylinder draws i cubic metre of air at each stroke. The furnace is inclosed in a horizontal cast-iron cylinder; the grate is inclined, and has an area of .80 square metre, or 8.6 square feet. The greater portion of the air passes through the grate. The engine makes 23 turns per minute, giving a speed of pistons of 225 feet per minute. The temperature in the chimney is 480 F. The absolute pressure in the feeding cylinder, is raised to 1.94 atmo- spheres, for which the period of compression is 51.5 per cent, of the stroke. In the working cylinder, the initial pressure is 1.68 atmospheres; the air is cut off on the upper side at 39 per cent, of the stroke, and expanded exactly to atmospheric pressure at the end of the stroke; on the lower side, the admission is longer, to compensate for the weight of the piston about 2 tons. The difference of the pressures, (1.94- 1.68 = ) .26 atmosphere, or 3.8 Ibs. per square inch, represents the resistance in the furnace and the passages. The average effective pressures are, in the feeding cylinder, 9.4 Ibs., and in the working cylinder 7.13 Ibs. per square inch; yielding respectively 80.62 and 119.74 indicator horse-power. Thus, it is seen that two-thirds of the working indicator power is expended in supplying air to the working cylinder. Allowing only 10 per cent, of the indicator power for general resistances, and so reducing it to 107.77 horse-power, the net useful work is 107.77 - 80.62 = 27.15 horse-power, which is 22.67 per cent, of the indicator power. The quantity of coal consumed is 88 Ibs. per hour, being at the rate of .735 Ibs. per indicator horse-power, or 3.24 Ibs. per net horse-power, as at the brake. 2 GAS-ENGINES. Gas-engines are worked by the explosion of a mixture of coal-gas and air, which acts on a piston within a cylinder. They may be double-acting or single-acting, and the explosion may be effected by means of an electric battery, or of lighted jets of gas placed in communication with the mixture. 1 At the meeting of the Royal Agricultural Society at Birmingham ; Messrs. Eastons and Anderson, Engineers. 2 See the Annales du Conservatoire des Arts et Metiers, vol. vii., for full particulars of Belou's engines. The data above given are drawn from this source. GAS-ENGINES. 919 LENOIR'S DOUBLE-ACTING GAS-ENGINE. Two horizontal engines of this kind, fired by electricity, were tested by M. Tresca. 1 During a part of the stroke, the gas and air, in fixed propor- tions, are admitted into the cylinder, and then exploded by an electric spark. By the explosion, heat and pressure are generated, and the pressure acts on the piston during the remainder of the stroke. During the return- stroke, the gaseous products are exhausted into the atmosphere; whilst the explosive action takes place on the other face of the piston. The heat of the cylinder is reduced by a continuous current of cold water applied on the outside. In the first engine, the cylinder was 7.1 inches in diameter, with a stroke of 4 inches. The mixture of gas and air was cut off at half-stroke, and the maximum absolute pressure in the cylinder was a little less than 6 atmo- spheres. The average speed of the engine was 129 turns per minute, giving a speed of piston of 153 feet per minute. The power measured by the brake was .57 horse-power, and the quantity of gas consumed amounted to 1 1 2 cubic feet per brake horse-power per hour. The gas and air were mixed in proportions of i to 10. Fifty-three per cent, of the heat gener- ated in the cylinder was carried off by the water outside. The combustion of the gases was very nearly complete. For the second trial, the engine had a cylinder 9^ inches in diameter, with a stroke of 4^ inches. The weight of the engine complete was 14 cwts. The speed of the engine was 100 turns per minute, giving 158 feet of piston per minute. The period of admission was a little more than half- stroke, and the maximum absolute pressure was 5.36 atmospheres. The quantity of gas consumed amounted to 97 cubic feet per brake horse-power per hour; the power developed at the brake being about i horse-power. The gas and air were mixed in the proportion of i to 1 1 y z ; and the volume of gas admitted for each stroke was 24^ cubic inches, the heat of combus- tion of which is, according to M. Tresca, 96 English units. It is not sur- prising that the temperature and the pressure after explosion, are lowered, as is shown by diagrams, almost instantaneously by contact with the metal; and it is for this reason, probably, that the stroke is made so short in pro- portion to the diameter. The quantity of water consumed for cooling the cylinder amounted to 4^ cubic feet per horse-power per hour, the temper- ature being raised 140 F. M. Tresca has estimated the distribution of the heat generated in the cylinder as follows : Heat carried off by the water and the products of combustion, 69 per cent. Heat converted into work at the brake, 4 Losses, not estimated, 27 IOO The net efficiency at the brake is thus taken as 4 per cent. OTTO AND LANGEN'S ATMOSPHERIC GAS-ENGINE SINGLE-ACTING. In this engine, the cylinder is vertical, open to the atmosphere at the upper end; it has a " free piston," its principal feature, which is impelled upwards 1 Annales du Conservatoire des Arts et Metiers, vol. i., 1861, page 894. Q20 AIR MACHINERY. against the atmosphere, by the explosion of gas below it. The stress of the explosion of gas is intense, but momentary, and the free piston mounts instantly and quickly against the atmospheric resistance, whilst yet the explosive force continues. Thus the explosive force is utilized efficiently, and the power is derived from the pressure of the atmosphere, by which the piston is driven downwards against the partial vacuum formed under the piston by the collapse of the gaseous products. To cool and contract the gases, the lower half and the bottom of the cylinder are jacketed with cold water. The piston-rod is formed as a rack, and gears into a pinion loose on the fly-wheel shaft. The pinion turns loose with the rack during the ascent, but, during the descent it engages with and turns the shaft by means of a friction-clutch, making two revolutions during one descent. The routine of the engine is as follows: i. The piston is lifted through '/nth of the stroke to receive the charge of gas and air. 2. The mixture is fired by a gas-light. 3. The piston makes the up-stroke. 4. The plenum under the piston becomes a vacuum of 22 inches of mercury, at the beginning of the down-stroke. 5. The down-stroke is made under an effective pressure of 1 1 Ibs. per square inch, and the force is transmitted to the shaft. 6. When the piston arrives near to the bottom, the vacuum becomes a plenum, by compression of the gases; and, by the weight of the piston and rack^ the gaseous products are expelled from the cylinder. The intermittent motion is worked by a tappet on the rack to raise the piston for the next charge. According to Mr. Crossley, 1 for a ^ horse-power engine, the cylinder is 6 inches in diameter, with a stroke of 40 inches; and the explosions are made at various rates up to that of 30 per minute. The mixture consists of 6^ volumes of air to i volume of coal-gas. He takes the heating power of i Ib. of coal-gas, of density .40, at 24,000 units of heat; and, for a con- sumption of 1.05 cubic feet of gas per minute, the heat supplied to the engine is equivalent to 584,000 foot-pounds, of which 70,000 foot-pounds, or 12 per cent., is yielded at the brake. The power at the brake is (70,000 -=-33,000 = ) 2.12 horse-power, and the consumption of gas is at the rate of (1.05 x 60 -f 2.12 = ) 30 cubic feet per horse-power at the brake per hour. From indicator-diagrams, it appears that, in the down-stroke, the effective pressure varies from 1 1 Ibs. per square inch to zero at four-fifths of the stroke, averaging 9 Ibs. for four-fifths, or about 7 Ibs. for the whole of the stroke. M. Tresca 2 tested a 6-inch single-acting gas-engine, in which the power at the brake, making 81 turns per minute, was .456 horse-power. The gas consumed per hour was Per Minute. fc.lhj.g-.*-. For work in cylinder, 20.09 cubic feet. 44.06 cubic feet. For inflaming, 2.08 4.57 22.17 48.63 The water-jacket absorbed only 800 English units of heat per hour. M. Tresca allows a heating power of only 6000 French units per cubic 1 See a paper by Mr. F. W. Crossley, on "Otto and Langen's Gas-Engine," in the Proceedings of the Institution of Mechanical Engineers, 1875* P a g e IQ* "* Annales du Conservatoire des Arts et Metiers, vol. vii., page 628. GAS-ENGINES. Q2I metre of coal-gas, equivalent to (6000 x 4 x = ) 21,000 English units per O J pound of 30 cubic feet, in round numbers. The quantity of heat generated, according to this allowance, was, then, (2i,ooox = ) 14,000 units per hour. The work at the brake was (.456 x 33,000 x 60 -r 772 = ) 1 1 70 units per hour, which represents an efficiency of (1170 x IOO-T- 14,000 = ) 8.4 per cent. THE OTTO GAS-ENGINE. Whilst the Otto and Langen atmospheric gas-engine superseded the Lenoir gas-engine, it was in its turn superseded by the " Otto Silent Gas-engine," the invention of Dr. Otto, of Deutz: called silent in com- parison with other gas-engines, but now known as the Otto Gas-engine. A novel and important feature was the compression of the explosive mixture before being fired: effecting economy of gas by increase of efficiency, and facilitating the use of engines of greater power than before. Constructed by Crossley Brothers, Manchester, the Otto engine is horizontal in its disposition, resembling generally a steam-engine. It is single-acting, having the cylinder open at one end, with a trunk-piston. The cycle of operations is fourfold: in the first out-stroke a charge of gas and air in mixture, in the ratio of i to 16, including the burnt gases, is drawn in; and in the first in-stroke, following, the charge is compressed until it reaches a pressure of 35 Ibs. per square inch; at the beginning of the second out-stroke the compressed charge is ignited and exploded, and acts on the piston for propulsion, during this, the working stroke; by the second in-stroke the gaseous products of combustion are expelled from the cylinder. Thus there are one charge and one explosion for every four single-strokes, or two double-strokes or revolutions. Strictly, therefore, the engine is half single-acting; and a heavy fly-wheel is necessary, to work the piston through the negative part of the cycle. The cylinder serves alternately as a compression-pump and as a motor-cylinder. It is jacketed with cold water to prevent overheating, although it is estimated that a loss of 42 per cent is thus incurred. The ignition of the charge has been effected by means of a slide-valve, carrying a gas-jet, kept constantly burning. This form has recently been superseded by .a system of ignition within a tube opening to the cylinder, charged with a strong igniting mixture. Space is provided for the com- pressed charge by a prolongation of the closed end of the cylinder. The initial pressure in the cylinder varies from 120 Ibs. to 190 Ibs. per square inch above the atmosphere. The Otto engine is constructed with a single cylinder, of various nominal power, of from ^ H.P. to 20 H.P., indicating from 2 H.P. to 50 H.P. ; and with double cylinders, indicating from 16 H.P. to 100 H.P. The 12 N.H.P. engine has been proved to develop 28 I.H.P., and 23 H.P. at the break, or 82 per cent of the indicator power; with a consumption of 20 cubic feet of gas per indicator horse-power per hour, or 24.3 cubic feet per brake horse- power. The total consumption of gas was at the rate of 560 cubic feet per hour; when running without load, 100 cubic feet per hour. In a 4 H.P. engine, 23.3 cubic feet was consumed per brake horse-power. In the use 922 AIR MACHINERY. of Dowson gas instead of ordinary coal-gas, i y*> pounds of anthracite coal is consumed per indicator horse-power per hour. The consumption has occasionally been only i.i pounds. CLERK'S GAS-ENGINE. The gas-engine of Mr. Dugald Clerk is single-acting. A charge is exploded at every out-stroke, the mixture of gas and air being as i to 9, admitted for the first half of the stroke. During the second half-stroke pure air is admitted. In order to effect the explosion at every out-stroke a displacer cylinder is employed, and the charged air is compressed till it attains a pressure of 38 Ibs. per square inch, when it is exploded and burns during the out-stroke. The exhaust takes place near the end of the stroke, and as the piston returns, the pure air of the charge is exhausted through the pipe, which is cooled. GASEOUS FUEL. THE WILSON GAS-PRODUCER. The Wilson gas-producer, introduced by Mr. Bernard Dawson, is an upright cylindrical chamber of firebrick, having a solid hearth, kept nearly full of small bituminous coal. A mixture of air and steam, comprising about 20 parts of air to i part of steam by weight, is delivered into the lower part of the chamber, the air being induced by two small jets of steam from a steam boiler. The fuel is resolved into combustible gases, hydrogen, carbonic oxide, and hydrocarbons, in the manner of the ordinary gas- furnace; and the gases pass through a number of openings above the level of the fuel, into an annular flue, whence they are conducted by an under- ground conduit to the place of consumption. The fuel is charged in at the top, which is closed by a pendulous conical valve or plug. As applicable for supplying heat to steam boilers, the results of a test- trial, in November, 1886, at Apsley Paper Mill, Hemel-Hempstead, con- ducted by the author, may be noticed. Four Cornish boilers were fitted with two 4-cwt Wilson gas-producers, for generating steam by gas-firing. The boilers are each 5^ feet in diameter, with a 3-feet furnace-tube, and 21 feet long; having eight Galloway tubes in each. The producers stand side by side in an open yard adjacent to the boiler-house. Each producer is cylindrical, 8 feet in diameter, 9 feet high, of firebrick cased in plate-iron. The internal hearth is 5 feet in diameter, having 20 square feet of area, the fuel space above the hearth is 4^ feet deep, and the gases pass through openings into the annular flue surrounding the neck or upper part of the furnace, whence they are conducted underground to the boilers. Here the supply of gas to each boiler is regulated by means of a valve. The gas is delivered through the doorway, together with air, into the furnace-tube, and combustion takes place. The four steam boilers are set in a row. No. i boiler was separated from Nos. 2, 3, and 4, and was devoted to the generation of steam for supplying the blast injector attached to each gas-producer. The three other boilers were connected for the supply of steam to the factory. The feedwater was measured separately into No. i boiler. The coal used in the producers was cobbles from Wyken Colliery, broken up by hand; charged into each hopper about four times per hour. THE WILSON GAS-PRODUCER. 923 The leading results of the test-trial are as follows; and for comparison, the results of a six-days' test of the same boilers, which had previously been made with hand-firing, are prefixed. In this case, the fire-grates were 4 feet 8 inches long, presenting an area of 14 square feet for each boiler. Hand-firing. Gas-firing. Coal consumed per boiler in full steam per hour, 163 Ibs. 258.6 Ibs. Water evaporated do. do. do. !493 cub. ft. 24.94 cub. ft. Water per pound of coal from and at 212 F 6.79 Ibs. 7.16 Ibs. (net) It is shown that the boilers in full steam did two-thirds more evaporative duty by gas-firing than by hand-firing; and, with 5^ per cent, more evapo- rative efficiency net, after allowance made for steam consumed in blowing the producers. It is also shown that the weight of steam consumed by the producers was equal to 8.29 per cent, of the total quantity generated in the four boilers. The total evaporative efficiency of the boilers with gas-firing, if no deduc- tion be made for the demands of the producers, is expressed by 6.56 pounds of water per pound of coal, or an equivalent of 7.81 pounds from and at 212, which is (7.81 -6.79 = .98 pound, or 14.4 per cent, more efficiency than was obtained by hand-firing. This is an expression of the absolute difference of efficiency in favour of gas-firing. The practical difference, after making the needful allowance, is, as above stated, 5 */ per cent. At intervals no smoke was visible with gas-firing; and there is no good reason why, with good draught, gas-firing should not be conducted entirely without smoke. Comparative trials have been made at Plas Power Colliery, by Mr. John H. Darby, in which it was shown by the best result that a greater absolute evaporative efficiency was attained of 9.85 per cent, in favour of gas-firing. The following was the average composition of the gases produced: Carbonic acid 6.26 Oxygen o.oo Hydrogen 14.68 Carbonic oxide 23.98 Marsh gas 4.72 Nitrogen 5-36 100.00 THE DOWSON GENERATOR GAS. It is known that highly-heating non-luminous gases can be produced by decomposing steam in the presence of carbon: passing steam and air through a fire of incandescent fuel. Mr. J. Emerson Dowson practises this system; and, in addition, he employs special means of generating and superheating the steam. The steam producer and superheater consists of a long coil of tube, of such a form that nearly all of it is exposed to the action of gas flame. Water is forced, under a pressure of from 20 Ibs. to 25 Ibs. per square inch, into the coil, in which it is converted into superheated steam. The gas required for heating the coil is drawn from the gas-holder. The retort or generator is of iron, lined with ganister. The fuel, anthracite, rests on a grate, above an inclosed chamber, into 924 AIR MACHINERY. which a jet of superheated steam is directed through a small opening, carrying with it, by induction, a current of air into the furnace, for combus- tion. The gas produced contains by volume, approximately, 20 per cent, of hydrogen, 30 per cent, of carbonic oxide, 3 per cent, of carbonic acid, and 47 per cent, of nitrogen. The gross quantity of fuel consumed in working Otto Gas-engines averages, as before stated, 1.3 pounds per indicator horse- power. Professor Witz, of Lille, tested a 9 horse-power gas-engine on the Delamare-Debouteville system; and proved a consumption of 89 cubic feet of the Dowson gas, or 1.33 pounds of coal per brake horse-power per hour. FANS OR VENTILATORS. COMMON CENTRIFUGAL FAN. The ordinary fan consists of a number of blades fixed to arms, re- .volving on a shaft at high speeds. It appears from the results of Mr. Buckle's experiments, 2 that when the fan is revolved in its case without any air being discharged, the pressure generated at the circumference of the fan varies as the square of the velocity of the fan, and the horse-power required to maintain the speed varies as the cube of the velocity. It further appears that the pressure generated at the circum- ference is one-ninth greater than that which is due to the actual circumfer- ential velocity of the fan. To express the relation of the pressure and the -.2 velocity of an air-current, the height due to the velocity is h = h is also 64 equal to the height of a column of air equal in weight to the pressure. The velocity due to the pressure may thence be deduced by means of the ordinary relation v-^^Jh. Mr. Buckle recommends the following proportions for fans of from 3 to 6 feet in diameter, and for pressures ranging from 3 to 6 oz. per square inch : The width and length of the vanes equal to one-fourth of the dia- meter. The diameter of the inlet openings in the sides of the fan-chest equal to half the diameter of the fan. For higher pressures, of from 6 to 9 oz. and upwards, Mr. Buckle recommends that the vanes should be narrower and longer, and the inlet opening smaller, than are prescribed by the above proportions. He gives the following table of dimensions. The number of blades may be 4 or 6. The case is made of the form of an arithmetical spiral, widening the space between the case and the revolving blades, circumferentially, from the origin to the opening for discharge; and it appears that the upper edge of the opening should be level with the lower side of the sweep of the fan : 1 English Mechanic, December 29, 1876, page 387. 2 "Experiments Relative to the Fan-Blast," by Mr. Buckle; Proceedings of the Institu- tion of Mechanical Engineers, 1847. FANS OR VENTILATORS. 925 Table'No. 314. DIMENSIONS OF FANS. (Mr. Buckle.) Pressure, from 3 to 6 oz. per square inch. Vanes. Diameter of Diameter of Fans. Width. Length. Inlet Openings. feet, inches. feet. inches. feet, inches. feet. inches. 3 o o 9 o 9 i 6 3 6 10% 10% i 9 4 o I I 2 4 6 I IX i iX 2 3 5 o i 3 i 3 2 6 6 o i 6 i 6 3 o Pressure, from 6 to 9 oz. per square inch, and upwards. 3 o o 7 I 3 6 o 8> i# i 3 4 o o 9^ 3X i 6 4 6 10% 4X i 9 5 o I 6 2 6 o I 2 10 2 4 MlNE VENTILATORS. 1 GuiBAL's FAN. The blades are, for the most part of their length, straight; but they curve forwards at the outer ends. They are fixed on polygonal centres, and at a con- siderable backward inclination usually 45, to the radius. The wheel is closely surrounded for about two-thirds of the circumference, by a casing of brickwork; for the remaining third, the casing gradually opens out into the discharge vent, which expands upwards as an inverted cone. By so forming the vent, the velocity of the discharged air is reduced, and converted into outward pressure, by the action of which the velocity through the fan is in- creased, and the efficiency is raised. A Guibal fan, working at Staveley Colliery, is 30 feet in diameter, and 10 feet wide. It makes 60 revolutions per minute in the day. The following are particulars of its performance : Speed, in Turns per Minute. Draft in Inches of Water. Volume of Air Discharged per Minute. Efficiency, in parts of the Gross Indicator Power of the Engine. cubic feet. per cent. 32 .70 43,852 40.38 51 1.70 86,283 43-09 6 4 2.77 101,773 53-27 68 3.10 1 10,005 53.85 1 These particulars of mine-ventilators are derived from papers on "Ventilation of Mines," by Mr. J. S. E. Swindell, and Mr. W. Daniel, in the Proceedings of the Institution of Mechanical Engineers, 1869, 1875. 926 AIR MACHINERY. The advantage of surrounding the fan by a casing, and of adjusting, by means of a slide, the size of the opening into the vent, is shown by the following results of trials at a mine in Belgium : Gross Efficiency. Without casing, 22 per cent. With casing, 31 With casing and expanding vent, 57 With casing and expanding vent, and with ) , slide adjusted, \ These are the efficiencies in parts of " the gross power supplied from the boiler." An efficiency of 60 per cent, is generally obtained by this venti- lator; equivalent to 80 per cent, of the net power of the engine. COOK'S VENTILATOR. This is a positive ventilator, making a given discharge for each revolu- tion. It consists of a revolving eccentric within a circular case, against which a flap-valve is maintained constantly in contact, to separate the entering current from the outgoing current. Two ventilators working at Saltburn have casings of 22 feet in diameter, and n feet 6 inches wide; the eccentric has a diameter equal to two-thirds of that of the casing, and the eccentricity is one-fourth of its diameter. The period of inlet and discharge of air is 235, or about two-thirds of a revolution. Making from 26 to 29 turns per minute, with a draught of from i to 3.25 inches, the efficiency was found to be from 58.5 to 64 per cent, of the indicator horse- power. BLOWING ENGINES. Blowing engines of recent design are direct-acting, the steam-piston and the air-piston being fixed to one rod, and the steam- and air-cylinders in line. There is a pair of blowing cylinders, each of which is worked by a steam-cylinder; and the two steam-cylinders are either a pair or are arranged as compound cylinders. The clearance in the air-cylinders should be reduced to the smallest practicable limits. At Lackenby Iron- works it is only 3 per cent, at each end ; the total area of valve-opening at each end, for the inlet, is J / 6 th of the area of the piston, and for the outlet, ^th. These proportions are unusually liberal. The two air-cylinders are 80 inches in diameter, with 54 inches of stroke, having each a capacity of 157 cubic feet. They make 24 double strokes per minute, giving a speed of piston of 216 feet per minute; 190,000 cubic feet of atmospheric air are supplied per ton of iron made, and the supply is sufficient for the produc- tion of 800 tons of iron per week. The blast-main is 30 inches in diameter, and has a capacity 12^ times the united volumes of the cylinders. The pressure in the main is 4^ Ibs. per square inch above the atmosphere, and it is free from fluctuations. The ratio of compression is '- = 1.3, and the valves, therefore, open to the main when the air-piston has passed through 20 per cent, of the stroke, approximately, allowing for clearance; whilst the air is driven into the compressor during 80 per cent, of the stroke. The clearance is, proportionally, 3 x 1.3 = 4 per cent, of the volume of com- pressed air; and thus the effective charge of air is (100-4 = ) 9 6 per cent. BLOWING ENGINES. Q2/ of the quantity compressed. The steam-cylinders are 32 and 60 inches in diameter, and their indicator horse-power is 290 horse-power; whilst that of the air-cylinders is 258 horse-power, representing an efficiency of 89 per cent. 1 An instance of very low pressure of blast produced by a blowing engine is given by Mr. Briggs. A pair of 1 2-inch steam-cylinders drive directly a pair of 48-inch air-cylinders, with a stroke of 24 inches. The steam-valves cut off at ^ths, and they have "negative" lead and ample cover to the exhaust, for the purpose of counteracting the expansive force of the com- pressed air left in the clearance. But the clearance in the air-cylinders is very considerable; it is equivalent to JJ/2 inches of the stroke, or 31^ per cent. At 60 double strokes per minute, when the speed of piston was 240 feet per minute; the indicator diagrams showed an average effective pressure of 17.8 Ibs. per square inch for steam, and 0.8 Ibs. for air, repre- senting an efficiency of 72 per cent. 2 In French blowing engines, according to M. Claudel, the proportion of the air discharged is only 75 per cent, of the volume described by the piston. The stroke is usually equal to the diameter of the air-cylinder, and the speed of piston varies from 100 to 200 feet per minute. The area of the inlet-valve openings is from */ I5 to X / I2 of that of the piston for speeds of from 100 to 150 feet per minute, and from */^ to J / 9 * for higher speeds. The outlet openings are from Z / I5 to I / 20 of the piston, in area. In Belgium, Mr. Cockerell employs Woolf cylinders, with the beam, for driving blowing engines. In one example, the engine is of 160 horse- power; the cylinders are 2.79 and 3.94 feet in diameter, adapted for an expansion-ratio of 10, at regular work, to yield a pressure of 7 inches of mercury, or 3^ Ibs. per square inch. The engines usually expand nine times, with steam of 4 atmospheres. The intermediate fall of pressure between the first and second cylinders is uniform throughout the stroke, equal to 1.5 Ibs. per square inch. The expansion curves are the same as the "theoretical curve." The efficiency, by the indicator applied to the steam- and the air-cylinders, is 81 per cent, for a blast of 4 Ibs., and 83^ per cent, for a blast of 4^ Ibs. per square inch. 3 ROOT'S ROTARY PRESSURE-BLOWER. Root's rotary blower is positive in its action, and consists of two revolvers on parallel axles, within a close-fitting case, rectangular in section, with semicircular ends. The revolvers consist each of two arms, formed with a bulbous expansion at each end; and being geared together by a pair of spur-wheels on their shafts, outside the case, they necessarily revolve at the same speed ; and they work together in such a manner that the ends of the arms of one revolver enter or gear into the middle of the other revolver. Being very correctly fitted, little air is allowed to escape between the revolvers, or between them and the casing. By their harmonious revolu- tions, one being horizontal whilst the other is vertical, the spaces below and above the revolvers are alternately contracted and enlarged in such an 1 "Blowing Engines at Lackenby Ironworks," by A. C. Hill. See Proceedings of the Institution of Mechanical Engineers, 1871, 1872. 2 Journal of the Franklin Institztte, March, 1876. 3 Portefeiiille de John Cockerell, 1876, vol. iii. 928 AIR MACHINERY. order that whilst air is drawn into the case at the lower side, it is expelled at the upper side. Four discharges of air are thus performed for each revolution of the machine, and a steady current is maintained. For blowing air, these machines are made by Messrs. Thwaites & Carbutt, of from y% to 14 nominal horse-power, to supply from 150 to 10,800 cubic feet of air per minute, from delivery orifices of from 2^/2, to 19 .inches in diameter. According to the results of tests made by a committee of engineers, in the United States, the efficiency of the blowers amounted to from 65 to 80 per cent, of the horse-power expended and applied to the machine. As mine-ventilators, Root's blowers are constructed with revolvers of from 3 feet 10^ inches to 25 feet in diameter, and 3 feet 2 inches to 13 feet wide, making from 280 to 40 revolutions per minute, delivering a volume of air of from 45 to 5000 cubic feet per turn, or from 12,500 to 200,000 cubic feet per minute. The effective power expended in delivering these volumes of air, for an exhaustion at a 6-inch water-column, is, by formula ( 14 ), page 896, from 15.5 to 189 horse-power, and the dimensions of cylinder for a non-condensing engine to drive the ventilators, vary from 14 inches diameter with 18 inches stroke, to 28 inches diameter with 48 inches stroke. FLOW OF WATER. FLOW OF WATER THROUGH ORIFICES. The fundamental formula for the flow of water by the action of gravity is 0=8 V h ................................ (i) v the velocity in feet per second. // = the height in feet through which it freely falls. This is the basic formula ( 6 ), for the action of gravity, page 279. The quantity of water delivered per second, through an orifice in the side of a vessel, supposing that there is no contraction, is expressed by the formula, Q = 8tf>v/^~ .............................. (2) Q = the quantity in cubic feet per second. a = the normal sectional area of the orifice or the stream in square feet. But, in effect, the quantity is less than is here expressed, by reason of the contraction of the outflowing stream, and the equation becomes, for prac- tical use, in which m is a coefficient, less than i, the value of which varies with the conditions of the orifice. When the water flows through an orifice in a thin plate, the average value of m is about .62, irrespective of the form of the orifice, and the formula becomes, Q= 4.96 a *J h or, in round numbers, (4) in which 7z = the height in feet, measured to the centre of the orifice. With an approaching velocity v', the general formula ( 3 ) becomes (s) When adjutages or spouts are added to an orifice, the outflow is increased. If an internal tube be added, it is diminished. The average values of the 930 FLOW OF WATER. coefficient m, and the product 8 m, in formulas follows : 3 ) and ( 5 ), are as FOKM o, *,. VaJo'" ' Internal tube, .............................................. 50 Thin plate simply, ........................................ 62 Cylinder, at least 2 diameters in length, ........... 82 Converging cone, length = 2> diameters, ......... 95 Vena contracta, length = l / 2 diameter of orifice; smallest diameter = .785 diameter of orifice,... Diverging cone, length = 9 diameters, ............. 1.46 4.0 5.0 6.6 7.6 g 11.7 MR. J. F. BATEMAN'S EXPERIMENTS AT GODBY RESERVOIR, IN I852. 1 Three rectangular openings, 6 inches deep, and 6 feet long, were made in boards 2^ inches and 5 inches thick, to the sections shown in Figs. 353. The forms of the edges, horizontal and vertical, were quad- rantal, to a radius of 2^4 inches. In No. i the bell-mouth section was No. i. No. 2. No. 3. Figs. 3S3- Godby Reservoir. Flow of water through submerged openings in boards. Scale outwards, in No. 2 inwards, and in No. 3 both outwards and inwards. The openings were entirely submerged on the inside, at depths of from i to 4 feet to the centres of the openings, and there was a free discharge. The following are the values of the coefficient 8 m, for formula ( 3 ), the coeffi- cient for the whole velocity due to the height being expressed by 8. COEFFICIENTS OF VELOCITY OF DISCHARGE ( 8 m ), IN FORMULA ( 3 ). Deduced from the Results of Experiments at Godby Reservoir. HEAD. feet. 4 3-5 3 2.5 2 1-5 I 0.5 Average Coefficients (maximum limit, 8). No. i _ No. 2 No. 3 (2% inches thick). (2% inches thick). (5 inches thick). 5.78 7-04 7-60 5.66 7.04 7.60 5- 6 7 7-04 7-60 5.60 7.04 7.80 5.60 7.04 7.80 5.50 7.04 7.78 5.60 6.89 7.30 q.^o 6.00 6.;; Averages, omitting the last, 5.63 7.02 7.64 1 See Mr. Bateman's paper on the Manchester Water-works, Proceedings of the Institution of Mechanical Engineers, 1866. FLOW OF WATER THROUGH ORIFICES. 93! Thence the values of m and 8 m are FORM OF OPENING IN BOARDS Formulas (3) and (5). (Mr. Bateman). . Value of m. Value of 8 m. No. i. Quadrantal outwards, 70 5.6 No. 2. inwards, 875 7.0 No. 3. outwards and inwards, 94 7.6 It is seen that the values of the coefficients were little affected by the variations of head, except when the head was less than about i foot, or double the height of the aperture. MR. JAMES BROWNLEE'S EXPERIMENTS ON THE FLOW OF WATER THROUGH A SUBMERGED NOZZLE, CONVERGENT AND DIVERGENT. 1 Mr. Brownlee's experiments were made with a nozzle, the sectional contour of which may be described as a double trumpet-mouth. The entrance was i^ inches in diameter, and i^ inches long, converging to a diameter of .1982 inch at the throat; whence it diverged to a diameter of I s/ l6 inch at the other or discharging end, through a length of 5.95 inches, which was equal to thirty times the diameter of the throat. Putting h^ and h 2 for the heads in feet of water at the entrance to, and the exit from, the nozzle, and v for the velocity of the water passing through the throat, the generating head is (h t - /i 2 ), and the relations of this head and the velocity are : 1.61 (6) These formulas differ from the normal formula ( i ), in embodying the i. 6 1 power of the velocity instead of the 2d power, and they indicate that the velocity of discharge is greater than that normally due to the head. The additional velocity is generated in consequence of a vacuous additional pressure at the throat, the sum of which, and the generating head (h^ ^ a ), is the true head under which the discharge takes place. The table No. 315 contains, for illustration, a selection from the experimental results of Mr. Brownlee. It appears that, under a double generating head, the water is driven through the compound nozzle, with ( 1>6l \/ 2 =) 1.538 times the speed for a given generating head; and that a double velocity does not require four times, as by the normal formula, but only (2 1 - 61 = ) 3.05 times the pressure. Mr. Brownlee attributes the great augmentation of velocity of flow through the throat of the compound nozzle, to the great length of the divergent outlet, comparatively to the diameter of the throat. The principle of the action upon which the augmented flow is effected, is that the velocity of the outflowing water is, by the necessity of occupying the expanding capacity of 1 Transactions of the Institution of Engineers and Shipbuilders in Scotland, vol. xix., page 81. 932 FLOW OF WATER. the nozzle, rapidly reduced ; and that a vacuum or reduction of back-pres- sure is induced at the throat, which, added to the generating head, makes up the true head to which the velocity is due. Table No. 315. FLOW OF WATER THROUGH COMPOUND OR DOUBLE- CONICAL NOZZLES (Mr. Brownlee's Experiments). The Heads are expressed in feet of water. Generat- ing Head, \-k* Vacuum at Throat of Nozzle. True Head, or Sum of the Generating Head and Velocity due to the Generating TT J Velocity due to the True Head. Experi- mental Velocity. Velocity by Formula (6). the Vacuum. Head. feet. feet. feet. ft. $ second. ft. $1 second. ft. $ second. ft. & second. .25 .52 77 4-01 7.04 6.66 6.77 .50 1-3 1.8 5.67 10.76 10.23 10.42 75 2.4 3-15 6.95 14.24 13-6 13-39 3-5 4-5 8.02 17.02 16.34 16.02 2 8.2 IO.2 n-35 25.63 24.74 24.64 3 14.0 17 13-9 33-09 31-95 31-7 4 19.8 23-8 16.05 38.84 37-9 37-9 5 26.0 31 17.94 44.69 43-45 43-52 6 31-1 37-1 19.66 48.88 48.14 48.74 FLOW OF WATER OVER WASTE-BOARDS, WEIRS, &c. To find the discharge of water over waste-boards, &c., the general for- mula is, Q = H - Q = the quantity in cubic feet discharged per second. m = a coefficient. /= the width of the notch or overflow, in feet. H =the height in feet of still-water above the edge of the notch or board. h = the height in feet of still- water, above the level of the water as it flows over the board. When the coefficient m = .62, the equation becomes, by reduction, (9) FLOW OF WATER IN CHANNELS, PIPES, AND RIVERS. The friction of fluids upon solid surfaces is independent of the pressure. It is proportional to the area of the surface, or to the area of the sides and bottom directly, and to the volume of moving water inversely; or, in brief, it is as the length of the contour or wetted perimeter of the conduit, c t FLOW OF WATER IN CHANNELS, PIPES, AND RIVERS. 933 divided by the sectional area a, of the current, or to - . It is proportional a to the square of the velocity nearly, or to mv*. The accelerating force is equal to -j x g, in which h is the height, / the length of the slope of the channel, and g is gravity, or 32.2. Then, gx = xmv 2 -, and, 10 The quotient is the hydraulic mean depth, or mean radius, and the c i -- elocity is proportional to \/ . Mr. Downing deduces from experiment on the flow of water in pipes, the formula, Velocity of water in a channel or pipe. /ha /ha , V - \/ X X 10,000 = IOO A/ x ( II ) I c I c a In pipes, -- - - ; and, when pipes are filled with the flowing water, the c 4 formula ( 1 1 ) becomes, by substitution, Velocity of water in a full pipe. v = the velocity, in feet per second. h the head, in feet. / = the length, in feet. d= the diameter, in feet. c = the wetted perimeter, in feet. a = the sectional area of the current, in square feet. Q = the quantity of water discharged, in cubic feet per second. D = = the hydraulic mean depth. f the fall, in feet per mile. The formula (12) is nearly identical with the formula employed by Mr. Hawksley, for the flow of water in a smooth pipe of small and uniform diameter : -xd ................................ (13) Quantity of water discharged by a channel or a pipe. f^ /ha / h T^ /\ Q=ioo# V /~ x = 100 fl/y/ x D ...................... ( 14 ; 934 FLOW OF WATER. When the pipe runs full under pressure, Q = .7854 d 2 x 50 /y/ ; from which Mr. Downing deduces the formulas : Quantity discharged through a pipe running full under pressure. (Cubic feet per second) Q= 39.27 ^/-xd 5 ( 15 ) (Cubic feet per minute) Q= 2356 /y/ -L-*d s ( 16 ) (Cubic feet per minute Q= J 7 ^ ... . (17) and d in inches ) ^ V / (Cubic feet per minute, n _ d in inches, /in yards) (Gallons per minute, n d s k / N d and /as above....) Q= I? - 3 V ........................ ('9) .) Mr. Downing further deduces from formula ( 1 1 _ (Gallons per hour Q */*... . ( 20 ) d and /as above....) v _ * ( T * \ ~ <7 5 V ~~ ' (Feet per second ) z/= .92 >y/ 2/D : (21) (Feet per minute ) v= 55 \/ 2/D (22) . The average velocity in an open channel is about 4/ 5 ths of the maximum velocity, which is usually attained at the centre, near the surface. 1 Limits of Velocity at the Bottom of a Channel. Mr. Beardmore gives the limits of velocity at the bottom, thus : 30 feet per minute does not disturb clay, with sand and stones. 40 do. do. moves coarse sand. 60 do. do. moves fine gravel, size of peas. 120 do. do. moves i-inch rounded pebbles. 1 80 do. do. moves angular stones, about i^ inches in diameter. CAST-IRON WATER-PIPES. Water-pipes are made to resist incidental stress, in addition to the normal stress by internal pressure. The proper thickness of cast-iron pipes has been expressed by numerous empirical formulas, widely divergent. The following simple formula is here deduced from Mr. Bateman's practice. In sixteenths of an inch,... t= 4 + - ( 23 ) 600 In inches and decimals,... /= .25 + ( 24 ) 9600 Do. do / = . 25 + _^ ( 25 ) 4250 1 See Mr. Downing's work, "Elements of Practical Hydraulics," from which the above formulas have been derived. CAST-IRON WATER-PIPES. 935 / = the thickness of the pipe, in inches and decimals, or in sixteenths of an inch. H = the head of pressure, in feet of water. p = the interior pressure, in pounds per square inch. d= the inside diameter of the pipe, in inches. Note. The pressure in Ibs. per square inch, is equal to the product of the head in feet of water, by .443. Mr. Hawksley's formula for the thickness of pipes under pressure, is, / = thickness, F factor of strength, p pressure, ^-diameter, s- tensile strength of material, C = a practical constant correction for imperfections ot process, method, and workmanship. For the usual head of 300 feet of water, formula ( 24 ) becomes, in inches ...... ^=.25+ or /=. ( 27 ) (28) The following are special divisors and multipliers to be employed in formulas ( 27 ) and ( 28 ) for various heads: Head in Feet. 100 ISO 2OO 2 5 3 00 350 4OO 500 750 1000 Socket-end. For a series of water-pipes cast at Woodside Ironworks, it is calculated, from the sections, that the equivalent length of pipe, of equal weight, for a socket-end, varies from 7.2 inches for 2^-inch pipes, to 8.6 inches for 24-inch pipes. Hence the formula for the equivalent length of pipe for the socket for any diameter : Equivalent length in inches, = 7 + - ........................ ( 2 9 ) Divisor in (27). 96 Equivalent Pressure Multiplier in (28). in Ibs. per Square Inch. I OIO4. AAI The 64 016 66 5 4.8 O2 1 886 O26 j jo 7 22 I ^2 Q 27 O37 1 J^-V >> 155-0 177 2 24. O4.2 IQ *//* )5 078 332.2 Q.6 .104. in feet, =.6 + -7:; ibo (3) in which d is the diameter of the pipe in inches. The table No. 316 gives the thickness and the weight of cast-iron water- pipes of given diameters for a working head of 300 feet of water, or 133 Ibs. per square inch. The bursting strengths, taking the ultimate tensile resist- ance of the metal at 7 tons per square inch, and the factors of safety, are given in the last two columns. 936 FLOW OF WATER. Table No. 316. CAST-IRON PIPES: THICKNESS, WEIGHT, AND STRENGTH. Diameter. Thickness by Formula (24) or (25). Nearest Thickness in six- teenths of an Inch. Net Weight per Foot run, for Thickness in Column 3. Length of Pipe equal in Weight to the Socket. Weight of a o-feet Length of Pipe. Bursting Pressure per Square Inch, reckoned on Column 3. Factor of Safety, for Normal Pressure of 300 Feet of "Water, or 133 Ibs. per Square Inch. inches. inches. whole sixteenths. pounds. feet. cwts. Ibs. times. 2 31 S/i6 7.09 .60 (6 ft.) .418 4900 36 2^ 33 H 10.6 .61 (6 ft.) .625 3920 30 3 35 12.4 .62 1. 06 3920 30 4 375 3 /s 16.1 .62 1.38 2940 22 5 .41 7/i6 23-4 63 2.01 2744 21 6 45 7/x6 27.7 63 2.38 2290 17 7 47 M 36.8 .64 3-17 2240 17 8 5 % 41.7 .64 3-59 1960 15 9 53 9/i6 52.8 65 4-55 1960 15 10 .56 9/i6 58.3 .66 5-03 1764 13 ii 59 9/i6 63-9 .66 5-51 1604 12 12 .62 # 77-5 .67 6.69 1633 12 13 65 # 83.6 67 7.22 1508 II H .70 /i6 99.1 .68 8.56 1540 12 15 7i Afi 105.9 .68 9.15 1440 II cwts. 16 75 ^ 1. 10 .69 10.66 1470 II 18 .81 J 3/i6 1-43 .70 13-87 1415 10.6 20 .87 # i. 60 7i 15-54 1372 10.3 21 .90 # 1.68 .72 16.33 1307 10 24 99 2.19 73 21.31 1307 10 27 1.09 x/x6 2.61 75 25.45 1234 9-3 30 1.18 3/i6 3-24 77 31-65 1241 9-3 33 1.27 X 3-75 .78 36.67 1190 8.9 36 1.36 H 4.50 .80 44.10 1198 8-9 39 1-45 7/i6 5.11 .82 50.18 1156 8.7 42 i-55 9/i6 5.98 83 58.78 1107 8.8 45 1.65 K 6.67 85 65.70 H33 8.5 48 1.74 7.63 .87 75.31 H43 8.6 Note to table. Flanges. The additional weight for a pair of flanges is reckoned as equivalent to that of a lineal foot of pipe ; equal to 1 1 per cent, extra for Q-feet lengths. Gas-pipes. Mr. Thomas Box gives the following thickness for gas- Table No. 317. THICKNESS OF CAST-IRON GAS-PIPES. pipes: 1 Diameter. Thickness. Diameter. Thickness. Diameter. Thickness. Diameter. Thickness. inches. inches. inches. inches. inches. inches. inches. inches. 1% .27 5 37 10 . 4 6 24 .64 2 29 6 39 12 49 30 .6 9 2X 3 7 .41 15 53 36 75 3 32 8 43 IS 57 4 35 9 45 21 .6 Practical Hydraulics, 1867. WATER-WHEELS. The work of water-wheels is done by the force of gravity acting on water. The total work in a fall of water is expressed by the product of the weight of water, w, and the height of the fall, h; or by w h; and, in order that the whole of this work should be realized by the wheel, the water must enter the machine without shock, and leave it without velocity. But there is. unavoidably, a residual velocity v' t and the loss of work due to this z/ 2 velocity, is w = w h ', in which h ' is the head due to the residual velocity. The part of the head expended in effective work, and upon . ' 2 internal resistances, is therefore h Ji=h w . 2 ^ There are two classes of wheels ; first, those which turn on a horizontal .axis; second, those which turn on a vertical axis. WHEELS ON A HORIZONTAL AXIS. UNDERSHOT-WHEELS, WITH RADIAL FLOATS OR BUCKETS. These are constructed from 10 to 25 feet in diameter; the floats are from 14 to 1 6 inches apart at the circumference, and from 24 to 28 inches deep. Putting v and v' for the initial and final velocities of the water, which flows under the wheel, v' is sensibly equal to the velocity of the middle of the float, and the maximum effect is obtainable when v' = , when the effici- 2 ency is 50 per cent. Smeaton tried velocities v' of from .34?' to .52 v t mean .43 v\ and obtained an efficiency, with models, of from 29 to 35, mean 33 per cent. By the best modern experiments, the efficiency is usually from 27 to 30 per cent. Experimentally, 40 per cent, is the maxi- mum. Probably, it cannot be exceeded, because the final velocity of the water is not reduced to y 2 v. PONCELET'S UNDERSHOT-WHEEL. The floats are curved, usually a portion of a circle, and so placed that the hollow of the curve is presented to the entering water, the edge of the float being set at an angle of 30 to the circumference of the wheel. There are 36 floats in wheels of from 10 to 13 feet in diameter, and 48 floats for diameters of from 20 to 23 feet. If the water could enter the wheel without shock, tangentially to the floats, the velocity of the floats being half the velocity of the water, the water would ascend the float, and would then descend by the force of gravity, and drop into the tail-race 938 WATER-WHEELS. with a final forward velocity = o. The efficiency, under these circumstances, would be 100 per cent. But the conditions of practice do not admit of a tangential entrance, and the efficiency is not more than 65 per cent, for falls of 4 feet and less, 60 per cent, for falls of from 4 feet 3 inches to 5 feet, and from 55 to 50 per cent, for falls of from 6 feet to 6^ feet. These efficiencies are materially greater than that of the undershot-wheels with radial floats ; and the experience of the Poncelet-float conspicuously demonstrates the essential importance of providing graduated entrances, and avoiding shocks, concussions, or eddies in the water. The most favourable ratio of the velocity of the floats to that of the water, is 55 per cent. The distance between the inner and outer circumferences that limit the floats should be at least a fourth of the head; Poncelet advises a third of the head. PADDLE-WHEEL IN AN OPEN CURRENT. Experience indicates that the most suitable ratio of the velocity of the floats to that of the current is 40 per cent. The depth of the floats should be from ^th to I / 5 ih of the radius; it should not be less than 12 or 14 inches. The diameter is usually from 13 to i6ft feet, with 12 floats; but it is thought that there might be an advantage in applying 18 or even 24 floats. The floats should be completely submerged at the lower side, but not more than 2 inches under water. BREAST-WHEEL. This wheel receives the water at a level a little below that of the axis. In practice, the efficiency is 70 per cent, when the height of the fall approaches 8 feet, and 50 per cent, for a fall of 4 feet. For a well constructed wheel, slow-moving, M. Morin found an exceptional efficiency of 93 per cent. Sir William Fairbairn states that the efficiency of high breast-wheels is 75 per cent, moving at the rate of 5 feet per second at the periphery. The usual velocity adopted by him for high and low falls, was from 4 to 6 feet per second; for a minimum velocity, 3 feet 6 inches per second, for falls of from 40 to 45 feet; for a maximum velocity, 7 feet per second, for falls of 5 or 6 feet. The water should be delivered to the wheel at a low velocity; or, if the velocity is considerable, the delivery should be at a tangent to the edge of the float. The most suitable velocity of the floats is 4 jj( feet per second ; the velocity should not exceed the limits of from 3 to 6^4 feet per second. The depth of water over the sliding gate should be from 8 to 10 inches, measured from still water. The diameter should be at least 11^2 feet; it is seldom more than from 20 to 23 feet. These diameters are suitable for falls of from 3 to 6 or even 8 feet. The distance apart of the floats should be i y$ to i y^ times the head over the gate, for slow wheels; for quick wheels, a little more. The depth of the floats should be a little more than 2:3 feet. Normally, the interior capacity between two floats should be nearly double the volume of the water there. OVERSHOT-WHEEL. The water is delivered nearly at the top of the wheel. The chief causes of loss of head are, first, the relative velocity of the water when it enters the wheel, and the velocity which it possesses at the moment it falls to WHEELS ON A VERTICAL AXIS. 939 the level of the tail-race. Such wheels answer well for heads of from 13 to 20 feet. For heads of less than 10 feet, breast-wheels are preferred. The velocity of the floats should not be less than 3 feet per second; it may be 6^ feet per second for small wheels, and 10 feet for larger wheels, without sensibly affecting the efficiency. The efficiency at a low speed may rise tc 80 per cent.; but ordinarily, with velocities of from 3 to 6^ feet, the effi- ciency varies from 70 to 75 per cent. At higher speeds, and when the buckets are more than two-thirds filled, the efficiency is only 60 per cent. For wheels employed in driving hammers, of from 10 to 13 feet in diameter, having a velocity of from 13 to 16 feet per second, the efficiency occasion- ally falls to 37 per cent. This low efficiency is due to the impetuosity of the fall of the water into the quick-moving buckets, whence the water is thrown out, partly by reaction, partly by centrifugal force. The capacity of the buckets should be three times the volume of the charge of water; they may be 10 or n inches deep, and 12 or 14 inches apart. With a velocity of 4 feet per second, i cubic foot of water per foot of breadth of the wheel, may be consumed per second. The stream, as it leaves the gate, is rarely more than 4 or 5 inches deep ; it is often less than 2 inches, deep. WHEELS ON A VERTICAL AXIS. TUB-WHEEL. The old-fashioned spoon-wheel or tub-wheel, consists of a number of paddles fixed on a vertical axis, revolving within a cylindrical well of masonry, with very little clearance. The paddles are slightly concave, and are struck in the hollow side by a horizontal current from a reservoir at a considerable head; the current enters the well tangentially, and, after having expended its force, it falls between the open paddles to the bottom of the well. The maximum efficiency is calculated to be that due to a velocity of the centre of the paddles, when they are struck, equal to one- third of the velocity of the current; and the efficiency is 30 per cent. In practice, the efficiency varies from 1 5 to 40 per cent. WHITELAW'S WATER-MILL. Mr. James Whitelaw, of Paisley, developed the principle of Barker's rnill,- and produced an efficient motor. Barker's mill consisted of two hollow radial arms revolving on a central pipe, through which water under pressure passed to the extremities of the arms, and was ejected through an orifice at the end of each arm, in an opposite direction; and so producing rotatory motion. In Whitelaw's mill, the arms taper from the centre towards the cir- cumference, and they are curved in such a manner as to allow the water to pass from the central openings to the orifices, in directions nearly straight and radial, when the machine runs at its proper speed; so that very- little centrifugal force is imparted to the water by the revolution of the arms, and that, therefore, a minimum of frictional resistance is opposed to the motion of the water. A model, 15 inches in diameter, measured to the centres of the orifices, with a central opening 6 inches in diameter, and two orifices of discharge, each 2.4 inches by 0.6 inch, was tested under 940 WATER-WHEELS. .a head of 10 feet, making 387 revolutions per minute. The efficiency .amounted to 73.6 per cent. At 324 revolutions, the efficiency was 71 per cent. According to the results of tests of another model mill, made by competent engineers, the efficiency amounted to 76 per cent., when the speed of the orifices was equal to that due to the height of the fall. A water-mill, onWhitelaw's system, 9.55 feet in diameter, having circular orifices 4.944 inches in diameter, with a fall of 25 feet, was erected on the Chard Canal in 1842, for the purpose of hauling boats up an inclined plane. The net work done by the machine represented an efficiency of 67.3 per cent., with the resistance of the gearing in addition. It was estimated that .the actual duty of this mill amounted to 75 per cent. 1 TURBINES. Turbines, like Whitelaw's mill, apply the force of water by impact and reaction combined; but they comprise an additional feature not embodied in Whitelaw's mill, the employment of guide-blades to change the direc- tion of the water descending under a head, so as to cause it to enter tangentially between the blades of the wheel. The blades of the wheel are so curved as to receive the water without shock, and to discharge it hori- zontally. Turbines are of three kinds; outward flow, downward flow, and inward flow. OUTWARD-FLOW TURBINES. Fonrneyron Turbine. This turbine acts with an outward flow; that is to say, the water enters from above through a central opening, and is guided by curved blades, to be discharged laterally at the base of a circular chamber, or well, equally at all parts of the circumference, into the buckets or curved blades of the wheel. The wheel is annular, and closely surrounds the circular chamber; thus it receives the whole of the water at its maximum velocity, and it is the function of the curved blades to receive and transmit the force of the water, and to discharge the water at the outer circumference of the wheel, with the least possible residual velocity. When the supply of water is insufficient for working the turbine at its full power, the exit openings from the well are partially closed by a cylin- drical sluice which is lowered upon them to the required extent. The efficiency is reduced in proportion as the sluice is lowered, for the -action of the water on the wheel is less favourably exerted. M. Morin tested a Fourneyron turbine, 2 metres, or 6.56 feet in diameter, and he found that, in this way, the efficiency varied from a maximum of 79 per cent, to 24 per cent., when the supply of water was reduced to a fourth of the full supply. In practice, the radial length of the buckets or floats of the wheel is a fourth of the radius, for falls not exceeding 6 */ feet, three-tenths for falls of from 6^ to 19 feet, and two-thirds for higher falls. Boy den Turbine. Mr. Boyden, of Massachusetts, designed an outflow- turbine of 75 horse-power, which realized an efficiency of 88 per cent. The peculiar features, as compared with Fourneyron's turbine, are, ist, and most important, the conducting of the water to the turbine through a vertical trun- cated cone, concentric with the shaft. The water, as it descends, acquires a gradually increasing velocity, together with a spiral movement in the direction 1 See Description of Whitelaw & Stirrafs Patent Water-mill, 1843. TURBINES. 941 of the motion of the wheel. The spiral movement is, in fact, a continua- tion of the motion of the water as it enters the cone. 20!. The guide-plates at the base are inclined, so as to meet tangentially the approaching water. 3d. A " diffuser," or annular chamber surrounding the wheel, into which the water from the wheel is discharged. This chamber expands outwardly, and thus the escaping velocity of the water is eased off and reduced to a fourth when the outside of the diffuser is reached. The effect of the diffuser is to accelerate the velocity of the water through the machine ; and the gain of efficiency is 3 per cent. The diffuser must be entirely submerged. RULES FOR OUTWARD-FLOW TURBINES. Lieutenant F. A. Mahan, U.S., has deduced, from the practice of Mr.. Boyden, the following rules for proportioning outward-flow turbines. He has also deduced the formulas which follow from the results of Mr. Francis' experiments on Mr. Boyden's turbines at the Tremont Mills, Lowell. 1 Rtiles for Proportioning Outward-flow Turbines. For falls of from 5 feet to 40 feet, and diameters not less than 2 feet : i st. The sum of the shortest distances between the buckets should be equal to the diameter of the wheel. 2d. The height of the orifices at the circumference of the wheel should be equal to one- tenth of the diameter of the wheel. 3d. The width of the crowns should be four times the shortest distance between the buckets. 4th. The sum of the shortest dis- tances between the curved guides, taken near the wheel, should be equal to the interior diameter of the wheel. For falls greater than 40 feet, the 2d rule should be modified : the height of the orifices should be smaller in proportion to the diameter of the wheel. On the basis of these rules, an efficiency of 75 per cent, may be obtained. Formulas for the proportions and performance of Outward-flow Turbines. 3 ( 4 ) H-.ioD ....................................... ( 5 ) N = 3 (D + io) ................................. ( 6 ) -8 ............. ............................... < 7 > ^ = .50 N to .75 N ........................... ( I0 ) ;' "-J ........................................... <> 1 Water- Wheels and Hydraulic Motors. New York, 1876. 94 2 WATER-WHEELS. D = the exterior diameter of the wheel. vol. xlvii., page 267. It is due to the author of this paper to say that he has deduced other conclusions than those drawn in the text, from the comparative results of the experiments with Appold's fan and Rankine's fan. 948 MACHINES FOR RAISING WATER. the efficiency is from 50 to 66 per cent. For higher lifts, an efficiency of from 70 to 80 per cent, may be realized. WATER-WORKS PUMPING ENGINES. The indicator-power of the single-acting beam-engines at the East London Water-works, according to Mr. Greaves' experiments, was effective in actual " pumping duty," or efficiency, or quantity of water raised, to the extent of 8 1 per cent. The difference, 19 per cent., was absorbed in nearly equal proportions by the friction of the engine, including that of the pole, on the one part, and the friction and resistance of the pump on the other part; being each nearly 10 per cent, of the "total load," or indicator-power. 1 Mr. David Thomson gives data for the "pumping duty" of the double- acting compound-cylinder rotative beam-engines of the Chelsea Water- works, in which the cylinders are placed vertically side by side, under one end of the beam, acting on the Woolf system. It appears that the duty is 80 per cent, of the indicator-power. 2 A pair of compound rotative beam-engines, designed by Mr. E. D. Leavitt, jun., have been erected at the St. Lawrence Water-works, Mass., U.S. 3 The first and second cylinders of each engine, are connected one to each end of the beam. They are placed together under the main centre, and conse- quently are directed obliquely each to its proper end of the beam; and, whilst the lower ends are close together, the upper ends lie apart from each other. The connecting-rod to the fly-wheel shaft is connected to the first- cylinder end of the beam; and the rod to the pump, to the second-cylinder end. The cylinders are fitted with gridiron valves, having a large area of opening, and small movement. The pumps are of the bucket-and-plunger type, first introduced by Mr. David Thomson, in England. The effective pressure in the boiler is 90 Ibs. per square inch. The cylinders are 18 inches and 38 inches in diameter, with 8 feet of stroke. They are entirely steam- jacketted. The clearance averages, for the first cylinder, 2.43 per cent., and for the second cylinder, 1.68 per cent, of the capacities of the cylinders respectively. The volume of the connecting pipe between their upper ends is 9.92 per cent, of that of the cylinder. The pump-barrel is 26^ inches, and the plunger is 18 inches, in diameter, with a stroke of 8 feet. The fly- wheel is 30 feet in diameter, and weighs 16 tons. From the results of the trials, it appears that the steam was cut off at about 30 per cent, of the stroke, with an initial absolute pressure of about 100 Ibs. per square inch. The engines made 16% turns per minute, and yielded about 195.5 m di- cator horse-power. The water delivered per stroke amounted to 95 per cent, of the capacity of the pump. The total quantity of coal (Cumberland) consumed was equivalent to 1.69 Ibs. per indicator horse-power per hour; and to 2.06 Ibs. per horse-power of duty at the pump. The efficiency of the engine was 81.94 per cent. The duty per 100 Ibs. of coal amounted to 96,200,000 foot-pounds. The quantity of water evaporated was 8.27 Ibs. per pound of coal, and (8.27 x 1.69 = ) 14 Ibs. of steam was consumed per indicator horse-power. 1 Article, "Steam Engine," by the author, in the Encyclopedia Britannica, 8th edition. 2 "On Double-Cylinder Pumping Engines," by Mr. David Thomson, in the Proceedings of the Institution of Mechanical Engineers, 1862, page 268. z Journal of the Franklin Institute, November, 1876, page 312. Report by Messrs. W. E. Worthen, J. C. Hoadley, and J. P. Davis; with illustrations. HYDRAULIC RAMS. 949 The pumping duties, at the pump, of large pumping engines, whether single-acting or double-acting, are thus seen to average about 81 per cent, of the indicator horse-power. HYDRAULIC RAMS. Hydraulic rams are used where there is a considerable flow of water with a moderate fall, to raise a small portion of the flow to a height greater than the fall. The outflow of water falling through a pipe, when the lower end of the pipe is suddenly closed, is suddenly arrested, and the momentum of the current in the pipe is expended in forcing a portion of itself through another pipe, the delivery-pipe, into an elevated reservoir. When the momentum is expended, the upward current ceases to flow, the valves in the delivery-pipe are closed against the return of the elevated water; and the valve by which the supply current was arrested, opens as it is relieved of the momentary excessive pressure, and the outflow is resumed. Thus, by a succession of impulses, the water is lifted. Mr. C. L. Hett recommends Daubuisson's formula for the efficiency : . (12) d=the quantity of water used, in gallons per minute. //! = the quantity of water raised, in gallons per minute. h = the head used, in feet. h t = the lift, in feet. Mr. Hett 1 gives the following table, calculated by means of Daubuisson's formula, showing the efficiency or percentage of duty due to proportions of lift to fall, of from 4 to 26 : Lift, when the Fall = i. Efficiency. Lift, when the Fall = i. Efficiency. Lift, when the Fall = i. Efficiency. ratio. per cent. ratio. per cent. ratio. per cent. 4 86 12 45 20 17 5 79 13 4i 21 H 6 73 H 37 22 II 7 68 15 34 23 8 8 63 16 30 24 5 9 58 17 27 25 2 10 53 18 23 26 ii 49 19 20 Mr. Hett recommends, to adopt, allowing for contingencies, only five- sixths of the efficiencies here given. For the diameter of the driving pipe, he gives Eytelwein's formula as sufficient for all practical purposes : Diameter of pipe in inches = .058 \f quantity of water in gallons. ... ( 13 ) 1 The Engineer , January 7, 1876, page 3. 950 HYDRAULIC MOTORS. HYDRAULIC MOTORS. HYDRAULIC PRESS. The action of the hydraulic press depends on the principle that fluids press equally in all directions ; and if the pressure applied to the plunger of the force-pump be multiplied in the ratio of the sectional areas or of the squares of the diameters of the plunger and the ram, the product is the pressure applied to the ram. The ram is packed with a leather collar, and according to the results of experiments made by Mr. John Hick, M.P., 1 the friction of the collar increases directly with the pressure and with the diameter; and it is independent of the depth of the collar. The friction is equivalent to i per cent, of the pressure for a 4-inch ram, y z per cent, for an 8-inch ram, and % per cent, for a 1 6-inch ram. The following formula is deduced: Leather, new or badly lubricated, . . . f .047 1 dp, ( I 4) Leather in good condition, f= .0314^/5 ( 15 ) f the total frictional resistance of a leather collar. d = the diameter of the ram, in inches. p = the pressure, in pounds per square inch. ARMSTRONG'S HYDRAULIC MACHINES. In a paper by Mr. Henry Robinson, 2 the following data are communi- cated, on the authority of Mr. Percy Westmacott, giving the coefficients of effect obtained in hydraulic machines of ordinary make, ; . Direct acting, 93 per cent. 2 to r, 80 4 to i 76 6 to 8 to 10 to 12 tO 14 to 72 .67 .63 59 54 16 to i, 50 These coefficients are based on the use of ordinary hemp-packing, and with sheaves and wrought-iron pins, and with no exceptional arrangements for lubrication. But, where special precautions have been taken, with large sheaves and small hard steel pins, the efficiency, multiplying 20 to i, with a load of 17^ tons, was as high as 66 per cent. Well made cupped leathers, used instead of hemp-packing, increase the efficiency. It is considered that the coefficient of effect obtained from a steam- engine pumping into an accumulator, may be taken at 91.7 per cent., the loss by friction amounting to 8.3 per cent. It is found by experiment that the difference of pressure with the accumulator (at 700 Ibs.) rising or falling is about 30 Ibs., representing .022 of effect. The compounded efficiency will therefore be ascertained by combining the efficiency of the engines with the above varying rates of efficiency. 1 Spans 1 Dictionary of Engineering, page 1992. 2 " On the Transmission of Motive-power to Distant Points," in the Proceedings of the Institution of Civil Engineers, 1876-77, vol. xlix. FRICTIONAL RESISTANCES. INTERNAL RESISTANCE OF STEAM-ENGINES. It may be assumed that, taken generally, the efficiency of steam-engines, that is to say, the ratio of the work transmitted through the engine to the fly-wheel shaft, to the effective work done by the steam in the cylinder, in- creases with the power of the engine. When engines are in good working order, and are doing their full duty, the efficiency varies from 80 per cent, for smaller engines, to 90 per cent, for larger engines. In one exceptional instance, a Corliss engine in France, the efficiency amounted to 93 per cent. RESISTANCE OF TOOLS. 1 The following are results of Dr. Hartig's experiments on the resistance of tools : Single-acting Shearing Machines. The power necessary to drive such tools when empty is expressed by the formula, P = horse-power. t = maximum thickness of plate to be cut. n = the number of cuts per minute. a = the area of surface cut or punched per hour, in square inches. F = (ii66+ 1691 /), a factor expressing the work required to produce a cut or sheared surface of I square inch. The power required to do the work itself, in addition to that required to drive the tool when empty, is c 33,000 x 60 1,980,000 For example, a shearing machine, cutting 4648 square inches of surface per hour, in plates 0.4 inch thick, would absorb 0.68 horse-power empty, and 4.3 horse-power in effective work; total, say, 5 horse-power. 1 The above particulars of the resistance of tools are abstracted from a notice of the results of Dr. Hartig's experiments, vs\ Engineering for October and December, 1874. 952 FRICTIONAL RESISTANCES. Plate-bending Machines. The net work required to bend a plate or a bar, is expressed by the formula, F = 85ooo J* 8 ^ for cold wrought-iron plates ( 3 ) F - i3^_l^l/ 5 for r ed-hot iron plates (4) fr, /, and / = the breadth, thickness, and length of the plate, in inches, r=the radius of curvature, in inches. F = the net work of bending the plate. The power required to drive large plate-bending rolls, when empty, is between 0.5 and 0.6 horse-power. Circular Saws. The horse-power required to drive circular saws running empty, is *'-< (5) 32,000 d = the diameter of the saw, in inches. n = the number of revolutions per minute. The net power required to cut with a circular saw, is proportional to the cubic contents of the material removed. For a saw for cutting hot iron, moving at a circumferential speed of 7875 feet per minute, and making a cut 0.14 inch wide, the power is expressed by the formulas P c = 0.702 A, for red-hot iron ( 6 ) P c = 1.013 A, for red-hot steel ( 7 ) A = the sectional area of surface cut through, in square feet. Work of Ordinary Cutting Tools, in Metal. Materials of a brittle nature, as cast-iron, are reduced most economically in power consumed, by heavy cuts ; whilst materials which yield tough curling shavings are more economi- cally reduced by thinner cuttings. The following formulas apply to light cutting work : The power required to plane away cast-iron is, Planing cast-iron. P = W (.01515+ ) .. .. (8) v o:> 1 1, 000^ W = the weight of cast-iron removed per hour, in pounds. s = the average sectional area of the shavings, in square inches. For planing steel, wrought-iron, and gun-metal, with cuts of an average character, Planing steel, P = o.ii2\Y, (9) wrought-iron, P= .052 W, ( 10 ) gun-metal, P= .0127 W, ( n ) For turning off metals, the power required is less than for planing RESISTANCE OF TOOLS. 953 them off, and it was found that the power was greater for smaller diameters than for larger diameters. Turning cast-iron, ......... P = .o3i4W, ......... (12) wrought-iron,. . . . P = .o32yW, ......... (13) steel, .............. P = .o 4 7W, ........... (14) For drilling metals, the power required to remove a given weight of material is greater than in planing. Volume is taken instead of weight in the formulas: applicable to holes of from 0.4 to 2 inches in diameter: Drilling cast-iron, dry, .............. P = P _/Kz> 33,000 956 FRICTIONAL RESISTANCES. RESISTANCE OF COLLIERY WINDING ENGINES. For working shafts from 246 to 580 yards deep, in the county of Durham, the following are particulars of dimensions and performance of engines and winding gear: 1 DIMENSIONS. No. DIRECT-ACTING ENGINE. Cylinder. Speed of Piston. Steam in Boiler. Drums. Mean Dia- meter. in. in. ft. per minute. Ibs. feet. I i cyl., vertical, condensing 65x84 I 7 6 1 9 flat 26 2 I 11 11 11 68x84 224 20 2S% 3 2 horizontal, non-condensing 40x72 253 conical 21 4 2 11 11 11 34x72 291 40 17;^ 5 I 11 11 11 48x72 2 3 2 flat 21$ PERFORMANCE. No. Gross Engine Power per Minute. Duty, Coal Raised, per Minute. Effi- ciency. Ropes. Average Speed of Ropes. foot-pounds. foot-pounds. per cent. in. in. feet per minute. I 10,342,350 7,729,344 75 flat, iron, 6 l /z x 7 /& IO2O 2 15,403,000 9,744,000 63 11 11 6 x # 1180 3 7,470,000 4,077,000 55 round, steel, 1% 1689 4 4,375>59 6 2,700,000 62 round, iron, i^ 1302 5 5,281,107 3,956,178 75 1300 RESISTANCE OF WAGGONS IN COAL PITS. The average resistance of waggons used in the Midland coal pits, having four 1 5-inch wheels at 2i-inch centres, is Vsoth of the weight, or 45 Ibs. per ton. The bodies of the waggons are 4 feet 3 inches by 3 feet wide, and 20 inches deep. On a roadway having an average fall of i in 30, worked by an endless chain, passed over a 7^-feet grooved pulley at the end of the course, the gravitation of full waggons descending, supplies sufficient hauling power to overcome all the wheel-friction and take up the empty waggons. The greatest traverse is from 5000 to 6000 feet; the speed is 3 miles per hour, and the tubs or waggons are attached to the rope at intervals of from 20 to 25 yards. 2 1 See paper by Mr. G. H. Daglish, on Winding Engines, in the Proceedings of the Institution of Mechanical Engineers, 1875, page 217. 2 Mr. G. Fowler, Proceedings of the Institution of Mechanical Engineers, 1870. RESISTANCE OF FLAX-MILL MACHINERY. 957 RESISTANCE OF MACHINERY OF FLAX MILLS. M. E. Cornut, in 1871-72, tested, by the indicator, the resistance of the steam-engines, shafting, and machinery of the flax mill at Hamegicourt. 1 There were two Woolf engines, with vertical cylinders and beam, condens- ing, which made, at regular speed, 25 turns per minute. First cylinder of each engine, 12.9 inches diameter, 44.3 inches stroke. Second 22.0 59.8 The power required to drive the machinery was very variable. It varied 15 or 20 per cent, according to the lubricant employed, and the mode of lubrication : Atmospheres. With vegetable oil and hand oiling, steam pressure required, 5 to 5X- With mineral oil and continuous oiling, 4^ maximum. Making a difference in pressure of at least 15 per cent. In accordance with this result, it was found, by direct test, that, in lubricating with vege- table oil (huile grasse), 2.90 horse-power per 100 spindles (wet system), was consumed, and only 2.44 horse-power with mineral oil, making a difference of 1 6 per cent. But the modes of lubrication are not distinguished. Finally, with vegetable oil, the belts, if tight, were broken at starting on Monday mornings, after a day's stoppage; but, with mineral oil, there were no breakages. The quantity of mineral oil consumed per day, in lubricating the same three pedestals, was, By hand oiling, 29.0 grains. By continuous oiling, 16.2 Showing a reduction of 44 per cent, by continuous oiling. To test the increase of resistance by want of lubrication, the engines,, shafting, and belts, were, one day in March, 1871, started empty (a vide), at 4 A.M. At 6 A.M., 30.08 horse-power was indicated. All the lubricators were then removed, except two next the engines, and the oil that remained was cleared off the journals. The engines and shafting continued in motion, and observations on the power expended were made at intervals. Time Observation. Elapsed after Remov- ing the Indicator Horse- Power. Augmentation of Resistance. Lubricators. hours. per cent. ISt 30.08 2d 2 31.60 $X 3d 3 3347 10.9 4th 4 35-34 1745 5 th 6 37-33 30.78 1 Essais Dynamometriques, Lille, 1873. These experiments were carefully and intelli- gently conducted, and M. Cornut's conclusions appear to be worthy of confidence. 958 FRICTIONAL RESISTANCES. At this time, the journals began to heat, and the experiment was ended. Taking the first three observations to represent ordinary practice in hand-oiling, a variation is manifested, of from 5 to 10 per cent, of re- sistance. From comparative tests made at 6 A.M. and at n A.M., respectively ^ hour and 5 y 2 hours after starting in the morning, the total resistance of the engines and machinery in ordinary work, for the second test, was from 8 to 9 per cent, less than for the first test. The temperature in the workshops had risen 10 or 11 F. in the interval. The resistance of the engines and machines, separately, was tested during the period from August, 1871, to February, 1872: The engines, shafting, and belts only, making 25 turns of the engine per minute, expended an average of 30.41 indicator horse-power. The maxi- mum power was 32.10 H.P., in December, 1871. Four cards consumed from 9.10 to 7.40 ; average 8.42 H.P. 14 drawing-frames of 29 heads, or 156 slivers, consumed from 6.82 to 8.04; average 7.19 H.P. 6 roving-frames, of 330 spindles, 9.04 to 8.25; average 8.67 H.P., or 2.627 H.P. per 100 spindles. 4 combing-machines, 2.228 H.P. 20 spinning-frames, dry, 1480 spindles, 47.50 H.P. 20 spinning-frames, wet, 2080 spindles; spinning Nos. 25 to 30, 46.59 H.P., or 2.24 H.P. per 100 spindles; Nos. 40 to 70, 35.82 H.P., or 1.72 H.P. per 100 spindles. For several particular numbers, the power was determined to be as follows : No. 16, 3.200 H.P. per 100 spindles. 20, 2.760 -25, 2.262 28, 2.190 30, 2.140 40, 1.917 M. Cornut deduced from those data, that the horse-power per 100 spindles varied inversely as the square root of the number. The average indicator-power that would have been required for driving the whole of the machinery in full work, was 148.42 horse-power. The actual total average power required was only 131.23 horse-power, or 88 per cent, of the power for full work; being 17.19 horse-power less than for the whole of the machines. This reduction represents the power for the proportion of machines at rest. The power required to drive the machines when empty was also mea- sured. The table No. 318 contains particulars of the horse-power required for each machine, at work, and empty. The indicator-power may be divided thus : Steam-engine, shafting, and belts, 30 I. H.P. or 20 per cent. Preparing and spinning machinery, &c., 120 or 80 150 100 RESISTANCE OF WOOLLEN-MILL MACHINERY. 959 Table No. 318. FLAX MILL AT HAME'GICOURT HORSE-POWER REQUIRED TO DRIVE THE ENGINES, SHAFTING, AND MACHINERY. (From M. Cornut's data.) Indicator Horse-Power. Efficiency DESCRIPTION. Total at Work. For One Machine at Work. For One Machine Empty. of the Machines. Engines, shafting, and belts,.. A Cards . H.-P. 30.41 8 42 H.-P. 2 IOC H.-P. T A^.'i. per cent. M 1 4 Drawing frames (29 heads ) or 156 slivers), j 7.19 .0934 per sliver i4Zj .0794 15 4 Combing machines 2 22 C C C T C T 7C 6 Roving frames (330 spindles) 20 Spinning frames, dry (1480 spindles), 778 47.50 O33 2.627 ? i oo spindles 3-21 .151 2.434 2.515 / j 7-3 21.6 20 Spinning frames, wet (2080 spindles) 46.59 2.24 1.613 19 Total horse-power, all at } work, calculated from j- separate experiments, ) Total hor.-power that would } have been actually ex- > pended, all at work ) 151.00 148.42 Estimation of Horse-power required for a Flax Spinning-mill. Let the power be expressed in terms of the number of spindles driven by i indi- cator horse-power. M. Cornut gives the following data : M. Feray d'Essone, 55 spindles per H.P. for No. 40. 20 for No. 6. English estimate \ 25 spindles per H.P., for No. 35 long staple, No. 1 8 tow. 40 spindles per H.P., for Nos. 35 to 51 long staple, Nos. 1 6 to 29 tow. 26 spindles per H.P., for Nos. 25 to 30 long staple, Nos. 14 to 23 tow. (2000 to 1 2,000 spindles), Dr. Hartig, Do M. Cornut: 2080 spindles, wet, 34.4 spindles per H.P., long fibre. 640 dry, 20.1 840 14-5 tow. 3560 23-7 RESISTANCE OF MACHINERY OF WOOLLEN MILLS. Dr. Hartig, of Dresden, tested the machinery of a woollen mill, by means of a dynamometer, which was applied directly to each machine. The prin- cipal results, showing the actual horse-power expended in driving each machine, when empty and when at work, are given in table No. 319. 960 FRICTIONAL RESISTANCES. Table No. 319. WOOLLEN MACHINERY HORSE-POWER REQUIRED TO DRIVE THE MACHINES. (From Dr. Hartig's data. ) DESCRIPTION. Ordinary Speed in Turns per Minute. Indicator Horse-power. Empty. At Work. Effi- ciency. At Work, Resistance of Shafting included. ' H.P. | 1. 00 1-75 .70 I.OO 1 75 .003 .007 075 13 55 2.25 1.50 2.75 3-25 2-75 1.70 75 75 2-75 45 .90 .60 25 .40 .90 Washing machine, two cylinders, .... Centrifugal pump for this machine, ... Hydro-extractor, turns. 35 300 IOOO to I2OO 350 to 500 300 350 to 450 no 2500 I IOO 17 40 to 45 40 IOO 45 no IOO 45 blows. 125 116 turns. 90 IOO IOO IOO 650 IOOO 250 250 H.P. 75 1.47 47 .42 34 .27 .005 13 .19 17 30 .16 53 43 .19 .20 .11 17 52 25 37 H.P. .223 77 i. 20 'is 95 58 43 .0027 .006 .071 50 2-54 i-59 2-74 3-40 3.26 1.64 1.99 73 2.03 2.45 .61 31 1.03 & cent. 39 17 29 55 36 13 74 93 89 73 95 78 74 74 86 94 93 14 22 64 Burring machine . Scutcher, continuous feed, Opening machine, Scribbler card, 40 inches wide, Intermediate card, ,, Finishing card, , , i spindle, spinning, . I spindle, doubling, Sizing and warping machine (with- ) out ventilator), .... .. ( Loom 7 54 feet wide Scouring machine, for 2 pieces, Fulling mill, i cylinder, fornouveautes, Do. 3 cylinders, for cloth, ... Double fulling mill, for noiiveautes,... Do. do. for cloth, Single fulling stock, two stamps (Dobb's), Double fulling stock, two stamps (Spranger's), Gig (Laineur), single, without spreader, Do. double, Machine for dressing the reverse ) side, with cylinders, \ Machine for dressing the reverse side, with eccentrics, Wringing machine, lengthwise, Do. do. across,. Brushing machine, one cylinder, Do. do. two cylinders, RESISTANCE OF MACHINERY FOR CONVEYANCE OF GRAIN. Conveyance of Grain horizontally by Screws and by Bands. A 1 2-inch screw, having 4 inches pitch, turning in a trough, with a clearance of ^ inch, revolving with the speed of maximum effect, 60 turns per minute, discharged 6^ tons of grain per hour, expending .04 horse-power per foot run. The sectional area of the body of grain moved was 49 per cent, of that of the screw. At speeds above 60 turns per minute, the grain did not advance, but revolved with the screw. RESISTANCE TO TRACTION ON COMMON ROADS. 961 A i2-inch screw, having a 1 2-inch pitch, delivered 34 tons per minute, at 70 turns per minute, expending .125 horse-power per lineal foot, or 3 7 per cent, less power for equal weights of grain. The sectional area of the grain was 72 per cent, of that of the screw. An endless band, 18 inches wide, travelling at about 9 feet per second, delivered 70 tons of grain per hour; power expended, .014 horse-power per foot run. To convey 50 tons per hour through 100 feet, the power expended by the screw was 18.38 H.P. ; by the endless band, 1.02 H.P. Grain has been raised by a dredger 30 feet long; efficiency, 50 per cent 1 Mr. Davison states that 95 feet of horizontal Archimedian screw, 15 inches in diameter, with an elevator lifting 65 feet, convey 40 quarters of malt per hour, for an expenditure of 3.13 indicator horse-power. RESISTANCE TO TRACTION ON COMMON ROADS. From the results of recent and carefully conducted experiments made by M. Dupuit, he made the following deductions as to the resistance to traction on macadamized roads and on uniform surfaces generally : 1. The resistance is directly proportional to the pressure. 2. It is independent of the width of tyre. 3. It is inversely as the square root of the diameter. 4. It is independent of the speed. M. Dupuit admits that, on paved roads, which give rise to constant concus- sion, the resistance increases with the speed; whilst it is diminished by an enlargement of the tyre up to a certain limit. M. Debauve 2 has deduced from experiment, that the advantage of a pave- ment over a metalled road is considerable for waggons, is less for stage- coaches, and is nearly nothing for voitures de luxe, or private carriages. He summarizes the resistances as follows: V _ HI _. RESISTANCE TO TRACTION. On Metalled Roads. On Paved Roads. Waggon, 67 Ibs. per ton 38 Ibs. per ton. Stage-coach,... 67 45 Cabriolet, 81 76 to 83 M. Tresca tested the resistance of a tramway omnibus on Loubat's system, adapted with wheels for running on a common road. The experi- ments were made on an inclined street in Paris, in good condition, having ascending gradients of i in 55, one part of which was paved, and another part macadamized. The frictional resistance, after the gravitation on the incline was eliminated, was as follows : i- Gross Weight. Speed. Frictional Resistance, tons. miles per hour. Ibs. per ton. Macadam, 5.67 10.7 83 Pavement, 5.67 10.1 66 1 The above data are derived from Mr. Westmacott's paper on " Corn- Warehousing Machinery," Proceedings of the Institution of Mechanical Engineers, 1869, page 208. 2 Manuel de Flngenieur des Fonts et Chaussees, 1873 ; 9 me fascicule, page 32. 61 9 62 FRICTIONAL RESISTANCES. RESISTANCE OF CARTS AND WAGGONS ON COMMON ROADS AND ON FIELDS. The resistance to traction of agricultural carts and waggons was tested at Bedford in July, 1874, by means of a horse-dynamometer designed by Messrs. Eastons & Anderson. 1 The first course was a piece of hard road 200 yards in length, rising i in 430; it was dry and in fair condition, largely made of gravel. The surface was in many places somewhat loose. The second course was along an arable field, growing oats, on a rising gradient of i in 1000; it was very dry, and was harder than in average condition. The fore-wheels of the waggons averaged 3 feet 3 inches, and the hind- wheels 4 feet 9 inches in diameter; the width of tyres was from 2^ to 4 inches. The weight, empty, averaged about a ton, and it was nearly equally divided between the front and hind wheels. The cart wheels were, say, 4 feet 6 inches high, with tyres 3}^ and 4 inches wide. The weight of the empty carts averaged 10 cwt. The following results arise out of the published data : ON ROAD. Pair-horse Waggon, with- out Springs, Loaded with Roots. Waggon with- out Springs, Loaded with Maize. Waggon with Springs, Loaded with Roots. Cart without Springs, Loaded with Roots. Load, 44 cwt. 80 cwt. 44 cwt. 20 cwt. Gross weight drawn, about Average speed per hour,... Maximum draft, 64 2^/2 miles 320 Ibs 100 2.60 miles 400 Ibs 64 2.47 miles 300 Ibs 5 *4 2.65 miles 1 80 Ibs Average draft, ICQ , 2C.I 1 3^ , AQ A Horse-power developed, ") at 33,000 foot-pounds > per minute... .. \ 1.06 H.P. 1.74 H.P. .88 H.P. .35 H.P. Draft per ton gross, on level, ON FIELD. Load, ( 43-5 Ibs., \ orI / 52 A A CWt 44-5 Ibs., or 1/50 80 cwt 34.7 Ibs., or 1/65 AA CWt. 28 Ibs., or 1/80 / 20 cwt Gross weight drawn, about Average speed per hour,... Maximum draft, 64 2.35 miles 1000 Ibs 100 2.52 miles 1 200 Ibs 64 2.35 miles 1000 Ibs 30 2. 6 1 miles 400 Ibs Average draft 7OO QQ7 7IO . 212 Horse-power developed, "i at 33,000 foot-pounds > per minute, ) 4.36 H.P. 6.70 H.P. 445 H.P. 1.48 H.P. Draft per ton gross, on level, ( 210 Ibs., \ or i/i i 194 Ibs., or i/ I2 210 Ibs., or !/ 140 Ibs., or i/ z 6 From these data it appears that, on the hard road, the resistance is only from i^th to '/eth of the resistance on the field. The lowest resistance is that of the cart on the road 28 Ibs. per ton; due, no doubt, as observed in Engineering, to the absence of small wheels like those of the waggons. 1 See a report of the trials in Engineering, July 10, 1874, page 23. RESISTANCE TO TRACTION ON ROADS AND FIELDS. 963 The highest resistance is 210 Ibs. per ton on the field. The addition of springs reduced the resistance 26 per cent, on the road; but, on the field, the resistance was not reduced by the addition of springs. Also, that the horse-power, of 33,000 foot-pounds per minute, developed, varied from y$ H.P. to 7 H.P. Allowing a pair of horses for the first and third columns above, two pairs for the second column, and one horse for the last column, the following are the total works done per horse, in tech- nical horse-power: HORSE-POWER PER HORSE. On Road. On Field. Pair-horse waggon, without springs, 53 H.P 2.18 H.P. Two pair-horse waggon, without springs, .44 1.68 Pair-horse waggon, with springs, 44 2.22 One-horse cart, without springs, 35 1.48 Total averages, 44 I. Averages, without springs, 44 1.7 Taking the average power exerted without springs, .44 horse-power, on the road, as the average for a day's work, it represents "44 x 33,000 = 14,520, say 15,000, foot-pounds per minute, for the power of a horse on a hard road. The resistance of a smooth well-made granite tramway, like the tram- ways in the City of London and Commercial Road, made with stones 5 or 6 feet in length, is from 12^ Ibs. to 13 Ibs. per ton of weight. Experiments on the tractional resistance for a loaded omnibus, on various kinds of roads, were made by a committee of the Society of Arts i 1 Ibs. Weight of omnibus, 2480 Load, 22 sacks of oats, at 149 Ibs., 3278 Total weight, 2.57 tons, or 5758 The loaded omnibus was drawn to and fro over each trial surface, and the mean result was taken as the resistance for an exact level : RESISTANCE. Average Speed. Total. Per Ton. miles per hour. Ibs. Ibs. Granite pavement, sets 3 to 4 inches wide, 2.87 44-75 1 7A 1 Asphalte roadway, 3.56 69.75 27.14 Wood pavement, 3.34 106.88 41.60 Good gravelly Macadam road, 3.45 114.32 44-4$ Granite Macadam, newly laid, 3.51 259.80 101.09 There is a want of consistency here, in the excessive resistance on an asphalte pavement, compared with that on a granite pavement. There can be no doubt that asphalte pavement, properly made, is, of all pave- ments, the least resistant; and that its resistance cannot be greater than the resistance of a granite tramway. 1 Report of the Committee, Journal of the Society of Arts, June 25, 1875. 964 FRICTIONAL RESISTANCES. Sir John Macneil gives the tractive force necessary to move a waggon weighing 21 cwt, at 2^ miles per hour, on roads of the following descrip- tions : Total Resistance Resistance, per Ton. Ibs. Ibs. Well-made pavement, 33 ... 31.2 Road made with 6 inches of broken stone of great hardness, " on a foundation of large stones set in the form of a pavement, or upon a bottoming of concrete, Old flint road, or a road made with a thick coating of broken stone, laid on earth, Made with a thick coating of gravel, laid on earth, 147 ... 140 46 ... 44 65 ... 62 Sir John Macneil made a series of experiments on the tractive resistance of a stage-coach, on a section of the Holyhead Road. The weight of the coach, empty, was 18 cwt, and the weight of seven passengers in addition, allowing i^ cwt. for each passenger, was 10^ cwt.; total weight 28^ cwt. The experimental gradients ranged from i in 20 to i in 600, and the speeds were 6, 8, and 10 miles per hour. It was found that, by some un- explained cause, the net frictional resistance at equal speeds varied considerably according to the gradient. The resistances were a maximum for the steepest gradient, and a minimum for gradients of i in 30 to i in 40; for these they are less than for i in 600. The mode of action of the horses on the carriage may have been an influential element. The averages show, FOR A STAGE-COACH. ON A METALLED ROAD. At 6 miles per hour, ............... 62 Ibs. per ton, frictional resistance. At 8 ............... 73 At 10 ............... 79 With these may be associated the resistance, by Sir John Macneil's experiments, of a waggon on a good road, namely, 44 Ibs. per ton, at 2^ miles per hour. Plotting the resistances for the above four speeds, the following approximate formula is deduced: Frictional Resistance to Traction of a Stage-coach on a Metalled Road in good condition. R the frictional resistance to traction per ton. v = the speed in miles per hour. Note. The formula is applicable to waggons at low speeds. It is simpler than the formulas deduced by Sir John Macneil. 1 M. Charie-Marsaines made observations of a general character, on the performances of Flemish horses drawing loads upon the paved and the macadamized roads in the north of France, where the country is flat, and the loads are considerable. 1 Sir John Macneil's formulas are given in Sir Henry Parnell on Roads, page 464. RESISTANCE ON RAILWAYS. 965 Table No. 320. PERFORMANCE OF HORSES ON ROADS IN FRANCE. (M. Charie-Marsaines.) Season of the Year. Description of Road. Weight Horse. Speed in Miles per Hour. Work Done per Hour in Tons Drawr One Mile. Ratio of Paved Road to Macadamized Road. Winter, | Pavement tons. 1.306 miles. 2.05 ton-miles. 2.677 i 644 to i Summer / Macadam 1 Pavement .851 1-395 I. 9 I 2.17 1.625 3.027 I 22Q tO I \ Macadam I.I4I 2.l6 2.464 The average daily work of a Flemish horse in the north of France is, on the same authority, as follows: Winter, 21.82 ton-miles per day in winter. Summer, 27.82 in summer. Mean for the year, say, ... 25.00 It has already been stated, page 720, that a good horse can draw a weight of i ton at 2^ miles per hour, for from 10 to 12 hours a day equivalent to (1x2^x10 = ) 2 5 tons drawn one mile per day. This is the same amount of performance as is above given from M. Charie-Marsaines. Conclusion. With the exception of Messrs. Eastons and Anderson at Bedford, the authorities on the tractional resistance to vehicles on common roads, ignore, with remarkable unanimity, the influence of sizes of the wheels and other essential particulars. It is better, therefore, to refrain from attempting to draw general conclusions, and to leave the figures " to speak for themselves." RESISTANCE ON RAILWAYS. The Author 1 deduced from experimental data, the following formulas for the resistance of locomotives and trains, under these conditions : the per- manent way in good order; the engine, tender, and train in good order; a straight line of rails; fair weather, and dry and clean rails; an average side wind, of average strength, varying (in the experiments) from slight to VERY strong: Resistance of Engine, Tender, and Train. 171 Resistance of Train alone. 24O R = total resistance of engine, tender, and train, in Ibs. per ton gross; R' = resistance of train alone, in Ibs. per ton; v = speed, in miles per hour. 1 Railway Machinery, 1855, pages 297, 298. 966 FRICTIONAL RESISTANCES. For ordinary practice, to meet the unfavourable conditions, which may occur in combination, of frequent quick curves under one mile radius, and strong side and head winds, the Author estimated from his own observations that the resistance, as calculated by means of the foregoing rules, should be increased 50 per cent., or one-half more. On this basis, for speeds of 5, 10, 15, 20, 30, 40, 50, 60 miles per hour, the frictional resistances per ton of engine, tender, and train, are, 12.2, 13, 14, 15.5, 20, 26, 34, 43.5 Ibs., and the frictional resistances per ton of the train alone are, 9.15, 9.6, 10.5, 11.4, 14.6, 19.0, 24, 31.5 Ibs. RESISTANCE ON STREET TRAMWAYS. The rails of street tramways are rolled with a groove for the guidance of the wheels by the flanges. The wheels of cars, therefore, do not run so freely as those of carriages or waggons on railways. The average frictional resistance of vehicles on tramways is 30 Ibs. per ton, although an occasional maximum of 60 Ibs. per ton may be reached, and, on the contrary, a mini- mum of, say, 15 Ibs. per ton, when the rails are wet and clean, straight and new. The resistance due to clogging of the grooves of rails was brought into evidence by Mr. J. Arthur Wright, who found that, on a dusty day, on the steam lines of the Rouen tramways, when, despite every effort to keep the rails clear, the grooves became filled with dust and dirt, the engines consumed about 2^ Ibs. of coke per mile more than they did under more favourable circumstances, when the consumption averaged about 1 2 Ibs. per mile. The excess is nearly 20 per cent, over. 1 1 These data are derived from Tramways, their Construction and Working, 1882, by D. Kinnear Clark, p. 180. APPENDIX. DR. SIEMENS' WATER-PYROMETER. [Appended to " Pyrometers" page 326.] This apparatus belongs to the second class of pyrometers described at page 327. In it, Dr. Siemens has reduced the water-pyrometer to a form complete and exact for purposes of scientific observation. The water is contained in a copper vessel capable of holding rather more than a pint. The sides and bottom of the vessel are fitted with an outer casing or jacket, the interspace of which is filled with felt, by which any radia- tion of heat from the vessel is prevented. A mercurial thermometer is fixed in the vessel, and immersed in the water. It is fitted with a small sliding scale in addition to the ordinary scale, which is graduated and figured with 50 degrees to i degree of the ordinary scale. Six solid copper cylinders are supplied with the pyrometer, each of which is accurately adjusted so that its capacity for absorbing heat shall be one-fiftieth of that of a pint of water. In using the pyrometer, a pint of water is measured into the copper vessel, and the sliding scale is set with its zero at the temperature of the water as indicated by the ordinary scale of the ther- mometer. A copper cylinder is put into the furnace or the hot-blast current, of which the temperature is to be measured, where it is left for a space of time of from 2 to 10 minutes, according to the intensity of heat to be measured. Having been thus raised to the temperature of the furnace or current, the cylinder is withdrawn and quickly dropped into the water; the temperature of the water is raised at the rate of i for each 50 of the temperature of the copper cylinder. The rise of the temperature may then be read off direct on the pyrometer scale. Add to the temperature thus noted, the observed initial temperature of the water, and the sum is the exact temperature required. For very high temperatures, cylinders of platinum may be employed. ATMOSPHERIC HAMMERS. {Appended to "Air Machinery" page 915.] In the hammers designed by M. Chenot Aine, 1 atmospheric air is employed as a spring, for the purpose of accumulating and of applying the motive power to the hammer. The hammer is a cylinder turned from end to end, and bored out to two different diameters. It is divided into two 1 Revue Industrielle, December, 1876, page 521 968 APPENDIX. chambers by a diaphragm at the middle. The lower end is thus completely inclosed, whilst the upper end is open. Two pistons fixed to one rod, passing through the diaphragm, play in the upper and lower chambers; and they receive a reciprocating motion from a crank overhead, driven by a band passed over a pulley fixed on the crank-shaft. The cylinder-hammer is thus floated on the pistons by means of air-cushions, of which there is one above the diaphragm, one below it, and a third below the lower piston ; and it is impressed with a reciprocating movement following the reciproca- tions of the pistons, by the agency of these cushions of air. The height of the fall and the force of the blow, are regulated by the speed at which the machine is driven. There is no sensible heating or cooling of the working parts, and M. Chenot estimates that the efficiency of the machine amounts to 75 per cent, of the power communicated to the driving pulley. The chief feature of interest in this machine is the employment of air compressed and expanded to two or three times its normal volume, without any inconvenience by either heat or cold. It is obvious that during the momentary actions and reactions, time is not afforded for the heating and cooling effects of changes of temperature in the air to take place. Hence the high efficiency. BERNAYS' CENTRIFUGAL PUMP. {Appended to " Pumps" page 944.] Mr. Joseph Bernays constructs the discs of his pump with a double joint, the inner one being the joint universally employed around the suction- openings, by which the water is admitted from the suction-passages into the disc. The second, or outer, joint is at the extreme diameter of the disc, and it prevents the pressure of the water in the delivery-pipe from reacting on the outer faces of the revolving disc. A saving of power is thus effected, in reducing the loss by friction on the disc. The form of the vanes of Mr. Bernays' encased pump, may be roughly described as semi-elliptical, or the half of a flat ellipse divided at its longest diameter; the concave surface being presented to the water in the direction of the motion. By the adoption of such a form, it is designed that the blade should scoop up the water arriving at the centre of the pump by its inner edge, and should project the water forward in a direction as nearly tangential as possible by its outer edge, at the circumference. When the pump is not encased, and the water is delivered into an open well or reservoir, the outer end of the vane is curved backwards for the purpose of facilitating the discharge radially. In this case, the blades acquire an ogee form, like Rankine's fan. Mr. Bernays has supplied the following note on centrifugal pumps : " The only parts of a centrifugal pump which, when at work, absorb more power by friction than is due to the mere velocity of the water passing through the pump, are the outer faces of the revolving disc. These outer faces are surrounded by water quite or nearly stationary, and as they them- selves revolve at a speed proportionate to the height to which the water is to be lifted, more or less independent of the quantity that passes through the pump, they tend to carry the surrounding water round with them. STEAM-VACUUM PUMP. 969 According to the greater or less pressure under which the pump works, the friction produced will be greater or less, just as there is greater skin resist- ance in a vessel of greater draft than in one of light draft, both having an equal extent of surface. The saving of power effected by the removal of the pressure of the water in the delivery pipe of Mr. Bernays' pump, as above explained, is all the more necessary, since, with a centrifugal pump, the power required for driving it increases rapidly, and in a greater ratio than the heights of delivery to which such pump may have to be applied. This is a point to which the attention of engineers and users of centrifugal pumps has never been called, or, if it has ever been mentioned, it has been, and is, constantly lost sight of. Nevertheless the fact is clear, and the explanation very simple indeed. The only working parts of a centrifugal pump which, irrespective of the friction previously mentioned, actually propel the water and absorb the power applied to it, are the arms or vanes radiating from the centre to the outside of the disc. The shape of these arms may for a moment be left out of consideration, as their more or less perfect form accounts for a mere percentage only of the whole power used for driving a centrifugal pump. The main power is used in driving or pushing the arms against the water at a speed calculated to produce a pressure equal to or rather in proportion to the height to which the water is raised. Now, the speed at the outside diameter of the pump disc is approximately equal to that of a body falling from the height to which the water is lifted, or it is directly proportionate to the square root of that height. And as the direct resistance which the arms meet with in their rotation, is simply proportional to the height to which the water is to be lifted (the same as in common reciprocating pumps), it follows that the amount of power necessary for working the pump, is a function of the height multiplied by its square root, or h ^/ h = >J h\ Thus, a pump requiring 20 H.P. to raise water 10 feet high, will, if the height be increased to 40 feet, not merely require 4 times the power for the same quantity delivered, but 4 multiplied by *J 4, or 8 times, that is, 1 60 H.P. And it evolves from this, that although centrifugal pumps are an exceedingly useful and simple mechanical appliance for raising large quantities of liquids to moderate height, and it may here be added, variable quantities to variable heights, they should not be made use of for great heads of delivery, where the cost of the power employed to work them is any consideration." STEAM- VACUUM PUMP. [Appended to ' ' Pumps, " page 944.] The steam-vacuum pump belongs to the class of pumps on Savary's old system, in which steam from the boiler is admitted into direct contact with the surface of the water to be forced. Experiments were conducted by Mr. J. F. Flagg, at the Cincinnati Exposition of 1875, on such a pump. 1 The water was drawn directly from a canal, through a 3-inch pipe, 155 feet 1 Journal of the American Society of Civil Engineers, December, 1876, page 381. 970 APPENDIX. long, with a lift of 10.83 f ee t The head of pressure was regulated by means of a cock applied to the discharge pipe. The proportion of primed water in mixture with the steam was ascertained, and allowed for : ist trial. ad trial. Effective pressure in boilers, Ibs. per sq. inch, ......... 72 Ibs .......... 57 Ibs. Temperature of water in canal, Fahrenheit, ......... 60 ......... 6i.5 Do. effluent water, ......... 86.9 ......... 73.! Pressure at the gauge, Ibs. per sq. inch, ......... 35.3 Ibs ....... 16.3 Ibs. Do. feet of head, ......... 81.8 ft .......... 37.7 ft. Steam consumed per horse-power, per hour, ......... 477.5 Ibs ....... 390.7^3. Coal do. do. (allowing 9 Ibs. water per pound of coal), ......... 53.1 Ibs ....... 43.4 Ibs. Duty per 100 Ibs. of coal, foot-pounds, ......... 3,732,260 ...... 4,561,200 INDEX. ACCELERATED Accelerated and retarded motion, 282, 286. Accelerating forces, 282. Acre, 137; equivalent value in French measures, i53, .156. Adhesion of leather belts, 744, 748. Air; as a standard for weight and measure, 127; pressure of air, 127; measures of pressure of air, 127; weight and volume of air, 127; weight of air compared with that of water, or its specific gravity, 128; specific heat of air, 128. Air and aqueous vapours, mixture of, 394, 396. Air, expansion of, 343, 345; relations of the pressure, volume, and temperature of air and other gases, 346, 351; special rules for one pound weight of a gas, 349; table of the volume, density, and pres- sure of air at various temperatures, 351; specific heat of air, 354, 358, 363 ; comparative density and volume of air and saturated steam, 391. Air, ascension of, by difference of temperature, 897. Air consumed in the combustion of fuels, 400, 405: coals, 427; coke, 435; wood, 443; wood-charcoal, 45 2 - Air and other gases, flow of, 891. See Flow of Air, page 974. Air, dry, or other gas, work of, compressed or ex- panded, 898. See Work of Dry Air, page 984. Air, hot, and stoves, heating by, 488. Air, resistance of, to the motion of flat surfaces, 897. Air, work of compression of, at constant tempera- ture, 899; adiabatically, 903. Air, work of expansion of, at constant temperature, 899; adiabatically, 904. Air machinery, 915; machinery for compressing air, and for working by compressed air, 915; compres- sion of air by water, 915; by direct-action steam- Bimps, 915; compressed-air machinery at Powell- uffryn Collieries, 916. Hot-air engines, 917; Rider's hot-air engine, 917; Belou's hot-air engine, 918. Gas-engines, 918 ; Lenoir's double-acting gas- engine, 919; Otto and Langen's atmospheric gas-engine, 919; Otto engine, 921; Clerk's engine, 922. Fans or ventilators, 924 ; Common centrifugal fan, 924; mine- ventilators: Guibal's fan, 925; Cook's ventilator, 926; blowing -engines, 926; Root's rotary pressure-blower, 927. Air-pyrometer, 327. Air-thermometers, 325; Regnault's, 326. Alloys, melting points of, 363; table, 366. Alloys, specific gravity of alloys of copper, 200, 201, 626, 627; of gold and other metals, 201. Aluminium-bronze, tensile strength of, 627. American coals, 418. See Coal, page 972. Anemometer, 892. Animal substances, weight and specific gravity of, 212. Annealed and unannealed wrought-iron plates, com- parative strength of: Krupp and Yorkshire plates, 583-585; Prussian plates, 586. Annealed and unannealed steel plates, comparative strength of : Fagersta,6o6, 607, 609-611; Siemens' steel, 613. Annealed and unannealed wire, phosphor bronze, copper, brass, steel, iron, tensile strength of, 629. BOLTS Anthracite: British, 409, 413; American, 418, 419; French, 421, 422; Russian, 422; manufacture of coke with, 432. Anthracitic coke, 432. Applications of heat, 459: Transmission of heat through solid bodies, 459; warming and ventila- tion, 477; heating of water by steam in direct con- tact, 490; spontaneous evaporation in open air, 491; desiccation, 493; heating of solids, 497. Aqueous vapour, mixture of, with air, properties of, 394, 396. Are, 149; equivalent value in English measures, i53- Armstrong's hydraulic machines, 950. Asphalte: composition, 437; heat of combustion, 437, 438; weight and specific gravity, 207, 437. Ass, work of, in carrying loads, 721. Atmospheric gas-engine, Otto and Langen's, 922. Atmospheric hammers, by M. Chenot Aine, 967. Australian coal, 423, 424. See Coal, page 972. Axles, railway, proportions of, 767. B Balls, cast-iron, weight of; multipliers for other metals, 258 ; diameters of, for given weights, 258. Barker's water-mill, 939. Beams, flanged, transverse strength of: cast-iron, 647; wrought-iron, 653. Beams, forms of, of uniform strength, 517. Beams, uniform, supported at three or more points, 533; distribution of weight on the points of sup- port, 533; deflection, 534. Beams, homogeneous, transverse strength of, 503. See Transverse Strength, page 981. Belt-pulleys and belts, 742: Speeds, 742; tensile strength of belts, 742; horse-power transmitted by leather belts, 743; adhesion and power of belts, 744; M. Morin's experiments; M. Claudel's data, es .... table of the driving power of belts, 749. India-rubber belting, 750. Weight of belt-pulleys, 750. Belts, 742. See Belt-pulleys and Belts. Belou's hot-air engine, 918. Bending strength of wrought-iron plates, 586. Bernays' centrifugal pump, 968. Bevil-wheels. See Toothed Wheels, page 981. Birmingham wire-gauges, 130, 131; metal-gauge or plate-gauge, 131. Bitumen, weight and specific gravity of, 207. Blower, Root's, 927. Blowing engines, 926 Boghead coal, 417. Boilers, strength of stayed surfaces of, 685. See Eva- porative Efficiency, Evaporative Performance. Boiling points of liquids, 368; of saturated solutions of salts, 369; of sea- water, 370; boiling points at various pressures, 370. Bolts and nuts, standard sizes of: Whitworth's 972 INDEX. BOLTS COALS system, 681, 682; American system, 683; Armen- gaud's French system, 683, 684. Bolts and nuts, screwed, tensile strength of, 680. See Tensile Strength, page 980. Boyden turbine, 940. Brass, tensile strength of, 627, 628; brass tubes, 627; brass wire, 627, 629. Brass, weight of: tabulated weights, 219-221; rule for the length of brass wire, 224; multiplier for the weight of brass bars, plates, sheets, &c., 226; special tables of the weight of brass tubes and sheets, 252, 266-268; multiplier for the weight of brass balls, 258. See Weight of Iron and other Metals, page 983. Breakage of coal, 409, 410. Breast water-wheel, 938. Bricks, cemented, adhesion of, 630. Bricks, crushing strength of, 631. Brickwork, crushing strength of, 631. British coals, 412. See Coal. Bronze, tensile strength of, 627, 628; aluminium- bronze, 627. Buckled iron plates, strength of, 660. Builders' measurement: superficial, 137; cubic, 137. Building, measures relating to, 144. Bulging strength of wrought iron: Sir Wm. Fair- bairn's experiments, 569; Mr. Kirkaldy's experi- ments on Krupp and Yorkshire iron, 585; on Prussian iron, 586. Bulging strength of steel: Fagersta steel, 611 ; Sie- mens' steel, 612. Bulk of coal, &c. See Weight and Bulk of Coal, &c., page 983. Buoyancy, 277. Bursting pressure, resistance to, 687. See Strength of Hollow Cylinders, page 979. Bushel measures: Standard bushel, 139; sundry bushel measures for coal, 142; equivalent value in French measures, 154. Camel, work of, in carrying loads, 721. Carbon, constituent, influence of, on the tensile strength of steel, 621; on the transverse strength of steel rails, 664. Carbon, process of combustion of, 399. Carree's cooling apparatus, 373- Cast iron, strength of, 553. See Strength of Cast Iron, page 979. Cast iron, weight of: Data for the weight, 217, 218; rules for the weight, 223; tabulated weights, 219-221 ; multiplier for the weight of cast-iron bars, plates, &c., 226; special tables of weight of cast-iron pipes, cylinders, and balls, 251, 253-258; and of weight of cast-iron water-pipes, 936; and gas-pipes, 936. See Weight of Iron and other Metals, page 983. Cast-iron columns, strength of, 643-645. Cast-iron flanged beams, transverse strength of, 647: Experiments by Mr. Hodgkinson, by Mr. Berkley, by Mr. Cubitt, and others, 647; tabu- lated results, 649; formulas and rules, 651. Elastic strength and deflection, formulas and rules, 652. Catenary, 273. Cement, strength of: Tensile, 630; crushing, 632. Central forces, 294. Centres, mechanical, 287. See Mechanical Prin- ciples, page 977. Centrifugal force, 274, 294; rules, 295. Centrifugal pumps, 946; centrifugal pump by Mr. J. Bernays, 968. Chain, endless, pump, 947. Chains, tensile strength of, 677. See Tensile Strength, page 980. Chains, weight of, 678, 679. Channels, flow of water in, 932; limits of velocity, 934. Charbon de Paris, 449. Charcoal, brown, manufacture of, 449. Circles, properties of, 21; mensuration of, 24. Circular arcs, 35; tables, 95, 97. Circumferences of circles, &c., 35; tables, 66, 87. Clerk's gas-engine, 922. Cloth measure, 130. Coal, 409; classification of coals, 409; small coal, 409; utilization of small coal, 410; deterioration of coal by exposure, 412. British coals, 412; composition of bituminous coals Dr.- Richardson's analyses, 412; weight and composition of British and foreign coals by Messrs. Delabeche and Playfair, 413; variations of chemical composition, 415; average composi- tion, 415; Welsh coals, analysis by Mr. G. J. Snelus; patent fuels, 416; weight and bulk of British coals, 416; hygroscopic water in British coals, 416; Torbanehill or Boghead coal, 417; its composition, 417; air chemically consumed in the combustion of coal, 428. American and foreign coals, 418; Professor W. R. Johnson's analyses, 418; composition, 418; weight and bulk, 418, 419. French coals, 420; classification according to behaviour in furnace, and according to size, 420; utilization of small coal, 420; composition and heating power, 420, 422; weight and volume, 420; lignites, 422. Indian coals, 423; comparative composition of Australian, Nerbudda, Nagpore, and English coals, 424; composition of Indian coals, 425. Combustion of coal, 426. See Combustion of Coal, page 973. Coal, best, and inferior fuels, equivalent weights of, 820. Coal, brown. See Lignite, page 976. Coals, volume, weight, and specific gravity of, 206, 207. Coals: evaporative performance of English coals by Messrs. Delabeche and Playfair, 770; of Hindley Yard, Lancashire, coal, in stationary boilers, 771; of South Lancashire and Cheshire coals, in a marine boiler at Wigan, 781; of Newcastle and Welsh coals in the Wigan marine boiler, 784; of Newcastle coals, in a marine boiler at Newcastle- on-Tyne, 785; of Newcastle and Welsh coals in the Newcastle boiler, 787; of Welsh and New- castle coals, in a marine boiler at Keyham, 790; of American coals in a stationary boiler, 791 ; and in a marine boiler, 795; coal in locomotives, 800; Llangennech coal in portable-engine boilers, 801. Grate-area and heating surface, relation of, to evaporative performance in steam-boilers, 802 ; experiments by Mr. Graham, 802 ; by Messrs. Woods & Dewrance, 803 ; by M. Paul Havrez, 803. Formulas, deduced from the results of experi- ments, 804-821 ; table of evaporative performance by formulas, 819 ; table of equivalent weights of best coal and inferior fuels, 820. Heating surface and grate-area, relations of, to evaporative performance in steam-boilers, 802. Evaporative performance of steam-boilers, ex- perimental, influence of various circumstances and various treatment on: proportion of oxygen in coals, 771; area of grate, 772, 781, 796; coking fires and spreading fires, 773, 779, 780, 782 ; thick- ness of fire, 773, 779, 780, 782 ; admission of air above the grate, 772-774, 781, 794. Green's economizer, 772, 775, 778, 779 ; water- tubes, 775, 776 ; volume of air supply, 778 ; steam of higher pressure, 779 ; D. K. Clark's steam- induction apparatus, 779 ; self-feeding firegrates (Vicars'), 780; calm and windy weather, 780; forcing the draught, 781, 784; inverted bridge, 782 ; reduction of the flue surface, 775, 783, 795, 796; prolonged firing, 782; C. W. Williams' INDEX. COALS CRANES 973 smoke preventer, 788 ; soot in the flues, 794 ; level of the grate, 795. American marine boiler, varying rate of com- bustion, varying area of grate, reduction of heat- ing surface, 796. Coal-gas, its composition, 457 ; heat of combustion, Coal measure, 139 ; coal weight, 142. Coal-pits, waggons in, resistance of, 956. Cockle-stove, 488. Coins, current weight of, English, 141, 190; French, 190; German, 191; Austrian, 192; Egyptian, 194 ; Indian, 195 ; Japanese, 195. Coke, 430 ; residuary coke in coals, by laboratory analysis, 430; quality dependent on proportion of hydrogen in the coal, 431 ; anthracitic coke, 432; proportion of coke yielded by coal, 432; weight and bulk, 432 ; composition, 433; moisture in coke, 434 ; loss of combustible matter in the conversion of coal into coke, 434 ; air chemically consumed in combustion of coke, 435 ; gaseous products of combustion, 435 ; heating power of coke, 436 ; temperature of combustion, 436 ; gas- coke, 439. See also Coke, Proportion of, in Coals. Coke, American, 419. Coke, proportion of, in coals, British, 414, 415, 417, 425 ; American, 418, 419 ; French, 421 ; Indian, 423-425 ; Australian, 424 ; sundry, 430, 432. Coke of lignite, 436 ; proportion of, 437, 438 ; quality according to constituent hydrogen m lignite, 437, 438. Cold, greatest degree of, 373, 377 ; sources of, 373 ; Siebe's ice-making machine, 373; Carree's cooling apparatus, 373 ; frigorific mixtures, 373 ; cold by evaporation, 376. Cold rolling, influence of, on the density of wrought iron, 578. Collapsing pressure, resistance to, 694. See Strength of Holloiu Cylinders, page 979. Colliery winding engines, resistance of, 956. Columns, strength of, 643 ; leading principles, Mr. Hodgkinson's investigations, 643 ; short flexible columns, long columns, 644; Mr. F. W. Shields on hollow cast-iron columns ; rules by Mr. Gordon, Mr. Stoney, Mr. Unwin, Mr. Baker, 645 ; timber columns, 646 ; Mr. Brereton on timber piles, 646 ; Mr. Laslett on columns of wood, 647. Combustible elements of fuel, 398 ; gases concerned in the combustion of fuel, 398 ; process of com- bustion, 399 ; air consumed in combustion, 400, 405 ; gaseous products, 400 ; heat evolved by combustion, 402 ; heating powers of combustibles, 404406 ; temperature of combustion, 407. Combustibles, chemical composition of, 403. See Combustible Elements of Fi4el. Combustion, 398. Combustible elements, 398 ; gases concerned in combustion, 398 ; process of combustion, 399. Air consumed in the combustion of fuels, 400, 405. Quantity of gaseous products of combustion, 400. Surplus air, 402, 407. Heat evolved by the combustion of fuel, 402. Table of the heating powers of combustibles, 404, 405. Temperature of combustion, 407. Combustion, heat of, 402, 404, 405 ; English coals, 414, 428, 430; French coals, 421, 422; coke, 436. Combustion, heat of, gas-coke, 402 ; lignites, 423, 437, 438 ; asphalte, 437, 438 ; of wood, 444 ; of wood-charcoal, 452 ; of peat, 455 ; of peat-char- coal, 455 ; of tan, 455 ; liquid fuels, 456 ; coal-gas, 457- Combustion of coal, 426 ; process, 426 ; summary of the products of decomposition in the furnace, 426 ; quantity of air chemically consumed in the com- plete combustion of coal, 427; table, showing composition, heat of combustion, and air consumed by British coals, 428 ; gaseous products of the complete combustion of coal, 428; surplus air, 429 ; total heat of combustion, 430. Compass, points of, 37, 117. Composition of coals, British, 412, 413, 415-417, $24, 428 ; American, 418 ; French, 420, 422 ; ndian, 423-425 ; Australian, 423, 424 ; coke, 433 ; lignite, 422, 436, 438 ; asphalte, 437 ; wood, 440. Composition of coke, 433. Composition, chemical, of steel, 603. Compound steam engine, 849 ; Woolf engine, ideal diagrams, 849 ; receiver-engine, ideal diagrams, 852 ; intermediate expansion in the Woolf engine, 855 ; and in the receiver-engine, 857 ; work of the Woolf engine, with clearance, 859 ; and of the receiver-engine, 862 ; comparative work of steam in the Woolf engine and the receiver-engine, 867. Formulas and rules for calculating the expansion and the work of steam in compound engines, 869. To find the work done in the two cylinders of compound engines, 875. Compound units, English, comparison of: velocity, 144; volume and time, pressure and weight, weight and volume, power, 145. Compressed air, flow of, through pipes, 896. Compressed-air engines, efficiency of, 909 ; table of corresponding ratios of temperatures and pressures, when the air is admitted for the whole of the stroke, 908 ; table of comparative final temper- atures and efficiencies, when the air is expanded adiabatically, and when it is admitted for the whole of the stroke, 908. Machinery for working by compressed air at Powell Duffryn collieries, 916. Compressed steel, 614. Compressibility of water, 126. Compressing air, machinery for, 915 ; by water, 915; by a direct-action steam-pump, 915, 916. Compression of gases, 345. Compression, of a gas, work of, at constant temper- ature, 899 ; adiabatically, 903. Compression of steam in the cylinder, 878. Compressive strength of cast iron. See Tensile Strength, page 980. Compressive strength of steel, 595, 596, 599, 602, 605, 609. Compressive strength of timber : Mr. Laslett's ex- periments, 539, 541 ; Mr. Kirkaldy's, 546, 547. Compressive strength of wrought iron : Mr. Edwin Clark's experiments, 570 ; the Steel Committee's, 579 ; Swedish hammered bars, 581 ; Mr. J. Tangye's experiments, 582. Concrete, crushing strength of, 632. Condensation of steam and vapours, 462, 472, 475. Constructions, elementary, strength of, 633. See Strength of Elementary Constructions, page 979. Contraction of wrought iron under tensile stress : bars, 572, 580; notched bars, 574; plates, 578; wire, 587. Cooke's ventilator, 926. Cooling apparatus, Carree's, 373. Copper, alloys of, specific gravity, 200, 201, 626, 627; tensile strength of, 626, 627. See Tensile Strength of A Hoys of Copper, page 980. Copper, tensile strength of, 626; wire, 628, 629. Copper, weight of: tabulated weights, 219-221 ; multipliers for the weight of copper bars, plates, sheets, &c., 220, 226 ; rule for the length of copper wire, 224; special tables of the weight of copper in sheets, pipes, and cylinders, 251, 261-265. See Weight of Iron and other Metals, page 983. Cord of wood, 186. Corde of wood, 154. Cotton ropes for transmission of power, by Mr. Ramsbottom, 755. Cranes, stress in, 697; power of men at, 718, 719. 974 INDEX. CRUSHING FRENCH Crushing resistance. See Compressive Strength, page 973. Crushing strength of bricks and brickwork, 631. Crushing strength of concrete, 632. Curvilineal figures, mensuration of, 25. Cycloid and epicycloid, problems on, 19 ; mensura- tion of, 25. Cylinders, hollow, strength of, 687. See Strength of Hollow Cylinders, page 979. D Daniell's pyrometer, 327. Deflection of beams and girders, 527. See Trans- verse Deflection, page 981. Deflection, torsional, 536. See Torsional Deflec- tion, page 981. Deflection, transverse, of shafts, 756 ; formulas for deflection, 756 ; for diameter and side, 757 ; for distributed weight, 757 ; over-hung shafts, 757. Formulas for gross distributed weight, length of span, 757. Deflection of cast iron. See Transverse Strength, page 981; and Torsional Strength, page 981. Deflection, transverse, of flanged beams: cast- iron, 652 ; wrought-iron, 657, 660. Deflection of steel bars. See Transverse Strength, page 981; and Torsional Strength, page 981. Deflection of timber. See Transverse Strength, page 982. Deflection of wrought iron. See Transverse Strength, page 982; and Torsional Strength, page 981. Density, specific, of steam, 384, 385. Desiccation, 493 ; drying chambers, 494 ; drying by contact with heated metallic surfaces, 496 ; drying grain, 496 ; drying wood, 496. Deterioration of coal by exposure, 412, 425. Dew-point, 392. Diagonal rivet-joints in iron plates, strength of, 638. Diamond weight, 141. Distillation of wood, 449. Drilled holes, influence of, on strength of iron plates, 584. Drilled wrought-iron plates and punched plates, tensile strength of: Krupp and Yorkshire iron, 584 ; Staffordshire bar, 633. Drilled steel plates and punched steel plates, tensile strength of, 610, 611 ; elongation, 611. Dry measure, English, 139 ; French, 149. Ductility. See Elongation. Elasticity, coefficient of, 503. Elements, mechanical, 296. See Mechanical Prin- ciples, page 977. Ellipse, problems on, 13 ; mensuration of, 25. Elongation of cast iron, under tensile stress, 558, 560. Elongation of steel, under tensile stress: bars, 593-596, 598, 601, 605-611, 613-615, 624. Elongation of timber, under tensile stress, 545, 546. Elongation of wrought iron, under tensile stress : bars, 57 2 -577 580, 581, 624; plates, 577, 578, 583; holes in plates, 584, 615. Engines, pumping, water-works, 947. Evaporation, cold by, 376. Evaporation, spontaneous, 491. Evaporative efficiency of steam-boilers, 768. Evaporative performance of steam boilers, 768; nor- mal standards, 768; heating power of fuels, 769; stationary boilers at Wigan, 771, 811; performance of a marine-boiler at Wigan, 781, 809, 816-821; performance of a marine-boiler at Newcastle-on- Tyne, 785, 807, 816-821; performance of a marine- boiler at Keyham Factory, 790; performance of American coals in a stationary boiler, 791; and in a marine-boiler, 795, 810; stationary boilers in France, 796, 812. Locomotive boilers, 798, 805, 813, 817-821; portable steam-engine boilers, 801, 814, 817-821. Relations of grate-area and heating-surface to evaporative performance, 802; general formulas for practical use, 816. Evaporative power of coal English, 414. Expansion by heat, 335; linear expansion of solids, with table, 335; expansion of liquids, 338; expan- sion of water, with table, 338-341; Rankine's for- mula for expansion of water, 340; table of the ex- pansion of liquids, 342; expansion of gases, 342; expansion of air, 343-345; table of the expansion of air and other gases, 343. Expansion of air, work of, at constant temperature, 899; adiabatically, 904. Explosive force, resistance of steel and iron to, 622. Extension of iron under stress. See Elongation. Extension of timber under stress, 545, 546. Extension of steel under stress. See Elongation. Factors of safety for cast iron, wrought iron, steel, 625; iron chains, 678, 679; timber, foundations, mason-work, 625; ropes, 626, 674; dead load, live load, 626; screwed bolts and nuts, 68 1; cast-iron water-pipes, 936. Fans or ventilators, 924. See Air-machinery, page 971. Fires, open, heating by, 488. Fire-wood: French wood-measure, 149, 154, 443; American measure, 186, 443; moisture in fire- wood, 439, 441 ; composition, 441, 442. Flax-mills, machinery of, resistance of, 957; horse- power required, 959. Floatation, axis of, 277; plane of, 277. Flow of air and other gases, 891; discharge of gases through orifices, 891; anemometer, 892; outflow of steam through an orifice, 893. Flow of air through pipes and other conduits, 894; flow of compressed air through pipes, 896. Ascension of air by difference of temperature, 897. Flow of water, 929; flow through orifices, 929; co- efficients of discharge, 930; Mr. Bateman's experi- ments, 930. Flow through a submerged nozzle, Mr. Brown- lee's experiments, 931. Flow over waste-boards, weirs, &c., 932. Flow in channels, pipes, and rivers, 932; limits of velocity at the bottom of a channel, 934. Cast-iron water-pipes, 934. Cast-iron gas-pipes, 936. Flue-tubes, large, resistance of, to collapsing pres- sure, 696. Fluid bodies, 276. Fluid-compressed steel, 614. Fluids, pressure of, 276. Fontaine's turbine, 942. Foot and its multiples, 129; equivalent value in French measures, 1511, 153, 154, 156. Foot, square: decimal parts, in square inches, 138. Forces in equilibrium, 271. See Mechanical Prin- ciples, page 976. Form of specimen, influence of, on the tensile strength of iron, 574, 584. Fourneyron turbine, 940. Framed work, strength of, 697. See Strength o} Elementary Constructions, page 979. French coals, 420. See Coal, page 972. INDEX. FRICTION HEAT 975 Friction of solid bodies, 722: journals, 722; flat surfaces in contact, 723; friction on rails, 724; work absorbed by friction, 725; horse-power absorbed by friction, 726. Friction of leather belts, 744, 748. Frictional resistance of rivetted plates, 570. Frictional resistance of shafting, 763: resistance of journals of horizontal shafting, 722-726, 763; up- right shafting, 763; table based on Mr. Webber's data, 764. Ordinary data: by Mr. Tweddell, 763; by Mr. Westmacott, Mr. Walker, M. Cornut, Mr. R. Davison, Mr. Webber, 766. Frictional resistance, 951: steam-engines, 951; tools, 951; wood-cutting machines, 954; grind- stones, 955 ; colliery winding-engines, 956; wag- gons in coal pits, 956; machinery of flax-mills, 957; machinery of woollen mills, 959; conveyance of grain, 960; traction on common roads and on fields, 961 ; resistance on railways, 965 ; on street tramways, 966. Frictional wheel-gearing, 741. Frigorific mixtures, 373. Fuel, artificial, 411, 416, 420, 432, 449. Fuel, combustible elements of, 398. See Combustible Elements, page 973. Fuels, 409: coal, 409; coke, 418, 430; lignite and asphalte, 436; wood, 439; wood-charcoal, 444; peat, 452; peat- charcoal, 455; tan, 455; straw, 456; liquid fuels, 456; coal-gas, 457; gaseous fuel, 922. Fuels, heating power of, 269. Fuels, inferior, equivalent weights of best coal and, 820. Fuels in France, weight and specific gravity of, 207. Fuels, liquid, 456. See Liquid Fuels, page 976. Fuels patent, 411, 416; Warlich's, 411; Wylam's, 411; Mezaline's, 411; Barker's, 411; Holland's, 411. Fusibility of solids, 363. Fusion of solid bodies, latent heat of, 367. Fuss, German, values of, 161. Gallon: definition of, volume of, relative weight of water and its volume in gallons, 125; imperial standard measure of capacity, 128; French equiv- alent, 154, 157; American gallon, 186. Galloway boiler, trials of, 771-777. Gaseous fuel, 922; Wilson's, 922; Dowson's, 92.3. Gaseous products of combustion, quantity of, by weight, 400; by volume, 401; surplus air, 402, 407; specific heat of products, 408; products for coal, 426, 428; for coke, 435; for wood, 443; for wood-charcoal, 452. Gaseous steam, specific heat of, 353, 384; total heat of, 384; density, 384, 385. Gases, air and other, flow of, 891. See Flow of A ir, page 974. Gases, expansion of, 342; of air, 343-345; table, 343; compression of, 345; relations of the pressure, volume, and temperature of gases, 346, 351; special rules for one pound weight of a gas 349 specific heat of, 354, 358, 363. Gases concerned in the combustion of fuel, composi- tion and combining equivalents of, 398, 399; vol- ume of, 399. Gases and vapours, weight and specific gravity of, 216. Gases and vapours, mixture of, 392: hygrometry, 392; hygrometers, 393; properties of saturated mixtures of air and aqueous vapour, with table, 394, 396. Gases, liquefaction and solidification of, 372. Gas-coke, heat of combustion, 402. Gas-engines, 918. See Air Machinery, page 971. Gas-pipes, cast-iron, thickness of, 936. Gas-thermometers, 325. | Gearing. See Mill-gearing, page 977. Gearing by ropes, 753. See Rope-gearing, page 978. Geometrical problems, i : on straight lines, i; straight lines and circles, 5; circles and rectilineal figures, 8; ellipse, 13; parabola, 17; hyperbola, 18; cycloid and epicycloid, 19; catenary, 20; circles, 21; plane trigonometry, 21; mensuration of surfaces, 23; mensuration of solids, 27; men- suration of heights and distances, 30; Girders, strength of: warren-girder, 699; lattice- girder, 708; strut-girder, 708. Glass, tensile strength of, 629; crushing strength, 632. Gold, weight of, 219-221; rule for the weight of gold wire, 224. Gold wire, tensile strength of, 628. Goods carried on the Bombay, Baroda, and Central India Railway, weight and volume of, 213. Grain (weight), 140; equivalent value in French measures, 155-157. Gram, apparatus for conveyance of, resistance of, 960. Gramme, 150; equivalent values in English mea- sures, 155, 157. Graphite: weight and specific gravity, 207; heat of combustion, 402. Gravity, 277; action on inclined planes, 285; centre of, 287; work done by it, 315. See Mechanical Principles, page 977. Guibal's fan, 925. Gun-metal, weight of: tabulated weights, 219-221; multiplier for the weight of gun-metal balls, 258,- Gun-metal, tensile strength of, 626, 627. Gyration, centre of, radius of, 288. H Hammers, atmospheric, 967. Hardened or tempered steel, tensile strength of, 594, 602, 603, 613. Head of pressure, 276. Heat, 371: Thermometers, 317; air-thermometers, 325. Pyrometers, 326, 966; air-pyrometer, 327. Luminosity at high temperatures, 328. Radiation of heat, 329. Conduction of heat, 331. Convection of heat, 331. Mechanical theory of heat, 332. Mechanical equivalent of heat, 718. Expansion by heat, 335; solids, 335; liquids, 338; gases, 342. Compression of gases, 345. Relations of the pressure, volume, and tempera- ture of air and other gases, 346. Special rules for one pound weight of a gas, 349- Table of the volume, density, and pressure of air at various temperatures, 351. Specific heat, 352; of solids, 358, 359; of water, 353, 354; of air and other gases, 354, 363. Fusibility or melting points of solids, 363. Latent heat of fusion of solid bodies, 367. Boiling points of liquids, 368. Latent heat and total heat of evaporation of liquids, 370. Liquefaction and solidification of gases, 372. Sources of cold, 373; frigorific mixtures, 374. Cold by evaporation, 376. Heat evolved by the combustion of fuel, 402, 404, 405. Temperature of combustion, 407. Heat, transmission of, through solid bodies from water to water through plates, 459; heating and evaporation of liquids by steam through metallic surfaces, 461; cooling of hot water in pipes, 469; 9/6 INDEX. HEAT MECHANICAL cooling of hot wort on metal plates in air, 470; cooling of hot wort by cold water in metallic refrigerators, 471; condensation of steam in pipes exposed to air, 472; condensation of vapours in pipes or tubes by water, 475. Heat, applications of, 459. See Applications of Heat, page 971. Heating of solids, 497: cupola furnace, 497; plaster ovens, 497 ; metallurgical furnaces, 497 ; blast furnace, 498. Heating of water by steam in direct contact, 490. Heating power. See Combustion, Heat of, page 973- Heating power of fuels, 769. Hectare, 149; equivalent value in English measures, J ? 3 ' Heights and distances, mensuration of, 30. Hemp ropes for gearing, 753. Horses, labour of, 720; work of, in carrying loads, 721. Horses, performances of, on roads in France, 964. Horse-power, 718; absorbed by friction, 726. Horse-power of round shafting, 760, 762; absorbed by friction, 763. Holtzapffel's Birmingham wire-gauge, 131; his Bir- mingham metal-gauge, or plate-gauge, 131 ; his Lancashire gauge, 132 ; values in parts of an inch of his wire-gauge, 134. Horse-power of toothed wheels, 737. Horse-power transmitted through leather belts, 743- 747- Horse-power required for a flax-spinning mill, 959 ; for a woollen mill, 960. Hot-air engines, 917 ; (first-class), Rider's, 917 ; (second-class), Belou's hot-air engine at Cusset, 918. Hydraulic machines, Armstrong's, 950. Hydraulic press, 950 ; strength of, 687. Hydraulic rams, 949. Hygrometers, 393 ; Daniell's, 393 ; Regnault's, 393 ; Mason's wet and dry thermometers, 394. Hygrometry, 392 ; dew-point, 392. Hygroscopic water in British coals, 416. Hyperbola, problems on, 18 ; mensuration of, 25. Ice, its volume, weight, melting point, specific heat, 127 ; specific heat of, 127, 353 ; specific gravity of, 127 ; melting point of, 364 ; latent heat^ of fusion of, 367; frigorific mixtures with, 375. Ice-making machine, Siebe's, 373. Inches, their multiples, 129; their decimal values in parts of a foot, 135 ; fractional parts and deci- mal equivalents, 135 ; equivalent value in French measures, 151, 153, 154, 156. Inclined plane, action of gravity upon, 285 ; prin- ciple of, 306 ; identity of it with the lever, 308 ; work done with it, 314. India-rubber belts, strength of, 680, 750. Indian coals, 423. See Coal, page 972. Inertia, moment of, 288. Iron, weight and specific gravity: of wrought iron, 202, 217, 218 ; of cast iron, 203, 217, 218. Iron and other metals, tables of weight of, 217. See Weight of Iron and Other Metals, page 983. Iron, wrought, weight of; data for the weight, 217, 218 ; rules for the weight, 223 ; tabulated weights, 219-221 ; weight of French galvanized wire, 225 ; special tables of weight of wrought-iron bars, plates, sheets, hoops, wire, >and tubes, 226-250 ; multipliers for wrought - iron balls, 258. See Weight of Iron and Other Metals, page 983. Iron, hammered ; multipliers for weight of bars, plates, &c., 220. Iron wire ropes, strength and weight of, 674-677. j Jonval turbine, 942. Joule's equivalent, 332. Journals of shafts, 766 ; of railway axles, 767. Journals, friction of, in their bearings, 722 ; coeffi- cients, 724 ; work of, 725 ; horse-power, 726. K Kilogramme, standard, 146, 150; equivalent value in English measures, 155, 157. Kilometre, 147; equivalent value in English mea- sures, 150, 156. L Labour, 718. See Work, page 984. Lama, work of, in carrying loads, 721. Lancashire wire-gauge, 132. Land measure, English, lineal, 130 ; superficial, 137; French, superficial, 149. Latent heat of fusion of solid bodies, 367 ; of eva- poration of liquids, 370, 373 ; of steam, 380. Lattice-girder, parallel, strength of, 708. Lead, weight of; tabulated weights, 219-221 ; multi- plier for the weight of lead bars, plates, &c., 220; special table of the weight of lead pipes, 269. See Weight of Iron and OtJter Metals, page 983. Lead, tensile strength of, 627. Leather-belting, tensile strength of, 679. See Ten- sile Strength, page 980. Lenoir's gas-engine, 919. Lever, 296; work done with it, 313. Lignite : composition, 422, 436, 438 ; heat of com- bustion, 423, 437, 438 ; hygrometric moisture, 436; weight and specific gravity, 207, 436, 437. Lineal measure, English, 129 ; French, 147. Liquefaction and solidification of gases, 372. Liquid fuels, 456 ; petroleum, 456 ; petroleum-oils, 456 ; schist-oil, 457 ; pine-wood oil, 457. Liquid measure, English, 138 ; French, 149. Liquids, expansion of, 338; of water, 338-341; table, 342 ; specific heat of, 362 ; boiling points of, 368 ; latent heat and total heat of evaporation of, 370, 372. Liquids, weight arid specific gravity, 215. Litre, defined, 147, 149 ; equivalent value in English measures, 154, 157. Logarithms of numbers, 32 ; table, 38. Logarithms, hyperbolic, of numbers, 35 ; table, 60. Lubrication, 957. Luminosity at high temperatures, 328. M Machines for raising water, 944. See Water, page 982. Malleable cast iron, strength of, 561. Mass, 287. Materials, strength of, 500. See Strength of Mater- ials, page 979. Mathematical tables, 32. Measurement, compound units of, equivalents of French and English, 157 ; weight, pressure, and measure, 157 ; volume, area, and length, 158; work, 158 ; heat, speed, money, 159. Measurement, principal units of, 129. Measures, English and French, approximate equi- valents of, 156. Mechanical centres, 287. See Mechanical Prin- ciples, page 977. Mechanical elements, 296. See Mechanical Prin- ciples,^^ 977. Mechanical equivalent of heat, 332. Mechanical theory of heat, 332; Joule's equivalent, 332; illustrations, 333. Mechanical principles, fundamental, 271. Forces in equilibrium: solid bodies, parallel- ogram of forces, polygon of forces, moments of forces, 271; the catenary, 273; centrifugal forces, INDEX. MELTING PRESSURE 977 274; parallel forces, 275; parallelepiped of forces, 276. Fluid bodies: Pressure of fluids, head of pres- sure, buoyancy, axis of floatation, plane of float- ation, stability, metacentre, 276. Motion: uniform motion, velocity, accelerated and retarded motion, 277. Gravity: falling bodies, time, velocity, and height of fall, 278; rules for the action of gravity, 279; table of velocity due to height, 280; table of height due to the velocity, 281; table of height of fall and velocity, for the time, 282. Accelerated and retarded motion in general: general rules for accelerating forces, 282 ; Action of gravity on inclined planes, special rules for the descent on inclined planes, 285. Average velocity of a moving body uniformly accelerated or retarded, 286. Mass, 287. Mechanical centres: centre of gravity, 287; centres of gyration, radius of gyration, moment of inertia, 288; centre of oscillation, 290; the pen- dulum, 291 ; centre of percussion, 294. Central forces: centripetal force, centrifugal force, rules, 294. Mechanical elements: lever, 296; pulley, 302; wheel and axle, 305; inclined plane, 306; identity of the inclined plane and the lever, 308 ; wedge, 309; screw, 311. Work: definitions, 312; work done with the lever, 313; with the pulley, 313; with the wheel and axle, 314; with the inclined plane, 314; with the wedge, 315; with the screw, 315; by gravity, 315; work accumulated in solid bodies, 315; work done by percussive force, 316. Melting points of solids, 363; table, 364. Men, labour of, 718; work of, in carrying loads, 720. Mensuration of surfaces, 23; solids, 27; heights and distances, 30; circle, 24; ellipse, parabola, hyper- bola, cycloid, and epicycloid, curvilineal figures, 25. Metacentre, 277. Metals, melting points of, 363; table, 365. Metals, weight and specific gravity of, 202, 626, 627. Metre, standard, 146, 147; equivalent value in English measure, 150, 156. Metric system, French; countries in which it is legalized, 146; mile, 129; equivalent value in French measures, 151, 156. Mill-gearing, 727; toothed gear, 727; frictional- wheel- gearing, 741; belt-pulleys and belts, 742; rope-gearing, 753; shafting, 756. Mine ventilators, 925. See Air Machinery, page 971. Mineral substances, sundry, weight and specific gravity of, 205. Mitre wheels. See Toothed WJieels, page 981. Moisture in coal, 416; in coke, 434; in wood, 439; in wood-charcoal, 451. Moments of forces, 271. Money, 190: Great Britain and Ireland: value, weight, and composition of coins, 190. France: value, weight, and composition of coins, and equivalent value in English money, 190. Germany: currency established in 1872, old currency, 191. Hanse towns: Hamburg, Bremen, Lubeckjigi. Austria, 192. Russia, 192. Holland, Belgium, Denmark, Sweden, Nor- way, 192. Switzerland, 193. Spain, Portugal, Italy, Turkey, Greece and Ionian Islands, Malta, 193. Egypt, Morocco, Tunis, Arabia, Cape of Good Hope, 194. Indian Empire, 195. China, Cochin-china, Persia, Japan, Java, 195. United States of America, 195. Canada British North America, 196. Mexico, 196. Central America and West Indies : British West Indies, Cuba, Guatemala, Honduras, Costa Rica, St. Domingo, 196. South America : Colombia, Venezuela, Ecua- dor, Guiana, Brazil, Peru, Chili, Bolivia, Argen- tine Confederation, Uruguay, Paraguay, 196. Australasia, 197. Mortar, strength of: tensile, 629; crushing, 632. Motion, 277; accelerated and retarded motion, 282. Mules, work of, in carrying loads, 721. Muntz's metal, tensile strength of, 627. Music-wire gauge, 132. N Nautical measure, 130. Needle-gauge, 132. Noria, 947. North Moor Foundry turbine, 942. Notched form of specimen of iron, comparative strength of, 574; of steel, 622. o Oscillation, centre of, 290. Otto & Langen's gas-engine, 919; Otto engine, 921. Outflow of steam through an orifice, 893. Overshot water-wheel, 938. P Paddle water-wheel, 938. Palladium wire, tensile strength of, 628. Parabola, problems on, 17; mensuration of, 25. Parallel forces, 275. Parallelogram of forces, 271. Parallelepiped of forces, 276. Peat, volume, weight, and specific gravity of, 207. Peat, 453: its origin, 453; composition, 453, 454; moisture, 453; air-dried peat, 454; products of distillation, 454; heat of combustion, 455. Peat-charcoal, 455 ; heat of combustion, 455. Pendulum, 291. Percussion, centre of, 294. Percussive force, work done by, 316. Phosphor-bronze, tensile strength of, 628, 629. Piles, timber, strength of, 646. Pipes, flow of air through, 894, 896. Pipes, flow of water in, 932. Pipes, heating of water by steam- in, 463; cooling of water in, 469; condensation of steam in, 472, 475; heating of rooms by hot water in, 481. Pipes, gas, cast-iron, thickness of, 936. Pipes, water, cast-iron, dimensions, weight, and strength of, 934. Pipes, weight of, 251, 253-258, 262, 269, 936. Piping, screwed iron, Whitworth's standard pitches for, 683. Pitch of toothed wheels, 728. Pivots, friction of, in their bearings, work of, 725; horse-power, 726. Plaster of Paris, tensile strength of, 629, 630. Platinum wire, rule for weight of, 224. Platinum wire, tensile strength of, 628. Poncelet's undershot water-wheel, 937. Pound (weight), 140, 141; avoirdupois and troy com- pared, 140; equivalent in French measures, 155. Power transmitted by shafting, 760; work for one turn, horse-power, 760, 762. Power. See Work, 984. Press, hydraulic, 949. Pressure of fluids, head of pressure, 276. INDEX. PRESSURE STEAM Pressure of vapours at 212 F., 370. Pressure of water, measures of pressure, 126. Products of combustion, 400. See Gaseous Pro- ducts, page 975. Pulley, 302; work done with it, 313. Pulleys, belt, 742. See Belt-pulleys, page 971. Pumps, reciprocating, 944; centrifugal, 946; end- less-chain, 947; steam- vacuum, 969. Pumping engines, water-works, 948. Punched holes, iron plates and bars with, strength of, 584, 633. Punched steel plates, tensile strength of, 610, 611, 642. Punching strength of wrought iron, 587. Pyrometers, 326; Wedgewood's, 327; DanielPs, 327; air-pyrometer, 327; Wilson's pyrometer, 327; Sie- mens' pyrometer, 967. R Rails, railway, transverse strength and deflection of, 661. See Transverse Strength, page 981. Rails, friction on, 724. Railway axles, proportions of, 767. Railway rails, transverse strength and deflection of, 661. See Transverse Strength, page 981. Railways, resistance on, 965. Rams, hydraulic, 948. Reciprocals of numbers, 37; table, 118. Reciprocating pumps, 944. Regnault's air-thermometer, 326. Resistance, frictional, 763, 951. See Frictional Resistance, page 975. Resistance of air to the motion of flat surfaces, 897. Resistance of materials. See Strength of Materials, page 979. Resistance on railways, 965. Resistance on street tramways, 966. Rider's hot-air engine, 917. Rivers, flow of water in, 932. Rivets and rivet iron, shearing strength of, 570, 588. Rivet-joints, tensile strength of, 633. Rivet-joints, proportions of, 641. Rivetted plates, frictional resistance of, 570. Rivetted wrought-iron joists, transverse strength of, 653- Roads, resistance to traction on, 961, 962. Roofs, strength of, 713. Root's blower, 927. Rope-gearing, 753: transmission of power by hemp- ropes, 753; use of wire-ropes, by M. Him, 754; cotton-ropes, by Mr. Ramsbottom, 755. Ropes, tensile strength and weight of, 673. See Tensile Strength, page 980. Rylands Brothers, wire-gauge, 133, 247. Safety, factors of, 625. See Factors of Safety, p. 974. Salts, saturated solutions of, boiling points of, 369. Screw, 311; work done with it, 315. Screw-threads, 68 1. Sea-water; its volume, weight and composition, 126. Sea-water, composition of, 126; weight of, 126; specific gravity of, 126; boiling point of, 369, 370. Section of land, 189. Ser, 1 80. Shafting, 756; transverse deflection, 756; overhung shafts, 757; torsional strength and deflection of round shafts, 758; power transmitted, 760; net weight/of shafting, 761; table of the strength of round Vrought-iron shafting, with multipliers for cast-iron and steel, 762. Frictional resistance of shafting, 763. Shearing strength of cast iron, 561. Shearing strength of steel, 595, 605; deduction, 617. Shearing strength of timber, 551. gases, 372. Shearing strength of wrought iron, 587: Mr. Edwin Clark, 570; Swedish iron, 581; Mr. Little's ex- periments, 587; chief-engineer Shock's, 587; con- clusion, 588. Shearing stress, definition of, 500; stress in beams and plate-girders, 525. Shearing stress, torsional, of steel bars, 620. Ships, measure for, 144. Siebe's ice-making machine, 373. Siemens' pyrometer, 967. Silver, weight of, 219-221 ; rule for the weight of silver wire, 224. Silver wire, tensile strength of, 628. Sines, cosines, &c., of angles, 30, 36; tables, 103, no. Small coal, 409, 420; utilization of, 410, 420; wash- ing of, 411. Snow, its volume and weight, 127. Snow, frigorific mixtures with, 375. Solder, soft, tensile strength, 627. Solidification and liquefaction of gases, Solids, mensuration of, 27. Solids, weight and specific gravity of, 199-214. Solids, expansion of, 335; specific heat of, 359; fusi- bility of, 363; latent heat of fusion of, 367. Solids, heating of, 497. See Heating of Solids, page 976. Specific gravity, 198. See Weight and Specific Gravity, page 982. Specific gravity of coals: English, 414, 416, 417, 424; American, 418, 419; French, 421; Indian, 424; Australian, 424. Specific gravity of timber, 539-543, 545~547- Specific gravity of cast iron, 554, 557. Specific gravity of wrought iron, 568, 578, 603. Specific gravity of wrought iron, influence of wire- drawing on specific density, 247; influence of cold- rolling, 578; influence of stretching, 578. Specific gravity of steel, 603. Specific heat, 352: specific heat of water, ice, steam, 127, 353; of water at various temperatures, 354; of air and other gases, 128, 354; of gases for equal volumes, 358; table of specific heat of solids, 359; of liquids, 362; of gases, 363; specific heat of gaseous steam, 384. Speeds of toothed wheels, 727. Spontaneous evaporation in open air, 491. Springs, steel, strength of, 671. See Transverse Strength, page 981. Spur wheels. See Toothed Wheels, page 981. Stability of floating bodies, 277. Staybolts, screwed, and flat surfaces, strength of, 685. Steam, 378. Physical properties of steam, 378. Saturated steam: relation of the temperature and pressure, 378; total heat, 379; latent heat, 380; appropriation of its constituent heat at 212, 380; volume and density, 381; relative volume, 382. Gaseous steam, 383; its total heat, 384; specific heat, 384. Specific density of gaseous steam, 384; of satu- rated steam, 384. Density of gaseous steam, 385. Table of the properties of saturated steam, from 32 to 212, 385, 386. Table of the properties of saturated steam for high pressures, 385, 387. Table of comparative density and volume of air and saturated steam, 391. Steam, expansive working of, principles of, 822- 877. Practice of, 879: actual performance of steam in the steam-engine, 879; data of the practical performance of steam in single-cylinder condens- ing engines, 880; in compound condensing engines, 88 1 ; in single-cylinder non-condensing engines, INDEX. STEAM STRENGTH 979 French 883; in American marine engines, stationary engines, 886. General deductions from the data of the actual performance of steam, 886: single-cylinders with steam jackets, condensing, 886; non-condensing, 886; without steam jackets, condensing, 887; non- condensing, 887 ; compound-cylinders with steam jackets, condensing, 887; proportional ratios of expansion in the first and second cylinders, 887; compound-engines without steam in jackets, con- densing, 888. Conclusions on the actual performance of steam, 888 ; table of the practical performance of steam- engines, 890. Steam, gaseous. See Gaseous Steam, page 975. Steam in the cylinder, compression of, 878. Steam, outflow of, through an orifice, 893. Steam-boilers, evaporative performance of, 768. Steam-engine: Action of steam in a single-cylinder, 822: pres- sure of steam during expansion in a cylinder, 822; work of steam by expansion, 824; clearance in steam-cylinders, 827; formulas for the work of steam in the cylinder, 828 ; initial pressure in the cylinder, 829; average total pressure in the cylin- der, 830; average effective pressure in the cylin- der, 830; period of admission and the actual ratio of expansion, 830; relative performance of equal weights of steam worked expansively, 832; pro- portional work done by admission and expansion, 833; influence of clearance in reducing the per- formance of steam in the cylinder, 834. Table of ratios of expansion of steam, with rela- tive periods of admission, pressure, and total per- formance, 835, 836; total work done by one pound of steam expanded in a cylinder, 838; consump- tion of steam worked expansively per horse-power of total work per hour, 840; table of the total work done by one pound of steam of 100 Ibs. total pressure per square inch, 840, 841 ; net cylin- der capacity relative to the steam expended and work done in one stroke, 843; table of relations of net capacity of cylinder to steam admitted and work done, 844, 846. Compression of steam in the cylinder, 878. Steam-engines, internal resistance of, 951, 957. Steam-engine, compound. See Compound Steam- engine, page 973. Steam-vacuum pump, 969. Steel, weight and specific gravity of, 202, 217. Steel, weight of: data for the weight, 217; rules for the weight, 223; tabulated weights, 219-221; multiplier for weight of steel bars, plates, &c., 226; special tables of weight of steel bars and chisel-steel, 251, 259-261; multiplier for steel balls, 258. See Weight of Iron and other Metals, page 983- Steel, strength of, 593. See Strength of Steel. Steel columns, strength of, 644, 646. Steel springs, strength of, 671. See Transverse Strength, page 981. Steel wire, tensile strength of, 617, 629. Steel wire-ropes, strength and weight of, 674, 675, 677. Stere, 149; equivalent value in English measures, i54- Sterro-metal, tensile strength of, 627. Stones, specific gravity of, 203; of precious stones, 202. Stones, strength of: tensile strength, 629; crushing strength, 631. Stoves and hot air, heating by, 488. Strain, definition of, 500. Straw, composition of, 456. Strength of materials, 500: kinds of stress, 500; elastic strength, ultimate or absolute strength, 500; work of resistance of material, 501; coeffi- cient of elasticity, 503; transverse strength of homogeneous beams, 504 ; forms of beams of uni- form strength, 517; shearing-stress in beams and plate - girders, 525; deflection of homogeneous beams and girders, 527; uniform beams supported at three or more points, 533; torsional strength and deflection of shafts, 534. Strength of timber, 537; strength of cast iron, 553; strength of wrought iron, 567; strength of steel, 593; recapitulation of data on the direct strength of iron and steel, 623. Working strength of materials factors of safety, 625. Strength of copper and other metals, 626; tensile strength of wire of various metals, 628. Strength of stone, bricks, &c., 629. Strength, transverse, of homogeneous beams, 503. See Transverse Strength, page 981. Strength, uniform, forms of beams of, 517: rectan- gular semi-beams, 518520; flanged semi-beams, 519, 521; semi-beams of circular or elliptical sec- tions, 519, 521; rectangular beams, 52 1-523; flanged beams, 523, 524; stress in curved flange, 525. Strength of framed work, 697: illustrations of stress in framed work, 697 ; elementary truss, 698 ; framed girders the Warren-girder, 699-708; parallel lat- tice-girder, 708; parallel strut-girder, 708-713; roofs, 713-717. Strength of cast iron, 553: Mr. Hodgkinson's ex- periments on tensile and compressive strength, 553; Dr. Anderson's, 555; strength as affected by the mass of the metal, 555; as affected by cold- blast and hot-blast, 555; increased by remelting, Mr. Bramwell's experiments, 556; Sir Wm. Fair- bairn's 557; Major Wade's, 557. Elastic tensile and compressive strength of cast iron, 558. Shearing strength of cast iron, 561. Malleable cast iron, 561. Transverse strength of cast iron: Mr. Barlow's experiments, 561; Mr. Edwin Clark's data, 562; Mr. Hodgkinson's data, 563; test bars, 564. Transverse deflection and elastic strength of cast iron; formulas, 564. Torsional strength and deflection of cast iron; Mr. Dunlop's data; formulas, 565. Recapitulation of data on the direct strength of cast iron, 623. Factors of safety, 625. Strength of cement: Portland, 630, 632; Roman, 630. Strength of elementary constructions, 633 ; rivet- joints, 633 ; pillars or columns, 643 ; cast-iron flanged beams, 647 ; wrought-iron flanged beams or joints, buckled iron plates, 660 ; railway rails, 661 ; steel springs, 671 ; ropes, 673 ; chains, 677 ; leather belting, 679 ; bolts and nuts, 680 ; iron piping, 683 ; screwed stay-bolts and stayed sur- faces, 685. Hollow cylinders: tubes, pipes, boilers, &c., 687. Framed work: cranes, girders, roofs, &c., 697. Strength of glass: tensile, 629; crushing, 632. Strength of hollow cylinders, 687: resistance to internal or bursting pressure, 687; hydraulic press, 687; its bursting strength, with formulas, 689, 690; longitudinal resistance, 692; resistance of wrought- iron tubes, 692, 695; a Lancashire boiler, 693; a cylindrical marine boiler, 693 ; cast-iron pipes, 694 ; lead pipes, 696. Resistance to collapsing pressure, 694: solid- drawn tubes, with formulas, 694; large flue-tubes, 696. Strength of steel, 593: Sir. Wm. Fairbairn's experi- ments, 568; Mr. Kirkaldy's early experiments on tensile strength of bars and plates, 593; strength of hematite steel, 594; strength of Krupp steel, 595- Experiments of the Steel Committee of Civil Engineers, 596; tensile strength of tempered steel; experiments at Woolwich Dockyard, 602 ; strength 980 INDEX. STRENGTH TENSILE of Fagersta steel, 604; Siemens'-steel plates and tyres, 612; Whit worth's fluid-compressed steel, 614. Chernoff's experiments on the influence of tem- perature on the structure of steel, 616. Tensile strength of steel wire, 617, 629. Shearing strength of steel, 617. Transverse strength and deflection of steel: data and formulas, 617, 618. Torsional strength and deflection of steel bars, 619. Strength of steel relatively to the proportion of constituent carbon, 621, 664. Resistance of steel and iron to explosive force, 622. Recapitulation of data on the direct strength of steel, 623. Factors of safety, 625. Strength of stones: tensile, 629; crushing, 631. Strength of timber, 537: general conditions of strength, 538; Mr. Laslett's experiments on trans- verse, tensile, and compressive strength, 538, 647; Mr. Fincham's experiments on the transverse strength of soft woods, 542; Mr. Barlow's experi- ments on transverse strength, 547. Transverse strength of beams of large scantling, Mr. Maclure's experiments, 542 ; Mr. Edwin Clark's experiments, 544; Mr. G. Graham Smith's experiments, 544; Mr. Baker's data, 544; MM. Chevandier and Wertheim's experiments, 545. Elastic strength and deflection of timber, 545; experiments on tensile strength by MM. Chevan- dier and Wertheim, 546 ; and by Mr. Laslett, 546 ; experiments on compressive strength by Mr. Kirkaldy, 546 ; Mr. Barlow's experiments on transverse strength, 547. Rules for the strength and deflection of timber, 548; analysis of Mr. Laslett's experimental results, 548; and of Mr. Fincham's results, 549; calculated tensile strength of timber of large scantling, 549; formulas for the transverse strength of timber of large scantling, 550, 551 ; formulas for the trans- verse deflection of timber beams of uniform rec- tangular section, 550; shearing strength of timber, 55 1 - Strength of toothed wheels, 735. Strength of wrought iron, 567; Mr. Telford's experi- ments on tensile strength, 567; Mr. Barlow's, 567; Sir William Fairbairn's, 567; Mr. Thomas Lloyd's, 569; Mr. Edwin Clark's, 570; Mr. Kirkaldy's 57 1 - Experiments of the Steel Committee of Civil Engineers, 579; hammered iron bars (Swedish), 581 ; Mr. J. Tangye's experiments on compressive resistance; Krupp and Yorkshire iron plates, 583; Prussian iron plates, 586; Sir Joseph Whitworth's experiments, 615. Tensile strength of iron wire, 247, 586, 628, 629, 676. Shearing and punching strength of iron, 587. Transverse strength of iron: Swedish bars, 582; Mr. Barlow's data, 588; Mr. Edwin Clark's data, 588; formulas; transverse deflection and elastic strength of wrought iron, 590. Torsional strength and deflection, 590. Resistance of iron to explosive force, 622. Recapitulation of data on the direct strength of iron, 623. Factors of safety, 625, 679. Strength of round wrought-iron shafting, table of, with multipliers for cast iron and for steel, 762. Stress, kinds of, 500. Stress, working, for screwed bolts, 681. Stretching, influence of, on the density of wrought- iron, 578. Strut-girder, parallel, strength of, 708. Sulphur, process of combustion of, 399. Surfaces, mensuration of, 23. Surplus air, in the combustion of coal, 770, 778, 794. Swaine turbine, 943. Tan, heat of combustion of, 455. Tangential water-wheels, Girard's, 943. Teeth of wheels. See Toothed Wheels, page 981. Temperature, difference of, ascension of air by, 897. Temperature of combustion, 407; coal, 407, 408; coke, 408, 436; wood, 444. Temperature, influence of, on the strength of metallic wires, 628. Temperature, influence of, on the structure of steel, 616. Temperatures, standard, 124, 198. Temperatures, high, luminosity at, 328. Tempered or hardened steel, tensile strength of, 594, 602, 603, 613. Tensile strength of brass, 627, 628; brass tube, 627; brass wire, 627, 629. Tensile strength of bronze, 627, 628. Tensile and compressive strength of cast iron: Mr. Hodgkinson's experiments, 553; Dr. Anderson's, 555; affected by mass of metal, 555; by cold blast and hot blast, 555; by remelting, 556. Elastic strength, 558. Tensile strength of chains, 677: stud-link chain - cable, 678; open-link chains, 678. Tensile strength of copper, 626; copper wire, 628, 629. Tensile strength of alloys of copper, 626: alloyed with phosphorus, 626; gun-metal, 626, 627; alloys of copper and tin, aluminium-bronze, yellow brass, brass tube, Muntz's metal, sterro-metal, 627; phos- phor-bronze, bronze, and brass, 628; brass wire, 629. Tensile strength of gold wire, 628. Tensile strength of gun-metal, 626, 627. Tensile strength of leather belting, 679, 742: Mr. Towne's experiment, 679; Messrs. Norris & Co.'s belting, Spill's machinery belting, untanned leather belts, india-rubber belts, 680, 750. Tensile strength of lead, 627. Tensile strength of Muntz's metal, 627. Tensile strength of palladium wire, 628. Tensile strength of phosphor-bronze,'628, 629. Tensile strength of platinum wire, 628. Tensile strength of plaster of Paris, 629, 630. Tensile strength of rivet joints, 633: In iron plates, 633: perforated iron plates, 633; experiments on rivet-joints by Sir Wm. Fairbairn, 633; by Mr. Bertram, 634; by Mr. J. G. Wright on diagonal joints, 637; by Mr. L. E. Fletcher, 638; by Messrs. John Elder & Co., 638; by Mr. Brunei, 638; shearing strength of rivets, 640; con- clusions on the strength of rivet-joints in iron plates, 640; proportions of rivet-joints, 641. In steel plates, 642: perforated steel plates, 642; rivetted joints, 642. Tensile strength of ropes: hemp rope, 673- 675; influence of twist and of moisture on the strength, 674; American hemp rope, 676: iron- wire rope, 674, 675; American iron-wire rope, 676; French iron- wire rope, 677; steel-wire rope, 674, 675, 677; cable fencing strands and solid fencing wire, 676. Tensile strength of screwed bolts and nuts, 680: Mr. Brunei's experiments, 680; working stress, 681, 684. Screwed stay-bolts and flat stayed surfaces of locomotive boilers, 685; of marine boilers, 686; rules, 686. Tensile strength of silver wire, 628. Tensile strength of soft solder, 627. Tensile strength of steel: Sir Wm. Fairbairn's ex- periments on steel plates, 568; Mr. Kirkaldy's early experiments on steel bars, 593; on steel INDEX. TENSILE TRANSVERSE plates, 594; on hardened steel; hematite steel, I 595; crank shafts of Krupp steel, 595. Experiments of the Steel Committee, 596: tables of results for tensile strength, first series, 598; second series, 601 ; chemical analysis and specific gravity of the steel bars, 603. Tensile strength of tempered steel, 603. Fagersta steel hammered bars, 604; ingots and hammered bars, 606 ; hammered and rolled bars, 607 ; bars reduced by hammering and by rolling, 608; plates of different thicknesses, 608; plates of dif- ferent forms, 608, 610; plates with drilled holes and punched holes, 610. Siemens'-steel plates and tyres: plates of dif- ferent thicknesses, 613; hardened plates, 613; tyres, 614. Whitworth's fluid-compressed steel, 614, 615. Steel wire, 617, 629. Influence of constituent carbon on tensile strength of steel, 621. Resistance of steel to explosive force, 622. Notched specimen, comparative strength of, 622. Tensile strength of steel wire, 617, 629. Tensile strength of sterro-metal, 627. Tensile strength of timber: Mr. Laslett, 538-540; MM. Chevandier and Wertheim, 545, 546; Mr. Laslett, 546; calculated from Mr. Laslett's experi- ments on transverse strength, 548; and from Mr. Fincham's experiments, 549; calculated for tim- ber of large scantling, 549. Tensile strength of tin, 627. Tensile strength of wrought iron : Mr. Telford's experiments, 567; Mr. Barlow's experiments, 567; Sir Wm. Fairbairn's experiments on plates and bars, 567; influence of cold-rolling, 569; Mr. Thomas Lloyd's experiments, 569; successive fracture of the same bars, 569; bars of different lengths, 570; Mr. Edwin Clark on strength of plates and bars, 570. Experiments on the tensile strength and elonga- tion of wrought iron by Mr. Kirkaldy, 571 ; speci- men bars, 571; tensile strength and elongation of iron bars, 571; contraction of the sectional area of fracture, 572; strength of bars as affected by the diameter, 571 ; by rolling, by turning, by forg- ing, by reheating, by intense cold, 573, 576; by notching, by screwing, 574; by welding, by sud- den stress, 575; by hardening, by case-hardening, 576; by cold-rolling, 577; hammered iron, 576; strength as affected by additional hammering, 576; by removing the skin, 576. Tensile strength of angle-iron, ship-strap, and beam iron (Mr. Kirkaldy), 577. Tensile strength of iron plates (Mr. Kirkaldy), 577; fractured sectional area, 578; strength as affected by cold-rolling and by galvanizing, 578. Specific gravity of the irons tested by Mr. Kirkaldy, 578. ' Sir Joseph Whitworth's experiments on best irons, 615. Experiments by the Steel Committee, 580; speci- fic gravity of iron tested, 603; Swedish hammered bars, 581; Krupp and Yorkshire plates, entire, drilled, and punched, 583; Prussian plates, 586, Iron wire, 586, 628, 629, 676. Tensile strength and elongation of wrought iron, in- fluence of various treatment on, Mr. Kirkaldy's experiments, 573: iron bars, rolling down, turning down, forging, reheating, intense cold, 573; notch- ing, screwing, 574; welding, sudden stress, 575; frost, hardening, case-hardening, 576; cold-roll- ing* 577- Hammered iron: additional hammering, re- moving the skin, 576. Iron plates : cold-rolling, galvanizing, 578 ; annealing, 583-586; drilled holes, 584; punched holes, 584. Tensile strength of zinc, 627. Tensile stress, contraction of wrought iron under: bars, 572; notched bars, 574; plates, 578; wire, 587. Test-bars of cast iron, 564. Thermometers, 317; equivalent temperatures by Fahrenheit, Centigrade, and Reaumur scales, 318; tables of equivalent temperatures by Fahrenheit and Centigrade scales, 319, 323; air- or gas-ther- mometers, 325; Regnault's, 326; Mason's wet and dry bulbs, 394. Timber, strength of, 537. See Strength of Tim- ber, page 980. Timber columns, strength of, 644, 646, 647. Tin, tensile strength of, 627. Tin, weight of: tabulated weights, 219-221; multi- plier for the weight of tin bars, plates, &c., 220; special tables of the weight of tin plates and pipes, 252, 268, 269. See Weight of Iron and other Metals, page 983. Ton, 140; iron-ton, 141; equivalent weight in French measure, 155, 157; New York ton, 187; Canadian ton, 187. Tools, resistance of, 951; work of, 952; resistance of wood-cutting machines, 954; grindstones, 955. Toothed wheels, 727; speed, 727; pitch, 728; table of multipliers for number of teeth and diameter, 729; table of diameters of toothed wheels, 730; form of the teeth of wheels, 731; strength, 735; work- ing strength, 736; breadth, 737; horse-power, 737; weight of toothed wheels, 739. Torsional deflection, general investigation of, 536; round shaft, 536; square shaft, 537; hollow shaft, Torsional strength of shafts, general investigation of, 534; solid round shafts, 534; hollow round shafts, 535 ; square shafts, 535 ; torsional deflection of round and square shafts, 536. Torsional strength and deflection of cast iron, 565. Torsional strength and deflection of round shafting, 758, 759, 762. Torsional strength of steel bars, 595, 596, 600, 604 ; formulas, 619. Torsional shearing stress and deflection of steel bars, formula, 620. Torsional strength and deflection of wrought iron : Swedish bar, 582; formulas, 590. Torsional stress, definition of, 500. Traction on common roads, resistance to, 961. Tramways, street, resistance on, 966. Transverse deflection of homogeneous beams and girders: general investigation, 527; beams of rectangular section, 529; double-flanged or hollow rectangular beams, 530; table of relative deflec- tions of beams, variously proportioned and loaded, S3 2 - Transverse strength of homogeneous beams : gen- eral investigations, 503 ; symmetrical solid beams, 503; formulas for rectangular beams, 507; gener- alized formula for solid beams (without overhang) of symmetrical section, 509; flanged or hollow beams of symmetrical section, 510; flanged beams not symmetrical in section, 513. Transverse strength and deflection of cast iron : Mr. Barlow's experiments, 561; Mr. Edwin Clark's data, 562; Mr. Hodgkinson's data, 563; test-bars, 564. Transverse deflection and elastic strength, 564. Transverse strength of flanged beams : cast iron, 647; wrought iron, 653. Transverse deflection of shafts, 756. See Deflection, page 974. Transverse strength and deflection of steel : bars, S95 596, 599, 604; formulas, 617619; railway rails, 661, 665, 668. Transverse strength of steel springs, 671; formulas and rules for laminated springs, 671; for helical springs, 672. Transverse strength and deflection of railway rails, 661; rails of symmetrical section, or double-headed 982 INDEX. TRANSVERSE WEIGHTS rails, 661; general formula, 662; Mr. Price Wil- liams' data, 662; influence of constituent carbon on the strength, 664; Mr. J. T. Smith's data, 665. Rails of unsymmetrical section, 665; general formula and rule, 665; steel flange-rail by Mr. John Fowler, 666; Mr. Kirkaldy's test of Mr. Fowler's rail, 667; influence of holes in the flanges, 668; wrought-iron flange- rails, 668. Deflection of rails, 668; double-headed rails, 668; formulas, 669; flange rails, 669; formulas, 670. Transverse strength and deflection of timber : Ex- periments described by M. Morin, 537; Mr. Laslett's experiments, 538-540; Mr. Fincham's, 542, 543; Mr. Maclure's, 542, 543; Mr. Edwin Clark's, 543, 544; Mr. G. Graham Smith's, 543, 544; Mr. Baker's, 544; MM. Chevandier and Wertheim's, 545; Mr. Barlow's, 547. Formulas for the transverse strength and de- flection of timber of large scantling, 550-552. Transverse strength and deflection of wrought iron: hammered iron bars (Swedish) 582, 589; Mr. Barlow's data, 588; Mr. Edwin Clark's data, 588; formulas, 589; railway rails, 661, 665. Transverse deflection and elastic strength: data, 590; formulas, 590; railway rails, 668. Traversers, rope-gearing for working, 755. Trigonometry, plane, 21; tables, 103, no. Tubes, weight of, 248, 250, 266. Tubes, wrought-iron, strength of, 692, 695; large flue-tubes, 677. Tub water-wheel, 939. Turbines, 940; Fourneyron's, Boyden's, 940; rules for outward-flow turbines, 941; Fontaine's, Jon- val's, North Moor Foundry's, 942; vortex wheel, Swaine turbine, 943. u let's, 937. of heat, mechanical equivalents, 332. Undershot water-wheels: with radial floats, Ponce- let's Units V Vapours. See Gases and Vapours, page 975. Vapours and gases, mixture of, 392. See Gases and Vapours, page 975. Vapours, pressure of, at 212 F., 370; boiling-points at various pressures, 371. Vegetable substances, weight and specific gravity of, 212. Velocity, definition of, 277. Ventilation, 477. See Warming and Ventilation. Ventilators, or fans, 924. See A ir Machinery, page 971. Vortex turbine, 943. w Waggons in coal pits, resistance of, 956. Warming and ventilation, 477 ; ventilation, 477 ; ventilation of mines by heated columns of air, 479 ; cooling action of window glass, 480 ; heating rooms by hot- water, 481 ; heating rooms by steam, 486 ; heating by ordinary open fires, 488 ; heating by hot-air and stoves, 488. Warren-girder, strength of, 699. Warrington wire-gauge, 133. Washing small coal, 411. Waste-boards, flow of water over, 932. Water, as a standard for weight and measure, 124: notable temperatures, weight and volume, 124, 125. The gallon and other measures of water, rela- tive weights and volumes of water, 125. Pressure of a column of water, 126. Compressibility of water, 126. Sea-water, its volume, weight, and composi- tion, 126. Ice and snow, their volume and weight, 127. French and English measures of water, 127. Water, expansion of, with table, 338-341; specific heat of, 353, 354. Water, flow of, 929. See Flow of Water, page 974. Water, hygroscopic, in coals, 416. Water, machines for raising, 944; reciprocating pumps, 944; centrifugal pumps, 946; endless-chain pump, 947; noria, 947; water -works pumping engines, 947; hydraulic rams, 948. Water-mill: Barker's, Whitelaw's, 939. Water-pipes, cast-iron, dimensions, weight, and strength of, 934. Water-tube boiler, trials of, 771, 777. Water-wheels, 937 : Wheels on a horizontal axis, 937; undershot, 937; Poncelet's, 937; paddle water-wheel, breast- wheel, overshot-wheel, 938. Wheels on a vertical axis, 939; tub-wheels, Whitelaw's water-mill, 939; turbines, outward- flow, 940; downward-flow, 942; inward -flow, 943. Tangential wheels, Girard turbine, 943. Water-works pumping engines, 947. Wedge, 309; work done with it, 315. Wedgewood's pyrometer, 327. Weight and specific gravity, 198. Standard temperatures, rules for specific gravity, comparative weights of various solids, liquids, and gases, 198. Specific gravity of allo}^ of copper, alloys having a greater density than the mean, alloys having a less density than the mean, 200. See also 626, 627. Weight and specific gravity of solid bodies: metals, 202, 578, 626, 627 ; precious stones, 203 ; stones, 204 ; sundry mineral substances, 205 ; coals, 206 ; peat, 207 ; fuel in France, 207 ; woods, Indian woods, colonial woods, 208 ; wood-char- coal, 211 ; animal substances, vegetable sub- stances, 212. Weight and volume of various substances, by Tred- gpld, 213. Weight and volume of goods carried on the Bombay, Baroda, and Central Indian Railway, 213. Weight and specific gravity of liquids, 215. Weight and specific gravity of gases and vapours, 216. Weights and measures, 124. Water as a standard for weight and measure, 124 ; air as a standard, 127. Great Britain and Ireland, imperial weights and measures, 128. Measures of length, 129 ; land measure, nautical measure, cloth measure, 130. Wire -gauges, 130; Birmingham wire -gauge (Holtzapffel's), Birmingham metal-gauge or plate- gauge (Holtzapffel's) for sheet metals, brass, gold, silver, &c., 131 ; Lancashire gauge (Holtzapffel's) for round steel wire and for pinion wire, needle- gauge, music wire-gauge, 132 ; Warrington wire- gauge (Rylands Brothers), Birmingham wire- gauge for iron sheets chiefly (South Stafford- shire), 133; Sir Joseph Whitworth & Co. 's stand- ard wire-gauge ; imperial standard wire-gauge, 134- Inches, their equivalent decimal values in parts of a foot, fractional parts of an inch, and their decimal equivalents, 135; sixteenths and thirty- seconds, 136. Measures of surface, 136 ; superficial measure, 136 ; builders' measurement, land measure, 137 ; decimal parts of a square foot in square inches, 138- Measures of volume, 137; solid or cube measure, builders' measurement, 137. Measures of capacity, 138 ; liquid measure, 138; dry measure, standard bushel, coal measure, old wine and spirit measure, 139 ; old ale and beer measure, apothecaries' fluid measure, 140. INDEX. - WEIGHTS WIRE Measures of weight, 140 ; avoirdupois weight, 140 ; troy weight, diamond weight, apothecaries' weight, old apothecaries' weight, weight of current coins, 141 ; coal weight, sundry bushel measures, wool weight, hay and straw weight, corn and flour weight, 142. Miscellaneous tables, 143; drawing papers, commercial numbers and stationery, measures relating to building, 143; sundry commercial measures, measures for ships, 144. Compound units, comparison of, 144 ; measures of velocity, 144 ; volume and time, pressure and weight, weight and volume, power, 145. France, the metric standards of weights and measures, 146; countries in which they are adopted, 147. Measures of length, 147 ; old measures, 147 ; French wire-gauges, 148. Measures of surface, land, 149. Measures of volume, 149 ; cubic measure, wood measure, 149. Measures of capacity, 149 ; liquid measure, dry measure, 149. Measures of weight, 150. Equivalents of British imperial and French metric weights and measures, 150: length, 150; millimetres and inches, 151; inches and milli- ' metres, 152; surface, 153; cubic measure, 154; wood measure, 154; capacity, 154; weight, 155. Approximate equivalents of English and French measures, 156. Equivalents of French and English compound units of measurement, 157; weight, pressure, and measure, 157; volume, area, length, and work, 158 ; heat, speed, money, 159. German Empire, weights and measures, 160. Old weights and measures of the German States, 161 ; German fuss, 161 ; kingdom of Prussia, 162; Bavaria, 164; Wiirtemberg, 165; Saxony, 166; Baden, 167 ; Hanse Towns, Hamburg, 168 ; Bre- men, Lubec, 169 ; German Customs Union, 169. Austrian Empire, 170. Russia, 171. Holland, Belgium, 173. Norway, Denmark, Sweden, 173. Switzerland, 175. Spain, Portugal, Italy, Turkey, Greece and Ionian Islands, Malta, 176. Egypt, Morocco, Tunis, Arabia, Cape of Good Hope, 179. Indian Empire, Bengal, Madras, Bombay, Ceylon, 180. Burmah, China, Cochin-China, Persia, Japan, Java, 183. United States of America, 186. British North America, 187. Mexico, 187. Central America and West Indies, British West Indies, Cuba, Guatemala and Honduras, British Honduras, Costa Rica, St. Domingo, 187. South America: Colombia, Venezuela, Ecua- dor, Guiana, Brazil, Peru, Chili, Bolivia, Argen- tine Confederation, Uruguay, Paraguay, 188. Australasia, 189. Weight of iron and other metals, table of, 217 ; data for wrought iron, 217, 218 ; data for steel, 217 ; data for cast iron, 217, 218 ; notice of the tables, 218. Table, weight of given volumes of metals, 219. Table, volume of given weights of metals for given weights, 219. Table, weight of i square foot of metals, 220. Table, weight of metals of a given sectional area, per lineal foot and per lineal yard, 221. Rules for the weight of wrought iron, cast iron, and steel, 223. Rule for the length of one hundredweight of wire of different metals of a given thickness, 224. Table, weight of French galvanized iron wire, 225. Weight of wrought-iron bars, plates, &c., special tables, 226 ; multipliers for other metals, 226. Table, weight of flat bar iron, 227. Table, weight of square iron, 239. Table, weight of round iron, 240. Table, weight of angle-iron and tee-iron, 242. Table, weight of wrought-iron plates, 243. Table, weight of sheet iron, 244. Table, weight of black and galvanized iron sheets, 245. Table, weight of hoop iron, 246. Table, weight and strength of Warrington iron wire (Rylands Brothers), 247. Table, weight of wrought-iron tubes, by internal diameter, 248. Table, weight of wrought-iron tubes, by exter- nal diameter, 250. Weight of cast iron, steel, copper, brass, tin, lead, and zinc, special tables, 251. Table, weight of cast-iron cylinders, by internal diameter, 253. Table, weight of cast-iron cylinders, by external diameter, 255. Table, volume and weight of cast-iron balls, for given diameters, multipliers for other metals, 258. Table, diameter of cast-iron balls, for given weights, 258. Table, weight of flat bar steel, 259. Table, weight of square steel, 260. Table, weight of round steel, 260. Table, weight of chisel steel: hexagonal, octagonal, and oval-flat, 261. Table, weight of one square foot of sheet copper, 261. Table, weight of copper pipes and cylinders, by internal diameter, 262. Table, weight of brass tubes, by external dia- meter, 266. Table, weight of one square foot of sheet brass, 268. Table, size and weight of tin plates, 268. Table, weight of tin pipes, 269. Table, weight of lead pipes, 269. Table, dimensions and weight of sheet zinc, 270. Weight of belt pulleys, 750. Weight of chains, 678, 679. Weight and bulk of coal: British, 206, 414, 416; American, 418, 419; French, 422. Weight and bulk of coke, 206, 432; of lignite, 207; of wood, 442. Weight of ropes: hemp, 674, 675; iron, 674, 675, 677; steel, 674, 675, 677. Weight of round wrought-iron shafting: net and gross, 761. Weight of toothed wheels: spur, 739; mortise, 741, bevel and mitre, 741. Weight of pure water, 124. Of sea-water, 126. Of ice and snow, 127. Weirs, flow of water over, 932. Welded joints in iron plates, tensile strength of, 634, 635- Wheel and axle, 305; work done with it, 314. Wheels, toothed, 727. See Toothed Wheels, p. 981. Whitelaw's water-mill, 939. Whit worth's standard wire-gauge, 134; bolts and nuts, 682; screwed piping, 683. Williams' system of smoke prevention, 785. Wilson's pyrometer, 327. Wire, weight of: rule for the length of one cwt. of wire of various metals of a given thickness, 224; INDEX. WIRE ZINC table of the weight of galvanized iron wire (French), 225; table of the weight and strength of Warrington iron wire, 247. Wire, tensile strength of: iron, 247, 586, 628, 629, 676; steel, 617, 629; copper, 628, 629; brass, 627, 629; phosphor-bronze, 629; gold, 628; silver, 628; platinum, 628; palladium, 628. Wire -gauges, English, 130; French, 148. See Weights and Measures, page 982. Wire-ropes, iron, strength and weight of, 674-677. Wire-ropes, steel, strength and weight of, 674, 675, 677. Wire-ropes for transmission of power, by M. Him. Wire-drawing, influence of, on the density of iron wire, 247. Woods, weight and specific gravity of, 208; Indian woods, 209; colonial, 209. Wood, 439 : classification, 439 ; constituent mois- ture, 439; composition, 440, 441 ; weight and bulk, 442; quantity of air consumed in combustion, 443; gaseous products of combustion, 443 ; heat of com- bustion, 444; temperature of combustion, 444; dis- tillation of, 449. Wood measure, French, 149. Wood, strength of. See Strength of Timber, page /ood-charcoal, weight and specific gravity of, 211. Wood-charcoal: wood for making it, 443; process of carbonization, 444, 448; yield of charcoal, 445, 447, 448 ; composition, 446, 447 ; charbon de Paris, 449; weight and bulk of charcoal, 450; moisture, 451; air consumed in combustion, 452; gaseous products of combustion, 452; heat of combustion, 452. Woollen mills, machinery of, resistance of, 959; horse-power required, 960. Work, definition of, 312; accumulated work in solid bodies, 315. See Mechanical Principles, page Work, or labour, 718; units of work, 718; labour of men, 718; labour of horses, 720; work of animals in carrying loads, 720; work absorbed by friction, Work of animals in carrying loads, 720. Work of tools, in metal, 952. Work that may be done for one turn of a shaft 760, 762; absorbed by friction, 763. Work of resistance of material, 501. Work of dry air or other gas, compressed or ex- panded, 898: general formulas, 898; work of compression of air at constant temperature, isothermally, without clearance, 899; with clear- ance, 900; work of expansion of air .at constant temperature, 900. Work of dry air in a non-conducting cylinder, adiabatically, 901; adiabatic compression of a gas, 901 ; table of compression or expansion of air with- out receiving or giving out heat, 902; work ex- pended in compressing dry gas, 903, 904; adia- batic expansion of gases, 904-909; table of corre- sponding ratios of pressures and temperatures, when air is admitted for the whole stroke, 908; table of comparative final temperatures and effici- encies of air expanded adiabatically and air ad- mitted for the whole of the stroke, 908. Efficiency of compressed-air engines, 909; com- pression and expansion of moist air, 912; work in expansion, 913; temperature in expansion, 914. Wrought iron, strength of, 567. See Strength of Wrought Iron, page 980. Wrought-iron columns, strength of, 644, 645. Wrought-iron flanged beams, transverse strength of, 653. Solid wrought-iron joists: tables of dimensions, weight, and strength, 653, 654; experiments by Mr. Kirkaldy, 655; rules and formulas, 656; elastic strength and deflection, formulas and rules, 657. Rivetted wrought-iron joists, 657; Mr. Davies' experiments, 658; rules, 659, 660. Wrought iron, resistance of, to explosive force, 622. Yard, imperial standard, 128. Zinc, weight of: tabulated weights, 219-221; multi- plier for the weight of zinc bars, plates, &c., 220; special table of the size and weight of sheet zinc, 270. See Weight of Iron and other Metals, page 983. Zinc, tensile strength of, 627. THE END. GLASGOW : W. G. BIACKIB AND CO., PRINTERS, VILLAFIEtD. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. 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