'.' n v ; '.. niumitg of LOWELL HYDRAULIC EXPERIMENTS. BEING A SELECTION FROM EXPERIMENTS ON HYDRAULIC MOTORS, ON THE FLOW OF WATER OVER WEIRS, IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION, AND THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. I MADE AT LOWELL, MASSACHUSETTS. BY JAMES B. FRANCIS, t CIVIL ENGINEER, MEMBER OP THE AMERICAN SOCIETY OP CIVIL ENGINEERS AND ARCHITECTS, FELLOW OF THE AMKRIOAN ACADEMY OP ARTS AND SCIENCES, MEMBER OF THE AMERICAN PHILOSOPHICAL SOCIETY, BTO. SECOND EDITION, KEVISED AND ENLARGED, WITH MANY NEW EXPERIMENTS, 3Lnb Illustrate!) WITH TWENTY-THREE COPPEP V -PLATE ENGRAVINGS. NEW YORK: D. VAN NOSTRAND, 192 BROADWAY. LONDON: TRUBNER& CO. 1868. Entered according to Act of Congress, in the year 1868, by JAMES B. FRANCIS, in the Clerk'R Office of the District Court of the District of Massachusetts. PREFACE TO THE SECOND EDITION. SINCE the first edition of this work appeared, in 1855, the manufacturing corpora- tions at Lowell, lessees of the water-power furnished by the Merrimack River at that point, have surrendered their leases and taken others containing new provisions for the purpose of more fully protecting all parties in the enjoyment of their respective rights ; this has rendered necessary a new and elaborate series of experiments for the purpose of perfecting the method of gauging the flow of water in open channels by the use of loaded tubes. Some experiments had been made on this subject at Lowell before the publication of the first edition, the principal results of which were given ; the later ex- periments are, however, so much more complete, and have been made under circum- stances so much more favorable, that it has been found necessary to rewrite, entirely, the chapter on that subject. The general use at Lowell of the Dlffuser, an apparatus for utilizing the power usually lost in turbines, from the water leaving them with a considerable velocity, has created much interest in Venturi's tube, the action in which involves the same principles as the Diffuser. Experiments on Venturi's tube had been previously made only when discharging into the air ; it appeared highly probable that greater results might be ob- tained if the tube was submerged, so as to discharge under water. Experiments made under these circumstances, and detailed at length in this edition, indicate a considerably greater flow than had been previously obtained. The author takes this opportunity of acknowledging his obligations to Mr. Uriah A. Boyden of Boston, for useful suggestions during the last twenty-five years, on almost every subject discussed in this volume. Also to Mr. John Newell, now of Detroit, Michigan, to whom he is much indebted for assistance in the execution and reduction of some of the most important series of experiments, and to whose fidelity the precision attained in the results is in no small degree due. Also to Mr. Joseph P. Frizell, now of Davenport, Iowa, to whom he is indebted for assistance in some points involving the higher mathematics. LOWKLI,, MASS., March, 18G8. TABLE OF CONTENTS. INTRODUCTION. PART I. EXPERIMENTS ON HYDRAULIC MOTORS. Number of the Article. EXPERIMENTS UPON THE TKEMONT TURBINE, 1 1-17. Introduction, 1 1835. Description of the Turbine, ............7 3647. Description of the Apparatus used in the Experiments, 14 4853. Mode of conducting the Experiments, . . . . . . . . . .19 54-74. Description of Table II., containing the Experiments upon the Turbine at the Tremont Mills, 25 75-82. Description of the Diagram representing the Experiments, . . . . . . 36 83-88. Path described by a Particle of Water in passing through the Wheel, . . . .39 89-98. RULES FOR PROPORTIONING TURBINES, 44 99-109. EXPERIMENTS ON A MODEL OF A CENTRE-VENT WATER-WHEEL, WITH STRAIGHT BUCKETS, 55 110-119. EXPERIMENTS ON THE POWER OF A CENTRE- VENT WATER-WHEEL, AT THE BOOTT COTTON-MILLS, 61 PART II. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, AND IN SHORT RECTANGULAR CANALS. i EXPERIMENTS ON THE FLOW OF WATKR OVER WEIRS, 71 120-125. Introduction, .71 126-135. Experiments made at the Tremont Turbine, on the Flow of Water over Weirs, . 76 VI CONTENTS. 136. Experiments on the Flow of Water over Weirs, made at the Centre-Vent Wheel for moving the Guard Gates of the Northern Canal, ....... 96 137. Experiments on the Effect produced on the Flow of Water over Weirs, by the Height of the Water on the Downstream Side, ......... 99 138-147. Experiments on the Flow of Water over Weirs, made at the Lower Locks, Lowell, . 103 148-159. Description of Table XIII., containing the Details of the Experiments on the Flow of Water over Weirs made at the Lower Locks, Lowell, in October and November, 1852, 112 160163. Comparison of the proposed Formula with the Results obtained by previous Experimenters, 126 164. Precautions to be observed in the application of the proposed Formula, . . . 133 165166. Experiments on the Discharge of Water over a Dam, of the same Section as that erected by the Essex Company, across the Merrimack River at Lawrence, Massachusetts, . 136 167-175. Experiments to ascertain the Effect of taking the Depths upon a Weir, by means of Pipes opening near the Bottom of the Canal, ........ 137 176. Formula for the Discharge over Weirs in which the crest is not horizontal. Formula for the Discharge over Weirs for any Latitude or Height above the Sea . . 143 A METHOD OF GAUGING THE FLOW OF WATER IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION, AND OF SHORT LENGTH. 177-179. Arrangements at Lowell for the Distribution of the Water-Power among the several Lessees 146 180. Method of Gauging the Water drawn at one of the Cotton Mills of the Hamilton Manu- facturing Company in 1830 ........... 148 181. Experiments of Messrs. Baldwin, Whistler, and Storrow in 1841 and 1842 . . 148 182-198. Method of Gauging the Flow of Water in Open Canals by means of Loaded Poles or Tubes 155 199-225. Experiments made to determine a Formula of Correction for Gaugings in Open Canals, by means of Loaded Poles or Tubes . . . . . . . . 169 226-238. Formula of Correction for Gaugings made with Loaded Poles or Tubes . . . 191 239-246. Application of the Method of Gauging Streams of Water by means of Loaded Poles or Tubes 201 EXPERIMENTS ON THE FLOW OF WATER THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 247-250. Former Experiments on this Subject 209 251-254. Description of the Apparatus used in the New Experiments . . . . . 212 255-257. Mode of conducting the Experiments ......... 216 258-260. Description of the Experiments 222 261-269. Deductions from the Experiments . . . 223 270. Description of a Turbine Water- Wheel of 700 Horse Power 230 \ - CONTENTS. vii TABLES. PART I. Number of p the Table. I. Weight of a Cubic Foot of pure Water, at different Temperatures, ..... 29 II. Experiments upon the Turbine at the Tremont Mills, in Lowell, Massachusetts, . . .32 III. Successive Steps in the Calculation for the Path of the Water in Experiment 30 on the Tremont Turbine, .............. 41 IV. Table for Turbines of different Diameters, operating on different Falls, . . . . .53 V. Experiments on a Model of a Centre- Vent Water- Wheel, 58 VI. Experiments on the Boott Centre-Vent Water- Wheel, \. \ 66 VII. Successive Steps in the Calculation for the Path of the Water in Experiment 30, on the Boott Centre-Vent Water- Wheel, 70 PART II. VIII. Experiments made at the Tremont Turbine, for the purpose of testing the Method of reduc- ing the Depths on the Weir to a uniform Fall, 83 IX. Experiments made at the Tremont Turbine, which were repeated under identical circumstances, 84 X. Experiments on the Flow of Water over Weirs, made at the Tremont Turbine, . . 88 XI. Experiments on the Flow of Water over Weirs, made at the Centre- Vent Wheel for mov- ing the Guard Gates of the Northern Canal, at Lowell, Massachusetts, . . . .98 XII. Experiments on the Effect produced on the Flow of Water over Weirs, by the Height of the Water on the Downstream Side, 102 XIII. Experiments on the Flow of Water over Weirs, made at the Lower Locks, Lowell, in Octo- ber and November, 1852, 122 XIV. Comparison of the proposed Formula with the Experiments of Poncelet and Lesbros, . 128 XV. Comparison of the proposed Formula with the Experiments of Castel, 130 XVI. Experiments on the Discharge of Water over a Dam of the same Section as that erected by the Essex Company, across the Merrimack River at Lawrence, Massachusetts, . . 137 XVII. Experiments made at the Lower Locks, to determine the Corrections to be applied to the Readings of the Hook Gauges, . 140 XVIH. Experiments to ascertain the Effect of taking the Depths upon a Weir, by means of Pipes opening near the Bottom of the Canal, .......... 142 A. B. C. Experiments of Messrs. Baldwin, Whistler, and Storrow, made for the Purpose of finding the Ratio between the Mean and Surface Velocities in certain Open Canals . . .152 XIX. Data and Computed Results of four Experiments with Loaded Tubes . . ... 168 XX. Observations in Experiment No. 1, Table XXII 176 Vlll CONTENTS. XXI. Comparison of the Height of the Tops of the Weirs with the Point of the Stationary Hook 179 XXII. Experiments from which the Formula of Correction for Flume Measurements is determined 186 XXIII. Mean Results of the Experiments in Table XXII. arranged according to Velocities . .192 XXIV. Mean Results of the Experiments in Table XXII. arranged according to Lengths of Tubes 195 XXV. Miscellaneous Experiments at the Tremont Measuring Flume ..... 198 XXVI. Gauge of the Quantity of Water passing the Boott Measuring Flume, May 17, 1860 . . 204 XXVII. Experiments on the Flow of Water through Submerged Tubes and Orifices . . 218 XXVIII. Velocities of Floats in Measuring Flumes 233 XXIX. Corrections for Flume Measurements ......... 241 XXX. Velocities due Heads for every 0.01 Foot up to 49.99 Feet 242 INTRODUCTION. THE northern regions of the United States of North America, probably possess a greater amount of water-power than any other part of the world of equal extent, and the active and inventive genius of the American people, combined with the very high price of labor, has had a powerful influence in bringing this power into use. Nevertheless, the water-power is so vast, compared with the pop- ulation, that only a small portion of it has, up to this time, been applied to the purposes of man. It was estimated, not long since, that the total useful effect derived from water-power in France, was about 20,000 horse-power. An amount of power far exceeding this, is already derived from the Merrimack Eiver and its branches, in Massachusetts and New Hampshire. What must be the amount of the population and wealth of the Northern States, when the other rivers that water them are equally improved? One of the earliest and most successful efforts to bring into use, in a sys- tematic manner, one of the larger water-powers, was made at Lowell in Massa- chusetts ; where, in 1821, a number of farms situated near Pawtucket Falls on the Merrimack River, were purchased by several capitalists of Boston, who obtained a charter from the State of Massachusetts under the name of The Merrimack Manufacturing Company. In 1826, the property was transferred to the Proprietors of ike Locks and Canals on Merrimack River, a corporation chartered in 1792 for the purpose of improving the navigation of the Merrimack River. Previously to the transfer, the Merrimack Manufacturing Company had erected a dam of about 950 feet in length, at the head of Pawtucket Falls, and had also enlarged the Pawtucket Canal, which was originally constructed, previously to the year 1800, by the Proprietors of the Locks and Canals on Merrimack River, for the purposes x INTRODUCTION. of navigation. Subsequently to the enlargement, however, this canal has been used both for purposes of navigation, and to supply water to the wheels of numerous manufacturing establishments. The dam at the head of Pawtucket Falls, in the ordinary state of the river, deadens the current of the river for about 18 milee), forming, in low water, a reservoir of about 1120 acres; this extensive reservoir is of great value in very low stages of the river, as it affords space for the accumulation of the flow of the river during the night, when the manufactories are not in operation. This accumulation is subsequently drawn off, together with the natural flow of the river, during the usual working hours. The total fall of the Merrimack River at Pawtucket Falls, in ordinary low water, is about 35 feet, of which about 2 feet is lost in consequence of the descent in the canals, leaving a net fall of about 33 feet. About of the water is used on the entire fall, and the remainder is used twice over, on falls of about 14 and 19 feet respectively. The water-power has been granted by the Propri- etors of the Locks and Canals on Merrimack Elver, in definite quantities called Mill Pmvers, which are equivalent to a gross power of a little less than 100 horse-power each. Grants have been made to eleven manufacturing companies, who have an aggregate capital, somewhat exceeding thirteen millions of dollars. ' Thus, to the Merrimack Manufacturing Company, there have been granted 24-f mill powers, each of which consists of the right to draw, for 15 hours per day, 25 cubic feet of water per second on the entire fall. Up to this time, there have been granted at Lowell 139-ij-J- mill powers, or a total quantity of water equal to 3595.933 cubic feet per second. A large portion of this water is used on turbines of a very superior description, and nearly all the remainder, on breast wheels of good construction, a. portion of which, however, do not use quite the whole of the fall on which they are placed. We may, however, assume that, upon an average, a useful effect is derived, equal to f of the total power of the water expended. Calling the fall 33 feet, and the weight of a cubic foot of water 62.33 pounds, we shall have for the effective power derived from the water-power granted by the Proprietors of the Locks and Canals on Merrimack River at Lowell, 3595.933 X 62.33 X 33 X 3 550" == 8965.4 horse-power. INTRODUCTION. xi s In consequence of the success attending the improvement of the water-power at Lowell, several other extensive water-powers in New England have been brought into use in a similar manner. Some of these undertakings have been quite suc- cessful, whilst with others, as yet only partially developed, the success has not been so decided. The great abundance of water-power in this country has had a strong ten- dency to encourage its extravagant use ; the machines used in the manufactories are usually great consumers of power ; the ability of a machine to turn off the greatest quantity of work with the least manual labor, and in the least time, has been the point mainly considered; and whether it required a greater or less amount of power, has been a secondary consideration. The engineering operations connected with the water-power at Lowell, have frequently demanded more definite information on certain points in hydraulics, than was to be found in any of the publications relating to that science ; and hence has arisen the necessity, from time to time, of making special experiments to supply the required information. Whenever such emergencies have arisen, the officers who have the general care of the interests of the several corporations, with a liberality founded on enlarged views of the true interests of the bodies they represent, have always been willing to defray such expenses as were neces- sary, in order that the experiments might be made in a satisfactory manner. The experiments recorded in the following pages, are a selection from those made by the author, in the discharge of his duty, as the Engineer of the Cor- porations at Lowell. They may be divided into two classes, namely, first, those on hydraulic motors, and, second, those on the flow of water over weirs, and in short rectangular canals. Combined with the description of the experiments, there are also given some other investigations, which may appear somewhat out of place, but which, from their utility or novelty, will be found interesting to many persons who have cultivated the science of hydraulics. The unit of length adopted in this work, is the English foot according to a brass standard measure made by Gary of London, now in the possession of the Lowell Machine Shop. HYDRAULIC EXPERIMENTS, PART I. EXPERIMENTS ON HYDRAULIC MOTORS. EXPERIMENTS UPON THE TREMONT TURBINE. 1. UNTIL within a few years, the water-wheels in use in the principal manufac- turing establishments in New England, were what are there generally called breast wheels, sometimes known also by the name of pitch back ivheels. They are the same in principle as the overshot-ivheel, the useful effect being produced, almost entirely, by the simple weight of the water in the buckets, and differing only from the ovcrsJiot-ivheel in this, that the water is not carried entirely over the top of the wheel, but is let into the buckets near the top, but on the opposite side from that adopted for the overshot-wheel. An apron, fitting as closely as practicable to the wheel, is used to prevent the water leaving the buckets, until it reaches very nearly the bottom of the wheel. In Lowell, these wheels have been constructed principally of wood, many of them of very large dimensions. Those in the mills of the Merrimack Manufacturing Company, for instance, are thirty feet in diameter, with buckets twelve feet long. Four of the mills belonging to this company, have two such wheels in each of them. Until the year 1844, the breast wheel, as above described, was considered here the most perfect wheel that could be used. Much prejudice existed here, as elsewhere, against the reaction wheels ; a great number of which had, however, been used throughout the country, in the smaller mills, and with great advantage ; for, although they usually gave a very small effect in proportion to the quantity of water expended, their cheapness, the small space required for them, their greater velocity, being less 1 2 EXPERIMENTS UPON THE TREMONT TURBINE. impeded by backwater, and not requiring expensive wheelpits of masonry, were very important considerations ; and in a country where water power is so much more abundant than capital, the economy of money was generally of greater importance than the saving of water. A vast amount of ingenuity has been expended by intelligent millwrights, on these wheels ; and it was said, several years since, that not less than three hundred patents relating to them, had been granted by the United States Government. They continue, perhaps as much as ever, to be the subject of almost innumerable modifica- tions. Within a few years, there has been a manifest improvement in them, and there are now several varieties in use, in which the wheels themselves are of simple forms, and of single pieces of cast-iron, giving a useful effect approaching sixty per cent, of the power expended. 2. The attention of American engineers was directed to the improved reaction water-wheels in use in France and other countries in Europe, by several articles in the Journal of the Franklin Institute ; and in the year 1843, there appeared in that journal, from the pen of Mr. Ellwood Morris, an eminent engineer of Pennsylvania, a translation of a French work, entitled, Experiments on water-ivlieels having a vertical axis, catted turbines, by Arthur Morin, Captain of Artillery, etc. etc. In the same journal, Mr. Morris also published an account of a series of experiments, by himself, on two turbines constructed from his own designs, and then operating in the neighborhood of Phila- delphia. The experiments on one of these wheels, indicate a useful effect of seventy-five per cent, of the power expended, a result as good as that claimed for the practical effect of the best overshot-wheels, which had, heretofore, in this country, been considered unapproachable, in their economical use of water. 3. In the year 1844, Uriah A. Boyden, Esq., an eminent hydraulic engineer of Massachusetts, designed a turbine of about seventy-five horse-power, for the Pick- ing House of the Appleton Company's cotton-mills, at Lowell, in Massachusetts, in which wheel, Mr. Boyden introduced several improvements, of great value. The performance of the Appleton Company's turbine, was carefully ascertained by Mr. Boyden, and its effective power, exclusive of that required to carry the wheel itself, a pair of bevel gears, and the horizontal shaft carrying the friction pulley of a Prony dynamometer, was found to be seventy-eight per cent, of the power expended. 4. In the year 1846, Mr. Boyden superintended the construction of three tur- bines of about one hundred and ninety horse-power each, for the same company. By the terms of the contract, Mr. Boyden's compensation depended upon the perform- ance of the turbines, and it was stipulated that two of them should be tested. The contract also contained the following clause, "and if the mean power derived from EXPERIMENTS UPON THE TREMONT TURBINE. 3 these turbines be seventy-eight per cent, of the power of water expended, the Apple- ton Company to pay me twelve hundred dollars for my services, and patent rights for the apparatus for these mills; and if the power derived be greater than seventy- eight per cent., the Appleton Company to pay me, in addition to the twelve hundred dollars, at the rate of four hundred dollars for every one per cent, of power, obtained above seventy-eight per cent." In accordance with the contract, two of the turbines were tested, a very perfect apparatus being designed by Mr. Boyden for the purpose, consisting, essentially, of a Prony dynamometer to measure the useful effects, and a weir to gauge the quantity of water expended. 5. A great improvement in the mode of conducting hydraulic experiments was here adopted, in making each set of observations continuous, the time of each obser- vation being noted ; thus, the observer who noted the height of the water above the wheel, recorded regularly, say every thirty seconds, the time and the height ; and so with the other observers, the recorded times furnishing the means of afterwards identi- fying simultaneous observations. 6. The observations were put into the hands of the author, for computation, who found that the mean maximum effective power of the two turbines tested, was eighty-eight per cent, of the power of the water expended. According to the terms of the contract, this made the compensation for engineering services, and patent rights for these three wheels, amount to fifty-two hundred dollars, which sum was paid by the Appleton Company without objection. 7. These turbines have now been in operation about eight years, and their per- formance has been, in every respect, entirely satisfactory. The iron-work for these wheels was constructed by Messrs. Gay and Silver, at their machine shop at North Chelmsford, near Lowell ; the workmanship was of the finest description, and of a deli- cacy and accuracy altogether unprecedented in constructions of this class. 8. These wheels, of course, contained Mr. Boyden's latest improvements, and it was evidently for his pecuniary interest that the wheels should be as perfect as possible, without much regard to cost. The principal points in which one of them differs from the constructions of Fourneyron, are as follows. 9. The wooden flume, conducting the water immediately to the turbine, is in the form of an inverted truncated cone, the ivater being introduced into the upper part of the cone, on one side of the axis of the cone (which coincides with the axis of the turbine) in such a manner, that the water, as it descends in the cone, Jias a gradually increasing velocity, and a spiral motion ; the horizontal component of the spiral motion being in the direction of the motion of the ^vheel. This horizontal motion is derived from the necessary velocity with which the water enters the truncated cone ; and the arrangement is such that, if perfectly propor- tioned, there would be no loss of power between the nearly still water in the principal 4 EXPERIMENTS UPON THE TREMONT TURBINE. penstock and the guides or leading curves near the wheel, except from the friction of the water against the walls of the passages. It is not to be supposed that the construction is so perfect as to avoid all loss, except from friction ; but there is, without doubt, a distinct advantage in this arrangement over that which had been usually adopted, and where no attempt had been made to avoid sudden changes of direction and velocity. 10. The guides, or leading curves, are not perpendicular, lid a little inclined lachvards from the direction of the motion of the ivheel, so that the water, descending with a spiral motion, meets only the edges of the guides. This leaning of the guides has also another valuable effect; when the regulating gate is raised only. a small part of the height of the wheel, the guides do not completely fulfil their office of directing the water, the water entering the wheel more nearly in the direction of the radius, than when the gate is fully raised ; by leaning the guides, it will be seen that the ends of the guides, near the wheel, are inclined, the bottom part standing further forward, and operating more efficiently in directing the water, when the gate is partially raised, than if the guides were perpen- dicular. 11. In Fourneyron's constructions, a garniture is attached to the regulating gate, and moves with it, for the purpose of diminishing the contraction ; this, considered apart from the mechanical difficulties, is probably the best arrangement ; to be perfect, however, theoretically, this garniture should be of different forms for different heights of gate ; but this is evidently impracticable. In the Appleton Turbine, the garniture is attached to the guides, the gate (at least the lower part of it] being a simple thin cylinder. By this arrangement, the gate meets with much less obstruction to its motion than in the old arrangement, unless the parts are so loosely fitted as to be objectionable ; and it is believed that the coefficient of effect, for a partial gate, is proportionally as good as under the old arrangement. 12. On the outside of the wheel is fitted an apparatus named, ly Mr. Boydcn, the Diffuser. The object of this extremely interesting invention, is to render useful a part of the power otherwise entirely lost, in consequence of the water leaving the wheel with a considerable velocity. It consists, essentially, of two stationary rings or discs, placed concentrically with the wheel, having an interior diameter a very little larger than the exterior diameter of the wheel ; and an exterior diameter equal to about twice that of the wheel ; the height between the discs, at their interior circumference, is a very little greater than that of the orifices in the exterior circumference of the wheel, and at the exterior circumference of the discs, the height between them is about twice as great as at the interior circumference ; the form of the surfaces connecting the interior and exterior circumferences of the discs, is gently rounded, the first elements of the curves, near the interior circumferences, being nearly horizontal. There is con- EXPERIMENTS UPON THE TREMONT TURBINE. 5 sequently, included between the two surfaces, an aperture gradually enlarging from the exterior circumference of the wheel, to the exterior circumference of the diffuser. When the regulating gate is raised to its full height, the section, through which the water passes, will be increased by insensible degrees, in the proportion of one to four, and if the velocity is uniform in all parts of the diffuser at the same distance from the wheel, the velocity of the water will be diminished in the same proportion ; or its velocity on leaving the diffuser, will be one fourth of that at its entrance. By the doctrine of living forces, the power of the water in passing through the diffuser must, therefore, be diminished to one sixteenth of the power at its entrance. It is essential to the proper action of the diffuser, that it should be entirely under water ; and the power rendered useful by it, is expended in diminishing the pressure against the water issuing from the exterior orifices of the wheel; and the effect produced, is the same as if the available fall under which the turbine is acting, is increased a certain amount. It appears probable that a diffuser of different proportions from those above indicated, would operate with some advantage without being submerged. It is nearly always inconvenient to place the wheel entirely below low-water-mark ; up to this time, however, all that have been fitted up with a diffuser, have been so placed ; and, indeed, to obtain the full effect of a fall of water, it appears essential, even when a diffuser is not used, that the wheel should be placed below the lowest level to which the water falls in the wheelpit, when the wheel is in operation. The action of the diffuser depends upon similar principles to that of diverging conical tubes, which, when of certain proportions, it is well known, increase the discharge ; the author has not met with any experiments on tubes of this form, discharging under water, although, there is good reason to believe, that tubes of greater, length and divergency would operate more effectively under water, than when discharg- ing freely in the air ; and that results might be obtained, that are now deemed impossible by most engineers. Experiments on the same turbine, with and without a diffuser, show a gain in the coefficient of effect, due to the latter, of about three per cent. By the principles of living forces, and assuming that the motion of the water is free from irregularity, the gain should be about five per cent. The difference is due, in part at least, to the unstable equilibrium of water, flowing through expanding apertures ; this must interfere with the uniformity of the velocities of the fluid streams, at equal distances from the wheel. 13. Suspending the wheel from the top of the vertical shaft, instead of running it on a step at the bottom. This had been previously attempted, but not with such success as to warrant its general adoption. It has been accomplished with complete success by Mr. Boyden, whose mode is, to cut the upper part of the shaft into a series of necks, G EXPERIMENTS UPON THE TREMONT TURBINE. and to rest the projecting parts upon corresponding parts of a box. A proper fit is secured by lining the box, which is of cast-iron, with babbitt metal, a soft metallic composition consisting, principally, of tin ; the cas1>iron box is made with suitable pro- jections and recesses to support and retain the soft metal, which is melted and poured into it, the shaft being at the same time in its proper position in the box. It will readily be seen that a great amount of bearing surface can be easily obtained by this mode, and also, what is of equal importance, it may be near the axis ; the lining metal, being soft, yields a little if any part of the bearing should receive a great excess of weight. The cast-iron box is suspended on gimbals, similar to those usually adopted for mariners' compasses and chronometers, which arrangement permits the box to oscillate freely in all directions, horizontally, and prevents, in a great measure, all danger of breaking the shaft at the necks, in consequence of imperfections in the workmanship, or in the adjustments. Several years' experience has shown, that this arrangement, carefully constructed, is all that can be desired ; and that a bearing thus constructed, is as durable, and can be as readily oiled, and taken care of, as any of the ordinary bearings in a manufactory. 14. The buckets are secured to the crowns of the wheel in a novel, and much more perfect manner, than had been previously used ; the crowns are first turned to the required form, and made smooth ; by ingenious machinery devised for the purpose, grooves are cut with great accuracy in the crowns, of the exact curvature of the buckets ; mortices are cut through the crowns, in several places in each groove ; the buckets, or floats, are made with corresponding tenons, which project through the crowns, and are riveted on the bottom of the lower crown, and on the top of the upper crown ; this construction gives the requisite strength and firmness, with buckets of much thinner iron than was necessary under any of the old arrangements ; it also leaves the passages through the wheel entirely free from injurious obstructions. 15. Mr. Boy den has also designed a large number of turbines for different man- ufacturing establishments in New England, many of them under contracts similar to that with the Appleton Company, and has accumulated a vast number of valuable ex- periments and observations upon them, which, it is to be hoped, he will find time to prepare for publication ; as such opportunities but rarely occur to engineers so able to profit by them. 16. In the year 1849, the Manufacturing Companies at Lowell purchased of Mr. Boyden, the right to use all his improvements relating to turbines and other hydraulic motors. Since that time it has devolved upon the author, as the chief engineer of these companies, to design and superintend the construction of such turbines as might be wanted for their manufactories, and to aid him in this important undertaking, Mr. Boyden has communicated to him copies of many of his designs for turbines, together EXPERIMENTS UPON THE TREMONT TURBINE. 7 with the results of experiments upon a portion of them ; he has communicated, how- ever, but little theoretical information, and the author has been guided, principally, by a comparison of the most successful designs, and such light as he could obtain from writers on this intricate subject. 17. The first designs, prepared by the author, after the arrangement with Mr. Boyden was entered into, were for four turbines of essentially the same dimensions; namely, two for the Suffolk Manufacturing Company, and two for the Tremont Mills, for the purpose of furnishing power for the cotton-mills of these companies at Lowell. These turbines 'were constructed at the Lowell Machine Shop, and were completed in January, 1851. For the purpose, principally, of estimating the success of these turbines, one of them was fitted up with a complete apparatus for measuring its power, and gauging the quantity of water discharged ; the gauging apparatus was afterwards used to make the experiments on the discharge of water over weirs of different proportions, for the purpose of determining, practically, some of the relations required to be known, in order to compute the flow of water through such apertures. DESCRIPTION OF THE TUUBINE. 18. The water is conducted from the principal feeder to the mills at Lowell, called the Northern Canal, by an arched canal, or penstock, about ninety feet in length. The forebay, inside the wheel-house, is constructed of masonry, and has a general width of twenty feet, and a depth of water of fourteen feet ; the channels through which the water passes, are so capacious, that the loss of fall in passing from the Northern Canal to the forebay, is scarcely sensible. During the experiments, however, the head of the penstock was partially closed by gates, so that there was a sensible fall at that time. The entrance of the arched canal is protected by a coarse rack, or grating, for the purpose of preventing large floating substances from entering the forebay ; each turbine is also separately guarded by a fine rack, placed in the forebay, which prevents the entrance into the turbine of all floating substances that might be injurious. Both racks are made of large extent, to avoid sensible loss of head to the water in passing through them. The extreme rigor of the New England winter renders it necessary to afford to water-wheels of all descriptions, complete protection from the cold. The result is, thai less interraption from frost is experienced, than in many milder climates. The wheel- house, in which these turbines are placed, is a substantial brick building, well warmed in the winter by steam. g EXPERIMENTS UPON THE TREMONT TURBINE. After passing the turbines, the water is conducted by an arched canal, or raceway, about nine hundred feet in length, to the lower level of the Western Canal, which serves as a feeder to the Mills of the Lawrence Manufacturing Company. 19. Plate I. is a vertical section through the centre of the turbine, and the axis" of the supply pipe. Plate II. is a plan of the turbine, and wheelpit. Plate III., Figure 1, is a plan of nearly one fourth part of the disc and wheel. Figure 2 is a plan of the whole wheel, the guides, and garniture. Figure 3 is a ver- tical section through both crowns of the wheel. The same letters indicate the same parts, in all these three plates. 20. A, the forcbay, in which the level of the water is nearly the same as in the Northern Canal ; it is represented at the usual working height. 21. JB, the surface of the water in the wheelpit, represented at the lowest height at which the turbine is intended to operate. 22. C, the masonry of the ivheelpit. The faces towards the wheel, are of granite ashlar work, in blocks containing, generally, from ten to forty cubic feet. The backing is of hard mica slate. The capping course, shown particularly on Plate II., is neatly dressed on its upper surface. The whole is compactly laid in hydraulic cement. 23. D, the floor of the wheelpit. This floor sustains the weight of part of the sup- ply pipe, and of part of the water in it, and all the rest of the apparatus, excepting the wheel itself and the vertical shaft, which are supported by beams and braces, directly from the side walls of the wheelpit. It was necessary that the floor should have suffi- cient stiffness to resist the great upward pressure which takes place when the wheelpit is kept dry by pumps, in order to permit repairs to be made. The walls of the wheel- pit are built upon the floor; there was, consequently, no danger of the whole floor being pressed upwards, but the great width of the pit, (twenty-four feet,) would allow the floor to yield in the centre, unless it had great stiffness. To meet these requirements, three cast-iron beams are placed across the pit, the ends extending about a foot under the walls, on each side ; on these are laid thick planks which are firmly secured to the castriron beams, by bolts. To protect the thick planking from being worn out by the constant action of the water, they are covered with a flooring of one inch boards, which can be easily renewed when necessary. 24. E, the wrought iron supply pipe. This is constructed of plate iron, f inch thick, riveted together in a similar manner to steam boilers. The horizontal part is nine feet in diameter, the curved part gradually diminishes in diameter, to its junction with the upper curb. The upper end of the supply pipe is terminated by a cast-iron ring F, turned smooth on the face, to receive the wooden head gate. The supply pipe is also furnished with the man hole and ventilating pipe G, and the leak box H. The use of EXPERIMENTS UPON THE TREMONT TURBINE. 9 the latter is, to catch the leakage of the head gate, whenever it is closed for repairs of the wheel; at such times, the leakage is carried off into the raceway, below the wheelpit, by a six inch pipe, furnished with a valve which can be opened and shut at pleasure. 25. I, the cast-iron curls. These conduct the water from the wrought iron sup- ply pipe, to the disc K. The curbs are made in four parts, for the convenience of the founder. The surfaces at which they are joined, are turned true in a lathe, packed with red lead, and bolted together with bolts one and a half inches diameter, placed about six inches apart. The general thickness of the iron is one and a quarter inches. The flanges are two inches thick. The upper curb has a projection cast on it, to receive the disc pipe. The lower curb is finished on all sides ; the outside, to permit the regulating gate to be moved up and down easily ; the inside, to present a smooth surface to the water, and to match accurately with the garniture L. The curbs are supported from the wheelpit floor by four columns, two of which are shown at N ' N, plate L, resting on the cast-iron beam 0; this is placed on the floor, for the purpose of distributing the weight. The centres of the columns are thirteen inches from the outside circumference of the wheel. The beams N' rest immediately upon the columns, and the curb upon the beams, the latter projecting over the columns far enough for that purpose. The beams N' also act as braces from the wheelpit wall to the curb, and are strongly bolted at each end. 26. K, the disc. This is of cast-iron, one and a half inches thick, and is turned smooth on the upper surface, and also on its circumference. It is suspended from the upper' curb, by means of the disc pipes MM. The disc carries on its upper surface thirty-three guides, or leading curves, for the purpose of giving the water, entering the wheel, proper directions. They are made of Kussian plate iron, one tenth of an inch in thickness, secured to the disc by tenons, passing through corre- sponding mortices, cut through the disc, and are riveted on the under-side. The upper corners of the guides, near the wheel, are connected by the garniture L, which is intended to diminish the contraction of the streams entering the wheel, when the regulating gate is fully raised. The garniture is composed of thirty-three pieces of cast-iron, or one to fill each space between the guides; these pieces of cast-iron are, necessarily, of irregular form ; for a top view of them see L, plate III., figure 2. They are also shown in section at plate I. They are carefully fitted to fill the spaces between the guides ; above the top of the guides, the adjoining pieces are in contact ; they are strongly riveted to the guides, and to each other. After they were all fitted and riveted, the disc was put in a lathe, and the top, the periphery, and a part of the inside of the garniture, were turned off, so that it would fit accu- 10 EXPERIMENTS UPON THE TBEMONT TURBINE. rately, but easily, to the corresponding part of the lower curb. The disc is not fast- ened to the lower curb, but is retained in its place, horizontally, by the latter. 27. MM, the disc pipe. The disc is fastened to the bottom of the disc pip by fifteen tap screws, one and a quarter inches in diameter. As there is a vertical pressure on the disc, due to the pressure of the whole head, on its horizontal area, the disc pipe and its fastenings require to be very strong. The pipe is eight and a half inches diameter, inside, or one and a half inches larger than the shaft passing through it, and is one and a quarter inches thick. The upper flange is furnished with adjusting screws, by which the weight is supported upon the upper curb, and which afford the means of adjusting the height of the disc. The escape of water between the upper curb and the upper flange of the disc pipe, is prevented by a band of leather on the outside, which is retained in its place by the wrought iron ring P. This ring is made in two segments. The top of the disc pipe, just below the upper flange, has two projections, or wings, which fit into corresponding recesses in the top of the curb; these are to prevent the disc from rotating in the opposite direction to the wheel, to which there is a powerful tendency, arising from the reaction of the water issuing from the guides. 28. R R, the regulating gate. This is represented on the section, at plate I., as fully raised, and in this position the wheel would be giving its full power. The gate is of cast-iron, the cylindrical part is one inch thick, the upper part of the cylinder is stiffened by a rib, to which are attached three brackets, one of which is shown at S, plate I., and the two others at S S, plate II. To these brackets are attached wrought iron rods, by which the gate is raised and lowered. The brackets are attached to the gate at equal distances, and therefore the rods support equal parts of its weight. To one of the rods is attached the rack V. The other two rods are attached, by means of links, to the levers T T, plate II. The other ends of these levers carry geared arch heads, into which, and into the rack V, work three pinions, W, of equal pitch and size, fastened to the same shaft. As the fulcrums of the levers T T, plate II., are exactly in the middle, between the pitch lines of the arch heads and the points to which the rods are attached, it will be seen, that by the revolution of the pinion shaft, the gate must be moved up or down, equally on all sides. The shaft on which the pinions are fastened, is driven by the worm wheel X, plates I. and II. ; this is driven by the worm 0, either by the governor Y, or the hand wheel Z. The shaft on which the worm a is fastened, is furnished with movable couplings, which, when the speed gate is at any intermediate points between its highest and lowest positions, are retained in place by spiral springs ; in either of the extreme positions, the couplings are sep- arated by means of a lever, moved by pins in the rack F; by this means both the EXPERIMENTS UPON THE TREMONT TURBINE. H regulator and hand wheel are prevented from moving the gate in one direction, when the gate has attained either extreme position. If, however, the regulator or hand wheel should be moved in the opposite direction, the couplings would catch, and the gate would be moved. The weight of the gate is counterbalanced by weights attached to the levers T T, and by the intervention of a lever to the rack F; by this arrange- ment, both the governor and hand wheel are required to operate, with only the force necessary to overcome the friction of the apparatus. 29. bb The wheel. This consists of a central plate of cast-iron, and of two crowns, cc, of the same material, to which the buckets are attached. The central plate and the crowns are turned accurately in a lathe, for the purpose of balancing them, and also to diminish, as much as possible, the resistances in moving rapidly through the water. The lower crown is fastened to the central plate, as shown at figures 1 and 3, plate III. These figures also show, at cc, the form of the crowns; the upper and lower crowns are precisely alike ; they are nine and a half inches wide. At the inner edge, and at the circumference, the thickness is 0.625 inches, and at 5.5 inches from the inner edge, where they have the greatest thickness, they are one inch thick. By reference to figure 1, plate III, it will be seen that the buckets do not extend to the circumference of the crowns. In the direction of the radius, the ends of the buckets are 0.25 inches from the circumference. This is for the purpose of permitting the wheel to be handled with less danger of injuring the ends of the floats; as these are filed down to an edge, they would be very likely to be damaged during the construction of the wheel, if they were not guarded by the slight projection of the crowns. This construction also enables the grooves in the crowns to afford more perfect support to the ends of the buckets, and also permits a tenon to be nearly at the extremity of the bucket. The buckets are forty-four in number, and are of the form represented on plate III., figure 1. They are made of plate iron of excellent quality, imported from Russia for the purpose, they are -% of an inch in thickness, and are secured to the crowns in the following manner. The crowns having been first turned to the required form, grooves are cut in them of the exact form of the buckets, to the depth e e, figure 3, plate III. ; this depth is 0.1 inches at the edges and 0.5 inches near the middle. These grooves are cut in a machine contrived for the purpose, in which the cutting tool is guided by a cam. Three mortices for each bucket are then cut through each crown ; corresponding tenons are left on the buckets; the latter are bent to the required form, by means of a pair of dies, prepared for the punpose, the plate iron having been first moderately heated. The tenons of all the buckets are then entered into the mortices in both 12 EXPERIMENTS UPON THE TREMONT TURBINE. crowns, the latter are then drawn together, by means of a number of screws applied to different parts of the circumference, and when the edges of the buckets are drawn into the bottom of the grooves, the tenons are riveted on the opposite sides. This construc- tion gives great stability to the buckets, and permits the use of very thin iron. - 30. dd The vertical shaft. This is of wrought iron, and is accurately turned in every part The diameters are as follows : Below the hub of the wheel, 7 inches. In the hub of the wheel, 7 " Between the top of the hub and the lower bearing, 7 " Between the bottom of the lower bearing and the hub of the bevel gear, . 8 " In the hub of the bevel gear, 8 " From the top of the hub of the bevel gear to the suspeusion box, 8 " By reference to plate I., it will be seen that the shaft does not run upon a step at the bottom, but upon a series of collars, resting upon corresponding projections in the suspension box e'. The part of the shaft on which the collars are placed, is made separate from the main shaft, and is joined to it at /, by means of a eocket in the top of the main shaft, which receives a corresponding part of the collar piece. The collars are made of cast steel ; they are separately screwed on, and keyed to a wrought iron spindle. 31. e' The suspension lox. This is made in two parts, to admit of its being taken off, and put on the shaft ; it is lined with babbit metal, a soft composition con- sisting principally of tin. It is found that bearings thus lined will carry from fifty to a hundred pounds to the square inch, with every appearance of durability. 32. /'/', The upper and lower bearings. These are of cast-iron, lined with babbit metal ; they are retained in position, horizontally, by means of adjusting screws ; ver- tically, their weight is sufficient. The parts of the shaft inside the hubs of the wheel and the bevel gear, are made slightly tapering, about -fa of an inch in diameter in the length of the hubs ; the hubs are bored out with the same taper, but a very little smaller in diameter ; they are then drawn on by a powerful screw purchase, and in this manner are made to fit very tight. To prevent danger of bursting the hubs, they are before being drawn on or bored out, strongly hooped with wrought iron hoops, driven on hot. 33. The suspension box e' (art. 31,) rests upon the gimbal g, plates I. and II. The gimbal itself is Supported on the frame hh, by adjusting screws, which give the means EXPERIMENTS UPON THE TREMONT TURBINE. Jg of raising and lowering the suspension box, and, with it, the vertical shaft and wheel. It will be perceived, by the arrangement of the bearings above and below the bevel gear, that no lateral strain can be thrown upon the suspension box. The construction of the shaft will evidently not admit, with safety, of lateral strain at the suspension box, and it is accordingly so arranged that this box is free to oscillate horizontally in any direction, a small quantity, in case any irregularity in the form of the shaft should require it. The lower end of the shaft is fitted with a cast steel pin i, plate I. This is retained in its place by the step, which is made in three parts, and lined with casehardened wrought iron. The step is furnished with adjusting screws, by means of which the shaft can be moved horizontally in any direction, a small distance. The weight of the wheel, upright shaft, and bevel gear, is supported by means of the suspension box e,' on the frame k, which rests upon the long beams m, reaching across the wheelpit, and supported at the ends by the masonry, and also at intermediate points by the braces nn. From economical considerations the dijfuser, described at art. 12, was omitted at the Tremont Turbine ; a large majority of the turbines in use at Lowell, however, are fitted up with that apparatus. 34. The following are some of the dimensions of the turbine, carefully taken after the parts were finished : Height between the upper and lower crowns, at the outer extrem- ities of the buckets, a mean of 44 measurements, .... 0.9314 feet. Height between the upper and lower crowns, at the inner extrem- ities of the buckets, a mean of 44 measurements, .... 0.9368 " Height between the crowns, at a point 5.5 inches from the inner edges of the crowns, (designed to be 0.75 inches less than at the inner edges,) . .' 0.8743 Shortest distance between the outer extremities of the buckets and the next adjacent buckets, a mean of 132 measurements, 0.18757 " Shortest horizontal distance between two adjacent guides, taken at the top of the circumferential part of the disc, a mean of 33 measurements, 0.1960 a Do. do. at the bottom of the garniture, 0.2117 " Do. do. half-way up between the disc and the garniture, . . 0.2044 " The shortest distance between the guides, by a mean of the whole 99 measurements, 0.20403 " Height from the top of the circumferential part of the disc to the bottom of the garniture, a mean of 33 measurements, . . 0.97090 " 14 EXPERIMENTS UPON THE TREMONT TURBINE. 35. The following are some of the most important dimensions of the apparatus ; they are taken from the original designs, which were very closely followed in the construction. Diameter of the exterior circumference of the crowns of the wheel, 8.333 feet. " " outer extremities of the buckets, 8.292 " " " interior edges of the crowns, and inner edges of the buckets, 6.750 " " outside of the cylindrical part of the regulating gate, 6.729 " " " inside of the cylindrical part of the regulating gate, 6.562 " a " of the outside of the lower curb, taken below the flange, 6.542 " " inside of the lower curb, taken at the top, . . . 6.333 a " " inside of the lower curb, taken at the top of the guides, 6.167 lower part of the disc, 6.729 DESCRIPTION OF THE APPARATUS USED IN THE EXPERIMENTS ON THE TREMONT TURBINE. 36. The details of this apparatus are represented on plate IV. The useful effect was measured with a Prony dynamometer, represented in sectional elevation at figure 1, and in plan at figure 2. 37. The friction pulley A is of cast-iron 5.5 feet in diameter, two feet wide on the face, and three inches thick. It is attached to the vertical shaft by the spider B, the hub of which occupies the place on the shaft intended for the bevel gear. The friction pulley has, cast on its interior circumference, six lugs, C C, correspond- ing to the six arms of the spider. The bolt holes in the ends of the arms are slightly elongated in the direction of the radius, for the purpose of allowing the friction pulley to expand a little as it becomes heated, without throwing much strain upon the spider. When the spider and friction pulley are at the same temperature, the ends of the arms are in contact with the friction pulley. The friction pulley was made of great thick- ness for two reasons. When the pulley is heated, the arms cease to be in contact with the interior circumference of the pulley, consequently they would not prevent the pressure of the brake from altering the form of the pulley. This renders great stiff- ness necessary in the pulley itself. Again, it is found that a heavy friction pulley insures more regularity in the motion, operating, in fact, as a fly-wheel, in equalizing small irregularities. EXPERIMENTS UPON THE TREMONT TURBINE. 15 38. Tlic brakes E and F are of maple wood ; the two parts are drawn together by the wrought iron bolts G G, which are two inches square. 39. T/te bell crank F' carries at one end the scale I, and at the other the piston of the hydraulic regulator K; this end carries also the pointer L, which indicates the level of the horizontal arm. The vertical arm is connected with the brake F, by the link M, figure 3. 40. The hydraulic regulator K, figures 1, 2, and 5, is a very important addition to the Prony dynamometer, first suggested to the author by Mr. Boyden in 1844. Its office is to control and modify the violent shocks and irregularities, which usually occur in the action of this valuable instrument, and are the cause of some uncertainty in its indications. The hydraulic regulator used in these experiments, consisted of the cast-iron cylinder K, about 1,5 feet in diameter, with a bottom of plank, which was strongly bolted to the capping stone of the wheelpit, as represented in figure 1. In this cylin- der, moves the piston N, formed of plate iron 0.5 inches thick, which is connected with the horizontal arm of the bell crank by the piston rod 0. The circumference of the piston is rounded off, and its diameter is about ^ inch less than the diameter of the interior of the cylinder. The action of the hydraulic regulator is as follows. The cylinder should be nearly filled with water, or other heavy inelastic fluid. In case of any irregularity in the force of the wheel, or in the friction of the brake, the tendency will be, either to raise or lower the weight ; in either case the weight cannot move, except with a corresponding movement of the piston. In consequence of the inelas- ticity of the fluid, the piston can move only by the displacement of a portion of the fluid, which must evidently pass between the edge of the piston and the cylinder, and the area of this space being very small, compared to the area of the piston, the motion of the latter ' must be slow ; giving time to alter the tension of the brake screws before the piston has moved far. It is plain that this arrangement must arrest all violent shocks, but, however violent and irregular they may be, it is evident that, if the mean force of them is greater in one direction than in the other, the piston must move in the direction of the preponderating force, the resistance to a slow movement being very slight. A small portion of the useful effect of the turbine must be expended in this instrument ; probably less, however, than in the rude shocks the brake would be sub- ject to without its use. 41. For the purpose of ascertaining the velocity of the wheel, a counter was attached to the top of the vertical shaft, so arranged that a bell was struck at the end of every fifty revolutions of the wheel. 42. To lubricate the friction pulley, and at the same time to keep it cool, water was let on to its surface in four jets, two of which are shown in figure 2, plate IV. IQ EXPERIMENTS UPON THE TREMONT TURBINE. These jets were supplied from a large cistern, in the attic of the neighboring cotton- mill, kept full, during the working hours of the mill, by force-pumps. The quantity of water discharged by the four jets was, by a mean of two trials, 0.0288 cubic feet per second. In many of the experiments with heavy weights, and consequently slow velocities, oil was used to lubricate the brake, the water, during the experiment, being shut off. It is found that, with a small quantity of oil, the friction between the brake and the pulley, is much greater than when the usual quantity of water is applied ; consequently, the requisite tension of the brake screws is much less with the oil, as a lubricator, than with water. This may not be the whole cause of the phenomenon, but, whatever it may be, the ease of regulating in slow velocities is incomparably greater with oil as a lubricator, than with water applied in a quantity sufficient to keep the pulley cool. The oil was allowed to flow on in two fine continuous streams; it did not, however, prevent the pulley from becoming heated sufficiently to decompose the oil, after running some time, which was distinctly indicated by the smoke and peculiar odor. When these indications became very apparent, the experiment was stopped, and water let on by the jets, until the pulley was cooled. As the pulley became heated, the brake screws required to be gradually slackened. In the experiments, in table II., the lubricating fluid was as follows. In the first twenty-six experiments, water alone was used. In the four experiments numbered from 27 to 30, three gallons of linseed oil were used. In all the experiments requiring a lubricator, and numbered from 31 to 48, inclusive, linseed oil was used. In experiments 49 and 50, resin oil was used. In experiments numbered from 51 to 60, inclusive, water alone was used. In experiment 61, resin oil was used. In experiment 62, resin oil and a small stream of water were used; in the latter part of the experiment, a good deal of steam was generated by the heat of the friction pulley. In experiment 63, resin oil alone was used. In experiments numbered from 66 to 72, inclusive, water alone was used. In experiments numbered from 73 to 79, inclusive, resin oil and a small stream of water were used. In experiments numbered from 81 to 84, inclusive, water alone was used. In experiments 85 and 86, resin oil and a small stream of water were used. In experiment 87, resin oil alone was used. In experiments 90 and 91, water alone was used. EXPERIMENTS UPON THE TREMONT TURBINE. 17 In experiment 92, resin oil and a small stream of water were used. 43. A special apparatus was provided to indicate the direction in which the water left the wheel. For this purpose the vane P, figures 1, 6, and 7, plate IV.. was placed near the circumference of the wheel, and was keyed on to the vertical shaft Q, which turned freely on a step resting on the wheelpit floor. The upper end of the shaft carried the hand E, figures 1 and 4, and directly under the hand Avas placed the graduated semicircle S, divided into 180. When the vane was parallel to a tan- gent to the circumference of the wheel, drawn through the point nearest to the axis of the vane, and the vane was in the direction of the motion of the wheel, the hand pointed at 0, and, consequently, wnen the vane was in the direction of the radius of the wheel, the hand pointed at 90. To prevent sudden vibrations of the vane, a modification of the hydraulic regulator was attached to the lower part of the vane shaft This apparatus is represented in detail by figures 6 and 8. 44. The quantity of water discharged by the wheel was gauged at a weir erected for the purpose at the mouth of the wheelpit. It is represented on plate V. Figure 1 is a plan, and figure 2 a section, showing the relative positions of the turbine A, the grating B, the gauge box C, and the two divisions or bays of the weir, D, and E. As the water issued from the orifices of the turbine with considerable force, particularly when the velocity of the wheel was much quicker or slower than that corresponding to the maximum coefficient of effect, there were often such violent commotions in the wheelpit, that, \mless some mode was adopted to diminish them before the water reached the weir, or even the place where the depths on the weir were measured, it would have been impossible to make a satisfactory gauge of the water. For this purpose the grating B, figures 1 and 2, was placed across the wheelpit. This grating presented numerous apertures, nearly uniformly distributed over its entire area, through which the water must pass. In the experiments with a full gate, the fall from the upper to the lower side of the grating was generally from three to four inches. The combined effect of this fall and of the numerous small apertures, was, to obliterate almost entirely the , whirls and commotions of the water above the grating. About 4.5 feet in length of the grating between F and G, figure 1, was so nearly closed, that but little water passed through that part of the grating; this made it very quiet in the vicinity of the gauge box C. Figure 3, plate V., is an elevation of the weir. The two bays D and E were of nearly equal length, the crest of the weir was almost exactly horizontal, and the extreme variation did not exceed 0.01 inch. The crest of the weir was of cast- iron, planed on the upper edge H, figures 2 and 4, and also on the upstream face, to a point 1.125 inches below the top; below this, at I, figure 4, there was a small bevel, 3 18 EXPERIMENTS UPON THE TREMONT TURBINE. also planed, the slope of which, on an average, was T 8 g inch in a height of f inch ; the remainder of the casting was unplaned. The crest of the weir H was f inch thick, and was horizontal. The upstream edge, at H, was a sharp corner. The ends of the weir K, figures 1, 2, and 3, were of wood, and of the same form as the crest H, except that there was no bevelled part corresponding to I, figure 4. The crest of the weir H was about 6.5 feet above the floor of the wheelpit. The ends of the weir K pro- jected from the walls of the wheelpit, and also from the central pier, a mean distance of 1.235 feet. The length of the bay D, was 8.489 feet, and of the bay E, 8.491 feet, making the total length of the weir 16.98 feet. 45. The depth of the water on the weir was taken in the gauge box C, figures 1 and 2, plate V., by means of the hook gauge L, which is represented in detail by figures 9, 10, and 11, plate IV. The hook gauge is the invention of Mr. Boyden,* and is an instrument of inesti- mable value in hydraulic experiments. All other known methods of measuring the heights of the surface of still water, are seriously incommoded by the effects of capillary attraction ; this instrument, on the contrary, owes its extraordinary precision to that phenomenon. At figure 10, plate IV., the point of the hook A, is represented as coinciding with the surface of the water. If the point of the hook should be a very little above the surface, the water in the immediate vicinity of the hook, would, by capillary attraction, be elevated with it, causing a distortion in the reflection of the light from the surface of the water. The most convenient method of observing with this instrument, according to the experience of the author, is, first, to lower the point of the hook, by means of the screw, to a little distance below the surface ; then to raise it again slowly by the same means, until the distortion of the reflection begins to show itself, then to make a slight movement of the screw in the opposite direction, so as just to cause the distortion to disappear ; the point will then be almost exactly at the level of the surface. With no particular arrangements for directing light on the surface, differences in height of 0.001 feet are very distinct quantities; but by special arrangements for light and vision, differences of 0.0001 feet might be easily appreciated. As this instrument cannot be efficiently used in a current, it was placed in the box C, in which the communication with the exterior was maintained by the hole M, * In Versuche iiber den avsflitss des wassers dvrch schieber, hahne, Happen und ventile, by Julius Weisbach, Leipzig, 1842, page 1, is described an instrument for observing heights of water, having a slight resemblance to the hook gauge; it was however used by Boyden in a more perfect form, several years previous to the publication of that work. EXPERIMENTS UPON THE TREMONT TURBINE. 19 when, by partially obstructing this communication, the extent of the oscillations could be diminished at will. For the most perfect observations, it is essential that the surface of the water should be at rest. If, however, it should oscillate a little, a good mean may be obtained by adjusting the point of the hook to a height at which it will be visible above the surface of the water only half the time. The movable rod to which the hook was attached, was of copper, and graduated to hundredths of feet, but, by means of the vernier, thousandths were measured, and in some cases ten thousandths were estimated. In later, and more perfect forms of this instrument, the point of the hook is immediately under the graduation. 46. The heights of the water in the forebay, and in the wheelpit, were taken by means of gauges, placed in the gauge boxes p and q, plate II. These boxes were similar to the box C, plate V., in which the hook gauge was placed. Both gauges were graduated to feet and hundredths, and both had the same zero point, viz., the level of the crest of the weir, so that the difference in the readings at the two gauges, gives, at once, the fall acting upon the wheel ; and the difference between the depths of the water on the weir, as observed at the hook gauge, and the reading at the gauge q, gives the fall at the grating. In consequence of want of space in plate II., the gauge box p is not represented in its true position, it was actually in front of the head gate, and about six feet distant. 47. The heights of the regulating gate were taken at the rack V, plate I. The weights used for measuring the useful effect, were pieces of pig-iron of various sizes, each of which had been distinctly marked with its weight by Mr. 0. A. Richardson, the official sealer of weights and measures for the City of Lowell. MODE OF CONDUCTING THE EXPERIMENTS. 48. A separate observer was appointed to note each class of data; the time of each observation was also noted, which gave the means of identifying simultaneous observations. To accomplish this, each observer was furnished with a watch having a second hand ; the watch by which the speed of the wheel was observed, was taken as the standard ; all the others were frequently compared with it, and when the vari- ations exceeded ten or fifteen seconds, they were either adjusted to the standard, or the difference noted. This mode of observing must, evidently, lead to more precise results than that in which a single observer, however skilful, undertakes to note all the phenomena, or 20 EXPERIMENTS UPON THE TREMONT TURBINE. even several of them. By the method adopted, a regular record is made of the state of things at very short intervals, furnishing the data for a mean result for any required period, and also the means of detecting, in most cases, the causes of apparent discrepancies. It also relieves the experimenter from the distraction of having numerous exact observations to make in a very short time, and leaves him much more at liberty to exercise a vigilant watch over the general course of the experiment. 49. As it may be useful to experimenters, not accustomed to this mode of observing, and, at the same time, afford the reader some means of judging of the accuracy of the results obtained in these experiments, the following extracts are given from the original note-books. The extracts include the data observed for experiment numbered 30 in table II. This experiment is selected, simply, because it gave the maximum coefficient of effect. WEIGHT IN THE SCALE. Extract from the note-look of the author, who superintended the experiments. 1,498 Ibs. 10i oz. 4 h , 43', added Weight for the next experiment, 26 1,524 Ibs. lOfoz. SPEED OF THE WHEEL. Extract from the note-book of Mr. Charles Leonard. Times at which the bell struck. Differences. Times at which the bell struck. Differences. Times at which the bell struck. Differences. H. inin. sec. Seconds. H. min. sec. Seconds. H. min. sec. Seconds. 4 55. 58.00 5. 0. 52.00 59.00 5. 4. 47.00 59.00 56. 56.50 58.50 1. 50.75 58.75 5. 45.50 58.50 57. 55.25 58.75 2. 49.50 58.75 6. 44.25 58.75 58. 54.25 59.00 3. 48.00 58.50 7. 43.00 58.75 59. 53.00 58.75 NOT& The bell struck once in every fifty revolutions of the wheel. EXPERIMENTS UPON THE TBEMONT TURBINE. 21 ELEVATION OF THE POINTER ON THE BELL CRANK. Time. Height of pointer, in feet. Time. Height of pointer, in feet. Time. Height of pointer, in feet. H. min. sec. H. min. sec. H. min. sec. 4. 55. 0. 0.19 4. 59. 30. 0.20 5. 4. 0. 0.17 30. 0.13 5. 0. 0. 0.18 30. 0.18 56. 0.13 30. 0.19 5. 0.24 30. 0.14 1. 0.21 30. 0.18 57. 0.15 30. 0.17 6. 0.19 30. 0.19 2. 0.20 30. 0.19 58. 0.20 30. 0.19 7. 0.16 30. 0.19 3. 0.19 30. 0.14 59. 0.21 30. 0.19 NOTE. The extremity of the pointer was 6.5 feet from the fulcrum of the bell crank, horizontal arms of the bell crank were level, the height of the pointer was 0.20 feet. When the HEIGHT OF THE WATER ABOVE THE WHEEL. Taken in the forebay by Mr. John Newell. Time. Height, in feet. Time. Height, in feet. Time. Height, in feet. H. min. | sec. H. min. sec. H. min. sec. 4. 55. 0. 15.100 4. 59. 30. 15.110 5. 4. 0. 15.120 30. 15.100 ' 5. 0. 15.115 30. 15.120 56. 15.100 30. 15.120 5. 15.120 30. 15.100 1. 15.120 30. 15.115 57. 15.110 30. 15.110 6. 15.115 30. 15.115 2. 15.105 30. 15.110 58. 15.110 30. 15.100 7. 15.110 30. 15.100 3. 15.115 30. 15.110 59. 15.105 30. 15.125 NOTE. The top of the weir is the zero point of the gauge in the forebay. HEIGHT OF TrfE WATER AFTER PASSING THE WHEEL. Taken in the wheelpit by Mr. Lloyd Hixon. Time. Height, Time. Height, Time. Height, in feet. in feet. in feet. H. min. sec. H. min. sec. H. min. sec. 4. 56. 0. 2.20 5. 0. 0. 2.21 5. 4. 0. 2.22 30. 2.21 30. 2.21 30. 2.21 57. 2.21 1. 2.21 5. 2.21 30. 2.21 30. 2.21 30. 2.21 58. 2.21 2. 2.21 6. 2.21 30. 2.21 30. 2.21 30. 2.20 59. 2.20 3. 2.20 7. 2.22 30. 2.21 30. 2.20 30. 2.20 NOTE. The top of the weir is the zero point of the gauge in the wheelpit. 22 EXPERIMENTS UPON THE TREMONT TURBINE. HEIGHTS OF THE WATER ABOVE THE WEIR BY THE HOOK GAUGE. Observed by Mr. Daniel Haejffely. Time. Height, Time. Height, Time. Height, in feet. in feet. in feet. H. mill. sec. H. min. sec. H. min. sec. 4. 57. 5. 1.8710 5. i. 10. 1.8690 5. 4. 35. 1.8730 58. 15. 1.8710 i. 45. 1.8700 0. 50. 1.8725 58. 50. 1.8720 2. 15. 1.8720 6. 25. 1.8725 59. 20. 1.8730 2. 50. 1.8720 6. 55. 1.8725 59. 50. 1.8715 3. 15. 1.8715 7. 20. 1.8720 5. 0. 15. 1.8715 3. 40. 1.8715 7. 45. 1.8715 0. 45. 1.8705 4. 5. 1.8730 I NOTE. The zero of the hook gauge was 0.002 feet below the top of the weir. DIRECTION OF THE WATER LEAVING THE WHEEL. Observed at the vane index by Mr. John C. Woodward. Time. Direction. Time. Direction. Time. Direction. H. min. sec. deg. | min. H. min. sec. deg. min. H. min. sec. deg. min. 4. 57. 0. 59. 0. 5. 1. 0. 57. 0. 5. 5. 0. 58. 0. 30. 57. 0. 30. 59. 30. 30. 59. 30. 58. 59. 0. 2. 58. 0. 6. 59. 30. 30. 58. 0. 30. 57. 0. 30. 57. 0. 59. 58. 0. 3. 60. 0. 7. 59. 0. 30. 58. 30. 30. 58. 0. 30. 57. 30. 5. 0. 57. 0. 4. 59. 0. 8. 59. 0. 30. 57. 30. 30. 56. 0. NOTE. When the vane pointed in the direction of the radius of the wheel, the reading of the index was 90. was in the direction of the motion of the wheel. 50. Previously to the commencement of the experiments, the apparatus for measuring the useful effect was carefully adjusted. The bell crank was balanced when there were no weights in the scale. For this purpose the link M, figure 3, plate IV., was removed, and the chamber of the hydraulic regulator filled with water ; weights were then applied to the top of the bell crank, near the end to which the hydraulic regulator was attached, until the whole was in equilibrium; the final adjustment was made, by placing a weight of about two pounds at the extremity of one of the horizontal arms of the bell crank, the arm was retained horizontally until a signal was given, when it was left at liberty to descend, and the time occupied in descending a certain distance was noted ; the weight was then removed to the extremity of the other arm, and the same process repeated. The balance weights were altered until the times of descent EXPERIMENTS UPON THE TREMONT TURBINE. 23 were equal. To overcome, as much as possible, the friction of the fulcrum, the pin forming it was lubricated with sperm oil, and, during the descent, the head of the pin was struck lightly and rapidly with a small hammer. After the bell crank was satisfactorily balanced, the link M was reattached, and the brake adjusted, by means of the screw which formed the connection between the link and the brake. It was adjusted so that a line upon the brake was perpendicular to the axis of the link, when the horizontal arm of the bell crank was horizontal. The length of the brake was then measured upon this line. The length of the brake as thus measured was found to be . . 9.745 feet. The effective length of the vertical arm of the bell crank was 4.500 " And the effective length of the horizontal arm to which the scale was hung, was 5.000 " Consequently, the effective length of the brake was 9 - 74 ^X 5 _ ^327773 " 51. The gauges in the forebay, and in the wheelpit, were carefully adjusted by levelling from the top of the weir. This was repeated by different persons, so as to remove all chance of error. 52. The hook gauge was compared with the weir, by a different method. When the regulating gate of the turbine was shut down as tight as possible, it was still found that a quantity of water leaked into the wheelpit, exceeding, a little, the quantity that leaked out of the wheelpit, so that a small quantity con- tinued to run over the weir. The principal leak into the wheelpit was between the regulating gate and the lower curb, the leather packing not being perfectly adjusted. The hook gauge was firmly attached to a post, placed in the wheelpit for that purpose, and at a height known to be nearly correct. The regulating gate was closed, and after the water had arrived at a uniform state, the height of the water at the hook gauge was noted, and, at the same time, the depths of the water on the weir were measured directly with a graduated rule. To per- form this accurately, a board, about four inches long, was held by an assistant on the crest of the weir, at the place where" it was intended to measure the depth ; the author then applied the rule, previously well dried, vertically, on the top of the weir, in front of the board. On first immersing the rule, the water in contact with it did not stand at the true level of the surface, but formed a little hollow around the rule ; it immediately commenced rising, however, and after a few moments came to a level, which was indicated by the reflection of a light from the surface, a lamp being held by an assistant, in a proper position, for that purpose. 24 EXPERIMENTS UPON THE TREMONT TURBINE. The depths on the weir, tciken in the manner just described, February 20th, 1851, were as follows. Depths bay on the westerly of the weir. Inches. Depths on the easterly bay of the weir. Inches. 0.37 0.36 0.37 0.37 0.36 0.36 0.36 0.36 Means . . 0.3675 .... 0.36 Or in feet . 0.0306 .... 0.0300 While the heights given in the preceding table were being measured, the depth by the hook gauge was constantly 0.0318 feet; consequently, by this com- parison, the zero of the hook gauge was 0.0012 feet below the mean height of the top of the weir, in the westerly bay, and 0.0018 feet below the mean height in the easterly bay, or 0.0015 feet below the mean height in both bays. A similar comparison was made February 22d, 1851, when the zero of the hook gauge was found to be 0.0024 feet below the mean height of the weir. The mean of the two comparisons, or 0.0020 was adopted as the correction to be sub- tracted from the reading of the hook gauge, to give the mean depth upon the weir. 53. During the experiments, the levels of the water in the upper and lower canals, were maintained nearly uniform. The height of the lower canal, at the place where the water, passing the weir, fell into it, varied a little, depending upon the quantity of water discharged by the wheel. It was highest when the wheel was running with the regulating gate fully raised, and the brake removed ; under these circumstances the surface of the water was from 0.3 feet to 0.4 feet below the top of the weir. In the other experiments with the regulating gate fully raised, the fall from the top of the weir to the surface of the water in the lower canal, was from 0.4 feet to 0.6 feet. . The brackets N and the planks 0, figure 2, plate V., were not put on until after the turbine experiments were concluded, so that the water passing the weir, met with no obstruction until it struck the water in the lower canal. It will be seen by the experiments on the weir, (art. 127,) that the obstruction, caused by the planks, was scarcely appreciable, which renders it certain that the effect of the lower canal, in obstructing the flow over the weir, must have been entirely inappreciable. EXPERIMENTS UPON THE TREMONT TURBINE. 25 DESCRIPTION OF TABLE II., CONTAINING THE EXPERIMENTS TJPON THE TURBINE AT THE TREMONT MILLS. 54. The data obtained by direct observation, and the results deduced from them by calculation, are arranged together, for convenience of reference, in table II The columns numbered ], 2, and 3, require no further explanations than are contained in the several headings. 55. COLUMN 4. Height of the regulating gate. The three first experiments were made under circumstances identical in every thing, except that the height of the regulating gate was varied a little, for the purpose of ascertaining the height giv- ing the maximum coefficient of effect. The mean height between the crowns of the wheel, at the inner edges of the buckets, was 0.9368 feet, or 11.2416 inches; the curvature of the disc and garniture, however, rendered it- necessary to raise the gate rather more than this, in order to present the most favorable aperture. By a comparison of the first three experiments, it appears that the most favorable result was obtained, with the gate raised to a height of 11.50 inches, or a little less; the succeeding experiments, numbered from 4 to 50, inclusive, were made with the regulating gate raised to the full height, or to 11.50 inches, nearly. A comparison of the first three experiments will show that there could be no appreciable differ- ence in the results, that could be attributed to the small differences in the heights of the gate, in the experiments numbered from 4 to 50, inclusive, and they are accordingly all classed together, as experiments with a full gate, the small dif- ference in the heights being accidental. The experiments numbered from 51 to 64, inclusive, were made with the gate raised 8.55 inches, or three-fourths of the full height, nearly. Those num- bered from 65 to 76, inclusive, were made with the gate at very nearly half of the full height. Those numbered from 77 to 79, inclusive, were made with the gate at about seven eighths of the full height. Those numbered from 80 to 89, inclusive, were made with the gate at about one fourth of its full height. And the last three experiments were made with the gate raised one inch. 56. COLUMN 5. Time. The times entered under the heads beginning, and ending. of the experiments, are taken from the notes of the "speed of the wheel," and indicate the times at which the bell, attached to the counter, was struck, which, by a comparison of the various note-books, appeared, by the regularity of the observations, to be the most suitable for the commencement and termination of the experiment. 4 26 EXPERIMENTS UPON THE TREMONT TURBINE. 57. COLUMN 6. Duration of the experiment, is obtained by taking the differences of the times of the" beginning, and ending of the experiment, as given in column 5. 58. COLUMN 7. Total number of revolutions of the wheel during the experiment. This is obtained from the note-book of the " speed of the wheel," by counting the number of observations of the times at which the bell was struck; this number, less one, multiplied by 50, which is the number of revolutions of the wheel to each stroke of the bell, gives the number of revolutions during the experiment. 59. COLUMN 8. Number of revolutions of the wheel per second, is obtained by dividing the total number of revolutions of the wheel, by the duration of the experiment. 60. COLUMN 9. The weight in the scale, requires no explanation. 61. COLUMN 10. Useful effect, or the friction of the brake, in pounds avoirdupois, raised one foot per second. This is obtained by multiplying together the weight in the scale, and the velocity that the point of application of the weight, tends to take. Or, in other words, the product of the weight into the velocity that the weight would actually take, if, for an infinitely short time, the brake, and the apparatus connecting it with the weight, were rigidly connected with the friction pulley. The effective length of the brake, including the leverage due to the different lengths of the arms of the bell crank, was 10.827778 feet (art. 50). The cir- cumference of a circle of this radius is 68.0329 feet. This circumference is a constant for all the experiments in which any useful effect was produced, and column 10 was obtained by the product of this constant, the weight, and the number of revolutions of the wheel per second. The computation was performed by logarithms, and if the results given in the tables should be verified by actual multiplications, minute differences Avould, no doubt, be detected. 62. COLUMNS 11 and 12. Heights of the water in the forebay and in the wheelptt. These heights are all referred to the top of the weir, consequently, the differences give the fall acting upon the wheel. 63. COLUMN 13. Total fall acting upon the tvheel. These are the differences referred to in the last sentence. In experiments 27 and 28, observations were taken in the ventilating pipe G, plate I., for the purpose of estimating the loss of fall to this part of the supply pipe ; it was not convenient, however, to measure these heights with complete accuracy. In experiment 27, the height of the water in the ventilating pipe was 0.106 feet below the level in the forebay ; in experi- ment 28 the difference was found to be 0.102 feet; in experiment 30, which gave the maximum coefficient of effect, the quantity of water discharged by the wheel, was a little less than in either experiment 27, or 28. We may, therefore. EXPERIMENTS UPON THE TREMONT TURBINE. 27 conclude, that when the regulating gate was fully raised, and the wheel running with the velocity giving the maximum coefficient of effect, the fall acting .upon the wheel being 12.903 feet, the loss of fall from the forebay to the ventilating pipe, was very nearly 0.10 feet. 64. COLUMN 14. Depth of water on the weir. The depths on the weir were observed with the hook gauge, described at art. 45. 65. COLUMN 15. Quantity of water passing the weir. These quantities have been calculated by the formula Q = 33$(l O.lnh)$, in which Q = Quantity, in cubic feet per second. / = The total length of the weir, in feet. n = The number of end contractions in the weir. h = The depth on the weir, in feet. It is unnecessary here to discuss the reasons that have induced the author to adopt this formula, sa different from any that has been used heretofore, as the subject is fully considered in another part of this work. A small quantity of water entered the wheelpit without passing through the wheel ; there was also a small quantity that leaked out by passing through the floor of the wheelpit; the latter quantity, when the depth on the weir was 0.496 feet, was estimated at 0.0409 cubic feet per second; see art. 130. As these quantities were very minute, and tended to compensate each other, they have been neglected, and the quantity computed as passing the weir is taken for the quantity discharged by the wheel. 66. COLUMN 16. Total power of the ivater. This column is obtained by multi- plying together the total fall acting upon the wheel, the quantity of water passing the weir per second, and the weight of a cubic foot of water. The temperature of the water was constantly at 32 Fahrenheit, it was nearly pure, and the weight of a cubic foot was taken at 62.375 pounds avoirdupois. The water of the Merrimack Eiver is always remarkably free from impurities, held in solution, flowing, as it does, from, and through a primitive formation, cov- ered with a sterile soil. In midwinter, at which season these experiments were made, it is more than ordinarily pure, as at that season the surface of the coun- try is usually covered with snow, and the soil frozen to a considerable depth ; the river itself, wherever it flows with a moderate current, is frozen over, so that heavy carriages can often pass with safety, and at the time when these experi- ments were made, the river for about eighteen miles before it reached the turbine, 28 EXPERIMENTS UPON THE TREMONT TURBINE. was covered with a solid coating of ice, with scarcely an opening in the whole distance. When the river is thus frozen, the water flows along under the ice, entirely free from floating particles of ice, even in the most severe weather. As the author had frequently felt the want of a table of the absolute weights of a cubic foot of water at different temperatures, he, several years since, com- puted the following table. In the Encyclopedia Britannica, seventh edition, vol. 21, page 846, is given the following extract from the British act of Parliament, establishing the standards fpr weights and measures. "Provided always, and be it enacted, that in all cases of dispute respecting the correctness of any measure of capacity, arising in a place where recourse cannot conveniently be had to any of the aforesaid verified copies or models of the standard measures of capacity, it shall and may be lawful, to and for, any justice of the peace, or magistrate, having jurisdiction in such place, to ascertain the con- tent of such measure of capacity by direct reference to the weight of pure or rain water which such measure is capable of containing; ten pounds avoirdupois weight of such water, at the temperature of 62 by Fahrenheit's thermometer, being the standard gallon ascertained by this act, the same being in bulk equal to 277.276, 1822 (1823, 277.274) -cubic inches, and so in proportion," etc. 277.274 cubic inches was taken, as it appeared to be the latest determination. In the first volume of the Traite de Chinue, by J. J. Berzetius, second French edition, Paris, 1846, there is given a table of the specific gravities of pure water, at different temperatures of the centigrade scale, deduced from Haellstroem's experi- ments. From these two authorities were derived the data for the following table. EXPERIMENTS UPON THE TREMONT TURBINE. 29 TABLE I. WEIGHT OF A CUBIC FOOT OF PURE WATER AT DIFFERENT TEMPERATURES. Temperature, in degrees of Fahrenheit's thermometer. Weight in air, of a cubic foot of pure water. Founds avoirdupois. Temperature, in degrees of Fahrenheit's thermometer. Weight hi air, of a cubic foot of pure water. Pounds avoirdupois. Temperature, in degrees of Fahrenheit's thermometer. Weight in air, of a cubic foot of pure water. Pounds avoirdupois. 32 62.375 50 62.368 69 62.278 33 62.377 51 62.365 70 62.272 34 62.378 52 62.363 71 62.264 35 62.379 53 62.359 72 62.257 36 62.380 54 62.356 73 62.249 37 62.381 55 62.352 74 62.242 38 62.381 56 62.349 75 62.234 39 ,.., 62.382 57 62.345 76 62.225 39.38 62.382 58 62.340 77 62.217 40 62.382 59 62.336 78 62.208 41 62.381 60 62.331 79 62.199 42 62.381 61 62.326 80 62.190 43 62.380 62 62.321 81 62.181 44 62.379 63 62.316 82 62.172 45 62.378 64 62.310 83 62.162 46 62.376 65 62.304 84 62.152 47 62.375 66 62.298 85 62.142 48 62.373 67 62.292 86 62.132 49 62.371 68 62.285 67. COLUMN 17. Ratio of the useful effect to the power expended. This column is obtained by dividing the numbers in column 10 by those in column 16. 68. COLUMN 18. Velocity due to the fall acting upon the wheel. The numbers in this column have been calculated by the formula V= the velocity in feet per second. ^ = the velocity acquired by a body at the end of the first second of its fall in a vacuum. h = the fall acting upon the wheel j this is given in column 13. The value of g has been calculated by the formula given in the second edition of the Trait6 D'HydrauKque, by D'Aubuisson, page 5, viz. : g = 9!" 8051 (1 0.00284 cos. 2 f) (1 if ) ; /being the latitude of the- place; e, its elevation above the level of the sea; and r, the radius of the terrestrial spheroid, at the level of the sea, and at the place ; { r= 6366407" 1 (1 + 0.00164 cos. 21.)}. 30 EXPEEIMENTS UPON THE TREMONT TURBINE. The latitude of Lowell, as given in the American Almanac, is 42, 38', 46", and the height above the sea is known to be about 25 metres. With these data, the above formula gives, in feet, ^ = 32.1618. 69. COLUMN 19. Velocity of the inferior circumference of the wheel. The diameter of the circle inscribing the inner edges of the buckets, is 6.75 feet; see art. 35. Consequently the interior circumference of the wheel is 21.20575 feet. The product of this number into the number of revolutions per second, given in column 8, gives the numbers in column 19. 70. COLUMN 20. Ratio of the velocity of the interior circumference of the wheel, to the velocity due to the fall acting on the wheel. This column is obtained by dividing the numbers in column 19 by the corresponding numbers in column 18. This column indicates the relative velocities of the wheel, in the different experiments, eliminated from the effects of the variations in the fall acting upon the wheel. 71. COLUMN 21. Quantity of ivater which passed the wheel, reduced to a uniform fall of thirteen feet. The numbers in this column are obtained from those in col- umn 15, in the following manner. Let IT= the observed fall acting upon the wheel. Q = the observed quantity. (X=the quantity that would have passed the wheel, if the fall had been thirteen feet, instead of H, all other circumstances being the same. As the quantity of water discharged by the wheel, all other things being equal, will vary as the square root of the fall acting upon the wheel, we have V/JT: Q =: ViT : S 0.006 25 15.118 2.219 12.899 1.8775 139.0291 111859.4 0.78840 28.8047 18.7662 0.65150 139.5724 1.00623 50 87 40.004 26 15.102 2.209 12.893 1.8750 138.7601 111591.0 0.78641 28.7980 18.5527 0.64424 139.3347 1.00452 51 lo 0.057 27 15.116 2.214 12.902 1.8758 138.8489 111740.3 0.78997 28.8080 18.6616 0.64779 139.3752 1.00481 51 18 0.018 28 15.117 2.211 12.906 1.8760 138.8711 111792.9 0.78934 28.8125 18.5141 0.64257 139.3759 1.00482 52 22 0.018 29 15.118 2.212 12.906 1.8727 138.5134 111504.9 0.79225 28.8125 18.3732 0.63768 139.0169 1.00223 53 52 0.017 30 15.111 2.208 12.903 1.8697 138.1892 111218.1 0.79375 28.8092 18.0474 0.62645 138.7076 1.00000 58 10 0.018 31 15.084 2.545 12.539 2.0891 162.3283 126960.2 0. 28.3999 37.9521 1.33635 165.2853 1.19161 32 15.119 2.204 12.915 1.8704 138.2668 111384.0 0.79294 28.8225 17.7330 0.61525 138.7211 1.00010 61 54 0.011 33 15.134 2.200 12.934 1.8701 138.2335 111521.1 0.79243 28.8437 17.2475 0.59796 138.5858 0.99912 66 5 0.008 34 15.129 2.188 12.941 1.8687 138.0869 111463.0 0.78903 28.8515 16.6254 0.57624 138.4013 0.99779 86 12 0.004 35 15.123 2.184 12.939 1.8652 137.7076 111139.7 0.78340 28.8493 15.7371 0.54549 138.0318 0.99513 99 25 0.014 30 15.128 2.184 12.944 1.8539 136.4917 110200.9 0.77916 28.8549 14.7319 0.51055 136.7866 0.98615J115 is 0.001 37 15.117 2.177 12.940 1.8412 135.1415 109077.1 0.76978 28.8504 13.6922 0.47459 135.4545 0.97655 131 18 0.013 38 15.036 2.536 12.500 2.0834 161.6944 126071.1 0. 28.3557 37.8674 1.33544 164.8966 1.18881 39 15.143 2.180 12.963 1.8431 135.3423 109433.4 0.76886 28.8761 13.7205 0.47515 135.5354 0.97713 130 51 0.019 40 15.136 2.163 12.973 1.8380 134.7976 109077.0 0.75277 28.8872 12.7235 0.44045 134.9378 0.97282 139 45 0.010 41 15.136 2.159 12.977 1.8282 133.7538 108265.8 0.72499 28.8916 11.2882 0.39071 133.8723 0.96514 147 25 0.005 42 15.133 2.185 12.948 1.8252 133.4330 107764.7 0.67887 28.8593 9.6302 0.33370 133.7007 0.96390 0.015 43 15.116 2.319 12.797 1.8460 135.6536 108280.4 0. 28.6906 0. 0. 136.7253 0.98571 44 15.096 2.322 12.774 1.8457 135.6205 108059.4 0. 28.6648 0. 0. 136.8150 0.98635 45 15.012 2.541 12.471 2.0864 162.0237 126034.8 0. 28.3228 37.8168 1.33521 165.4245 1.19261 46 15.156 2.202 12.954 1.8737 138.6244 112009.3 0.79104 28.8660 17.6447 0.61126 138.8703 1. 001 17 60 52 0.012 47 15.134 2.202 12.932 1.8726 138.5023 111720.6 0.79097 28.8415 17.3179 0.60045 138.8660 1.00114 63 16 0.008 48 15.142 2.194 12.948 1.8723 138.4690 111832.0 0.78904 28.8593 17.0327 0.59020 138.7468 1.00028 66 27 0.01 6 49 15.1 44! 2.193i 12.951 1.8714 138.3692 111777.2 0.78859 28.8627 16.7394 0.57997 138.6307 0.99945 81 IS 0.007 50 1 5.1 44! 2.1 92J 12.952 1.8694 138.1559 111613.5 0.78723 28.8638 16.3058 0.56492 138.4117 0.99787 89 . 11 0.010 34 EXPERIMENTS UPON THE TREMONT TURBINE. TABLE EXPERIMENTS UPON THE TURBINE AT THE 1 a 3 4 5 6 7 8 9 1O Temperature of TIME. Total the atmosphere in number Useful effect, No. degrees of Height Duration of Number of Weight In or the of the DATE, Fahrenheit's thermometer. of the of the revolu- tions revolutions the scale, friction of the brake, experi- ment. 1851. External air in In the regulat- ing gate, Beginning of the experiment. Ending of the experiment. experi- ment, of the wheel during the of the wheel per second. in pounds avoirdupois. in pounds avoirdupois, raised one foot per the wheelpit. second. shade. ment. H. mln. sec. H. mln. sec. 51 February 21, P.M. 34.75 36.50 8.55 2 17 11.50 2 23 56.00 404.50 600 1.48331 390.95 39452.4 52 ii tt u 34.50 36.25 it 2 24 31.00 2 32 28.00 477.00 600 1.25786 775.61 66373.6 53 (t it a 34.50 36.50 a 2 33 10.00 2 41 55.50 525.50 600 1.14177 963.30 74827.2 54 t( It U 34.50 36.50 ti 2 41 55.50 2 50 25.00 509.50 550 1.07949 1069.05 78512.0 55 tf it 11 34.50 37.00 u 2 50 25.00 2 58 33.00 488.00 500 1.02459 1150.77 80215.4 56 11 it It 34.50 36.75 a 3 8 34.00 3 16 20.50 466.50 450 0.96463 1242.98 81572.6 57 tl tt n 34.50 37.00 u 3 17 13.50 3 25 17.00 483.50 450 0.93071 1293.63 81911.6 58 u 11 tt 34.25 37.00 u 3 25 17.00 3 33 37.00 500.00 450 0.90000 1345.47 82382.7 59 it u u 34.25 37.25 u 3 33 37.00 3 42 17.00 520.00 450 0.86538 1396.11 82195.5 60 u u u 34.25 36.75 tt 3 43 15.50 3 52 14.25 538.75 450 0.83527 1444.03 82057.9 61 u a ( 34.25 36.50 tt 4 10 26.00 4 19 45.50 559.50 450 0.80429 1494.68 81786.2 62 ti u 34.00 36.00 tl 4 31 37.50 4 40 15.50 518.00 400 0.77220 1548.69 81360.6 63 u a a 34.00 36.00 It 4 51 21.50 4 59 36.00 494.50 350 0.70779 1656.98 79788.1 64 u u a tt 5 8 37.00 5 19 17.25 640.25 1100 1.71808 0. 0. 65 February 22, A.M. 5.65 8 57 1.00 9 7 47.00 646.00 1000 1.54799 0. 0. 66 ft U 35.50 35.75 it 9 15 39.00 9 21 12.00 333.00 450 1.35135 316.03 29054.7 67 U < U 35.50 36.50 tt 9 21 51.25 9 28 0.50 369.25 450 1.21869 519.69 43087.9 68 U (C 11 35.50 36.75 u 9 28 45.00 9 36 26.00 461.00 500 1.08460 720.20 53142.4 69 (( U ( 35.50 37.25 tt 9 36 26.00 9 43 4.50 398.50 400 1.00376 832.26 56834.2 70 U tt t 35.75 37.25 tt 9 43 57.00 9 50 16.00 379.00 350 0.92348 934.74 58727.1 71 tt tt t 35.75 37.25 it 9 51 14.00 9 58 9.00 415.00 350 0.84337 1033.30 59287.8 72 tt t( t 36.75 37.25 tt 9 59 12.50 10 7 48.75 516.25 400 0.77482 1115.02 58776.2 73 tt tt t 39.00 35.75 u 10 21 10.00 10 30 50.25 580.25 450 0.77553 1115.02 58830.1 74 tt It t 38.25 36.00 it 10 42 44.50 10 51 14.50 510.00 350 0.68627 1204.84 56253.1 75 tt tt t 38.25 36.25 u 10 58 28.00 11 6 35.00 487.00 300 0.61602 1277.98 53559.4 76 tt tt t 38.25 36.25 it 11 13 59.00 11 21 17.00 438.00 200 0.45662 1482.56 46056.1 77 February 22, A.M. 37.50 35.75 9.96 11 33 20.00 11 40 9.50 409.50 350 0.85470 1482.56 86207.6 78 tt tt tt 38.00 36.00 ft 11 46 21.00 11 53 27.00 426.00 350 0.82160 1544.87 86351.4 79 " " P.M. 40.75 35.75 U 42.50 8 6.00 443.50 350 0.78918 1604.85 86164.4 80 February 22, P.M. 42.75 36.00 2.875 2 33 11.00 2 38 31.25 320.25 400 1.24902 0. 0. 81 ft It U 43.75 36.50 tt 2 42 3.00 2 47 52.00 349.00 400 1.14613 118.59 9247.0 82 tt tt tt 44.00 37.00 U 2 48 41.00 2 54 45.50 364.50 350 0.96022 325.39 21256.6 83 tl U tt 44.25 37.25 it 2 55 46.00 3 2 14.50 388.50 300 0.77220 519.86 27310.9 84 tt tt tt 43.75 37.25 tt 3 2 14.50 3 9 41.00 446.50 300 0.67189 612.22 27985.1 85 It tl tt 43.50 37.00 It 3 11 21.50 3 18 48.00 446.50 250 0.55991 704.44 26833.8 86 ft If tl 43.50 36.75 It 3 20 34.50 3 27 47.00 432.50 200 0.46243 777.58 24462.9 87 It U tt 43.25 36.50 It 3 33 10.50 3 44 17.00 666.50 200 0.30007 882.02 18006.4 i 88 tt tt tl 42.25 36.25 tt 4 2 0. 0. 1195.06 0. 89 It 11 U It 4 10 0. 0. 1054.25 0. 90 February 22, P.M. 42.00 37.75 1.00 4 27 3.50 4 33 47.00 403.50 250 0.61958 118.59 4998.8 91 a it a a 4 35 1.00 4 42 16.50 435.50 300 0.68886 73.14 3427.7 92 it tt 11 41.00 37.25 a 4 45 42.00 5 2 54.00 1032.00 400 0.38760 296.39 7815.6 EXPEEIMENTS UPON THE TREMONT TURBINE. 35 II. CONTINUED. TREMONT MILLS, IN LOWELL, MASSACHUSETTS. 11 12 13 14 15 10 17 18 19 20 21 22 23 24 Height of the Height of the Total Quantity Total power Ratio Velocity Velocity Ratio of the Quantity of Ratio of the Direction Mean No. water water after fall Depth of of water of the water of the due to the of the velocity of the interior passed the reduced of the water elevation of the above the passing water I ' h in pounds fall acting interior circumrnce wheel, quantity in leaving of the experi- ment. wheel, taken in the the wheel, taken in the acting upon th wheel, on the weir, the weir, in cubic feet avoirdupois raised one useful effect to the power on the wheel, in circumfnce of the wheel, in of the wheel to the velocity reduced to a uniform fall of 13 feet, in column 21 to the reduced the wheel, as indicated by the pointer on the bell forebay, wheel- in feet. in feet. per second. foot per expended. feet per feet per due to the cubic feet quantity in vane. crank. in feet. in, feet second. second. second. . per second. experiment wheel. 30. deg. m. Feet. 51 15.095 2.337 12.758 1.9173 143.3319 114060.7 0.34589 28.6468 31.4548 1.09802 144.6849 1.04309 12 - -0.002 52 15.128 2.255 12.873 1.8792 139.2094 111778.7 0.59379 28.7756 26.6739 0.92696 139.8945 1.00856 17 82 - -0.001 53 15.134 2.225 12.909 1.865G 137.7518 110917.6 0.67462 28.8159 24.2121 0.84023 138.2365 0.99660 22 19 - -0.007 54 15.134 2.194 12.940 1.8660 137.7962 111219.8 0.70592 28.8504 22.8914 0.79345 138.1153 0.995.73 25 - -0.013 55 15.139 2.189 12.950 1.858b 137.0026 110664.7 0.72485 28.8616 21.7272 0.75281 137.2668 0.98961 28 56 0.011 56 15.138 2.187 12.951 1.8450 135.5434 109494.5 0.74499 28.8627 20.4557 0.70873 135.7996 0.97903 33 58 -^0.016 57 15.143 2.178 12.965 1.8408 135.0974 109252.2 0.74975 28.8783 19.7365 0.68344 135.2797 0.97529 37 47 0.014 58 15.144 2.168 12.976 1.8336 134.3304 108724.1 0.75772 28.8905 19.0852 0.66060 134.4546 0.96934 42 43 0.011 59 15.151 2.152 12.999 1.8240 133.3014 108082.5 0.76049 28.9161 18.3511 0.63463 133.3066 0.96106 47 ;;o 0.005 60 15.153 2.139 13.014 1.8186 132.7344 107746.9 0.76158 28.9328 17.7125 0.61219 132.6630 0.95642 54 :{? 0.004 61 15.155 2.129 13.026 1.8117 131.9960 107246.3 0.76260 28.9461 17.0556 0.58922 131.8642 0.95066 59 ;!'.) 0.042 62 15.162 2.122 13.040 1.8022 130.9913 106544.4 0.76363 28.9617 16.3751 0.56541 130.7903 0.94292 76 :!(> 0.021 63 15.162 2.134 13.028 1.8013 130.8932 106366.6 0.75012 28.9484 15.0091 0.51848 130.7525 0.94265 94 I 0.022 64 15.079 2.359 12.720 1.9742 149.5470 118652.1 0. 28.6041 36.4332 1.27370 151.1840 1.08995 65 15.139 1.969 13.170 1.7160 121.9685 100194.6 0. 29.1057 32.8262 1.12783 121.1788 0.87363 66 115.148 2.071 13.077 1.6829 118.5511 96699.5 0.30046 29.0028 28.6564 0.98806 118.2016 0.85216 6 30 4-0.001 67 15.159 2.025 13.134 1.6590 116.0987 95112.0 0.45302 29.0659 25.8432 0.88912 115.5050 0.83272 8 80 +0.002 68 15.164 1.988 13.176 1.6409 114.2599 93904.9 0.56592 29.1123 22.9997 0.79003 113.4942 0.81823 12 82 0.020 69 15.171 1.956 13.215 1.6309 113.2448 93346.0 0.60885 29.1554 21.2856 0.73007 112.3198 0.80976 16 5 0.027 70 15.173 1.920 13.253 1.6139 111.5197 92188.4 0.63703 29.1973 19.5831 0.67072 110.4502 0.79628 21 17 0.006 71 15.179 1.897 13.282 1.5959 109.7130 90893.3 0.65228 29.2292 17.8844 0.61187 108.5420 0.78252 29 56 0.014 72 15.183 1.872 13.311 1.5793 108.0452 89707.1 0.65520 29.2611 16.4306 0.56152 106.7756 0.76979 39 2 4-0.001 73 15.179 1.869 13.310 1.5783 107.9493 89620.7 0.65643 29.2600 16.4457 0.56205 106.6848 0.76913 40 18 0.029 74 15.159 1.833 13.326 1.5541 105.5341 87720.9 0.64127 29.2776 14.5530 0.49707 104.2353 0.75147 69 27 0.028 75 15.174 1.812 13.362 1.5371 103.8516 86555.6 0.61879 29.3171 13.0631 0.44558 102.4352 0.73850 95 0.024 76 15.183 1.771 13.412 1.5034 100.5410 84110.0 0.54757 29.3719 9.6830 0.32966 98.9847 0.71362 144 56 0.029 77 15.079 2.196 12.883 1.8620 137.3618 110380.8 0.78100 28.7868 18.1246 0.62961 137.9842 0.99478 55 52 0.037 78 15.079 2.183 12.896 1.8583 136.9694 110176.6 0.78375 28.8013 17.4226 0.60492 137.5206 0.99144 64 0.029 79 15.087 2.175 12.912 1.8544 136.5469 109973.0 0.78350 28.8192 16.7351 0.58069 137.0114 0.98777 74 28 0.018 80 14.774 1.427 13.347 1.2914 80.4534 66979.0 0. 29.3006 26.4865 0.90396 79.4007 0.57243 30 81 14.769 1.400 13.369 1.2737 78.8433 65746.8 0.14065 29.3248 24.3046 0.82881 77.7476 0.56051 1 30 +0.008 82 14.772 1.377 13.395 1.2492 76.6213 64018.1 0.33204 29.3.533 20.3622 0.69369 75.4831 0.54419 4 32 +0.010 83 H4.783 1.348 13.435 1.2206 74.0590 62062.0 0.44006 29.3971 16.3751 0.55703 72.8501 0.52521 11 80 +0.002 84 14.793 1.315 13.478 1.1960 71.8750 60424.6 0.46314 29.4441 14.2480 0.48390 70.5889 0.50890 20 9 0.001 85 14.806 1.293 13.513 1.1748 70.0063 59006.4 0.45476 29.4823 11.8733 0.40273 68.6646 0.49503 41 84 0.022 86 14.820 1.264 13.556 1.1497 67.8158 57342.0 0.42661 29.5292 9.8061 0.33208 66.4105 0.47878 81 40 0.025 87 14.803 1.244 13.559 1.1113 64.5053 54554.9 0.33006 29.5324 6.3633 0.21547 63.1616 0.45536 0.026 88 14.762 1.246 13.516 1.0623 60.3593 50886.6 0. 29.4856 0. 0. 59.1959 0.42677 89 14.771 1.240 13.531 1.0630 60.4190 50993.4 0. 29.5020 0. 0. 59.2216 0.42695 90 14.806 0.821 13.985 0.7798 38.2210 33340.8 0.14993 29.9928 13.1386 0.43806 36.8505 0.26567 +0.004 91 14.815 0.814 14.001 0.7846 38.5699 33683.5 0.10176 30.0099 14.6079 0.48677 37.1655 0.26794 +0.030 92 i 14.832 0.812 14.020 0.7653 37.1733 32508.0 0.24042 30.0303 8.2193 0.27370 35.7956 0.25806 0.006 36 EXPERIMENTS UPON THE TEEMONT TURBINE DESCRIPTION OF THE DIAGRAM REPRESENTING THE EXPERIMENTS. 75. For the purpose of presenting a general view of the experiments, the coefficients of effect, at different velocities, are plotted at figure 1, plate VI., on a system of coordinates. The ratios of the velocities of the interior circumference of the wheel, to the velocities due the fall acting upon the wheel, given in column 20, table II., are taken to represent the velocities; these ratios are here called the velocities, and are taken on the axis of abscissas AX; the correspond- ing coefficients of effect given in column 17, table II., are taken upon the axis of ordinates AY. 76. The line CD represents the experiments made with the regulating gate fully raised ; to avoid confusion a portion of the experiments are omitted ; the experiments represented are those numbered from 4 to 42, inclusive, which were made in regular sequence, with gradually increasing weights. It will be observed in the table of experiments, that several trials were made with the brake entirely removed; these were made, generally, after the wheel had been left for some time, for the purpose of seeing if it was in as good running order as usual ; if any material change had taken place, it would have been indicated by a change in the velocity of the wheel. The experiments thus made, omitting experiment 12, in which the height in the wheelpit was not observed, are collected together in the following table. Number of the experiment. Ratio of the velocity of the interior cir- cumference of the wheel, to the velocity due the fall acting upon the wheel. 13 23 31 38 45 1.33366 1.33567 1.33635 1.33544 1.33521 Mean . . . 1.33527 The greatest variation in these velocities is in experiment 13, which is part below the mean ; the running condition of the wheel must, consequently, have been nearly uniform. In all the experiments with the brake removed, the coefficient of effect, of course, is nothing, and they would be represented on the diagram by points on the axis' of abscissas; for the sake of distinctness, only one of those tried when Ihe gate was at its full height, is represented on the diagram. EXPERIMENTS UPON THE TREMONT TURBINE. 37 There is a small irregularity in the line CD, at numbers 26 and 27; both these experiments were made with the same weight in the scale, and under sim- ilar circumstances, except that in 26, water was used to lubricate the friction pulley, and in 27 oil was used. It has been stated, that, with heavy loads, the brake operates much more steadily with oil as a lubricator, than with water, and the change in the lubrica- tor at experiment 27, was made in consequence of the difficulty experienced by the operator, in regulating the tension of the brake screws. In experiment 26, nearly his whole strength, applied to the extremity of a wrench about three feet long, was required to move the nuts, whereas, in experiment 27, the same opera- tion was performed with great ease. Experiment 26 was of much shorter dura- tion than experiment 27, and a portion of the discrepancy may be due to a proportionally less perfect observation of the data in 26. The line CD shows that, with a velocity of the interior circumference of the wheel not less than 44 or more than 75 per cent, of that due to the fall, the useful effect is 75 per cent, or more, of the total power expended. Beyond these points, the change in the coefficient of effect is nearly equal for equal and opposite variations of speed ; thus, the diagram indicates that the coefficient of effect is 70 per cent, of the power expended, at the velocities 0.360 and 0.834. 0.436 0.360 = 0.076 0.834 0.750 = 0.084. Taking the mean of the extreme velocities, that is, of 0, when the wheel was still, and 1.335, when the brake was removed, we have 1.335 + = 0j66m SB which is not far from the velocity giving the maximum coefficient of effect; that is to say, when the gate is fully raised, the coefficient of effect is a maximum when the wheel is moving with about half its maximum velocity, 77. Experiments 43 and 44 were both made with the gate fully raised, but the wheel at rest, the brake being screwed up sufficiently tight to prevent the wheel from revolving ; they were made for the purpose of ascertaining the total effort that could be exercised by the wheel. By reference to column 9, of the table of experiments, it will be seen that, in experiment 43, the weight sustained was 4213.38 pounds, and in 44, the weight was 3946.38 pounds. These experiments were made under circumstances nearly identical, except that in 43, the weight preponderated, and in 44, the power of 38 EXPERIMENTS UPON THE TREMONT TURBINE. the wheel preponderated. In 43, the weight was the least that would cause the scale to lower when the bell crank was placed horizontally, and then left free ; on the other hand, in experiment 44, the weight was the greatest that would allow the scale to be raised under the same circumstances; that is to say, in 43, the weight represents the force exercised by the water against the wheel, plus the friction of the entire apparatus, and in 44, the weight represents the same thing, minus the friction; the difference of the weights, or 4213.38 3946.38 = 267 pounds, represents double the friction, and the true force exercised by the water against the wheel, is represented by the weight 4213.38 + 3946.38 __ ^Q gg pounc } s 2 This weight acted at a distance from the centre of the wheel, equal to the effective length of the brake, or 10.827778 feet (art. 50). The radius of the turbine, at the outer extremities of the buckets, is 4.146 feet (art. 35), consequently, the equivalent force acting tangentially at the outer extremities of the buckets, was 4079.88 X 10.827778 ^^ dg 4.146 78. The line E F represents the experiments numbered 77, 78, and 79, made with the gate raised 9.96 inches, or about 87 per cent, of the full height. By a reference to the table of experiments, it will be seen that, although the regu- lating gate was lowered 13 per cent., the quantity of water discharged by the wheel was diminished less than one per cent. 79. The line GH represents the experiments numbered from 51 to 64, inclu- sive, made with the gate raised 8.55 inches, or about three fourths of the full height. 80. The line IK represents the experiments numbered from 65 to 76, inclu- sive, made with the gate raised 5.65 inches, or nearly a half of the full height. 81. The line LM represents the experiments numbered from 80 to 87, inclu- sive, made with the gate raised 2.875 inches, or one fourth of its full height. Experiments 88 and 89 were made with the same height of gate, but with the wheel held fast by the brake; the force exerted by the wheel at the distance 10.827778 feet, independent of friction, was 1195.06 -f 1054.25 -if" ' = EXPERIMENTS UPON THE TREMONT TURBINE. 39 82. The line NO represents the three experiments numbered 90, 91, and 92, made with the regulating gate raised one inch. An examination of the diagram will show that the velocity corresponding to the maximum coefficient of effect, diminishes with the height of the gate. For heights not less than one fourth of the whole height, this diminution is sufficiently regular ; for heights less than one fourth, the experiments are not sufficient to indicate the velocity giving the best effect, but the diminution is evidently more rapid than for greater heights of gate. PATH DESCRIBED BY A PARTICLE OF WATER IN PASSING THROUGH THE WHEEL. 83. As in many other problems in hydraulics, resort is here had to a par- ticular hypothesis, which, at best, is only an approximation to the truth, neverthe- less, it may be the means of throwing some light upon the mode in which the water acts upon the wheel. The particular hypothesis here assumed is this; every particle of water contained in the wheel, situated at the same distance from the axis, moves in the same direction relative to the radius, and writh the same velocity. According to this hypothesis, the successive sections in which the same particles of water are found, are in cylindrical surfaces, concentric with the wheel. Applying this hypothesis to experiment 30, on the Tremont Turbine, let us suppose ($' = the mean quantity of water discharged through each aperture of the wheel, in cubic feet per second. to = the angular velocity of the wheel. R = the radius of the circle inscribing the inner edges of the buckets, or A, figure 3, plate VI. R = the radius B. t = the time occupied by a particle of water in passing from the section A D to the section B O, or, which is the same thing, through the radial distance R R. A = the area of AS CD, in square feet. H= the mean height, in feet, between the crowns of the wheel, between the sections A D and B 0. We have AH=the volume of water contained between the sections AD and B C. 40 EXPERIMENTS UPON THE TREMONT TURBINE. t is the time occupied by a particle of water in passing from the section AD to the section B C, and it will evidently be the time required for the dis- charge of the volume AH. We find t by the proportion If the wheel was at rest, a particle of water at A would arrive at B in the time t, but the wheel is moving with the angular velocity to, therefore the point B, in the time t, will have advanced to E, and consequently, a particle of water at A, instead of being at B, at the end of the time t, will have arrived, by some path, at the point E. In this manner, by taking successive values of R, sufficiently near to each other, the entire path of a particle of water, from its entrance into the wheel, up to the moment of its discharge, may be traced; and as, by the hypothesis, all the particles at the same distance from the axis move with the same velocity, and in the same relative direction, the path of the entire stream, from its entrance into the wheel to its discharge, will be determined. In experiment 30, we have the total quantity discharged by the wheel equal to 138.1892 cubic feet per second; as the wheel has forty-four apertures, q,_ 138.1892 _3 14066 cubic feet per gecond. 44 The velocity of the interior circumference of the wheel was 18.0474 feet per second, and the interior radius of the wheel being 3.375 feet, we have 0, = ^^^= 5.3474 feet per second, o.o/O consequently, BE _wuiKAu. _ j 702 g RAH 3.14066 84. The successive steps in the calculation for the entire path, are given in table III. The arcs of circles F G, HI, etc. are drawn on a plan of the buckets, figure 2, plate VI., with the radii contained in the first column. COLUMN 2 contains the entire areas of these circles. EXPERIMENTS UPON THE TREMONT TURBINE. 41 COLUMN 3 contains the areas of the rings comprised between these circles, which are obtained by taking the differences of the successive areas in column 2. COLUMN 4 contains the areas reduced to square feet, of that part of each ring corresponding to a single aperture in the wheel, including also the area occupied by the thickness of the corresponding part of one bucket. COLUMN 5. Corrections for the thickness of the buckets ; these are deduced from measurements taken on a full sized plan of the buckets. COLUMN 6. True areas of the partial rings, being the differences of the cor- responding areas in columns 4 and 5. COLUMN 7. Mean heights of the partial rings; these are also taken from a full sized drawing of the wheel. COLUMN 8. Volumes of the partial rings, or the products of the corresponding numbers in columns 6 and 7. COLUMN 9. Volumes between the radius R, and the successive values of the radius tf. These are obtained by adding together the volumes of the partial rings, up to the corresponding radius ; they are the successive values of A H. COLUMN 10. The ordinates; these are successive values of 1.7026 RAH, the successive values of K being taken in feet, instead of inches, as they are given in column 1. TABLE III. 1 a 3 4 5 6 ,7 8 9 10 Value of K, and successive values of R'. Inches. Areas in square inches, of circles of the radii in the last column. Areas in square inches of the complete rings. A of the areas of the rings in the last column. Square feet. Correction for the thickness of the bucket, in square feet. True areas of the partial rings, in square feet. Mean height of the partial rings, infect. Volumes of the partial rings, in cubic feet. Volumes between K and the successive values of R f . Cubic feet. Ordinates in feet, to be measured on arcs of the corresponding radii hi column 1. 40.5 5152.997 41.5 5410.G08 257.611 0.04066 0.00091 0.03975 0.9264 0.03682 0.03682 0.2168 42.5 5674.502 263.894 0.04165 0.00099 0.04066 0.9080 0.03692 0.07374 0.4447 43.5 5944.679 270.177 0.04264 0.00106 0.04158 0.8940 0.03717 0.11091 0.6845 44.5 6221.139 276.460 0.04363 0.00115 0.04248 0.8840 0.03755 0.14846 0.9373 45.5 6503.882 282.743 0.04462 0.00128 0.04334 0.8775 0.03803 0.18649 1.2039 46.5 6792.909 289.027 0.04562 0.00146 0.04416 0.8755 0.03866 0.22515 1.4854 47.5 7088.218 295.309 0.04661 0.00174 0.04487 0.8800 0.03949 0.26464 1.7835 48.5 7389.811 301.593 0.047'60 0.00212 0.04548 0.8920 0.04057 0.30521 2.1003 49.0 7542.964 153.153 0.02417 0.00138 0.02279 0.9055 0.02064 0.32585 2.2654 49.25 7620.129 77.165 0.01218 0.00078 0.01140 0.9145 0.01042 0.33627 2.3498 49.50 7697.687 77.558 0.01224 0.00087 0.01137 0.9210 0.01047 0.34674 2.4352 49.75 7775.638 77.951 0.01230 0.00081 0.01149 0.9277 0.01066 0.35740 2.5228 6 42 EXPERIMENTS UPON THE TREMONT TURBINE. 85. The arcs FG, HI, etc., figure 2, plate VI, are taken equal to the ordi- nates 0.2168, 0.4447 etc., in column 10 of the table ; the points Q, G, I, etc. K, are joined by a line, which is the limit of the stream on one side. The limit on the other side is found by making the arcs GL = FN, IM=HO, etc.; the points R, L, M, etc. P, being joined by a line, give the limits of the stream on this side. 86. By an inspection of the figure, it is plain that, in experiment 30, the path of the water through the wheel must have been a continuation of the direction given to it by the fixed guides VW, and that there was no sudden change of direction or velocity, up to a point near where the water was dis- charged from the wheel. The abrupt change at this point, indicated by the figure, could not, in reality, have taken place, as we know by the direction assumed by the vane, which is represented at ST in its mean position during the experiment. 87. The foregoing hypothesis will evidently lead to results more nearly cor- rect, the nearer the buckets are to each other, until, in the case in which the spaces between them are infinitely small, it will give the path accurately. In applications like the above, where the spaces are very considerable, it is assumed by the hypothesis that the water passes through in curved laminse, superimposed on each other, the first of which, in contact with the concavity of the bucket, is constrained by it and the rotation of the wheel, to move in a particular path; this, in its turn, constrains the next lamina to move in a similar path; and so on. By an inspection of figure 2, plate VI., it is reasonable to suppose, that a lamina, far removed from the concavity of the bucket, will take a path differing from that of a lamina near it ; the abruptness in the curve near its extremity, will be diminished, somewhat in proportion to the distance of the lamina from the concavity of the bucket, the water passing out from the wheel more nearly in the direction in which it was moving, during its approach to the circumference of the wheel. These views go far to explain the discrepancy between the path determined by the hypothesis, and the direction assumed by the vane. 88. Whatever objection may be made to the method by which the path, given in figure 2, plate VI., is obtained, it cannot be denied that its general course must have been nearly as represented ; this being admitted, it is difficult to see how centrifugal force can operate in the important manner that is com- monly assigned to it. The path is concave to the axis only in a very slight degree, and through a part only of its course ; nevertheless, it is only in con- EXPERIMENTS UPON THE TREMONT TURBINE. 43 sequence of a concavity in the path, that centrifugal force can have any exist- ence. With the gate only partially raised, this force may act powerfully in increasing the discharge, and a similar effect may be produced, at high velocities, with the gate fully raised ; but in experiment 30, giving the maximum coefficient of effect, it can have had only a slight action. RULES FOR PROPORTIONING TURBINES, 89. IN making the designs for the Tremont, and other turbines, the author has been guided by the following rules, which he has been led to by a com- parison of several turbines designed by Mr. Boyden, which have been carefully tested and found to operate well. Rule 1st. The sum of the shortest distances between the buckets, should be equal to the diameter of the wheel. Rule 2d. The height of the orifices at the circumference of the wheel, should be equal to one tenth of the diameter of the wheel. Rule 3d. The width of the crowns should be four times the shortest dis- tance between the buckets. Rule 4th. The sum of the shortest distances between the curved guides, taken near the wheel, should be equal to the interior diameter of the wheel. The turbines, from a comparison of which the above rules were derived, varied in diameter from twenty-eight inches to nearly one hundred inches, and operated on falls from thirty feet to thirteen feet. The author believes that they may be safely followed for all falls between five feet and forty feet, and for all diameters not less than two feet, and, with judicious arrangements in other respects, and careful workmanship, a useful effect of seventy-five per cent, of the power expended, may be relied upon. For falls greater than forty feet, the second rule should be modified, by making the height of the orifices smaller in proportion to the diameter of the wheel. 90. Taking the foregoing rules as a basis, we may, by aid of the experi- ments on the Tremont Turbine, establish the following formulas. Let D = the diameter of the wheel at the outer extremities of the buckets. d=fhe diameter of the wheel, at the interior extremities of the buckets. -T=the height of the orifices of discharge, at the outer extremities of the buckets. W= the width of the crowns occupied by the buckets. RULES FOR PROPORTIONING TURBINES. 45 N= the number of buckets. n = the number of guides. P the horse-power of the turbine ; a horse-power being 550 pounds avoir, raised one foot per second. h = the fall acting upon the wheel. Q = the quantity of water expended by the turbine, in cubic feet per second. V= the velocity due the fall acting upon the wheel. V = the velocity of the water passing the narrowest sections of the wheel. v = the velocity of the interior circumference of the wheel : all the veloci- ties being in feet per second. C= the coefficient of V, or the ratio of the real velocity of the water passing the narrowest sections of the wheel, to the theoretical velocity due the fall acting upon the wheel. The unit of length is the English foot. It is assumed that the useful effect is seventy-five per cent, of the total power of the water expended. According to rule 1, we have the sum of the widths of the orifices of dis- charge, equal to D. Then the sum of the areas of all the orifices of discharge, is equal to DH. By the fundamental law of hydraulics we have therefore We can find the value of C in the last equation by experiment 30, on the Tremont Turbine. In that wheel we have for the sum of the widths of the orifices of discharge, 44 X 0.18757 = 8.25308 feet, and the height of the orifices of discharge = 0.9314 feet. Then we have, for the sum of the areas of all the ori- fices of discharge, = 8.25308 X 0.9314 = 7.68692 square feet By experiment 30, we have Q = 138.1892 cubic feet per second, h = 12.903 feet, = 8.0202 feet, 46 RULES FOR PROPORTIONING TURBINES. consequently, 138.1892 = 7.68692 X 8.0202 V12.903 C, or C= 0.624. By rule 2, we have H 0.10 D : then HD= 0.10 D*, and Q = JIDV f = Q.lQIPC^2gF, or Q = 0.5 If- \Th. Calling the weight of a cubic foot of water 62.33 pounds avoir, we have p _ 0.75 X 62-33 o , -550- -U' 1 ' or P = 0.085 Qh; or, substituting the value of Q just found, P= 0.0425 1? h v/T, from which we may deduce 91. The number of buckets is, to a certain extent, arbitrary, and would usually be determined by practical considerations : some of the ideas to be kept in mind are the following. The pressure on each bucket is less, as the number is greater ; the greater number will therefore permit of the use of the thinner iron, which is important, in order to obtain the best results. The width of the crowns will be less for a greater number of buckets : a narrow crown appears to be favorable to the useful effect, when the gate is only partially raised. As the spaces between the buckets must be proportionally narrower for a larger number of buckets, the liability to become choked up, either with anchor ice, or other substances, is increased. The amount of power lost by the friction of the water against the surfaces of the buckets, will not be materially changed, as the total amount of rubbing surface on the buckets, will be nearly constant for the same diameter: there will be a little less on the crown, for the larger number. The cost of the wheel will probably increase with the number of buckets. The thickness and quality of the iron, or other metal intended to be used for the buckets, will sometimes be an element. In some waters, wrought iron is rapidly corroded. RULES FOR PROPORTIONING TURBINES. 47 The author is of opinion that a general rule cannot be given for the num- ber of buckets; among the numerous turbines working satisfactorily in Lowell, there are examples in which the shortest distance between the buckets ia as small as 0.75 inches, and in others as large as 2.75 inches. As a guide in practice, to be controlled by particular circumstances, the fol- lowing is proposed ; to be limited to diameters of not less than two feet ; Taking the nearest whole number for the value of JV! The Tremont Turbine is 8 feet in diameter, and, according to the proposed rule, should have fifty-five buckets, instead of forty-four. With fifty-five buckets, the crowns should have a width of 7.2 inches, instead of 9 inches ; with the narrower width, it is probable that the useful effect, in proportion to the power expended, would have been a little greater when the gate was partially raised. 92. By the 3d rule, we have for the width of the crowns, " N and for the interior diameter of the wheel By the 4th rule, d is also equal to the sum of the shortest distances between the guides, where the water leaves them. 93. The number n, of the guides, is, to a certain extent, arbitrary; the practice at Lowell has been, usually, to have from a half to three fourths of the number of the buckets; exactly half would probably be objectionable, as it would tend to produce pulsations, or vibrations. 94. The proper velocity to be given to the wheel, is an important consid- eration. Experiment 30, on the Tremont Turbine, gives the maximum coefficient of effect for that wheel; in that experiment the velocity of the interior circum- ference of the wheel, is 0.62645 of the velocity due to the fall acting upon the wheel. By reference to the other experiments with the gate fully raised, it will be seen, however, that the coefficient of effect varies only about two per cent. from the maximum, for any velocity of the interior circumference, between fifty per cent, and seventy per cent, of that due to the fall acting upon the wheel. By reference to the experiments in which the gate is only partially raised, it will be seen that the maximum corresponds to slower velocities; and as turbines, 48 RULES FOR PROPORTIONING TURBINES. to admit of being regulated in velocity for variable work, must, almost necessarily, be used with a gate not fully raised, it would appear proper to give them a velocity such, that they will give a good effect under these circumstances. With this view, the following is extracted from the experiments in table II. Ratio of the velocity of the interior cir- Number of the experiment. Height of the regulat- ing gate, in inches. cumference of the wheel, to the velocity due the fall acting upon the wheel, cor- responding to the maximum coefficient of effect. 30 11.49 . 0.62645 62 8.55 0.56541 73 5.65 0.56205 84 2.875 0.48390 By this table it would appear, that, as turbines are generally used, a velocity of the interior circumference of the wheel, of about fifty-six per cent, of that due to the fall acting upon the wheel, would be most suitable. By reference to the diagram at plate VI., it will be seen that, at this velocity when the gate is fully raised, the coefficient of effect will be within less than one per cent, of the maximum. Other considerations, however, must usually be taken into account, in deter- mining the velocity ; the most frequent is the variation of the fall under which the wheel is intended to operate. If, for instance, it was required to establish a turbine of a given power, on a fall liable to be diminished to one half, by backwater, and, that the turbine should, be of a capacity to give the requisite power at all times; in this case, the dimensions of the turbine must be deter- mined for the smallest fall; but if it has assigned to it a velocity, to give the maximum effect at the smallest fall, it will evidently move too slow for the greatest fall; and this is the more objectionable, as, usually, when the fall is greatest, the quantity of water is the least, and it is of the most importance to obtain a good effect. It would then be usually, the best arrangement, to give the wheel a velocity corresponding to the maximum coefficient of effect, when the fall is the greatest. To assign this velocity, we must first find the propor- tional height of gate, when the fall is greatest; this may be determined approxi- mately by aid of the experiments on the Tremont Turbine. We have seen that P = 0.085 Qh. Now, if h is increased to 2h, the velocity, and, consequently, the quantity of water discharged, will be increased in the proportion of ^h to y/2A; that is to say, the quantity for the fall 2 h, will be ~ RULES FOR PROPORTIONING TURBINES. 49 V Calling P 1 the total power of the turbine on the double fall, we have F=Q.Q85 \f2Q2h, or P' = 0.085 X 2.8284 Qh. Thus, the total power of the turbine is increased 2.8284 times, by doubling the fall ; on the double fall, therefore, in order to preserve the effective power uniform, the regulating gate must be shut down to a point that will give only S.'OST P ar t f the total power of the turbine. In experiment 15, the fall acting upon the wheel was 12.888 feet, and the total useful effect of the turbine was 85625.3 pounds raised one foot per second ; s-'ST'ST P ar t f this * s 30273.4 Ibs. ; consequently, the same opening of gate that would give this last power, on a fall of 12.888 feet, would give a power of 85625.3 Ibs. raised one foot per second, on a fall of 2 X 12.888 feet = 25.776 feet To find this opening of gate, we must have recourse to some of the other experiments. In experiment 73, the fall was 13.310 feet, the height of gate 5.65 inches, and the useful effect 58830.1 pounds. In experiment 83, the fall was 13.435 feet, the height of gate 2.875 inches, and the useful effect, 27310.9 pounds. Reducing both these useful effects to what they would have been, if the fall was 12.888 feet, the useful effect in experiment 73, 58830.1 (j||f) = 56054.5, 3 " 83, 27310.9 (JJ^I) = 25660.1. By a comparison of these useful effects with the corresponding heights of gate, we find, by simple proportion of the differences, that a useful effect >f 30273.4 pounds raised one foot high per second, would be given when the height of the regulating gate was 3.296 inches. By another mode : as 25660.1 : 2.875 :: 30273.4 : 2.875 X 8 2 ^' 4 = 3.392 inches, 2obo0.1 a little consideration will show, that the first mode must give too little, and the second, too much; taking a mean of the two results, we have for the height of the gate, giving ^-.g-J^ of the total power of the turbine, 3.344 inches. Referring to table II., we see that, with this height of gate, in order to obtain the best coefficient of useful effect, the velocity of the interior circumference of 7 50 RULES FOR PROPORTIONING TURBINES. the wheel, should be about one half of that due to the fall acting upon the wheel ; and by comparison of experiments 74 and 84, it will be seen that, with this height of gate, and with this velocity, the coefficient of useful effect must be near 0.50. This example shows, in a strong light, the well-known defect of the turbine, viz., giving a diminished coefficient of useful effect, at times when it is important to obtain the best results. One remedy for this defect would be, to have a spare turbine, to be used when the fall is greatly diminished ; this arrange- ment would permit the principal turbine to be made nearly of the dimensions required for the greatest fall. As at other heights of the water, economy of water is usually of less importance, the spare turbine might generally be of a cheaper construction. 95. To lay out the curve of the buckets, the author makes use of the following method. Keferring to plate III., figure 1, the number of buckets, N, having been deter- mined by the preceding rules, set off the arc gi=^ Let (a=ffh, the shortest distance between the buckets; t = the thickness of the metal forming the buckets. Make the arc gjc=.5(a. Draw the radius Ok, intersecting the interior cir- cumference of the wheel at I; the point I will be the inner extremity of the bucket. Draw the directrix Im tangent to the inner circumference of the wheel. Draw the arc on, with the radius w-f-tf, from i, as a centre; the other directrix, gp, must be found by trial, the required conditions being, that, when the line ml is revolved round to the position gt, the point m being constantly on the directrix gp, and another point at the distance mg = rs, from the extremity of the line describing the bucket, being constantly on the directrix ml, the curve described shall just touch the arc no. A convenient line for a first approxima- tion, may be drawn by making the angle Ogp = 11. After determining the directrix according to the preceding method, if the angle Ogp should be greater than ]2, or less than 10, the length of the arc g k should be changed, to bring the angle within these limits. The curve gss's"l, described as above, is nearly the quarter of an ellipse, and would be precisely so, if the angle gml was a right angle; the curve may be readily described, mechanically, with an apparatus similar to the elliptic tram- mel ; there is, however, no difficulty in drawing it by a series of points, as is sufficiently obvious. RULES FOR PROPORTIONING TURBINES. 51 96. The trace adopted by the author, for the corresponding guides, is as follows. The number n having been determined, divide the circle, in which the extremities of the guides are found, into n equal parts, vw, wx, etc. Put o/ for the width between two adjoining guides, and t for the thickness of the metal forming the guides. We have by rule 4, a/ = . With w as a centre, and the radius (a'-\-if, draw the arc yz; and with # as a centre, and the radius 2 (a/ -(- t'], draw the arc a' b'. Through v draw the portion of a circle v c', touching the arcs y z and a' V ; this will be the curve for the essential part of the guide. The remainder of the guide, c'd', should be drawn tangent to the curve c'v ; a convenient radius is one that would cause the curve c'd', if continued, to pass through the centre 0. This part of the guide might be dispensed with, except that it affords great support to the part c'v, and thus permits the use of much thinner iron than would be necessary, if the guide ter- minated at c', or near it. 97. Collecting together the foregoing formulas for proportioning turbines, which, it is understood, are to be limited to falls not exceeding forty feet, and to diameters not less than two feet; we have for the horse-power, P= 0.0425 2? h \JJ', for the diameter, for the quantity of water discharged per second, for the velocity of the interior circumference of the wheel, when the fall is not very variable, v = 0.56 \l2gh, or, v 4.491VJ; x for the height of the orifices of discharge, 52 RULES FOR PROPORTIONING TURBINES. for the number of buckets, for the shortest distance between two adjacent buckets, -=> for the width of the crown occupied by the buckets, W * D - "W for the interior diameter of the wheel, ,7 n 8 - d = D- ; for the number of guides, = 0.50^ to 0.75 N; for the shortest distance between two adjacent guides, Table IV. has been computed by these formulas. For falls greater than forty feet, the height of the orifices in the circum- ference of the wheel, should be diminished ; the foregoing formulas may, however, still be made use of; thus, supposing that for a high fall, it is determined to make the orifices three fourths of that given by the formula ; divide the given power, or quantity of water to be used, by 0.75, and use the quotient in place of the true power, or quantity, in determining the dimensions of the turbine ; no modi- fication of the dimensions will be necessary, except that -fa of the diameter of the turbine should be diminished to -fa of the diameter, to give the height of the orifices in the circumference. 98. It is plain, from the method by which the preceding formulas have been obtained, that they cannot be considered as established, but should only be taken as guides in practical applications, until some more satisfactory are pro- posed, or the intricacies of the turbine have been more fully unravelled. The turbine has been an object of deep interest to many learned mathematicians, but, up to this time, the results of their investigations, so far as they have been published, have afforded but little aid to Hydraulic Engineers. RULES FOR PROPORTIONING TURBINES. 53 TABLE IV. Table for Turbines of different diameters, operating on different falls; assuming that the useful effect is seventy-Jive per cent, of the power expended ; also that the velocity of the interior circumference is fifty-six per cent, of the velocity due the fall; and also that the height between the crowns is fa of the outside diameter. ' Outside diameter 2.000 feet. Outside diameter 3.000 feet. Outside diameter 4.000 feet. Outside diameter 5.000 feet. Outside diameter 6.000 feet. Inside " 1.656 " Inside " 2.385 " Inside " 3.238 " Inside " 4.111 " Inside " 5.000 " Fall Number of backets 36. Number of buckets 89. Number of buckets 42, Number of buckets 45. Number of buckets 48. in Quantity Quantity Quantity Quantity of water Number of water Number of water Number of water Number Quantity Number feet. dis- Number of dis- Number of dis- Number of dis- Number of of water Number of charged of revolu- charged of revolu- charged of horse- revolu- charged of horse- revolu- discharged of horse- revolu- in cubic horse- tions in cubic horse- tions in cubic power. tions in cubic power. tions in cubic power. tions feet per power. per feet per power. per feet per per feet per per feet per per second. minute. second. minute. second. minute. second. minute. second. minute. 5 4.47 1.90 123.3 10.06 4.28 80.4 17.88 7.60 59.2 27.95 11.88 46.7 40.25 17.11 38.4 6 4.90 2.50 135.1 11.02 5.62 88.1 19.60 9.99 64.9 30.62 15.61 51.1 44.09 22.49 42.0 7 5.29 3.15 145.9 11.91 7.08 95.2 21.17 12.59 70.1 33.07 19.68 55.2 47.62 28.34 45.4 8 5.66 3.85 156.0 12.73 8.66 101.7 22.63 15.39 74.9 35.35 24.04 59.0 50.91 34.62 48.5 9 6.00 4.59 165.4 13.50 10.33 107.9 24.00 18.36 79.5 37.50 28.69 62.6 54.00 41.31 51.5 10 6.32 5.38 174.4 14.23 12.10 113.7 25.30 21.50 83.8 39.53 33.60 66.0 56.92 48.38 54.2 11 6.63 6.20 182.9 14.92 13.95 119.3 26.53 24.81 87.9 41.46 38.76 69.2 59.70 55.82 56.9 12 6.93 7.07 191.0 15.59 15.90 124.6 27.71 28.27 91.8 43.30 44.17 72.3 62.36 63.60 59.4 13 7.21 7.97 198.8 16.23 17.93 129.7 28.84 31.87 95.5 45.07 49.80 75.2 64.90 71.72 61.9 14 7.48 8.90 206.3 16.84 20.04 134.6 29.93 35.62 99.1 46.77 55.66 78.1 67.35 80.15 64.2 15 7.75 9.88 213.5 17.43 22.22 139.3 30.98 39.50 102.6 48.41 61.72 80.8 69.71 88.88 66.4 16 8.00 10.88 220.5 18.00 24.48 143.9 32.00 43.52 106.0 50.00 68.00 83.5 72.00 97.92 68.6 17 8.25 11.92 227.3 18.55 26.80 148.3 32.99 47.66 109.2 51.54 74.47 86.0 74.22 107.24 70.7 18 8.49 12.98 233.9 19.09 29.21 152.6 33.94 51.93 112.4 53.03 81.14 88.5 76.37 116.84 72.8 19 8.72 14.08 240.3 19.61 31.68 156.8 34.87 56.32 115.5 54.49 87.99 90.9 78.46 126.71 74.8 20 8.94 15.21 246.6 20.12 34.21 160.9 35.78 60.82 118.5 55.90 95.03 93.3 80.50 136.84 76.7 21 9.17 16.36 252.7 20.62 36.81 164.8 36.66 65.44 121.4 57.28 102.25 95.6 82.49 147.24 78.6 22 9.38 17.54 258.6 21.11 39.47 168.7 37.52 70.17 124.2 58.63 109.64 97.9 84.43 157.88 80.5 23 9.59 18.75 264.4 21.58 42.19 172.5 38.37 75.01 127.0 59.95 117.20 100.1 86.32 168.76 82.3 24 9.80 19.99 270.1 22.04 44.97 176.2 39.19 79.95 129.8 61.24 124.92 102.2 88.18 179.89 84.0 25 10.00 21.25 275.7 22.50 47.81 179.8 40.00 85.00 132.4 62.50 132.81 104.3 90.00 191.25 85.8 26 10.20 22.54 281.1 22.95 50.71 183.4 40.79 90.15 135.1 63.74 140.86 106.4 91.78 202.84 87.5 27 10.39 23.85 286.5 23.38 53.66 186.9 41.57 95.40 137.6 64.95 149.06 108.4 93.53 214.65 89.1 28 10.58 25.19 291.8 23.81 56.67 190.3 42.33 100.75 140.2 66.14 157.42 110.4 95.25 226.69 90.8 29 10.77 26.55 296.9 24.23 59.73 193.7 43.08 106.20 142.6 67.31 165.93 112.4 96.93 238.94 92.4 30 10.95 27.93 302.0 24.65 62.85 197.0 43.82 111.74 145.1 68.46 174.59 114.3 98.59 251.41 94.0 31 11.14 29.34 307.0 25.05 66.02 200.3 44.54 117.37 147.5 69.60 183.39 116.2 100.22 264.08 95.5 32 11.31 30.77 311.9 25.46 69.24 203.5 45.25 123.09 149.8 70.71 192.33 118.0 101.82 276.96 97.0 33 11.49 32.23 316.7 25.85 72.51 206.6 45.96 128.91 152.2 71.81 201.42 119.9 103.40 290.04 98.5 34 11.66 33.70 321.5 26.24 75.83 209.7 46.65 134.81 154.5 72.89 210.64 121.7 104.96 303.33 100.0 35 11.83 35.20 326.2 26.62 79.20 212.8 47.33 140.80 156.7 73.95 220.00 123.4 106.49 316.81 101.5 36 12.00 36.72 330.8 27.00 82.62 215.8 48.00 146.88 158.9 75.00 229.50 125.2 108.00 330.48 102.9 37 12.17 38.26 335.4 27.37 86.09 218.8 48.66 153.04 161.1 76.03 239.13 126.9 109.49 344.34 104.3 38 12.33 39.82 339.9 27.74 89.60 221.7 49.32 159.29 163.3 77.05 248.89 128.6 110.96 358.40 105.7 39 12.49 41.40 344.3 28.10 93.16 224.6 49.96 165.62 165.4 78.06 258.78 130.3 112.41 372.64 107.1 40 12.65 43.01 348.7 28.46 96.77 227.5 50.60 172.03 167.5 79.06 268.79 132.0 113.84 387.06 108.5 54 KULES FOR PROPORTIONING TURBINES. TABLE IV. CONTINUED. Outside diameter 7.000 feet. Outside diameter 8.000 feet. Outside diameter 9.000 feet Outside diameter 10.000 feet. Inside " 5.902 " Inside " 6.815 " Inside " 7.737 " Inside " 8.667 " Number of buckets 51. Number of buckets 54. Number of buckets 57. Number of buckets 60. Fall in Quantity of Number Quantity of Number Quantity of Number Quantity of Number feet water Number of water Number of water Number of water Number of discharged, of revolu- discharged, of revolu- discharged, of horse- revolu- discharged of horse- revolu- in cubic horse- tions in cubic horse- tions in cubic power. tions in cubic power. tions feet per power. per feet per power. per feet per per feet per per second. minute. second. minute. second. minute. second. minute. 5 54.78 23.28: 32.5 71.55 30.41 28.1 90.56 38.49 24.8 111.80 47.52 22.1 6 60.01 30.61 35.6 78.38 39.97 30.8 " 99.20 50.59 27.2 122.47 62.46 24.2 7 64.82 38.57 38.4 84.67 50.37 33.3 107.15 63.76 29.3 132.29 78.71 26.2 8 69.30 47.12 41.1 90.51 61.55 35.6 114.55 77.90 31.4 141.42 96.17 28.0 9 73.50 56.23 43.6 96.00 73.44 37.8 121.50 92.95 33.3 150.00 114.75 29.7 10 77.47 65.86 46.0 101.19 86.02 39.8 128.07 108.86 35.1 158.11 134.40 31.3 11 81.26 75.97 48.2 106.13 99.23 41.7 134.32 125.59 36.8 165.83 155.05 32.8 12 84.87 86.57 50.3 110.85 113.07 43.6 140.30 143.10 38.4 173.21 176.67 34.3 13 88.34 97.61 52.4 115.38 127.49 45.4 146.03 161.36 40.0 180.28 199.21 35.7 14 91.67 109.09 54.4 119.73 142.48 47.1 151.53 180.33 41.5 187.08 222.63 37.0 15 94.89 120.98 56.3 123.94 158.02 48.7 156.86 199.99 42.9 193.65 246;90 38.3 16 98.00 133.28 58.1 128.00 174.08 50.3 162.00 220.32 44.3 200.00 272.00 39.6 17 101.02 145.97 59.9 131.94 190.65 51.9 166.99 241.29 45.7 206.16 297.89 40.8 18 103.94 159.03 61.7 135.76 207.72 53.4 171.83 262.89 47.0 212.13 324.56 42.0 19 106.79 172.47 63.3 139.48 225.27 54.9 176.53 285.10 48.3 217.94 351.98 43.1 20 109.57 186.26 65.0 143.11 243.28 56.3 181.12 307.91 49.6 223.61 380.13 44.3 21 112.27 200.41 66.6 146.64 261.75 57.7 185.60 331.28 50.8 229.13 408.99 45.4 22 114.91 214.89 68.2 150.09 280.67 59.0 189.96 355.23 52.0 234.52 438.55 46.4 23 117.50 229.71 69.7 153.47 300.03 60.4 194.23 379.72 53.2 239.79 468.79 47.5 24 120.02 244.85 71.2 156.77 319.81 61.7 198.41 404.76 54.3 244.95 499.70 48.5 25 122.50 260.31 72.7 160.00 340.00 62.9 202.50 430.31 55.4 250.00 531.25 49.5 26 124.93 276.09 74.1 163.17 360.60 64.2 206.51 456.39 56.5 254.95 563.44 50.5 27 127.30 292.17 75.5 166.28 381.61 65.4 210.45 482.97 57.6 259.81 596.26 51.4 28 129.64 308.55 76.9 169.33 403.00 66.6 214.31 510.05 58.7 264.58 629.69 52.4 29 131.93 325.22 78.3 172.32 424.78 67.8 218.09 537.61 59.7 269.26 663.72 53.3 30 134.19 342.19 79.6 175.27 446.94 68.9 221.83 565.66 60.7 273.86 698.35 54.2 31 136.41 359.44 80.9 178.17 469.47 70.1 225.50 594.18 61.7 278.39 733.55 55.1 32 138.59 376.97 82.2 181.02 492.37 71.2 229.10 623.16 62.7 282.84 769.33 56.0 33 140.74 394.78 83.5 183.82 515.63 72.3 232.66 652.59 63.7 287.23 805.67 56.9 34 142.86 412.86 84.7 186.59 539.24 73.4 236.16 682.48 64.6 291.55 842.57 57.7 35 144.94 431.21 86.0 189.31 563.21 74.5 239.60 712.82 65.6 295.80 880.02 58.5 36 147.00 449.82 87.2 192.00 587.52 75.5 243.00 743.58 66.5 300.00 918.00 59.4 37 149.03 468.69 88.4 194.65 612.17 76.6 246.35 774.77 67.4 304.14 956.51 60.2 38 151.03 487.82 89.6 197.26 637.15 77.6 249.66 806.40 68.3 308.22 995.55 61.0 39 153.00 507.20 90.8 199.84 662.47 78.6 252.92 838.44 69.2 312.25 1035.11 61.8 40 164.95 526.83 91.9 202.39 688.12 79.6 256.15 870.89 70.1 316.23 1075.17 62.6 55 EXPERIMENTS ON A MODEL OF A CENTJIE-VENT WATER-WHEEL, WITH STRAIGHT BUCKETS. 99. THE author was led to this design by the consideration of the path of the water in passing through the wheel, according to the hypothesis in art. 83. It is a wheel well suited for low falls, in which the water, over the wheel, may stand at its natural height, without requiring a vertical shaft of great length. Its simplicity and cheapness, combined with its other good qualities as a hydraulic motor, must recommend it for many such situations. 100. Plate VII., figure 1, is a general plan, and figure 2, a vertical section of the apparatus. Figure 3 is a vertical section through the apertures in the guides and wheel; the guides and buckets are omitted to avoid confusion in the figure. Figure 4 is a horizontal section of part of the guides and buckets, showing, also, the path of the water in experiment 3, according to the hypothesis in art. 83. A is the wheel ; the exterior diameter is 22|- inches ; the interior diameter is 19 inches; the height between the crowns, or B C, figure 3, is 2|| inches; it carries thirty-six buckets, EE, figure 4, of steel, about -fa of an inch in thick- ness, fastened to the wheel by means of the wooden cushions F F, figure 3 ; the upper cushions are screwed to the disc D, and the lower ones to the crown G. The disc D is of castriron, f inch thick, with a suitable hub by which it is connected with the vertical shaft. HH are guides of cast-iron, which direct the water into the wheel, and also support the plate I, which protects the wheel from pressure on its upper surface ; the contraction of the streams entering the apertures between the guides, is dimin- ished by the curved wooden garniture K; there are twenty-four guides. The mean shortest distance between the buckets at ab, figure 4, is 0.0339 feet; the mean shortest distance between the guides cd, figure 4, is 0.0437 feet; and the height of both is 2| | inches 0.2344 feet ; we have, therefore, for the sum of the areas of the smallest sections between the guides, 0.0437 X 0.2344 X 24 = 0.24584 square feet. 56 EXPERIMENTS ON A MODEL OF A Similarly, the sum of the areas of the smallest sections between the buckets is 0.0339 X 0.2344 X 36 = 0.28606 square feet. | The water is admitted into the forebay L, by the pipes MM; the diaphragm N is to diminish the agitation of the water. 101. The apparatus for gauging the water discharged by the wheel, con- sisted of the weir 0, which had sharp edges; the depth on the weir was measured by a hook gauge, in the box P, which communicated, by a small aperture, with the surrounding water; the height of the water above the wheel was taken at a gauge in the box Q; this box was made sloping on one side, in order to permit a better view of the gauge. The zeros of both gauges were at the level of the top of the weir; consequently, the difference in the read- ings of the gauges gave at once the fall acting upon the wheel. 102. The apparatus for measuring the power, consisted of the Prony dynamom- eter R, attached to the upper part of the vertical shaft ; the weights were applied by means of the bell crank S, figures 1, 2, and 5 ; the oscillations of the brake were diminished by the hydraulic regulator T, and the extent of the oscillations was limited by the stops UU. The speed of the wheel was obtained by means of a counter, driven by the worm V, attached to the top of the upright shaft; this was so arranged as to strike a bell once in fifty revolutions of the wheel. In order to diminish the passive resistances, the weight, bearing upon the step W, was counterbalanced, in part, by other weights, one of which is represented at y, figure 2 ; these were attached to the brakes at the points 2&X, by vertical cords passing over pulleys ; the weight, resting on the step when the wheel was immersed, and the dynamometer attached, was found to be 170 pounds; the coun- terbalance was 160 pounds, leaving 10 pounds bearing upon the step. The entire apparatus for measuring the power, was in equilibrium when there were no weights in the scale. 103. In all the experiments, except experiment 10, the brake was lubricated with oil ; in experiment 10 water was used for this purpose ; experiments 9 and 10 were identical in all other respects. It was noticed in experiment 10 that the whole apparatus trembled very much ; this must have consumed some power, which is perceptible in the coefficients of effect. Experiment 9, in which oil was used, and in which the trembling of the apparatus was very slight, gives a coeffi- cient of effect of 0.6922; while experiment 10, in which water was used to lubricate the brake, and in which the trembling of the apparatus was very dis- tinct, gave 0.6886 as the coefficient of effect. CENTRE-VENT WATER-WHEEL, WITH STRAIGHT BUCKETS. 57 \ 104. All the apparatus was constructed with great care and precision; the surfaces of the cast-iron guides were ground smooth ; and the cast-iron disc and lower crown of the wheel were turned true, and polished, in order to diminish, as much as possible, the resistance of the water to the motion of the wheel. 105. In table V., the quantity of water discharged has been calculated by the formula # = 3.33(7 Q.lnh)$, in which Q = ihe quantity in cubic feet per second; l = the length of the weir = 3.003 feet ; n = the number of end contractions 2 ; h = the depth upon the weir. The weights were obtained for the purpose from Mr. 0. A. Richard- son, the official sealer of weights and measures for the City of Lowell. The effective length of the lever of the dynamometer, was two feet. The tempera- ture of the water was 63 Fahrenheit. Temperature of the air at 8 h , 35' A. M., 63 Fahrenheit. The weight of a cubic foot of water is taken at 62.3128 pounds, which is deduced from table I. If, in any experiment, the brake touched, even momentarily, either of the stops UU, it was rejected; with the use, however, of a regular and sufficient quantity of oil to lubricate the brake, and a properly constructed hydraulic reg- ulator, there is seldom any difficulty from this cause, except at very low velocities. 8 58 EXPERIMENTS ON A MODEL OF A r4 PQ 5 i g H fe O p H ft O S X o 02 PS co 03 00 CO CO co |*li*f|all -.Sggo-2g, s a a S x _r a. i S-'Sgg-"^^,i s v, -a s: s v. % i*l S-S^o & 3 o -g 5 fe fl y TT -3 -r d S 1 1 1 1 1 S i j i i * i S -s fa o & *0 a! '1 O -^ -*S C O * I s I* H o < - JS o, ja c g I g * 3 a 3 " 1 1 a i i-g ^ *% ll|l & F 1 1 s B i - |S f * I 5 1 = i S ^ Tt< O t OO Cl CO O *-O 05 OO CO CO OO O O CO (M OJ CO i O -^ |> CO -*J< OOOOOO OOOOO OOCO OOOOO O O O < O O O O O O i I i O O O O rH rH rH i i I-H O (>i rH -^t* i-H t> 00 ^ t-- CO rH CO t CO CM O t O CO COGOOOCO ^ i O i-H i-H I-H O O O O i-H i-H rH ,-H r-< o CO i I CO O O CO O rH O~ CO 00 l>- 1O O CO - O O COCOCO OOOOO OOOOO O rH i-5 t-- 00 O O ~-^~rH t~CO~~O~~ GO O OO COOC3OOO* O*O OOO* rHCOCO i-H r-( Ift (M O O CD CO O C5 G^ Tf< CO COOOOOG<1O (NiO(N-^~o~!-H~''* co -^ o t- o co co o i> rH rH i-H C5 O5 C5 O5 ^4 ^-1 ^-1 ff^ ff^ ^H ^J ^4 C*'! rH rH rH rH O^ ^^ ^^ ^>1 ^^ C^ O - O rH CO CO CO CO CO CO CO CO CO CO CO CO CO OOOOO OOOOO OOO OO O l> O -^ OlOt>-OCO rHrHt^ ^H CO O> ^H C^ "^ t^" Ci ^^ ^^ O> CO ^-1 " ~) CO 1O CO CO CO " (M CO O CO GO rH rH >O I C^ !> CO ^5 C^ rH rH rH OJ rH rH CN rH rH rH rH O O ic >ra ra t~ t- oo -g OOO S S <3> O *O t>- t> lO O O 10 C3 * cs i i co co COQOOO^I-I cooo^ rHlOlOrHrH (NrHrHOC^ CO * I l>- co *o ^o ^^ ^^ co *^ co ^^ COCOCOCOOO COOOOICOOO COrHO (jq^HrHff^kO (NkO^rHrH U^rHliO i-l O O < CO i ICO -^OOOCNO rHOC75 i i^ ^-J *O ' CO *O ^^ rH rH rH rH CO CO CO ' C5 t~ O O 1 1O i-l O O ^ C*i !> CO rH O ^ ^ CO O -O rH I C5 l-H o >O co 10 10 , the wheel; E, the guides ; F, the regulating gate, the apparatus for moving which, is not rep- resented ; G, the disc, which relieves the wheel from the vertical pressure of the water, and which also supports the lower bearing of the vertical shaft. The leather packing of the regulating gate F, slides against the circumference of the disc, which is turned smooth and cylindrical for that purpose, and the disc itself is supported by means of four brackets, two of which are represented at HH, by the columns II. The vertical shaft K is of wrought iron, and it passes through the stuffing box L, and is supported by the box M, which has a series of recesses lined with babbit metal, fitted to receive a corresponding series of projections in the vertical shaft. The wheel, the vertical shaft, and the bevel gear usually on the latter, have a total weight of about 15,200 pounds ; the bearing surface in the box M is about 331 square inches, consequently, the weight, per square inch, of bearing surface, is about 46 pounds. Figures 3 and 4, plate VIII., represent the wheel and guides on a larger scale. The buckets and guides are equal in number, there being forty of each ; the buckets are of plate iron, ^ of an inch in thickness ; the guides are of the same material, T 8 g of an inch in thickness. The following dimensions were taken after the parts were finished : Mean shortest distance between adjacent buckets, or a I figure 4, 0.1384 feet. AT THE BOOTT COTTON-MILLS. C3 Mean height between the crowns, at the inner extremities of the 'buckets, or cd, figure 3, 1.2300 feet. Mean height between the crowns, at the outer extremities of the buckets, or ef, figure 3, 0.9990 Mean shortest distance between the adjacent guides, or gh, figure 4, 0.1467 " Mean height of the orifices between the guides, or ik, figure 3, 1.0066 " Diameter of the wheel at the outside of the buckets, .... 9.338 " Diameter of the wheel at the inside of the buckets, .... 7.987 " 113. Several of the peculiar features of this design are covered by patents issued by the Government of the United States to TJ. A. Boyden. His patents cover the arrangement of the regulating gate, by placing it between the guides and the wheel, and having it detached from the garniture ; making the height between the crowns of the wheel greater where the water is discharged, than where it enters; they also cover the self-adjusting apparatus on which the box M is supported. . 114. Returning to figures 1 and 2, -plate VIII., N is the friction pulley of the dynamometer, which is attached to the part of the shaft intended to receive the hub of the bevel gear, for the transmission of the power ; 0, the brake of maple wood ; P, the bell crank, and Q, the hydraulic regulator ; the friction pulley and the brake were subsequently used in the experiments on the Tremont Turbine, in the account of which they are more particularly described, (see arts. 37 and 38). R, the weir at which the water discharged by the wheel was gauged; S, a grating for the purpose of equalizing the flow of the water towards the weir ; T, the gauge box in which the depths on the weir were observed. The communication between the water inside the box, and that surrounding it, was maintained by means of an aperture in the bottom of the box, (which extended 1.06 feet below the top of the weir,) and which was 4.12 feet from the weir. It may be thought, at first sight, that the depths on the weir were taken so near it, as to be affected by the curvature in the surface, caused by the discharge over the weir, but the experiments at the Lower Locks, (art. 173,) prove, conclusively, that when the communication between the water inside the box, and that outside of it, is maintained, by means of a pipe opening near the bottom of the canal, the depths are not affected in any appreciable degree, by the curvature in the surface. If any such effect was produced in this case, it must have been very slight. U and V are the gauge boxes at which the heights of the water, below and above the wheel, were observed, in order to 64 EXPERIMENTS ON THE BOOTT CENTRE-VENT WATER-WHEEL. obtain the fall acting upon the wheel. The velocity of the wheel was obtained by means of the counter W. The apparatus for lubricating the brake is not represented on the plate ; in some of the experiments, water was used, and in others, linseed oil. The experiments were made according to the method of continuous observa- tions, which has been sufficiently described in the account of the experiments on the Tremont Turbine. 115. The experiments on the Boott centre- vent water-wheel, are given in detail in table VI., which will be intelligible, without much further explanation than is contained in the respective headings of the several columns. 116. COLUMN 10. Useful effect, or the friction of the brake, in pounds avoirdupois raised one foot per second. The brake was connected with the vertical ' arm of the bell crank, by a link, which was horizontal when the brake was in its normal position. When in this position, the length of a perpendicular, from the centre of the vertical shaft, to the line joining the points of the brake and bell crank to which the link was attached, was 9.743 feet ; the effective length of the vertical arm of the bell crank, was 4.5 feet, and of the horizontal arm to which the scale was attached, 5 feet; consequently, the effective length- of the brake was 9.743 X 5 ^ 1Q826 feet 4.5 117. COLUMN 15. Quantity of water passing the wheel, in cubic feet per second. This quantity was gauged at the weir. The length of the weir was 13.998 feet; the width of the raceway on the upstream side of the weir, was 17 feet ; the crest of the weir was 11.14 feet above the bottom of the raceway. The quantity has been computed by the formula determined from the experiments made, in 1852, at the Lower Locks. (See art 258.) In this formula Q = ihe quantity in cubic feet per second. J = the length of the weir =13.998 feet. n = the number of end contractions = 2. h = the depth on the weir, given in column 14. 66 EXPERIMENTS ON A CENTRE-VENT WATER-WHEEL, TABLE EXPERIMENTS ON THE BOOTT 1 a 3 4 5 6 7 8 9 1O TIME. m-i-i ature of J-Otal Dumber Useful effect, the Height nf or the No. of the DATE, 1849. water in de- grees of Fahren- heit's ther- of the regu- lating gate, in nches. Beginning of the experiment. Ending of the experiment. LJUraiion of the experi- ment, in seconds. OI revolu- tions of the wheel during the iMimDcr 01 revolutions of the wheel per second. *v eignc 1 11 the scale, hi pounds avoirdupois. friction of the brake, in pounds avoirdupois, raised one foot per experi- mome- experi- second. ment. ter. H. luin . sec. H. lpin. sec. ment. 1 October 17. A.M. 54 3 10 13 19 10 17 32 253 150 0.59289 575.56 23211.8 2 u 11 30 3.5 11 36 42 398.5 350 0.87829 202.09 12073.5 3 it it tt n It 11 46 11 11 54 15 484 350 0.72314 407.25 20032.3 4 " 29, " 49.5 tl 11 2 17 11 19 7 1010 550 0.54455 606.00 22447.2 5 u u u U tl 11 33 41 11 45 22 701 350 0.49929 666.34 22630.5 6 (i a (t u ft 11 59 24 6 49 445 200 0.44944 720.50 22026.8 7 November 5, p. M. 44 tt 4 14 47 4 21 20 393 100 0.25445 931.87 16129.1 8 " 7, A.M. 45 tt 9 40 19 9 48 32 493 500 1.01420 0. 0. 9 October 17, P.M. 53 6 2 34 15.5 2 44 59 643.5 650 1.01010 334.06 22952.9 10 u u u tt 2 56 6 3 5 39 573 550 0.95986 441.22 28807.9 11 ft 11 it u (t 3 17 rr 6 3 26 1 f\ 56 OO \ 590 A GO K. 550 A x.r\ 0.93220 A OAAOA 501.72 Pi co J^n 31814.1 Q A A *? {* A 12 13 tt it it it u 4 50 36 5 10 4 fii& 13.5 4JU.O 817.5 401) 700 U.UUUSJU 0.85627 Obi.oU 656.59 O44/D.U 38243.0 14 " 29, " ' 50 tt 2 4 10 2 14 58 648 450 0.69444 955.50 45135.3 15 a it it U u 2 41 41 2 52 50 669 400 0.59791 1140.94 46402.8 16 November 7, A.M. 45 it 9 29 27 9 37 57 510 600 1.17647 0. 0. 17 October 29, P.M. 50 9 3 20 41 3 27 28 407 450 1.10565 263.00 19779.8 18 tl U U it tt 3 33 18 3 35 49 151 150 0.99338 531.75 35931.0 19 u u a it tt 3 36 44 3 44 8 444 400 0.90090 786.75 48212.7 20 tt tt tt u tt 3 45 5 3 54 10.5 545.5 450 0.82493 1001.47 56195.7 21 U U It u a 3 55 14 4 6 53.5 699.5 550 0.78628 1107.37 59226.4 22 It It U ti a 4 21 14 4 80 10 536 400 0.74627 1205.00 61168.8 23 tt U ft u (C 4 31 A 1 19 4 40 C -I 31 4 FT 552 C /i) E 400 A AA 0.72464 1259.16 62065.4 24 25 tt ft tt u a 4 4 41 54 42 35 4 5 51 4 .5 7.5 502.5 572.5 400 400 0.71111 0.69869 1297.31 1329.78 62752.2 63199.3 26 November 7, A.M. 45 tt 9 19 24 9 27 22.5 478.5 600 1.25392 0. 0. 27 November 5, A. M. 44 12 9 4 34.5 9 13 58.5 564 400 0.70922 1554.22 74979.3 28 tt tt tt u ti 9 15 10 9 21 7.5 357.5 250 0.69930 1584.00 75347.2 29 tt tt it tt ti 9 33 15 9 39 23 368 250 0.67935 1613.94 74580.9 30 It tl tt u ti 9 40 37 9 48 3.5 446.5 300 0.67189 1644.37 75153.2 31 It It It It tt 10 3 10 7 37.5 454.5 300 0.66007 1675.06 75208.3 32 U It It It tt 10 8 54.5 10 16 37 462.5 300 0.64865 1705.47 75249.2 33 tt tf tt It ti 10 32 31 10 41 41 550 350 0.63636 1735.94 75142.9 34 ft It tt tl u 10 43 10 51 0.5 480.5 300 0.62435 1768.41 75103.3 35 It It tt u 11 11 1 53 11 10 2 489 300 0.61350 1802.06 75202.0 36 tt tt tt u 11 11 11 24 11 18 26 422 250 0.59242 1836.19 73993.4 37 " 6, P.M. u 11 3 31 12 3 35 20 248 0. 3155.34 0. 38 a tt u u tt 3 40 16 3 42 22 126 0. 2797.27 0. 39 " 7, A.M. 45 tl 9 10 57 9 18 5 428 550 1.28505 0. 0. 1 In experiments Nos. 8, 16, 26, and 39, the brake was removed. In experiment No. 37, the weight preponderated. In No. 38, the wheel preponderated (art. 77.) AT THE BOOTT COTTON-MILLS. 67 VI. CENTRE- VENT WATEIMVHEEL 11 13 13 14 15 16 17 18 19 20 No. Height of Ratio of the of the exper- Height of the water above the the water below the wheel, taken in the Total fall acting upon the wheel. Depth of water on the weir. Quantity of water passing the wheel, in Total power of the water, in pounds avoirdupois, Ratio of the useful effect to the power Velocity due to the fall acting on the wheel, Velocity of the exterior circumference of the wheel, velocity of the exterior circumference of the wheel, to the velocity iment. wheel. wheelpit. cubic feet per second. raised one foot per second. expended. hi feet per second. in feet per second. due to the fall acting on the wheel. Feet. Feet. Feet. Feet. 1 16.013 1.410 14.603 1.2964 67.532 61493.4 0.37747 30.648 17.393 0.56750 2 16.036 1.364 14.672 1.2619 64.887 59364.4 0.20338 30.721 25.766 0.83871 3 15.955 1.387 14.568 1.2821 66.432 60347.0 0.33195 30.612 21.214 0.69301 4 15.558 1.400 14.158 1.2845 66.614 58821.6 0.38161 30.178 15.975 0.52937 5 15.607 1.410 14.197 1.2899 67.029 59351.7 0.38129 30.219 14.647 0.48470 6 15.563 1.420 14.143 1.2881 66.889 59002.2 0.37332 30.162 13.185 0.43714 7 15.604 1.360 14.244 1.2943 67.368 59858.1 0.26946 30.269 7.465 0.24661 8 15.573 1.273 14.300 1.2115 61.083 54486.4 0. 30.329 29.753 0.98101 9 15.956 1.668 14.288 1.5145 84.998 75732.8 0.30308 30.316 29.633 0.97746 10 15.930 1.704 14.226 1.5308 86.355 76608.0 0.37604 30.250 28.159 0.93086 11 15.914 1.717 14.197 1.5395 87.080 77093.2 0.41267 30.219 27.347 0.90496 12 15.923 1.730 14.193 1.5467 87.685 77607.2 0.44424 30.215 26.429 0.87470 13 15.944 1.750 14.194 1.5539 88.285 78143.8 0.48939 30.216 25.120 0.83134 14 15.581 1.803 13.778 1.5762 90.166 77480.4 0.58254 29.770 20.372 0.68433 15 15.481 1.875 13.606 1.5943 91.697 77812.1 0.59634 29.584 17.540 0.59291 16 15.451 1.506 13.945 1.4180 77.112 67076.7 0. 29.950 34.513 1.15237 17 15.408 1.890 13.518 1.6418 95.762 80736.3 0.24499 29.488 32.436 1.09997 18 15.323 1.950 13.373 1.6734 98.490 82145.2 0.43741 29.329 29.142 0.99362 19 15.352 1.983 13.369 1.6955 100.418 83728.2 0.57582 29.325 26.429 0.90125 20 15.413 2.017 13.396 1.7184 102.421 85571.4 0.65671 29.354 24.200 0.82442 21 15.426 2.047 13.379 1.7230 102.825 85800.0 0.69029 29.336 23.066 0.78629 22 15.418 2.076 13.342 1.7308 103.517 86138.0 0.71013 29.295 21.893 0.74731 23 15.424 2.102 13.322 1.7337 103.769 86218.8 0.71986 29.273 21.258 0.72620 24 15.465 2.131 13.334 1.7328 103.689 86229.3 0.72774 29.286 20.861 0.71232 25 15.464 2.160 13.304 1.7389 104.229 86483.7 0.73077 29.253 20.497 0.70067 26 15.417 1.715 13.702 1.5981 92.018 78648.0 0. 29.688 36.785 1.23907 27 15.398 1.998 13.400 1.8316 112.525 94057.5 0.79716 29.359 20.806 0.70868 28 15.434 2.003 13.431 1.8367 112.987 94662.2 0.79596 29.393 20.515 0.69796 29 15.321 1.990 13.331 1.8320 112.562 93603.9 0.79677 29.283 19.929 0.68058 30 15.369 1.991 13.378 1.8368 112.996 94296.4 0.79699 29.335 19.711 0.67193 31 15.367 1.981 13.386 1.8377 113.071 94415.2 0.79657 29.343 19.364 0.65990 32 15.369 1.986 13.383 1.8387 113.164 94471.1 0.79653 29.340 19.029 0.64856 33 15.336 1.980 13.356 1.8379 113.090 94219.0 0.79753 29.311 18.668 0.63692 34 15.362 1.981 13.381 1.8443 113.673 94881.9 0.79154 29.338 18.316 0.62431 35 15.385 1.980 13.405 1.8511 114.293 95571.2 0.78687 29.364 17.998 0.61291 36 15.292 1.971 13.321 1.8476 113.969 94703.1 0.78132 29.272 17.379 0.59371 37 15.442 1.905 13.537 1.8087 110.454 93270.1 0. 29.508 0. 0. 38 15.477 1.902 13.575 1.8072 110.325 93422.4 0. 29.550 0. 0. 39 15.415 1.819 13.596 1.6884 99.795 84635.0 0. 29.573 i 37.698 1.27477 68 EXPERIMENTS ON A CENTRE-VENT WATER-WHEEL, 118. The results of the experiments in table VI., are represented by a sys- tem of coordinates at figure 1, plate IX. ; the relative velocities, given in column 20, are taken for the abscissas, and the corresponding ratios of the useful effects to the powers expended, given in column 17, are taken for the ordinates. The numbers on the figure refer to the experiments in table VI., which the several points represent ; the points not numbered represent some experiments not reported, in consequence of an imperfection in the gauge of the quantity of water discharged, owing to a defective arrangement of the grating. These experi- ments have been corrected by a comparison with those that are reported ; not- withstanding this correction, however, they ought not to be considered as of equal value with those reported in table VI. In the figure, the points repre- senting the latter experiments, are connected by full lines; the points representing the experiments considered imperfect, are connected by broken lines. The line AB represents the experiments reported, that were made with the regulating gate fully raised ; the line CD, the experiments with the gate raised three quarters of its full height ; EF, the experiments with the gate raised a half, and GIf, the experiments with the gate raised one quarter of its full height. It will be seen that the maximum coefficient of effect, with the gate fully raised, is given, when the outside of the wheel is moving with a velocity equal to about sixty- seven per cent, of that due to the fall acting upon the wheel, at which velocity, the useful effect is very nearly eighty per cent, of the total power of the water. The coefficient of effect diminishes rapidly as the regulating gate is lowered, ;md the maximum is also found at a slower speed ; thus, when the gate is raised three inches, or one quarter of its full height, the maximum coefficient of effect is thirty-eight per cent, of the power expended ; which is given when the outside of the wheel is moving with a velocity about one half of that due to the fall acting upon the wheel. 119. AB CD, figure 2, plate IX., represents the path of the water as it passed through one of the apertures of the wheel, in experiment 30, according to the hypothesis in art. 83 ; the steps in the calculation for which, are given in table VII. In the formula we have for this case, = the ordinate measured on the arc of a circle the radius of which is R ; its several values are given in column 10. R = the distance from the centre of the wheel for which the ordinate is AT THE, BOOTT COTTON-MILLS. 69 computed ; its several values are given in inches, in column 1 ; to compute the value of in feet, R must be taken in feet. w = the angular velocity. In experiment 30, the velocity of the outside of the wheel was 19.711 feet per second, and the radius of the out- side of the wheel is 4.669 feet, consequently, _ 19.711 _ 49917 -T669 - 42217 - AH= the volume of that part of the space between two adjacent buckets, included between the outside of the wheel and the radius R' ; its several values are given in column 9. (>" = the quantity of water discharged, per second, by each orifice in the wheel. In experiment 30, we have, by table VI., the total quantity dis- charged = 112.996 cubic feet per second, and as there are forty orifices, we have 112.996 = 2.8249. In figure 2, plate IX., the buckets and guides are drawn to a scale one fourth the full size; the radius of the arc AB = R = 5Q.Q28 inches. To find the limit of the stream on the side B C, the arcs IF, KH, etc., N C, are drawn with the radii 55 inches, 54 inches, etc., 47.922 inches; the arcs E F, G H, etc., C, being taken from column 10, equal to 0.415 feet, 0.796 feet, etc., 2.748 feet ; the points B, F, If, etc., C, being connected by suitable lines, determine the limit of the stream on that side. The limit of the stream on the other side is found by making the arcs FL = UI, HM= G K, etc., CD= ON; the points A, L, M, etc., D, being connected by suitable lines, determine the limit of the stream on that side. By an examination of figure 2, it will be seen, that the section of the stream just after it has entered the wheel, is sensibly greater than the section of the stream as it leaves the guides, and that, consequently, if the stream flowed according to the hypothesis, there must have been a sudden change in the velocity of the water, causing a shock, which, according to the common theory, implies a loss of power. This indicates a defect in the design; nevertheless, the success attending this first essay, on a large scale, of a centre-vent water-wheel, in which due regard has been paid to accuracy of construction and perfection of workmanship, guided by such light as the present imperfect theories can afford, ought to encourage us to hope, that, when it has received the same degree of attention as the turbine, it will not be much behind that celebrated motor, in its economical use of water. 70 EXPERIMENTS ON A CENTRE-VENT WATER-WHEEL. TABLE VII. 1 a 3 4 5 6 7 8 9 1O 1 Ordinate* ^JT in feet, to Value of Areas in Areas in of the areas of Correction Corrected Mean Volumes be meas- A and square inches, square the complete for the areas of the height Volumes of between R ured on successive of circles of inches, of the rings in the thickness of partial rings, of the the partial and the arcs of the values of the radii in the complete preceding the bucket, in in square partial rings, In successive corre- .ft', in inches. preceding column. rings. column, in square feet. square feet. feet. rings, in feet. cubic feet. values of R^ in cubic feet. sponding radii in column 1. 56.028 9861.890 55.000 9503.318 358.572 0.06225 0.00168 0.06057 1.001 0.06063 0.06063 0.415 54.000 9160.884 342.434 0.05945 0.00210 0.05735 1.008 0.05781 0.11844 0.796 53.000 8824.734 336.150 0.05836 0.00227 0.05609 1.021 0.05727 0.17571 1.160 52.000 8494.866 329.868 0.05727 0.00262 0.05465 1.042 0.05695 0.23266 1.507 51.000 8171.282 323.584 0.05618 0.00304 0.05314 1.070 0.05686 0.28952 1.839 50.000 7853.982 317.300 0.05509 0.00386 0.05123 1.105 0.05661 0.34613 2.155 49.000 7542.964 311.018 0.05400 0.00561 0.04839 1.147 0.05550 0.40163 2.451 48.750 7466.191 76.773 0.01333 0.00168 0.01165 1.177 0.01371 0.41534 2.522 48.500 7389.811 76.380 0.01326 0.00181 0.01145 1.190 0.01363 0.42897 2.591 48.250 7313.824 75.987 0.01319 0.00202 0.01117 1.204 , 0.01345 0.44242 2.659 47.922 7214.723 99.101 0.01721 0.00252 0.01469 1.221 0.01794 0.46036 2.748 PART II. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, AND IN SHORT RECTANGULAR CANALS. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 120. THE laws governing the flow of water over weirs, have received the attention of several distinguished engineers and men of science, among whom may be named Smeaton and Brindley in England; Du Buat, Navier, D'Aubuisson, Castel, Poncelet, Lesbros, and Boileau, in France ; and Eytelwein and Weisbach in Germany. A great number of experiments have been made and recorded ; the earlier ones rude and imperfect ; the later ones, particularly those by Poncelet, Lesbros, and Boileau, with a perfection of apparatus previously unknown. There has been in this branch of hydraulics, as well as in others, a steady advance with the accumulation of experiments and the improvement of the means of observation ; the result, however, of these numerous labors, is far from satisfac- tory to the practical engineer. On a careful review of all that has been done, he finds that the rules given for his use, are founded on the single natural law governing the velocity of fluids, known as . the theorem of Torricelli ; omit- ting, in consequence of the extreme complexity of the subject, all consideration of many other circumstances, which, it is well known, materially affect the flow of water through orifices. He finds also that it has been attempted to correct the theoretical expression thus found, by coefficients obtained by comparing the results derived from it, with those furnished by experiment ; but when he comes to investigate these experiments, even after rejecting all excepting those made with the greatest care, and with apparatus capable of insuring the greatest precision, he finds such discordances in the resulting coefficients, that he loses all hope of arriving at correct results when he applies them on the great scale. They will undoubtedly furnish sufficiently accurate results, if the apparatus used is a repro- 72 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. duction, both in form and dimension, of that used in the experiments ; but this is seldom attainable, the experiments having been made on such a minute scale. Boileau,* in discussing the various formulas that have been proposed, points out many of their defects, and has himself proposed a new one, coupled, however, with some special conditions in the form of the weir, and the mode of taking the depth upon the sill. No correct formula for the discharge of water over weirs, founded upon natu- ral laws, and including the secondary effects of these laws, being known, we must rely entirely upon experiments, taking due care in the application of any formula deduced from them, not to depart too far from the limits of the experiments on which it is founded. Engineers have generally agreed that the most convenient form of weir for gauging streams of water, is one which is cut in a vertical plane side of a reser- voir, the sill being horizontal, the sides vertical, and the contraction complete. In order that the contraction may be complete, the sill and sides of the weir must be so far removed from the bottom and lateral sides of the reservoir, that they may produce no more effect upon the discharge, than if they were removed a distance indefinitely great ; also, the aperture must be effectively the same, as if cut in a plate having no sensible thickness. The condition relating to the dis- tance of the bottom and sides of the reservoir, can seldom be strictly complied with, when gauging large streams of water ; it is found, however, that, when the sill is at a height above the bottom of the reservoir not less than twice the height of the water above the sill, and the sides are removed a distance at least equal to the height above the sill, a correction free from serious error can usually be made for the effect of the velocity of the .water approaching the weir. The condition that the aperture shall be effectively the same as if cut in a plate having no sensible thickness, is usually more easily complied with. The effect of the contraction is such, that the water has a strong tendency to leave the bottom and sides of the aperture for a certain distance, and to touch the aperture only at the upstream edge ; if, however, the thickness of the plank or other material, exceeds a certain amount, (depending upon the depth flowing over,) the water will follow the top of the plank ; in this case, all that is requisite is, to cut away the downstream side of the weir at an angle of, say, forty-five or sixty degrees with the horizontal ; leaving horizontal, only a small part of the thick- * Jaugeage des cours d"eau a faille ou a moyenne section by M. P. Boileau (Paris : 1850) ; or Journal de I'Ecok Polytechnique, No. xxxiii. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 73 ness of the sill. It is essential, however, that the corners of the sill and sides of the weir presented to the stream, should be full and sharp, and not rounded or bevelled in any degree. 121. Two modes present themselves for studying, experimentally, the laws governing the discharge of water over weirs. First, that which has been uni- formly adopted heretofore, namely, to obtain by direct measurement the quantity of water discharged in a given time, through an aperture of known dimensions ; this is evidently the only mode of resolving the question completely. To per- form the experiments, however, upon a scale of magnitude corresponding to the ordinary practical applications, usually requires an apparatus of great cost, and such as is beyond the reach of most experimenters. The great difficulty is, to obtain a suitable basin, in which to make the direct measurement of the quantity dis- charged by the weir. The second mode dispenses with a direct measurement of the quantity. If we have two weirs of the same form, but of different lengths, and we know that the quantities of water discharged by them, in certain circumstances, are equal ; knowing also the depth upon the sill of each weir, we have the data for an equation by which one unknown quantity may be determined. Neither the coefficient of contraction, nor the absolute discharge can, however, be obtained by such an equation. ' 122. The discharge over weirs is commonly assumed to vary as the square root of the third power of the depth ; let us suppose it to be unknown, and equal to a. Suppose also I the length, and h the depth, on one of the weirs ; and I' and // the corresponding dimensions for the other weir ; O, a constant coefficient ; Q, the quantity which, by hypothesis, is the same for both weirs. Assuming, accord- ing to the common formula, that the quantity is proportional to the length of the weir, we have Q= Cl'h' a ; consequently, Clh= Ol'K'i taking the logarithms, we have a (Log. h Log. A') = Log. I' Log. I ; 10 74 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. therefore, _ Log. /' Log. I "Log. h Log. h" We can thus, by means of two experiments, determine the power of the depth which will lead to identical quantities in the computed discharge of the two weirs. 123. It is assumed in the above equations, that the quantity discharged by a weir is directly proportioned to its length; this, in weirs having complete con- traction, is, however, known not to be true, in consequence of the contraction which takes place at the ends of the weir. This contraction diminishes the dis- charge. When the weir is of considerable length in proportion to the depth of the water flowing over, this diminution is evidently a constant quantity, whatever may be the length, provided the depth is the same ; we may, therefore, assume that the end contraction effectively diminishes the length of such weirs, by a quantity depending only upon the depth upon the weir. It is evident that the amount of this diminution must increase with the depth ; we are unable, how- ever, in the present state of the science, to discover the law of its variation ; but experiment has proved that it is very nearly in direct proportion to the depth. As it is of great importance, in practical applications, to have the for- mula as simple as possible, it is assumed in this work that the quantity to be subtracted from the absolute length of a weir having complete contraction, to give its effective length, is directly proportional to the depth. It is also assumed that the quantity discharged by weirs of equal effective lengths, varies according to a constant power of the depth. There is no reason to think that either of these assumptions is perfectly correct ; it will be seen, however, that they lead to results agreeing very closely with experiment. 124. The formula proposed for weirs of considerable length in proportion to the depth upon them, and having complete contraction, is Q=C(l bnh)h;* in which Q = the quantity discharged in cubic feet per second. C= a constant coefficient. l=ihe total length of the weir in feet. b = a constant coefficient. * This formula was first suggested to the author by Mr. Boyden, in 1846. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 75 H = the number of end contractions. In a single weir having complete con- traction, n always equals 2, and when the length of the weir is equal to the width of the canal leading to it, n = 0. h = the depth of water flowing over the weir, taken far enough upstream from the weir, to be unaffected by the curvature in the surface caused by the discharge. a = a constant power. The coefficient C can be determined only from experiments in which the actual discharge is known ; the constants, a and b can, however, be determined without knowing the actual discharge in any particular case. It has been stated that the proposed formula is applicable only to weirs having a considerable length in proportion to the depth of water running over them. It is found by experiment that, when the length equals or exceeds three times the depth, the formula applies ; but in lengths less than this in proportion to the depth, the formula cannot be used with safety; the error increasing as the relative length of the weir diminishes. It is evident, from the construction of the formula, that it cannot be of gen- eral application. The factor I bnh represents the effective length of the weir; if l=bnh this effective length becomes 0, and the formula would give for the discharge, which is absurd; similarly, if Inky I, the discharge given by the formula would be negative. In weirs of very short length in proportion to the depth, the effect of the end contraction cannot be considered as independent of the length. The end contraction influences the discharge to a certain distance, A, from the end of a weir ; if the whole length of the weir is greater than 2 A, the effect of the end contraction is independent of the length ; but if the length is less than 2 A, the whole breadth of the stream is affected in its flow by the end contractions, and, consequently", the proposed formula would not apply. In practical applications, this will seldom be an inconvenience, as it is nearly always practicable so to proportion the weir, that the length may not be less than three times the depth upon it; if, however, there is no end contraction, the proportion of the length to the depth is not material. 125* The author has made numerous experiments on the discharge of water over weirs, according to each of the methods described above. First, those at the Tremont Turbine, and at the centre-vent water-wheel for moving the guard gates of the Northern Canal. In none of these experiments has any attempt been made to measure the absolute quantities flowing over the weirs ; but simply to cause quantities of water known to be equal, to pass over 76 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. weirs of different dimensions, noting the depth of water and length of weir in each case. From these data, as is explained above, certain factors in the formula can be determined. Second, those at the Lower Locks, in which the absolute quantities passing over weirs of known dimensions, were measured directly. As each of these three sets of experiments were made with different appara- tus, they will be described separately. EXPERIMENTS MADE AT THE TREMONT TURBINE, ON THE FLOW OF WATER OVER WEIRS. 126. The apparatus constructed to gauge the water discharged by the Tremont Turbine, with some modifications, was used for the experiments on the discharge over the weir; for a general description of this apparatus, see arts. 44, 45, and 46. The experiments consisted in allowing a quantity of water, of unknown vol- ume, to enter the wheelpit, through the turbine, the regulating gate of which was sufficiently opened for the purpose. This volume of water was then caused to flow over weirs of different dimensions, and the corresponding depth on the weir, assumed by the water in each experiment, was noted after the water had arrived at a uniform state. The experiments are divided into series, in each of which the regulating gate was unchanged throughout, so that the apertures through which the water entered the wheelpit remained constant during each series. Some variations necessarily occurred in the head acting upon these orifices ; they were small, however, when compared to the whole head. The depths on the weir have been reduced, according to well-known principles, to what they would have been if the head had been constant. The leakage of the wheelpit also rendered another small correction necessary. After the corrections are made, we have in each series a collection of experiments in which the quantity dis- charged is the same, and we have also the requisite dimensions of the different weirs. These data, if perfectly accurate, are sufficient to enable us to determine, in the proposed formula for the discharge, the values of the constants a and b. It is not to be presumed, however, that the data are perfectly correct, but we can, at any rate, find the values of a and b that will give the most uniform results to the computed discharges in all the experiments in a series ; the actual discharge being, by hypothesis, a constant quantity. 127. Some additions to the apparatus used in the experiments on the tur- bine were made for the weir experiments. The partitions, represented by figures EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 77 5, 6, and 7, plate V., were provided for the purpose of shortening or subdividing the weir. They were made of wood, faced on part of one side with plates of sheet-iron a, T 3 g of an inch in thickness ; the width b c was about 1.5 feet ; the iron plate was two inches less. One side of the timber P, figure 2, was in the same vertical plane as the upstream edge of the weir H. When the partitions were placed upon the weir, the top of them was supported by the timber P, and the bottom by the plate of iron a, which rested against the weir.. Flash- boards, represented by figures 8, 9, and 10, plate V., were also provided to close up portions of the weir; these, together with the partitions, were maintained in their respective positions, simply by the pressure of the water against them. Wherever leaks appeared at the joints of the partitions or flashboards, they were stopped with great ease and effect, by a little dough made of unbolted Indian meal, a handful of which was drawn over the upstream side of the joints ; of course the orifices closed in this manner were very minute. In plate X., all the modifications of the weir produced by changing the partitions and flashboards, are represented ; the several figures are referred to in column 8, table X. In the greater number of the experiments, two or more spaces were used at the same time ; they were always of very nearly equal length, so that the length of each may be obtained by dividing the whole length of the weir given in column 6 by half the number of end contractions given in column 7. ,The brackets N, figures 1 and 2, plate V., were placed on the downstream side of the weir, to support a board on which to stand for the purpose of adjusting the partitions and flashboards. The top of the board was about 9.5 inches below the top of the weir. In some of the experiments, a part of the sheet of water fell upon this board ; in experiment 50 it was moved nearer to the weir, so that the entire sheet of water fell upon it, but without producing any sensible effect upon the discharge. In experiment 51, a three inch plank was placed on the top of the board, as is represented by the dotted lines at 0, figure 2, plate V. ; the effect of this obstruction, as indicated by the increased depth on the weir as measured by the hook gauge, was, to diminish the discharge, with the same depth on the weir, about foVo^ It is to be regretted that the casting forming the sill of the weir, was not planed on its whole height on the side II Q, figure 4, plate V. When the weir was erected no, thought was entertained of using it for these experiments, requiring, as they do, to be of value, to be free from all disturbing causes. The disturbance caused by the projection at I, can, however, have been scarcely sensible. 128. The data furnished by observation, together with the necessary reduc- tions, and the results deduced from them, are contained in table X. Most of 78 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. the columns are sufficiently explained by the respective headings ; several of them, however, require further explanation. 129. COLUMN 11. Fall affecting the leakage of the wheelpit. This is obtained by adding together the corresponding numbers in columns 9 and 10. 130. COLUMN 12. Depth of water on the weir corrected for the leakage of the wheelpit. This is obtained in the following manner. It was clear, from the construction of the wheelpit, (art. 23,) that nearly the whole of the leakage passed through the wooden flooring, and that all the orifices through which it passed were constantly below the surface of the lower canal. In the construction of the wheelpit, no particular precautions were taken to pre- vent a free communication from the bottom of the wooden flooring to the lower canal ; and as the amount of the leakage was very small, and the material, fine sand free from large springs, it is clear that the water could have had no appreciable obstruction after passing through the flooring, except from the pressure of the water in the lower canal. This being the case, the amount of the leak- age would depend upon the head; or, in other words, upon the height from the surface of the water in the wheelpit, to the surface of the water in the lower canal. Let j = the quantity of water leaking out of the wheelpit, in cubic feet per second. A, A', A', etc. = the areas of the several orifices through which the water passed. C,C',C", etc. := the corresponding coefficients of contraction. A:=the head, or the height from the surface of the water in the wheel- pit, to the surface of the water in the lower canal. This head applies to all the orifices, as they are all below the surface of the water in the lower canal. L = CA \l 2 ff h + C'A \l 2 g h -f C"A'\f2gh + etc. ; L = ( CA -f- C'A' + C"A"+ etc.,) V2gJ. The areas A, A, A", etc., are constant, as are also the coefficients C, C', C", etc., the variations in the head not being very great. Let c = CA + C'A' + C"A" + etc. : then EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 79 The factor c \Zg, being constant, can be determined by an experiment in which L and h are known. To determine this constant, the following experiment was made. The weir was closed up by the flashboards, and made tight in the usual manner, so that no appreciable quantity passed over the weir ; the head gate was closed, and the small quantity leaking through it was caught in the leak box and carried over the weir in the leak pipe (art. 24). The water in the wheelpit having then no supply, its surface began to lower, in consequence of the leakage through the floor ; while thus falling, the following observations were made. February 5, 1851, at 10", 20', 30", A.M., the height of the water in the wheelpit above the top of the weir, was ..... 0.596 feet. And at ll h , 1', 46", A.M., the height was ......... 0.396 Consequently the surface of the water in the wheelpit lowered in 2476" ................... 0.200 feet. The area of the surface of the water in the wheelpit, after making the proper deductions, was about 506 square feet ; consequently, T 506X0.2 ,. , = 2476 == O-O^O^ cubic feet per second. During the interval of 2476 seconds, the mean height of the water in the lower canal was 1.2316 feet below the top of the weir, and the mean height in the wheelpit, during the same period, was 0.496 feet above the top of the weir, then h = 1.2316 + 0.4960 = 1.7276 feet. ^ Substituting these values of L and h in the equation we have = 0.03112: consequently, Z = 0.03112 \fh. To find the depth on the weir, corrected for the leakage of the wheelpit, let h' = the depth on the weir by observation, 80 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. h" = ihe depth on the weir corrected for the leakage, 1= length of the weir, Q = the quantity passing over the weir, the dimensions being all in feet. We have Q-\-L = ihe total quantity entering the wheelpit, and which would have passed over the weir, if there had been no leakage out of the wheelpit. To determine the corrected depth, it is necessary to assume some formula giving nearly the relations between the quantities h', I, and Q. Let us use that given by Lesbros* for a depth of 0.20 metres and complete contraction, which, when reduced to the English foot as the unit, and adopting our own notation, is we shall have also by subtraction from which we derive or substituting for L its value 0.03112 \^, we have , (,& . 0.03112^X3 3.12/ / ' By this formula, the reduced heights given in column 12 have been obtained. 131. COLUMN 15. Fall from the surface of water in the forebay, to the surface of the water in the wheelpit. This is obtained by taking the difference of the cor- , responding numbers in columns 13 and 14. 132. COLUMN 16. Uniform fall from the forebay to the wheelpit, to which the depths on the tceir in each series are reduced. The fall in the same series given in column 15, which is the nearest to the mean fall in all the experiments in the series, is assumed for this purpose ; it is unimportant what fall is taken, provided it is near the mean. 133. COLUMN 17. Depth on the weir corrected for the leakage of the wheelpit, and the variation in the fall. It must be recollected that all the experiments of each * Experiences Hydrauliques sur les lots de I'ecoulement de I'eau, by M. Lesbros, Paris : 1851. Table XXXIX. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 81 series, were made with the same opening of the regulating gate of the turbine ; that is, the areas of the orifices through which the water entered the wheelpit, were the same in each. In all the experiments, a small quantity of the water entering the wheelpit, passed between the gate and the lower curb, in consequence of the leather packing not being perfectly adjusted ; this did not affect the results, however, as these orifices were also submerged in the wheelpit. Under these cir- cumstances, if the head had been constant, the quantity of water entering the wheelpit, would also have been constant; but the head was subject to a varia- tion, comparatively small certainly, but sufficient to produce a material change in the quantity of water entering the wheelpit, and, consequently, in the depth on the weir. To clear the results from this source of irregularity, it will be necessary to ascertain what the depths on the weir would have been, if the head had been constant. For this purpose, let ff= the constant head to which the depths on the weir, in any particular series, are to be reduced, and which varies but little from the actual heads in the same series; J7' the actual head in the particular experiment to be reduced; H" the depth on the weir, corrected for the variation of the head, or corresponding to the constant head H; A" the depth on the Aveir corresponding to the head H', and which is the depth given by observation, corrected for the leakage of the wheelpit ; whence, we derive LHI _ -uit ff\^ h h -, . By this last formula, the corrected depths given in column 17 have been computed. By an inspection of column 13, it will be seen that the level of the water above the wheel was maintained throughout each series with great uniformity, excepting in a few experiments in which it was intentionally altered, as will be seen presently. The height of the water in the wheelpit necessarily varied with the depth upon the weir, and this is the principal cause of the variations in the fall. Several of the experiments given in table X., were made for the express purpose of testing the accuracy of the method of reduction just described. Thus, in experiments 41 and 42, the weir was in the same state as in experiment 40, but the height of the water above the wheel was lowered, and the differences in the observed depths upon the weir, given in column 9, are to be attributed entirely to the diminution in the quantity of water entering the wheelpit, in con- sequence 6f the diminished head. If the method of reduction is accurate, how- ever, the corrected depths in these three experiments, given in column 17, should be the same. In table VIII, are collected all the experiments made for this object, together with the other experiments forming part of the corresponding series, with which they may be compared, the weir having been in the same state. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 83 TABLE VIII. Number of the experiment. Fall from the forebay to the wheelpit. Feet. Corrected depth upon the weir, in Feet. Variation in the fall from the initial experiment. Feet. Variation in the cor- rected depth, from the initial experiment. Feet. 40 14.088 0.79096 41 13.554 0.79049 0.534 0.00047 42 13.149 0.78976 0.939 0.00120 49 13.904 0.95477 52 13.436 0.95380 0.468 0.00097 53 12.962 0.95097 0.942 0.00380' 63 13.719 1.13177 64 12.806 1.12508 0.913 0.00669 72 13.816 0.92170 73 13.315 0.92145 0.501 0.00025 74 ,12.665 0.92153 1.151 0.00017 It will be perceived that the variations in the fall, to which the method ol reduction is applied in these experiments, are, nearly all of them, much greater than any that occur in the regular experiments. This was arranged for the pur- pose of applying an extreme test to the method. Several of the variations in the corrected depths, are not within the limits of ordinary observation ; several of them, however, are sensible, and being all in the same direction, they cannot be attributed entirely to errors of observation, but, in part at least, to either a slight defect in the method of reduction, or to the instability of the apparatus. It was observed during the course of the experiments, that the quantity of water entering the wheelpit, sometimes diminished sensibly, although no change had been made in the height of the regulating gate ; the precaution having been taken to fix, in a secure manner, the apparatus by which the gate was moved. At the time the experiments were made, this change was attributed to a minute lowering of the gate, taking place very slowly, and arising from a defect in the stiffness of the apparatus, aided by a slight, but not totally insensible vibration of the whole apparatus, caused by the passage of the water through the aper- tures. To show how minute a change in the height of the regulating gate, would produce the observed changes in the quantity, let us take the two first experiments given in table IX. The regulating gate was raised to a height not 84 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. exceeding 0.01 feet; supposing it to have been at just that height, and that any change in its height would have produced an equal proportional change in the discharge, the observed proportional change in the quantity was 0.00046 ; consequently, the absolute change in the height of the gate must have been 0.0000046 feet. In order to prevent this source of irregularity from affecting the experi- mentS, the regulating gate was usually set some hours before the experiments were made. This probably obviated the difficulty in part, but not entirely, as will be seen by table IX., in which are collected all the experiments that were repeated under identical circumstances. TABLE IX. Number of the experiment. Number of the series. Corrected depth upon the weir, in feet. Variation in the depth from the initial experi- ment, in feet. Proportional change in the quantities that entered the wheelpit. Time that had elapsed when the experiment was made, since the gate was Bet. Hours. Minutes. 3 7 I. a 0.19583 0.19577 0.00006 0.00046 8 12 ii. a 0.23386 0.23505 4- 0.00119 4- 0.00764 1 ' 22 31 16 20 24 m. tt tt 0.29223 0.29166 0.29210 0.00057 0.00013 0.00292 0.00067 4 5 6 33 39 39 26 30 IV. u 1.06532 1.06548 -f-0.00016 + 0.00023 16 17 58 51 35 40 V. u .0.79190 0.79096 0.00094 0.00178 2 3 25 34 44 49 VI. tl 0.95656 0.95477 0.00179 0.00281 5 6 20 40 55 58 63 VII. u u 1.13356 1.13306 1.13177 0.00050 0.00179 0.00066 0.00237 2 3 4 26 31 39 66 69 VIII. ft 1.06358 1.06272 0.00086 0.00121 3 3 08 46 Mean pr the sig Dportional change in the quantity, neglecting 0.00208 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 85 Although the variations in the depths given in the preceding table are very small, the fact that they are nearly all negative precludes the idea that they are entirely due to errors of observation ; we must, therefore, attribute to some other cause a portion of the irregularity. 134. COLUMN 19. Combination of experiments used to determine the value of a. It has been shown (art. 122) how, by means of two experiments in which the quantities passing over different weirs are equal, we may determine a in the formula Q = Clh*. We now propose to show how, by means of two such experiments, the value of a may be found in the proposed formula Q = C(lbnh}h\ In this equation, we have I and a constant quantities to be determined; we have also C a constant, which we may here consider as indeterminate; the same may be said of Q, as limited to the experiments in the same series. Let I, n, and h, represent the length of the weir, the number of end con- tractions, and the depth upon the weir in one experiment; and li, n i} and hi, the corresponding quantities in another experiment of the same series; we have Q=C(l bnh)h a ; . and since for the same series Q is constant, we have (lbnh}h*=(l l bn l h l }h l a : taking the logarithms, , a Log. h -\- Log. (I bnh] = a Log. hi-\- Log. (^ whence we derive _ Log. (/! Jjij^j) Log. (I bnfi) Log. h Log. AI This equation is still indeterminate, but can be rendered determinate, by assuming a value for b. If the formula represents the true law, and the experiments from which the values of the constants are to be derived are perfectly accurate, the particular combination of experiments to be used is evidently unimportant. As such an 86 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. assumption would be very unreasonable, we have combined the experiments, with a view of obtaining the best approximation from imperfect data ; and this we have accomplished by selecting experiments the most remote from each other in the values of the respective data they furnish; thus, in series I., the combina- tions are made by combining experiment 6, in which I has the least, and, con- sequently, h the greatest value, with each of the others, omitting entirely all the experiments which, for any reason, appear to be unsuitable. Generally, in each series, one experiment has been repeated as a test, in order to show if any change had taken place in the apparatus; thus, in series III., experiments 16, 20, and 24, were made, so far as is known, under identical circumstances ; in such cases, means deduced from the repeated experiments have been used instead of making a separate combination with each. 135. COLUMNS numbered 20 to 25. Values of a when = 0.07, I = 0.065, etc. The object is, to find the values of a and I, in the formula Q= C(llnh}h", that will give to the computed discharges in each series the most uniform results. For this purpose, successive values of b are assumed, and the corresponding values of a, determined. The value of I leading to values of a, having the least vari- ation among themselves, will evidently be that most nearly fulfilling this condition. To aid in the selection of the proper value of I, the table gives the differences between the values of a deduced from each combination, and the mean value of a deduced from all the combinations, with the same value of b, and the sums of these differences (having no regard to the sign) are also given. If will be seen that the sum of the differences is least when the value of b = 0.05, the corre- sponding mean value of a being 1.46994, or 1.47 very nearly ; consequently, to represent the whole of the experiments with the most uniformity, the formula becomes Q=C(l 88 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE X. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE TREMONT TURBINE. 1 3 3 4 5 6 7 8 9 1O 11 Temperature of the atmosphere TIME. No Height in degrees of Dura- Total of Reference Depth of of the Fall affect- Number of the series and of the experiment. Date of the experiment 1861. thermometer. Beginning of the experiment. Ending of the experiment. tion of the experi- ment, n min- length of the weir, in feet. I the end con- trac- tions. to the figures on plate X. water on the weir by obser- vation ; In feet. In the lower canal, below the top of the ing the leakage of the wheel- pit; in feet. External ilr in the Near the weir. utes. hi weir, A. shade. H. inin. H. tiiin. in feet. Series I. Exp. 1 January 30, A.M. 10 10 12 12 6.987 2 Fig. 1 0.3125 1.16 1.47 2 II 11 II 10 18 10 25 7 13.978 4 2 0.1948 1.16 1.35 3 11 II II 6.50 10 39 10 46.5 7.5 13.978 8 " 3 0.1952 1.16 1.36 4 II II << 6.25 31.50 11 2 11 6 4 10.482 6 " 4 0.2389 1.16 1.40 5 II II II 5.75 31.00 11 20 11 26 6 7.000 4 51 0.3149 1.17 1.48 6 a u a 6.25 11 40 11 45 5 3.500 2 6 0.5028 1.25 1.75 " 7 u u a 5.75 11 52 11 55 3 13.978 8 3 0.1951 1.22 1.42 Series II. Exp. 8 January 30, P.M. 30.75 2 22 2 26 4 13.978 8 Fig. 3 0.2330 1.10 1.33 " 9 u u u 4.50 30.50 2 35.5 2 41 5.5 10.482 6 ii 4 0.2842 1.10 1.38 " 10 u u u 4.50 30.50 2 54 3 6 7.000 4 5 0.3738 1.10 1.47 11 II II (( 4.25 3 15 3 21 6 3.500 2 6 0.5973 1.20 1.80 " 12 II ti II 4.00 30.75 3 31 3 38 7 13.978 8 3 0.2341 1.27 1.50 13 11 U U 3.50 30.50 3 53 3 59 6 13.978 4 " 2 0.2330 1.22 1.45 14 u u u 4 11 4 16.5 5.5 6.987 2 1 0.3719 1.10 1.47 15 II 11 11 2.75 30.75 4 24 4 32 8 16.980 4 " 7 0.2046 1.09 1.29 Series III. Exp. 16 January 31, P.M. 5.00 31.00 2 23.5 2 32 8.5 13.978 4 Fig. 2 0.2916 1.17 1.46 " 17 11 11 11 5.00 31.25 2 41 2 49.5 8.5 6.987 2 " 1 0.4652 1.17 1.64 18 U 1 11 5.00 31.25 2 56.5 3 5 8.5 13.978 8 " 3 0.2932 1.16 1.45 " 19 11 1 11 5.00 31.00 3 12 3 18 6 10.484 6 11 4 0.3564 1.13 1.49 20 11 1 11 5.00 30.75 3 29.5 3 35.5 6 13.978 4 " 2 0.2910 1.12 1.41 21 1C 1 11 4.50 30.50 3 46 3 53 7 6.989 4 5 0.4684 1.15 1.62 22 U 1 1C 4.50 4 2.5 4 8.5 6 3.500 2 " 8 0.7478 1.12 1.87 23 11 II 11 4.25 31.00 4 14 4 20 6 16.980 4 11 7 0.2548 1.12 1.37 " 24 II 11 II 4.00 4 29.5 4 35.5 6 13.978 4 " 2 0.2914 1.16 1.45 Series IV. Exp. 25 February 1, A.M. 5.00 31.00 9 15 9 21 6 13.978 4 Fig. 2 0.4071 1.14 1.55 " 26 ii ti s ii 7.00 9 38.5 9 46 7.5 3.496 2 " 8 1.0447 1.12 2.16 27 u u it 8.50 30.00 9 52 9 57 5 6.989 4 " 5 0.6577 1.15 1.81 28 II 1C 1 10.00 10 4 10 11 7 10.484 6 11 4 0.4977 1.10 1.60 29 II II 1 10.00 30.50 10 16 10 21 5 13.978 8 " 3 0.4096 1.15 1.56 30 II II 1 10.50 10 31.5 10 39.5 8 3.496 2 8 1.0456 1.17 2.22 31 II II I 14.25 31.50 10 57 11 1 4 3.496 2 " 8 1.0452 1.12 2.17 32 II 11 1 15.00 11 19 11 24.5 5.5 6.987 2 " 1 0.6494 1.17 1.82 33 II 11 II 15.50 30.75 11 29.5 11 36 6.5 16.980 4 " 7 0.3576 1.22 1.58 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 89 TABLE X CONTINUED. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE TREMONT TURBINE. 13 13 14 15 16 17 18 Number of the scries and of the experiment. Depth of water on the weir, corrected for the leakage of the wheelpit, in Height of water above the wheel, taken in the forcbay j Height of the water in the wheelpit ; Fall from the surface of the water in the forcbay, to the surface of the water in the wheelpit ; Uniform fall from the forehay to the wheelpit, to which the depths on the weir in each series are Depth on the weir, corrected for the leakage of the wheelpit, and the varia- tion in the fall ; REMARKS. feet. A//. in feet. infect. in feet. a'. reduced ; in feet. H, in feet. A'". Series I. Exp. 1 0.31456 14.869 0.320 14.549 14.549 0.31456 In experiments 2, 3, and 7, the contrac- " 2 0.19605 14.896 0.201 14.695 ft 0.19540 tion was incomplete, as the water fol- 3 0.19645 14.894 0.205 14.689 It 0.19583 lowed the top of the weir. a 4 0.24043 14.881 0.247 14.634 11 0.23997 5 0.31696 14.892 0.320 14.572 It 0.31679 6 0.50634 14.876 0.510 14.366 tt 0.50848 " 7 0.19638 14.886 0.200 14.686 tt 0.19577 Series II. Exp. 8 0.23414 14.910 0.240 14.670 14.619 0.23386 In experiment 15 the contraction was 9 0.28560 14.909 0.290 14.619 C4 0.285CO incomplete, as the water followed the top 10 0.37568 14.915 0.380 14.535 u 0.37640 of the weir. " 11 0.60059 14.916 0.610 14.306 It 0.60494 " 12 0.23530 14.906 ! 0.240 14.666 u 0.23505 13 0.23419 14.912 0.240 14.672 u 0.23390 14 vO.37379 14.910 0.380 14.530 (( 0.37455 15 0.20558 14.918 0.210 14.708 11 0.20517 Series III. Exp. 16 0.29266 14.897 0.300 14.597 14.532 0.29223 " 17 0.46698 14.900 0.470 14.430 tt 0.46808 18 0.29426 14.897 0.300 14.597 (i 0.29382 19 0.35770 14.897 0.365 14.532 0.35770 20 0.29205 14.890 0.300 14.590 0.29166 " 21 0.47017 | 14.887 0.477 14.410 u 0.47149 22 0.75080 14.883 0.760 14.123 u 0.75798 23 0.25571 14.878 0.260 14.618 cc 0.25520 " 24 0.29246 14.886 0.300 14.586 0.29210 Series IV. Exp. 25 " 26 27 28 0.40803 1.04743 0.65928 0.49884 14.904 14.877 14.871 14.877 0.420 1.060 0.670 0.510 14.484 13.817 14.201 14.367 14.537 it tt tt 0.40852 1.06532 0.66444 0.50080 In experiments 26 and 30, the water flowing over the weir fell upon a board placed upon the brackets N, figures 1 and 2, plate V. ; in experiment 31 the board was removed. So far as is known the 29 0.41053 14.893 0.420 14.473 " 30 1.04837 14.908 1.060 13.848' " 31 1.04794 ; 14.886 1.060 13.826 tt tf tt 0.41113 1.06548 1.06560 three experiments were identical in all ; other respects. By comparing the cor- rected depths upon the weir given in col- " 32 0.65099 14.904 0.660 " 33 0.35842 : 14.907 0.370 14.244 14.537 (I tt 0.65543 0.35842 umn 1 7, it appears that the board offered no appreciable obstruction to the discharge. 12 90 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE X CONTINUED. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE TREMONT TUBBING. 19 20 31 33 6 = 0.07 6 = 0.065. 6 = 0.06. Number of the series and of the experiment. Combination of experi- ments used to determine the value of a. Values Differences from the mean value of a, or from 1.47797. Values Differences from the mean value of a, or from 1.47695. Values Differences from the mean value of a, or from 1.47394. of a. of a. of a. + - + - + - Series I. Exp. 1 1 and 6 1.4691 0.00887 1.4669 0.00905 1.4648 0.00914 2 3 " 4 4 6 1.4753 0.00267 1.4742 0.00175 1.4731 0.00084 5 5 " 6 1.4814 0.00343 1.4801 0.00415 1.4789 0.00496 6 " 7 Series II. Exp. 8 8, 12, and 11 1.4768 0.00117 1.4756 0.00035 1.4744 0.00046 " 9 9 " 11 1.4787 0.00073 1.4775 0.00155 1.4763 0.00236 10 10 " 11 1.4805 0.00253 1.4791 0.00315 1.4777 0.00376 " 11 " 12 8, 12, " 11 1.4768 0.00117 1.4756 0.00035 1.4744 0.00046 13 13 " 11 1.4781 0.00013 1.4766 0.00065 1.4751 0.00116 " 14 14 11 1.4774 0.00057 1.4748 0.00115 1.4723 0.00164 " 15 15 " 11 1.4800 0.00203 1.4786 0.00265 1.4772 0.00326 Series III. , Exp. 16 16,20, 24, and 22 1.4778 0.00017 1.4759 0.00005 1.4740 0.00006 " 17 17 " 22 1.4784 0.00043 1.4752 0.00075 1.4721 0.00184 " 18 18 22 1.4811 0.00313 1.4797 0.00375 1.4782 0.00426 " 19 19 " 22 1.4827 0.00473 1.4811 0.00515 1.4795 0.00556 20 16,20,24, " 22 1.4778 0.00017 1.4759 0.00005 1.4740 0.00006 " 21 21 " 22 1.4814 0.00343 1.4796 0.00365 1.4778 0.00386 22 23 23 22 1.4752 0.00277 1.4734 0.00255 1.4716 0.00234 " 24 16, 20, 24, " 22 1.4778 0.00017 1.4759 0.00005 1.4740 0.00006 Series IV. Exp. 25 25 and 26, 30 1.4827 0.00473 1.4800 0.00405 1.4773 0.00336 " 26 27 27 " 26,30 1.5023 0.02433 1.4997 0.02375 1.4971 0.02316 28 28 " 26, 30 1.4857 0.00773 1.4834 0.00745 1.4812 0.00726 29 29 " 26, 30 1.4838 0.00583 1.4817 0.00575 1.4796 0.00566 30 81 " 32 32 " 26,30 1.4878 0.00983 1.4832 0.00725 1.4786 0.00466 33 33 " 26,30 1.4853 0.00733 1.4828 0.00685 1.4802 0.00626 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 91 TABLE X CONTINUED. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE TREMONT TURBINE. 33 24 35 6-0.055 6 = 0.05. 6 = 0.045. Number of the series and of the Differences from the Differences from the Differences from the experiment. Values mean value of a, or from 1.47194. Values mean value of a, or from 1.46994. Values mean value of a, or from 1.46795. of a. of a. of a. + - + - + - Series I. Exp. 1 1.4626 0.00934 1.4605 0.00944 1.4584 0.00955 2 3 4 1.4721 0.00016 1.4710 0.00106 1.4700 0.00205 " 5 1.4778 0.00586 1.4765 0.00656 1.4754 0.00745 6 a 7 Series H. Exp. 8 1.4733 0.00136 1.4722 0.00226 1.4710 0.00305 " 9 1.4750 0.00306 1.4737 0.00376 1.4725 0.00455 V 10 1.4762 0.00426 1.4749 0.00496 1.4734 0.00545 11 12 1.4733 0.00136 1.4722 0.00226 1.4710 0.00305 13 1.4735 0.00156 1.4721 0.00216 1.4706 0.00265 a 14 1.4697 0.00224 1.4671 0.00284 1.4646 0.00335 15 1.4758 0.00386 1.4744 0.00446 1.4730 0.00505 Series III. Exp. 16 1.4721 0.00016 1.4702 0.00026 1.4683 0.00035 " 17 1.4688 0.00314 1.4656 0.00434 1.4624 0.00555 " 18 1.4768 0.00486 1.4753 0.00536 1.4739 0.00595 19 1.4780 0.00606 1.4764 0.00646 1.4748 0.00685 20 1.4721 0.00016 1.4702 0.00026 1.4683 0.00035 " 21 1.4760 0.00406 1.4742 0.00426 1.4725 0.00455 22 23 1.4699 0.00204 1.4681 0.00184 1.4663 0.00165 " 24 1.4721 0.00016 1.4702 0.00026 1.4683 0.00035 Series IV. Exp. 25 1.4746 0.00266 1.4720 0.00206 1.4693 0.00135 " 26 27 1.4945 0.02256 1.4920 0.02206 1.4895 0.02155 " 28 1.4789 0.00696 1.4767 0.00676 1.4745 0.00655 " 29 1.4776 0.00566 1.4755 0.00556 1.4735 0.00555 " 30 " 31 " 32 1.4741 0.00216 1.4695 0.00044 1.4651 0.00285 " 33 1.4777 0.00576 1.4752 0.00526 1.4728 0.00485 92 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE X CONTINUED. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE TREMONT TURBINE. 1 a 3 4 5 6 *y 8 9 10 11 Temperature of the atmosphere TIME. Dura- Total No. of Depth of Heij-ht of the Fall Number of the series and of the experiment. Date of the experiment, 1851. in degrees of Fahrenheit's thermometer. tion of the experi- ment, in min- length of the weir, in feet. ;. the end con- trac- tions. Reference to the figures on plate X. water on the weir by obser- vation ; in feet. water in the lower canal, below the top of the affect- ing the leakage of the wheel- pit; in feet. Beginning of the experiment. Ending of the experiment. External iti r in thi 1 Near the shade. weir. H. mln. n. min. utes. n. h'. weir, in feet h. Series V. Exp. 34 February I,P.M. 20.75 31.50 2 11 2 15.5 4.5 13.978 4 Fig. 2 0.4937 1.12 1.61 " 35 it it a 20.00 31.50 2 25 2 33 8 6.987 2 " 1 0.7908 1.16 1.95 " 36 ti it it 21.50 31.50 2 39 2 43 4 13.978 8 " 3 0.4981 1.17 1.67 " 37 a it it 22.25 31.50 2 49.5 2 54 4.5 10.484 6 u 4 0.60CO 1.23 1.84 " 38 tl it it 20.50 3 3.5 3 14 10.5 6.989 4 " 5 0.8000 1.21 2.01 39 it u u 21.50 30.00 3 19 3 23.5 4.5 16.980 4 " 7 0.4337 1.22 1.65 40 a a a 21.00 31.50 3 34 3 48 14 6.987 2 " 1 0.7896 1.24 2.03 41 a a 18.50 31.25 4 16 4 26 10 6.987 2 " 1 0.7790 1.25 2.03 42 u u u 18.00 4 52 5 8 6.987 2 1 0.7704 1.35 2.12 Series VI. Exp. 43 February 3, P. M. 38.25 2 6 2 11 5 13.978 4 Fig. 2 0.5977 1.10 1.70 " 44 it tt 38.25 32.25 2 20.5 2 30 9.5 6.987 2 1 0.9561 1.17 2.13 " 45 a tt tt 38.25 2 37 2 43 6 6.987 4 " 9 0.9636 1.17 2.13 46 u tt tt 37.50 32.00 2 48.5 2 56 7.5 13.978 8 3 0.6023 1.16 1.76 47 It tt tt 37.50 3 6.5 3 13 6.5 10.488 6 " 4 0.7308 1.11 1.84 " 48 tt tt tt 37.25 32.00 3 16.5 3 22.5 6 16.980 4 " 7 0.5238 1.13 1.65 " 49 tl It tt 37.00 3 40 3 45 5 6.987 2 1 0.9533 1.13 2.08 " 50 tt u tt 3 47 3 59 12 6.987 2 " 1 0.9531 1.13 2.08 " 51 tt tt tt 4 2 4 4 2 6.987 2 1 0.9539 1.13 2.08 " 52 tt tt tl 4 23 4 31 8 6.987 2 " 1 0.9415 1.13 2.07 " 53 tt tl 11 33.75 4 46.5 5 13.5 6.987 2 " 1 0.9275 1.14 2.07 Series VII. Exp. 54 February 4, A.M. 26.00 31.75 9 6 9 12.5 6.5 16.980 4 Fig. 7 0.5233 1.12 1.64 55 it ti tt 26.50 9 26.5 9 37 10.5 5.487 2 " 10 1.1278 1.14 2.27 " 66 it tt ti 31.00 31.75 9 44 9 57 13 6.987 2 " 11 0.9544 1.15 2.10 57 ti it u 30.00 31.75 10 17 10 22 5 8.489 2 " 12 0.8375 1.14 1.98 " 58 it it u 31.25 31.75 10 31 10 36 5 5.487 2 " 10 1.1269 1.17 2.30 " 59 ti ti n 33.50 31.75 10 42 10 46.5 4.5 6.987 4 " 13 0.9609 1.13 2.09 " 60 ti it ti 34.00 31.75 10 56.5 11 1 4.5 13.978 8 " 3 0.6017 1.13 1.73 " 61 11 11 It 35.00 11 10 11 15 5 10.489 6 " 4 0.7303 1.12 1.85 62 11 tl 11 37.50 31.75 11 20 11 26 6 13.978 4 " 2 0.5971 1.15 1.75 " 63 11 It tl 38.00 31.75 11 39.5 11 55 15.5 5.487 2 " 10 1.1256 1.14 2.27 " 64 " " P.M. 40.75 16 25 9 5.487 2 " 10 1.0935 1.13 2.22 Series VIII. Exp. 65 February 4, P.M. 38.25 32.00 3 10.5 3 14 3.5 16.980 4 Fig. 7 0.2316 1.19 1.42 66 It U tl 37.50 3 33.5 3 40 6.5 1.829 2 14 1.0581 1.17 2.23 " 67 11 11 U 37.00 3 45 3 52 7 3.658 4 " 15 0.6650 1.18 1.84 68 tl tl 11 36.75 3 58 4 2 4 5.487 6 16 0.5066 1.24 1.75 " 69 It 11 tl 4 11.5 4 16 4.5 1.829 2 " 14 1.0574 1.21 2.27 " 70 11 It It 36.25 32.00 4 21 4 25 4 8.489 2 " 12 0.3706 1.19 1.56 " 71 tl tl U 4 32 4 37 5 5.487 2 10 0.4980 1.19 1.09 Series IX. Exp. 72 February 4, P.M. 34.25 4 53 5 2 9 16.980 4 Fig. 7 0.9206 1.18 2.10 " 73 it ti u 34.25 5 17.5 5 24 6.5 16.980 4 " 7 0.9091 1.02 1.93 74 It It 11 33.75J 5 31 5 43 12 16.980 4 " 7 0.8941] 1.17 2.06 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 93 TABLE X CONTINUED. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE TREMONT TURBINE. 12 13 14 15 16 17 18 Number of the Depth of water on the weir, Height of water Height of Fall from the surface of the water in the Uniform fall from the Depth on the weir, corrected corrected for above the water forebay, to the wheelpit, lor tne leakage series and of the experiment. the leakage of the wheel, taken in the in the the surface of the water to which the depths on the weir in each of the wheelpit, and the varia- REMARKS. wheelpit, in forebay ; wheelpit ; in the wheelpit ; series are tion in the fall ; feet. A". in feet. infect. in feet. IF. reduced ; in feet. H. in feet. U". Series V. Exp. 34 0.49456 14.845 0.510 14.335 14.079 0.49160 In experiments 41 and 42 the weir was " 35 0.79229 14.910 0.810 14.100 tt 0.79190 in the same state as in experiment 40 ; " 36 0.49897 14.891 0.520 14.371 0.49557 the height of the water above the wheel " 37 0.60710 14.891 0.620 14.271 tt 0.60437 was reduced for the purpose of testing " 38 0.80151 14.899 0.820 14.079 tt 0.80151 the method of reduction. " 39 0.43446 14.908 0.450 14.458 tt 0.43063 " 40 0.79112 14.897 0.809 14.088 tt 0.79096 " 41 0.78054 14.352 0.798 13.554 U 0.79049 " 42 0.77198 13.939 0.790 13.149 tt 0.78976 G(l>Jpa Y" T Ot^l lea A. Exp. 43 " 44 45 " 46 " 47 0.59850 0.95752 0.96501 0.60311 0.73181 14.903 14.929 14.915 14.894 14.887 0.620 0.980 0.992 0.630 0.755 14.283 13.949 13.923 14.264 14.132 13.907 u tt tl tt 0.59320 0.95656 0.96464 0.59804 0.72790 In experiments 50 and 51 the weir was in the same state as in experiment 49, except- ing that in 50 a board was placed on the brackets N, figs. 1 and 2, plate V., on which the water fell ; and in exp. 51, the plank 0, fig. 2, plate V., was placed in the position 48 0.52449 14.889 0.550 14.339 tl 0.51917 represented: the top of the plank was 6.5 " 49 0.95471 14.884 0.980 13.904 u 0.95477 inches below the top of the weir. In exps. 50 0.95451 14.886 0.980 13.906 It 0.95453 52 and 53, the weir was in the same state as " 51 0.95530 14.887 0.980 13.907 tl 0.95530 in exp. 49 ; the height of the water above 52 0.94291 14.406 0.970 13.436 tt 0.95380 the wheel was lowered for the purpose of 53 0.92892 13.914 0.952 12.962 U 0.95097 testing the method of reduction. Series VII. Exp. 54 0.52399 14.889 0.550 14.339 13.882 0.51837 In experiments 63 and 64 the weir was " 55 1.12952 14.884 1.150 13.734 1.13356 in the same state ; in experiment 64 the " 56 0.95581 14.882 0.980 13.902 a 0.95535 height of the water above the wheel was " 57 0.83870 14.875 0.865 14.010 tt 0.83614 lowered for the purpose of testing the " 58 1.12863 14.872 1.152 13.720 tt 1.13306 method of reduction. " 59 0.96230 14.872 0.990 13.882 tt 0.96230 " 60 0.60251 14.868 0.629 14.239 it 0.59743 " 61 0.73131 14.865 0.754 14.111 it 0.72733 " 62 0.59791 14.860 0.620 14.240 tt 0.59286 " 63 1.12732 14.869 1.150 13.719 it 1.13177 " 64 1.09523 13.926 1.120 12.806 it 1.12508 Series VIII. Exp. 65 0.23257 14.894 0.240 14.654 13.839 0.22817 In experiments 66, 67, 68, and 69, the " 66 1.06337 14.899 1.068 13.831 U 1.06358 lengths of the several bays of the weir " 67 0.66802 14.895 0.676 14.219 11 0.66202 were deemed to be too short relative to " 68 0.50885 14.902 0.515 14.387 11 0.50231 the depth flowing over, for the proposed " 69 1.06272 14.905 1.066 13.839 U 1.06272 formula to apply. " 70 0.37221 14.905 0.380 14.525 11 0.36625 71 0.50023 14.909 0.505 14.404 It 0.49360 Series IX. 72 0.92119 14.864 1.048 13.816 13.839 0.92170 Experiments 72, 73, and 74 were made " 73 0.90967 14.350 1.035 13.315 " 0.92145 for the express purpose of testing the " 74 0.89469 13.678 1.013 12.665 u 0.92153 mode of reduction. 94 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE X CONTINUED. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE TREMONT TURBINE. 19 2O 21 22 i = 0.07 b = 0.066. 6 = 0.08. Number of the Combination of experi- series and of ments used to determine Differences from the Differences from the Differences from the the experiment the yaluc of a. Values mean value of a, or from Values mean value of a. or from Values mean value of a, or from 1.47797. 1.47595. 1.47394. of a. of a. of a. + - + - + - Series V. Exp. 34 34 and 38 1.4645 0.01347 1.4611 0.01485 1.4577 0.01624 " 35 35,40, " 39 1.4737 0.00427 1.4726 0.00335 1.4716 0.00234 36 36 " 38 1.4679 0.01007 1.4660 0.00995 1.4G41 0.00984 " 37 37 38 1.4651 , 0.01287 1.4630 0.01295 1.4610 0.01294 " 38 39 39 " 38 1.4700 0.00797 1.4670 0.00895 1.4640 0.00994 40 35, 40, " 39 1.4737 0.00427 1.4726 0.00335 1.471G 0.00234 " 41 " 42 Series VI. Exp. 43 43 and 45 1.4827 0.00473 1.4785 0.00255 1.4744 0.00046 " 44 44,49, " 48 1.4706 0.00737 1.4694 0.00655 1.4681 0.00584 45 " 46 46 " 45 1.4821 0.00413 1.4798 0.00385 1.4774 0.00346 " 47 47 " 46 1.4890 0.01103 1.4870 0.01105 1.4850 0.01106 " 48 48 " 47 1.4879 0.00993 1.4834 0.00745 1.4788 0.00486 " 49 44,49, " 48 1.4706 0.00737 1.4694 0.00655 1.4681 0.00584 50 51 " 52 " 53 Series VII. Exp. 54 54 and 55, 58, 63 1.4715 0.00647 1.4696 0.00635 1.4677 0.00624 " 55 " 56 56 and 54 1.4699 0.00807 1.4686 0.00735 1.4674 0.00654 " 57 57 and 55, 58, 63 1.4881 0.01013 1.4843 0.00835 1.4806 0.00666 58 59 59 and 54 1.4850 0.00703 1.4814 0.00545 1.4778 0.00386 60 60 and 55, 58, 63 1.4695 0.00847 1.4689 0.00705 1.4683 0.00564 " 61 61 " 55,58,63 1.4619 0.01607 1.4620 0.01395 1.4620 0.01194 62 62 " 55,58,63 1.4711 0.00687 1.4691 0.00685 1.4671 0.00684 " 63 " 64 SeriesVIII. Exp. 65 65 and 71 1.4755 0.00247 1.4747 0.00125 1.4739 0.00004 " 66 67 " 68 69 70 70 " 71 1.4845 0.00653 1.4829 0.00695 1.4813 0.00736 71 Series IX. Exp. 72 " 73 " 74 Sums of the differences and 0.26767 0.25085 0.23672 mean values of a. 1.47797 1.47595 1.47394 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE X CONTINUED. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE TREMONT TURBINE 33 34 35 Number of the 6 = 0.055 6 = 0.05. 6 = 0.045. series and of the Differences from the Differences from the Differences from the experiment. Values mean value of , or from Values mean value of a, or from Values mean value of a, or from 1.47194. 1.46994. 1.46795. ota. of a. of a. + - + + - Series V. Exp. 34 1.4543 0.01764 1.4510 0.01894 1.4476 0.02035 " 35 1.4706 0.00134 1.4695 0.00044 1.4684 0.00045 " 36 1.4622 0.00974 1.4602 0.00974 1.4584 0.00955 " 37 1.4589 0.01304 1.4567 0.01324 1.4547 0.01325 38 " 39 1.4610 0.01094 1.4581 0.01184 1.4551 0.01285 40 1.4706 0.00134 1.4695 0.00044 1.4684 0.00045 " 41 42 Series VI. Exp. 43 1.4703 0.00164 1.4662 0.00374 1.4622 0.00575 " 44 1.4668 0.00514 1.4656 0.00434 1.4643 0.00365 " 45 " 46 1.4751 0.00316 , 1.4728 0.00286 1.4705 0.00255 a 47 1.4830 0.01106 1.4811 0.01116 1.4790 0.01105 48 1.4744 0.00246 1.4699 0.00004 1.4654 0.00255 49 1.4668 0.00514 1.4656 0.00434 1.4643 0.00365 " 50 51 < 52 " 53 Series VII. Exp. 54 1.4657 0.00624 1.4638 0.00614 1.4619 0.00605 " 55 56 1.4661 0.00584 1.4649 0.00504 1.4636 0.00435 57 1.4769 0.00496 1.4733 0.00336 1.4696 0.00165 58 " 59 1.4742 0.00226 1.4706 0.00066 1.4670 0.00095 " GO 1.4677 0.00424 1.4672 0.00274 1.4666 0.00135 " 61 1.4620 0.00994 1.4621 0.00784 1.4621 0.00585 " 62 1.4652 0.00674 1.4633 0.00664 1.4613 0.00665 " 63 64 Series VIII. Exp. 65 1.4731 0.00116 1.4722 0.00226 1.4714 0.00345 66 67 " 68 " 69 * " 70 1.4797 0.00776 1.4782 0.00826 1.4765 0.00855 71 Series IX. Exp. 72 " 73 " 74 0.23124 0.22900 0.23945 1.47194 1.46994 1.46795 i 96 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE CENTRE- VENT WHEEL FOR MOVING THE GUARD GATES OF THE NORTHERN CANAL. 136. This centre-vent wheel usually operates under about ten feet fall, and is of about sixty horse-power under this fall. It was constructed from nearly the same designs as the model centre-vent wheel, described in art. 100, and rep- resented on plate VII. For a general description of the Guard Gates, see vol. I., page 775, Appletoris Dictionary of Machines, Mechanics, etc., New York : D. Appleton & Co., 1852. A set of experiments upon the power of this wheel was made in 1848, in which the water discharged by the wheel was gauged at a weir constructed for the purpose, below the wheel. The following experiments were made with the same apparatus. The total length of the weir was 18.02 feet, which, for the purposes of these experiments, was diminished to 16.02 feet by two movable planks or par- titions, one foot wide each, the upstream faces of which, when placed upon the weir, were in the same plane as the upstream face of the weir. The form of the weir was such as to give complete contraction; it was constructed of wood, with the upstream face vertical. The crest of the weir was formed of southern hard pine plank, four inches in thickness ; the top was 0.53 inches wide, and bevelled off on the downstream side, at an angle of 40 with the vertical ; the ends of the weir and the sides of the partitions were of the same form. The bottom of the canal or basin, measured near the weir, was about 6.72 feet below the top of the weir. The water discharged by the wheel passed to the basin through an irregular and contracted channel, cut in rock, and confined by cement masonry. This basin was specially excavated in the rock, of large dimensions, in order that the water might reach the weir in a sufficiently quiet state to permit a satisfactory measurement to be made ; and also, for the same object, two gratings were placed across the basin, parallel to each other, and about six feet apart, the downstream grating being about seventeen feet from the weir. The effect of these several precautions was such that, although the water escaped from the wheelpit in a rapid and turbulent current, in the basin between the downstream grating and the weir, the water was tranquil and free from perceptible irregularities in its motion towards the weir. The depths upon the weir were measured by the hook gauge, described at art. 45, and represented by figures 9, 10, and 11, plate IV.; this was placed in EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 97 the basin about eight feet from the weir, in a box, in which the communication with the surrounding water was maintained by a small aperture in the bottom ; the box and hook gauge were firmly attached to a timber strongly bolted to the masonry forming one side of the basin. The quantity of water discharged by the wheel is usually regulated by the head gate, admitting the water from the river into the forebay above the wheel. When it is desired to diminish the quantity discharged by the wheel, this gate is partially closed, the effect of which is to diminish the fall acting upon the wheel ; but this method was unsuitable for these experiments, on account of the great agitation in the forebay, produced by the fall at the head gate. During these experiments, the head gate was fully opened, and the quantity of water discharged by the wheel was diminished by closing up a portion of the spaces between the guides, with pieces of wood. The wheel was prevented from revolving by the brake of the Prony dyna- mometer. The entire apparatus about the wheel remained unchanged throughout the four experiments, except that the head gate was closed on several occasions, to enable the partitions on the weir to be moved. This gate was large (five feet square,) and care was taken to keep it open to its full extent, in all these experiments. The apertures through which the water entered the wheelpit being the same, the quantity of water discharged must have been uniform, if the head acting upon the orifices had been constant; small variations, however, unavoidably occurred in the head, for which it was necessary to correct the depths upon the weir. This has been done in a manner precisely similar to that adopted in .the experi- ments upon the weir at the Tremont Turbine, described at art. 133. The apertures in the wheel and between the guides, were entirely submerged. The effective height of the water in the wheelpit was measured in a chamber constructed for the purpose, in the masonry. A free communication was main- tained between the water in the wheelpit and in the chamber by an iron pipe about 3.5 inches diameter. The surface of the water in the chamber was, in all the experiments, above the level of the top of the apertures between the guides. The height above the wheel was taken in the forebay nearly over the wheel, the gauge being placed in a box in the usual manner ; the zeros of the gauges, at which both these heights were taken, were at the same level, consequently, the difference in the readings gave the fall acting upon the apertures. 13 98 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. W O d O H PS H C P5 O N -, 03 H - W gg W 03 > = P5 H o oT K ^ fc H (2 O O " W r. O S CQ H Jz; H . i. -" CS , d 55 s J _0 9 B, a .2 to j .t; c g"S I I -e 1 | X t; e 1 *J p u - 2 2 -C 2 tq H" -^ o 3 o o M ^ 3 J3 ^ o ^- to R 9 10 & O V 00 * S ,- GO i "I .a" Jl sf ^* S J? rt ^ ro * s u g ti g S K a c "3 u H 8,H f a ^3 Q | g g -a > >3 H a . ll '-4-' O, g 3. ' v '7 ^ j I g | Bg 1 a~ fjjg t| & -8 s fe 2 g rH q O q CO co rH q rH CO S rH O EC ^ :S d - u p (M rH (N B ^ t -~'3'o -- S 5 S c s 5 Se 5 S ^^ s||fj|Jfll* 06 CO 00 CO CO CO CO i i s & o IN t- rH CO t- CO t i S f * S rH q q i s * s - a no * . o g a S a S A CO (N -* CM 95 ^ S 1 = a | O *^ / .3 _a 00 CO CO CO (2 = |" 1 S S S i -o . ^ 10 iO CO ^ fifljlll] s S! l^ Cj3 o- oC*sO**"" s ** ^ *" *n c'* j ^'* a go g * P. 1,3 S co !>. fn CO CO 1 i ^ 5 || co CO CO CO C J3 "5 "c *" o o o g *" ^ 5 ^3 .9 rH rH S * it fl ja o ' 3 s O * 00 03 "& -d h " d co rH co o r* " S 's S | 1 * CO IN * t- t^ i ' >0 (M I 1 M CO CO rH o 10 *" IS rH rH * si So i o H o t> Q O g H O B B H K O 02 H & H S HH H & B 2 2 i It ^1! 1 * IJD tj *E **" w **- ^- Q if, II 111 1 } 1 1 rS 's 1= 1 1 _g B. .a S ^,0 > <0 Jj -S 3 S -5 s - h S e s ~ . B o 5 -2 .S - -3 p S .S | * -|| o .b -| , T Y 1 r s {filijtiP ^ f^ f O? *ti r* <^? *^2 t~* O> t* t~ t' t 1 * t* t* t 00 t* odd o oddd o 3 g I .S | . 2 iiiiii i , j _g x 1 - 1 (M t>- O CO tc o o o o c o o o o c do o do ++ 1 4 1O CS O rH d d -H- If B Pi *O *O 'O ^^ O 1 *O -nod d dodo r-5 1 4- 4- 4H-4-4- 1 10 i III 1 O >O i I >O t^ C5 O OO CO !-HO (M to t*- O CNfMT* OO i 1 CO O tO ^ oooooo oo cooocccn oo odd d ddo'd d CD 5= B- * 3:! 1 * 3 ||rB - (NOT-H to i 1 t>. . t- 000000 OO 06060606 CO O CO (M C5 >O i 1-5 rH rH i-H r-5 r- OO -* CO <>l GO i-H 1Q O 'O i 1 t 1 rH 1 (MCOCO >O tO 00 O i 1 X5 TO tffPl OOO O OOOO O iCCOO CO COOI"-O to !H co co co co i"^ s^i co co i OOO O OOOO O co co co co co oo B 5 rH Considering the care with which these comparisons were made, and the per- fection of the method, the differences cannot be attributed to errors of observa- tion, but, rather, to a want of stability in some parts of the apparatus. The corrections determined October 26th, were used in reducing all the experiments made from October 20th, to November 7th, both inclusive ; for all subsequent experiments, the corrections found November 8th were used. The twenty-three experiments numbered from 11 to 33, in table XIII., were made under circumstances as nearly identical as practicable. They were made at different times throughout the course, for the purpose of neutralizing errors of the same class as that just described, the resulting effects of which ought to be shown by the variation in the coefficients deduced from experiments made at different times. These experiments are collected together in the following table: Differences from the DATE, Number of mean deduced from all experi- Mean coefficients. the experiments, or 1862. ments. from 3.3223. Oct. 20th, P.M., and Oct. 21st, A.M. 6 3.3186 0.0037 Oct. 21st, p. M., and Oct. 22d, A.M. 8 3.3216 0.0007 Oct. 29th, P.M. 6 3.3278 + 0.0055 Nov. llth, P.M. 3 3.3207 0.0016 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. The extreme variation is between the experiments of October 20th and 29th, in which it amounts to T . The greatest difference from the mean deduced from all the 23 experiments, is in the coefficient deduced from the experiments of October 29th, in which it amounts to F | T . It is fair to presume that simi- lar irregularities, not in any case much exceeding the above, and arising princi- pally from want of stability in the apparatus, exist in other parts of this series of experiments. 144. The capacity of the gauging chamber was obtained by measuring its dimensions. For this purpose, horizontal lines were traced on the sides of the chamber at every foot in height ; the widths were then measured at right angles to the sides, at points two feet apart; from these widths, and other neces- sary measurements, the total area was obtained at each horizontal section. When these measurements were made, the chamber was of course empty, but when filled with water, its dimensions would evidently be somewhat larger, in conse- quence of the sides and bottom yielding to the pressure. To ascertain what allowance to make for this, a systematic measurement was made in the spaces between the planking and the walls, both when the chamber was empty and when filled to the usual height; similar measurements were made for the bottom, by placing poles vertically, resting upon, and fastened to the bottom ; the eleva- tions of the tops of these poles were taken with a levelling instrument, both when the chamber was empty, and when filled. It was thus ascertained that the capacity of the chamber, when filled with water to the usual height, was 11.11 cubic feet greater than when empty. Two persons made independent measurements of the capacity of the chamber, the results of which differed only about of a cubic foot, a coincidence which must of course be considered as accidental. The capacity finally determined upon for 9.5 feet in height, (which was nearly the depth filled, in each experiment,) and including the enlargement resulting from the pressure, was 12138.18 cubic feet. 145. The chamber was not quite water-tight, but the amount of the leakage was determined by noting the rate at which the surface of the water lowered, when none was admitted from the weir, and the waste gate was closed ; this was repeated with the water in the chamber at different depths. It was thus found that the mean leakage was 0.035 cubic feet per second ; that is, the product of 0.035 multiplied by the number of seconds that the water flowing over the weir continued to enter the chamber during an experiment, must be added to the quantity in the chamber at the moment the water was diverted, in order to give the true quantity that passed over the weir in the same time. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. HI 146. It was not convenient to empty the chamber entirely after each experi- ment, but the heights of the water in the chamber at the beginning and end- ing of each, were ascertained with great accuracy by means of hook gauges, placed in the boxes X and Y, figures 1, 2, and 3, plate XII., which were fastened to a post strongly bolted to the wall A. A communication was established, at will, between the water in the chamber and either of the boxes, by pipes and cocks. The operation of taking the heights was as follows : the chamber having been sufficiently emptied, the waste gate K was closed, the communication of the lower box with the chamber was established, and when the oscillations in * the surface had ceased, the height of the water was taken; the cock was then shut, and a signal made for opening the swing gate. When the chamber had been filled, and the flow of water into the chamber diverted by closing the swing gate, the communication with the upper box was opened ; when the oscil- lations had ceased, observations of the water were taken at short and regular intervals, for some minutes, the time and height being noted. In consequence of the leakage of the chamber, the surface lowered slowly, and the continued obser- vations were made for the purpose of being able to infer the exact height at which the water stood in the chamber at the instant that the swing gate was shut, the very slow rate at which the surface of the water in the chamber lowered, permitting this to be done with great precision. For the success, how- ever, of this operation, it was essential that the timekeeper used should agree with the chronometer, by which the times of opening and shutting the swing gate were noted ; it was accordingly frequently compared, and any difference noted. 147. Plate XIV. represents the different forms of weir on which experiments were made. All the figures are on the same scale, namely, five feet to an inch, or -fa the full size. Figure 1 is a longitudinal section, figure 2, a plan, and figure 3, an eleva- tion of what we call the regular weir, that is, a weir in which the contraction is complete, both on the ends and on the bottom. Figure 4 is an elevation of a weir of precisely the same form as that last described, excepting that it is divided into two equal parts or bays by the par- tition, which is two feet wide. The upstream side of the partition is in the same vertical plane as the remainder of the weir, having no bolt heads or other pro- jection below the level of the surface of the water. Figures 5, 6, and 7, represent a weir of precisely the same form as that first above described, excepting that the depth of the canal approaching the weir is diminished. 112 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. Figures 8, 9, and 10, represent the same weir as first above described, modi- fied so that the contraction at the ends is suppressed, that is, the canal leading to the weir is of the same width as the weir. These figures also show the apparatus used to ascertain the effect of taking the depths upon the weir at different distances from it, by means of pipes opening near the bottom of the canal. Figures 11 and 12 represent the upper part of a dam, of the same section as that erected by the Essex Company, in 1846-8, across the Merrimack Eiver at Lawrence, (about nine miles below Lowell). This magnificent work has an overfall 900 feet in length, the perpendicular fall being about 24 feet. This form was experimented upon, in order to obtain a formula for computing the flow of the river over this dam. DESCRIPTION OF TABLE XIII. Containing the details of the experiments on the flow of water over weirs, made at the Lower Locks, Lowell, in October and November, 1852. 148. The columns numbered from 1 to 5, require no further explanation than is contained in the respective headings. 149. COLUMN 6. Duration of the experiment. This is the interval of time during which the water flowed into the chamber ; it is obtained by taking the difference of the corresponding times in column 5. 150. COLUMN 7. Mean depth upon the weir ly observation. It was found imprac- ticable in many cases, to maintain the canal at a uniform height throughout an experiment, although every endeavor was made. For instance, no experiments were made when the mills were in operation, nor until some hours after the usual time when they ceased drawing water; this rendered it necessary to per- form the experiments either during the night, or on Sunday ; in consequence of the lateness of the season, advantage was taken of both these opportunities. When any change was made in the level of the water in the canal, for the purpose of varying the depths upon the weir, a considerable time was allowed to elapse before the experiments were resumed, in order that the level of the water might get well established. In spite of all precautions, however, variations fre quently occurred in the depths upon the weir, which, with the ordinary mode of taking an arithmetical mean of the several observations of the depth, would have materially affected the accuracy of the results ; this difficulty was obviated in a EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. H3 great degree, by the use of a novel mode of obtaining the mean depth, which will now be explained. Let h,h',h",Qic.h n represent the several observed depths upon the weir, the suc- cessive values not differing greatly from each other. t,t',t",etc.t n , the corresponding intervals of time between the several observations; T, the sum of all the intervals of time ; Q, the total volume of water actually flowing over the weir in the time T; If, the mean depth upon the weir that would discharge the volume Q, in the time T; I, the length of the weir ; C, a constant coefficient : we shall have, evidently, very nearly, "*+ etc. + we have also whence we derive, by substituting the value of Q previously found, As an example of the application of this method, let us take the observations made at the north hook gauge during experiment 74 ; this is selected, simply because the variations in the depths upon the weir were greater than in any other experiment. 15 114 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. EXTRACT FROM THE NOTES TAKEN AT THE NORTH HOOK GAUGE. Commencement of the experi- ment by the time of this watch. 9* 12' 12.4". Ending of the experiment by the time of this watch, 9*24' 49.9". October 24th, 1852, A.*., 9, V, watch 12" fast. TIME. Beading of the hook gauge. 9* 9' 15" 0.6360 10 50 0.6320 11 45 0.6325 12 45 0.6310 1 13 15 0.6310 1 14 20 0.6300 1 14 50 0.6365 3 15 20 0.6290 1 16 30 0.6300 1 17 5 0.6335 2 17 55 0.6380 3 18 35 0.6480 5 19 20 0.6500 6 20 0.6470 5 20 55 0.6470 5 21 25 0.6445 4 22 10 0.6530 7 22 35 0.6550 7 23 5 0.6480 5 23 45 0.6580 8 24 35 0.6605 9 Arithmetical } 0>6428 mean reading, ) For the purpose of simplifying the operation of finding the mean, it is assumed that we can, without sensible error, use an arithmetical mean of all depths not varying more than 0.002 feet from each other ; accordingly an arith- metical mean has been taken of all the readings marked 1 in the margin of the above table, and similar means have been taken of the other readings marked with the same number in the margin. It will be perceived that it was noted at 9 h 5', that the watch was 12" fast ; by another comparison with the chronom eter made at 10 h 47', the watch was 22" fast; from these two comparisons it is inferred that, at the middle of the experiment, the watch was 13.3" fast. Instead of changing the times of all the observations, the time of the commencement and ending of the experiment has been changed to conform to this watch, but for the purpose of this reduction only. By the method adopted, it is assumed that the height of the water did not change until half the interval of time between two consecutive observations had elapsed ; accordingly, we find that the time cor- EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 115 responding to the first mean depth, is from the beginning of the experiment to 9 h 14' 35", or 142.6", and from 9" 15' 5" to 9 h 16' 47.5", or 102.5", making 245.1". The several mean readings and the corresponding times, given in the following table, are obtained in this manner; the depths upon the weir corresponding to the several mean readings, are also given, which are found by subtracting 0.03072 feet from each mean reading, (see art. 143). Time correspond- Mean depths upon Number Mean reading of the ing to each mean the weir, deduced of the hook gauge. reading. from the several mean mean readings. reading. Feet. Seconds. Feet. i 0.63020 245.1 0.59948 2 0.63350 42.5 0.60278 3 0.63725 75.0 0.60653 4 0.64450 37.5 0.61378 5 0.64750 167.5 0.61678 6 0.65000 42.5 0.61928 7 0.65400 62.5 0.62328 8 0.65800 45.0 0.62728 9 0.66050 39.9 0.62978 The quantities in the third column of this table are the values of *-, -i- , etc., i & ' in the expression given above for H; the quantities in the fourth column are the corresponding values of h, h', etc. The value of T being 757.5, all the quantities in the second member of the equation are known; by substituting these values we find H= 0.6113. The arithmetical mean of the eighteen observations is 0.6428 ; deducting the correction 0.03072, we find the mean depth to be 0.6121 ; the difference by the methods is 0.0008. A similar computation on the observations at the south hook gauge gives H= 0.6099. By taking the arithmetical mean of the observations, we find the depth, by the south hook gauge to be 0.6096. The mean of the above values of H, or 0.6106, is adopted as the depth on the weir in experiment 74. A similar reduction has been made of the observations at each hook gauge, in all the experiments ; the arithmetical mean of the two results obtained for each experiment, is given in column 7. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. Notwithstanding the advantage attending this mode of reduction, it cannot be denied that, for the most perfect experiments, the depth on the weir should be invariable throughout, and that, cceteris paribus, the experiments will be the less valuable, the greater the variation. To enable the reader to judge of the rela- tive value of the experiments, as far as it depends upon this variation, the small figures to the left and above the several depths in column 7 are given; they indicate the highest number of values of h, h', h", etc. used in the reduction of the observations, at either of the hook gauges, in the corresponding experiments. 151. COLUMN 8. Mean velocity of tlie water approaching the weir. This is obtained by dividing the corresponding quantity of water flowing over the weir, given in column 14, by the area of the section of the canal, at the hook gauge boxes. In the weir having contraction at the ends, this^would strictly include all the space under the gauge boxes, although, from the form of the walls, it is evident that the current could flow only in a small part of this space ; consequently, the portion in which the current could not flow is not included in the areas used. 152. COLUMN 9. Head due to the velocity in column 8. This is sufficiently explained in the heading. 153. COLUMN 10. Depth upon the weir, corrected for the velocity of the water approaching the weir. In the common formula for the discharge of water over weirs, The second member may be separated into three factors, namely: C, the coefficient of contraction; I, the length of the weir; and ffft^ZglT, the theoretical discharge for the unit of length. According to a well-known elementary theorem in hydrau- lics, the latter factor may be represented by the area of a segment of a parab- ola, of which the parameter is 2y; thus, in figure 5, plate XII., if AB = H, and BC=\j2gir, and the curve A MO is a parabola, of which the vertex is A, we shall have the area of the segment AS C^H^ZgH; also, the velocity of the fluid at any point P will be represented by the ordinate PM. The factor Hl^ZgH may also be decomposed into two others: H=AJB, and WZgH, which equals the mean value of all the ordinates of the parabola between A and C, and represents the mean velocity of the fluid for the whole height of the ori- fice. In demonstrating this theorem, it is assumed that the water in the reser- voir is at rest ; we can, however, easily establish an analogous theorem, in which it is assumed that the water in the reservoir has a velocity approaching the weir, in the direction perpendicular to the plane of the weir. Suppose h to be EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. the head due this velocity; and in figure 6, plate XII., let AB H, and AD=: h, we shall have for the velocity v', at any point P in the height of the orifice, but this value of v' is the ordinate corresponding to the abscissa, AP-\-h = DP, of a parabola whose parameter is 2^. We have also We can, consequently, represent the discharge for the unit of length, by the area of the surface ABCG, which is a portion of the segment BCD; the area of ABCG is the difference of the areas of the segments BOD and AGD; the area of B CD is IBD X B 0=$(H+h)\/2ff(ir+h), and the area of ADG is consequently, the area of ABCG is and for the total discharge we have The formula (^4) may be put under the form Q= Cll^TgH 1 . Suppose H' to represent a depth upon the weir that would give the discharge Q f by the formula ( C ), we shall have substituting the value of Q' in (B~), and reducing, we find 118 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. The equation (B), from which this value of H' is derived, does not agree with that given for a similar case by most writers on hydraulics, who seem generally to have followed Du Buat ; * it agrees, however, with the expression given by Weisbach,f who appears to have been the first to point out the error. The formula (D) was communicated to the author, in 1849, by Mr. Boyden, accompanied by a demonstration somewhat resembling the above. The values of IT, given in column 10, have been computed by the formula (D) from the corresponding values of H and h in columns 7 and 9. 154. COLUMNS 11, 12, and 13 are sufficiently explained by their respective headings. 155. COLUMN 14. Quantity of water passing the weir per second. The quantities in this column are obtained by dividing the total quantities given in column 13, by the corresponding intervals of time in column 6. 156. COLUMN 15. Value of C in the formula Q having the corresponding values in column 14. In the formula proposed at art. 124, namely: Q= C(llnh)h*, the values of the constants a and b are to be determined by experiment. The values adopted in the formula by which the coefficients in this column have been computed, namely : a = f , b = 0.1, were determined upon after many trials of other values; in consequence of their giving results according the most nearly with all the experiments, and at the same time having a convenient degree of simplicity. It is quite likely that many other values of a and b (probably an unlimited number) might be found that would accord somewhat nearer with the experiments; a closer approximation than is given by the use of the values adopted, could have, however, but little practical value ; much less, it was thought, than would be derived from the use of the simple values adopted. The use of a fractional power, such as a =1.47, deduced from the experiments at the Tre- mont Turbine (art. 135), is very inconvenient, and, to persons not well skilled in the use of logarithms, offers great difficulty. * Principes cT Hydraulique, etc., by M. Du Buat. Paris: 1816. Vol. 1, page 201. t Attgemeine Maschinen Encyclopiidie. Leipzig : 1841. Vol. 1, page 489. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 157. COLUMNS 16, 17, and 18, are, for the purpose of obtaining correct mean results of the experiments, made under circumstances nearly identical. In conse- quence of the variations in the height of the canal (art. 150), it was impracti- cable to repeat the experiments with precisely the same depth upon the weir ; by the method adopted for obtaining these mean results, all inconvenience from this source is obviated. As the formula by which the values of C, in column 15, are obtained, is such as to give results agreeing very nearly with experiment, even when the depths differ considerably, it is plain that the values of C deduced from experiments having nearly the same depths, cannot be affected by small variations in the depths, and will be subject to no greater irregularities than if, in the several experiments from which they are deduced, the depths had been precisely the same. We can consequently take a mean coefficient with the same confidence that we could take a mean quantity, if the depths had been precisely the same. These mean coefficients are given in column 16. In column 17 are given depths on the weir, nearly a mean of those in the experiments from which the corresponding mean coefficients have been deduced. In column 18, are given what may be called the mean quantities of water actually found by experiment to be discharged with the corresponding depths in column 17. A method similar to the above was used to reduce the quantities discharged in the experiments of Castel, reported in the Aniiales de chimie et de Physique, vol. 62. Paris: 1836; reprinted in the first volume of the Annales des Pords et Chaussees for 1837. 158. COLUMN 19. Quantity of water passing the weir, calculated by the formula H" having the corresponding values in column 17. The coefficient 3.33 is derived from the arithmetical mean of all the coeffi- cients in column 15, which is 3.3318, the two final decimals being omitted for the sake of simplicity. The largest coefficient in column 15, is that deduced from experiment 34, which is 3.3617, exceeding the coefficient adopted by T |^ part; the smallest coefficient is that deduced from experiment 4, which is 3.3002, being less than the coefficient adopted, by ^^ part; that is, the formula by which the quantities in column 19 are computed, will represent every experi- ment in the table, within one per cent. 159. COLUMN 20. Proportional difference, or the absolute difference of the quantities in columns 18 and 19, divided by the quantity in column 18. The greatest propor- tional difference is that deduced from experiments 34 and 35, which is 0.0090, or a little less than one per cent. In these experiments there were two weirs, 120 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. about four feet long each, separated by a partition two feet wide ; the near neighborhood of the two orifices appears to have affected the discharge. The next largest proportional difference is that deduced from experiments 36 to 43, which is 0.0068, or about f of one per cent. ; in these experiments, the depth of the water in the canal leading to the weir, was only about three times the depth upon the weir. The experiments with the diminished depth in the canal were made for the purpose of testing the method of correcting the depths, upon the weir, for the velocity of the water approaching the weir (art. 153). They indicate that the method is not strictly accurate, as might have been anticipated, omitting, as it does, all consideration of the effect produced by this velocity, in modifying the contraction. It is well understood that such an effect is produced,* but it is of such a complicated nature, that the investigations hith- erto undertaken have thrown but little light upon it. It will be perceived by referring to column 4, that the experiments 51 to 55 were made under the same circumstances as experiments 44 to 50, excepting that the sheet of water, after passing the weir, was prevented from expanding laterally for a certain distance. This was accomplished by placing boards at the ends of the sheet, as represented by the broken lines at A, figures 8 and 9, plate XIV. By referring to column 16, it will be seen that the effect of these boards was to diminish the coefficient from 3.3409 to 3.3270, corresponding to a diminution of the quantity discharged by the weir, with the same depth, of %%$, or about four-tenths of one per cent. ; in other words, the effect of the boards upon the discharge was the same as would be produced by shortening the weir ?ITP or \ inch, at each end. By reference to figure 8, plate XIV., it will be perceived that these boards did not affect the free communication between the atmosphere and the air under the sheet of water ; if this communication had been obstructed, so that the pressure of the air under the sheet had been dif- ferent from that of the atmosphere, it would have affected the discharge. * Jaugeage des cours d'eau, etc., by M. P. Boikau, page 40. Paris: 1850. 122 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE 1 a 3 4 5 6 7 8 _ Mean Temperatures by velocity of Fahrenheit's the water thermometer. Tune of the commencement and conclusion of the experiment, approach- ing the Num her of Date of the Reference to the figures on plate XIV., and particular as indicated by the telegraphic signals. Dura- tion of the Mean depth upon the weir by weir at the transverse sect, thro' the holes in the the expert experiment. description of the weir. experi- observation. hook gauge ment. 1852. Of the all Of the ment. boxes, or in the shade. water. Commencement. Conclusion. H. six feet from the weir. V. H. niiti sec. H. tnin sec. we. Feet. Feet. 1 Oct. 27, P.M Figures 1, 2, and 3. 10 15 0.8 10 18 13.6 192.8 2 1.52430 0.7682 2 3 a it tt tt a tt Width of the canal on the upstream side of the weir, 13.96 feet. Mean depth of the canal opposite the hook gauge boxes 6.048 feet below the top of the weir. 11 11 20 54 1.3 1.3 11 11 23 57 14.2 10.6 192.9 189.3 2 1.55045 2 1.55930 0.7813 0.7882 4 " 28, A.M 43.75 46.5 26 17.9 29 20.1 182.2 'i.segio 0.7889 5 Oct. 24, P.M 52 48.5 f i 0.5 c 8 20.5 260.0 "1.23690 0.5904 6 It U it C 33 2.3 ( 37 21.5 259.2 2 1.24195 0.5933 7 It ft It Figures 1, 2, and 8. 10 1.7 10 4 18.4 256.7 2 1.24795 0.5971 8 11 U tt Same as the preceding. 10 31 2.1 10 35 20.6 258.5 2 1.25085 0.5944 9 tl ft U 11 2.3 11 4 16.1 253.8 "1.25290 0.6000 10 U tl tt 11 30 1.8 11 34 21.4 259.6 2 1.25490 0.5987 11 Oct. 20, P.M 10 1 0.8 10 7 22.0 381.2 "0.96711 0.4256 12 U U tl . 10 30 0.9 10 .'So 44.0 343.1 "1.02755 0.4594 13 ft tl It 11 12 0.7 11 17 47.4 346.7 "1.03395 0.4629 14 tl 11 il 11 48 0.6 11 53 43.3 342.7 2 1.03315 0.4634 15 " 21, A. M. 24 59.5 30 37.5 338.0 "1.04060 0.4680 16 11 It U 43 49 1 0.0 1 5 37.4 337.4 "1.03735 0.4666 17 " " P.M. c 48 8.0 c 54 36.4 388.4 "0.96325 0.4233 18 it u ft 10 23 1.2 10 29 9.9 368.7 "0.97590 0.4304 19 it u tt 42 48.75 10 52 0.4 10 58 8.7 368.3 "0.97950 0.4318 20 It ft 11 11 23 1.4 11 29 4.2 362.8 "0.98885 0.4377 21 It tl tl Figures 1. 2, and 3. 11 53 1.5 11 59 3.5 362.0 "0.99460 0.4418 22 " 22, A. M. Same as the preceding. 43 0.0 49 48.8 408.8 "0.91570 0.3951 23 11 ft 11 42 1 12 0.7 1 18 40.9 400.2 2 0.92800 0.4015 24 u u a 1 42 0.3 1 48 26.8 386.5 2 0.94625 0.4126 25 " 29, P.M. 9 2 3.5 9 7 54.2 350.7 2 1.01275 0.4517 26 It It U 51.5 48.25 9 35 2.7 9 40 51.6 348.9 4 1.01160 0.4520 27 It tl U 10 5 1.2 10 10 59.7 358.5 00.99495 0.4429 28 U U 11 10 34 59.8 10 40 39.7 339.9 n.03360 0.4653 29 il it H 11 3 0.2 11 8 29.6 329.4 "1.05565 0.4779 30 U il il 11 32 1.8 11 37 27.0 325.2 2 1.06920 0.4863 31 Nov. 11, P.M. 34 41.25 8 56 59.5 9 3 6.0 366.5 20.98370 0.4352 32 u u u 9 30 0.6 9 36 7.8 367.2 "0.97820 0.4320 33 u u u 10 0.3 10 6 19.8 379.5 *0.96700 0.4236 Figure 4. 34 35 Nov. 3, P.M. il it 45 48 Width of the canal on the upstream side of the weir, 18.% feet. Mean depth of the canal opposite the hook gauge boxes, 6.048 feet below top of the wet. Two 9 9 12 59 1.3 59.6 9 10 19 7 37.8 18.4 456.5 438.8 6 1.01025 6 1.02625 0.3527 0.3596 equal bays separated by a partition 2 feet wide. 36 Oct. 31, A.M. - 7 17 15.8 7 22 52.9 337.1 "1.02805 0.9496 37 It it tl 7 47 59.6 7 53 29.9 330.3 a 1.03720 0.9589 38 u u a 46 48.75 Figures 6, 6, and 7. 9 46 0.3 9 51 34.9 334.6 "1.04455 0.9684 39 40 41 u u a it it it u u u Width of the canal on the upstream side of the weir, 18.96 feet. Mean depth of the canal opposite the hook gauge boxes, 2.014 feet below the top of the weir. Bot- tom of the canal horizontal for 23 feet on the upstream side of the weir. 10 10 11 14 41 10 1.4 0.7 7.8 10 10 11 19 46 15 23.3 28.2 32.5 321.9 327.5 324.7 2 1. 04495 2 1.04600 5 1.05130 0.9693 0.9691 0.9756 42 li 11 11 11 39 59.4 11 45 8.3 308.9 "1.07945 1.0049 43 " " P.M. 15 1.4 20 15.7 314.3 2 1.07115 0.9958 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 123 XIII. LOWER LOCKS, LOWELL, IN OCTOBER AND NOVEMBER, 1852. 9 10 11 IS 13 14 15 16 17 18 19 ao Depth upon the Quantity of water Head due to weir, corrected Total quan- Approx- that would have Proportional Num- ber of the experi- ment. the Telocity n column 8, or the values of k by the formula A- + for the velocity of the water ap- proaching the weir, or the val- ues of H' by the formula H'= 32 Length of the weir I. No. of end con- trac- tions. n. tity of water that passed the weir dur- ing each ex- periment, as measured in the lock chamber. Quantity of water pass- ing the weir per second. Value of C in the formula C(l0.1nH')H'% Q having the corresponding values in col- umn 14. Mean value of C for each particular descrip- tion of weir. imate mean depth upon the weir, for each particu- lar de- scription of weir. passed the weir with the depth in column 17, calcula- ted by the formula Q= 8 qj 0.1nH'/)H"2 C having the cor- responding value in column 16. Quantity of water passing the weir, calculated by the formula Q= % 3.33(l-Q.lnIP>)H"* difference, or the absolute difference of the quantities in columns 18 and 19, divided by the quan- tity in column 64.3236. [(H+A)5-]3 W. 18. Feet. Feet. Feet. Cubic feet. Cubic feet Feet. Cubic feet per Cubic feet per per second. second. second. 1 0.00917 1.53300 9.997 2 11815.19 61.2821 3.3318 2 0.00949 1.55945 u U 12069.49 62.5686 3.3174 3 0.00966 1.56845 a tt 11964.90 63.2060 3.3230 3.3181 1.56 62.6147 62.8392 + 0.0036 4 0.00968 1.57828 it tt 11542.52 63.3508 3.3002 5 0.00542 1.24208 9.997 2 11723.21 45.0893 3.3412 6 0.00547 1.24718 u tt 11753.09 45.3437 3.3398 7 0.00554 1.25325 tt tt 11725.57 45.6781 3.3405 8 0.00549 1.25610 it tt 11760.23 45.4941 3.3159 3.3338 1.25 45.4125 45.3608 0.0011 9 0.00560 1.25825 a tt 11658.13 45.9343 3.3396 -> 10 0.00557 1.26022 tt tt 11903.41 45.8529 3.3260 11 0.00282 0.96983 9.997 2 11872.75 31.1457 3.3265 12 0.00328 1.03071 U u 11645.35 33.9416 3.3129 13 0.00333 1.03716 U u 11869.83 34.2366 3.3110 14 0.00334 1.03636 tt It 11745.20 34.2725 3.3182 15 0.00341 1.04388 u tt 11713.35 34.6549 3.3196 16 0.00339 1.04061 tt tt 11651.45 34.5330 3.3233 17 0.00279 0.9 6593 It tt 12023.62 30.9568 3.3261 18 0.00288 0.97868 tt it 11627.93 31.5377 3.3234 19 0.00290 0.98229 tt tt 11659.61 31.6579 3.3179 20 0.00298 0.99172 it It 11661.78 32.1438 3.3216 21 0.00303 0.99752 It tt 11754.36 32.4706 3.3265 3.3223 1.00 32.5486 32.6240 + 0.0023 22 0.00243 0.91804 tt tt 11721.88 28.6739 3.3218 23 0.00251 0.93042 tt It 11682.99 29.1929 3.3155 24 0.00265 0.94881 tt tt 11629.93 30.0904 3.3198 25 0.00317 1.01580 tt tt 11678.40 33.3003 3.3211 26 0.00318 1.01466 It tt 11623.48 33.3147 3.3281 27 0.00305 0.99789 tt tt 11670.68 32.5542 3.3333 28 0.00337 1.03684 tt tt 11697.19 34.4136 3.3297 29 0.00355 1.05906 u tt 11685.17 35.4741 3.3263 30 0.00368 1.07274 tt tt 11764.18 36.1752 3.3283 31 0.00294 0.98653 tt tt 11702.07 31.9293 3.3251 32 0.00290 0.98099 tt tt 11629.66 31.6712 3.3259 33 0.00279 0.96969 tt tt 11762.21 30.9940 3.3111 34 0.00193 1.01212 7.997 4 11863.64 25.9883 3.3617 35 0.00201 1.02820 U if 11655.85 26.5630 3.3586 3.3601 1.02 26.2686 26.0333 0.0090 36' 0.01402 1.04098 9.997 2 11747.38 34.8484 3.3519 37 0.01430 1.05039 U It 11657.37 35.2933 3.3498 38 0.01458 1.05799 a it 11953.56 35.7249 3.3548 39 0.01461 1.05842 tt tt 11513.09 35.7660 3.3567 40 0.01460 1.05946 tt tt 11715.10 35.7713 3.3523 3.3527 1.06 35.8026 35.5602 0.0068 41 0.01480 1.06494 tt tt 11712.43 36.0716 3.3548 42 0.01570 1.09390 tt tt 11579.82 37.4873 3.3509 43 0.01542 1.08535 tt tt 11645.21 37.0513 3.3505 L 124 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. j TABLE EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS, MADE AT THE 1 3 a 4 5 6 7 8 Mean Temperatures by velocity of Fahrenheit's Tune of the commencement and the' water approach- thermometer. conclusion of the experiment, ing the weir at the Dura- Mean depth transverse Num telegraphic signals. sect, thro' tion of upon the the holes ber o: Date of the Reference to the figures on plate XIV., and particular the weir by in the the experi experiment. description of the weir. experi- observation. hook gauge 1852. Of the ail Of the ment. boxes, or in the water. Commencement. Conclusion. a. six feet from the shade weir. V. H. min sec. H. pun sec. nee. Feet. 44 Nov. 7, A.M 44 44 Figures 8, 9, and 10. 7 50 1.0 7 55 51.2 350.2 "0.98675 0.5455 45 46 a u u ii 11 tt Mean width of the canal for 20 feet on the upstream side of the weir, 9.992 feet. Mean depth of the canal opposite the hook gauge boxes, 5.048 feet below the top 10 38 11 0.0 59.2 j 10 43 17 53.7 57.4 353.7 358.2 4 0.98490 "0.97450 0.5446 0.5376 47 11 U 11 of the weir. 10 43 59.6 10 49 59.0 359.4 "0.97620 0.5385 48 U It tl 11 15 0.0 11 21 0.4 360.4 "0.97600 0.5387 49 U 11 11 11 48 4.4 11 54 0.7 356.3 4 0.97775 0.5394 50 " " P.M. 21 0.2 26 58.3 358.1 "0.97690 0.5390 51 Nov. 7, P.M. 42.25 43.75 Figures 8, 9, and 10. 8 23 7.6 8 28 52.0 344.4 8 1.00505 0.5589 52 11 11 U Width and depth same as the preceding. The sheet 8 55 59.7 9 1 46.9 347.2 2 1.00600 0.5581 53 11 11 11 of water after passing the weir, was prevented from expanding in widthj for a certain distance, by boards at q B 28 3.1 9 33 51.3 348.2 "1.00520 0.5574 54 11 It 11 each end of the weir, placed in the same planes as the c 59 59.8 10 5 52.9 353.1 4 0.99265 0.5480 55 U tl tl sides of the canal leading to the weir. 10 31 1.5 10 36 54.0 352.5 "0.99240 0.5477 56 Oct. 24, P.M. Figures 1, 2, and 8. ~2 24 59.3 2 32 53.8 474.5 *0.81860 0.3405 57 U H u Width of the canal on '.he upstream side of the wen-, 3 3 0.4 3 11 11.1 490.7 ^.80755 0.3338 58 59 11 11 U tl U 11 13.96 feet. Mean depth of the canal opposite the hook gauge boxes, 5.048 ieet below the top of the wea. 3 4 40 17 0.3 4.7 3 4 48 25 19.9 44.9 499.6 520.2 "0.77690 0.3272 0.3170 60 It 11 11 4 55 1.7 5 3 18.6 496.9 "0.80125 0.3306 61 11 11 U 5 29 1.8 5 37 27.6 505.8 "0.79400 0.3258 62 Oct. 31, P.M. Figures 5, 6, and 7. ~2 20 0.3 2 28 40.5 520.2 "0.77115 0.6694 63 U tl tl Width of the canal on the upstream side of the weir, 3 1.3 3 8 32.4 511.1 '0.78725 0.6872 64 65 U tt tt It It It 47 48.75 18.96 feet. Mean depth of the canal opposite the hook gauge boxes, 2.014 feet below the top of the weir. Bot- tom of the canal horizontal, for 23 feet on the upstream 3 4 38 14 4.4 0.2 3 4 46 21 7.9 5.5 483.5 425.3 "0.80455 "0.87960 0.7052 0.7870 66 11 U tl side of the weir. 4 47 58.0 4 54 56.2 418.2 "0.88865 0.7963 67 Nov. 7, P.M. Figures 8, 9, and 10. 2 7 2.7 2 16 14.7 552.0 "0.73620 0.3659 68 11 tl It Mean width of the canal for 20 feet on the upstream 2 43 1.0 2 51 6.8 485.8 *0.80195 0.4122 69 It tt 11 side of the weir, 9.992 feet. Mean depth of ibe canal opposite the hook gauge boxes, 5.048 feet below the top 3 17 59.7 3 25 56.0 476.3 "0.80950 0.4176 70 It 11 tl of the weir. 3 51 59.9 3 59 51.7 471.8 *O.SU95 0.4213 71 tl 11 11 4 25 0.0 4 32 53.9 473.9 '0.81325 0.4192 72 Oct. 24, A.M. Figures 1, 2, and 3. 7 12 2.5 7 25 9.4 786.9 "0.59190 0.2182 73 11 It tt 46.5 47.75 Width of the canal on the upstream side of the weir, 7 49 59.8 8 2 52.7 772.9 "0.59240 0.2186 74 It tl tt 13.96 feet. Mean depth of the canal opposite the hook gauge boxes 5.048 feet below the top of the weir. 9 11 59.1 9 24 36.6 757.5 "0.61060 0.2279 75 It It tl 10 33 59.7 10 45 14.4 674.7 "0.65525 0.2509 76 It It It 59.5 48 11 8 1.4 11 19 27.9 686.5 6 0.64305 0.2449 77 11 It tl 11 50 0.3 12 1 35.3 695.0 4 0.63795 0.2419 78 " P.M. 64.5 48.5 24 58.5 36 42.9 704.4 "0.63370 0.2396 79 Oct. 31, P.M. Figures 5, 6, and 7. ~7 ~6 59.6 7 18 8.6 669.0 "0.65150 0.5405 80 u n it Width of the canal on the upstream side of the weir, 7 46 0.3 7 57 9.3 669.0 "0.65590 0.5455 81 it u tt 45.25 48.75 13.96 feet. Mean depth of the canal opposite the hook gauge boxes, 2.014 feet below the top of the weir. Bot- 8 24 0.0 8 34 56.9 656.9 "0.65985 0.5496 82 u it 11 tom of the canal horizontal for 23 feet on the upstream 9 59.4 9 12 42.4 703.0 4 0.63135 0.5193 83 it it n side of the weir. 9 40 1.0 9 51 27.8 686.8 "0.64250 0.5309 84 n u n 10 23 1.7 10 34 11.0 669.3 "0.65460 0.5439 85 Oct. 31, P.M. Figures 5, 6, and 4. 11 43 0.7 11 56 42.8 822.1 6 0.66940 0.4382 86 87 NOV. 1, A.M. tt It H Width, depth, and bottom of the canal same as the preceding. Two equal bays separated by a partition 2 feet wide. 1 21 59.9 3.8 1 35 13 26.0 17.9 806.1 794.1 "0.67900 "0.68360 0.4459 0.4496 8 It 11 11 1 39 59.8 1 53 9.5 789.7 "0.68815 0.4526 EXPEEIMENTS ON THE FLOW OF WATER OVER WEIRS. 125 X 1 1 1 CONTINUED. LOWER LOCKS, LOWELL, IN OCTOBER AND NOVEMBER, 1852. 1 9 .10 11 12 13 14 15 1G 17 18 19 20 Depth upon the Quantity of water Head due tc weir corrected Total quan- Appro* that would have Proportional Num- ber of the experi- ment. the yelocit 1 in column 8 or the values of h by the formula A- 7 " 1 for the velocity of the water ap- proaching the weir, or the val- ues of H f by the formula #'= Length of the weir. 1. No. of end con- trac- tions n. tity of water that passed the weir dur- ing each ex- periment, as measured in the lock Quantity of water pass- ing the well per second. Value of C in the formula Q= 5 C(l 0.1H')H/2 Q having the corresponding values in col- Mean value of C for each particular descrip- tion of weir. imatc* mean depth upon the weir for each particu- lar de- scription of weir passed the weir with the depth in column 17, calcula- ted by the formula. Q= 8 qj-o.inH"jw2 C having the cor- responding value in column 16. Quantity of water passing the weir, calculated by the formula Q= J 3.33(Mnn.H")/f'' 5 difference, or the absolute difference of the quantities in columns 1 and 19, divided by the quan- 64.323b . , i 48 chamber. umn 14. tity in column [(H+/ift A5J3 Sf'. 18. Feet. Feet. Feet. Cubic feet. Cubic feet. Feet. Cubic feet per Cubic feet per second. second. 44 0.00463 0.99117 9.995 11524.62 32.9087 3.3366 45 0.0046 0.98930 u tt 11616.54 32.8429 3.3394 46 0.00449 0.97879 tt tt 11592.18 32.3623 3.3437 47 0.00451 0.98051 it tt 11655.28 32.4299 3.3418 3.3409 0.98 32.3956 32.2899 0.0033 48 0.0045 0.98031 tt tt 11689.79 32.4356 3.3434 49 0.00452 0.98207 it it 11576.77 32.4916 3.3402 50 0.00452 0.98122 it tt 11623.93 32.4600 3.3413 51 0.00480 1.00968 9.995 11646.88 33.8179 3.3349 52 0.00484 1.01061 tt tt 11725.23 33.7708 3.3257 53 0.00483 1.00980 tt tt 11743.85 33.7273 3.3254 3.3270 1.00 33.2534 33.2833 + 0.0009 54 0.00467 0.99710 U tt 11683.48 33.0883 3.3249 55 0'.00466 0.99684 it tt 11656.76 33.0688 3.3243 56 0.0018C 0.82034 9.997, 2 11539.45 24.3192 3.3287 57 0.00173 0.80923 U tt 11675.86 23.7943 3.3234 58 0.00166 0.79726 tt tt 11628.76 23.2761 3.3237 3.3246 0.80 23.4011 23.4391 + 0.0016 59 0.00156 0.77842 It tt 11694.31 22.4804 3.3261 60 0.00170 0.80290 U tt 11698.38 23.5427 3.3268 61 0.00165 0.79560 M tt 11719.40 23.1700 3.3188 62 0.00697 0.77768 9.997 ~2 11718.53 22.5270 3.3376 63 0.00734 0.79412 it tt 11887.02 23.2577 3.3406 64 0.00773 0.81178 U tt 11610.17 24.0128 3.3383 3.3403 Q.83 24.8313 24.7548 0.0031 65 0.00963 0.88855 it tt 11695.06 27.4984 3.3435 66 0.00986 0.89782 It tt 11671.63 27.9092 3.3417 67 0.00208 0.73821 9.995 11676.57 21.1532 3.3368 68 0.00264 0.80449 tt a 11709.76 24.1041 3.3422 69 0.00271 0.81211 tt tt 11645.16 24.4492 3.3424 3.3393 0.80 23.8821 23.8156 0.0028 70 0.00276 0.81760 it tt 11647.58 24.6876 3.3410 71 0.00273 0.81588 It tt 11638.23 24.5584 3.3341 72 0.00074 0.59262 9.997 2 11803.57 15.0001 3.3284 73 0.00074 0.59312 tt tt 11614.60 15.0273 3.3303 74 0.00081 0.61139 tt tt 11902.13 15.7124 3.3284 75 0.00098 0.65621 tt tt 11760.381 17.4305 3.3237 3.3275 0.62 16.0382 16.0502 + 0.0008 76 0.00093 0.64395 tt tt 11659.42 16.9839 3.3306 77 0.00091 0.63883 tt It 11648.02 16.7597 3.3259 78 0.00089 0.63456 tt tt 1 1 685.54 16.5894 3.3250 79 0.00454 0.65579 9.997 2 11657.19 17.4248 3.3258 80 0.00463 0.66027 u It 11783.30 17.6133 3.3278 81 0.00470 0.66429 tt tt 11674.77 17.7725 3.3278 3.3262 0.65 17.1990 17.2187 + 0.0011 82 0.00419 0.63532 tt tt 1 1 682.49 16.6181 3.3249 83 0.00438 0.64664 tt tt 11715.28 17.0578 3.3244 84 0.00460 0.65894 tt tt 11748.86 17.5540 3.3266 85 0.00299 0.67226 7.997 4 11690.02 14.2197 3.3382 86 0,00309 0.68195 It tt 11703.92 14.5192 3.3378 87 0.00314 0.68660 tt tt 11644.82 14.6642 3.3378 3.3368 0.68 14.4541 14.4247 0.0020 88 0.00318 0.69118 it It 11678.11 14.7880 3.3333 1 i 126 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. COMPARISON OF THE PROPOSED FORMULA WITH THE RESULTS OBTAINED BY PREVIOUS EXPERIMENTERS. 160. We find on record a great number of experiments on the discharge of water over weirs ; in the present state of the science of hydraulics, however, a large proportion of them can be considered only in the light of first approxi- mations; of great value undoubtedly, at the respective epochs at which they were made; but it could serve no useful purpose to compare them with the results obtained with the more perfect apparatus used of late years. Three sets of experiments have been made in France within the last thirty years, on a comparatively minute scale, it must be admitted, but with complete apparatus, and conducted with great care. They were made by Poncelet and Lesbros at Metz, in 1827 and 1828; by Castel at Toulouse, in 1835; and by Boileau at Metz, in 1846. It will be recollected that the application of the proposed formula to the discharge over weirs in which the contraction at the ends is complete, is limited to depths on the weir, not exceeding one third of the length of the sheet; this limitation permits the comparison to be made with only a portion of the results obtained by Poncelet and Lesbros, and by Castel. Boileau operated on weirs in which the end contraction was suppressed, and to which form the limitation does not apply. 161. Comparison of the proposed formula, with the results obtained by Poncelet and Lesbros. These experiments are to be found among the magnificent series made at the expense of the French Government, and recorded at length in Experiences hydrauliques sur les lois de I'ecoulement de I'eau by M. M. Poncelet and Lesbros, Paris : 1832; and in the continuation under the same title by M. Lesbros, Paris: 1851. In table XXXIX., of the last mentioned work, are given the coefficients for computing the discharge over weirs of a variety of forms, and of certain lengths, and with certain depths of water, by the formula in which d is the discharge, m the coefficient, I the length, h the depth, and g= 9.8088 metres, or 32.1817 feet. The comparison can be usefully made with only one of the forms experimented upon, namely : that in which the orifice was made in a thin plate, in the plane side of a reservoir; the orifice being at a great distance from the bottom and lateral sides, and the- discharge made freely into the air. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 127 In table XIV. are given the quantities computed according to Lesbros, for all the depths for which he gives values of m, determined by experiment, and which are within the limitation required by the proposed formula, namely : that the depth shall not exceed one third of the length. The quantities are also given as computed by the proposed formula. It will be perceived by the final column of the table, that the proportional differences are nearly constant, and that the quantities by the proposed formula are too small by a little more than two per cent. If the coefficient of the proposed formula was changed from 3.33 to 3.41, the computed results would agree very nearly. It should be recol- lected that the constants in the proposed formula have been determined from experiments in which the depths upon the weir were from 0.6 to 1.6 feet, or about eight times the depths in the experiments by Poncelet and Lesbros. It is the general result of all the precise experiments on the discharge through openings of a variety of forms, in a thin plate, that, for very small heads, the coefficients require to be increased; which proves that the law of the discharge varying as the square root of the head, does not hold good for very small heads. The comparison in table XIV. affords the same indications; and the constancy of the proportional differences, indicates that the correction of the length, to compensate for the effect of the end contraction, is practically correct, both for large and small depths upon the weir. It would not be difficult so to determine the values of the constants in the formula Q= C(llnh)h tt , as to represent the experiments both of Poncelet and Lesbros and the Lower Locks experiments with nearly the same degree of exactness that the latter are represented, with the constants that have been adopted. This would undoubt- edly be an advantage in some particular cases in practice, but if it was intended to make the formula general, the sacrifice of simplicity would be more than an equivalent disadvantage. 128 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE XIV. The length of the weir is constant, and equal to 0.6562 feet. 1 a 3 4 6 Quantity of water dis- Value of the charged by the formula Quantity of water dis- Proportional difference, Depth on coefficient m Q = m/A v' 2 ? *> charged by the formula or the absolute dif- the weir. according to m haying the correspond- a. ference divided by the Lesbros. ing value in the pre- Q = 3.33( L0.\nH)H . quantity in column 8. ceding column. Feet. Cubic feet per second. Cubic feet per second. 0.06562 0.417 0.0369 0.0360 0.0245 0.08202 0.414 0.0512 0.0500 0.0225 0.09843 0.412 0.0670 0.0655 0.0228 0.11483 0.409 0.0838 0.0820 0.0207 0.13124 0.407 0.1019 0.0997 0.0209 0.14764 0.405 0.1210 0.1184 0.0212 0.16404 0.404 0.1413 0.1379 0.0239 0.18045 0.402 0.1622 0.1583 0.0243 0.19685 0.401 0.1844 0.1794 0.0271 0.21326 0.399 0.2069 0.2012 0.0274 162. Comparison of the proposed formula with the results obtained ly Castel. An abstract of these experiments may be found in the Annaks de Ckimie et de Physique, vol. 62. Paris : 1836 ; and in the Annaks des ponts et chaussees, vol. 1, for 1837. Paris. It appears to have been a leading idea in these experiments, to imitate, as nearly as possible, the forms and proportions of the weirs ordinarily used in practice for gauging streams of water; in fact, to reproduce them on a small scale, anticipating that the rules deduced from precise experiments upon them might be applied, without modification, to gaugings on a large scale. The weir was formed by damming up a wooden canal, 2.4279 feet in width, by a thin plate of copper, in which the weir was formed, the crest being 0.5578 feet above the bottom of the canal; the width of the weir varying from about i of a foot to 2i feet. The latter width is so near that of the canal, that the end contraction must have been sensibly modified, so that any comparison of the results obtained from it would be of little use; they have consequently been omitted. In the abstract referred to, a table is given of the coefficients deduced from the experiments, for a variety of widths and depths. In table XV. are given the quantities computed with these coefficients, for all the widths and depths to which the proposed formula is applicable ; also the quantities as com- puted by the proposed formula. In consequence of the small dimensions of the canal, the water approaching the weir had a sensible velocity; in table XV. EXPERIMENTS ON THE FLOW OF WATEK OVER WEIRS. 129 the depths on the weir, for which the quantities have been computed by the proposed formula, have been corrected for this velocity. It will be seen by referring to the final column, that the proportional differences are considerably greater, and have less uniformity than in the comparison with the experiments of Poncelet and Lesbros; nevertheless, there is a certain harmony in the results of both comparisons, and they serve to show how unsafe it is, in the present state of the science of hydraulics, to apply rules to gauging streams of water passing over weirs, of which the dimensions differ greatly from those in the experiments from which the rules have been deduced. 17 130 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE XV. Width of the canal leading to the weir 2.4279 feet ; height of the crest of the weir above the bottom of the canal 0.5578 feet. 1 2 3 4 5 6 7 8 _ Quantity of water Head due the Depth on the weir, Value of the coeffi- cient m, in the discharged by the formula mean velocity of the water in the corrected for the Telocity of the water Quantity of water discharged by the Proportional differ- ence, or the abso- Length of the Depth on the formula r canal leading to in the canal by the formula lute difference of weir. weir. ITfpJn^Tf Q=m$LHy2gJ1, the weir by the formula Q= the quantities in " * V*s i m having the corre- formula 1V= 3 columns 4 and 7, according to Castel sponding value in the h - 11 W__LfcW' 7 V I"^ B,23(L-Q.lnH')&? divided by the quantity in col- 64.373 1 \fl~^-fl) JJ n j A umn 4. Feet. Feet. Cubic feet per second. Feet. Feet. Cubic feet per second. 0.3281 0.09843 0.618 0.0335 0.00001 0.09844 0.0317 , 0.0537 0.6562 0.19685 0.604 0.1852 0.00016 0.19701 0.1796 0.0302 0.16404 0.611 0.1425 0.00010 0.16414 0.1380 0.0311 (t 0.13124 0.619 0.1033 0.00006 0.13130 0.0998 0.0339 u 0.09843 0.624 0.0676 0.00003 0.09846 0.0655 0.0318 0.9843 0.32809 0.604 0.5976 0.00120 0.32924 0.5778 0.0331 u 0.26247 0.606 0.4290 0.00072 0.26316 0.4189 0.0237 u 0.19685 0.610 0.2805 0.00036 0.19720 0.2755 0.0176 u 0.16404 0.616 - 0.2155 0.00023 0.16426 0.2109 0.0211 u 0.13124 '0.623 0.1559 0.00014 0.13138 0.1519 0.0257 u 0.09843 0.631 0.1026 0.00006 0.09849 0.0993 0.0322 1.3124 0.39371 0.621 1.0769 0.00337 0.39687 1.0266 0.0468 (I 0.32809 0.621 0.8192 0.00225 0.33022 0.7876 0.0386 0.26247 0.620 0.5852 0.00134 0.26375 0.5682 0.0291 ft 0.19685 0.622 0.3813 0.00067 0.19749 0.3720 0.0245 a 0.16404 0.626 0.2920 0.00043 0.16446 0.2842 0.0266 u 0.13124 0.632 0.2109 0.00025 0.13148 0.2042 0.0320 a 0.09843 0.636 0.1379 0.00012 0.09855 0.1332 0.0341 1.6404 0.32809 0.631 1.0405 0.00363 0.33147 1.0003 0.0386 u 0.26247 0.632 0.7457 0.00218 0.26452 0.7192 0.0355 " 0.19685 0.632 0.4843 0.00108 0.19788 0.4692 0.0312 (t 0.16404 0.633 0.3690 0.00069 0.16470 0.3578 0.0304 u 0.13124 0,636 0.2653 0.00039 0.13161 0.2566 0.0327 u 0.09843 0.642 0.1740 0.00019 0.09861 0.1671 0.0393 1.9685 0.32809 0.644 1.2743 0.00545 0.33308 1.2174 0.0446 u 0.26247 0.644 0.9118 0.00326 0.26549 0.8725 0.0431 " 0.19685 0.645 0.5931 0.00163 0.19838 0.5675 0.0432 0.16404 0.644 0.4505 0.00103 0.16502 0.4320 0.0410 ii 0.13124 0.645 0.3229 0.00058 0.13179 0.3094 0.0417 H 0.09843 0.651 0.2117 0.00027 0.09869 0.2012 0.0495 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 131 163. Comparison of the proposed formula, imth that obtained by Boileau. The experiments from which Boileau deduced his formula, are given at length in Jaugcage des cours d'eau a foible ou a moyenne section, by M. P. Boileau. Paris : 1850. Boileau has particularly studied the discharge in the form of weir in which the contraction at the ends is suppressed ; that is to say, the form in which the weir occupies the whole width of the canal conducting the water to it. The proposed formula is applicable to this case, by making n = 0. Boileau experi- mented on three weirs of this form ; one of them was 5.30 feet in length, with the crest 1.54 feet above the bottom of the canal ; the other two were 2.94 feet in length, the crest in one being 1.12 feet above the bottom of the canal ; and in the other 1.61 feet above the bottom ; the depths on the weir varying from 0.19 feet to 0.72 feet. By a train of reasoning combined with the results of his experiments, Boileau has arrived at the following formula for weirs of this form : in which Q = the discharge. $i=the height of the crest of the weir, above the bottom of the canal, which is supposed to be horizontal for a short distance, upstream from the weir. JET= the depth on the weir, taken before the sheet begins to curve in con- sequence of the discharge. Z:=the width of the canal, and also the length of the weir. g = 9.8088" 1 . The coefficient 0.417 is determined from a mean of 14 experiments. Adopting the English foot as the unit, and reducing, we have For this form of weir, the proposed formula becomes : (B) H' being the depth upon the weir, corrected for the velocity of the water approaching the weir. These formulas differ so essentially that they can be conveniently compared 132 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. only by applying them to particular cases. In the formula (.4), as & increases O I TT relatively to H, the factor ,--j--==== approaches unity, which is the limit when 8 is infinitely greater than H; in the ' latter case we have also, H' =: H; the formulas (A) and (B) then become identical, excepting the coefficients, that in (B) being v fa less than in (A). Hence we may conclude that for any length of weir, and for any depth upon it, providing that the depth of the canal leading to the weir, is very great relatively to the depth on the weir, the quantities computed by the formulas (A) and (B) will differ ^|^ only. In practice, however, S is seldom very great, relative to H. Let us take an example conforming more nearly to the usual cases that occur in practice. Let ff=l foot, #=3 feet, L = 10 feet, by the formula (A), Q= 34.552 cubic feet per second. In the formula (B\ H' is the depth on the weir, corrected for the mean velocity of the water approaching the weir ; this velocity is equal to the quotient of the area of the section of the canal, divided by the quantity. But the quantity itself depends on this velocity. The formula (.Z?), if put under a form to give the quantity directly from the measured depth upon the weir, would become very complicated ; it will be equally exact and much easier, to find the quantity by successive approximations as follows. 1st approximation. Assume H' = 1, then Q = 33.3. 2nd approximation. If Q = 33.3, the mean velocity of the water in the canal leading to the oo o weir is . '. . = 0.8325 ; and for the head due this velocity we have / =1.0103; Q = 33.810. A third approximation in a similar manner gives Q = 33.817. The proportional difference of the quantities by the two formulas is about r , or a little over two per cent. Boileau, in establishing his formula, assumes that the living force in the entire section of the canal is expended in increasing the discharge over the EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 133 weir; in the method adopted in this work for correcting the depth on the weir for the velocity of the water in the canal, it is assumed that the living force in the part of the section of the canal equal to the area of the orifice of dis- charge only, is expended in increasing the discharge ; as applied to a weir of the form under consideration, it is clear that neither of these assumptions is strictly true ; the latter, however, appears to be the most rational, and to agree the best with experiment. PRECAUTIONS TO BE OBSERVED IN THE APPLICATION OF THE PROPOSED FORMULA. 164. Q=:3.33(L in which Q = the discharge, in cubic feet per second ; L = the length of the weir ; n = the number of end contractions ; H the depth on the weir ; the English foot being the unit of measure. When the contraction is complete at each end of the weir, n = 2 ; when the weir is of the same width as the canal conducting water to it, the end con- traction is suppressed, and n = 0. This formula is only applicable to rectangular weirs, made in the side of a dam, which is vertical on the upstream side, the crest of the weir being hori- zontal, and the ends vertical ; also, the edges of the orifice presented to the cur- rent must be sharp; for, if bevelled or rounded off in any perceptible degree, a material effect will be produced on the discharge ; it is essential, moreover, that the stream should touch the orifice only at these edges, after passing which it should be discharged through the air, in the same manner as if the orifice was cut in a thin plate. See fig. 3, plate XVIII. The formula is not applicable to cases in which the depth on the weir exceeds one third of the length ; nor to very small depths. In the experiments from which it has been determined, the depths have varied from 7 inches to nearly 19 inches, and there seems no reason why it should not be applied with safety to any depths between 6 inches and 24 inches. The height of the surface of the water in the canal, above the crest of 134 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. the weir, is to be taken for the depth upon the weir; this height should be taken at a point far enough from the weir to be unaffected by the curvature caused by the discharge ; if more convenient, it may be taken by means of a pipe opening near the bottom of the canal near the upstream side of the weir, which pipe may be made to communicate with a box placed in any convenient situation ; and if the box and pipe do not leak, the height may be observed, in this manner, very correctly (art. 175). However the depth may be observed, it may require to be corrected for the velocity of the water approaching the weir. The end contraction must either be complete, or entirely suppressed; the necessary distance from the side of the canal or reservoir to the end of the weir, in order that the end contraction may be complete, is not definitely deter- mined; in experiments 1 to 4, table XIII., the depth on the weir was about 1.5 feet, and the distance from the side of the canal to the end of the weir, about 2 feet; the proposed formula applies well to all these experiments. In cases where there is end contraction, we may assume a distance from the side of the canal to the end of the weir equal to the depth on the weir, as the least admissible, in order that the proposed formula may apply. As to the fall below the weir, requisite to give a free discharge to the water, it is not definitely determined ; a comparison of experiments 49, 50, and 51, table X., indicates that, when the depth on the weir is 1 foot, and the entire sheet, after passing the weir, strikes a solid body at about 0.5 feet below the crest of the weir, the discharge, with the same depth, is diminished about j^nr- By experiments 1 and 2, table XII., it appears that, when the sheet passing the weir, falls into water of considerable depth, the depth on the weir being about 0.85 feet, no difference is perceptible in the discharge, whether the water is 1.05 feet or 0.235 feet below the crest of the weir ; it is very essential, however, in all cases, that the air under the sheet should have free communication with the external atmosphere. With this precaution it appears that, if the fall below the crest of the weir is not less than half the depth upon the weir, the discharge over the weir will not be perceptibly obstructed. If the sheet is of very great length, however, more fall will be necessary, unless some special arrangement is made to supply air to the space under the sheet at the places that would otherwise not have a free communication with the atmosphere. In respect to the depth of the canal leading to the weir, experiments 36 to 43, table XIII., show that, with a depth as small as three times that on the weir, the proposed formula agrees with experiment, within less than one per cent. ; this proportion may be taken as the least admissible, when an accurate gauging is required. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 135 It not unfrequently happens that, in consequence of the particular form of the canal leading to the weir, or from other causes, the velocity of the water in the canal is not uniform in all parts of the section; this is a frequent cause of serious error, and is often entirely overlooked. If great irregularities exist, they should be removed by causing the water to pass through one or more gratings, presenting numerous small apertures equally distributed, or otherwise, as the case may require, through which the water may pass under a small head ; these gratings should be placed as far from the weir as practicable. If the canal leading to the weir has a suitable depth, it will be requisite only when great precision is required, to correct the depth upon the weir for the velocity of the water in the canal by the formula (Z>) (art. 153) ; thus, in experiment 42, table XIII., the water in the canal had a mean velocity of about 1 foot per second, the effect of which was to increase the discharge about two per cent. ; in experiment 82, in which the velocity was about 0.5 feet per second, the discharge was increased about one per cent. ; these examples will enable the operator to judge, in each case, of the necessity of going through the troublesome calculation for correcting the depth on the weir. MISCELLANEOUS EXPERIMENTS ON THE FLOW OF WATER, MADE AT THE LOWER LOCKS, IN NOVEMBER, 1852. On the discharge of water over a dam of the same section as that erected by the Essex Company, across the Merrimack River at Lawrence, Massachusetts. 165. As these experiments cannot be usefully compared with those on weirs of more regular form, they have not been included in table XIII. ; and as they are of less general interest, they will not be given with much detail. The form of the dam is represented by figures 11 and 12, plate XIV. (art. 147) ; the other apparatus was the same as that used for the experiments in table XIII. The end contraction was suppressed by making the canal leading to the overfall of the same width as the overfall itself. The water in the hook gauge boxes communicated only with the water contained in the spaces between the masonry and the wood-work forming the sides and bottom of the canal leading to the overfall; as there was a free communication between the water at A, figures 11 and 12, and that near the hook gauge boxes, and as the water between these places was sensibly at rest, we may consider that the height of the water was taken at A. 166. In table XVI. these experiments are exhibited in sufficient detail to be intelligible. COLUMNS 1 and 2 require no explanation. COLUMN 3. The heights contained in this column are above the mean level of the crest of the dam, which was very nearly horizontal for a distance of 2.95 feet from C to D. These heights have not been corrected for the velocity of the water approaching the weir; indeed, from the manner in which they were observed, no correction was necessary. , COLUMN 4. The quantities in this column have been obtained in the manner described in the explanation of table XIII. (art. 155). COLUMN 5. Quantity of water passing over the dam, calculated by the formula .53 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 137 This formula was arrived at by trial of various powers of h, and was adopted as representing, the most nearly, the results of the five experiments in the table ; it should be distinctly understood, however, that it is not applicable to depths much greater or less than in the experiments from which it is deduced. In April, 1852, the depth of water flowing over the dam at Law- rence, was 10 feet ; if the quantity then passing over the dam was com- puted by this formula, it is probable that it would be greatly in error. x COLUMN 6. Proportional difference. It will be observed that the greatest pro- portional difference is 0.0085, or less than one per cent. ; we may therefore say with confidence, that we can compute the flow of water over the Lawrence dam, when free from ice or other obstruction, for any depth not greater than 20 inches or less than 7 inches, without being liable to an error exceeding one per cent. TABLE XVI. Time, from November 10th, 8A, 57', P.M., to November llth, OJ. 11', A. H. Temperature of the air at 10, 5W, P. M., 34.50 Fahrenheit. " " water " " " 41.75" " The air culm. 1 3 3 4 5 6 Number of the experi- ment Length of the ova-full. I. Mean height of the surface of the water in the hook gauge boxes, above the top of the horizontal crest of the dam. Feet. Quantity of water passing over the dam, as measured in the lock cham- ber. In cubic feet per second. Quantity of water passing over the dam calculated by the formula Q = 3.01208 lli M In cubic feet per Proportional difference, or the absolute differ- ence of the quantities in columns 4 and 5, divided by the quantity in column 4. h. second. 89 9.995 0.58720 13.385 13.332 0.0040 90 a 0.79035 20.892 21.005 4-0.0054 91 a 0.97670 28.914 29.039 4-0.0043 92 a 1.32520 46.183 46.317 4-0.0029 93 u 1.63380 64.346 63.804 0.0085 EXPERIMENTS TO ASCERTAIN THE EFFECT OF TAKING THE DEPTHS UPON A WEIR AT DIFFERENT DISTANCES FROM IT, BY MEANS OF PIPES OPENING NEAR THE BOTTOM OF THE CANAL. 167. It is often a matter of great doubt and uncertainty, to know at what distance from the weir the depth of the water upon it should be observed; very often also it becomes a matter of necessity to observe the depth at a dis- tance from the weir so small that, according to some, the quantity of water passing the weir, computed in the usual manner, would be liable to sensible 18 138 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. error. For the purpose of obtaining some light upon this point these experi- ments were undertaken, and, as they were made with all the precautions for insuring accuracy that could be devised, they will be described with some detail. 168. Figures 8, 9, and 10, plate XIV., represent the form of the weir, and the system of pipes used for these experiments. The canal leading to the weir was of the same width as the weir, so that the end contraction was suppressed. The pipes were of lead, about three fourths of an inch interior diameter, the lower extremities of which, numbered from 1 to 8, were about three inches above the bottom of the canal, and terminated in holes in the board CO; the side of the board at which they opened was vertical, and in the axis of the canal; the ends of the pipe did not project through the board; the other extremi- ties of the pipes were fastened by small flanges to the bottoms of the hook gauge boxes ; holes were made in the bottoms of the boxes corresponding to each pipe, and communication between the boxes and the pipes could be con- trolled at pleasure, by plugging up these holes. It will be readily perceived that heights of the water observed by this apparatus are not necessarily the true elevations of the surface of the water immediately over the orifices of the pipes, but that they are the elevations of the surface in the hook gauge boxes; an elevation which is due to the statical pressure on the orifice of the pipe. 169. In order to obtain the heights at different distances from the weir, observations were necessarily made with both hook gauges at the same time, one of which was always in communication with a pipe opening at 6 feet from the weir, the apertures in the bottom of the box, communicating with all the other pipes, being plugged up ; at the other hook gauge, either pipe might be in communication with the box, all the other apertures being plugged up ; thus, the depth at six feet from the weir was observed in each experiment, to be used as a standard with which the depth observed simultaneously at any other distance might be compared ; this mode of proceeding was rendered necessary, in consequence of the impossibility of maintaining the level of the water uniform for any considerable length of time. 170. In considering the sources of error to which the observations with the hook gauges were liable, it appeared that four kinds required to be specially guarded against, namely : First, imperfect comparison of the gauges, with the top of the weir. Second, defective stability, in consequence of which the relative ele- vations of the gauges and the weir might not be constant. Third, errors in the graduation of the gauges. Fourth, the difference in the habit of observers, in making the point of the hook coincide with the surface of the water ; or, what we may call, the personal error. In relation to the first, we must bear in mind EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 139 that the requirement here is not so much that the absolute height above the top of the weir should be exactly determined, as that the difference of the heights at two points, at different distances from the weir, should be determined correctly ; if then we know the relative heights of the two gauges, the object can be attained, even if we do not know precisely the height of either of them, relatively to the weir. The heights of the gauges relative to each other, could easily be ascertained at any time, by closing up all the apertures in each box, except those communicating with pipes, numbers 4 and 5, which, it will be seen by reference to figure 9, had a common orifice at their lower extremities ; con- sequently, the surface of the water in both boxes must have been at the same level. The correction to be applied to the reading of one of the hook gauges, was taken as previously determined for the experiments on the discharge over the weir, and the correction for the other gauge, was deduced from simultaneous observations on both gauges, when the boxes communicated with a common ori- fice, in- the manner just described. The second source of error was guarded against as much as practicable, by making the observations for the correction just described, at nearly the same time as the experiments to which it was to be applied. The danger of error from the tUrd source was much diminished by making the observations for the correction, with nearly the same depth upon the weir as in the experiments to which it was to be applied. The fourth source of error was eliminated by determining the correction separately for each pair of observers. In short, these four sources of error were reduced to a minimum by determining for each session of the experiments, and for each pair of observers, the relative corrections to be applied to the readings of the hook gauges, to give the depths upon the weir; the depths, when the observations for these corrections were made, being nearly the same as in the experiments to which they were to be applied. 171. In table XVII. are given the results of the observations made for the purpose of obtaining the relative corrections for the gauges, for each session of the experiments, and for each pair of observers. In computing the depth upon the weir by the north hook gauge, the correction 0.03072 is applied to the mean reading of the gauge, (art. 143) ; the mean reading of the south hook gauge is given ; as the water in both boxes is at the same height, the differ- ence between the depth upon the weir, as determined by the north hook gauge, and the mean reading of the south hook gauge, must give the correction for the last named gauge. 140 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE XVII. DATE, 1862. Time of beginning the observation. North hook gauge, in communi- cation with pipe No. 5 opening near the bottom of the canal at 6 feet from the weir. South hook gauge, in communication with pipe No. 4, opening near the bottom of the canal at 6 feet from the weir. Obserrer. Arithmetical mean depth on the weir. Feet. Observer. Arithmetical mean reading of the gauge. Feet. Correction to be applied to the mean reading to give the depth on the weir. Feet. Mean correction for each cession, and each pair of observers. Feet. November 3. tt tt 9 13' P.M. 10 " Francis K 1.01180 1.02617 Avery a 1.03760 1.05375 0.02580 0.02758 0.02669 November 3. 11* 16' P.M. Haeffely 1.00739 Newell 1.03377 0.02638 0.02638 November 3. tt tt 4. 9* 29' P.M. 10 45 1 47 A.M. Francis " u 1.01984 1.01073 1.04532 Newell a tt 1.04625 1.03716 1.07169 0.02641 0.02643 0.02637 0.02640 November 3. 4. 11" 0' P.M. 1 58 A.M. Francis ft 1.00807 1.04734 Haeffely tt 1.03431 1.07350 0.02624 0.02616 0.02620 November 7. u a (t a u u u u u u tt u M t( 7" 50' A.M. 9 38 " 2 7 P.M. 8 22 " 8 56 " 9 28 " 10 10 31 Francis u tt it U 11 it a 0.98775 0.98555 0.73665 1.00696 1.00677 1.00580 0.99338 0.99294 Avery a u tt tt u u tt 1.01362 1.01195 0.76357 1.03287 1.03311 1.03244 1.01973 1.01961 0.02587 0.02640 0.02692 0.02591 0.02634 0.02664 0.02635 0.02667 0.02639 November 7. u u u u 8 4' A.M. 9 49 2 26 P.M. Francis a a 0.98932 0.98019 0.78315 Newell tt 1.01478 1.00597 0.80906 0.02546 0.02578 0.02591 0.02572 November 7. 9* 20' A.M. Haeffely 0.99305 Newell 1.01997 0.02692 0.02692 172. It will be perceived, by an examination of table XVII., that there are greater irregularities in the comparisons by some observers, than in those by others; this is to be attributed, principally, to the different degrees of experi- ence and skill in the observers. EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. 141 173. In table XVIII. are given the details of the experiments, to ascertain the effect of observing the depths upon the weir, at different distances from the weir, by means of pipes opening near the bottom of the canal. In order to obtain the depth upon the weir by the north hook gauge, the correction 0.03072 has been applied to the mean readings of this gauge. The correction for the south hook gauge is taken from the final column of table XVII., for the corresponding session and pair of observers. From want of time, pipes num- ber 6 and 7 were not made use of. It will be perceived, by referring to the final column of table XVIII., that the differences in the heights, at the different distances tried, are very inconsid- erable, and such as could be detected only by the most delicate means of observation. 174. Two comparisons were made in a similar manner, of the heights, when one gauge box communicated with a pipe opening near the bottom of the canal, and the other with a pipe opening through the side, at about 4.2 feet above the bottom, the orifices of both being at 6 feet from the weir, as repre- sented at B, figures 8, 9, and 10, plate XIV. ; the following are the results. First comparison, made November 7th, beginning at 3 h , 52', P.M. Francis, at north hook gauge, with pipe No. 5, depth on weir 0.81616 feet. Avery, at south hook gauge, with pipe B 0.81641 " Difference -f 0.00025 feet, Second comparison, made November 7th, beginning at 4 h , 5', P.M. Francis, at north hook gauge, with pipe No. 5, depth on weir 0.81775 feet. Newell, at south hook gauge, with pipe B " " 0.81776 " Difference +0.00001 feet. These differences are so minute that we may conclude that the depth was the same whether the pipe opened near the bottom of the canal or at 4.2 feet above. 175. These experiments, taken in connection with those of Boileau,* who has arrived at similar results, leave no doubt as to the propriety, whenever convenience requires it, of observing the depths upon the weir by means of a pipe opening into the dead water, near the bottom of the canal on the upstream side of the weir. ' Jaugeage des cours d'eau, by M. P. Boikau. Paris : 1850. 142 EXPERIMENTS ON THE FLOW OF WATER OVER WEIRS. TABLE XVIII. DATE, 1852. Time of beginning the observation. North hook gauge. South hook gauge. Difference in the depths upon the weir, the pipe opening at 6 feet from the weir being the standard. Mean difference in the depths upon the weir, the pipe opening at 6 feet from the weir being the standard. Pipe No.5opensat 6fec " "6 "8 " " 7 " 10 " a .1 g 12 " t from the weir. t< tt a u Pipe No.l opens at 1 inch from the weir. " " 2 " 2 feet " " u u 3 "4 " " " Num- ber of the pipe. Observer. Corrected depth upon the weir. Num- ber of the pipe. Observer. Corrected depth upon the weir. November 3 4 7 u u f< 11* 53' P.M. 27 A.M. 10 33 " 10 44 " 10 56 " I 5 5 5 Francis u Haeffely Francis tf 1.01267 1.01439 0.97530 0.97644 0.97658 1 1 1 1 1 Newell Haeffely Newell Avery Newell 1.01321 1.01459 0.97547 0.97683 0.97695 + 0.00054 + 0.00020 -- 0.00017 -- 0.00039 - - 0.00037 + 0.00033 November 4 U U U 0* 37' A. M. 1 23 " 1 34 " 5 5 5 Haeffely Francis Haeffely 1.02189 1.04220 1.04472 2 2 2 Newell Haeffely Newell 1.02286 1.04263 1.04481 + 0.00097 + 0.00043 + 0.00009 + 0.00050 November 7 11* 48' A.M. 21 P.M. 5 5 Francis u 0.97829 0.97734 3 3 Avery u 0.97883 0.97800 + 0.00054 + 0.00066 + 0.00060 November 4 u u 7 o o ,H" "S H c W r-1 P5 S <5 S H f-< W K H PH o 1/5 I CO 03 00 O to . g 'E3 HP a "a 33 O O O "O OOOOOOOOO OOOOOOOOOO 1 1 I+++I++ OOOOOOC3tOOOflOCCO500 fill I oocooooooooooocooooo OOOOOOOOOO Ii - 1 * 1 i * 1 "Oc6o6i>t^(> rH CO rH -l O)r-ICOCOr-ieO(Nt-l a *a 333 -0 13 t3 3 3 ) O OO-HOi ICO-H oooooooooo ooo'oooooo'o'oo o >t5CT>COt>-Ot>- oooocooooot^cooooot-0000 ooo'oo'ooo'odoo OOCCOOOOOOOOOOOOt>l>l>00 OOiOOOOOOiOOOO ^* co ^^ T^ co co ^H ^* co co I S = I criptlon f the riments. tn CM (N > 60 'g-S p ^ < rH r* rH S - .1 * ho 2 * PN c IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 153 co & 1 oqoooooooo dodddddddd o o d d o'o o 00 cocosrs .[~. (o fMOCOr-(OOO ooooooo CO CO CO rH O ^3 co ++7 i i +++++++7+ i i 71++ of g a Cubic feet per Second. r- i> i Ol OS O b- O CO ^Ht^.i OCN 3 ^^T^TjI-^l^f^JI^TiOiO^^^ O feet on cc i>- oo co >ra (N co 10 IN co oo >dsMtDod^c6dtocoodcdodo5^'rato CO O o o> w S a 03 oocqoqcocooqoqcocooo oodddododd oq oq o o co oq d o cqt^cooqco;GCGqoccqi--.oqo6cccqt~.oq ddddddddo'o'dddddddddo'd 00 111] I <-# >> > Cu per O-Hh-COOr-'^OO -OO''^OO-^fM '7^ C^C^t"i O ^ w P * 02 H S PQ W O 03 S CO * cocococococococococococo OJ 5M CO CO 1 ^f CO O "^ CO . -H -. O S^l -H 03 oooooooooooooo 1ti^rHi ti^CMi < li q c; -H oq ' d d co i--; d ' O CD CO lO CO OOl-Ji-Hi-;rHrHi-Hi-H5OCOCOaqi>; coc6o6odo6cdodo6o6t^i>i>t^c^ i>. t>- co t- oo oocooooooooooooocooococooocooccoi^-t>t>-i>.ir- s 93 3 Q co OOOOO>OOOOOOOOOOOKi't5iOOO i-H^I CO lf t) i ' ^ CO COCO r- (CO CO i I "^ , I^Hi li lOO'OOOOOOOOOO>raiOO>OO Oi lOi-li I ON CO CO O5 O CO (M O CO > t 's- cc cc O5 (M 02 >-> 3 fib 3 bO 3 CO SO 6D 3D to bJD CO O 2 O CM * 5 DA ** I to . g-5-a w * J OJ 2 g w V ' If the body is a cylinder, put D for its diameter; then for one foot in length of the cylinder we have A = JD, we have also w = n IP N. Substituting these values in (1.) and reducing, we have '~ ngDN In the case of a floating body we have N = n, and consequently Then (see Button's Mathematics, "On the Motion of Fluids"), giving dv the nega- tive sign, because v diminishes as s increases, and hence = ^ d s, (2.) v n D which, by integration, gives ? ' (0 Equation (2.) may be put under the form dt *\ Multiplying both sides by -7- and reducing, we have CL S d*s -, . 2k j. -^(11= =^ d t as* n D Integratuig, remembering that -y- or - is equal to -p> when t = o, we have Returning to the real case and denoting by s' the distance traversed by the tube in the time t, V being the velocity of the current, and v t that of the tube at the expiration of the time t, we shall have s' = Vt s. Substituting the values of s and t by (3.) and (4.) and also V ^ for v, we have i n 8 = IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 167 195. By equation (5.) we see that, theoretically, the tube never quite attains the velocity of the current ; and that the distance it must float in order to at- tain the velocity of the current, within a given fractional part, is proportional to the diameter of the tube and is independent of the velocity of the current. In the following experiments, s' was about 20 feet, and D =. 2 inches =: foot. Substituting these values in (5.) we find V v, _ _ 1 V ~ 64 nearly, That is to say : a tube 2 inches in diameter, after floating 20 feet from the point where it is put into the current, acquires a velocity equal to about ff that of the current. 196. Observation teaches us that floating bodies move faster than the stream in which they are floating; this is undoubtedly the reason why vessels moving with the current in a calm can be steered ; they not only partake of the motion of the water, but they have an independent motion due to the inclination of the surface of the water; the constant intermingling of the upper and lower parts of a stream prevents the water at and near the surface from attaining a velocity as great as it otherwise would. Navier* has investigated this subject; assuming that the velocity of the water is uniform to the depth to which the body is immersed, he finds, adopting our own notation, in which V, = the excess of the velocity of the floating body over that of the water. g = the velocity imparted by gravity in one second. Q = the volume of water displaced by the floating body. / = the slope of the surface. k = a coefficient depending on the form of the body. A = the area of the greatest transverse section of the body. In these experiments the floating bodies are cylinders with the axes vertical, for which case k is nearly 0.77 (art. 194). Put L for the length of the im- mersed part of such a cylinder and D for the diameter ; then Q = | 7i I? L, and A = D L. Substituting these values, and also the values of g and k, in the above equation, and reducing, we have V e = 8.1 ^D~I. (7.) * Architecture Hydraulique, par BELIDOR. Paris, 1819, page 358. 1G8 A METHOD OF GAUGING THE FLOW OF WATER The value of / can be determined from Eytelwein's formula for the motion of water in open channels, which when the English foot is the unit is * R 1= 0.000 024 265 1 v -f 0.000 111 415 5 v 2 . (8.) In which R is the mean radius, / the descent in the unit of length, and v the mean velocity. Formula (7.) indicates that the excess of velocity is proportional to the square root of the diameter of the tube, and also to the square root of the slope. Except in very small velocities, the velocity of the current is nearly proportional to the square root of the slope ; consequently, the excess of the velocity of the floating body over that of the fluid in which it is floating is nearly proportional to- the velocity of the current, except when the latter velocity is very small. In experiment 1 we have R = 5.5656 and v = 2.6719 (art. 192) ; substituting these values in (8.) we find 1= 0.000 154 56, we have also Z>=; substituting these values in (7.) we find V t = 0.0411 feet per second, which is about g^ of the mean velocity of the water. Neglecting the small effect this excess of velocity would have on the velocity deduced from the equality of the pressures on the up- stream and down-stream sides of the tube, we find for the computed velocity of the tube in experiment 1, 2.6750 0.0426 + 0.0411 = 2.6735 feet per second; which differs 0.0015 feet per second, or TT Vg> from the mean velocity of the water for a depth equal to the length of the immersed part of the tube, determined by the formulas of Humphreys and Abbot. The mean velocity of the tube by experiment was 2.6830 feet per second, which exceeds the computed velocity by 0.0095 feet per second, or ^| T . Similar computations have been made for exper- iments 7, 43, and 47, table XXII., which are selected as giving a wide range of conditions. The data and results are given in the following table. TABLE XIX. 1 a 3 4 5 6 7 8 9 1O 11 Depth of the Parameter of No. of the Depth of water in Mean Radius. Length of the immersed Mean velocity of the water deduced from Assumed value of axis of the parabola, representing the scale of the parabola, representing Maximum velocity of Velocity of the water at Velocity of the water at !>.]'. the flume. part of the weir / velocities, below the velocities. the water. the surface. the bottom. the tube. measurement. surface of the water. D R d. V d. B V a A V V D Feet. feet. Feet per Second. Feet. Feet per Second. Feet per Second. Feet per Second. 1 9.533 5.5656 9.482 2.6719 0.5 1.5973 0.01283 2.8979 2.8652 2.0899 7 9.530 5.5645 8.530 2.6539 0.1 1.7306 0.01280 2.8686 2.8302 2.0902 43 8.172 5.07-23 7.120 0.4961 0.0 1.6079 0.00777 0.5871 0.5670 0.2521 47 8.165 5.0696 8.122 0.4842 0.3 1.5158 0.00770 0.5777 0.5600 0.2374 A Treatise on Water- Works, by CHARLES S. STORROW. Boston, 1835. IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 109 TABLE XIX. CONTINUED. 13 13 14 15 16 17 18 19 No. of the Exp. Mean velocity of the water for a depth equal to the length of the immersed part Velocity of the tube dedured from the formula founded on the equality of the pressures on the up-stream and down-stream sides Difference between the velocities in column 12 and column 13. Slope of the surface of the water in the flume, deduced from Eytelwein's formula for the motion of water in open channels. Excess of the velocity of the tube over that of the water in which it is fl oatinp, Computed Telocity of the tube. Mean velocity of the tube by experiment. Difference between the velocity of the tube by computation and by experiment. o cue. of the tube. deduced from Absolute Proportional Navier's difference. difference. formula. ' mt v t Feet per / V. Feet per Sec. Feet per Second. Second. Feet per Sec. Feet per Sec. Feet per Sec. 1 2.6750 2.6324 0.0426 0.000 154 56 0.0411 2.6735 2.6830 + 0.0095 + 0.0036 7 2.7088 2.6752 0.0336 0.000 152 60 0.0409 2.7161 2.7260 -j- 0.0099 + 0.0036 43 0.5247 0-5108 0.0139 0.000 007 78 0.0092 0.5200 0.5190 0.0010 0.0019 47 0.5111 0.4669 0.0442 0.000 007 47 0.0090 0.4759 0.4950 + 0.0191 + 0.0401 197. It will be seen, by column 19 in the preceding table, that the differ- ences between the computed and observed velocities are not very regular ; perhaps as much so, however, as could be anticipated, considering the wide difference in the conditions in the experiments of Humphreys and Abbot and in the exper- iments at the Tremont measuring flume, and that their data for determining formulas (1.) and (2.) are not of a character to afford much confidence in their application to cases where the conditions are so different. 198. From the preceding investigation we infer, that in rectangular channels, in which the natural .scale of velocities at different depths is established, and the surface velocity not very much retarded by the wind, the tube is retarded on account of the pressures on the tube being as the squares of the relative velocities of the water and tube at different parts of its length, and is accelerated by the independent motion of the tube due to the slope of the surface of the water, and that the retardations and accelerations compensate each other to a greater or less degree under different circumstances. Taking a mean of the four experiments in table XIX., the computed velocity of the tube is about Jg- less than the observed velocity; and assuming this rela- tion to be of general application, we might, evidently, by a process the reverse of that by which table XIX. is computed, from the observed velocity of the tube, arrive at the mean velocity of the water in the flume. It would, however, involve lengthy computations, and the result would not be free from uncertainty, on account of the doubtful applicability of the formulas of Humphreys and Abbot; and how- ever interesting such an investigation might be as a scientific matter, it will be safer, in practice, to rely upon rules deduced from suitable experiments, even if such rules are empirical. 199. In arranging the programme of these experiments, it was designed to make them under the various circumstances which occur in the gaugings in the 22 170 A METHOD OF GAUGING THE FLOW OF WATER several measuring flumes at Lowell, and as nearly as practicable on the same scale ; the only material deviation from what was desired in the latter respect, was in the width of the channel; this was necessarily limited to the width of the canal in which the experimental flume was placed. A series of experiments with tubes of seven different lengths, and with velocities varying from 2.7 to 0.5 feet per second, was made with a flume of as great a width (26.745 feet) as could con- veniently be made in the canal, and another series, similar in respect to length of tubes, but with velocities varying from 5.0 to 1.4 feet per second, was made with a flume of half the width of the preceding. 200. The experiments consisted in making a gauge of the quantity of water passing the measuring flume, by observing the velocity of loaded tubes floating down different parts of the section of the flume, and from these observations de- ducing the mean velocity of the tubes for the whole section ; this mean velocity is provisionally assumed to be the mean velocity of the water in the flume, and when multiplied into the area of the section gives the quantity of water passing the flume according to the flume measurement. After leaving the flume, the same volume of water is made to pass over a weir, and the depth on the crest being observed, the quantity is computed by means of a formula determined from the experiments made at Lowell, in 1852, and previously described in this work. The quantity thus computed (with a minute correction for leakage in the experiments on the narrow flume) is taken as the true quantity passing through the measur- ing flume, and the comparison of this quantity with that obtained by the flume measurement determines the correction in that particular experiment. 201. Figures 1 and 2, plate XVI., are a general plan and longitudinal section of the entire apparatus used in the experiments with the wide flume. A. is the Northern Canal, through which the principal supply of water is primarily con- ducted from the Merrimack River to the manufacturing establishments. JB, the Tremont Gates, through which water is at times drawn, to make up any deficiency in the supply in the lower level of the Western Canal. C is a grating put across the canal for the purpose of equalizing the flow of the water in different parts of the section of the canal. D is a raft or float for the purpose of destroying the oscillations of the surface, caused by the admission of the water at the gates B, and which oscillations were partially propagated through the grating C. With- out the float the oscillations of the surface extended into the measuring flume, and imparted corresponding vertical oscillations to the tubes, causing those extend- ing nearly to the bottom to touch occasionally, which would of course tend to retard them. E is the measuring flume. F the Tremont Wasteway, over which the occasional supply from the Tremont Gates passes into the lower level of the IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. Western Canal, W; on this wasteway is erected the weir for gauging the water after it has passed through the measuring flume. Figures 1 and 2, plate XV., are a plan and transverse section of the wide flume. The original section of the canal is lined, from A to B, with planks about 2.25 inches in thickness, planed on the surface in contact with the current, and fastened to timbers which are securely bolted to the side walls and to stones sunk in the bottom of the canal for the purpose. The lining plank is connected with an old piling, C D, put in for another purpose, which extends through the side walls of the canal and into the earth on each side, effectually preventing any flow of water outside of the plank lining. E F represents an obstruction in the canal, used in a portion of the experiments for the purpose of creating irreg- ularities in the flow through the measuring flume. G is a float of timber and plank for the purpose of destroying the oscillations of the surface of the water caused by the obstruction E F. The obstruction and float were used only in exper- iments 123 to 140, which do not form any part of the series from which the formula of correction is deduced. 202. Figures 3 and 4, plate XV., represent the same measuring flume as figures 1 and 2, with the changes made for the purpose of narrowing the flume. The partition A B was placed near the middle of the flume ; the dam C pre- vented any flow of water through the part of the flume shut off by the partition. In order to make the flow through the narrow flume more nearly like that through a long canal of uniform section, and in this respect, more like the flow through the wide flume, the partition was extended above the flume from A to D, a distance of about 100 feet. This extension of the partition was constructed of planks, the lower ends of which were set in the earth forming the bottom of the canal, and the upper ends were secured to timbers and stayed as represented in figure 3. The part of the partition from A to D was intended to be as nearly impervious to the passage of water through it as it could be conveniently made without jointing the planks ; the partition from A to B was made with more care and was intended to be water tight; the lining of the flume was also in- tended to be water tight; neither lining nor partition were, however, quite tight. In the experiments with the wide flume, no difficulty was experienced from this cause ; in the experiments with the narrow flume it was necessary to ascertain the correction to be applied on account of the leakage. It would occupy much space to give an intelligible description ,of the operations performed to arrive at the correction to be made on this account, and as it was found to be very small, less than -j^V^ part of the quantity passing the flume in any experiment, further mention of it is unnecessary. 172 A METHOD OF GAUGING THE FLOW OF WATER 203. The whole length of the measuring flume was about 100 feet, only 70 feet, however, was included between the upper and lower transit stations H and I ; the principal part of the remainder, A H, being about 28.5 feet, was used as an entrance or mouth-piece to the part used for ascertaining the velocity, in order that the eddies and other irregularities incident to the small change in the form and dimensions of the canal, might be, to some extent, obliterated, before reach- ing the part of the flume used for ascertaining the velocity. This space was also serviceable by giving opportunity for the tubes to become free from considerable oscillations and to attain, sensibly, the velocity of the current. 204. Figures 5 and 6, plate XV., represent two of the loaded tubes, used for ascertaining the velocity of the water in the flume. Figure 5 represents the tube used in experiment 1, in which it extended as nearly to the bottom, E E, as appeared to be safe and not touch during its passage. Figure 6 represents the tube used in experiment 7, in which the space between the bottom of the tube and the bottom of the canal was about one foot. The tubes are cylinders, two inches in diameter, made of tinned plates, soldered together, with a piece of lead, C B, of the same diameter, soldered to the lower end, and of sufficient weight to sink the tube nearly to the required depth, which was such as to leave about four inches of its length above the surface of the water. The required depth of immersion was marked with red paint at A. In order to adjust it precisely, the tube was placed in a tank made for the purpose, and small pieces of lead were dropped into the top of the tube ; these rested on the mass of lead, C B, and were added until the tube was sunk to the required depth ; the orifice D was then closed with a cork. The tubes were allowed to remain floating in the tank for some time after they were adjusted, in order to ascertain whether they leaked or not ; if they did they were taken out of the tank and filled with water, in order to ascertain the position of the leak, which was then stopped with solder and the operation of adjustment repeated. The centres of gravity of the tubes thus adjusted were at G, G B in figure 5 being about 1.90 feet, and in figure 6 about 1.78 feet. The centres of gravity being so low, the tubes had a strong tendency to maintain a vertical position. The velocity of the current being, how- ever, generally more rapid near the surface than near the bottom, the upper parts of the tubes must, of course, generally, have had an inclination down stream; no special observations were made of the amount of inclination ; in the small part projecting above the surface of the water none was apparent, and as it was evidently very small, it has been assumed in all these experiments that the tubes constantly maintained a vertical position. Tubes of thirty-three different lengths, from six feet to ten feet, six of each IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 173 length, had been previously provided for the ordinary measurements of the water used by the manufacturing companies. From this stock three or four of each length required for these experiments were selected and specially adjusted for each experiment. The tubes were put into the water by an assistant standing upon the bridge K, figure 1. plate XV. ; it is done by a manoeuvre requiring a little practice to perform it satisfactorily. The assistant stands with his face up stream, with the tube in hand, the loaded end directed downwards, but up stream, at an angle with the horizon, greater or less, depending on the velocity of the current. At a signal, he pushes the tube rapidly into the water at the angle at which he previously held it, until the painted mark near the upper end of the tube reaches the surface of the water, he retains his hold of the upper end of the tube until the current has brought it to a vertical position, when he abandons it to the current; he then turns round and observes, at its passage under the transit timber H, how far the tube is from the left side of the flume, the up-stream face of the timber being, for this purpose, graduated in feet, and distinctly marked and numbered. He also observes its passage under the middle timber L, and the lower transit timber / in a similar manner. As he makes the observations he calls the distances, which are recorded by another assistant. The mean obtained by adding together the observed distances at the upper and lower transit timbers, and twice the observed distance at the middle timber, and dividing the sum by four, is taken as the mean distance of the tube from the left side of the flume during its passage. 205. The up-stream sides of the timbers H and / are vertical, and 70 feet apart, and form the upper and lower transit stations. The times when the tube passes the transit stations are noted by an observer at .2V, who has a marine chronometer on a table before him. The passage of the tube at the transit stations is observed by assistants who are seated at M and 0. The signals of the transits are communicated to the observer of the times by means of an electric telegraph erected for the purpose ; connected with the telegraph are two break- circuit keys which are conveniently placed within reach of the assistants at M and 0, and a telegraphic call is placed on the table at .2V, near the chronometer. When the tube has been abandoned to the current by the assistant on the bridge K, the assistant at M puts one of his eyes in the vertical plane forming the upper transit station, and at the instant when the tube passes this plane he de- presses the key of the break-circuit, which causes a signal to be made at the call near the chronometer, the observer at N noting the time when the signal is made. The chronometer marks half seconds only, but the times are noted, by 174 A METHOD OF GAUGING THE FLOW OF WATER estimation, to tenths of seconds. (Art. 142.) The difference of the observed times of the transits at the two stations gives the time during which the tube passes the 70 feet; dividing the distance by the time, the quotient is the velocity in feet per second. Another assistant observed the depth of the water in the flume ; this was done during the passage of each tube ; the height of the water was observed in the box P, figure 1, plate XV., placed between the lining planks and the wall of the canal ; there was a communication between this box and the flume by means of a pipe, which opened into the flume near the timber L, and about four feet above the bottom of the flume. The box P contained a scale graduated to hundredths of feet, the zero point of which was at the mean elevation of the bottom of the part of the flume between the transit stations H and /. The bottom of the flume was very nearly horizontal, the elevations to obtain the mean were taken at 32 points, the extreme difference observed was 0.027 feet. 206. Printed forms, bound up in books, were prepared, in which the obser- vations were entered. Table XX. compiled from three of these books, contains the observations made in experiment No. 1, together with some of the steps towards obtaining the quantity of water. The distances given in column 1 were arranged and entered previous to commencing the experiment, and were called in order, for the information of the assistant who put in the tubes, by the assistant who observed the times of the transits, as he became ready to make the observations. The intervals of time, given in column 4, are the differences of the times of the tran- sits given in column 3. The velocities of the tubes given in column 5, are taken from table XXVIII., which has been computed, for the purpose of facilitating the ordinary measurements of the water used by the manufacturing companies at Lowell. 207. To find the mean velocity of the tubes, all the observed velocities are plotted on section paper, engraved for the purpose; reduced copies of several of these diagrams are given in plate XVII. The ordinates of the irregularly curved line are intended to represent the mean velocities of the tubes at the corre- sponding points in the width of the flume ; this line is drawn on the original diagram by the eye, which it is plain cannot lead us much astray. The area of the figure A B C D, experiment 1, divided by the width of the flume, will evidently give the mean velocity of the tubes. The areas in experiment 1 for each foot in width, excepting the last, are given in column A, table XX.; the sum of these areas is 71.768, which being divided by 26.746, the width of the flume, gives 2.6833 feet per second for the mean velocity of the tubes. This last quantity, (assuming it to be the same as the mean velocity of the water,) mul- tiplied by the area of the transverse section of the stream, which in this experiment IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 175 is 26.746 X 9.533 = 254.97 square feet, gives 684.16 cubic feet per second, as the quantity of water passing, according to the flume measurement. 208. It will be perceived, by reference to the diagrams in plate XVII., that the observed velocity at the same part of the section is constantly varying; this is not due, in any sensible degree, to errors of observation, but to actual changes in the velocity, due to the unstable condition of the current. In all these exper- iments, the area of the section, and the quantity of water flowing, were sensibly constant throughout an experiment; the mean velocity must, consequently, have been nearly constant, and the only explanation of the observed variations in the velocity is, that there was a constant interchange of place of currents of different velocities. 209. The water after leaving the measuring flume passed to the weir erected on the Tremont Wasteway, F, figures 1 and 2, plate XVI. This weir was in two divisions, each having about 40 feet in length of water-way; the Westerly divis- ion, and a part of the Easterly division, are represented on an enlarged scale by figures 3 and 4. Figure 5 is a sectional elevation of the weir and some of the apparatus connected therewith. A is a grating for the purpose of equalizing the flow towards the weir, and for obliterating the irregularities in the direction of the currents approaching the weir, which it is obvious, from an inspection of the form of the approaches, would have otherwise existed. The whole length of the grating was 88 feet; the vertical slats were 4 inches wide, in the direction of the current, and one inch thick, the spaces between the slats, for the passage of the water, were about 1.125 inches wide. To equalize the flow still further, horizontal slats 1.5 inches wide were placed on the up-stream side of the grating; they were placed principally at the Westerly part of the grating, on which the current from the measuring flume impinged most directly. The whole length of the grating being divided into five nearly equal parts, the Westerly part had eight horizontal slats, the next part had six slats, the next four, the next two, and the next, or most Easterly part, had none. The effect of this grating was to obliterate all sensible lateral currents ; it did not, however, entirely equalize the flow, except in a small portion of the experiments. In experiment 1, in which the discharge over the weirs was 681.25 cubic feet per second, the mean depth on the Easterly division of the weir was 0.0387 feet less than on the Westerly division ; in experiments 43 to 49, in which the mean discharge was 106.05 cubic feet per second, the mean depth on the Easterly division of the weir was 0.00026 feet greater than on the Westerly division. In computing the discharge the mean of the observed depths on the two divisions of the weir is taken, the small in- equalities in the depths on the two divisions produce inappreciable effects on the results. 176 A METHOD OF GAUGING THE FLOW OF WATER 9 2.3 A ;> a 3 ,S .. 'Sic;- ,111141 ^-X*?jes o - * 1 3 3^ S T: 2 :. ~ ti a - 3 ** = -"^= .= >- 7 J isfllf*! s? " gj.si'SBX CO O OS 'O O CO i CM 00 i CO CO I 1-1 t~. S ii s 7 III * S?3 III ! 14 II | 111 & sll PQ o >o (> c "O lO'tcos^-H 10^ ,1 IOCOO J o ci ci 06 c -H 40 (Mi i I li til J I = Ijlll !lil j ! 3I**|| 3=33 (N oo IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 177 210. The up-stream face of the weir F P, figure 5, was a vertical plane, 6 feet in height and 88 feet long; the crest of the weir was of the form represented by figure 3, plate XVIII., and was horizontal for a width of 0.5 inches ; the up-stream edge presented to the current was as sharp as could be conveniently maintained in wood ; the down-stream side of the cr*est was chamfered off at an angle of 45 with the vertical. The two divisions of the weir were separated by a space B four feet wide, and at each of the ends C there was a space of two feet; the up-stream faces of these spaces were in the same vertical plane as the up-stream face of the weir, and were deemed to be ample to insure complete contraction at the ends of the sheets of water. The dam or wasteway on which the weir was erected was of a form adapted to the convenient discharge of water over its crest, and for the regu- lation of the flow over the same; this was, however, not the form to which the ordinary formula for computing the flow over a weir applies, and it was therefore necessary to make such changes in the form of the crest as would permit of such application. It was not deemed admissible to take down the top of the existing dam, and to reconstruct it of suitable form ; all that could be done was to make additions which could be removed when the experiments were completed. 211. In order to preserve a sufficient depth of flow over the weir, the crest could not be raised more than one foot above the wasteway. The standards D, figures 3 and 5, which formed part of the wasteway and were required to support the flash-boards used in regulating the flow over the wasteway, it was necessary to leave undisturbed ; in order that they should not obstruct the flow over the weir, the crest of the latter was placed at a certain distance up stream ; this was accom- plished by fastening the large timber E, figure 5, to the up-stream face of the wasteway, the plank F, figures 3 and 5, forming the crest of the weir, was fastened to this timber. As thus arranged, the sheet of water passing over the weir fell vertically, and with very slight obstructions, to the cap of the wasteway, and passed horizontally, a distance of about 1.4 feet from the up-stream face of the weir plank F, before it struck the standards D. 212. The weir was made in two divisions for the purpose of facilitating the passage of air under the sheet, former observations having shown that air thus situated is rapidly carried away by the water, and unless sufficient means are pro- vided for renewing it, its place will be speedily taken by water, which will materially affect the flow over the weir and prevent the correct application of the formula for computing the discharge. This precaution proved, however, to be in- sufficient to prevent the space under the sheet from becoming filled with water; it was evident that a portion of the water striking the top of the wasteway flowed back towards the weir and filled the space which ought to be kept free; to prevent 23 178 A METHOD OF GAUGING THE FLOW OF WATER this, the board G, figures 3 and 5, was put on; its width was sufficient to reach from the top of the timber E very nearly to the underside of the sheet ; this remedied the difficulty in a great degree, but, unless the width of the board was properly adjusted to the sheet, it failed to operate satisfactorily; if too low, the water flowed back over the top, if too high, the sheet of water struck the board, in either case very soon filling up the space between the board and the weir plank; at first the only escape of the water from the trough formed by the board G and the weir plank was at the ends, and the trough being forty feet. long, the escape from the central parts was very slow. This difficulty, however, was remedied by attaching leaden pipes, two inches in diameter, to the board G; these pipes were about sixteen feet long and were laid on the inclined surface of the apron of the wasteway, the lower ends of the pipes being about five feet below the upper ends. The Easterly division was first fitted up with twenty-six of such pipes ; upon trial this proved to be a much greater number than was necessary to afford escape for the water flowing back over the top of the board G, and the Westerly division, which is that shown on figure 3, was provided with only half the number, which proved to be amply sufficient. It was necessary to readjust the height of the board G, whenever a material change was made in the depth of water on the weir. It is represented in figure 5, as it was in experiments 1 to 7, in which the depth on the weir was near the maximum. The top of the board G, in these seven experiments was about 0.105 feet below the top of the weir. 213. The depth on the weir was observed at each division separately, by means of hook gauges, similar to that represented by figures 2, 3, and 4, plate XIII. A gauge acting on the same principles is described in article 45. The gauge for the Westerly division was placed in the box "ff, figures 3, 4, and 5, plate XVI. ; this box was carefully made so that no water passed into or out of it, except through the pipes in the bottom, and it was strongly fastened to the post /, which was firmly set in the earth at the bottom and supported by the braces K at the top. When observations were being made with the hook gauge for the depth on the weir, the three pipes L L L formed the only communication between the water in the box and the water in the basin between the grating and the weir; the surface of the water in the box was assumed to be at the height giving the mean depth on this division of the weir ; subject, however, to a small correction to be described hereafter. 214. The small box was firmly secured to the planking forming the interval between the two divisions of the weir; it had no communication with the water outside of it, except by means of the pipes JW and Q, which furnished the means IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 179 of connecting it with either of the hook gauge boxes when desired. The box O contained a stationary hook, the point of which was formed by a portion of a sphere of about half an inch in diameter; the coincidence of the level of the sur- face of the water with the highest part of the spherical surface could be as defi- nitely ascertained, as if the hook had terminated in a sharp point, as in the hook gauges, whilst the spherical surface permitted a levelling-rod to be placed upon it for the purpose described presently. For convenience in using the hook gauges, their zero points were placed several inches above the top of the weir. In order to ascertain the precise elevation of the zero point of one of these gauges relatively to the mean height of the top of the corresponding division of the weir, the water was adjusted to a depth of about one foot on the weir, the three pipes L were closed, and the pipe N opened. The pipe N then furnished a free communication between the boxes H and 0, neither of which at this time had any other orifice for the passage of water in or out. Water was then put into or taken out of these boxes until its surface coincided with the highest part of the spherical surface which formed the point of the stationary hook in the box 0; when this was done and the water in the boxes free from oscillations, the height of the surface of the water in the box H was observed by means of the hook gauge, which evidently gave the height of the point of the stationary hook in the box 0, by the scale of the -hook gauge in the box M. The height of the point of the stationary hook in the box O above the mean height of the top of the weir was obtained by levelling with a Troughton and Simms dumpy level ; this was done with great care and with all the precautions necessary for insuring accuracy ; it was done three times during the course of the experiments, with the results given in the following table. TABLE XXI. Height of the point of the Height of the point of the Hook in the Box above Hook in the Box above Date. the mean height of the the mean height of the top of the Westerly divis- top of the Easterly divis- ion of the Weir. ion of the Weir 1856. Feet. Feet. October 7. 1.0087 1.0112 " 17. 1.0090 1.0111 November 19. 1.0089 1.0127 215. From the observations in the preceding table it is evident that the rela- tive elevations of the weir and the point of the stationary hook were not subject to sensible change. Comparisons between the hook gauges and the stationary hook were made every day, with a depth of about one foot on the weir, and the cor- 180 A METHOD OF GAUGING THE FLOW OF WATER rection determined and used in all the experiments of that day. The relative heights of the hook gauges and stationary hook were subject to greater changes than were observed between the stationary hook and the top of the weir. The experiments extended from October 7 to November 13; the difference of height of the stationary hook and the zero of the Westerly hook gauge was greatest on October 8, when it was 0.4402 feet, and least on October 23, when it was 0.4352 feet, the change, which was not abrupt, being 0.0050 feet. The corresponding change at the Easterly hook gauge was 0.0066 feet, the sign and dates being the same as at the Westerly hook gauge. These differences are not very great, and as the correc- tions were determined daily, no appreciable errors can result therefrom. 216. The experiments of 1852, described in a former part of this work, from which the formula for computing the quantity of water flowing over the weir in these experiments is deduced, were made upon a weir of great simplicity of form, in which the sheet of water passing over the weir had an unobstructed fall of not less thon three feet; see figure 1, plate XIII. Other experiments indicated that the sheet of water may meet with great obstructions soon after passing the weir, with- out its flow over the weir being sensibly affected thereby (see ante, page 134), and it was thought, that in these experiments the obstructions to the flow of the water after passing the weir, would affect the discharge over the weir to so small an extent as to be inappreciable. It was highly important, however, to avoid all ques- tion on this point; and to determine the matter, a special series of experiments was undertaken. For this purpose two weirs were erected in the upper chamber of the Lower Locks in Lowell, K, figure 1, plate XL The upper weir was constructed of a form to which the formula for computing the discharge could be applied without objec- tion. The lower weir in a portion of the experiments was of the same form as the upper weir, and in the other portion the form was the same as the weir at the Tremont Wasteway. The experiments consisted in causing the same volume of water to flow over both weirs, and observing the depth assumed by the water on each weir, when the flow had become permanent, the differences in the depths, if any, being due to differences in the forms and conditions of the two weirs. The Lock chamber is twelve feet wide, and the weirs were each eight feet long, leaving a space of two feet at each end to insure complete contraction. The up- stream faces of the weirs were vertical planes, and the crests and ends were of the same form as the weir at the Tremont Wasteway. The bottoms of the channels on the up-stream sides of both weirs were six feet below the tops of the weirs. The water entered the Lock chamber through the head gates, and under a head of several feet, which caused a great commotion in the water at the upper end of IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 181 the chamber. The upper weir was placed about sixty feet from the upper end of the chamber, and to obliterate the disturbance in the water before it reached the weir, three gratings, at right angles to the sides, were placed across the chamber at intervals of about twelve feet; each grating contained about one half of the aperture per square foot, for the passage of water, as the grating used at the Tremont Wasteway. The lower grating was about fourteen feet from the weir. The surface of the water between the two upper gratings was nearly all covered by a float of planks, for the purpose of obliterating the oscillations of the surface. The second or lower weir was about thirty-five feet from the upper weir, and similar arrangements were made for obliterating disturbances in the water, as were provided for the upper weir, except that there were only two gratings, the disturbances caused by the fall of the water from the upper weir into the basin below it being much less than were caused by the entrance of the water at the upper end of the chamber. The lower grating was about fourteen feet from the lower weir. The depths of the water on the weirs were observed by means of hook gauges similar to that represented on plate XIII. The difference of the leakages into and out of the part of the chamber included between the two weirs was ascertained, and a cor- rection applied for the same ; and also for the rise or fall, if any, of the surface of the water in the same space during the time occupied by an experiment. In arranging the apparatus, it was designed to make the immediate approach of the water to the two weirs precisely alike. It was not certain, however, that the precautions taken to insure uniformity would produce the desired result. To avoid doubts on this point, the lower weir in part of the experiments, as stated above, was made of the same form as the upper weir, in which case any difference in the depths on the two weirs, the quantity of water flowing being the same at both, and there being no obstructions below, must be due to differences in the immediate approach of the water to the weirs. A series of experiments was made under these circumstances, with different quantities of water flowing, from which it was ascer- tained, that when the depth on the upper weir was about 0.5 feet, the depth on the lower weir was 0.0008 feet greater; when the depth was about a foot on the upper weir, it was the same on the lower weir; when about 1.5 feet on the upper weir, it was 0.0040 feet less on the lower weir; and when about 2 feet in depth on the upper weir it was about 0.0094 feet less on the lower weir. These differences were probably due to small differences in the relative velocities of the water imme- diately approaching the weirs, at different depths, and might, doubtless, have been partially remedied by suitable modifications of the gratings. It would have required much time, however, and was not essential to our arriving at correct results, the ex- periments with the two weirs alike having been sufficiently numerous and varied to enable a table of corrections to be made. 182 A METHOD OF GAUGING THE FLOW OF WATER 217. Another series of experiments was made with the lower weir like that erected at the Tremont Wasteway, the apron, trough, pipes, standards, etc., being re- produced, as nearly as the length of the weir would permit. The height of the board, forming the down-stream side of the trough, was of course varied in the dif- ferent experiments, to conform to the corresponding changes at the Tremont -weir. The upper weir remained unchanged throughout all the experiments. Water being admitted at the upper end of the chamber, and the flow become permanent, or as nearly so as practicable, observations were made of the depth which the water as- sumed at the two weirs. It would occupy much space to describe all the exper- iments made; it will perhaps be sufficient to state some of the results arrived at. After correcting the depth on the lower weir for the differences described in the pre- ceding section, which did not depend on the forms of the weirs, the following dif- ferences were found. When the depth on the upper weir was about 0.8 feet, the depth on the lower weir was 0.0007 feet less ; when the depth on the upper weir was about 1.5 feet, the depth on the lower weir was the same ; when the depth on the upper weir was about 2 feet, the depth on the lower weir was 0.0085 feet greater. This last difference corresponds to a diminution of flow over the loAver weir, with the same depth on the weir, of T ^^. 218. The effect of what appear to be obstructions to the flow over a weir is, generally, to increase the depth on the weir over what it would be if the flow was free ; sometimes, however, it has the contrary effect. (See article 137.) The exper- iments at the Lower Locks described in the preceding section furnished the data for a table of corrections of the depths of water on the Tremont weir, due to the obstructions to the flow of the sheet after passing the crest of the weir. In exper- iment 1, table XXII., in which this correction has nearly its greatest value, it is 0.0058 feet 219. Another small correction was also applied. In the experiments of 1852 (art. 173), it was found that there was no sensible difference in the observed depth upon the weir, whether the external orifice of the pipe, forming the communication between the water approaching the weir and the hook gauge box, was close to the plane of the weir or six feet up stream from that plane, the external orifice of the pipe being at a considerable depth below the top of the weir. In arrang- ing the apparatus at the weir at the Tremont Wasteway, it was thought that there would be less liability to errors in the observed depths, from currents acting on the external orifices of the pipes, if they were very near the plane of the weir, and at the bottom of the canal, and they were accordingly so arranged. In the experiments of 1852, however, on which the formula for computing the flow over the weir is founded, the orifice in the hook gauge box was six feet IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 183 from the weir, and in order to ascertain whether any difference could be detected in the observed depths on the weir at the Tremont Wasteway, with the external orifice of the pipe at different distances from the weir, some special experiments were made. For this purpose an apparatus of pipes similar to that represented in figures 8 and 9, plate XIV., was placed at the bottom of the canal, on the up-stream side of the weir at the Tremont Wasteway. The orifices of the pipes were pro- tected from the action of lateral currents, if any existed, by a second board, placed parallel to the board in which the lower ends of the pipes were inserted, and three inches distant; these boards were placed at right angles to the weir, and the space between them was open at the top and the up-stream end, so that the current flowing towards the weir, flowed through the trough formed by the two boards, by the open ends of the pipes, which, to avoid eddies, did not project beyond the plane of the board. With this apparatus, observations were made of the differences in the depths on the weir, when the different pipes were in communication with the hook gauge box ; substantially the same precautions being taken to secure precision in the results as are described in article 170. Taking the observations made with the pipe opening at 0.52 feet from the weir, as represented at B, figures 3, 4, and 5, plate XVI., as the standard ; when the depth on the weir was about 0.76 feet, the differences in the depths observed by means of the other pipes were as follows : By the pipe opening at 2 feet from the plane of the weir, difference = 0.0003 feet. 4 0.0003 " a e 0.0004 " a it g " " " " = 0.0001 " " 10 " " " " " = 0.0003 " a " 12 " " " " " 0.0012 " When the depth on the weir was about 1.44 feet, the differences observed were as follows : By the pipe opening at 2 feet from the plane of the weir, difference = -f- 0.0020 feet. a u u 4 Q 0009 " 6 " " " " " == 0.0013 u u u u g _ 0.0054 " 10 = 0.0089 " 12 " " " " " = 0.0124 " Up to six feet from the weir, these differences are very small; it was thought best, however, to take account of them. 184 A METHOD OF GAUGING THE FLOW OF WATER By a discussion of the whole of the experiments a table was formed, for cor- recting the observed depths on the weir, to what they would have been if observed with the pipe opening at G feet from the weir. When the depth on the weir is 0.5 feet, this correction is 0.0002 feet. a 0.8 " " 0.0004 " " 1.0 0.0006 " * 1.5 " " 0.0014 " " 2.0 " " 0.0023 " 220. By table XVIII., containing the results of similar experiments at the Lower Locks, made about four years previously, it will be seen, that the differ- ences between the depths on the weir, observed by means of a pipe opening at six feet from the plane of the weir, and by a pipe opening at one inch from the plane of the weir (changing the signs to conform to the experiments at the Tremont weir), were as follows: When the depth on the weir was about 0.80 feet, difference = 0.00060 feet. " 1.00 " " - 0.00033 " The small differences in these results from those obtained at the Tremont Wasteway weir may be explained by the different forms of the approaches to the weirs, and the different arrangement of the apparatus. 221. The formula for computing the quantity of water flowing over weirs, de- duced from the experiments made at the Lower Locks in 1852, viz. : Q = 3.33 (L 0.1 n H} H%, (A.) is adapted to weirs of widely differing proportions, including all the forms on which experiments are given in table XIII. By reference to column 16 in that table, it will be seen, however, that the experiments on each particular description of weir generally give a coefficient differing slightly from the mean value deduced from the whole of the experiments. In case any of those particular forms should be reproduced, it is evident, that the quantity of water flowing over the same could be more accurately computed, by using the corresponding coefficient given in column 16, than by using that given in formula (A.), which is a mean, deduced from the whole of the experiments. In determining the formula by which to com- pute the flow over the weir at the Tremont Wasteway, it was apparent that results more exact could be attained by deducing a new formula from a selection of the experiments given in table XIIL, in which the circumstances were most IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 185 nearly like those at the Tremont Wasteway weir. For this purpose 53 experiments were selected, and the formula deduced from them is Q = 3.318 (L 0.08 n H) H%. (B.} As applied to the weir at the Tremont Wasteway, When the depth is 1 foot, the discharge by formula (A.) = 265.09 cubic feet per sea And by formula (B.) 264.40 " Difference ^ = 0.69 When the depth is 2 feet, the "discharge by formula (A.) = 746.02 " " And by formula (B.) = 744.84 Difference ?fa = 1.18 " " When the depth is 3.5527 feet, both formulas give the same discharge. 222. In making these experiments, there were several objects in view, which may be classed under two heads, viz. : 1st. To determine a formula for correcting the quantity passing a measuring flume, as deduced from the mean velocity of the tubes; there being no unusual disturbing causes. 2d. To ascertain the degree of uniformity in measurements made under like circumstances ; and to determine the magnitude of the errors to which we are liable, when measurements are made under exceptionable circumstances, such as high winds and great irregularities in the motion of the water. The experiments adapted to the first object were necessarily made under the normal conditions of freedom from high wind, and from great irregularity in the currents. Table XXII. contains 105 experiments selected as being suitable for this purpose, and table XXV. contains 35 experiments made for the purposes in- cluded in the second class. 24 186 TABLE EXPERIMENTS MADE AT THE TREMONT WEIR AND MEASURING FLUME, 1 3 3 Weir Measurement. Flume Tempera- 4 5 6 7 8 9 1O 11 IS 13 14 15 tureVin degrees of Fahrenheit's Quantity of water Corrected quantity Difference between the depth Date. thermometer Total Observed Observec Mean Corrected passing over the Correc- passing the flume. Mean Length of water in the No. length of the weirs. depth of water on the depth of water on the observed depth of water depth of water on the weirs, com- puted tion for the leak- age into deduced from the weir Mean width of the flume. depth of water in the of the im- mersed flume and the length of the the 1866. of the Westerly weir. Easterly weir. on the weirs. weirs. by the formula the Hume measure- ment. fluuie. part of the immersed part of atmos- of the tube. the tube, Exp phere water. H Q Q' divided by shade. 3.318 (t O.OSn//)//"' the depth of water in Feet. Feet. Feet. Feet. Feet. Cubic ft. Cubic ft. Cubic fl. Feet. Fet. Feet. the flume. per sec. per sec. per sec. D 1 Oct. 7 A.M. 52.5 57.0 80.007 1.8972 1.8585 .8778 1.8839 681.25 681.25 26.746 9.533 9.482 0.005 2 it 11 u 53.5 57.0 tt 1.8750 1.8395 .8572 1.8634 670.U2 670.22 tt 9.515 9.430 0.009 3 11 a u 60.5 57.0 tt 1.8556 1.8214 .8385 1.8448 660.26 660.26 tt 9.496 9.380 0.012 4 tt tt tt 64.5 57.5 tt 1.8720 1.8362 .8541 1.8604 668.61 668.61 tt 9.510 9.330 0.019 i 5 " " P.M. 65.0 58.0 ti 1.8906 1.8548 .8727 1.8787 678.45 678.45 tt 9.531 9.230 0.032 6 tt tt u 65.0 58.0 ti 1.8896 1.8539 .8717 1.8777 677.91 677.91 tt 9.532 9.130 0.042 7 ti it ti 64.0 58.0 it 1.8872 1.8507 .8689 1.8750 676.45 676.45 tt 9.530 8.530 0.105 8 " 8 A.M. 51.5 57.0 80.008 1.7846 1.7527 1.7686 1.7752 623.43 623.43 tt 9.422 9.360 0.007 9 u it ti 60.5 57.0 it 1.7882 1.7570 1.7726 .7790 625.42 625.42 tt 9.426 9.320 0.011 10 tl tt ft 63.5 57.0 it 1.7825 1.7495 1.7660 .7726 622.07 ; 622.07 it 9.421 9.280 0.015 11 " "" P.M. 66.6 57.0 tt 1.7810 1.7490 .7650 .7716 621.54 621.54 it 9.421 9.220 0.021 12 fl tf tt 66.1 57.0 tt 1.7720 1.7416 .7568 .7633 617.20 617.20 tt 9.412 9.120 0.031 13 tf tt tt 62.9 57.2 tt 1.7713 1.7394 .7553 .7618 616.4l| 616.41 it 9.410 9.020 0.041 14 ft tt tt 60.0 58.0 tt 1.7626 1.7333 .7479 .7546 612.66 612.66 tt 9.402 8.410 0.106 15 " 9 A.M. 53.0 58.0 80.009 1.5061 1.4864 .4962 .5027 486.08 486.08 tt 9.141 9.080 0.007 16 tt it it 59.5 58.0 tf 1.5045 1.4854 .4949 .5015 485.50 485.50 tt 9.140 9.030 0.012 17 ti it tt 70.5 58.0 H 1.5046 1.4849 .4947 .5013 485.40 485.40 ti 9.138 8.980 0.017 18 " " P.M. 80.5 58.0 tt 1.5038 1.4842 .4940 .5006 485.06 485.06 ti 9.138 8.930 0.023 19 tt tt tt 79.5 58.5 ft 1.5033 1.4833 .4933 .4999 484.72 484.72 tt 9.137 8.830 0.034 20 tt tt it 79.5 59.0 tt 1.4925 1.4737 .4831 .4895 479.71 479.71 u 9.126 8.730 0.043 21 tt it tt 74.0 59.0 tt 1.4839 1.4639 .4739 .4804 475.34 475.34 tt .9.118 8.120 0.109 22 " 11 A.M. 72.5 59.0 80.010 1.2091 1.1995 .2043 .2086 351.03 351.03 26.745 8.838 7.830 0.114 23 tt it it 78.5 59.0 tt 1.2131 1.2031 .2081 .2124 352.68 352.68 fl 8.842 8.430 0.047 24 " " P.M. 78.0 60.0 tt 1.1968 1.1899 .1933 .1975 346.22 346.22 tf 8.830 8.530 0.034 25 ft ft tt 77.5 60.0 1.1942 1.1857 .1899 .1941 344.75 344.75 ft 8.827 8.630 0.022 26 " 13 A.M. 57.5 60.0 80.011 1.1985 1.1886 .1935 .1977 346.31 346.31 tf 8.827 8.780 0.005 27 tt tt tt 59.0 60.0 tt 1.1854 1.1759 .1806 .1847 340.70 340.70 ft 8.815 8.730 0.010 28 tt tt tt 64.5 60.0 tt 1.1820 1.1726 .1773 .1813 339.24 339.24 ft 8.810 8.680 0.015 29 " vl P.M. 69.0 60.0 tt 1.3715 1.3555 .3635 .3691 422.96 422.96 If 8.997 7.980 0.113 30 tt tt tt 68.0 60.0 tt 1.3533 1.3382 .3457 .3512 414.72 414.72 ft 8.981 8.600 0.042 31 it tt ft 67.0 60.0 it 1.3580 1.3427 .3503 .3559 416.87 416.87 ft 8.985 8.700 0.032 32 tt tt tt 65.5 60.0 tt 1.3524 1.3387 .3455 .3510 414.63 414.63 ft 8.978 8.800 0.020 33 " 14 A.M. 40.0 59.0 80.012 1.3752 1.3618 .3685 .3742 425.32 425.32 ft 9.006 8.850 0.017 34 tt it tt 40.5 59.0 it 1.3772 1.3635 .3703 .3760 426.15 426.15 (f 9.009 8.900 0.012 35 tt tt tt 42.0 59.0 tt 1.3681 1.3552 .3616 .3673 422.13 422.13 tt 8.997 8.960 0.004 36 " " P.M. 45.0 58.0 tt 0.9792 0.9756 0.9774 0.9801 256.58 256.58 ft 8.609 8.230 0.044 37 tt ti tt 43.5 58.0 tt 0.9840 0.9818 0.9829 0.9856 258.74 258.74 (t 8.615 8.330 0.033 38 tt tt tt 41.5 58.0 tt 0.9746 0.9732 0.9739 0.9766 255.21 255.21 tt 8.604 8.430 0.020 39 " 15 A.M. 34.5 56.0 u 1.0016 0.9971 0.9993 1.0022 265.29 265.29 tt 8.631 8.480 0.017 40 tt tt tt 39.0 56.0 tt 0.9916 0.9872 0.9894 0.9922 261.34 261.34 if 8.620 8.530 0.010 41 tf tf ft 47.5 56.0 tt 0.9841 0.9805 0.9823 0.9850 258.51 258.51 ft 8.611 8.570 0.005 42 " " P.M. 52.5 56.0 tt 0.9942 0.9900 0.9921 0.9950 262.44 262.44 (t 8.626 7.620 0.117 43 tt tt tt 50.0 56.0 tt 0.5504 0.5502 0.5503 0.5513 108.43 108.43 (f 8.172 7.120 0.129 44 " 16 A.M. 38.5 54.0 80.014 0.5445 0.5444 0.5444 0.5454 106.70 106.70 tf 8.167 7.720 0.055 45 it it u 47.0 54.0 tl 0.5403 0.5406 0.5404 0.5414 105.53 105.53 (f 8.164 7.920 0.030 46 if ti tt 55.5 54.0 ft 0.5394 0.5395 0.5394 0.5404 105.24 105.24 (f 8.163 8.070 0.011 47 " " P.M. 60.5 54.0 fl 0.5410 0.5412 0.5411 0.5421 105.74 105.74 tf 8.165 8.122 0.005 48 ft ft it 58.5 54.0 tt 0.5342 0.5350 0.5346 0.5356 103.84 103.84 ft 8.159 8.020 0.017 49 .i 21 " 71.0 53.5 tt 0.5447 0.5454 0.5450 0".5460 106.88 106.88 ft 8.171 8.070 0.012 50 " 27 A.M. 35.0 47.0 ft 1.9105 1.8662 1.8883 1.8943 686.93 686.93 46 87 it it ii it 1.3702 1.3402 1.3552 1.3608 419.15 0.35 418.80 ii 8.960 8.650 0.035 XX tt tt ft tt 1.3678 1.3419 1.3548 1.3604 418.96 0.35 418.61 ft 8.956 8.900 0.006 89 ft it it tt 1.3820 1.3505 1.3662 1.3719 424.26 0.35 423.91 ti 8.971 8.850 0.013 90 it it tt tt 1.3794 1.3491 1.3642 1.3698 423.30 0.35 422.95 ft 8.967 7.950 0.113 91 ft ft tt 40.0 42.0 it 1.3788 1.3488 1.3638 1.3694 423.11 0.35 422.76 tf 8.962 8.750 0.024 92 " 12 A.M. 34.0 40.0 ii 1.6555 1.5954 1.6254 1.6323 550.05 0.41 549.64 it 9.213 9.150 0.007 93 tt tt ft tt 1.6472 1.5888 1.6180 1.6249 546.32 0.41 545.91 it 9.210 9.100 0.012 94 tt ii tf tt 1.6480 1.5830 1.6155 1.6223 545.02 0.41 544.61 tl 9.207 9.050 0.017 95 tl ft fi tt 1.6363 1.5795 1.6079 1.6147 541.21 0.41 540.80 ct 9.201 8.800 0.044 96 ti it tt ti 1.6519 1.5890 1.6204 1.6273 547.53 0.41 547.12 tt 9.215 8.900 0.034 97 tf ii ft it 1.6464 1.5862 1.6163 1.6231 545.42 0.41 545.01 ft 9.208 8.200 0.109 98 tt ft tf 36.0 40.0 tt 1.6460 1.5900 1.6180 1.6249 546.32 0.41 545.91 U 9.208 9.000 0.023 99 " " P.M. 42.0 40.0 tt 1.8508 1.7655 1.8081 1.8144 644.14 0.48 643.66 tt 9.392 9.350 0.004 100 tf tt ii it 1.8396 1.7588 1.7992 1.8056 639.48 0.48 639.00 ti 9.383 9.300 0.009 101 tt ft tt tt 1.8486 1.7674 1.8080 1.8144 644.14 0.48 643.66 tt 9.386 9.100 0.030 102 ft tt it 43.0 40.0 tt 1.8582 1.7721 1.8151 1.8214 647.85 0.48 647.37 tt 9.396 9.000 0.042 103 tt tt tf ti 1.8687 1.7819 1.8253 1.8316; 653.27 0.48 652.79 ft 9.407 8.400 0.107 104 tt tt it tt 1.8446 1.7565 : 1.8005 1.8069 640.17 0.48 639.69 tt 9.382 9.250 0.014 105 ft tt ft 1-2.1 40.0 " 1.8375 1.7538 1.7956 1.8022 637.68^ 0.48 637.20 it 9.381 9.200 0.019 XXI I CONTINUED. FROM WHICH THE FORMULA OF CORRECTION C =0.116 (/B- 189 0.1) IS DETERMINED. Measurement. 18 19 20 21 22 23 1 ft Difference between the Proportion- al differ- Quanti- ty of Proportion- al differ- Remarks on the Force and Direction J.O 1 / quantity of ence, or water ence of the of the Wind at the Flume, during Quanti- water pass- the differ- passing corrected the Experiments. Mean ty of water ing the Hume ence in the the quantity as No. velocity of tho passing the mean column, deduced in the of tubes the ve ocity of divided by from the flume given tho through- out the flume, deduced the tubes, and the the quan- tity de- mean velocity in the preceding General Remarks. flume from the quantity duced from of the column, Exp. by tho mean velocity deduce, from the weir the flume measure- tubes correc'd and the weir meas- Force. Direction. g m. of the measure- ment. by the urement. tubes. r\lt ment. formula Feet Cubic ft. Q" Q' Cubic ft. Q" Q'" = V per sec. per sec. per nee. -0.1) ). 57 1.991 487.27 4.15 0.0085 488.54 o.iiu.59 Very gentle, 1 Down stream. 58 1.969 481.00 2.31 0.0048 481.00 0.0048 variable. | U It 11 (( 59 1.978 483.14 0.17 0.0004 478.88 0.0092 it ii ti t< 60 2.029 496.76 + 4.36 +0.0088 490.44 0.0040 Very gentle. Irregular. - 61 2.013 493.07 0.21 0.0004 491.33 0.0040 Calm. 62 2.034 498.21 + 5.27 +0.0106 497.66 +0.0096 ( Very gentle, ) \ variable. Up stream. 63 2.040 498.97 +12.47 +0.0250 485.65 0.0017 Down stream. 64 65 1.360 1.376 151.08 152.69 0.33 -- 1.97 0.0022 - -0.0129 150.42 151.10 0.0065 - -0.0025 | Moderate, but ) } variable. ( Irregular. Reduced copies of the diagrams, constructed for experiments, 67, 68, and 69, are given in plate XVII 66 1.385 153.65 -- 3.33 - -0.021 7 151.53 - -0.0080 (C U (i 67 1.349 149.69 -- 0.06 - -0.0004 150.08 - -0.0030 ( More moderate, ) i but variable. ) it 68 1.370 151.99 -- 3.11 40.0205 151.02 - -0.0144 i * 69 1.381 153.12 + 5.02 +0.0328 148.74 - -0.0043 11 U " 70 1.373 152.32 + 2.23 +0.0146 152.15 +0.0137 ( Moderate, but ) { variable. I Down stream. 71 2.023 230.23 + 3.62 +0.0157 227.23 +0.0027 72 2.020 229.87 + 3.97 +0.0173 223.49 0.0107 73 1.990 226.55 _ 1.04 0.0046 226.30 0.0057 Moderate. Down stream. 74 2.029 231.09 + 2.82 .+0.0122 229.79 +0.0067 Calm. 75 2.031 231.29 + 2.95 +0.0128 229.03 +0.0030 Very gentle. Up stream. 76 2.029 231.24 + 1.61 +0.0070 230.22 +0.0026 Moderate. ( Generally up \ stream. 77 2.001 227.94 0.93 0.0041 228.37 0.0022 Very gentle. Irregular. 78 2.691 313.28 -- 7.52 +0.0240 304.59 0.0038 Gentle. Down stream. 79 2.663 310.20 -- 4.44 +0.0143 306.00 +0.0008 Hardly perceptible. Uncertain. 80 2.638 306.91 -- 1.65 +0.0054 303.81 0.0047 " " H 81 2.638 307.09 -- 2.12 +0.0069 305.13 +0.0005 It U H 82 2.606 303.51 1.09 0.0036 302.31 0.0075 t( (t u 83 2.607 303.53 0.65 0.0021 303.19 0.0033 (( U U 84 2.526 293.64 5.33 0.0182 294.64 0.0145 85 3.484 417.20 0.81 -A).0019 415.73 0.0055 Calm. 86 3.534 423.27 + 3.79 +0.0090 417.65 0.0044 Very gentle. Irregular. 87 3.532 423.21 + 4.41 +0.0104 418.94 +0.0003 Calm. 88 3.454 413.70 4.91 0.0119 414.78 0.0091 U 1 89 3.510 421.06 2.85 0.0068 420.37 0.0084 U 90 3.623 434.38 +11.43 +0.0263 422.48 0.0011 U 91 3.531 423.17 + 0.41 +0.0010 420.47 0.0054 92 4.451 548.35 1.29 0.0024 549.39 0.0005 Calm. Reduced copies of the diagrams constructed 93 4.401 541.99 3.92 0.0072 541.39 0.0083 Hardly perceptible. Down stream. for experiments 92, 96, and 97, are given in plate 94 4.415 543.50 1.11 0.0020 541.59 0.0055 H U U XVII. 95 4.400 541.33 -- 0.53 --O.OdlO 534.44 0.0118 Calm. 96 4.454 548.87 -- 1.75 - -0.0032 543.50 0.0066 U 97 4.496 553.64 -- 8.63 - -0.0156 538.86 0.0113 98 4.446 547.41 -- 1.50 - -0.0027 544.13 0.0033 II 99 5.156 647.55 + 3.89 +0.0060 650.31 +0.0103 ( 100 5.079 637.28 1.72 0.0027 637.66 0.0021 a 101 5.209 653.73 +10.07 - -0.0154 648.18 - -0.0070 a 102 5.226 656.67 + 9.30 - -0.0142 648.68 - -0.0020 Hardly perceptible. Down stream. 103 5.346 672.45 +19.66 - -0.0292 654.74 - -0.0030 Calm. 104 5.138 644.59 + 4.90 - -0.0076 643.22 - -0.0055 " 105 5.136 644.21 + 7.01 - -0.0109 641.38 - -0.0066 '* Means for the braically Means for the v signs . wide flume, taken algu- +0.0088 0.0129 +0.0013 0.0080 ide flume, disregarding Means for the narrow flume, taken al- gebraically +0.0072' ... 0.0011 Means for the narrow flume, disregard- 0.0105 0.0057 Taking the whole 105 experiments. Means taken algebraically . . . +0.0082 . . . +0.0004 Means disregarding signs . 0.0120 0.0071 190 A METHOD OF GAUGING THE FLOW OF WATER DESCRIPTION OF TABLE XXII. 223. It will be seen that the experiments are divided into groups of seven. In all the experiments in the same group, the quantity of water passing was in- tended to be the same. Precise uniformity in the quantity was not essential for the attainment of the object in view, and as such uniformity would have required much time to bring about, it was not attempted. The width of the flume remained constant ; the depth of water in the flume depended upon the depth on the weir, which was determined by the quantity of water flowing, and which was, as before stated, nearly constant. We have then in each group seven experiments, in which the width of the flume, the depth of the water, the quantity of water passing, and the mean velocity of the water, are very nearly constant. The only ma- terial variation is in the length of the immersed part of the tube. For instance, in the first seven experiments, the length of the immersed part of the tube (column 14) varied from 9.482 feet to 8.530 feet, the depth of water in the flume (column 13) in the same experiments remaining nearly constant. 224. Experiments 1 to 63 are all with the wide flume, figures 1 and 2, plate XV.; the minute variations in the width, given in column 12, arise from the measures having been taken several times during the course ; and the same remark applies to the length of the weir, given in column 4. Experiments 64 to 105 are all with the narrow flume, figures 3 and 4, plate XV. 225. Table XXII will be understood from the headings of the several columns, together with what has been said previously, without much further ex- planation. The mean observed depth of water on the weir is given in column 8. As explained above, this observed depth is subject to several corrections, which it has not been thought necessary to give in detail in the table. It may be well, however, to indicate them for one of the experiments, say the first, in which they are as follows : Mean observed depth on the weir, 1.8778 feet. Correction for the difference in the observed depth, when the lower orifice of the hook gauge box pipe is at a point 6 feet from the plane of the weir, instead of 0.52 feet, as in the exper- iment, . 0.0021 1.8757 " Correction for the velocity of the water approaching the weir. See section 153, -)_ 0.0140 " 1.8897 IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 191 Correction for the obstruction to the flow over the weir, by the apron, trough, &c ' . 0.0058 feet. Corrected depth on the weir, as given in column 8 1.8839 " The correction for the leakage into the flume is required only in the experi- ments with the narrow flume, as is previously explained. FORMULA OF CORRECTION FOR GAUGINGS MADE WITH LOADED POLES OR TUBES. 226. The absolute difference in the quantities deduced from the weir measure- ment and from the mean velocity of the tubes is given in column 18, Table XXII, and the proportional difference of the same quantities is given in column 19. The quantity deduced from the weir measurement, given in column 11, is taken as the true quantity passing the flume. By reference to columns 15 and 19 it will be seen, that, when the tube extends nearly to the bottom of the flume, the differences are small, generally less than one per cent. In each group there is one exper- iment in which the tube does not extend to the bottom within about one foot; in these the differences in the quantities obtained by the two methods are greater, as might be expected ; in these, however, the differences are, generally, less than three per cent; in one experiment only (43) does it exceed four per cent. 227. It was anticipated, when the programme of the experiments was arranged, that the differences would be found to vary with the velocity of the water in the flume. If any such relation exists, it should be indicated by the mean values of the proportional differences in the several groups. Table XXIII., arranged according to velocities, and for each width of flume separately, gives these mean values. 192 A METHOD OF GAUGING THE FLOW OF WATER TABLE XXIII. Numbers of the Mean Telocity of the experiments Width of the flume : the tubes : in feet per Mean proportional constituting the in feet. difference. second. group. 43 to 49 26.75 0.499 - 0.0262 36 " 42 (t 1.136 - 0.0079 22 " 28 U 1.476 - 0.0074 29 35 ft 1.756 - 0.0044 15 21 a 1.983 - 0.0024 57 " 63 u 2.008 - 0.0043 8 " 14 Si 2.481 - 0.0079 1 " 7 u 2.670 f- 0.0097 50 " 56 It 2.690 - 0.0092 Means, 1.855 + 0.0088 64 to 70 13.37 1.371 - 0.0144 71 " 77 tl 2.018 - 0.0080 78 " 84 . 2.627 - 0.0038 85 " 91 II 3.524 - 0.0036 92 98 u 4.438 - 0.0016 99 " 105 a 5.184 - 0.0115 Means, 3.194 + 0.0071 Mean proportional difference for all the experiments. 4- 0.0082 228. By the preceding table it does not appear that the difference depends on the velocity. In both the wide and narrow flume, however, the difference is greatest when the velocity is least, although the velocities in the two cases are very different. Whether this is accidental or depends on some principle is a question I have no means of answering. 229. In the wide flume, the mean proportional difference is 0.0088, or about J of one per cent. In the narrow flume, the mean proportional difference is 0.0071, or a little less than f of one per cent. Thus, on comparing the whole of the experiments in the two flumes, given in table XXIII., it appears that the pro- portional differences vary only 0.00.17, or about ^ of one per cent. 230. The proportional differences given in column 19 are very irregular, and of the nature of residual quantities, depending upon errors of observation, the in- stability of the currents and the numerous causes tending to produce differences in the results, derived from the mean velocity of the tubes and the weir measure- ment. I am unable to assign to each cause its legitimate effect ; all I can do is to find an empirical formula that will represent, with sufficient accuracy for practical purposes, the difference in the usual cases which occur in practice. In arranging the programme of experiments, it was designed to cover the usual range IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 193 of velocities and proportional depths of immersion of the tubes, and any application of the empirical formula founded on them will generally be free from the objection of being outside the range of the experiments on which it is founded. 23] . We have to seek for an expression or formula which will enable us to deduce the real quantity from that deduced from the velocity of the tubes, by assuming that they indicate the mean velocity of the water for the whole depth of the part of the stream in which they float. In the absence of experimental data it would be rational to assume that the formula of correction is a function of three quantities, viz. : 1. The width of the flume relatively to its depth. 2. The mean velocity of the current. 3. The depth to which the tube is immersed, relatively to the whole depth of the stream. 1. The sides of the flume must, of course, cause a retardation of the current similar to that produced by the bottom; by reference to the several diagrams on plate XVII. it will be seen that the velocity of the tubes is diminished near the sides. It is not practicable to measure the velocity, by means of the tubes, quite close to the sides, but in drawing the curves, representing the mean velocities of the tubes, it will be seen that the retarding effects of the sides are attempted to be allowed for. We have experiments only on flumes of two widths, one being twice the width of the other; the depths being nearly the same, the relative width in one will be about twice that in the other. By reference to table XXIII. it will be seen that in the wide flume the mean proportional difference is -|- 0.0088, the mean velocity being 1.855 feet per second. In the narrow flume, if we take the whole of the experiments, the mean velocity is much greater than in the exper- iments in the wide flume. If, however, we take the three first groups, which in- clude experiments No. 64 to 84, we have for the mean velocity 2.005 feet per second, and a mean proportional difference of -j- 0.0087. Comparing the results from the two flumes, it appears that by doubling the relative width, other cir- cumstances remaining nearly the same, the proportional difference has not been sensibly affected. We may, therefore, conclude that the relative width of the flume need not enter into the formula of correction, care being taken, in drawing the curves, representing the mean velocities in different parts of the width of the flume, to inflect the curve downwards at the sides, as has been done in reducing these experiments. 2. As depending on the mean velocity of the current. It results, from Navier's 25 194 A METHOD OF GAUGING THE FLOW OF WATER investigation, that, so far as it depends on the excess of the velocity of the tube above that of the water in which it is floating, the absolute difference is pro- portional to the velocity (art. 196); the proportional difference, which we are con- sidering, must therefore be constant, or independent of the velocity. By reference to table XXIII. it will be seen that the mean proportional differences in the several groups of experiments in each flume appear to have two maxima and one minimum ; the experiments in which the velocities are least and greatest having the greatest proportional difference, and some intermediate velocity having the least proportional difference. Comparing the whole of the experiments in both flumes, we find in the group having the least velocity the largest proportional difference ; but this result having, apparently, no connection with the results deduced from the great mass of the experiments, must, until explained, be considered anomalous. Comparing the results deduced from all the experiments, excepting those comprised in the first group, no connection can be traced between the velocities and the mean proportional differences. We must therefore conclude, that the correction is independent of the velocity. 3. As depending on the depth to which the tube is immersed, relatively to the whole depth of the stream. It is evident that, in the cases in which the nat- ural scale of velocities at different depths has become established, the difference in question must depend mainly upon this circumstance, and its magnitude may be computed by the formulas of Humphrey and Abbot together with those of Navier and Frizell, as has been previously shown (arts. 193, 196); but in these experiments, and in the cases which usually occur in practice, this natural relation is not estab- lished, and consequently these formulas do not apply; and there appears to be no alternative but to determine an empirical formula from the experiments, which will serve for practical purposes. 232. In determining the formula of correction, it is assumed that the pro- portional difference depends only upon the relative depth to which the tube is immersed. Instead of using this relative depth, it has been found more convenient to use a quantity depending directly upon it, viz. the difference between the depth of the water in the flume and the depth to which the tube is immersed, divided by the depth of the water in the flume; this we call D, and its value in each experiment in table XXII. is given in column 15. For the purpose of more convenient graphic representation, the data given in table XXII. are reduced, by taking means of the values of D within certain limits, and also of the corresponding values of the proportional differences ^"~ given in column 19. These means, arranged according to the values of D, are given separately for each width of flume, in table XXIV. IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 195 TABLE XXIV. Number of Greatest and least values of D Me in value of the Width of the experiments in the experiments from which proportional difference. flume. from which the the means are deduced. Mean value of D. means are Q-Q< Feet. deduced. Greatest. Least. Q" 26.746 9 0.007 0.004 0.0054 + 0.00129 (( 12 0.012 0.008 0.0107 - 0.00027 ff 8 0.017 0.015 0.0165 - 0.00400 H 7 0.023 0.019 0.0211 - 0.00251 u 9 0.034 0.029 0.0318 - 0.00856 U 9 0.055 0.041 0.0446 - 0.01577 It 9 0.129 0.104 0.1118 + 0.03033 13.372 6 0.007 0.004 0.0058 0.00503 tt 5 0.012 0.009 0.0114 0.00040 u 4 0.017 0.013 0.0152 0.00080 u 9 0.024 0.018 0.0213 + 0.00616 u 5 0.035 0.030 0.0336 -j- 0.00944 tt 7 0.048 0.036 0.0440 + 0.01269 tt 6 0.120 0.107 0.1132 4- 0.02420 233. In the diagram figure 2, plate XVIII. the abscissas represent the mean values of D in the preceding table and the ordinates the corresponding mean Qn Qi values of the proportional differences -' Q,, , the double circles representing the experiments with the wide flume, and the single circles the experiments with the narrow flume. As will be seen, the parabolic curve A B represents, nearly, the mean result of all the experiments. Calling the ordinates of jthe curve (7, and the abscissas D, its equation is v/Z> 0.1) (1.) C is the proportional difference to be deducted from the quantity directly deduced from the mean velocity of the tubes; Q" being the quantity thus deduced and Q'" being the corrected quantity, we have substituting the value of C in (1.), we have Q'" =.(!_. 0.116 (yTD 0.1)) Q". Table XXIX. gives the values of the coefficient 1 0.116 (y/~^ 0.1) (2.) 196 A METHOD OF GAUGING THE FLOW OF WATER for the values of D from 0.000 to 0.100, together with the logarithms of the same. 234. Column 20, table XXII, gives the values of Q" f t>y formula (2.), and column 21 the proportional differences between the values of Q'" and the quantities as measured at the weir. Taking the whole of the experiments together, it will be seen that the mean proportional difference, taken algebraically, is -f- 0.0004, or, dis- regarding the signs, 0.0071; the latter quantity is about f of one per cent, and is the mean error or discrepancy between the measurement by the weir and the cor- rected measurement in the flume. It will be observed that the largest discrepancies are in the group of experiments numbered from 43 to 49, in which the velocity was very slow; in one of these experiments, viz. No. 46, the corrected flume measurement is about -^ greater than the weir measurement. In experiment No. 47 the corrected flume measurement is about -fa greater than the weir measurement. In experiment No. 16 the corrected flume measurement is about T J less than the weir measurement. In all the other experiments, the difference is less than -fa, or two per cent 198 TABLE MISCELLANEOUS EXPERIMENTS AT THE 1 3 3 Weir Measurement. Flume Tempera- 4 5 O 7 8 9 10 11 ia 13 14 15 degrees of Fahrenheit's Quant it; of wate Corrects quantit between the depth Date. thermometer Total Observe* Observec Mean Corrects passing over the Correc- passing r helium Mean Length of water in the No. of length of the weirs. depth o: water on the depth of water on the observed depth of water depth ol water on the weirs, com- puted tion for the leak- age into deduce* from the weir Mean width ol the flume depth o water in the r of the im- mersed flume an< the length of the the 1856. of the Westerly weir. Easterly weir. on the weirs. weirs. by the formula the flume measure 1 1 iciit. flume. part of the immersed part of phere in shade. of th water L H 3.31 Q = !(-0.08n/ OH*' Q' tube. the tube, divided by the deptt Feet. Feet. Feet. Feet. Feet. Cubic ft. Cubic ft. Cubic ft Feet. Feet. Feet. of water in the flume. per sec. per sec. per sec. D 106 Oct. 22 A.M. 53.0 53.0 80.014 0.5509 0.5508 0.5508 0.5518 108.58 0.00 108.58 26.745 8.177 8.070 0.013 107 it it ti 57.5 53.0 u 0.5501 0.5501 0.5501 0.5511 108.38 0.00 108.38 tt 8.173 8.070 0.013 108 it 11 It 60.5 53.0 ft 0.5504 0.5505 0.5504 0.5514 108.46 0.00 108.46 tt 8.175 8.070 0.013 109 110 111 NOV. 13 A.M. ti if it U (( it 32.0 40.0 80.012 (t u 1.1583 1.1661 1.1634 1.1358 1.1435 1.1409 1.1470 1.1548 1.1521 1.1508 1.1587 1.1560 326.23 329.59 328.44 0.27 0.27 0.27 325.96 329.32 328.17 13.372 ft tt 8.760 8.766 8.765 8.650 8.650 8.650 0.013 0.013 0.013 112 (t ff U 36.0 40.0 H 1.1604 1.1373 1.1488 1.1527 327.04 0.27 326.77 tt 8.760 8.650 0.013 113 Oct. 23 P.M. 50.5 53.0 80.014 1.8580 1.8362 1.8471 1.8534 664.91 0.00 664.91 26.745 9.529 9.430 0.010 114 " 30 A.M. tt 1.3612 1.3509 1.3560 1.3616 419.51 0.00 419.51 26.746 9.003 8.600 0.045 115 it it ti 67.0 47.0 tt 1.3685 1.3588 1.3636 1.3692 423.02 0.00 423.02 tt 9.016 8.900 0.013 116 " " P.M. 67.0 47.0 ti 1.3391 1.3350 1.3370 1.3424 410.70 0.00 410.70 ft 9.015 8.000 0.113 117 it U it 66.0 47.0 tt 1.3291 1.3255 1.3273 1.3327 406.28 0.00 406.28 (f 9.010 8.850 0.018 118 tt it it 64.0 47.0 tt 1.2405 1.2221 1.2313 1.2358 362.92 0.00 362.92 ff 8.882 7.852 0.116 119 " 31 0.9921 0.9857 0.9889 0.9917 261.15 0.00 261.15 ft 8.628 8.220 0.047 120 121 Oct. 22 P.M. " 23 " 54.0 48.0 53.0 53.0 80.014 it 1.8994 1.8408 1.8539 1.8144 1.8766 1.8276 1.8826 1.8339 680.61 654.50 0.00 0.00 680.61 654.50 26.745 tt 9.526 9.500 9.429 9.430 0.010 0.007 122 it ti tt 45.5 53.0 ti 1.8244 1.8001 1.8122 1.8186 646.36 0.00 646.36 tt 9.479 9.430 0.005 123 124 Nov. 4 A.M. ti (I ti 52.0 53.0 47.0 47.0 80.014 tt 1.8837 1.8682 1.8450 1.8321 .8643 .8501 1.8704 1.8563 674.03 666.47 0.00 0.00 674.03 666.47 26.746 ft 9.527 9.518 9.430 9.430 0.010 0.009 125 tt tt tt 53.0 47.0 tt 1.8588 .8230 .8409 1.8472 661.60 0.00 661.60 " tt 9.509 9.431 0.008 126 (t if It 54.0 47.0 tt 1.8550 .8193 .8371 1.8433 659.51 0.00 659.51 it 9.506 9.432 0.008 127 tt tt {{ 54.0 47.0 tt 1.8378 .8023 .8200 .8263 650.46 0.00 650.46 ft 9.490 9.434 0.006 128 (( tt tt 56.0 47.0 u 1.8217 .7888 .8052 .8116 642.65 0.00 642.65 tt 9.476 9.436 0.004 129 tt tt (t 56.0 47.0 t 1.8692 .8328 .8510 .8572 666.95 0.00 666.95 ft 9.529 9.437 0.010 130 " " P.M. 56.0 47.0 t 1.8731 .8271 .8501 .8563 666.47 0.00 666.47 tt 9.555 9.439 0.012 131 tt tt tt 58.0 47.0 t 1.8670 1.8233 .8451 .8514 663.85 0.00 663.85 ft 9.551 9.445 0.011 132 tt tt ti 58.0 47.0 1.8603 1.8166 .8384 .8447 660.26 0.00 660.26 tt 9.549 9.440 0.011 133 it tt ti 58.0 47.0 t 1.8376 1.7950 1.8163 1.8226 648.49 0.00 648.49 tt 9.529 9.430 0.010 134 ft ft it 58.5 47.0 t 1.8780 1.8435 1.8607 1.8669 672.16 0.00 672.16 tt 9.562 9.430 0.014 135 136 137 138 139 140 Nov. 5 A.M. tt tt tt tt ff ft tl (( ft ft ft ft tt tt ff 39.0 40.0 40.0 41.0 48.0 48.0 48.0 48.0 80.014 tt tt tt tt tt 1.4879 1.4820 1.4823 1.4832 1.4855 1.4794 1.4695 1.4631 1.4659 1.4679 1.4724 1.4691 1.4787 1.4725 1.4741 1.4755 1.4789 1.4740 1.4852 1.4790 1.4806 1.4820 1.4854 1.4805 477.68 474.70 475.46 476.14 477.77 475.42 0.00 0.00 0.00 o.oo i 0.00 0.00 477.68 474.70 475.46 476.14 477.77 475.42 26.746 tt tt tt tt tt 9.164 9.159 9.168 9.175 9.176 9.174 9.040 9.040 9.040 9.040 9.040 9.040 0.014 0.013 0.014 0.015 0.015 0.015 XXV. TREMONT WEIR AND MEASURING FLUME. 199 Measurement. 18 19 ao SI 23 33 1 f\ 1 "Y Difference between the Proportion- al differ- Quanti- ty of Proportion- al differ- Remarks on the Force and Direction A.\J 1 1 quantity of ence, or water ence of the of the Wind at the Flume, during Quanti- water pass- the dilTer- passing corrected the Experiments. Mean ty of water ing the flume ence in the the quantity as No. velocity of the passing the mean column, deduced in the of tubes the velocity of divided by from the flume given the through- out the flume, deduced the tubes, and the the quan- tity de- mean velocity in the preceding General Remarks. flume from the quantity duced from of the column, Exp. by the mean velocity deduced from the weir the flume measure- tubes, correct and the weir meas- Force. Direction. e> Ul. of the measure- ment. by the urement. tubes. f\n ment. Q"-Q' formula Q"'-Q r Feet V Cubic ft. Cubic ft. ~Q" ~ Q'" = U.H6(v'l> Q' . -0.1)). per sec. per sec. per sec. * ^ 106 107 108 0.503 0.502 0.504 110.09 109.62 110.10 + 1.51 + 1.24 + 1.64 +0.0137 +0.0113 +0.0149 109.91 109.44 109.92 +(P. Ill >> +0.0098 +0.0135 Calm. u u 1 These three experiments were mode under circum- stances as nearly identical OH practicable, for the pur- pose of testing the degree of uniformity attained in the results. The greatest variation in the proportional differences in column 21, from the mean, is 0.0020, or TJoff* Reduced copies of the diagrams constructed for these experiments are given on plate XVII. 109 2.7.".!' 322.34 3.62 0.0112 321.81 0.0127 Calm. These four experiments were also made under cir- cumstances as nearly identical as practicable, for the 110 111 2.811 2.820 329.48 330.49 + 0.16 + 2.32 +0.0005 +0.0070 328.94 329.95 0.0012 +0.0054 Very gentle. It It Down stream. (I (1 same purpose as tho preceding. The greatest variation in the proportional differences in column 21, from the mean, is 0.0107 = -gV- 112 2.796 327.54 + 0.77 +0.0024 327.01 +0.0007 Calm. Reduced copies of the diagrams constructed for these experiments we given on plate XVII. 113 2.542 647.88 17.03 0.0263 647.88 0.0256 I Very strong, j variable. J ( Irreg., gen. | down stream. These seven experiments were made when the wind was blowing with considerable force in the direction 114 115 .771 .725 426.55 416.02 + 7.04 7.00 +0.0165 0.0168 421.00 415.34 +0.0036 0.0182 Brisk. u Down stream. u u of the current. It will he seen that the results are less regular than in the preceding seven experiments, and in the experiments in Table XXII., in none of which 116 .759 424.07 --13.37 +0.0315 112.45 +0.0043 u u u was there much wind. The menu proportional differ- ence in column 21, in these seven experiments, is 117 .695 408.53 -- 2.25 +0.0055 406.91 +0.0016 Strong. *( (1 O.MV.M, which would indicate that the wind blowing down stream had a small effect in diminishing the ve- 118 .563 371.37 + 8.45 +0.0228 361.01 0.0053 H (( U locity of the tubes, hut the irregularities are too great 119 .174 270.84 -- 9.69 +0.0358 267.17 +0.0231 ( Brisk, strong ) U U to permit this inference to he drawn. All that can be salelv inferred is, that the wind had no sensible effect ( at times. ) on the velocity of the tubes, except to increase the Irregularities. 120 121 122 2.720 2.494 2.531 693.06 633.68 641.63 +12.45 20.82 4.73 +0.0180 0.0329 0.0074 693.06 634.88 643.81 +0.0183 0.0300 0.0039 Very moderate. i Very strong, j ( variable. J it Up stream. ( Irreg., gen. '( up stream. In these three experiments the wind was blowing generally in the opposite direction to the current. The mean proportional difference in column 21 is 0.0*132, hut the results are too irregular to permit any inference to be drawn, except that the effect of the wind was insensible, except in increasing the irregu- larities. 123 2.595 661.35 12.68 0.0192 661.35 0.0188 Calm. These twelve experiments were miule with an ob- struction placed in the canal about 150 feet above the 124 125 2.611 2.599 664.76 661.00 1.71 0.60 0.0026 0.0009 665.16 661.81 0.0020 +0.0003 Hardly perceptible. Calm. Up stream. flume (K F, plate XV. and art. 201),whtch caused great disturbances in the motion of the water in the flume, every part of it betiig tilled with eddies, both horizon- 126 2.594 659.61 + 0.10 +0.0002 660.41 +0.0014 u tal and vertical. The mcun proportional difference in column 21, disregarding the signs, is 0.0)21. In table 127 128 2.613 2.488 663.27 630.54 +12.81 12.11 +0.0193 0.0192 665.00 633.23 +0.0224 0.0147 u XXII, the corresponding mean is 0.0071. Hence we infer, that the irregularities were greater when the cur- rent was disturbed in the manner described than when 129 2.539 647.09 19.86 0.0307 647.09 0.0298 Very gentle. Up stream. undisturbed, in the ratio of 17 to 10. The mean pro- portional difference in column 21, regarding the signs, 130 2.633 672.98 + 6.51 +0.0097 672.23 +0.0086 K a (t *i is 0.0021, from which, considering the irregularities, all that can be inferred is, that the disturbance of the 131 2.628 671.36 + 7.51 +0.0112 670.98 +0.0107 Hardly perceptible. if u current has no sensible effect on the results, except in increasing the irregularities. 132 2.566 655.29 4.97 0.0076 654.92 0.0081 Calm. In experiments 123, 124, 127, 128, 132, and 133, the 133 2.515 640.98 7.51 0.0117 640.98 0.0116 u tubes were put into the water in regular order, from the left to the right side of the flume, at intervals of 134 2.677 684.64 +12.48 +0.0182 683.18 +0.0164 " about one foot, passing once across in the usual man- ner. In experiments 125, 126, l^y, 130, 13!, and 134, they were put in in the same order, but at .intervals of about four feet, and passing across the flume ibur times, in each experiment, taking different points at each crossing. It was thought possible that the positions of the quick and slow currents might not remain constant throughout an experiment, which, with the ordinary mode of putting in the tubes, might lead to an errone- ous result. Comparing the results obtained by the two methods, and disregarding the signs, tbe mean propor- tional difference in column 21, in the six experiments in which the tubes were put in in the usual manner, is I 0.012!*. In the other six experiments the mean propor- tional difference is 0.0112. The small difference in the results,considering the irregularities, cannot be attrib- uted to the mode of putting in the tubes. Reduced copies of the diagrams constructed for ex- periments 123 and 124 are given in plate XVII. 135 136 2.045 1.947 501.32 477.06 +23.64 + 2.36 +0.0472 +0.0049 500.25 476.28 - -0.0472 - -0.0033 Strong. Down stream. U (t These six experiments were made under similar cir- cumstances to the twelve experiments next preceding) except that there was a high wind blowing in the di- rection of the current. The mean proportional differ- 137 138 139 2.032 1.960 1.923 498.21 480.85 471.86 +22.75 + 4.71 5.91 +0.0457 +0.0098 0.0125 497.15 479.59 470.63 - -0.0456 - -0.0072 0.0149 ( Strong, but I j variable. J " (t U ence in column 21 is +0.0207, which indicates that the joint effect of the disturbance of the current, and the strong wind blowing in the direction of the current, was to increase the velocity of the tubes about two per cent. 140 2.012 493.77 +18.35 +0.0372 492.48 +0.0359 Violent. " " In experiments 135, 136, 130, and 140 the tubes were put into the water in tho usual manner. In experi- ments l->7 and l-'X they were put in as described above in experiments 125, 126, Ac. The mean proportional difference in column 21, in the four experiments in which the tubes were put in in the usual manner, dis- regarding the signs, is 0.0203, and in the other two ex- periments the mean is 0.0264 ; indicating no sensible difference in the magnitude of the irregularities, de- pending on the manner in which the tubes were put into the water. Reduced copies of the diagrams constructed for ex- periments 138, 139, and 140 are given in plate XVII. 200 A METHOD OF GAUGING THE FLOW OF WATER DESCRIPTION OF TABLE XXV. 235. The experiments in this table were made like those in table XXII., and have been reduced in the same manner. The special purposes for which they were made are described in the final column of the table, headed "General Remarks." By referring to the table, it will be seen that the first seven experiments were made for the purpose of testing the degree of uniformity attainable in the results, when the circumstances under which the measurements were made were the same. This is a fundamental question in all kinds and methods of measuring, and is dis- tinct from the errors of observation to which all methods are liable. In geodesic and astronomical methods the difficulties arise principally from the instability of instruments and from atmospheric changes. In measuring the velocity of streams of water, the instability of the currents, mentioned in article 208, appeared to afford a peculiar liability to this trouble, and it was necessary to make special exper- iments to ascertain the magnitude of the irregularities due to it. In the three experiments, numbered 106 to 108, in which the circumstances were as nearly alike as practicable, the extreme variation is about ^{ 7 ; in the next group of four exper- iments, in which the circumstances were also alike, as nearly as practicable, the extreme variation is about -fa ; so far as is known, there was no want of care in the execution of any of these experiments, and the irregularities must be considered as inseparable from the method. In a greater number of trials the extreme variation would probably be greater. We must infer from these seven experiments, that any single measurement is liable to be erroneous to the amount of one per cent, or perhaps rather more ; and in any two experiments the errors may be in opposite directions, in which case they may vary from each other two per cent, or rather more. It is of course very desirable that the method should be free from this liability to error; except by accident, however, the quantity of water used at a manufacturing establishment or flowing in a stream will not be found twice alike. An approximation within one or two per cent of the truth is sufficient for most practical purposes; the errors are as liable to be one way as the other, and by re- peating the measurement several times and taking the mean, the probabilities are that the result will be very nearly as correct as if the method was free from this liability to error in a single measurement. 236. The seven experiments numbered from 113 to 119 were made, when the wind was blowing with considerable force down stream. Taking the mean, it would appear that the effect of the wind was to cause the corrected flume measurements to be about one quarter of one per cent less than the weir measurements. In these IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 201 experiments the length of the immersed parts of the tubes varied from 7.85 feet to 9.43 feet ; the length projecting above the water, in each case, was about 0.33 feet ; taking a mean, about ^ part of the length projected above the surface of the water, and was liable to be acted upon by the wind. The effect of the wind blowing down stream, with a velocity greater than that of the current, must be to give the tube a greater velocity than it would have in a calm or with the wind blowing up stream. By the mean result of the seven experiments the contrary effect would appear to have been produced. By comparing the differences in these seven experiments, given in column 21, with the corresponding differences in table XXII., it will be seen that the irregularities in the results of the measurements were much greater when the wind was blowing strongly than when it was calm, or nearly so. The extreme variation in the seven experiments is nearly five per cent; under these circumstances, it is apparent that, in order to detect with certainty so small a difference as one quarter of one per cent, a much larger number of exper- iments is necessary, and that, with the small number made, the real effects may easily be obscured by the irregularities. 237. In experiments 121 and 122 the wind was very strong, but variable, irregular in direction, but generally up stream; the mean result of the two exper- iments is, that the velocity of the tubes was retarded about g 1 ^ ; but the number of experiments is evidently insufficient to determine it definitely. We may infer from the ten experiments, numbered from 113 to 122, that, although measurements made when the wind is blowing strongly, either up stream or down stream, are subject to greater irregularities than measurements made when there is little or no wind, by making a considerable number of trials, the mean results will vary but little, whether the wind is blowing strongly or not. 238. In the twelve experiments, numbered 123 to 134, there was a great com- motion in the stream caused by an obstruction in the channel above, as is explained in the table. The irregularities are increased, but the mean result is not sensibly affected. In the six experiments numbered 135 to 140 there was a similar agitation in the stream, and also a high wind blowing down stream ; the effect was to increase the irregularities in the results, and the mean velocity of the tubes appears to have been increased about two per cent. APPLICATION OF THE METHOD OF GAUGING STREAMS OF WATER BY MEANS OF LOADED POLES OR TUBES. 239. As previously stated, this method is more generally adopted at Lowell, for gauging large volumes of water, than any other. Six measuring flumes have been 26 201} A METHOD OF GAUGING THE FLOW OF WATER constructed in the canals there ; all made in a similar manner to that described in article 201, and represented in figures 1 and 2, plate XV. Their principal dimen- sions and the quantities of water usually gauged in them are as follows : The Merrimack flume, about 100 feet long and 50 feet wide, intended to gauge about 1,500 cubic feet of water per second. The Appleton flume, about 150 feet long and 50 feet wide, intended to gauge about 1,800 cubic feet of water per second. The Lowell Manufacturing Company's flume, about 150 feet long and 30 feet wide, intended to gauge about 500 cubic feet of water per second. The Middlesex flume, about 150 feet long and 20 feet wide, intended to gauge about 260 cubic feet of water per second. The Prescott flume, about 180 feet long and 66 feet wide, intended to gauge about 2,000 cubic feet of water per second. The Boott flume, about 100 feet long and 42 feet wide, intended to gauge about 800 cubic feet of water per second. The depths of the water in these flumes are various, usually, however, between eight and ten feet; sometimes, when the river is low, the depth is diminished to about six feet. It will be seen that the widths of the flumes are not strictly in proportion to the quantities of water intended to be gauged in them ; the widths and depths have usually been determined by the dimensions of the canals in which they are placed. 240. Under the existing arrangements at Lowell, a daily account is usually kept of the excess of water, if any, drawn by each manufacturing company, over and above the quantity to which it is entitled under its lease. In ordinary times this is arrived at with sufficient exactness by means of occasional measurements, but when the flow of water in the river is too small to supply the wants of all, it is necessary to make frequent measurements of the -quantity of water drawn by those who habitually draw an excess. In the latter case the usual course of proceeding is this. A gauging party, consisting of one or more engineers and a sufficient number of assistants, is assigned to each flume where measurements are required. Arrangements are made so t that the observations for a single gauge occupy about half an hour. Several gauges are made during the day, the intervals between the times when the observations are made being occupied by the same party in working out the results, which, as soon as obtained, are communicated to the proper local authorities at the manufacturing establishments where the water is drawn. This is done to enable them to adjust the amount of machinery they run, so as to draw only the quantity of water to which they are entitled. If IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 203 they continue to draw an excess after due notice, they ure liable to heavy penal- ties. It is essential to the proper working of these arrangements that the results of the gaugings should be arrived at and communicated as speedily as possible ; with this view, as well as to reduce the expense, engraved diagrams and printed forms and tables have been prepared, and all the apparatus provided and prep- arations made which can in any way facilitate the operation. 241. The 'mode of making the observations for a gauge in a measuring flume is substantially the same as that practised in the experimental flume in the Tremont Canal, and fully described in articles 204 and 205. With the view, how- ever, of reducing the number of assistants required, a stop-watch beating quarter seconds is used instead of a marine chronometer, and the electric telegraph is dispensed with. The observer with the stop-watch takes his position at the upper transit station, and starts the watch when the tube passes it ; he then walks to the lower transit station and stops the watch when the tube passes it. By this method two observers are dispensed with. Another observer notes the depth of the water in the flume, and also records the distances of the tubes from the left side of the flume, which are observed and called by the assistants who put in and take out the tubes. One other assistant is required to carry back the tubes to the up- stream station, making five in the party. 242. Ordinarily, about an hour is occupied in making the observations for a measurement. The following measurement is given in detail as an example of the whole process. The flume in which it was made is situated a short distance below a bend of about ninety degrees in the canal, which produces a great irreg- ularity in the current, the velocity being much greater on the right-hand side of the flume than on the left-hand side ; sometimes there is no sensible current on the left-hand side. It being inconvenient to perform the measurement under such circumstances, the difficulty was remedied by placing an obstruction near the lower end of the bend; the up-stream face of this obstruction was an oblique plane, so placed as to direct a part of the current towards the leftrhand side of the flume. Although far from producing a uniform velocity in all parts of the flume, it removed all the trouble in making the measurement due to the original irregularity. The remaining irregularities in the velocity are indicated by the in- flections of the curved line A B on plate XIX. The mean width of the part of the flume between the upper and lower transits is 41.76 feet. 204 A METHOD OF GAUGING THE FLOW OF WATER. TABLE XXVI. GAUGE OF THE QUANTITY OF WATER PASSING THE BOOTT MEASURING FLUME, MADE BETWEEN 10 HOURS 30 MINUTES AND 11 HOURS 30 MINUTES, A.M., MAY I?TH, 1860. Time during Distance of the tube from the Distance from which the tube left-hand side of the flume during thi> left-hand Length passed from its passage. Bide of the of the the up-stream Mean flume at immeraec transit station Telocity of Depth of which the part of the to the the tube. At the At the water in the tube was put tube. down-stream upper lower flume. into the transit station transit transit Mean. water. a distance of station. station. 70 feet. Feet. Feet. Seconds. Feet per Feet. Feet. Feet. Feet. Second. 0.0 8.40 33.3 2.102 0.3 0.8 0.55 8.510 1.5 tt 31.0 2.258 1.8 1.6 1.70 8.481 3.0 ft 30.2 2.318 3.2 2.1 2.65 8.450 4.5 tt 28.3 2.473 4.4 4.5 4.45 8.470 6.0 u 29.5 2.373 6.2 5.4 5.80 8.445 7.5 u 27.0 2.593 8.2 10.1 9.15 8.438 9.0 tt 26.2 2.672 9.7 10.4 10.05 8.440 10.5 U 25.0 2.800 10.5 8.8 9.65 8.470 12.0 11 25.8 2.713 12.3 10.9 11.60 8.483 13.5 u 25.2 2.778 13.8 15.5 14.65 8.490 15.0 u 25.0 2.800 15.2 18.0 16.60 8.500 16.5 it 29.5 2.373 17.0 20.4 18.70 8.498 18.0 u 27.0 2.593 18.0 17.8 17.90 8.505 19.5 u 28.8 2.431 19.7 19.0 19.35 8.505 21.0 tt 30.7 2.280 21.1 20.9 21.00 8.522 22.5 tt 81.8 2.201 23.4 29.3 26.35 8.533 24.0 tl 33.7 2.077 23.7 22.1 22.90 8.510 25.5 If 33.8 2.071 26.5 29.7 28.10 8.495 27.0 tt 31.0 2.258 27.0 25.2 26.10 8.483 28.5 tt 31.0 2.258 28.6 26.5 27.55 8.495 30.0 tt 29.0 2.414 31.0 34.3 32.65 8.550 31.5 tt 28.0 2.500 32.1 30.0 31.05 8.630 33.0 u 31.0 2.258 32.5 28.1 30.30 8.610 34.5 u 26.2 2.672 34.6 36.7 35.65 8.625 36.0 u 28.8 2.431 . 36.5 35.0 35.75 8.632 37.5 tt 28.5 2.456 37.5 35.5 36.50 8.612 39.0 tt 28.0 2.500 40.1 40.5 40.30 8.578 40.0 tt 28.0 2.500 39.0 39.6 39.30 8.578 41.0 tt 29.2 2.397 41.2 40.6 40.90 8.560 0.0 tt 34.3 2.047 0.5 0.4 0.45 8.471 10.0 tt 26.5 2.642 9.8 8.7 9.25 8.580 20.0 tf 32.2 2.174 20.9 19.9 20.40 8.605 30.0 tt 30.8 2.273 31.5 33.8 32.65 8.635 41.0 tt 30.5 2.295 41.4 40.6 41.00 8.610 Mean 8.5294 Products of the mean Telocities into the widths, for each foot in width, excepting the last product, which is for a width of 0.76 feet ; commencing at the left-hand side of the flume. 2.264 X 0.76 2.073 2.193 2.284 2.359 2.422 2.478 2.529 2.577 2.623 2.666 2.705 2.744 2.776 2.801 2.8H 2.798 2.747 2.648 2.514 2.363 2.249 2.172 2.120 2.098 2.105 2.130 2.163 2.023 2.246 2.289 2.331 2.373 2.418 2.450 2.483 2.510 2.531 2.544 2.540 2.504 2.417 = 1.721 Sum, 101.523 Mean Telocity ) 101-623 _ of the tubes, I " ~ IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 205 243. The mean velocity of the tubes is obtained by means of a diagram, a copy of which, on the same scale as the original, is given in plate XIX. The small circles represent the several observations, the abscissa and ordinate of each being the mean distance from the left-hand side of the flume and the observed velocity of the tube as given in table XXVI. The curved line represents the mean and is drawn by the eye, giving due weight to each observation. The mean velocity is 2.4311 feet per second, and is found by taking a mean of the ordinates of the curve ; the process is given in column A, table XXVI. The mean depth of the water in the flume was ..... 8.5294 feet. The length of the immersed part of the tube was .... 8.4000 " Difference ........... - ...... . 0.1294 Then D (art. 232) = = 0.0152. The mean section of the water-way in the flume was 41.76 X 8.5294 = 356.188 square feet And the quantity of water passing, by the tube measurement, was 356.188 X 2.4311 865.929 cubic feet per second = #". This is to be corrected by formula (2.), art. 233. Substituting for D its value 0.0152, -we have for the coefficient of correction 1 _ 0.116 (v/^> 0.1) = 0.99730 (see table XXIX.) and the corrected quantity Q'" = 0.99730 X 865.929 = 863.59 cubic feet per second. 244. In the preceding example the entire volume of water flowing through the canal was gauged. It often happens that only a portion of the entire flow of the stream is to be gauged, namely, the quantity drawn out of the canal at a single orifice or branch canal. In this case a flume of suitable dimensions is constructed and connected with, the edges of the orifice or the sides and bottom of the branch canal, so that no water can enter the orifice or branch canal except through the measuring flume. A rough preliminary estimate of the quantity should be made by some other method ; this will enable the sectional area of the measuring flume to be determined, so that the velocity in it may be convenient for observation, say between one foot and three feet per second, although it may exceed these limits, in either direction, if the circumstances are such as to require it. It will generally be most convenient to place the flume so that its axis will be parallel, or nearly so, with the axis of the canal. Its 206 A METHOD OF GAUGING THE FLOW OF WATER length will usually be limited by local circumstances and economical considerations; a considerable length in which to measure the velocity of the loaded tubes is desirable, although not essential. If the means for observing the transits and the times of the same are good, a less length is necessary than in cases where the means of observing are less perfect. By means of the electric telegraph and a skilled observer of the chronometer, as in the experiments at the Tremont measur- ing flume (art. 205), an interval of a few seconds between the times of the transits at the upper and lower stations will enable a good gauge to be made. If the observations are made in the less perfect manner practised at the Boott measuring flume, and described in art. 241, a considerably longer interval is necessary in order to attain equally accurate results. There seems to be scarcely any limit to the shortness of the time admissible in the first case, if corresponding care and precautions are adopted in making the observations.* In the second case, it will depend much on the degree of skill of the observer. The method has not been used extensively enough, as yet, to enable a limit to be definitely fixed. A practised observer, with a stop-watch beating quarter seconds, the transit stations being twenty-five feet apart, has been able to observe both transits, when the time between them was ten seconds, and in some cases seven and a half seconds. 245. The distance between the transit stations is only a part of the length required for the flume ; a certain length above the upper transit station is neces- sary to give room for putting the tubes into the water, and to permit them to attain, sensibly, the same velocity as the water before they arrive at the transit station. By reference to art. 195 it will be seen that a tube two inches in diameter, floating twenty feet, attains ff of the velocity of the current. Twenty feet was about the distance the tubes floated before they reached the upper tran- sit station, in the experiments given in table XXII., from which the formula for the correction of flume measurements was determined, and the correction for the very small error, resulting from this distance being insufficient, is implicitly in- cluded in the formula. Twenty feet may therefore be taken as the proper distance, and if circumstances are such as to require a much less distance, the resulting error can be corrected by means of formula (5.), article 194. 246. The same method may be extended to gauging natural watercourses. A favorable place for the purpose should be selected; that is, one free from reverse currents, the bottom smooth, the section uniform for a sufficient distance, * Methods for making and recording observations of time are practised in some astronomical obser- vatories, by means of which the one-hundredth part of a second is estimated; these methods could un- doubtedly be adapted to our purpose if required. IN OPEN CANALS OF UNIFORM RECTANGULAR SECTION. 207 and with as long a reach above, free from bends, great irregularities of section and obstructions, as can be found. Two parallel sections, in planes at right angles to the direction of the current, or nearly so, should be carefully measured, so that the depth at every point may be known. The proper distance between the sections will depend much on the regularity of the channel ; it will usually be desirable that they should be far enough apart to permit the observations for the velocity to be made, without resorting to the use of the electric telegraph; except- ing in very large rivers, a distance of from fifty to one hundred feet, depending on the width, would usually permit this to be done with sufficient accuracy for most purposes, although a greater distance would usually be desirable. The loaded poles or tubes must not touch the bottom while passing from one transit station to the other. It will probably rarely occur that one hundred feet in length of the channel of a river will be found of such regularity that the poles could be immersed to an average depth of six inches from the bottom. By resorting to the more exact mode of observing the transits, the sections might be within twenty feet of each other, or even half that distance if necessary. There would seldom be any difficulty in finding a suitable place for a gauge made in this manner, in any river confined within regular banks. Something could be done, in so short a length, towards removing obstructions and filling up depressions. In making the observations, loaded tubes or poles, of lengths adapted to the different parts of the section, should be provided ; they may be put into and taken out of the water from boats or rafts. Theodolites should be placed in the planes of the sections, on the same bank ; the observer at each should have the key of a break-circuit within his reach, while observing the transit of the floating pole. The observations of the times of the transits may be made in the same manner as at the Tremont measur- ing flume (art. 205). If the sections are very near together, a separate "observer may be necessary for the transit at each station, both, however, using the same chronometer. The distance from fixed points on the bank, at which the floats pass the transits, corresponding to the distances from the left-hand side of the flume, in the flume measurements, can be observed by means of marked cords, stretched across the river, just over the water, and at short distances above and below the sections, and supported from the bottom at intervals, if necessary; or it may be done by means of a system of signals and triangulations. The section of the river not being rectangular, it will usually be most con- venient to divide it into several parts, finding the area of the section, the mean velocity of the poles, computing the quantity and making the correction by formula (2.), article 233, for each part separately. The sum will of course be the gauge of the whole river. 208 A METHOD OF GAUGING THE FLOW OF WATER. The degree of accuracy attainable in gauging a natural watercourse, by this method, will depend entirely upon the regularity and smoothness of the part of the channel selected for the operation, and of the immediate approach to the same. If the bottom is covered with large stones or sunken timber, it will prevent the attainment of much precision. In such cases, if the greatest attainable precision is desired, either the obstructions must be removed or the bed of the channel filled up with some sort of material suitable for the purpose, to the level of the top of the obstructions. In any case, the degree of precision attainable will depend on the degree of approximation in the channel to the regularity and smoothness of the measuring flumes. EXPERIMENTS ON THE FLOW OF WATER THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 247. DANIEL BERNOULLI proved, on the hypothesis that no force is lost, that the fluid in all parts of the same section has the same velocity, and remains in one mass; that the velocity of the discharge from a vessel, by an orifice of small area relatively to that of the vessel, is that due to the head above the orifice from which the fluid is finally discharged, whether such orifice is in the side or bottom of the vessel itself, or at the end of a tube projecting from the side or bottom of the vessel, the sides of the tube being either parallel, con- verging, or diverging.* This being established, it follows, if the conditions of the hypothesis can be complied with, that the velocity of discharge from a simple orifice in a vessel may be increased to any extent by the application of a tube with diverging sides; for the area of the orifice at the end of the tube from which the fluid is finally discharged may be as many times larger than the orifice in the side or bottom of the vessel as we please, and as the same quantity must pass through both orifices in the same time, the velocity through the orifice in the vessel will be as much greater than the velocity through the orifice at the end of the tube as its area is less. 248. The fact that the flow through an orifice could be increased by the application of a diverging tube appears to have been known to the ancient Romans. Experiments have been made upon them in modern times by Gravesande, Bernoulli, Venturi, and Eytelwein, and perhaps others. And experiments on the discharge of air between two discs, which afford an aperture similar in effect to a diverging tube, have been made by Thomas Hopkins.! Most of our exper- imental knowledge on the flow of water through diverging tubes is due to Venturi, whose experiments were made at Modena about the year 1791, and published in * Hydrodynamica. Strasburg, 1738. t Memoirs of the Literary and Philosophical Society of Manchester. Vol. V., Second Series. Lon don, 1831. 27 210 EXPERIMENTS ON THE FLOW OF WATER Paris in 1797, under the title, Recherches experimentales sur le Principe de la Com- munication laterale du Mouvement dans Us Fluids* Venturi experimented on many forms of diverging tubes; in pipes of regular form the maximum increase of velocity was obtained with a conical tube in which the sides diverged from each other at an angle of 4 27'; this tube was applied to a mouth-piece having nearly the form of the contracted vein; a certain volume of water under a constant head was discharged through the mouth-piece alone in forty-two seconds; when the diverging tube was applied to the mouth-piece, the same quantity of water was discharged, under the same head, in twenty-one seconds; increasing the velocity through the mouth-piece in the ratio of 1 to 2. In a similar tube of greater length the water did not fill the tube throughout its whole length unless a prominence was made on the inside of the tube, at the bottom, which caused the water to fly upward and fill the down-stream end of the tube; with this tube, the same volume of water was discharged in nineteen seconds, increasing the discharge through the mouth-piece in the ratio of 1 to 2.21. Eytelwein made some similar experiments with a mouth-piece and a tube whose sides diverged at an angle of 5 9'. He found that the application of the tube to the mouth-piece increased the velocity through the latter in the ratio of 1 to 1.69. 249. According to Bernoulli's theory, in Venturi's experiment, last above quoted, the velocity through the smallest section of the mouth-piece should be in- creased by the diverging tube, in the ratio of 1 to 3.03. In Eytelwein's exper- iment the increase should be in the ratio of 1 to 3.21. In both these experiments the water in the tube undoubtedly remained in unbroken masses. There must, con- sequently, have been considerable losses of force. The increased flow appears to be due to what is termed by Venturi the lateral communication of motion in fluids, and to the pressure of the atmosphere. According to the principle of Venturi, a column of water flowing through a mass of water at rest tends to communicate a portion of its velocity to the mass, and to cause it to move along with it; and if the column of water is moving in a pipe a little larger than itself, it will communicate motion to the entire shell of water surrounding it. If the water is flowing through a conical tube whose sides diverge at a small angle, the section of the pipe is continually enlarging by insensible degrees; but by the principle of Venturi the stream must fill each successive section, and the mean velocity -I * See a translation of Venturi's work, in Nicholson's Journal of Natural Philosophy, Vol. III. London, 1802. Also, in Tracts on Hydraulics, by Thomas Tredgold, 2d Edition. London, 1836. THROUGH SUBMERUED ORIFICES AND DIVERGING TUBES. 211 must diminish in the ratios that the areas of the sections increase. The pressure of the atmosphere on the surface of the water in the vessel and on the orifice from which the water escapes may for this purpose be called the same, and equal to a column of water thirty-four feet high. Supposing the mass of water flowing through the pipe to be divided into very thin slices, by planes at right angles to the direction of the current ; from its inertia, each slice will tend to retain its velocity, but on account of the enlarging sections it cannot do this, but tends to separate itself from the slice immediately following it; this is prevented by the pressure of the atmosphere, and the effect is to balance a portion of the pressure of the atmosphere on its down-stream side ; the entire pressure of the atmosphere remains on the up-stream side of the slice, and the difference between the effective pressures on the up-stream and down-stream sides accelerates the motion of the slice. All the slices are acted on in a similar manner, and the increased discharge is due to the sum of the actions upon them. In the experiment above quoted of Venturi, with a pipe of regular form, the discharge through the orifice took place under a head of 2.887 feet; the head being as the square of the velocity, the equivalent head, under which the discharge took place with the diverging tube, was 2.887 X 2 2 = 11.548 feet, which exceeds the actual head of water in the experiment by 8.661 feet, which is the portion of the total pressure of the atmosphere on the surface of the water in the reservoir ren- dered active in that experiment. 250. Venturi found no increased discharge by increasing the length of his diverging tube beyond 1.096 feet, on account of the water not filling the whole section of the part of the tube added beyond that length. This difficulty, however, can be obviated by submerging the diverging tube ; for in that case it must remain full of water, whatever may be its length or the angle of divergency of its sides. In these experiments the tubes were submerged, which distinguishes them from any previously recorded, and greater effects were produced. The diffuser applied by Mr. Boyden to turbine water-wheels, to increase their efficiency (art. 12), acts on the same principle as the diverging tube ; this apparatus has been extensively applied in Lowell, and it has thus become a matter of great interest to ascertain to what extent a conical diverging tube, discharging under water, could be made to increase the discharge through a simple orifice. For this purpose the following experiments were made. 212 EXPERIMENTS ON THE FLOW OF WATER DESCRIPTION OF THE APPARATUS. 251. The tube used in these experiments is represented by figures 1, 2, 3, and 4, plate XXI. It is composed of cast iron and is made in five pieces, A, S, C, D, and E, which when screwed together, as represented in figures 1 and 2, form a com- pound tube, consisting of a mouth-piece of a form to avoid contraction, and a diverg- ing tube, in which the diameter increases from about 0.1 foot, at its junction 6 with the mouth-piece, to about 0.4 foot at /. The part of the mouth-piece between a and g is formed by the revolution of a common cycloid about the axis of the tube ; from a to & it is cylindrical. The interior of the parts C, D, E are frustums of a cone; a portion of the part B is also a frustum of the same cone ; but, to avoid any angle in passing from the cylinder a 6 to the frustum of the cone, a portion of the part B is formed by the revolution of an arc of a circle of about 22.69 feet radius, the sides of the cylinder a 6 and of the cone both being tangent to this arc. The parts of the compound tube being screwed together could be readily taken apart and the mouth-piece used by itself, or with one or more of the conical parts attached. The interior of the mouth-piece and diverging tubes were first turned separately, they were then screwed together and ground on a mandril with emery until they became quite smooth, without, however, having a bright polish. This mode of finishing insured the smallest possible degree of irregularity at the junctions of the several parts. 252. For the purpose of making the experiments, the compound tube was mounted in a cistern (figures 1, 2, and 3, plate XX.) constructed for the purpose. The cistern was made of white-pine wood, very strongly framed, and supported on brick piers, which were built up several feet in height from a solid foundation The cistern consists of three compartments; the upper compartment, E, is the reservoir supplying the mouth-piece M, and the diverging tube attached to it. F, the middle compartment, receives the water discharged through the tube. G is the lower compartment, in the end of which is placed the weir, W, at which the quantity of water discharged was gauged. The supply of water for the experiments was obtained from the main pipes laid down by the manufacturing companies at Lowell for conveying water from an elevated reservoir to their several establishments mainly for the purpose of extin- guishing fire. For these experiments, it was important that the supply of water flowing into the reservoir E should be as nearly uniform as possible, but the effec- tive pressure in the main pipes was subject to some irregularity, which of course THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 213 caused a corresponding irregularity in the discharge from the orifice through which the supply of water was drawn. To eliminate this source of irregularity, the water was first drawn into the cistern I, figures 2 and 3, plate XX., in con- siderably greater volume than was required to be admitted into the reservoir E ; the excess passed over a weir in the side of the cistern /, and from thence was discharged through the waste-pipe K. The supply for the reservoir E was drawn from the cistern / through the pipe H, the quantity being regulated by the cock L. By this arrangement, it will be seen that, so long as the water was admitted into the cistern I in excess of that admitted into the reservoir E, the head acting on the cock L must have been subject to only very small variations, and con- sequently the discharge through a constant orifice in the cock L must have been very nearly uniform. It was important that the water in the part of the reservoir E, near the side containing the mouth-piece, should be as nearly quiescent as pos- sible. The water was admitted under a head of about 18 feet, which necessarily produced a great commotion in the part of the reservoir where it entered, and to prevent this from extending to the side containing the mouth-piece, it was made to pass through six diaphragms, R, R, R, &c., figures 1 and 2, plate XX. The first two diaphragms were made of boards, about one inch thick, containing numerous holes about half an inch in diameter, as shown in figure 4; the other four diaphragms were of strainer-cloth, placed about two inches apart and stretched tightly in a frame. The strainer-cloth used was the well-known fabric sold under that name, made of flax or hemp, with about twenty threads to an inch in both warp and filling, the width of the spaces between the threads being from two to three times the thickness of the thread. The effect of these diaphragms was to prevent any sensible commotion in the part of the reservoir between the lower diaphragm and the side containing the mouth-piece. The part of the reservoir E, between the down-stream diaphragm and the mouth-piece, was about 2.34 feet long in the direction of the current, 3 feet wide, and 4.5 feet deep. The division F was about 6.75 feet long, 3 feet wide, and 3.35 feet deep; the water passed from this division to the division G through the diaphragm N, similar to the wooden diaphragms in division E, above described; and also through the diaphragm P, consisting of a single thickness of strainer- cloth. The dimensions of the part of the reservoir G, between the diaphragm P and the end containing the weir W, is about 3.6 feet long in the direction of the current, 3 feet wide, and 3.20 feet deep. The disturbance in the division F was slight, and as the apparatus was first designed, the weir was placed in the partition N~, but on trial the agitation was found to be too great to admit of an un- exceptionable gauge at the weir; the division G was then added, which, with the diaphragms, removed all difficulty from this cause. 214 EXPERIMENTS ON THE FLOW OF WATER 253. A weir was adopted to gauge the quantity of water passing through the tube, in preference to any other kind of orifice, because it admitted of greater variations in the quantity of water discharged, with any admissible variation in the height of the water in the reservoir in the side of which it is placed ; and by adopting a weir of the same dimensions and form as that used by Poncelet and Lesbros (art. 161), the quantity could be computed with great precision. The weir W, figures 1, 2, and 6, plate XX., is represented on a larger scale by figures 5, 6, and 7, plate XXL, and a section of the crest of the weir is given, full size, in figure 8, plate XXI. The crest and sides of the weir were made of plates of cast iron, planed and finished with great care, the up-stream edges presented to the current having sharp corners, or as nearly so as could be made with that metal. The only material variation from the weir used by Poncelet and Lesbros is in the thickness of the crest, which in their weir was an edge, whereas in our weir it had a thickness of about 0.02 inch ; this variation was made to enable the zero points of the several gauges, used for measuring the heights of the water in the different compartments of the apparatus, to be made in a particular manner, which will be described hereafter. This difference in the thickness of the crest of the weir could have affected the accuracy of the gauge in only a few of the experiments, namely, those in which the depth on the weir was less than 0.05 foot, as at this depth and all greater depths it was observed that the contraction was complete ; that is to say, at this depth the stream passing over the weir touched the orifice only at the up-stream edge, as repre- sented in figure 3, plate XFIII., and the flow was the same as if the crest of the weir had no sensible thickness. With depths on the weir less than 0.05 foot, the stream of water was in contact with the whole width of the crest of the weir ; which, if it had any sensible effect, would tend to increase the discharge, with the same depth on the weir, in consequence of an action similar to that produced by a short additional tube attached to the down-stream side of an orifice in a thin plate. The length of the curved part of the mouth-piece A, figure 2, plate XXL, measured on the axis a g, is 1.00 foot The length of the cylindrical part of the mouth-piece, measured on the axis a &, is 0.10 " The effective lengths of the parts S, C, D, and E, of the diverging tube, are each 1.00 " The diameter of the circle generating the semi-cycloid of the mouth-piece is . 0.635 THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 215 The diameters of the several parts of the mouth-piece and diverging tube are given in column 15, table XXVII. The angle of divergence of the sides of the conical part of the com- pound tube is , 5" 1'. 254. The elevations of the surface of the water in the several compartments of the cistern were measured by means of the hook gauges represented by figures 9, 10, and 11, plate XXL, and described in articles 45 and 143. They were placed in the hook gauge boxes A, B, C, D, figures 1 and 2, plate XX. Com- munication was established between the several hook gauge boxes and the cor- responding compartments of the cistern by means of the orifices 0, figures 1, 2, 5, and 6. The orifices affording communication with the compartments F and G were 0.10 foot in diameter ; the orifice affording communication with the compart- ment E was about five times as large; oscillations in the elevation of the surface being anticipated in this compartment, the amplitude of which it was desirable to measure. There is reason to think that the flow through a diverging tube is to a certain extent in a condition of "unstable equilibrium. In Venturi's exper- iments, the water discharging into the air from diverging tubes was observed to have great irregularity of motion, " and even eddies within the tube ; whence the jet comes forth by leaps, and with irregular scattering." * These irregularities are undoubtedly due, in part at least, to an unstable equilibrium, and there must be a corresponding irregularity in the exhausting power of the diverging tube, which would be indicated, in our experiments, by oscillations in the elevation of the surface of the water in compartment E, which would rise and fall as the exhausting power of the tube was less or greater. The elevations of the surface of the water in all the compartments is reckoned from the top of the weir. When no water was admitted to the reser- voir E, the water in all the divisions of the cistern would fall to the level of the crest of the weir. The comparison between the zero points of the several hook gauges and the crest of the weir was made in the manner described in article 143. Two ten-pointed instruments (figure 14, plate XXI.), of slightly dif- ferent dimensions, were used, which furnished independent results, a mean of which was taken. They were made of steel and magnetized, which enabled them to maintain their positions when placed on the crest of the weir. Small variations in the apparatus were expected to occur, resulting from changes of temperature and in the hygrometric condition of the wood of which the cistern was con- * Tracts on Hydraulics. 216 EXPERIMENTS ON THE FLOW OF WATER structed; comparisons were accordingly made on each day that experiments were made ; the results are given in the following table : Date. Corrections to be applied to the reading of the hook gauges, to give the elevations of the points of the hooks above the top of the weir. 1851. Gauge .1. Gauge B. Gauge C. Gauge D. September 20. 1.5535 1.5490 1.5451 0.3921 " 21 A.M. 1.5519 .5476 1.5439 0.3916 " 21 P.M. 1.5525 .5484 1.5449 0.3920 22. 1.5528 .5487 1.5447 0.3918 " 25. 1.5531 .5487 1.5454 0.3926 26. 1.5535 .5490 1.5458 0.3930 October 7. 1.5541 .5502 1.5474 0.3940 " 10. 1.5541 1.5502 1.5476 0.3938 12. 1.5541 1.5502 1.5476 0.3942 " 16. 1.5536 1.5500 1.5472 0.3935 MODE OF CONDUCTING THE EXPERIMENTS. 255. Water was admitted through the leathern hose Q into the cistern /, figures 2 and 3, plate XX., in excess of the supply required for the experiment. The index of the cock L, figures 2 and 3, was set hi the desired position. When it was supposed that the flow had become permanent throughout all the divis- ions of the cistern, observations of the elevations of the surface of the "water in the several compartments were commenced; they were taken by a separate observer at each hook gauge, every thirty seconds, and were continued until some minutes after the elevation of the surface in the compartments F and G had become stationary, which indicated that a permanent flow had been obtained. The watches used by the several observers were set to indicate the same time, and the time when each observation was made being recorded, a subsequent comparison of the records of the observations made at the several hook gauges enabled those to be selected in which the permanence of the flow was the most perfect. Not less than five, and usually more than ten, successive observations, made at the same times at each hook gauge, were used, from which the mean elevations in the several compartments during the experiment were deduced. 256. Experiments 1 to 18, table XXVII., were made with the mouth-piece A alone. Experiments 19 to 38 were made with the mouth-piece A and the first joint B of the diverging tube. Experiments 39 to 50 were made with the mouth- piece and the two joints B and C of the diverging tube. Experiments 51 to 64 THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 217 were made with the mouth-piece and the three joints B, C. and D of the diverging tube. Experiments 65 to 90 were made with the complete compound tube, repre- sented by figures 1 and 2, plate XXL, and in figures 1 and 2, plate XX. Exper- iment 91 was made with the mouth-piece alone. Experiment 92, with the complete compound tube. Experiments 93 to 101 were made with an orifice in a thin plate represented by figures 12 and 13, plate XXI. This plate is of cast iron 0.042 foot in thickness, but the orifice is chamfered off on the down-stream side, so that the effective thickness of the plate at the orifice is 0.0014 foot, or about one sixtieth of an inch. 257. The mouth-piece, diverging tubes, and plate were all of cast iron ; this metal was adopted instead of brass as being the cheapest, and experience having shown that oxidation of cast iron immersed in the water of Merrimack River pro- ceeds very slowly, and expecting to be able to find, readily, some substance, a coating of which, of imperceptible thickness on the surface of the metal, would entirely prevent it; no such substance was found, however. Drying oils of several kinds were tried, also a mixture of grease and mercury, also collodion, but without satis- factory effect. The plan finally adopted was to keep the interior of the orifices and tubes and the accessible parts of the weir, when not in use, covered with a thick coating of grease. Previous to each session of the experiments this was removed as completely as possible by rubbing with cotton-waste and woollen cloth, until on rub- bing with a clean white cloth no sensible mark was made on it. Of course the whole of the grease was not removed by this operation ; the quantity remaining, how- ever, must have been extremely small, but it was sufficient to protect the iron from oxidation for some time, or until it was partially washed off With this process, however, there must have been constant changes going on in the state of the interior surface of the tube, which might affect the flow of the water in some degree. I accordingly noted carefully the circumstances and indications relating to the application and removal of the grease; and under the head of Remarks in the table of experiments I have stated the essential parts of my observations on this matter. 28 218 TABLE EXPERIMENTS ON THE FLOW OF WATER 1 2 3 4 5 G 7 8 9 Time of making the obser- Temperature, Reference to Powtion Mean Value of Quantity Height of the surface of the vations from which the in degrees figure 2, of the depth of Cin the of water water in compartment F, No. iiH'iin heights given of Fahren- plate XXI., index of water on formula discharged, figures 1 and 2, Date. in this table are heit's ther- indicating the in- the weir. in the calculated plate XX. of deduced. mometer ; the parts of let cock. by gauge next col- by the the com- A. uinii. formula the tsu pound tube used. k Z> = KXJI loot. Beginning. Ending. Clh\/ '2(ih by gauge E. by gauge Mean. II. Min. Sec. H. Min. Sec. of the air. of the water. Degrees. Feet. Cubic feet per second. Feet. Feet. Feet. 1 Sept. 20, P.M. 3 37 15 3 50 45 64.6 A 32.50 0.0269 0.4219 0.00980 0.0269 0.0269 0.0269 2 it it li 3 57 3 59 64.6 tf 32.50 0.0270 0.4219 0.00985 0.0269 0.0270 0.0269 3 tt it U 4 22 30 4 26 45 64.6 tt 34.25 0.0388 0.4202 0.01690 0.0391 0.0392 0.0391 4 U it it 4 31 ;)ii 4 35 30 64.6 tf 34.25 0.0383 0.4203 0.01658 0.0380 0.0384 0.0382 5 ti it tt 4 53 15 4 58 15 64.6 fi 35.50 0.0467 0.4191 0.02226 0.0469 0.0471 0.0470 6 11 tt U 5 20 50 5 26 62.6 64.6 tt 36.50 0.0532 0.4182 0.02701 0.0536 0.0535 0.0535 7 tt tt tt 5 38 40 5 41 50 62.6 64.6 tf 37.50 0.0607 0.4172 0.03283 0.0612 0.0612 0.0612 8 " 21, A.M. 9 11 40 y 17 40 56.4 62.8 ft 37.50 0.0616 0.4170 0.03355 0.0614 0.0620 0.0617 9 it it it 9 41 40 9 45 40 58.5 62.9 ft 38.50 0.0680 0.4162 0.03884 0.0683 0.0686 0.0684 10 tt it it 10 H 30 10 20 60.5 63.2 tt 39.50 0.0739 0.4153 0.04391 0.0738 0.0746 0.0742 11 it ti ft 10 46 50 10 54 20 60.9 63.3 If 40.50 0.0803 0.4145 0.04964 0.0799 0.0802 0.0800 12 ti it ti 11 15 40 11 19 61.0 63.4 tt 41.50 0.0848 0.4138 0.05378 0.0846 0.0852 0.0849 13 " 21, P.M. 2 16 40 2 23 50 61.0 63.4 ft 42.50 0.0906 0.4130 0.05927 0.0905 0.0910 0.0907 14 it tt ti 2 39 30 2 45 20 62.0 63.6 tt 43.25 0.0945 0.4125 0.06306 0.0948 0.0951 0.0949 15 tt ti tt 3 5 3 9 62.9 63.7 tt 44.00 0.0991 0.4118 0.06761 0.0992 0.0999 0.0995 1C it tt tt 3 30 30 3 33 40 67.7 63.7 U 44.75 0.1037 0.4112 0.07227 0.1036 0.1045 0.1040 17 tt it tt 3 46 40 3 52 67.1 63.8 tf 45.50 0.1069 0.4108 0.07556 0.1071 0.1077 0.1074 18 " 22, A.M. 9 13 30 9 17 20 58.0 62.7 tf 45.67 0.1072 0.4107 0.07586 0.1063 0.1085 0.1074 19 it ti tt 10 29 15 ~10 Ii4 60.9 62.9 AB 54.50 0.1505 0.4049 0.12441 0.1531 0.1559 0.1545 20 " 25, A.M. 8 59 20 9 3 30 60.8 62.4 tt 32.50 00285 0.4216 0.01068 0.0284 0.0282 0.0283 21 it ti ti !i 16 9 19 61.0 62.3 ft 34.25 0.0393 0.4201 0.01722 0.0393 0.0392 0.0392 22 it tt tt 9 L'.S (i 9 31 61.3 62.3 ft 35.50 0.0472 0.4190 0.02261 0.0478 0.0476 0.0477 23 tt tt tt 9 45 30 9 45 50 61.7 62.3 ft 36.50 0.0555 0.4179 0.02876 0.0556 0.0557 0.0556 24 ti tt it 9 58 10 1 25 62.2 62.3 tf 37.50 0.0618 0.4170 0.03372 0.0621 0.0622 0.0621 25 it (i tt 10 14 10 10 17 63.8 62.4 ft 38.50 0.0678 0.4162 0.03867 0.0682 0.0686 0.0684 2G it tt ti 10 30 10 33 20 63.7 62.4 (t 39.50 0.0732 U.4154 0.04330 0.0740 0.0740 0.0740 27 28 tt tt tt 10 11 45 5 30 30 10 11 50 8 20 64.2 64.8 62.4 62.5 tt 40.50 41.50 0.0796 0.0849 0.4146 0.4138 0.04900 0.05387 0.0803 0.0860 0.0805 0.0865 0.0804 0.0862 29 tt tt tt 11 19 10 11 25 10 65.0 62.6 tt 42.50 0.0901 0.4131 0.05880 0.0911 0.0914 00912 30 tt tt tt 11 40 40 11 43 40 65.4 62.6 ft 43.25 0.094G 0.4125 0.06316 0.0957 0.0963 0.0960 31 " 25, P.M. 15 20 66.3 62.8 tf 54.67 0.1517 0.4048 0.12587 0.1547 0.1578 0.1562 32 tt tt ti 3 27 3 31 30 70.5 63.6 ft 54.67 0.1512 0.4048 0.12525 0.1549 0.1575 0.1562 33 tt t tt 3 49 10 3 53 70.7 63.6 tf 44.00 0.0998 0.4118 0.06833 0.1013 0.1017 0.1015 34 ti t tt 4 10 4 13 30 70.5 63.7 tt 44.75 0.1031 0.4113 0.07166 0.1050 0.1057 0.1053 35 it t tf 4 31 4 34 10 70.5 63.7 it 45.50 0.1080 0.4106 0.07669 0.1100 0.1105 0.1102 36 ti t tt 4 57 30 5 3 30 70.7 ( O Q fi 47.00 0.1155 0.4096 0.08461 0.1176 01 ) V 4 0.1184 01 OAA 0.1180 01 ono 38 tt t tt 5 5 20 41 30 5 5 23 47 70.0 70.2 o4.y 64.0 if 50.00 54.67 0.1260 0.1507 0.4081 0.4049 0.09606 0.12466 .12o4 0.1534 .loOO 0.1572 .1 2u 0.1553 39 " 26, P.M. ~2 32 10 2 35 ~30 74.7 64.6 ABC 60.00 0.1775 0.4023 0.15833 0.1889 0.1897 0.1093 40 tt t.. tt 3 29 3 32 75.0 64.6 tf 32.50 0.0292 0.4215 0.01107 0.0297 0.0296 0.0296 41 tt tt tt 3 38 30 3 41 40 75.1 64.6 tf 34.25 0.0396 0.4201 0.01742 0.0402 0.0401 0.0401 42 tt tt tt 3 52 30 3 56 25 75.1 64.6 ft 36.50 00557 0.4179 0.02891 0.0566 0.0567 0.0566 43 tt tt it 4 5 4 10 40 75.4 64.6 it 38.50 0.0677 0.4162 0.03858 0.0694 0.0690 0.0692 44 ft tt -it 4 28 40 4 32 40 75.5 64.7 tt 40.50 0.0795 0.4146 0.04891 0.0814 0.0812 0.0813 45 tt it it 4 44 10 4 46 50 76.0 64.8 tf 42.50 0.0914 0.4129 0.06004 0.0939 0.0940 0.0939 46 it it tt 4 59 30 5 3 20 76.6 64.9 ft 45.50 0.1067 0.4108 0.07535 0.1103 0.1103 0.1103 47 tf tt ft 5 17 40 5 22 40 75.6 65.0 tf 50.00 0.1271 0.4080 0.09729 0.1321 0.1319 0.1320 48 it tt tt 5 41 40 5 46 tf 54.67 0.1458 0.4054 0.11878 0.1529 0.1529 0.1529 49 tt tt tt 6 10 30 6 18 50 tt 0.1696 0.4031 0.14817 0.1804 0.1808 0.1806 50 Oct. 7, A.M. 9 16 30 9 19 58.2 59.5 it 60.00 0.1778 0.4023 0.15873 0.1897 0.1908 0.1902 XXVII. THROUGH SUBMERGED TUBES AND ORIFICES. 219 10 11 13 13 14 15 16 17 18 Mean Effective Velocity Mean ve- Ratio of Diameter Mean ve- Ratio of height of head pro- due the locity by the veloci- of the locity by the veloci- the sur- ducing head in experi- ty at the tube or experi- ty at the No. face of the the dis- the pre- ment smallest orifice at ment at final dis- water in charge. ceding through section to the place the final charge to of compart- column. the small- the veloci- of final discharge the veloci- Remarks. ment J?, est section ty due the discharge. from the ty due the the figures 1 of the head. tube. head. and 2, tube or Exp. plato XX, orifice. by gauge D. H V V ti ' V' T Feet. Feet. Feet per second. Feet per second. Feet. Feet per second. 1 0.0608 0.0339 1.4767 1.2035 0.8150 0.1018 1.2035 0.8150 On the completion of experiment 7, the water was drawn out 2 0.0609 0.0340 1.4789 1.2103 0.8183 tt 1.2103 0.8183 of the cistern, and the interior of the mouth-piece examined. Only slight traces of oxidation were observed. In order to pre- s 0.1389 0.0998 2.5337 2.0765 0.8195 tt 2.0765 0.8195 vent oxidation before the experiments were resumed, the inte- 4 5 0.1384 0.2117 0.1002 0.1647 2.5387 3.2549 2.0369 2.7347 0.8024 0.8402 a tt 2.0369 2.7347 0.8024 0.8402 rior was wiped dry, and smeared with a grease consisting of about 20 parts of beef tallow, 10 parts of fine sperm oil, and 1 part of beeswax. The cistern remained empty until the experi- 6 0.2835 0.2300 3.8464 3.3180 0.8626 tt 3.3180 0.8626 ments were resumed, September 21st, when, previous to experi- ment 8, the grease was removed by thoroughly rubbing the 7 0.3790 0.3178 4.5213 4.0341 0.8923 tt 4.0341 0.8923 surface with cloth and cotton-waste. 8 0.3735 o.siis 4.4784 4.1222 0.9205 tt 4.1222 0.9205 Experiment 8 was a repetition of experiment 7 ; the increased discharge observed in experiment 8 must be attributed to the 9 0.4838 0.4154 5.1691 4.7719 0.9232 u 4.7719 0.9232 change in the state of the surface, due to the greasing and wip- 10 0.6010 0.5269 5.8217 5.3945 0.9266 tt 5.3945 0.9266 ing previously described. At6 h 30 P.M., September 21st, the water wasdrawn outof the 11 12 0.7390 0.8616 0.6590 0.7767 6.5107 7.0682 6.0985 6.6070 0.9367 0.9348 tt tt 6.0985 6.6070 0.9367 0.9348 cistern and the interior of the mouth-piece examined. A large part of the surface at and near the smallest section, where the velocity of the water was greatest, was covered with oxida- 13 .0486 0.9579 7.8495 7.2822 0.9277 tt 7.2822 0.9277 tion ; this was rubbed off with a cloth, when the previous lus- 14 .1782 1.0833 8.3475 7.7481 0.9282 u 7.7481 0.9282 tre of the surface was observed to be tarnished. It was then greased anew. The water was left out of the cistern until the 15 .3322 1.2327 8.9046 8.3065 0.9328 tt 8.3065 0.9328 experiments were resumed September 22d, A.M., previous to 16 .5008 1.3968 9.4788 8.8786 0.9367 tt 8.8786 0.9367 which the grease was wiped off. Experiment 18 was a repetition of experiment 17, for the purpose of ascertaining the effect of 17 18 .6214 .6232 1.5140 1.5158 9.8684 9.8743 9.2837 9.3205 0.9407 0.9439 tt tt 9.2837 9 3205 0.9407 0.9439 the change in the state of the surface. There was no change in the discharge, however, that could be attributed to the change in the state of the surface. 19 1.6235 1.4690 9.7207 15.2853 1.5725 0.1454 7.4928 0.7708 After the conclusion of the experiments Septeinlmr 22d, the 20 0.0485 0.0202 1.1399 1.3116 1.1506 tt 0.6429 0.5640 water was drawn out of the cistern and the mouth-piece and the first joint of the diverging tube were greased. The cistern re- 21 22 0.0873 0.1204 0.0481 0.0727 1.7590 2.1625 2.1162 2.7781 1.2031 1.2847 u u 1.0374 1.3618 0.5897 0.6297 mained empty until 9 A.M., September 24th, when it was filled. September 25th, A.M., previous to the commencement of the ex- periments, the cistern was emptied and the grease wiped off the 23 0.1552 0.0996 2.5311 3.5329 1.3958 it 1.7318 0.6842 interior of the mouth-piece and first joint of the diverging tube. 24 0.1923 0.1302 2.8939 4.1423 1.4314 tt 2.0305 0.7017 25 0.2327 0.1643 3.2509 4.7508 1.4614 u 2.3288 0.7164 26 0.2745 0.2005 3.5912 5.3193 1.4812 tt 2.6075 0.7261 At 2 h 25 m P.M., September 25th, the cistern was emptied and 27 0.3286 0.2482 3.9956 6.0203 1.5067 tt 2.9511 0.73o6 the interior of the pipes examined. The mouth-piece was free from oxidation, the first joint of the diverging tube was oxidated 28 0.3836 0.2974 4.3738 6.6187 1.5133 tt 3.2445 0.7418 sufficiently to redden the fingers when rubbetl upon it; both the 29 30 0.4370 0.4920 0.3458 0.3960 4.7163 5.0470 7.2238 7.7604 1.5317 1.5376 tt tt 3.5410 3.8041 0.7508 0.7537 pipes were wiped clean and dry, then coated with grease which was afterwards wiped off as much as practicable by rubbing with a cloth. Experiment 32 was a repetition of 31, to ascertain the 31 1.6179 1.4617 9.6965 15.4647 1.5949 a 7.5807 0.7818 effect due to the state of the surface caused by cleaning and greasing. The change in the discharge, however, due to this 32 1.6023 1.4461 9.6446 15.3883 1.5955 tt 7.5432 0.7821 cause, was, if any, extremely small. 33 0.5470 0.4455 5.3531 8.3947 1.5682 tt 4.1150 0.7687 After the conclusion of the experiments September 25th, P.M. the cistern was emptied; the mouth-piece WHS found free from 34 0.5971 0.4918 5.6244 8.8038 1.5653 tt 4.3155 0.7673 oxidation, and the first joint of the diverging pipe was only 35 0.6660 0.5558 5.9792 9.4227 1.5759 tt 4.6190 0.7725 slightly oxidated; both pipes were greased and tho cistern filled with water. 36 0.7850 0.6670 6.5501 10.3957 1.5871 tt 5.0959 0.7780 37 0.9836 0.8544 7.4134 11.8017 1.5919 a 5.7851 0.7804 38 1.6257 1.4704 9.7253 15.3158 1.5748 tt 7.5077 0.7720 39 40 1.6040 0.0439 1.4147 0.0143 9.5393 0.9591 19.4523 1.3599 2.0392 1.4179 0.2339 tt 3.6847 0.2576 0.3863 0.2686 September 26 I 1 ' 25 m P.M. The cistern lias stood full of water since last evening i the water was now drawn off, and the grmse wiped off the mouth-piece and first joint of diverging pipe. 41 0.0710 0.0309 1.4098 2.1405 1.5183 ti 0.4055 0.2876 The second joint was then put on for the experiments of to-day. 42 0.1182 0.0616 1.9906 3.5520 1.7844 a 0.6728 0.3380 43 0.1667 0.0975 2.5043 4.7403 1.8929 0.8979 0.3586 44 0.2241 0.1428 3.0307 6.0090 1.9827 ' (t 1.1383 0.3756 October 7, A.M. The cistern has been kept full of water since 45 0.2993 0.2054 3.6348 7.3771 2.0296 tt 1.3974 0.3845 September 26th, excepting on two or three occasions, when it was emptied, to permit the tubes to be cleaned and greased anew. 46 0.4220 0.3117 4.4777 9.2576 2.0675 tt 1.7536 0.3916 This morning, on emptying the cistern and wiping off the grease, 47 0.6271 0.4951 5.6427 11.9537 2.1184 u 2.2643 0.4013 no oxidation was observed. 48 0.8673 0.7144 6.7788 14.5929 2.1527 a 2.7643 0.4078 49 1.2805 1.0999 8.4113 18.2043 2.1643 u 3.4483 0.4100 50 1.5018 1.3116 9.1851 19.5016 2.1232 it 3.6941 0.4022 220 TABLE EXPERIMENTS ON THE FLOW OF WATER 1 3 3 4 5 6 7 8 9 Time of making the obser- vations from which the Temperature, in degrees Reference to figure 2, Position of the Mean depth of Value of C in the Quantity of water Ileight of the surface of the water in compartment F t No. Date. mean heights given in this table are of Fahren- heit's ther- plate XXI., indicating index of the in- water on the weir. formula in the discharged, calculated figures 1 and 2, plate XX. of deduced. mometer ; the parts of let cock. by gauge next col- by the the com- A. umn. formula the pound tube 1854. used. h D = Exp. Beginning. Ending. G'l/il/ 'Igk by gauge B. by gauge C. Mean. H. Mln. Sec. H. Min. Sec. of the air. of the water. Degrees. Feet. Cubic feet per second. Feet. Feet. Feet. 51 Oct. 7, A.M. 10 59 11 1 30 66.0 60.5 ABCD 62.00 0.1874 0.4014 0.17137 0.2055 0.2058 0.2056 52 ti it tt 11 18 11 20 30 66.1 60.6 it 32.50 0.0284 0.4217 0.01062 0.0290 0.0288 0.0289 53 u u u 11 40 11 42 40 66.1 60.6 If 34.25 0.0394 0.4201 0.01729 0.0405 0.0404 0.0404 54 U (( It 11 51 11 53 40 66.1 60.6 It 36.50 0.0555 0.4179 0.02876 0.0575 0.0574 0.0574 55 " 7, P.M. 2 16 2 18 30 68.6 59.8 ff 38.50 0.0668 0.4163 0.03783 0.0701 0.0700 0.0700 56 It It it 2 27 2 29 69.0 59.6 It 40.50 0.0801 0.4145 0.04945 0.0846 0.0848 0.0847 57 ff It (1 2 35 40 2 29 69.1 59.5 tf 42.50 0.0908 0.4130 0.05947 0.0963 0.0962 0.0962 58 (t (t (t 2 47 10 2 51 30 69.1 59.4 tt 45.50 0.1083 0.4106 0.07701 0.1157 0.1157 0.1157 -59 ii 11 It 3 6 3 11 69.5 59.3 ii 50.00 0.1273 0.4079 0.09750 0.1372 0.1372 0.1372 60 (t it a 3 22 3 26 40 69.9 59.3 It 54.67 0.1462 0.4053 0.11924 0.1593 0.1595 0.1594 61 it U If 3 42 20 3 47 30 70.1 59.4 ft 60.00 0.1700 0.4030 0.14866 0.1875 0.1880 0.1877 e-2 it ft (i 4 17 4 22 30 70.9 59.5 ft 62.00 0.1880 0.4013 0.17215 0.2098 0.2102 0.2100 63 it U It 4 40 10 4 45 71.3 59.7 II 63.50 0.197-1 0.4004 0.18481 0.2215 0.2216 0.2215 64 Oct. 10, A.M. 8 43 8 47 61.2 59.0 ft 62.00 0.1895 0.4012 0.17417 0.2063 0.2066 (1.2(1114 ~65 ft tl U 9 51 30 3 ~55 30 65.0 59.2 ABCDE 62.50 0.1907 0.4010 0.17574 0.2100 0.2101 0.2100 66 11 U tt 10 55 30 11 5 30 63.8 59.0 tf 62.50 0.1893 0.4012 0.17390 0.2091 0.2094 0.2092 67 ft 11 tt 11 17 30 11 22 30 64.0 59.0 If 32.50 0.0292 0.4215 0.01107 0.0300 0.0298 0.0299 68 tt (f tt 11 44 11 47 41 34.25 0.0390 0.4202 0.01703 0.0404 0.0401 0.0402 69 " " P.M. 2 4 30 2 8 30 64.3 59.1 tt 35.50 0.0460 0.4192 0.02177 0.0481 0.0478 0.0479 70 tt tt tt 2 23 30 2 28 30 64.8 59.3 tf 36.50 O.C563 0.4178 0.02937 0,0589 0.0587 0.0588 71 tf ft U 2 43 30 o 46 30 ft 37.50 0.0621 0.4170 0.03396 0.0652 0.0649 0.0650 72 tt tt tt 2 58 3 2 30 65.0 59.5 II 38.50 0.0680 0.4162 0.03884 0.0716 0.0712 0.0714 73 tt tt u 3 33 30 3 38 65.3 59.6 tt 39.50 0.0745 0.4153 0.04444 0.0788 0.0785 0.0786 74 tt tt ti 3 51 30 3 57 30 65.6 59.7 It 40.50 0.0801 0.4145 0.04945 0.0849 0.0847 0.0848 75 ft tt tt 4 14 4 21 66.1 59.7 ff 41.50 0.0848 0.4138 0.05378 0.0901 0.0897 0.0899 76 it tt tt 4 34 4 40 66.5 59.8 ft 42.50 0.0916 0.4129 0.06024 0.0978 0.0975 0.0976 77 tt tt tt 4 57 5 1 66.2 59.8 If 43.25 0.0960 0.4123 0.06454 0.1025 0.1023 0.1024 78 it tt tt 5 40 5 42 ff 62.50 0.1931 0.4008 0.17898 0.2191 0.2184 0.2187 79 " 12.A.M. 8 33 8 37 30 62.8 59.5 ii 62.50 0.1906 0.4011 0.17565 0.2092 0.2090 0.2091 80 it if it 8 51 30 8 56 30 62.7 59.5 11 44.00 0.1003 0.4117 0.06882 0.1041 0.1042 0.1041 81 ft ft ft 9 9 30 9 17 30 62.8 59.6 (t 44.75 0.1042 0.4111 0.07277 0.1090 0.1087 0.1088 82 it 11 ft 9 29 9 35 63.1 59.6 ft 45.50 0.1128 0.4099 0.08172 0.1189 0.1184 0.1186 83 it (t if 9 51 30 9 57 30 63.2 59.6 ft 47.00 0.1150 0.4096 0.08406 0.1210 0.1206 0.1208 84 tt ft tt 10 12 10 17 64.0 59.6 tf 50.00 0.1275 0.4079 0.09773 0.1348 0.1346 0.1347 85 tt ft ft 10 38 30 10 44 65.0 59.7 ft 54.67 0.1471 0.4052 0.12031 0.1575 0.1575 0.1575 86 tf ft it 11 6 30 11 10 65.6 59.8 tl 60.00 0.1697 0.4031 0.14830 0.1846 0.1850 0.1848 87 tf ft tf 11 34 30 11 37 30 67.6 59.9 ft 62.00 0.1896 0.4012 0.17431 0.2095 0.2088 0.2091 88 tf tt tt 11 53 30 11 58 30 tf 62.50 0.1911 0.4010 0.17630 0.2114 0.2117 0.2115 89 " " P.M. 2 40 2 50 69.8 60.3 U 62.50 0.1917 0.4010 0.17713 0.2131 0.2136 0.2133 90 ft ft tt 3 3 3 11 30 69.8 60.3 II 62.50 0.1919 0.4009 0.17736 0.2136 0.2143 0.2139 91 ft tt it 4 20 4 25 69.3 60.6 A 45.50 (1.1(177 0.4107 0.07639 0.1124 0.1144 0.1134 92 it U ii 5 27 30 5 30 30 ~7h6 60.7 ABCDE 62.50 0.1917 0.4010 0.17713 0.2204 0.2209 0.2206 93 94 95 " 16, A.M. tt ti ii ft tt it 9 9 11 18 58 4 30 9 10 11 21 1 7 30 30 30 55.0 56.1 56.6 57.1 57.1 57.6 40.00 32.50 35.50 U.077S 0.0293 0.0494 0.4148 0.4215 0.4187 0.04737 0.01113 0.02419 0.0786 0.0299 0.0504 0.0793 0.0296 0.0503 0.0789 0.0297 0.0503 96 tt tt tt 11 39 11 45 80 59.2 58.0 36.50 0.0522 0.4184 0.02626 0.0541 0.0533 0.0537 97 " u P.M. 2 10 2 14 40.00 0.0777 0.4148 0.04728 0.0796 0.0798 0.0797 98 ft tl U 2 40 2 44 65.2 37.50 0.0618 0.4170 0.03372 0.0634 0.0639 0.0636 99 ft U It 2 59 3 3 65.6 57.5 38.50 0.0682 0.4161 0.03900 0.0703 0.0702 0.0702 100 tf ff II 3 l!i s 23 65.7 57.6 39.50 0.0744 0.4153 0.04435 0.0769 0.0769 0.0769 101 (t If II 4 11 4 14 61.9 57.8 40.00 0.0775 0.4148 0.04710 0.071)9 0.0804 0.0801 XXVII CONTINUED. THROUGH SUBMERGED TUBES AND ORIFICES. 221 1O 11 12 13 14 15 16 17 18 Mean Effective Velocity Mean ve- Ratio of Diameter Mean ve- Ratio of height of head pro- due the locity by the veloci of the locity by the veloci the sur- ducing head in experi- ty at the tube or experi- ty at the No. face of the the dis- the pre- ment smallest orifice at ment at final dis- water in charge. ceding through section to the place the final charge to of compart- column. the small the veloci of final discharge the veloci ment E, est section ty due the discharge from the ty due the Remarks. the figures 1 of the head. tube. head. and 2, tube or Exp plate XX orifice. by gauge D. H 7 V V r ' V T Feet Feet. Feet per Feet per Feet per second. scrnml. ee . second. 51 1.6327 1.4271 9.5810 21.055C 2.1976 0.3209 2.1189 0.2212 At 8 h 35 m A.M., October 7, the diaphragm of strainer cloth in 52 0.0427 0.0138 0.9422 1.3050 1.3850 0.1313 0.1394 the gauging basin was cleaned ; it had become obstructed by an accumulation of gummy matter, apparently an exudation from 53 0.0809 0.0405 1.6140 2.1243 1.3162 ti 0.2138 0.1325 the new pine planks of which the cistern was constructed. 54 55 0.1162 0.1581 0.0588 0.0881 1.9448 2.3805 3.5329 4.6472 1.8166 1.9522 ti 0.3555 0.4677 0.1828 0.1965 October 7, P.M. After the conclusion of experiment (33, the cistern was emptied and the three joints B, 0, and D of the di- verging tube taken off and examined ; all of them, together with 56 0.2173 0.1326 2.9205 6.0757 2.0804 u 0.6114 0.2094 the mouth-piece, were a little oxidated, the mouth-piece the least so, and the joints C and D the most ; they were then all wiped 57 0.2773 0.1811 3.4131 7.3064 2.1407 it 0.7353 0.2154 clean and coated anew with grease ; the diverging tube was not 58 0.3901 0.2744 4.2012 9.4620 2.2522 ti 0.9522 0.2267 put on again to-day. October 10, A.M. The cistern has been kept full of water 59 0.5740 0.4368 5.3006 11.9790 2.2599 ti 1.2055 0.2274 since October 7. This morning it was emptied, and the grease 60 0.7887 0.6293 6.3623 14.6494 2.3025 n 1.4743 0.2317 wiped off the mouth-piece ; the joints B, C, and D were put on, the grease having been first wiped off. 61 1.1048 0.9171 7.6806 18.2642 2.3780 it 1.8380 0.2393 62 1.3872 1.1772 8.7018 21.1509 2.4306 n 2.1286 0.2446 63 1.5827 1.3612 9.3572 22.7058 2.4266 u 2.2850 0.2442 64 1.5952 1.3888 9.4516 21.3992 2.2641 ** 21535 0.2278 65 1.6283 1.4183 9.5514 21.5920 2.2606 0.4085 1.3409 0.1404 At 9k October 1(1, the cistern was emptied and the joint E put on- 66 1.6165 1.4073 9.5143 21.3653 2.2456 ti 1.3268 0.1395 No change was made in the apparatus between experiments 67 68 0.0438 0.0687 0.0139 0.0285 0.9456 1.3540 1.3599 2.0925 1.4381 1.5455 u ti 0.0845 0.1300 0.0893 0.0960 65 and 66; the water flowed continuously from 9 h 20 TO until after the conclusion of experiment 66. October 10, P.M. After the conclusion of experiment 78 the 69 70 0.0858 0.1163 0.0379 0.0575 1.5614 1.9232 2.6741 3.6087 1.7126 1.8764 ti n 0.1661 0.2241 0.1064 0.1165 cistern was emptied, and the four joints of the diverging tube ta&en off. There were only a few slight streaks of oxidation on the mouth-piece ; the joints B and C of the diverging tube were 71 0.1374 0.0724 2.1580 4.1725 1.9335 n 0.2591 0.1201 oxidated in longitudinal streaks ; joints D and E were nearly cov- ered with oxidation, which was however rubbed off with ease, 72 0.1596 0.0882 2.3819 4.7719 2.0034 ti 0.2963 0.1244 leaving the surface, apparently, as smooth as before. The inte- 73 74 0.1884 0.2163 0.1098 0.1315 2.6576 2.9084 5.4603 6.0757 2.0546 2.0890 tt 0.3391 0.3773 0.1276 0.1297 rior of the mouth-piece and of the four joints of the diverging tube were wiped clean and coated with grease ; the diverging tube was not put on again to-day. 75 0.2423 0.1524 3.1310 6.6070 2.1102 it 0.4103 0.1310 October 12, A.M. The apparatus was prepared for the experi- ments of to-day by removing the grease from the interior of the 76 0.2848 0.1872 3.4701 7.4013 2.1329 it 0.4596 0.1325 mouth-piece and four joints of the diverging tube, and putting 77 0.3104 0.2080 3.6578 7.9294 2.1678 n 0.4924 0.1346 the latter in their places. At 3 h 15" P.M., October 12, the cistern was emptied and the 78 1.5010 1.2823 9.0820 21.9899 2.4213 it 1.3656 0.1504 tube examined ; the interior of the mouth-piece and all the 711 80 1.6176 0.3261 1.4085 0.2220 9.5184 3.7789 21.5804 8.4558 2.2672 2.2376 it ti 1.3402 0.5251 0.1408 0.1390 joints were oxidated, and in a little greater degree than after experiment 78 as noted above. The four joints of the diverging tube were taken off, and together with the mouth-piece were well 81 0.3539 0.2451 3.9706 8.9407 2.2517 u 0.5552 0.1398 rubbed with a cloth, which removed all the red oxide. 82 0.4248 0.3062 4.4380 10.0407 2.2624 tl 0.6236 0.1405 83 0.4397 0.3189 4.5291 10.3283 2.2804 It 0.6414 0.1416 84 0.5557 0.4210 5.2039 12.0072 2.3073 11 0.7457 0.1433 85 0.7987 0.6412 6.4222 14.7812 2.3016 tl 0.9180 0.1429 86 1.1483 0.9635 7.8725 18.2204 2.3144 11 1.1315 0.1437 87 1.5575 1.3484 9.3131 21.4161 2.2996 tt 1.3300 0.1428 88 1.5884 1.3769 9.4110 21.6600 2.3016 tl 1.3451 0.1429 89 1.5745 1.3612 9.3572 21.7621 2.3257 11 1.3515 0.1444 90 1.5588 1.3449 9.3010 21.7907 2.3428 It 1.3533 0.1455 91 1.6285 1.5151 9.8720 9.3858 0.9507 0.1018 9.3858 0.9507 92 1.5069 1.2863 9.0961 21.7621 2.3925 0.4085 1.3515 0.1486 At 4J> 30 P.M., October 12, the cistern was emptied again, and the four joints of the diverging tube re-attached. At 6 P.M. the cistern was emptied ; the wide part of the mouth-piece was much oxidated, but only slightly so at the smallest section. The diverging tube was oxidated in only a few spots. 93 94 1.5925 0.1213 1.5136 0.0916 9.8671 2.4274 5.8316 1.3695 0.5910 0.5642 0.1017 It 5.8316 1.3695 0.5910 0.5642 Orifice in a thin plate. The plate, Figs. 12 and 13, plate XXI., containing the orifice, was put on October 14 ; the accessible parts of it were greased, 95 0.4855 0.4352 5.2909 2.9783 0.5629 11 2.9783 0.5629 and the cistern filled with water, and so remained until October 16, A.M., when it was emptied, and the grease wiped off. No 96 0.5372 0.4835 5.5768 3.2328 0.5797 It 3.2328 0.5797 oxidation was observed. 97 98 1.5784 0.8400 1.4987 0.7764 9.8184 7.0G69 5.8203 4.1504 0.5928 0.5873 tt 5.8203 4.1504 0.5928 0.5873 At O 1 * 15" P.M., October 16, the cistern was emptied and the alate examined ; there was a thin coating of oxide over most of the surface ; all the accessible parts of tbe plate were wiped 99 1.0944 1.0242 8.1167 4.8012 0.5915 It 4.8012 0.5915 clean and greased anew. At l h 15' P.M., the grease was wiped 100 1.4004 1.3235 9.2267 5.4601 0.5918 It 5.4601 0.5918 off again. 101 1.5704 1.-I003 9.7909 5.7979 0.5922 It 5.7979 0.5922 222. EXPERIMENTS ON THE FLOW OF WATER DESCRIPTION OF TABLE XXVIL, CONTAINING THE EXPERIMENTS ON THE FLOW OF WATER THROUGH SUBMERGED TUBES AND ORIFICES. 258. The greater portion of this table will be intelligible from the headings of the several columns, without further explanation. As previously stated, the quantity of water flowing was gauged by means of a weir of substantially the same form and dimensions as that used by Fencelet and Lesbros, in their experiments made at Metz in 1827 and 1828. Table X., Experiences hydrauliques, &c., previously cited, contains the results, of the exper- iments made in 1828. The quantities E discharged by experiment with certain depths on the weir are given ; also the quantities with the same depths, com- . _ tr puted by the formula d=lh^2gh; also the values of -j. These last quantities are the values of the coefficient C, by means of which the real discharge can be deduced from the value of d. We can then compute the real discharge by the formula The value of C is not the same for all depths, as may be seen by the follow- ing table, which contains the principal results of table X. of Poncelet and Lesbros above cited, changing the unit from metres to English feet. The length of the weir I was 0.10 metres or 0.6562 foot. Depth of water on the weir, taken 11.48 feet up stream from the weir. h Discharge by experiment. E Discharge computed by the formula Value of C in the formula. d=lh^2gh. D=Clh)/2gh, Feet. Cubic feet per second. Cubic feet per second. 0.6821 1.1528 2.9656 0.3888 0.5351 0.8098 20608 0.3930 0.3376 0.4071 1.0327 0.3943 0.1985 0.1864 0.4655 0.4003 0.1463 0.1194 0.2947 0.4053 0.0771 0.0468 0.1127 0.4149 The values of C, given in column 7, are deduced from the values of C in the preceding table, by interpolation. The quantities of water discharged by the tube or orifice given in column 8 are computed by the formula D = C I h y/ 2 c h, in which C has the value given in column 7; the length of the weir I, by THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 223 measurement, = 0.6579 foot; h = the value given in column 6, and g = 32.1618, which is its value for the place where the experiments were made (art. 68). 259. As previously stated, according to the first design of the apparatus, the weir was intended to be placed in the partition JV, figures 1 and 2, plate XX., and the depth on the weir was intended to be measured by the hook gauge B ; on trial, however, it was found that the agitation in the compartment F was too great to admit of a satisfactory gauge being made with the weir in this position, and it was accordingly removed to the position represented in the figures. The hook gauge B was allowed to remain, and the height of the surface of the water in the compartment F was observed by means of both the gauges B and C, and the mean of the two is taken as the elevation of the surface of the water in this compartment. By comparing the heights taken at the two gauges, given in column 9, it will be seen that, when the quantity of water discharged was small, there was little or no difference in the indications of the two gauges; with the larger volumes, the height at gauge B was sensibly the greatest. The effective head producing the discharge given in column 11 is the dif- ference of the heights of the surface of the water in compartments E and F. The velocity given in column 12 is computed by the formula V =. \l 2 g h. 260. The smallest section of the compound tube is in the mouth-piece be- tween a and 6, figure 2, plate XXI., and was found, by careful and repeated measurements made by different persons, to be 0.1018 foot. The diameter of the orifice in the thin plate was found in a similar manner to be 0.1017 foot. The area of the orifice in the mouth-piece was consequently 0.0081393 square foot, and the area of the orifice in the thin plate was 0.0081233 square foot. The velocities given in column 13 are obtained by dividing the quantities given in column 8 by the area of the smallest section through which the water was discharged. DEDUCTIONS FROM THE EXPERIMENTS GIVEN IN TABLE XXVII. 261. Confining ourselves, for the present, to the velocities at the smallest section, we find by these experiments that in all the tubes and orifices used the ratio of the velocity at the smallest section to the velocity due the head is least when the heads are very small. Thus with the mouth-piece A alone, When the effective head is 0.0339 foot (experiment 1), the ratio is 0.8150 " 0.2300 " ( " 6), 0.8626 " " 0.9579 ( 13), 0.9277 " " " 1.5140 feet ( 17), 0.9407 224 EXPERIMENTS ON THE FLOW OF WATER With the mouth-piece A and the first joint B of the diverging tube, When the effective head is 0.0202 foot (experiment 20), the ratio is 1.1506 0.0996 ( 23), " " 1.3958 0.8544 ( 37), " " 1.5919 " 1.4704 feet ( 38), " 1.5748 With the mouth-piece .4 and the two first joints B and 'C of the diverging tube, When the effective head is 0.0143 foot (experiment 40), the ratio is 1.4179 0.0616 ( 42), " 1.7844 1.0999 feet ( 49), 2.1643 1.3116 " ( " 50), " " 2.1232 With the mouth-piece A and the three first joints B, C, and D of the diverg- ing tube, When the effective head is 0.0138 foot (experiment 52), the ratio is 1.3850 0.0588 ( 54), 1.8166 1.1772 feet ( 62), 2.4306 1.3612 ( 63), 2.4266 With the complete compound tube, When the effective head is 0.0139 foot (experiment 67), the ratio is 1.4381 " 0.0575 ( 70), , 1.8764 1.2823 feet ( 78), 2.4213 1.4085 ( 79), 2.2672 With the thin plate, When the effective head is 0.0916 foot (experiment 94), the ratio is 0.5642 " " 0.4835 " ( 96), 0.5797 1.0242 feet ( 99), 0.5915 1.4903 ( 101), " 0.5922 262. By the preceding extracts from table XXVII. it will be seen that the ratio of the velocity at the smallest section of the tube or orifice to the velocity due the head is the least when the effective head is the least, and in the cases of the mouth-piece and orifice in the thin plate, the ratio is the greatest when the effec- tive head is the greatest. THKOUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 225 In the case of the diverging tube, the value of the ratio is a maximum when the effective head is somewhat less than the greatest. It is the general result of the great number of experiments on record, on the flow of water through orifices in a thin plate, discharging freely into the air, that the coefficient of discharge (which in simple orifices is the same thing as the ratio of the velocity at the smallest section of the orifice to the velocity due the head) is greatest for very small heads. In these experiments where the discharge takes place under water, the coefficient of discharge is least with the very small heads. This result is so marked and uniform that there can be no doubt of the fact. 263. As to the value of the coefficient of discharge for the mouth-piece A, a mean of all the experiments in which the effective head is not less than 1.5 feet gives 0.9451, the mean effective head being 1.5150 feet. This is nearly the same as the greatest value of the coefficient of discharge found by Castel for the smallest section of an orifice in a converging conical tube, namely, 0.956, which is for a tube in which the sides converge at an angle of 13 40', and discharging freely into the air.* Michelotti, in one of his experiments, by employing a cycloidal tube, found it 0.983.1 Eytelwein found 0.9798. t Other experimenters have found from 0.96 to 0.98. We must, therefore, conclude that the coefficient of discharge for the mouth-piece A, when discharging under water, is about 3 per cent less than has been found for similar orifices when discharging freely into the air. 264. The value of the coefficient of discharge for the orifice in a thin plate, taking the mean of the three experiments in which the effective head is near 1.5 feet, is 0.5920, the mean effective head being 1.5009 feet. This is less than has been found for circular orifices in a thin plate discharging freely into the air. There are great numbers of these experiments on record, made with orifices of various diameters and under various heads. The general result for the coefficient of discharge is very nearly 0.62. We must, therefore, conclude that the flow through a submerged orifice in a thin plate is less than when the discharge takes place freely into the air, in the ratio of 0.59 to 0.62, or about 5 per cent less. 265. The values of the ratio of the velocity at the smallest section to the velocity due the head, for the several combinations of the mouth-piece and the diverging tube, taking the largest values found in these experiments, are as follows : * D'Aubuisson's Hydraulics, Bennett's translation, page 56. f MemoireS de 1' Academic Royale des Sciences de Turin, 1784-85. } Handbuch der Mechanik und der Hydraulik. 29 226 EXPERIMENTS ON THE FLOW OF WATER For the mouth-piece A alone (exp. 91) 0.9507 For the mouth-piece A and the first joint B of the diverg- ing tube ( 32) 1.5955 For the mouth-piece A and the first two joints B and C of the diverging tube ( 49) 2.1643 For the mouth-piece A and the first three joints B, C, and D of the diverging tube ( " 62) 2.4306 For the complete compound tube as represented by figure 2, plate XXI ( 78) 2.4213 The maximum effect was produced with the mouth-piece and first three joints of the diverging tube, the addition of the fourth joint caused a slight diminution. In experiment 62, giving the greatest effect, the increase in the velocity of the water in the smallest section due to the diverging tubes is in the ratio of 0.9507 to 2.4306, or as 1 to 2.5566. To produce this increased velocity in the smallest section without using the diverging tube the head must be increased in the ratio of 1 to (2.5S66) 2 or as 1 to 6.5364. The effective head in experiment 62 was 1.1772 feet. To give the .velocity in the same experiment, if the diverging tube had not been attached, would have required an effective head of 1.1772 X 6.5364 = 7.6947 feet. The difference in these heads is 7.6947 1.1772 = 6.5175 feet. A portion of the pressure of the atmosphere on the surface of the water in the upper division E of the cistern, figures 1 and 2, plate XX., equivalent to this head of water, is rendered active by the addition of the diverging tube to the mouth-piece. 266. According to Bernoulli's theory, the velocity of the water at its final discharge from the tube should be that due to the head;* in experiment 62 this * Call A the area of the section and V the velocity of the water at ab, figure 2, plate XX. B the area of the section and v the velocity at cd; A = the head or difference of height of the surface of the water in compartments E and F. The motion having become permanent, we have A V = Bv. The volume of water included between the sections ab and cd in the small time t will move .to a' b' c' d' ; the volume included between the sections a' b' and cd is common to both positions, every particle in one having its counterpart in the other, both in position and velocity. In finding the change in the living force in the two position?, we need only consider the volumes a a' bb' and cc' dd 1 . These volumes are equal, and assuming the water to be pure and at its maximum density, the weight of each is 62.382 A Vt. THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 227 velocity is 8.7018 feet per second ; the velocity at other parts of the compound tube would be inversely as the squares of the diameters ; at the smallest section the velocity must be greater than at the final discharge in the ratio of 1 to /03209\ 2 * ( Q ' J = 9.9367. To give this velocity at the smallest section without the diverg- ing tube would require the effective head of water to be increased from 1.1772 feet to 1.1772 X (9.9367) 2 = 116.24 feet; the increase being 115.06 feet; if the pressure of the atmosphere was great enough, its pressure, to this extent, would be rendered active. The total pressure of the atmosphere is usually about 34 feet, and this of course is the limit to which it can be rendered active. Abstracting from the effects of vaporization, whenever the exhausting effect of the diverging tube exceeds the pressure of the atmosphere, (added to the pressure due to the actual head of water at the smallest section,) breaks must occur in the mass of water in the compound tube, at or near the smallest section, and the flow through the smallest section will be the same as if the discharge took place in a vacuum. In experiment 62, the exhausting effect of the diverging tube, ,...., . ,, . , ,. . 62.382^1 Vt The living force of the volume a a' bb 1 is - cc , dd , is - 9 The increase of living force in passing from one position to the other being 68.882^ Fl ()) ,_^ (1) if This increase of living force is produced by the action of gravity on the volume of water A Vt descending through the height h, which is equivalent to an amount of work represented by 62.382^1 Vth. (2.) By the doctrine of living forces, the living force (1.) is equivalent to the amount of work represented by 62.382 A Vt (v *_ v ^ (3-) The amount of work in (2.) and (3.) must be equal ; we have, therefore, 62.382 AVth = from which we deduce h = - - If V is very small relatively to v, it may be neglected, and we have v' h = ~ , and v = \/2gh. y 228 EXPERIMENTS ON THE FLOW OF WATER according to Bernoulli's theory, exceeds three times the actual pressure at the smallest section, and if it had produced its full effect according to theory or even one third of that effect, breaks must have occurred in the mass of water near the smallest section. The ratio of the actual velocity of the water at its final discharge to the velocity according to Bernoulli's theory is given in column 17. In experiment 62 it is 0.2446, or about one quarter of the velocity due the head, indicating a loss of about fifteen sixteenths of the living force. It is difficult to see how so much can be lost. There are no abrupt changes in velocity, and the interior surfaces of the mouth-piece and diverging tube are smooth and free from sensible irregularity. The slight oxidation observable after some of the experiments appears to have produced no sensible loss, as in experiment 62, which gave the greatest result, there was considerable oxidation, while in other experiments giving a less effect there was no oxidation. The chief discrepancy between the hypothesis on which Bernoulli's theory is founded and the real conditions of the motion appears to be due to the retard- ing effects of the walls of the tube. According to the hypothesis, the velocity in all parts of the same section is the same ; Prony's well-known formula for the motion of water in pipes is founded upon the idea that the principal retardation is due to the sides ; whence it follows, that the velocity must be least at the sides and greatest at the centre. Darcy* made many experiments on the subject by means of Pitot's tube, and found that in long straight pipes there was a material variation in the velocities at different distances from the centre, and determined a formula expressing the law of the variation. It would not be safe to apply this formula to these experiments on account of the short length and varying diameter of the compound tube, but it is clear that variations in the velocity must exist to an extent which must greatly modify the results deduced from Bernoulli's theory. 267. As previously stated, Venturi, by adding a diverging tube increased the dis- charge of an orifice having nearly the form of the contracted vein, and discharging freely into the air, in the ratio of 1 to 2.21. In these experiments, in an orifice without, contraction discharging under water the discharge was increased by adding a diverging tube in the ratio of 1 to 2.56. Making the comparison with an orifice in a thin plate, the maximum coefficient of discharge with the thin plate is 0.5928, and with the month-piece of cycloidal form and diverging tube, the maximum coefficient * Recherches experimentales relatives au Mouvement de tEau dans las Tuyaux, par HENET DAKCT. Paris, 1857, THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 229 is 2.4306 ; the discharge with the same area of orifice and the same head being increased in the ratio of 1 to 4.12. 268. Considerable irregularities will be observed in the value of the ratio of the velocity in the smallest section to the velocity due the head, given in column 14. Thus, in the experiments with the complete compound tube, we have the following, which were intended to be identical, the repetitions being made for the purpose of detecting such variations, if any should occur. "In all these experiments the index of the inlet cock, L, figures 2 and 3, plate XX., was set at the same point, viz. 62.5, or as nearly so as practicable, in order to admit the same quan- tity of water. Number of the experiment ill Table XXVII. Quantity of water discharged ; in Cubic feet per second. Effective head pro- ducing the discharge ; iu feet. Ratio of the Telocity at the smallest section to the velocity due the head. 65 0.17574 1.4183 2.2606 66 0.17390 1.4073 2.2456 78 0.17898 1.2823 2.4213 79 0.17565 1.4085 2.2672 88 0.17630 1.3769 2.3016 89 0.17713 1.3612 2.3257 90 0.17736 1.3449 23428 92 0.17713 1.2863 2.3925 [n the preceding table, the small irregularities in the quantities of water dis- charged are due to corresponding small variations in setting the index of the inlet cock. The irregularities in the effective head are mainly due to changes in the efficiency of the diverging tube. The only known variation on which these changes could depend is in the state of the interior surface of the tube. Thus No. 65 was the second experiment made after the grease was wiped off. Twelve exper- iments were, made between Nos. 65 and 78, no change being made in the state of the surface, except that caused by the action of the water, which undoubtedly had washed off, before No. 78 was made, a part or the whole of the grease not removed by wiping. In the experiments made soon after wiping the surface, it is probable that the water was repelled from it by the grease, but after the water had run through the tube for some hours the grease was washed off sufficiently to permit the water to come in contact with the iron, which appears to have increased, materially, the exhausting effect of the diverging tube. 269. Previous to making the experiments, it was anticipated that when the diverging tube was used there would be sensible oscillations in the elevation of the surface of the water in compartment E, figures 1 and 2, plate XX., due to the unstable equilibrium of the stream. Although the amplitudes of the oscillations 230 EXPERIMENTS ON THE FLOW OF WATER of the surface were much less than was expected, they were quite sensible. Thus we find, by referring to the original notes, that with the mouth-piece alone, the amplitude of the oscillations, when the effective head was 0.10 foot, was about 0.0003 foot " " " " 1.00 " " " 0.0006 " " " " 1.40 feet " " 0.0007 " With the complete compound tube the amplitude of the oscillations, when the effective head was 0.10 foot, was about 0.0021 foot. " " " 1.00 " " " 0.0103 " " " 1.40 feet " " 0.0117 " The variation with heads from 1.00 foot to 1.40 feet being about 17 times as great with the complete diverging tube as with the mouth-piece alone. 270. As previously stated, the principles involved in the flow of water through a diverging tube find a useful application in Mr. Boyden's Diffuser. This inven- tion, applied to a turbine water- wheel 104.25 inches in diameter and about seven hundred horse power, is represented in plates XXII. and XXIII. This turbine is one of four of the same power constructed from the designs of the author for the cotton-mills of the Merrimack Manufacturing Company in Lowell. Plate XXII. is a sectional elevation through the axis, showing the lower parts of the apparatus, a, a, a, a is the wheel, carrying 60 floats of Russian sheet iron, 0.15 inch thick; 6 the main shaft, which is suspended from the top, in a similar manner to the Tremont turbines (plate I.); c, c is the disc, carrying 33 guides, c, c, c', c, of Russian sheet iron, 0.125 inch thick, which lean one horizontally to six vertically ; d, d, the disc pipe, which hangs at its upper end, upon a part of the curved pipe or curb e, e, not represented in the plate ; /, /, the garniture, which supports the upper part of the guides, and is curved at its lower edge, in order to afford a favorable aperture for the flow of the water entering the wheel ; g, g, the lower curb ; h, h, the speed gate, which is represented as raised to its greatest height; i, a gate rod, which with two others, not represented in the plate, enables the gate to be moved by the governor or by hand ; k, k, beams extending from the granite walls of the wheel-pit to the lower curb and supporting the latter; I, I, pillars resting upon granite blocks in the floor of the wheel-pit, and supporting the beams k, k; m, m, the diffuser, which is supported by the pillars I, I, by means of the curved beams n, n, n, n ; w, w, low water level of the surface of the water in the wheel-pit. The wheel is placed suf- THROUGH SUBMERGED ORIFICES AND DIVERGING TUBES. 231 ficiently low, to permit the diffuser to be submerged at all times when the wheel is in operation, that being essential to the most advantageous operation of the diffuser. Figure 1, plate XXIII., is a horizontal section through the wheel, showing also the disc, guides, and garniture, and also the lower part of the diffuser. Figure 2 is a horizontal section on a larger scale, showing part of the wheel, guides, and diffuser. Figure 3 is a vertical section, showing part of the wheel, diffuser, &c. When the speed gate is fully raised, and the wheel is moving with the velocity giving its greatest coefficient of useful effect, the water passes through the wheel in a path, which is nearly represented by " the dotted line a, b, figure 2, plate XXIII. On leaving the wheel it necessarily has considerable velocity, which would involve a corresponding loss of power, except for the effect of the diffuser, which utilizes a portion of it. When operating under a fall of 33 feet and the speed gate raised to its full height, this wheel discharges about 219 cubic feet of water per second. The area of the annular space o, o, o, o, plate XXII., where the water enters the diffuser, is 0.802 X 8.792 TT = 22.152 square feet; and if the 219 stream passes through this section radially, its mean velocity must be 22 52 = 9.886 feet per second, which is due to a head of 1.519 feet. The area of the annular space p, p, p, p, where the water leaves the diffuser, is 1.5 X 15.333 TT = 72.255 219 square feet, and the mean velocity = 3.031 feet per second, which is due to a head of 0.143 feet. According to this, the saving of head, due to the diffuser is 1.519 _ 0.143 = 1.376 feet, being 9 , L3 /S,o> or about 4 I P er cent of tne head OO. 1 .0 1 J available without the diffuser, which is equivalent to a gain in the coefficient of useful effect to the same extent. As previously stated (art. 12), experiments on the same turbine, with and without a diffuser, have shown a gain due to the latter, of about 3 per cent in the coefficient of useful effect. The diffuser adds to the co- efficient of useful effect by increasing the velocity of the water passing through the wheel, and it must of course increase the quantity of water discharged in the same proportion. If it increases the available head 3 per cent, the velocity, which varies as the square root of the head, must be increased about 1.5 per cent, and the quantity discharged must be increased in the same proportion. The power of the wheel, which varies as the product of the head into the quantity of water discharged, must be increased about 4.5 per cent. 232 EXPLANATION OF TABLES XXVIII., XXIX., AND XXX. These tables have been prepared in the office of the Proprietors of the Locks and Canals on Merrimack River, for the purpose of facilitating the computations connected with gauging the quantities of water drawn from their canals at Lowell. TABLE XXVIII. gives the velocities of floats for eight different distances be- tween the transit stations, and for times of passage between them for every tenth of a second, from 20 to 100 seconds. The use of the table may be extended to such other distances between the transit stations as are multiplies or submultiplies of the distances given in the table, by taking the time the same multiple or submultiple as the distance. TABLE XXIX. gives the values of the coefficient (l 0.116 (\I~T) 0.1)) for values of D for every 0.001 from 0.000 to 0.100, with the logarithms of the same. (See art. 233.) TABLE XXX. gives the velocities, in feet per second, due to every 0.01 foot head, from 0.00 to 49.99 feet, computed for Lowell, by the formulas given in art. 68. These formulas, reduced to the English foot as the unit, become g = 32.1695 (1 - - 0.00284 cos. 2 1) (i - r = 20887540 (1 + 0.00164 cos. 2 I). The values of g by these formulas for several latitudes and heights above the sea. are given in the following table : Height above the Sea. Feet. Latitude. 300 350 400 45 5O 55 600 100 200 300 32.1239 32.1236 32.1233 32.1229 32.1383 32.1380 32.1377 32.1374 32.1537 32.1534 32.1531 32.1528 32.1695 32.1692 32.1689 32.1686 32.1854 32.1851 32.1848 32.1845 32.2008 , 32.2005 32.2002 32.1998 32.2152 32.2149 32.2146 32.2143 400 500 600 700 32.1226 32.1223 32.1220 32.1217 32.1371 32.1368 32.1364 32.1361 32.1524 32.1521 32.1518 32.1515 32.1683 32.1680 32.1677 32.1674 32.1842 32.1839 32.1835 32.1832 32.1995 32.1992 32.1989 32.1986 32.2140 32.2137 32.2134 32.2131 800 900 1000 1100 32.1214 32.1211 32.1208 32.1205 32.1358 32.1355 32.1352 32.1349 32.1512 32.1509 32.1506 32.1503 32.1671 32.1668 32.1665 32.1662 32.1829 32.1826 32.1823 32.1820 32.1983 32.1980 32.1977 32.1974 32.2128 32.2125 32.2121 32.2118 233 TABLE XXVIII. TABLE OF VELOCITIES OP TUBES IN MEASURING FLUMES, IN FEET PER SECOND. THE TIME OCCUPIED IN PASSING FROM THE UPSTREAM TO THE DOWNSTREAM TRANSIT STATION, AND THE DISTANCE BETWEEN THEM, BEING GIVEN. TIME. DISTANCE BETWEEN THE TRANSIT STATIONS. IN FEET. TIME DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET. See's. 50. 1 60. 70. 80. 90. 100. 110. 120. See's. 50. 60. 70. 80. 90. 100. 110. 120. 20.0 2.500 3.000 3.500 4.000 4.500 5.000 5.500 6.000 25.0 2.000 2.400 2.800 3.200 3.600] 4.000 4.400 4.800 20.1 2.488 2.985 3.483 3.980 4.478 4.975 5.473 5.970 25.1 1.992 2.390 2.789 3.187 3.586| 3.984 4.382 4.781 20.2 2.475 2.970 3.465 3.960 4.455 4.950 5.446 5.941 25.2 1.984 2.381 2.778 3.175 3.571 3.968 4.365 4.762 20.3 2.463 2.956 3.448 3.941 4.433 4.926 5.419 5.911 25.3 1.976 2.372 2.767 3.162 3.557 3.953 4.348 4.743 20.4 2.451 2.941 3.431 3.922 4.412 4.902 5.392 5.882 25.4 1.969 2.362 2.756 3.150 3.543 3.937 4.331 4.724 20.5 2.439 2.927 3.415 3.902 4.390 4.878 5.366 5.854 25.5 1.961 2.353 2.745 3.137 3.529 3.922 4.314 4.706 20.6 2.427 2.913 3.398 3.883 4.369 4.854 5.340 5.825 25.6 1.953 2.344 2.734 3.125 3.516 3.906 4.297 4.687 20.7 2.415 2.899 3.382 3.865 4.348 4.831 5.314 5.797 25.7 1.946 2.335 2.724 3.113 3.502 3.891 4.280 4.669 20.8 2.404 2.885 3.365 3.846 4.327 4.808 5.288 5.769 25.8 1.938 2.326 2.713 3.101 3.488 3.876 4.264 4.651 20.9 2.392 2.871 3.349 3.828 4.300 4.785 5.263 5.742 25.9 1.931 2.317 2.703 3.089 3.475 3.861 4.247 4.633 21.0 2.381 2.857 3.333 3.810 4.286 4.762 5.238 5.714 26.0 1.923 2.308 2.692 3.077 3.462 3.846 4.231 4.615 21.1 2.370 2.844 3.318 3.791 4.265 4.739 5.213 5.687 26.1 1.916 2.299 2.682 3.065 3.448 3.831 4.215 4.598 21.2 2.358 2.830 3.302 3.774 4.245 4.717 5.189 5.660 26.2 1.908 2.290 2.672 3.053 3.435 3.817 4.198 4.580 21.3 2.347 2.817 3.286 3.756 4.225 4.695 5.164 5.634 26.3 1.901 2.281 2.662 3.042 3.422 3.802 4.183 4.563 21.4 2.336 2.804 3.271 3.738 4.206 4.673 5.140 5.607 26.4 1.894 2.273 2.652 3.030 3.409 3.788 4.167 4.545 21.5 2.326 2.791 3.256 3.721 4.186 4,651 5.116 5.581 26.5 1.887 2.264 2.642 3.019 3.396 3.774 4.151 4.528 21.6 2.315 2.778 3.241 3.704 4.167 4.630 5.093 5.556 26.6 1.880 2.256 2.632 3.008 3.383 3.759 4.135 4.511 21.7 2.304 2.765 3.226 3.687 4.147 4.608 5.069 5.530 26.7 1.873 2.247 2.622 2.996 3.371 3.745 4.120 4.494 21.8 2.294 2.752 3.211 3.670 4.128 4.587 5.046 5.505 26.8 1.866 2.239 2.612 2.985 3.358 3.731 4.104 4.478 21.9 2.283 2.740 3.196 3.653 4.110 4.566 5.023 5.479 26.9 1.859 2.230 2.602 2.974 3.340 3.717 4.089 4.461 22.0 2.273 2.727 3.182 3.636 4.091 4.545 5.000 5.455 27.0 1.852 2.222 2.593 2.963 3.333 3.704 4.074 4.444 22.1 2.262 2.715 3.167 3.620 4.072 4.525 4.977 5.430 27.1 1.845 2.214 2.583 2.952 3.321 3.690 4.059 4.428 22.2 2.252 2.703 3.153 3.604 4.054 4.505 4.955 5.405 27.2 1.838 2.206 2.574 2.941 3.309 3.676 4.044 4.412 22.3 2.242 2.691 3.139 3.587 4.036 4.484 4.933 5.381 27.3 1.832 2.198 2.564 2.930 3.297 3.663 4.029 4.396 22.4 2.232 2.679 3.125 3.571 4.018 4.464 4.911 5.357 27.4 1.825 2.190 2.555 2.920 3.285 3.650 4.015 4,380 22.5 2.222 2.667 3.111 3.556 4.000 4.444 4.889 5.333 27.5 1.818 2.182 2.545 2.909 3.273 3.636 4.000 4.364 22.6 2.212 2.655 3.097 3.540 3.982 4.425 4.867 5.310 27.6 1.812 2.174 2.536 2.899 3.261 3.623 3.986 4.348 22.7 2.203 2.643 3.084 3.524 3.965 4.405 4.846 5.286 27.7 1.805 2.166 2.527 2.888 3.249 3.610 3.971 4.332 22.8 2.193 2.632 3.070 3.509 3.947 4.386 4.825 5.263 27.8 1.799 2.158 2.518 2.878 3.237 3.597 3.957 4.317 22.9 2.183 2.620 3.057 3.493 3.930 4.367 4.803 5.240 27.9 1.792 2.151 2.509 2.867 3.226 3.584 3.943 4.301 23.0 2.174 2.C09 3.043 3.478 3.913 4.348 4.783 5.217 28.0 1.786 2.143 2.500 2.857 3.214 3.571 3.929 4.28C. 23.1 2.165 2.597 3.030 3.463 3.896 4.329 4.762 5.195 28.1 1.779 2.135 2.491 2.847 3.203 3.559 3.915 4.270 23.2 2.155 2.586 3.017 3.448 3.879 4.310 4.741 5.172 28.2 1.773 2.128 2.482 2.837 3.191 3.546 3.901 4.255 23.3 2.146 2.575 3.004 3.433 3.863 4.292 4.721 5.150 28.3 1.767 2.120 2.473 2.827 3.180 3.534 3.887 4.240 23.4 2.137 2.564 2.991 3.419 3.846 4.274 4.701 5.128 28.4 1.761 2.113 2.465 2.817 3.169 3.521 3.873 4.225 23.5 2.128 2.553 2.979 3.404 3.830 4.255 4.681 5.106 28.5 1.754 2.105 2.456 2.807 3.158 3.509 3.860 4.211 23.6 2.119 2.542 2.966 3.390 3.814 4.237 4.661 5.085 28.6 1.748 2.098 2.448 2.797 3.147 3.497 3.846 4.196 23.7 2.110 2.532 2.954 3.376 3.797 4.219 4.641 5.063 28.7 1.742 2.091 2.439 2.787 3.136 3.484 3.833 4.181 23.8 2.10112.521 2.941 3.361 3.782 4.202 4.622 5.042 28.8 1.736 2.083 2.431 2.778 3.125 3.472 3.819 4.167 23.9 2.092 2.510 2.929 3.347 3.766 4.184 4.603 5.021 28.9 1.730 2.076 2.422 2.768 3.114 3.460 3.806 4.152 24.0 2.083 2.500 2.917 3.333 3.750 4.167 4.583 5.000 29.0 1.724 2.069 2.414 2.759 3.103 3.448 3.793 4.138 24.1 2.075 2.490 2.905 3.320 3.734 4.149 4.564 4.979 29.1 1.718 2.062 2.405 2.749 3.093 3.436 3.780 4.124 24.2 2.066 2.479 2.893 3.306 3.719 4.132 4.545 4.959 29.2 1.712 2.055 2.397 2.740 3.082 3.425 3.767 4.110 24.3 2.058 2.469 2.881 3.292 3.704 4.115 4.527 4.938 29.3 1.706 2.048 2.389 2.730 3.072 3.413 3.754 4.096 24.4 2.049 2.459 2.869 3.279 3.689 4.098 4.508 4.918 29.4 1.701 2.041 2.381 2.721 3.061 3.401 3.741 4.082 24.5 2.041 2.449 2.857 3.265 3.673 4.082 4.490 4.898 29.5 1.695 2.034 2.373 2.712 3.051 3.390 3.729 4.068 24.6 2.033 2.439 2.846 3.252 3.659 4.065 4.472 4.878 29.6 1.689 2.027 2.365 2.703 3.041 3.378 3.716 4.054 24.7 2.024 2.429 2.834 3.239 3.644 4.049 4.453 4.858 29.7 1.684 2.020 2.357 2.694 3.030 3.367 3.704 4.040 24.8 2.016 2.419 2.823 3.226 3.629 4.032 4.435 4.839 29.8 1.678 2.013 2.349 2.685 3.020 3.356 3.691 4.027 24.9 2.008 2.410 2.811 3.213 3.614 4.016 4.418 4.819 29.9 1.672 2.007 2.341 2.676 3.010 3.344 3.679 4.013 234 TABLE XXVIII CONTINUED. TABLE OF VELOCITIES OF TUBES IN MEASURING FLUMES, IN FEET PER SECOND. THE TIME OCCUPIED IN PASSING FROM THE UPSTREAM TO THE DOWNSTREAM TRANSIT STATION, AND THE DISTANCE BETWEEN THEM, BEING GIVEN. TIME. See's. DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET. TIME. See's. DISTANCE BETWEEN TUB TRANSIT STATIONS, IN FEET. 50. 60. 70. 80. 90. 100. 110. 120. 50. 60. 70. 80. 90. 100. 110. 120. 30.0 1.667 2.000 2.333 2.667 3.000 3.333 3.667 4.000 35.0 1.429 1.714 2.000 2.286 2.571 2.857 3.143 3.429 30.1 1.661 1.993 2.326 2.658 2.990 3.322 3.654 3.987 35.1 1.425 1.709 1.994 2.279 2.564 2.849 3.134 3.419 30.2 1.656 1.987 2.318 2.649 2.980 3.311 3.642 3.974 35.2 1.420 1.705 1.989 2.273 2.557 2.841 3.125 3.409 30.3 1.650 1.980 2.310 2.640 2.970 3.300 3.630 3.960 35.3 1.416 1.700 1.983 2.266 2.550 2.833 3.116 3.399 30.4 1.645 1.974 2.303 2.632 2.961 3.289 3.618 3.947 35.4 1.412 1.695 1.977 2.260 2.542 2.825 3.107 3.390 30.5 1.639 1.967 2.295 2.623 2.951 3.279 3.607 3.934 35.5 1.408 1.690 1.972 2.254 2.535 2.817 3.099 3.380 30.6 1.634 1.961 2.288 2.614 2.941 3.268 3.595 3.922 35.6 1.404 1.685 1.966 2.247 2.528 2.809 3.090 3.371 30.7 1.629 1.954 2.280 2.606 2.932 3.257 3.583 3.909 35.7 1.401 1.681 1.961 2.241 2.521 2.801 3.081 3.361 30.8 1.623 1.948 2.273 2.597 2.922 3.247 3.571 3.896 35.8 1.397 1.676 1.955 2.235 2514 2.793 3.073 3.352 30.9 1.618 1.942 2.265 2.589 2.913 3.236 3.560 3.883 35.9 1.393 1.671 1.950 2.228 2.507 2.786 3.064 3.343 31.0 1.613 .935 2.258 2.581 2.903 3.226 3.548 3.871 36.0 1.389 1.667 1.944 2.222 2.500 2.778 3.056 3.333 31.1 1.608 .929 2.251 2.572 2.894 3.215 3.537 3.859 36.1 1.385 1.662 1.939 2.216 2.493 2.770 3.047 3.324 31.2 1.603 .923 2.244 2.564 2.885 3.205 3.526 3.846 36.2 1.381 1.657 1.934 2.210 2.486 2.762 3.039 3.315 31.3 1.597 .917 2.236 2.556 2.875 3.195 3.514 3.834 36.3 1.377 1.653 1.928 2.204 2.479 2.755 3.030 3.306 31.4 1.592 .911 2.229 2.548 2.866 3.185 3.503 3.822 36.4 1.374 1.648 1.923 2.198 2.473 2.747 3.022 3.29 V 7 31.5 1.587 .905 2.222 2.540 2.857 3.175 3.492 3.810 36.5 1.370 1.644 1.918 2.192 2.466 2.740 3.014 3.288 31.6 1.582 .899 2.215 2.532 2.848 3.165 3.481 3.797 36.6 1.366 1.639 1.913 2.186 2.459 2.732 3.005 3.279 31.7 1.577 1.893 2.208 2.524 2.839 3.155 3.470 3.785 36.7 1.362 1.635 1.907 2.180 2.452 2.725 2.997 3.270 31.8 1.572 1.887 2.201 2.516 2.830 3.145 3.459 3.774 36.8 1.359 1.630 1.902 2.174 2.446 2.717 2.989 3.261 31.9 1.567 1.881 2.194 2.508 2.821 3.135 3.448 3.762 36.9 1.355 1.626 1.897 2.168 2.439 2.710 2.981 3.252 32.0 1.562 1.875 2.187 2.500 2.812 3.125 3.437 3.750 37.0 1.351 1.622 1.892 2.162 2.432 2.703 2.973 3.243 32.1 1.558 1.869 2.181 2.492 2.804 3.115 3.427 3.738 37.1 1.348 1.617 1.887 2.156 2.426 2.695 2.965 3.235 32.2 1.553 1.803 2.174 2.484 2.795 3.106 3.416 3.727 37.2 1.344 1.613 1.882 2.151 2.419 2.688 2.957 3.226 32.3 1.548 1.858 2.167 2.477 2.786 3.096 3.406 3.715 37.3 1.340 1.609 1.877 2.145 2.413 2.681 2.949 3.217 32.4 1.543 1.852 2.160 2.469 2.778 3.086 3.395 3.704 37.4 1.337 1.604 1.872 2.139 2.406 2.674 2.941 3.209 32.5 1.538 1.846 2.154 2.462 2.769 3.077 3.385 3.692 37.5 1.333 1.600 1.867 2.133 2.400 2.667 2.933 3.200 32.6 1.534 1.840 2.147 2.454 2.761 3.067 3.374 3.681 37.6 1.330 1.596 1.862 2.128 2.394 2.660 2.926 3.191 32.7 1.529 1.835 2.141 2.446 2.752 3.058 3.364 3.670 37.7 1.326 1.592 1.857 2.122 2.387 2.653 2.918 3.183 32.8 1.524 1.829 2.134 2.439 2.744 3.049 3.354 3.659 37.8 1.323 1.587 1.852 2.116 2.381 2.646 2.910 3.175 32.9 1.520 1.824 2.128 2.432 2.736 3.040 3.343 3.647 37.9 1.319 1.583 1.847 2.111 2.375 2.639 2.902 3.166 33.0 1.515 1.818 2.121 2.424 2.727 3.030 3.333 3.636 38.0 1.316 1.579 1.842 2.105 2.368 2.632 2.895 3.158 33.1 1.511 1.813 2.115 2.417 2.719 3.021 3.323 3.625 38.1 1.312 1.575 1.837 2.100 2.362 2.625 2.887 3.150 33.2 1.506 1.807 2.108 2.410 2.711 3.012 3.313 3.614 38.2 1.309 1.571 1.832 2.094 2.356 2.618 2.880 3.141 33.3 1.502 1.802 2.102 2.402 2.703 3.003 3.303 3.604 38.3 1.305 1.567 1.828 2.089 2.350 2.611 2.872 3.133 33.4 1.497 1.796 2.096 2.395 2.695 2.994 3.293 3.593 38.4 1.302 1.563 1.823 2.083 2.344 2.604 2.865 3.125 33.5 1.493 1.791 2.090 2.388 2.687 2.985 3.284 3.582 38.5 1.299 1.558 1.818 2.078 2.338 2.597 2.857 3.117 33.6 1.488 1.786 2.083 2.381 2.679 2.976 3.274 3.571 38.6 1.295 1.554 1.813 2.073 2.332 2.591 2.850 3.109 33.7 1.484 1.780 2.077 2.374 2.671 2.967 3.264 3.561 38.7 1.292 1.550 1.809 2.067 2.326 2.584 2.842 3.101 33.8 1.479 1.775 2.071 2.367 2.663 2.959 3.254 3.550 38.8 1.289 1.546 1.804 2.062 2.320 2.577 2.835 3.093 33.9 1.475 1.770 2.065 2.360 2.655 2.950 3.245 3.540 38.9 1.285 1.542 1.799 2.057 2.314 2.571 2.828 3.085 34.0 1.471 1.765 2.059 2.353 2.647 2.941 3.235 3.529 39.0 1.282 1.538 1.795 2.051 2.308 2.564 2.821 3.077 34.1 1.466 1.760 2.053 2.346 2.639 2.933 3.226 3.519 39.1 1.279 1.535 1.790 2.046 2.302 2.558 2.813 3.069 34.2 1.462 1.754 2.047 2.339 2.632 2.924 3.216 3.509 39.2 1.276 1.531 1.786 2.041 2.296 2.551 2.806 3.061 34.3 1.458 1.749 2.041 2.332 2.624 2.915 3.207 3.499 39.3 1.272 1.527 1.781 2.036 2.290 2.545 2.799 3.053 34.4 1.453 1.744 2.035 2.326 2.616 2.907 3.198 3.488 39.4 1.269 1.523 1.777 2.030 2.284 2.538 2.792 3.046 34.5 1.449 1.739 2.029 2.319 2.609 2.899 3.188 3.478 39.5 1.266 1.519 1.772 2.025 2.278 2.532 2.785 3.038 34.6 1.445 1.734 2.023 2.312 2.601 2.890 3.179 3.468 39.6 1.263 1.515 1.768 2.020 2.273 2.525 2.778 3.030 34.7 1.441 1.729 2.017 2.305 2.594 2.882 3.170 3.458 39.7 1.259 1.511 1.763 2.015 2.267 2.519 2.771 3.023 34.8 1.437 1.724 2.011 2.299 2.586 2.874 3.161 3.448 39.8 1.256 1.508 1.759 2.010 2.261 2.513 2.764 3.015 34.9 1.433 1.719 2.006| 2.292 2.579 2.865 3.152 3.438 39.9 1.253 1.504 1.754 2.005 2.256 2.506 2.757 3.008 235 TABLE XXVIII CONTINUED. TABLE OF VELOCITIES OF TUBES IN MEASURING FLUMES, IN FEET PER SECOND. THE TIME OCCUPIED IN PASSING FROM THE UPSTREAM TO THE DOWNSTREAM TRANSIT STATION, AND THE DISTANCE BETWEEN THEM, BEING GIVEN. TIME See's. DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET. TIME. See's. DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET. 50. 60. 70. 80. 90. 100. 110. 120. 50. 60. 70. 80. 90. 100. 110. 120. 40.0 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 45.0 1.111 1.333 1.556 1.778 2.000 2.222 2.444 2.667 40.1 1.247 1.496 1.746 1.995 2.244 2.494 2.743 2.993 45.1 1.109 1.330 1.552 1.774 1.996 2.217 2.439 2.661 40.2 1.244 1.493 1.741 1.990 2.239 2.488 2.736 2.985 45.2 1.106 1.327 1.549 1.770 1.991 2.212 2.434 2.655 40.3 1.241 1.489 1.737 1.985 2.233 2.481 2.730 2.978 45.3 1.104 1.325 1.545 1.766 1.987 2.208 2.428 2.649 40.4 1.238 1.485 1.733 1.980 2.228 2.475 2.723 2.970 45.4 1.101 1.322 1.542 1.762 1.982 2.203 2.423 2.643 40.5 1.235 1.481 1.728 1.975 2.222 2.469 2.716 2.963 45.5 1.099 1.319 1.538 1.758 1.978 2.198 2.418 2.637 40.6 1.232 1.478 1.724 1.970 2.217 2.463 2.709 2.956 45.6 1.096 1.316 1.535 1.754 1.974 2.193 2.412 2.632 40.7 1.229 1.474 1.720 1.966 2.211 2.457 2.703 2.948 45.7 1.094 1.313 1.532 1.751 1.969 2.188 2.407 2.626 40.8 1.225 1.471 1.716 1.961 2.206 2.451 2.696 2.941 45.8 1.092 1.310 1.528 1.747 1.965 2.183 2.402 2.620 40.9 1.222 1.467 1.711 1.956 2.200 2.445 2.689 2.934 45.9 1.089 1.307 1.525 1.743 1.961 2.179 2.397 2.614 41.0 1.220 1.463 1.707 1.951 2.195 2.439 2.683 2.927 46.0 1.087 1.304 1.522 1.739 1.957 2.174 2.391 2.609 41.1 1.217 1.460 1.703 1.946 2.190 2.433 2.676 2.920 46.1 1.085 1.302 1.518 1.735 1.952 2.169 2.386 2.603 41.2 1.214 1.456 1.699 1.942 2.184 2.427 2.670 2.913 46.2 1.082 1.299 1.515 1.732 1.948 2.165 2.381 2.597 41.3 1.211 1.453 1.695 1.937 2.179 2.421 2.663 2.906 46.3 1.080 1.296 1.512 1.728 1.944 2.160 2.376 2.592 41.4 1.208 1.449 1.691 1.932 2.174 2.415 2.657 2.899 46.4 1.078 1.293 1.509 1.724 1.940 2.155 2.371 2.586 41.5 1.205 1.446 1.687 1.928 2.169 2.410 2.651 2.892 46.5 1.075 1.290 1.505 1.720 1.935 2.151 2.366 2.581 41.6 1.202 1.442 1.683 1.923 2.163 2.404 2.644 2.885 46.6 1.073 1.288 1.502 1.717 1.931 2.146 2.361 2.575 41.7 1.199 1.439 1.679 1.918 2.158 2.398 2.638 2.878 46.7 1.071 1.285 1.499 1.713 1.927 ;2.141 2.355 2.570 41.8 1.196 1.435 1.675 1.914 2.153 2.392 2.632 2.871 46.8 1.068 1.282 1.496 1.709 1.923 2.137 2.350 2.564 41.9 1.193 1.432 1.671 1.909 2.148 2.387 2.625 2.864 46.9 1.066 1.279 1.493 1.706 1.919 2.132 2.345 2.559 42.0 1.190 1.429 1.667 1.905 2.143 2.381 2.619 2.857 47.0 1.064 1.277 1.489 1.702 1.915 2.128 2.340 2.553 42.1 1.188 1.425 1.663 1.900 2.138 2.375 2.613 2.850 47.1 1.062 1.274 1.486 1.699 1.911 2.123 2.335 2.548 42.2 1.185 1.422 1.659 1.896 2.133 2.370 2.607 2.844 47.2 1.059 1.271 1.483 1.695 1.907 2.119 2.331 2.542 42.3 1.182 1.418 1.655 1.891 2.128 2.364 2.600 2.837 47.3 1.057 1.268 1.480 1.691 1.903 2.114 2.326 2.537 42.4 1.179 1.415 1.651 1.887 2.123 2.358 2.594 2.830 47.4 1.055 1.266 1.477 1.688 1.899 2.110 2.321 2.532 42.5 1.176 1.412 1.647 1.882 2.118 2.353 2.588 2.824 47.5 1.053 1.263 1.474 1.684 1.895 2.105 2.316 2.526 42.6 1.174 1.408 1.643 1.878 2.113 2.347 2.582 2.817 47.6 1.050 1.261 1.471 1.681 1.891 2.101 2.311 2.521 42.7 1.171 1.405 1.639 1.874 2.108 2.342 2.576 2.810 47.7 1.048 1.258 1.468 1.677 1.887 2.096 2.306 2.516 42.8 1.168 1.402 1.636 1.869 2.103 2.336 2.570 2.804 47.8 1.046 1.255 1.464 1.674 1.883 2.092 2.301 2.510 42.9 1.166 1.399 1.632 1.865 2.098 2.331 2.564 2.797 47.9 1.044 1.253 1.461 1.670 1.879 2.088 2.296 2.505 43.0 1.163 1.395 1.628 1.860 2.093 2.326 2.558 2.791 48.0 1.042 1.250 1.458 1.667 1.875 2.083 2.292 2.500 43.1 1.160 1.392 1.624 1.856 2.088 2.320 2.552 2.784 48.1 1.040 1.247 1.455 1.663 1.871 2.079 2.287 2.495 43.2 1.157 1.389 1.620 1.852 2.083 2.315 2.546 2.778 48.2 3.037 1.245 1.452 1.660 1.867 2.075 2.282 2.490 43.3 1.155 1.386 1.617 1.848 2.079 2.309 2.540 2.771 48.3 1.035 1.242 1.449 1.656 1.863 2.070 2.277 2.484 43.4 1.152 1.382 1.613 1.843 2.074 2.304 2.535 2.765 48.4 1.033 1.240 1.446 1.653 1.860 2.066 2.273 2.479 43.5 1.149 1.379 1.609 1.839 2.069 2.299 2.529 2.759 48.5 1.031 1.237 1.443 1.649 1.856 2.062 2.268 2.474 43.6 1.147 1.376 1.606 1.835 2.064 2.294 2.523 2.752 48.6 1.029 1.235 1.440 1.646 1.852 2.058 2.263 2.469 43.7 1.144 1.373 1.602 1.831 2.059 2.288 2.517 2.746 48.7 1.027 1.232 1.437 1.643 1.848 2.053 2.259 2.464 43.8 1.142 1.370 1.598 1.826 2.055 2.283 2.511 2.740 48.8 1.025 1.230 1.434 1.639 1.844 2.049 2.254 2.459 43.9 1.139 1.367 1.595 1.822 2.050 2.278 2.506 2.733 48.9 1.022 1.227 1.431 1.636 1.840 2.045 2.249 2.454 44.0 1.136 1.364 1.591 1.818 2.045 2.273 2.500 2.727 49.0 1.020 1.224 1.429 1.633 1.837 2.041 2.245 2.449 44.1 1.134 1.361 1.587 1.814 2.041 2.268 2.494 2.721 49.1 1.018 1.222 1.426 1.629 1.833 2.037 2.240 2.444 44.2 1.131 1.357 1.584 1.810 2.036 2.262 2.489 2.715 49.2 1.016 1.220 1.423 1.626 1.829 2.033 2.236 2.439 44.3 1.129 1.354 1.580 1.806 2.032 2.257 2.483 2.709 49.3 1.014 1.217 1.420 1.623 1.826 2.028 2.231 2.434 44.4 1.126 1.351 1.577 1.802 2.027 2.252 2.477 2.703 49.4 1.012 1.215 1.417 1.619 1.822 2.024 2.227 2.429 44.5 1.124 1.348 1.573 1.798 2.022 2.247 2.472 2.697 49.5 1.010 1.212 1.414 1.616 1.818 2.020 2.222 2.424 44.6 1.121 1.345 1.570 1.794 2.018 2.242 2.466 2.691 49.6 1.008 1.210 1.411 1.613 1.815 2.016 2.218 2.419 44.7 1.119 1.342 1.566 1.790 2.013 2.237 2.461 2.685 49.7 1.006 1.207 1.408 1.610 1.811 2.012 2.213 2.414 44.8 1.116 1.339 1.562 1.786 2.009 2.232 2.455 2.679 49.8 1.004 1.205 1.406 1.606 1.807 2.008 2.209 2.410 44.9 1.114] 1.336 1.559 1.782 2.004 2.227 2.450 2.673 49.9 1.002 1.202 1.403 1.603 1.804 2.004 2.204 2.405 236 TABLE XXVIII CONTINUED. TABLE OF VELOCITIES OF TUBES IN MEASURING FLUMES, IN FEET PER SECOND. THE TIME OCCUPIED IN PASSING FROM THE UPSTREAM TO THE DOWNSTREAM TRANSIT STATION, AND THE DISTANCE BETWEEN THEM, BEING GIVEN. TIME See's. DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET. TIME See's. DISTANCE BEHVEEN TIIE TRANSIT STATIONS, IN FEET. 50. 60. 70. 80. 90. 100. 110. 120. 50. 60. 70. 80. 90. 100. 110. 120. 50.0 1.000 1.200 1.400 1.60( 1.800 2.000 2.200 2.400 55.0 0.909 1.091 1.273 1.455 1.636 1.818 2.000 2.182 50.1 0.998 1.198 1.397 1.597 1.796 1.996 2.196 2.395 55.1 0.907 1.089 1.270 1.452 1.633 1.815 1.996 2.178 50.2 0.99G 1.195 1.394 1.594 1.793 1.992 2.191 2.390 55.2 0.906 1.087 1.268 1.449 1.630 1.812 1.993 2.174 50.3 0.994 1.193 1.392 1.590 1.789 1.988 2.187 2.386 55.3 0.904 1.085 1.266 1.447 1.627 1.808 1.989 2.170 50.4 0.992 1.190 1.389 1.587 1.786 1.984 2.183 2.381 55.4 0.903 1.083 1.264 1.444 1.625 1.805 1.986 2.166 50.5 0.990 1.188 1.386 1.584 1.782 1.980 2.178 2.376 55.5 0.901 1.081 1.261 1.441 1.622 1.802 1.982 2.162 50.6 0.988 1.186 1.383 1.581 1.779 1.976 2.174 2.372 55.6 0.899 1.079 1.259 1.439 1.619 1.799 1.978 2.158 50.7 0.986 1.183 1.381 1.578 1.775 1.972 2.17C 2.367 55.7 0.898 1.077 1.257 1.436 1.616 1.795 1.975 2.154 50.8 0.984 1.181 1.378 1.575 1.772 1.969 2.165 2.362 55.8 0.896 1.075 1.254 1.434 1.613 1.792 1.971 2.151 50.9 0.982 1.179 1.375 1.572 1.768 1.965 2.161 2.358 55.9 0.894 1.073 1.252 1.431 1.610 1.789 1.968 2.147 51.0 0.980 1.176 1.373 1.569 1.765 1.961 2.157 2.353 56.0 0.893 1.071 1.250 1.429 1.607 1.786 1.964 2.143 51.1 0.978 1.174 1.370 1.566 1.761 1.957 2.153 2.348 56.1 0.891 1.070 1.248 1.426 1.604 1.783 1.961 2.139 51.2 0.977 1.172 1.367 1.562 1.758 1.953 2.148 2.344 56.2 0.890 1.068 1.246 1.423 1.601 1.779 1.957 2.135 51.3 0.975 1.170 1.365 1.559 1.754 1.949 2.144 2.339 56.3 0.888 1.066 1.243 1.421 1.599 1.776 1.954 2.131 51.4 0.973 1.167 1.362 1.556 1.751 1.946 2.140 2.335 56.4 0.887 1.064 1.241 1.418 1.596 1.773 1.950 2.128 51.5 0.971 1.165 1.359 1.553 1.748 1.942 2.136 2.330 56.5 0.885 1.062 1.239 1.416 1.593 1.770 1.947 2.124 51.6 0.969 1.163 1.357 1.550 1.744 1.938 2.132 2.326 56.6 0.883 1.060 1.237 1.413 1.590 1.767 1.943 2.120 51.7 0.967 1.161 1.354 1.547 1.741 1.934 2.128 2.321 56.7 0.882 1.058 1.235 1.411 1.587 1.764 1.940 2.116 51.8 0.965 1.158 1.351 1.544 1.737 1.931 2.124 2.317 56.8 0.880 1.056 1.232 1.408 1.585 1.761 1.937 2.113 51.9 0.963 1.156 1.349 1.541 1.734 1.927 2.119 2.312 56.9 0.879 1.054 1.230 1.406 1.582 1.757 1.933 2.109 52.0 0.962 1.154 1.346 1.538 1.731 1.923 2.115 2.308 57.0 0.877 1.053 1.228 1.404 1.579 1.754 1.930 2.105 52.1 0.960 1.152 1.344 1.536 1.727 1.919 2.111 2.303 57.1 0.876 1.051 1.226 1.401 1.576 1.751 1.926 2.102 52.2 0.958 1.149 1.341 1.533 1.724 1.916 2.107 2.299 57.2 0.874 1.049 1.224 1.399 1.573 1.748 1.923 2.098 52.3 0.956 1.147 1.338 1.530 1.721 1.912 2.103 2.294 57.3 0.873 1.047 1.222 1.396 1.571 1.745 1.920 2.094 52.4 0.954 1.145 1.336 1.527 1.718 1.908 2.099 2.290 57.4 0.871 1.045 1.220 1.394 1.568 1.742 1.916 2.091 52.5 0.952 1.143 1.333 1.524 1.714 1.905 2.095 2.286 57.5 0.870 1.043 1.217 1.391 1.565 1.739 1.913 2.087 52.6 0.951 1.141 1.331 1.521 1.711 1.901 2.091 2.281 57.6 0.868 1.042 1.215 1.389 1.562 1.736 1.910 2.083 52.7 0.949 1.139 1.328 1.518 1.708 1.898 2.087 2.277 57.7 0.867 1.040 1.213 1.386 1.560 1.733 1.906 2.080 52.8 0.947 1.136 1.326 1.515 1.705 1.894 2.083 2.273 57.8 0.865 1.038 1.211 1.384 1.557 1.730 1.903 2.076 52.9 0.945 1.134 1.323 1.512 1.701 1.890 2.079 2.268 57.9 0.864 1.036 1.209 1.382 1.554 1.727 1.900 2.073 53.0 0.943 1.132 1.321 1.509 1.698 1.887 2.075 2.264 58.0 0.862 1.034 1.207 1.379 1.552 1.724 1.897 2.069 53.1 0.942 1.130 1.318 1.507 1.695 1.883 2.072 2.260 58.1 0.861 1.033 1.205 1.377 1.549 1.721 1.893 2.065 53.2 0.940 1.128 1.316 1.504 1.692 1.880 2.068 2.256 58.2 0.859 1.031 1.203 1.375 1.546 1.718 1.890 2.062 53.3 0.938 1.126 1.313 1.501 1.689 1.876 2.064 2.251 58.3 0.858 1.029 1.201 1.372 1.544 1.715 1.887 2.058 53.4 0.936 1.124 1.311 1.498 1.685 1.873 2.060 2.247 58.4 0.856 1.027 1.199 1.370 1.541 1.712 1.884 2.055 53.5 0.935 1.121 1.308 1.495 1.682 1.869 2.056 2.243 58.5 0.855 1.026 1.197 1.368 1.538 1.709 1.880 2.051 53.6 0.933 1.119 1.306 1.493 1.679 1.866 2.052 2.239 58.6 0.853 1.024 1.195 1.365 1.536 1.706 1.877 2.048 53.7 0.931 1.117 1.304 1.490 1.676 1.862 2.048 2.235 58.7 0.852 1.022 1.193 1.363 1.533 1.704 1.874 2.044 53.8 0.929 1.115 1.301 1.487 1.673 1.859 2.045 2.230 58.8 0.850 1.020 1.190 1.361 1.531 1.701 1.871 2.041 53.9 0.928 1.113 1.299 1.484 1.670 1.855 2.041 2.226 58.9 0.849 1.019 1.188 1.358 1.528 1.698 1.868 2.037 54.0 0.926 1.111 1.296 1.481 1.667 1.852 2.037 2.222 59.0 0.847 1.017 1.186 1.356 1.525 1.695 1.864 2.034 54.1 0.924 1.109 1.294 1.479 1.664 1.S4K 2.033 2.218 59.1 0.846 1.015 1.184 1.354 1.523 1.692 1.861 2.030 54.2 0.923 1.107 1.292 1.476 1.661 1.845 2.030 2.214 59.2 0.845 1.014 1.182 1.351 1.520 1.689 1.858 2.027 54.3 0.921 1.105 1.289 1.473 1.657 1.842 2.026 2.210 59.3 0.843 1.012 1.180 1.349 1.518 1.686 l.55 2.024 54.4 0.919 1.103 1.287 1.471 1.654 1.838 2.022 2.206 59.4 0.842 1.010 1.178 1.347 1.515 1.684 1.852 2.020 54.5 0.917 1.101 1.284 1.468 1.651 1.835 2.018 2.202 59.5 0.840 1.008 1.176 1.345 1.513 1.681 1.849 2.017 54.6 0.916 1.099 1.282 1.465 1.648 1.832 2.015 2.198 59.6 0.839 1.007 1.174 1.342 1.510 1.678 1.846 2.013 54.7 0.914 1.097 1.280 1.463 1.645 1.828 2.011 2.194 59.7 0.838 1.005 1.173 1.340 1.508 1.675 1.843 2.010 54.8 0.912 1.095 1.277 1.460 1.642 1.825 2.007 2.190 59.8 0.836 1.003 1.171 1.338 1.505 1.672 1.839 2.007 54.9 0.911 1.093 1.275 1.457 1.639 1.821 2.004 2.186 59.9 0.835 1.002 1.169 1.336 1.503 1.669 1.836 2.003 237 TABLE XXVIII CONTINUED. TABLE OF VELOCITIES OF TUBES IN MEASURING FLUMES, IN FEET PER SECOND. THE TIME OCCUPIED IN PASSING FROM THE UPSTREAM TO THE DOWNSTREAM TRANSIT STATION, AND THE DISTANCE BETWEEN THEM, BEING GIVEN. TIME. Sec'8. DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET TIME. See's. DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET. 50. 60. 70. 80. 90. 100. 110. 120. 50. 60. 70. 80. 90. 100. 110. 120. 60.0 0.833 1.000 1.167 1.333 1.500 1.667 1.833 2.000 65.0 0.769 0.923 1.077 1.231 1.385 1.538 1.692 1.846 60.1 0.832 0.998 1.165 1.331 1.498 1.664 1.830 1.997 65.1 0.768 0.922 1.075 1.229 1.382 1.536 1.690 1.843 60.2 0.831 0.997 1.163 1.329 1.495 1.661 1.827 1.993 65.2 0.767 0.920 1.074 1.227 1.380 1.534 1.687 1.840 60.3 0.829 0.995 1.161 1.327 1.493 1.658 1.824 1.990 65.3 0.766 0.919 1.072 1.225 1.378 1.531 1.685 1.838 60.4 0.828 0.993 1.159 1.325 1.490 1.656 1.821 1.987 65.4 0.765 0.917 1.070 1.223 1.376 1.529 1.682 1.835 60.5 0.826 0.992 1.157 1.322 1.488 1.653 1.818 1.983 65.5 0.763 0.916 1.069 1.221 1.374 1.527 .679 1.832 60.6 0.825 0.990 1.155 1.320 1.485 1.650 1.815 1.980 65.6 0.762 0.915 1.067 1.220 1.372 1.524 .677 1.829 60.7 0:824 0.988 1.153 1.318 1.483 1.647 1.812 1.977 65.7 0.761 0.913 1.065 1.218 1.370 1.522 .674 1.826 60.8 0.822 0.987 1.151 1.316 1.480 1.645 1.809 1.974 65.8 0.760 0.912 1.064 1.216 1.368 1.520 .672 1.824 60.9 0.821 0.985 1.149 1.314 1.478 1.642 1.806 1.970 65:9 0.759 0.910 1.062 1.214 1.366 1.517 .669 1.821 61.0 0.820 0.984 1.148 1.311 1.475 1.639 1.803 1.967 66.0 0.758 0.909 1.061 1.212 1.364 1.515 1.667 1.818 61.1 0.818 0.982 1.146 1.309 1.473 1.637 1.800 1.964 66.1 0.756 0.908 1.059 1.210 1.362 1.513 1.664 1.815 61.2 0.817 0.980 1.144 1.307 1.471 1.634 1.797 1.961 66.2 0.755 0.906 1.057 1.208 1.360 1.511 1.662 1.813 61.3 0.816 0.979 L142 1.305 1.468 1.631 1.794 1.958 66.3 0.754 0.905 1.056 1.207 1.357 1.508 1.659 1.810 61.4 0.814 0.977 1.140 1.303 1.466 1.629 1.792 1.954 66.4 0.753 0.904 1.054 1.205 1.355 1.506 1.657 1.807 61.5 0.813 0.976 1.138 1.301 1.463 1.626 1.789 1.951 66.5 0.752 0.902 1.053 1.203 1.353 1.504 .654 1.805 61.6 0.812 0.974 1.136 1.299 1.461 1.623 1.786 1.948 66.6 0.751 0.901 1.051 1.201 1.351 1.502 .652 1.802 61.7 0.810 0.972 1.135 1.297 1.459 1.621 1.783 1.945 66.7 0.750 0.900 1.049 1.199 1.349 1.499 .649 1.799 61.8 0.809 0.971 1.133 1.294 1.456 1.618 1.780 1.942 66.8 0.749 0.898 1.048 1.198 1.347 1.497 .647 1.796 61.9 0.808 0.969 1.131 1.292 1.454 1.616 1.777 1.939 66.9 0.747 0.897 1.046 1.196 1.345 1.495 .644 1.794 62.0 0.806 0.968 1.129 1.290 1.452 1.613 1.774 1.935 67.0 0.746 0.896 1.045 1.194 1.343 1.493 .642 1.791 62.1 0.805 0.966 1.127 1.288 1.449 1.610 1.771 1.932 67.1 0.745 0.894 1.043 1.192 1.341 1.490 .639 1.788 62.2 0.804 0.965 1.125 1.286 1.447 1.608 1.768 1.929 67.2 0.741 0.893 1.042 1.190 1.339 1.488 1.637 1.786 62.3 0.803 0.963 1.124 1.284 1.445 1.605 1.766 1.926 67.3 0.743 0.892 1.040 1.189 1.337 1.486 1.634 1.783 62.4 0.801 0.962 1.122 1.282 1.442 1.603 1.763 1.923 67.4 0.742 0.890 1.039 1.187 1.335 1.484 1.632 1.780 62.5 0.800 0.960 1.120 1.280 1.440 1.600 1.760 1.920 67.5 0.741 0.839 1.037 1.185 1.333 1.481 1.630 1.778 62.6 0.799 0.958 1.118 1.278 1.438 1.597 1.757 1.917 67.6 0.740 0.888 1.036 1.183 1.331 1.479 1.627 1.775 62.7 0.797 0.957 1.116 1.276 1.435 1.595 1.754 1.914 67.7 0.739 0.886 1.034 1.182 1.329 1.477 1.625 1.773 62.8 0.796 0.955 1.115 1.274 1.433 1.592 1.752 1.911 67.8 0.737 0.885 1.032 1.180 1.327 1.475 1.622 1.770 62.9 0.795 0.954 1.113 1.272 1.431 1.590 1.749 1.908 67.9 0.736 0.884 1.031 1.178 1.325 1.473 1.620 1.767 63.0 0.794 0.952 1.111 1.270 1.429 1.587 1.746 1.905 68.0 0.735 0.882 1.029 1.176 1.324 1.471 1.618 1.765 63.1 0.792 0.951 1.109 1.268 1.426 1.585 1.743 1.902 68.1 0.734 0.881 1.028 1.175 1.322 1.468 1.615 1.762 63.2 0.791 0.949 1.108 1.266 1.424 1.582 1.741 1.899 68.2 0.733 0.880 1.026 1.173 1.320 1.466 1.613 1.760 63.3 0.790 0.948 1.106 1.264 1.422 1.580 1.738 1.896 68.3 0.732 0.878 1.025 1.171 1.318 1.464 1.611 1.757 63.4 0.789 0.946 1.104 1.262 1.420 1.577 1.735 1.893 68.4 0.731 0.877 1.023 1.170 1.316 1.462 1.608 1.754 63.5 0.787 0.945 1.102 1.260 1.417 1.575 1.732 1.890 68.5 0.730 0.876 1.022 1.168 1.314 1.460 1.606 1.752 63.6 0.786 0.943 1.101 1.258 1.415 1.572 1.730 1.887 68.6 0.729 0.875 1.020 1.166 1.312 1.458 1.603 1.749 63.7 0.785 0.942 1.099 1.256 1.413 1.570 1.727 1.884 68.7 0.728 0.873 1.019 1.164 1.310 1.456 1.601 1.747 63.8 0.784 0.940 1.097 1.254 1.411 1.567 1.721 1.881 68.8 0.727 0.872 1.017 1.163 1.308 1.453 1.599 1.744 63.9 0.782 0.939 1.095 1.252 1.408 1.565 1.721 1.878 68.9 0.726 0.871 1.016 1.161 1.306 1.451 1.597 1.742 64.0 0.781 0.937 1.094 1.250 1.406 1.562 1.719 1.875 69.0 0.725 0.870 1.014 1.159 1.304 1.449 1.594 1.739 64.1 0.780 0.936 1.092 1.248 1.404 1.560 1.716 1.872 69.1 0.724 0.868 1.013 1.158 1.302 1.447 1.592 1.737 64.2 0.779 0.935 1.090 1.-246 1.402 1.553 1.713 1.869 69.2 0.723 0.867 1.012 1.156 1.301 1.445 1.590 1.734 64.3 0.778 0.933 1.089 1.244 1.400 1.555 1.711 1.866 69.3 0.722 0.866 1.010 1.154 1.299 1.443 1.587 1.732 64.4 0.776 0.932 1.087 1.242 1.398 1.553 1.708 1.863 69.4 0.720 0.865 1.009 1.153 1.297 1.441 1.585 1.729 64.5 0.775 0.930 1.085 1.240 1.395 1.550 1.705 1.860 69.5 0.719 0.863 1.007 1.151 1.295 1.439 1.583 1.727 64.6 0.774 0.929 1.084 1.238 1.393 1.548 1.703 1.858 69.6 0.718 0.862 l.OOC 1.149 1.293 1.437 1.580 1.724 64.7 0.773 0.927 1.082 1.236 1.391 1.546 1.700 1.855 69.7 0.717 0.861 1.004 1.148 1.291 1.435 1.578 1.722 C, 1,S 0.772 0.926 1.080 1.235 1.389 1.543 1.698 1.852 69.8 0.716 0.860 1.003 1.146 1.289 1.433 1.576 1.719 64.9 0.770 0.924 1.079 1.233 1.387 1.541 1.695 1.849 69.9 0.715 0.858 1.001 1.144 1.288 1.431 1.574 1.717 238 TABLE XXVIII CONTINUED. TABLE OF VELOCITIES OF TUBES IN MEASURING FLUMES, IN FEET PER SECOND. THE TIME OCCUPIED IN PASSING FROM THE UPSTREAM TO THE DOWNSTREAM TRANSIT STATION, AND THE DISTANCE BETWEEN THEM, BEING GIVEN. TIME Sec'B. DISTANCE BETWEEN TILE TRANSIT STATIONS, IN FEET. TIME. See's. DISTANCE BETWEEN TIIE TRANSIT STATIONS, IN FEET. 50. 60. 70. 80. 90. 100. 110. 120. 50. 60. 70. 80. 90. 100. 110. 120. 70.0 0.714 0.857 1.000 1.143 1.286 1.429 1.571 1.714 75.0 0.667 0.800 0.933 1.067 1.200 1.333 1.467 1.600 70.1 0.713 0.850 0.999 1.141 1.284 1.427 1.569 1.712 75.1 0.666 0.799 0.932 1.065 1.198 1.332 1.465 1.598 70.2 0.712 0.855 0.997 1.140 1.282 1.425 1.567 1.709 75.2 0.665 0.798 0.931 1.064 1.197 1.330 1.463 1.596 70.3 0.711 0.853 0.996 1.138 1.280 1.422 1.565 1.707 75.3 0.664 0.797 0.930 1.062 1.195 1.328 1.461 1.594 70.4 0.710 0.852 0.994 1.136 1.278 1.420 1.562 1.705 75.4 0.663 0.796 0.928 1.061 1.194 1.326 1.459 1.592 70.5 0.709 0.851 0.993 1.135 1.277 1.418 1.560 1.702 75.5 0.662 0.795 0.927 1.060 1.192 1.325 1.457 1.589 : 70.6 0.708 0.850 0.992 1.133 1.275 1.416 1.558 1.700 75.6 0.661 0.794 0.926 1.058 1.190 1.323 1.455 1.587 , 70.7 0.707 0.849 0.990 1.132 1.273 1.414 1.556 1.697 75.7 0.661 0.793 0.925 1.057 1.189 1.321 1.453 1.585 70.8 0.706 0.847 0.989 1.130 1.271 1.412 1.554 1.695 75.8 0.660 0.792 0.923 1.055 1.187 1.319 1.451 1.583 70.9 0.705 0.846 0.987 1.128 1.269 1.410 1.551 1.693 75.9 0.659 0.791 0.922 1.054 1.186 1.318 1.449 1.581 71.0 0.704 0.845 0.986 1.127 1.268 1.408 1.549 1.690 76.0 0.658 0.789 0.921 1.053 1.184 1.316 1.447 1.579 71.1 0.703 0.844 0.985 1.125 1.266 1.406 1.547 1.688 76.1 0.657 0.788 0.920 1.051 1.183 1.314 1.445 1.577 71.2 0.702 0.843 0.983 1.124 1.264 1.404 1.545 1.685 76.2 0.656 0.787 0.919 1.050 1.181 1.312 1.444 1.575 71.3 0.701 0.842 0.982 1.122 1.262 1.403 1.543 1.683 76.3 0.655 0.786 0.917 1.048 1.180 1.311 1.442 1.573 71.4 0.700 0.840 0.980 1.120 1.261 1.401 1.541 1.681 76.4 0.654 0.785 0.916 1.047 1.178 1.309 1.440 1.571 71.5 O.G99 0.839 0.979 1.119 1.259 1.399 1.538 1.678 76.5 0.654 0.784 0.915 1.046 1.176 1.307 1.438 1.569 71.6 0.698 0.838 0.978 1.117 J.257 1.397 1.536 1.676 76.6 0.653 0.783 0.914 1.044 1.175 1.305 1.436 1.567 71.7 0.697 0.837 0.976 1.116 1.255 1.395 1.534 1.674 76.7 0.652 0.782 0.913 1.043 1.173 1.304 1.434 1.565 71.8 0.696 0.836 0.975 1.114 1.253 1.393 1.532 1.671 76.8 0.651 0.781 0.911 1.042 1.172 1.302 1.432 1.562 71.9 0.695 0.834 0.974 1.113 1.252 1.391 1.530 1.669 76.9 0.650 0.780 0.910 1.040 1.170 1.300 1.430 1.560 72.0 O.G94 0.833 0.972 1.111 1.250 1.389 1.528 1.667 77.0 0.649 0.779 0.909 1.039 1.169 1.299 1.429 1.558 72.1 0.693 0.832 0.971 1.110 .248 1.387 1.526 1.664 77.1 0.649 0.778 0.908 1.038 1.167 1.297 1.427 1.556- 72.2 0.693 0.831 0.970 1.108 .247 1.385 1.524 1.662 77.2 0.648 0.777 0.907 1.036 1.166 1.295 1.425 1.554 72.3 0.692 0.830 0.968 1.107 .245 1.383 1.521 1.660 77.3 0.647 0.776 0.906 1.035 1.164 1.294 1.423 1.552 72.4 0.691 0.829 0.967 1.105 .243 1.381 1.519 1.657 77.4 0.646 0.775 0.904 1.034 1.163 1.292 1.421 1.550 72.5 0.610 0.828 0.966 1.103 .241 .379 1.517 1.655 77.5 0.645 0.774 0.903 .032 1.161 1.290 1.419 1.548 72.6 0.689 0.826 0.964 1.102 .240 .377 1.515 1.653 77.6 0.644 0.773 0.902 .031 1.160 1.289 1.418 1.546 72.7 0.688 0.825 0.963 1.100 .238 .376 1.513 1.651 77.7 0.644 0.772 0.901 .030 1.158 1.287 1.416 1.544 72.8 0.687 0.824 0.962 1.099 .236 .374 1.511 1.648 77.8 0.643 0.771 0.900 .028 1.157 1.285 1.414 1.542 72.9 0.686 0.823 0.960 1.097 1.235 .372 1.509 1.646 77.9 0.612 0.770 0.899 .027 1.155 1.284 1.412 1.540 73.0 0.685 0.822 0.959 1.096 1.233 .370 1.507 1.614 78.0 0.641 0.769 0.897 1.026 1.154 1.282 1.410 1.538 73.1 0.684 0.821 0.958 1.094 1.231 .368 1.505 1.612 78.1 0.640 0.768 0.896 1.024 1.152 1.280 1.408 1.536 73.2 0.683 0.820 0.956 1.093 1.230 .366 1.503 1.639 78.2 0.639 0.767 0.895 1.023 1.151 1.279 1.407 1.535 73.3 0.682 0.819 0.955 1.091 1.228 .364 1.501 1.637 78.3 0.639 0.766 0.894 1.022 1.149 1.277 1.405 1.533 73.4 0.681 0.817 0.954 1.090 1.226 1.362 1.499 1.635 78.4 0.638 0.765 0.893 1.020 1.148 1.276 1.403 1.531 73.5 0.680 0.816 0.952 1.088 1.224 1.361 1.497 1.633 78.5 0.637 0.764 0.892 1.019 1.146 1.274 1.401 1.529 73.6 0.679 0.815 0.951 1.087 1.223 1.359 1.495 1.630 78.6 0.636 0.763 0.891 1.018 1.145 1.272 1.399 1.527 73.7 0.678 0.814 0.950 .085 1.221 1.357 1.493 1.628 78.7 0.635 0.762 0.889 1.017 1.144 1.271 1.398 1.525 73.8 0.678 0.813 0.949 .084 1.220 .355 1.491 1.626 78.8 0.635 0.761 0.888 1.015 1.142 1.269 1.396 1.523 73.9 0.677 0.812 0.947 .083 1.218 .353 1.488 1.624 78.9 0.634 0.760 0.887 1.014 1.141 1.267 1.394 1.521 74.0 0.676 0.811 0.946 .081 .216 .351 1.486 1.622 79.0 0.633 0.759 0.886 1.013 1.139 1.266 1.392 1.519 74.1 0.675 0.810 0.945 .080 .215 .35(1 1.484 1.619 79.1 0.632 0.759 0.885 1.011 1.138 1.264 1.391 1.517 74.2 0.674 0.809 0.943 .078 .213 .34S 1.482 1.617 79.2 0.631 0.758 0.884 1.010 1.136 1.263 1.389 1.515 74.3 0.673 0.808 0.942 .077 .211 .34(1 1.480 1.615 79.3 0.631 0.757 0.883 1.009 1.135 1.261 1.387 1.513 74.4 0.672 0.806 0.941 .075 .210 .344 1.478 1.613 79.4 0.630 0.756 0.882 1.008 1.134 1.259 1.385 1.511 74.5 0.671 0.805 0.940 1.074 1.208 .342 1.477 1.611 79.5 0.629 0.755 0.881 1.006 1.132 1.258 1.384 1.509 74.6 0.670 0.804 0.938 1.072 1.206 .340 1.475 1.609 79.6 0.628 0.754 0.879 1.005 1.131 1.256 1.382 1.508 74.7 0.669 0.803 0.937 1.071 1.205 1.339 1.473 1.606 79.7 0.627 0.753 0.878 1.004 1.129 1.255 1.380 1.506 74.8 e.efis 0.802 0.93G 1.070 1.203 1.337 1.471 1.604 79.8 0.627 0.752 0.877 1.003 1.128 1.253 1.378 1.504 74.9 0.668 0.801 0.935 1.068 1.202 1.335 1.469 1.602 79.9 0.626 0.751 0.876 1.001 1.126 1.252 1.377 1.502 239 TABLE XXVIII CONTINUED. TABLE OF VELOCITIES OF TUBES IN MEASURING FLUMES, IN FEET PER SECOND. THE TIME OCCUPIED IN PASSING FROM THE UPSTREAM TO THE DOWNSTREAM TRANSIT STATION, AND THE DISTANCE BETWEEN THEM, BEING GIVEN. TIME. See's. DIS 50. rANCE BETWEEN THE TRANSIT STATIONS, IN FEET. TIME. See's. DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET. 60. 70. 80. 90. 100. 110. 120. 50. 60. 70. 80. 90. 100. 110. 120. 80.0 0.625 0.750 0.875 1.000 1.125 1.250 1.375 1.500 85.0 0.588 0.706 0.824 0.941 1.059 1.176 1.294 1.412 80.1 0.624 0.749 0.874 0.999 1.124 1.248 1.373 1.498 85.1 0.588 0.705 0.823 0.940 1.058 1.175 1.293 1.410 80.2 0.623 0.748 0.873 0.998 1.122 1.247 1.372 1.496 85.2 0.587 0.704 0.822 0.939 1.056 1.174 1.291 1.408 80.3 0.623 0.747 0.872 0.996 1.121 1.245 1.370 1.494 85.3 0.586 0.703 0.821 0.938 1.055 1.172 1.290 1.407 80.4 0.622 0.746 0.871 0.995 1.119 1.244 1.368 1.493 85.4 0.585 0.703 0.820 0.937 1.054 1.171 1.288 1.405 80.5 0.621 0.745 0.870 0.994 1.118 1.242 1.366 1.491 85.5 0.585 0.702 0.819 0.936 1.053 1.170 1.287 1.404 80.6 0.620 0.744 0.868 0.993 1.117 1.241 1.365 1.489 85.6 0.584 0.701 0.818 0.935 1.051 1.168 1.285 1.402 80.7 0.620 0.743 0.867 0.991 1.115 1.239 1.363 1.487 85.7 0.583 0.700 0.817 0.933 1.050 1.167 1.284 1.400 80.8 0.619 0.743 0.866 0.990 1.114 1.238 1.361 1.485 85.8 0.583 0.699 0.816 0.932 1.049 1.166 1.282 1.399 80.9 0.618 0.742 0.865 0.989 1.112 1.236 1.360 1.483 85.9 0.582 0.698 0.815 0.931 1.048 1.164 1.281 1.897 81.0 0.617 0.741 0.864 0.988 1.111 1.235 1.358 1.481 86.0 0.581 0.698 0.814 0.930 1.047 1.163 1.279 1.395 81.1 0.617 0.740 0.863 0.986 1.110 1.233 1.356 1.480 86.1 0.581 0.697 0.813 0.929 1.045 1.161 1.278 1.394 81.2 0.616 0.739 0.862 0.985 1.108 1.232 1.355 1.478 86.2 0.580 0.696 0.812 0.928 1.044 1.160 1.276 1.392 81.3 0.615 0.738 0.861 0.984 1.107 1.230 1.353 1.476 86.3 0.579 0.695 0.811 0.927 1.043 1.159 1.275 1.390 81.4 0.614 0.737 0.860 0.983 1.106 1.229 1.351 1.474 86.4 0.579 0.694 0.810 0.926 1.042 1.157 1.273 1.389 81.5 0.613 0.736 0.859 0.982 1.104 1.227 1.350 1.472 86.5 0.578 0.694 0.809 0.925 1.040 1.156 1.272 1.387 81.6 0.613 0.735 0.858 0.980 1.103 1.225 1.348 1.471 86.6 0.577 0.693 0.808 0.924 1.039 1.155 1.270 1.386 81.7 0.612 0.734 0.857 0.979 1.102 1.224 1.346 1.469 86.7 0.577 0.692 0.807 0.923 1.088 1.153 1.269 1.384 81.8 0.611 0.733 0.856 0.978 1.100 1.222 1.345 1.467 86.8 0.576 0.691 0.806 0.922 1.037 1.152 1.267 1.382 81.9 0.611 0.733 0.855 0.977 1.099 1.221 1.343 1.465 86.9 0.575 0.690 0.806 0.921 1.036 1.151 1.266 1.381 82.0 0.610 0.732 0.854 0.976 1.098 1.220 1.341 1.463 87.0 0.575 0.690 0.805 0.920 1.034 1.149 1.264 1.379 82.1 0.609 0.731 0.853 0.974 1.096 1.218 1.340 1.462 87.1 0574 0.689 0.804 0.918 1.033 1.148 1.263 .378 82.2 0.608 0.730 0.852 0.973 1.095 1.217 1.338 1.460 87.2 0.573 0.688 0.803 0.917 1.032 1.147 1.261 .376 82.3 0.608 0.729 0.851 0.972 1.094 1.215 1.337 1.458 87.3 0.573 0.687 0.802 0.916 1.031 1.145 1.260 .375 82.4 0.607 0.728 0.850 0.971 1.092 1.214 1.335 1.456 87.4 0.572 0.686 0.801 0.915 1.030 1.144 1.259 .373 82.5 0.606 0.727 0.848 0.970 1.091 1.212 1.333 1.455 87.5 0.571 0.686 0.800 0.914 1.029 1.143 1.257 .371 82.6 0.605 0.726 0.847 0.969 1.090 1.211 1.332 1.453 87.6 0.571 0.685 0.799 0.913 1.027 1.142 1.256 .370 82.7 0.605 0.726 0.846 0.967 1.088 1.209 1.330 1.451 87.7 0.570 0.684 0.798 0.912 1.026 1.140 1.254 1.368 82.8 0.604 0.725 0.845 0.966 1.087 .208 1.329 1.449 87.8 0.569 0.683 0.797 0.911 1.025 1.139 1.253 1.367 82.9 0.603 0.724 0.844 0.965 1.086 1.206 1.327 1.448 87.9 0.569 0.683 0.796 0.910 1.024 1.138 1.251 1.365 83.0 0.602 0.723 0.843 0.964 1084 1.205 1.325 1.446 88.0 0.568 0.632 0.795 0.909 1.023 1.136 1.250 1.364 83.1 0.602 0.722 0.842 0.963 1.083 1.203 1.324 1.444 88.1 0.568 0.681 0.795 0.908 1.022 1.135 1.249 1.362 83.2 0.601 0.721 0.841 0.962 1.082 1.202 1.322 1.442 88.2 0.567 0.680 0.794 0.907 1.020 1.134 1.247 1.361 83.3 0.600 0.720 0.840 0.960 1.080 1.200 1.321 1.441 88.3 0.566 0.680 0.793 0.906 1.019 1.133 1.246 1.359 83.4 0.600 0.719 0.839 0.959 l.O/'J 1.199' 1.319 1.439 88.4 0.566 0.679 0.792 0.905 1.018 1.131 1.244 1.357 83.5 0.599 0.719 0.838 0.958 1.078 .198 1.317 1.437 88.5 0.565 0.678 0.791 0.904 1.017 1.130 1.243 1.356 83.6 0.598 0.718 0.837 0.957 1.077 .196 1.316 1.435 *88.6 0.564 0.677 0.790 0.903 1.016 1.129 1.242 .354 83.7 0.597 0.717 0.836 0.956 1.075 .195 1.314 1.434 88.7 0.564 0.676 0.789 0.902 1.015 1.127 1.240 .353 83.8 0.597 0.716 0.835 0.95-5 1.074 .193 1.313 1.432 88.8 0.563 0.676 0.788 0.901 1.014 1.126 1.239 .351 83.9 0.596 0.715 0.834 0.954 1.073 .192 1.311 1.430 88.9 0.562 0.675 0.787 0.900 1.012 1.125 1.237 .350 84.0 0.595 0.714 0.833 0.952 1.H71 1.190 1.310 1.429 89.0 0.562 0.674 0.787 0.899 1.011 1.124 1.236 .348 84.1 0.595 0.713 0.832 0.951 1.070 1.189 1.308 1.427 89.1 0.561 ff.673 0.786 0.898 1.010 1.122 1.235 .347 84.2 0.594 0.713 0.831 0.950 I.ii69 1.188 1.306 1.425 89.2 0.561 0.673 0.785 0.897 1.009 1.121 1.233 .345 84.3 0.593 0.712 0.830 0.949 1.; 68 1.186 1.305 1.423 89.3 0.560 0.672 0.784 0.896 1.008 1.120 1.232 1.344 84.4 0.592 0.711 0.829 0.948 1.166 1.185 1.303 1.422 89.4 0.559 0.671 0.783 0.895 1.007 1.119 1.230 1.342 84.5 0.592 0.710 0.828 0.947 1.065 1.183 1.302 1.420 89.5 0.559 0.670 0.782 0,894 1.006 1.117 1.229 1.341 84.6 0.591 0.709 0.827 0.946 1.064 1.182 1.300 1.418 89.6 0.558 0.670 0.781 0.893 1.004 1.116 1.228 1.339 84.7 0.590 0.708 0.826 0.945 1.063 1.181 1.299 1.417 89.7 0.557 0.66 i 0.780 0.892 1.003 1.115 1.226 1.338 84.8 0.590 0.708 0.825 0.943 1.061 1.179 1.297 1.415 89.8 0.557 0.66 -i 0.780 0.891 1.002 1.114 1.225 1.336 84.9 0.589 0.707 0.824 0.942 1.060 1.178 1.296 1 1.413 89.9 0.556 0.667 0.779 0.890 1.001 1.112 1.224 1.335 240 TABLE XXVIII CONTINUED. TABLE OP VELOCITIES OF TUBES IN MEASURING FLUMES, IN FEET PER SECOND. THE TIME OCCUPIED IN PASSING FROM THE UPSTREAM TO THE DOWNSTREAM TRANSIT STATION, AND THE DISTANCE BETWEEN THEM, BEING GIVEN. TIME Sec DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET. TIME See's. DISTANCE BETWEEN THE TRANSIT STATIONS, IN FEET. 50. 60. 70. 80. 90. 100. 110. 120. 50. 60. 70. 80. 90. 100. 110. 120. 90.0 0.556 0.667 0.778 0.889 1.000 1.111 1.222 1.333 95.0 0.5261 0.632 0.737 0.842 ! 0.947 1.053 1.158 1.263 90.1 0.555 0.66G 0.777 0.888 0.999 1.110 1.221 1.332 95.1 0.526 0.631 0.736 0.841 ! 0.946 1.052i 1.157 1.262 90.2 0.554 0.665 0.776 0.887 0.998 1.109 1.220 1.330 95.2 0.525 0.630 0.735 0.840 0.945 1.050 1.155 1.261 90.3 0.554 0.664 0.775 0.886 0.997 1.107 1.218 1.329 95.3 0.525 0.630 0.735 0.839 0.944 1.049 1.154 1.259 90.4 0.553 0.664 0.774 0.885 0.996 1.106 1.217 1.327 95.4 0.524 0.629 0.734 0.839 0.943 1.048 1.153 1.258 90.5 0.552 0.663 0.773 0.884 0.994 1.105 1.215 1.326 95.5 0.524 0.628 0.733 0.838 0.942 1.047 1.152 1.257 90.6 0.552 0.662 0.773 0.883 0.993 1.104 1.214 .1.325 95.6 0.523 0.628 0.732 0.837 0.941 1.046 1.151 1.255 90.7 0.551 0.662 0.772 0.882 0.992 1.103 1.213 1.323 95.7 0.522 0.627 0.731 0.836 0.940 1.045 1.149 1.254 90.8 0.551 0.661 0.771 0.881 0.991 1.101 1.211 1.322 95.8 0.522 0.626| 0.731 0.835 0.939 1.044 1.148 1.253 90.9 0.550 0.660 0.770 0.880 0.990 1.100 1.210 1.320 95.9 0.521 0.626 0.730 0.834 0.938 1.043 1.147 1.251 91.0 0.549 0.659 0.769 0.879 0.989 1.099 1.209 1.319 96.0 0.521 0.625 0.729 0.833 0.937 1.042 1.146 1.250 91.1 0.549 0.659 0.768 0.878 0.988 1.098 1.207 1.317 96.1 0.520 0.624 0.728 0.832 0.937 1.041 1.145 1.249 91.2 0.548 0.658 0.768 0.877 0.987 1.096 1.206 1.316 96.2 0.520 0.624 0.728 0.832 0.936 1.040 1.143 1.247 91.3 0.548 0.657 0.767 0.876 0.986 1.095 1.205 1.314 96.3 0.519 0.623 0.727 0.831 0.935 1.038 1.142 1.246 91.4 0.547 0.656 0.766 0.875 0.985 1.094 1.204 1.313 96.4 0.519 0.622 0.726 0.830 0.934 1.037 1.141 1.245 91.5 0.546 0.656 0.765 0.874 0.984 1.093 1.202 1.311 96.5 0.518 0.622 0.725 0.829 0.933 1.036 1.140 1.244 91.6 0.546 0.655 0.764 0.873 0.983 1.092 1.201 1.310 96.6 0.518 0.621 0.725 0.828 0.932 1.035 1.139 1.242 91.7 0.545 0.654 0.763 0.872 0.981 1.091 1.200 1.309 96.7 0.517 0.620 0.724 0.827 0.931 1.034 1.138 1.241 91.8 0.545 0.654 0.763 0.871 0.980 1.089 1.198 1.307 96.8 0.517 0.620 0.723 0.826 0.930 1.033 1.136 1.240 91.9 0.544 0.653 0.762 0.871 0.979 1.088 1.197 1.306 96.9 0.516 0.619 0.722 0.826 0.929 1.032 1.135 1.238 92.0 0.543 0.652 0.761 0.870 0.978 1.087 1.196 1.304 97.0 0.515 0.619 0.722 0.825 0.928 1.031 1.134 1.237 92.1 0.543 0.651 0.760 0.869 0.977 1.086 1.194 1.303 97.1 0.515 0.618 0.721 0.824 0.927 1.030 1.133 1.236 92.2 0.542 0.651 0.759 0.868 0.976 1.085 1.193 1.302 97.2 0.514 0.617 0.720 0.823 0.926 1.029 1.132 1.235 92.3 0.542 0.650 0.758 0.867 0.975 1.083 1.192 1.300 97.3 0.514 0.617 0.719 0.822 0.925 1.028 1.131 1.233 92.4 0.541 0.649 0.758 0.866 0.974 1.082 1.190 1.299 97.4 0.513 0,616 0.719 0.821 0.924 1.027 1.129 1.232 92.5 0.541 0.649 0.757 0.865 0.973 1.081 1.189 1.297 97.5 0.513 0.615 0.718 0.821 0.923 1.026 1.128 1.231 92.6 0.540 0.648 0.756 0.864 0.972 1.080 1.188 1.296 97.6 0.512 0.615 0.717 0.820 0.922 1.025 1.127 1.230 92.7 0.539 0.647 0.755 0.863 0.971 1.079 1.187 1.294 97.7 0.512 0.614 0.716 0.819 0.921 1.024 1.126 1.228 92.8 0.53!) 0.647 0.754 0.862 0.970 1.078 1.185 1.293 97.8 0.511 0.613 0.716 0.818 0.920 1.022 1.125 1.227 92.9 0.538 0.646 0.753 0.861 0.969 1.076 1.184 1.292 97.9 0.511 0.613 0.715 0.817 0.919 1.021 1.124 1.226 93.0 0.538 0.645 0.753 0.860 0.968 1.075 1.183 1.290 98.0 0.510 0.612 0.714 0.816 0.918 1.020 1.122 1.224 J3.1 0.537 0.644 0.752 0.859 0.967 1.074 1.182 1.289 98.1 0.510 0.612 0.714 0.815 0.917 1.019 1.121 1.223 93.2 0.536 0.644 0.751 0.858 0.966 1.073 1.180 1.288 98.2 0.509 0.611 0.713 0.815 0.916 1.018 1.120 1.222 93.3 0.536 0.643 0.750 0.857 0.965 1.072 1.179 1.286 98.3 0.509] 0.610 0.712 0.814 0.916 1.017 1.119 1.221 93.4 0.535 0.642 0.749 0.857 0.964 1.071 1.178 1.285 98.4 0.508 0.610 0.711 0.813 0.915 1.016 1.118 1.220 93.5 0.535 0.642 0.749 0.856 0.963 1.070 .176 1.283 98.5 0.508 0.609 0.711 0.812 0.914 1.015 1.117 1.218 93.6 0.534 0.641 0.748 0.855 0.962 1.068 .175 1.282 98.6 0.507 0.609 0.710 0.811 0.913 1.014 1.116 1.217 93.7 0.534 0.640 0.747 0.854 0.961 1.067 .174 1.281 98.7 0.507 0.608 0.709 0.811 0.912 1.013 1.114 1.216 93.8 0.533 0.640 0.746 0.853 0.959 1.066 .173 1.279 98.8 0.506 0.607 0.709 0.810 0.911 1.012 1.113 1.215 93.9 0.532 0.639 0.745 0.852 0.958 1.065 .171 1.278 98.9 0.506 0.607 0.708 0.809 0.910 1.011 1.112 1.213 94.0 0.532 0.638 0.745 0.851 0.957 1.064 1.170 1.277 99.0 0.505 0.606 0.707 0.808 0.909 1.010 1.111 1.212 94.1 0.531 0.638 0.744 0.850 0.956 1.063 1.169 1.275 99.1 0.505 0.605 0.706 0.807 0.908 1.009 1.110 1.211 94.2 0.531 0.637 0.743 0.849 0.955 .062 1.168 1.274 99.2 0.504 0.605 0.706 0.806 0.907 1.008 1.109 1.210 94.3 0.530 0.636 0.742 0.848 0.954 .060 1.166 1.273 99.3 0.504 0.604 0.705 0.806 0.906 1.007 1.108 1.208 94.4 0.530 0.636 0.742 0.847 0.953 .059 1.165 1.271 99.4 0.503 0.604 0.704 0.805 0.905 1.006 1.107 1.207 94.5 0.529 0.635 0.741 0.847 0.952 .058 1.164 1.270 99.5 0.503 0.603 0.704 0.804 0.905 1.005 1.106 1.206 94.6 0.529 0.634 0.740 0.846 0.951 .057 1.163 1.268 99.6 0.502 0.602 0.703 0.803 0.904 1.004 1.104 1.205 94.7 0.528 0.634 0.739 0.845 0.950 .056 1.162 1.267 99.7 0.502 0.602 0.702 0.802 0.903 1.003 1.103 1.204 94.8 0.527 0.633 0.738 0.844 0.949 .055 1.160 1.266 99.8 0.501 0.601 0.701 0.802 0.902 1.002 1.102 1.202 94.9 0.527 0.632 0.738 0.843 0.948 1.054 1.159 1.264 99.9 0.501 0.601 0.701 0.801 0.901 1.001 1.101 1.201 100.0 0.500 0.600 0.700 0.800 0.900 1.000 1.100 1.200 241 TABLE XXIX. VALUES OF THE COEFFICIENT (l 0.11G (y^D 0.1)). D Value of the Coefficient. Logarithm of the Coefficient. D Value of the Coefficient. Logarithm of the Coefficient. 0.000 1.01160 0.0050088 0.050 0.98566 T.9937271 0.001 1.00793 0.0034304 0.051 0.98540 T.9936126 0.002 1.00641 0.0027749 0.052 0.98515 T.9935024 0.003 1.00525 0.0022741 0.053 0.98489 T.9933877 0.004 1.00426 0.0018462 0.054 0.98464 T.9932775 0.005 1.00340 0.0014741 0.055 0.98440 T.9931716 0.006 1.00261 0.0011320 0.056 0.98415 T.9930613 0.007 1.00189 0.0008200 0.057 0.98391 T.9929554 0.008 1.00122 0.0005295 0.058 0.98366 T.9928450 0.009 1.00060 0.0002605 0.059 0.98342 T.9927390 0.010 1.00000 0.0000000 0.060 0.98319 T.9926375 0.011 0.99943 T.9997524 0.061 0.98295 T.9925314 0.012 0.99889 T.9995177 0.062 0.98272 T.9924298 0.013 0.99837 T.9992915 0.063 0.98248 T.9923237 0.014 0.99787 1.9990740 0.064 0.98225 T.9922220 0.015 0.99739 T.9988650 0.065 0.98203 T.9921248 0.016 0.99693 T.9986647 0.066 0.98180 T.9920230 0.017 0.99648 T.9984686 0.067 0.98157 T. 9919213 0.018 0.99604 T.9982768 0.068 0.98135 T.9918239 0.019 0.99561 T.9980893 0.069 0.98113 T.9917266 0.020 0.99520 T.9979104 0.070 0.98091 T.9916292 0.021 0.99479 T.9977314 0.071 0.98069 1.9915317 0.022 0.99439 T.9975567 0.072 0.98047 T.99 14343 0.023 0.99401 T.9973908 0.073 0.98026 T.9913413 0.024 0.99363 T.9972247 0.074 0.98004 T.9912438 0.025 0.99326 T.9970629 0.075 0.97983 T.9911507 0.026 0.99290 T.9969055 0.076 0.97962 T.9910576 0.027 0.99254 T.9967480 0.077 0.97941 T.9909645 0.028 0.99219 T.9965948 0.078 0.97920 T.9908714 0.029 0.99185 T.9964460 0.079 0.97900 T.9907827 0.030 0.99151 T.9962971 0.080 0.97879 T.9906895 0.031 0.99118 1.9961525 0.081 0.97859 T.9906008 0.032 0.99085 1.9960079 0.082 0.97838 T.9905076 0.033 0.99053 T.9958676 0.083 0.97818 T.9904188 0.034 0.99021 T.9957273. 0.084 0.97798 T.9903300 0.035 0.98990 T.9955913 0.085 0.97778 T.9902411 0.036 0.98959 T.9954553 0.086 0.97758 T.9901523 0.037 0.98929 1.9953236 0.087 0.97738 T. 9900634 0.038 0.98899 T.9951919 0.088 0.97719 T.9899790 0.039 0.98869 T.9950601 0.089 0.97699 1.9898901 0.040 0.98840 T.9949327 0.090 0.97680 1.9898057 0.041 0.98811 T.9948053 0.091 0.97661 T.9897212 0.042 0.98783 T.9946822 0.092 0.97641 1.9896322 0.043 0.98755 1.9945591 0.093 0.97622 1.9895477 0.044 0.98727 T.9944359 0.094 0.97604 1.9894676 0.045 0.98699 T.9943128 0.095 0.97585 1.9893831 0.046 0.98672 T.9941939 0.096 0.97566 1.9892985 0.047 0.98645 T.9940751 0.097 0.97547 1.9892139 0.048 0.98619 T.9939606 0.098 0.97529 1.9891338 0.049 0.98592 T.9938417 0.099 0.97510 1.9890492 0.050 0.98566 T.9937271 0.100 0.97492 1.9889690 31 242 TABLE XXX. TELOCITIES, IN, FEET PER SECOND, DUE TO HEADS FROM TO 4.99 FEET. Head. O 1 a 3 4 5 6 7 8 9 0.0 0.000 0.802 1.134 1.389 1.604 1.793 1.965 2.122 2.268 2.406 .1 2.536 2.660 2.778 2.892 3.001 3.106 3.208 3.307 3.403 3.496 .2 3.587 3.675 3.762 3.846 3.929 4.010 4.090 4.167 4.244 4.319 .3 4.393 4.465 4.537 4.607 4.677 4.745 4.812 4.878 4.944 5.009 .4 5.072 5.135 5.198 5.259 5.320 5.380 5.440 5.498 5.557 5.614 .5 5.G71 5.728 5.783 5.839 ' 5.894 5.948 6.002 6.055 6.108 6.100 .6 6.212 6.264 6.315 6.366 6.416 6.466 6.516 6.565 6.614 6.662 .7 6.710 6.758 6.805 6.852 6.899 6.946 6.992 7.038 7.083 7.129 .8 7.173 7.218 7.263 7.307 7.351 7.394 7.438 7.481 7.524 7.566 .9 7.609 7.651 7.693 7.734 7.776 7.817 7.858 7.899 7.940 7.980 1.0 8.020 8.060 8.100 8.140 8.179 8.218 8.257 8.296 8.335 8.373 .1 8.412 8.450 8.488 8.526 8.563 8.601 > 8.638 8.675 8.712 8.749 .2 8.78G 8.822 8.859 8.895 8.931 8.967 9.003 9.038 9.074 9. 1 09 .3 9.144 9.180 9.214 9.249 9.284 9.319 9.353 9.387 9.422 9.450 .4 9.490 9.523 9.557 9.591 9.624 9.658 9.691 9.724 9.757 9.790 .5 9.823 9.855 9.888 9.920 9.953 9.985 10.017 10.049 10.081 10.113 .6 10.145 10.176 10.208 10.240 10.271 10.302 10.333 10.364 10395 10.426 .7 10.457 10.488 10.518 10.549 10.579 10.610 10.640 10.670 10.700 10.730 .8 10.760 10.790 10.820 10.850 10.879 10.909 10.938 10.967 10.997 11.026 9 11.055 11.084 11.113 11.142 11.171 1 1.200 11.228 11.257 11.285 11.314 2.0 11.342 11.371 11.399 1 1 .427 11.455 11.483 11.511 11.539 11.567 11.595 .1 11.622 11.650 11.678 11.705 11.733 11.760 11.787 11.814 11.842 11.869 .2 11.896 11.923 11.950 11.977 12.004 12.030 12.057 12.084 12.110 12.137 .3 12.163 12.190 12.216 12.242 12.269 12.295 12.321 12.347 12.373 12.3'J9 .4 12.425 12.451 12.477 12.502 12.528 12.554 12.579 12.605 12.630 12.656 .5 12.681 12.706 12.732 12.757 12.782 12.807 12.832 1 2.857 12.882 12.907 .6 12.932 12.957 12.982 13.007 13.031 13.056 13.081 13.105 13.130 13.154 .7 13.179 13.203 13.227 13.252 13.276 13.300 13.324 13.348 13.372 13.396 .8 13.420 13.444 13.468 13.492 13.516 13.540 13.563 13.587 13.611 13.634 .9 13.658 13.681 13.705 13.728 13.752 13.775 13.798 13.822 13.845 13868 3.0 13.891 13.915 13.938 13.961 13.984 14.007 14.030 14.053 14.075 14.098 .1 14.121 14.144 14.166 14.189 14.212 14.234 14.257 14.280 14.302 1 4.325 .2 14.347 14.369 14.392 14.414 14.436 14.459 14.481 14.503 14.525 14.547 .3 14.569 14.591 14.613 14.635 14.657 14.079 14.701 14.723 14.745 14.767 .4 14.789 14.810 14.832 14.854 14.875 14.897 14.918 14.940 14.961 1 4.983 .5 15.004 15.026 15.047 15.069 15.090 15.111 15.132 15.154 15.175 15.196 .6 15.217 15.238 15.J59 15.281 15.302 15.323 15.344 15.364 15.385 15.406 .7 15.427 15.448 15.469 15.490 15.510 15.531 15.552 15.572 15.593 15.614 .8 15.634 15.655 15.675 15.696 15.716 15.737 15.757 15.778 15.798 15.818 .9 15.839 15.859 15.879 15.899 15.920 15.940 15.960 15.980 16.000 16.020 4.0 16.040 16.060 16.080 16.100 16.120 16.140 16.160 16.180 16.200 '16.220 .1 16.240 16.259 16.279 1 6.299 16.319 1 6.338 1 6.358 16.378 16.397 16.417 .2 16.4:37 16.456 16.476 16.495 16.515 16.534 16.554 1 6.573 16.592 16.612 .3 16.631 16.650 16.670 16.689 16.708 16.727 16.747 16.766 16.785 16.804 .4 16.823 16.842 16.862 16.881 16.900 16.919 16.938 16.957 16.976 16.994 .5 17.013 17.032 17.051 17.070 17.089 17.108 17.126 17.145 17.164 17.183 .6 17.201 17.220 17.239 17.257 17.276 17.295 17.313 17.332 17.350 17.369 .7 17.387 17.406 17.424 17.443 17.461 17.480 17.498 17.516 17.535 17.553 .8 17.571 17.590 17.608 17.626 17.644 17.663 17.081 17.699 17.717 17.735 .9 17.753 17.772 17.790 17.808 17.826 17.844 17.862 17.880 1 7.898 17.916 1 243 TABLE X X X CONTINUED. VELOCITIES, IN FEET PER SECOND, DUE TO HEADS FROM 5 TO 9.99 FEET. Head. O 1 a 3 4 5 6 7 8 9 5.0 17.934 17.952 17.970 17.987 18.005 18.023 18.041 18.059 18.077 18.094 .1 18.112 18.130 18.148 18.165 18.183 18.201 18.218 18.236 18.254 18.271 .2 18.289 18.306 18.324 18.342 18.359 18.377 18.394 18.412 18.429 18.446 .3 18.464 18.481 18.499 18.516 18.533 18.551 18.568 18.585 18.603 18.020 A 18.637 18.655 18.672 18.689 18.706 18.723 18.741 18.758 18.775 18.792 .0 18.809 18.826 18.843 18.860 18.877 18.894 18.911 18.928 18.945 18.962 .6 18.979 18.996 19.013 19.030 19.047 19.064 19.081 19.098 19.114 19.131 .7 19.148 19.165 19.182 19.198 19.215 19.232 19.248 19.265 19.282 19.299 .8 19.31;) 19.332 19.348 19.365 19.382 19.398 19.415 19.431 19.448 19.464 .9 19.481 19.497 19.514 19.530 19.547 19.563 19.580 19.596 19.613 19.629 6.0 19.645 19.662 19.678 19.694 19.711 19.727 19.743 19.760 19.776 19.792 .1 19.808 19.825 19.841 19.857 19.873 19.889 19.906 19.922 19.938 19.954 .2 19.970 19.986 20.002 20.018 20.034 20.050 20.067 20.083 20.099 20115 .3 20.131 20.147 20.162 20.178 20.194 20.210 20.226 20.242 20.258 20.274 A 20.290 20.306 20.321 20.337 20.353 20.369 20.385 20.400 20.416 20.432 .5 20.448 20.403 20.479 20.495 20.510 20.526 20.542 20.557 20.573 20.589 .6 20.604 20.620 20.635 20.651 20.667 20.682 20.698 20.713 20.729 20.744 .7 20.760 20.775 20.791 20.806 20.822 20.837 20.853 20.868 20.883 20.899 .8 20.914 20.929 20.945 20.960 20.976 20.991 21.006 21.021 21.037 21.052 .9 21.067 21.083 21.098 21.113 21.128 21.144 21.159 21.174 21.189 21.204 7.0 21.219 21.235 21.250 21.265 21.280 21.295 21.310 21.325 21.340 21.355 .1 21.370 21.386 21.401 21.416 21.431 21.446 21.461 21.476 21.491 21.500 .2 21.520 21.535 21.550 21.565 21.580 21.595 21.610 21.625 21.640 21.655 .3 21.669 21.684 21.699 21.714 21.729 21.743 21.758 21.773 21.788 21.803 A 21.817 21.832 21.847 21.861 21.876 21.891 21.906 21.920 21.935 2 1 .950 .5 21.964 21.979 21.993 22.008 22.023 22.037 22.052 "22.066 22.081 22.096 .6 22.110 22.125 22.139 22.154 22.168 22.183 22.197 22.212 22.226 22.241 .7 22.255 22.270 22.284 22.298 22.313 22.327 22.342 22.356 22.370 22.385 .8 22.399 22.414 22.428 22.442 22.457 22.471 22.485 22.499 22.514 22.528 .9 22.542 22.557 22.571 22.585 22.599 22.614 22.628 22.642 22.656 22.670 8.0 22.685 22.699 22.713 22.727 22.741 22.755 22.769 22.784 22.798 22.812 .1 22.826 22.840 22.854 22.868 22.882 22.896 22.910 22.924 22.938 22.952 .2 22.966 22.980 22.994 23.008 23.022 23.036 23.050 23.064 23.078 23.092 .3 23.106 23.120 23.134 23.148 23.162 23.175 23.189 23.203 23.217 23.231 .4 23.245 23.259 23.272 23.286 23.300 23.314 23.328 23.341 23.355 23.369 .5 23.383 23.396 23.410 23.424 23.438 23.451 23.405 23.479 23.492 23.506 .6 23.520 23.534 23.547 23.561 23.574 23.588 23.602 23.615 23.629 23.643 .7 23.656 23.670 23.683 23.697 23.711 23.724 23.738 23.751 23.765 23.778 .8 23.792 23.805 23.819 23.832 23.846 23.859 23.873 23.886 23.900 23.913 .9 23.927 23.940 23.953 23.967 23.980 23.994 24.007 24.020 24.034 24.047 9.0 24.061 24.074 24.087 24.101 24.114 24.127 24.141 24.154 24.167 24.181 .1 24.194 24.207 2-4.220 24.234 24.247 24.260 24.274 24.287 24.300 24.313 .2 24.326 24.340 24.353 24.366 24.379 24.392 24.406 24.419 24.432 24.445 .3 24.458 24.471 24.485 24.498 24.511 24.524 24.537 24.550 24.563 24.576 .4 24.589 24.603 24.616 24.629 24.642 24.655 24.668 24.681 24.694 24.707 ,5 24.720 24.733 24.746 24.759 24,772 24.785 24.798 24.811 24.824 24.837 .6 24.850 24.803 24.876 24.888 24.901 24.914 24.927 24.940 24.953 24.966 .7 24.979 24.992 25.005 25.017 25.030 25.043 25.056 25.069 25.082 25.094 .8 25.107 25.120 25.133 25.146 25.158 25.171 25.184 25.197 25.209 25222 .9 25.235 25.248 25.260 25.273 25.286 25.299 25.311 25.324 25.337 25.349 244 TABLE XXX CONTINUED. VELOCITIES, IN FEET PER SECOND, DUE TO HEADS FROM 10 TO 14.99 FEET. Ilead. O 1 2 3 4 5 6 7 8 9 10.0 25.362 25.375 25.387 25.400 25.413 25.425 25.438 25.451 25.463 25.476 .1 25.489 25.501 25.514 25.526 25.539 25.552 25.564 25.577 25.589 25.602 .2 25.614 25.627 25.640 25.652 25.665 25.677 25.690 25.702 25.715 25.728 .3 25.740 25.752 25.765 25.777 25.790 25.802 25.815 25.827 25.839 25.852 .4 25.864 25.877 25.889 25.902 25.914 25.926 25.939 25.951 25.964 25.976 .5 25.988 26.001 26.013 26.026 26.038 26.050 26.063 26.075 26.087 26.099 .6 26.112 26.124 26.136 26.149 26.161 26.173 26.186 26.198 26.210 26.222 .7 26.235 26.247 26.259 26.272 .26.284 26.296 26.308 26.320 26.333 26.345 .8 20.357 26.369 26.381 26.394 26.406 26.418 26.430 26.442 26.454 26.467 .9 26.479 26.491 26.503 26.515 26.527 26.540 26.552 26.564 26.576 26.588 11.0 26.600 26.612 26.624 26.636 26.648 26.660 26.672 26.684 26.697 26.709 .1 26.721 26.733 26.745 26.757 26.769 26.781 26.793 26.805 26.817 26.829 .2 26.841 26.853 26.865 26.877 26.889 26.901 26.913 26.924 26.936 26.948 .3 26.960 26.972 26.984 26.996 27.008 27.020 27.032 27.044 27.056 27.067 .4 27.079 27.091 27.103 27.115 27.127 27.139 27.150 27.162 27.174 27.186 .5 27.198 27.210 27.221 27.233 27.245 27.257 27.269 27.280 27.292 27.304 .6 27.316 27.328 27.339 27.351 27.363 27.375 27.386 27.398 27.410 27.422 .7 27.433 27.445 27.457 27.468 27.480 27.492 27.504 27.515 27.527 27.539 .8 27.550 27.562 27.574 27.585 27.597 27.609 27.620 27.632 27.644 27.655 9 27.667 27.678 27.690 27.702 27.713 27.725 27.736 27.748 27.760 27.771 12.0 27.783 27.794 27.806 27.817 27.829 27.841 27.852 27.864 27.875 27.887 .1 27.898 27.910 27.921 27.933 27.944 27.956 27.967 27.979 27.990 28.002 .2 28.013 28.025 28.036 28.048 28.059 28.071 28.082 28.094 28.105 28.117 .3 28.128 28.139 28.151 28.162 28.174 28.185 28.196 28.208 28.219 28.231 .4 28.242 28.253 28.265 28.276 28.288 28.299 28.310 28.322 28.333 28.344 .5 28.356 28.367 28.378 28.390 28.401 28.412 28.424 28.435 28.446 28.458 .6 28.469 28.480 28.491 28.503 28.514 28.525 28.537 '28.548 28.559 28.570 .7 28.582 28.593 28.604 28.615 28.627 28.638 28.649 28.660 28.672 28.683 .8 28.694 28.705 28.716 28.727 28.739 28.750 28.761 28.772 28.783 28.795 .9 28.806 28.817 28.828 28.839 28.850 28.862 28.873 28.884 28.895 28.906 13.0 28.917 28.928 28.939 28.951 28.962 28.973 28.984 28.995 29.006 29.017 .1 29.028 29.039 29.050 29.061 29.073 29.084 29.095 29.106 29.117 29.128 .2 29.139 29.150 29.161 29.172 29.183 29.194 29.205 29.2 1 6 29.227 29.238 .3 29.249 29.260 29.271 29.282 29.293 29.304 29.315 29.326 29.337 29.348 .4 29.359 29.370 29.381 29.392 29.403 29.413 29.424 29.435 29.446 29.457 .5 29.468 29.479 29.490 29.501 29.512 29.523 29.533 29.544 29.555 29.566 .6 29.577 29.588 29.599 29.610 29.620 29.631 29.642 29.653 29.664 29.675 .7 29.686 29.696 29.707 29.718 29.729 29.740 29.751 29.761 29.772 29.783 .8 29.794 29.805 29.815 29.826 29.837 29.848 29.858 29.869 29.880 29.891 .9 29.901 29.912 29.923 29.934 29.944 29.955 29.966 29.977 29.987 29.998 14.0 30.009 30.020 30.030 30.041 30.052 30.062 30.073 30.084 30.094 30.105 .1 30.116 30.126 30.137 30.148 30.159 30.169 30.180 30.190 30.201 30.212 .2 30.222 30.233 30.244 30.254 30.265 30.276 30.286 30.297 30.307 30.318 .3 30.329 30.339 30.350 30.360 30.371 30.382 30.392 30.403 30.413 30.424 .4 30.435 30.445 30.456 30.466 30.477 30.487 30.498 30.508 30.519 30.529 .5 30.540 30.551 30.561 30.572 30.582 30.593 30.603 30.614 30.624 30.635 .6 30.64a 30.656 30.666 30.677 30.687 30.698 30.708 30.719 30.729 30.739 .7 30.750 30.760 30.771 30.781 30.792 30.802 30.813 30.823 30.833 30.844 .8 30.854 30.865 30.875 30.886 30.896 30.906 30.917 30.927 30.938 30.948 .9 30.958 30.969 30.979 30,990 31.000 31.010 31.021 31.031 31.041 31.052 245 TABLE X X X CONTINUED. VELOCITIES, IN FEET PER SECOND, DUE TO HEADS FROM 15 TO 19.99 FEET. Head. O 1 3 3 4 5 6 7 8 9 15.0 31.062 31.072 31.083 31.093 31.103 31.114 31.124 31.134 31.145 31.155 .1 31.165 31.176 31.186 31.196 31.207 31.217 31.227 31.238 31.248 31.258 .2 31.268 31.279 31.289 31.299 31.310 31.320 31.330 31.340 31.351 31.361 .3 31.371 31.381 31.392 31.402 31.412 31.422 31.433 31.443 31.453 31.463 .4 31.474 31.484 31.494 31.504 31.514 31.525 31.535 31.545 31.555 31.565 .5 31.576 31.586 31.596 31.606 31.616 31.626 31.637 31.647 31.657 31.667 .6 31.677 31.687 31.698 31.708 31.718 31.728 31.738 31.748 31.758 31.768 .7 31.779 31.789 31.799 31.809 31.819 31.829 31.839 31.849 31.859 31.870 .8 31.880 31.890 31.900 31.910 31.920 31.930 31.940 31.950 31.960 31.970 .9 31.980 31.990 32.000 32.011 32.021 32.031 32.041 32.051 32.061 32.071 16.0 32.081 32.091 32.101 32.111 32.121 32.131 32.141 32.151 32.161 32.171 .1 32.181 32.191 32.201 32.211 32.221 32.231 32.241 32.251 32.261 32271 .2 32.281 32.291 32.301 32.311 32.321 32.330 32.340 32.350 32.360 32.370 .3 32.380 32.390 32.400 32.410 32.420 32.430 32.440 32.450 32.460 32.470 .4 32.480 32.489 32.499 32.509 32.519 32.529 32.539 32.549 32.559 32.569 .5 32.579 32.588 32.598 32.608 32.618 32.628 32.637 32.647 32.657 32.667 .6 32.677 32.687 32.696 32.706 32.716 32.726 32.736 32.746 32.755 32.765 .7 32.775 32.785 32.795 32.804 32.814 32.824 32.834 32.844 32.854 32.863 .8 32.873 32.883 32.893 32.903 32.912 32.922 32.932 32.941 32.951 32.961 .9 32.971 32.980 32.990 33.000 33.010 33.019 33.029 33.039 33.049 33.058 17.0 33.068 33.078 33.088 33.097 33.107 33.117 33.126 33.136 33.146 33.156 .1 33.165 33.175 33.185 33.194 33.204 33.214 33.223 33.233 33.243 33.252 .2 33.262 33.272 33.281 33.291 33.301 33.310 33.320 33.330 33.339 33.349 .3 33.359 33.368 33.378 33.388 33.397 33.407 33.416 33.426 33.436 33.445 .4 33.455 33.465 33.474 33.484 33.493 33.503 33.513 33.522 33.532 33.541 .5 33.551 33.560 33.570 33.580 33.589 33.599 33.608 33.618 33.628 33.637 .6 33.647 33.656 33.666 33.675 33.685 33.694 33.704 33.713 33.723 33.733 .7 33.742 33.752 33.761 33.771 33.780 33.790 33.799 33.809 33.818 33.828 .8 33.837 33.847 33.856 33.866 33.875 33.885 33.894 33.904 33.913 33.923 .9 33!932 33.942 33.951 33.961 33.970 33.980 33.989 33.998 34.008 34.017 18.0 34.027 34.036 34.046 34.055 34.065 34.074 34.083 34.093 34.102 34.112 .1 34.121 34.131 34.140 34.149 34.159 34.168 34.178 34.187 34.197 34.206 .2 34.215 34.225 34.234 34.244 34.253 34.262 34.272 34.281 34.290 34.300 .3 34.309 34.319 34.328 34.337 34.347 34.356 34.365 34.375 34.384 34.393 .4 34.403 34.412 34.422 34.431 34.440 34.450 34.459 34.468 34.478 34.487 .5 34.496 34.505 34.515 34.524 34.533 34.543 34.552 34.561 34.571 34.580 .6 34.589 34.599 34.608 34.617 34.626 34.636 34.645 34.654 34.664 34.673 .7 34.682 34.691 34.701 34.710 34.719 34.728 34.738 34.747 34.756 34.766 .8 34.775 34.784 34.793 34.802 34.812 34.821 34.830 34.839 34.849 34.858 .9 34.867 34.876 34.886 34.895 34.904 34.913 34.922 34.932 34.941 34.950 19.0 34.959 34.968 34.978 34.987 34.996 35.005 35.014 35.024 35.033 35.042 .1 35.051 35.0GO 35.069 85.079 35.088 35.097 35.106 35.115 35.124 35.134 .2 35.143 35.152 35.161 35.170 85.179 35.188 35.198 35.207 35.216 35.225 .3 35.234 35.243 35.252 35.262 35.271 35.280 35.289 35.298 35.307 35.316 .4 35.325 35.334 35.344 35.353 35.362 35.371 35.380 35.389 35.398 35.407 .5 35.416 35.425 35.434 35.443 35.453 35.462 35.471 35.480 35.489 35.498 .6 35.507 35.516 35.525 35.534 35.543 35.552 35.561 35.570 35.579 35.588 .7 35.597 35.606 35.615 35.624 35.634 35.643 35.652 35.661 35.670 35.679 .8 35.688 35.697 35.706 35.715 35.724 35.733 35.742 35.751 35.760 35.769 .9 35.778 35.787 35.796 35.805 35.814 35.823 35.832 35.841 35.849 35.858 TABLE XXX CONTINUED. VELOCITIES, IN FEET PER SECOND, DUE TO HEADS FROM 20 TO 24.99 FEET. Head. 1 2 | 3 4 5 6 7 8 9 20.0 35.8C7 35.876 35.885 35.894 35.903 35.912 35.921 35.930 35.939 35.948 .1 35.957 35.966 35.975 35.9, 7 and 8 . Scale for FiQure* 9.1 and 11. Fio . (') APPARATUS USED IN GAUGING THI Fio \ Seftle to' 1 Fiourcs .1 ~ j I 1 1 1. '.!. ."). .'), (\ 7. rt. ;) a nil JO ,i t>~ s tor Fio 4 Fui (i Fiii .'> Fi.i Fio 4- YATER AT THE TREMONT TTRB1NK - Am //a<"//r'-/i/ u. i mini .u u i k'uu i LI i ui, i "UN. OT A iT rOTHE VENT WHEEL AT , 1WOTT dlTTOX Mfl.l.S. \ I-J... } . trmj: fffl t i-ir -r 4-f-i rtrf h Ukf-L EfS -H- 1-H- H -H-H-H- Fi I. Kid .Y Fid. (i. Fid H: Fid !."> Fio.i4. Fki 5. Fio 4.. Fid. . Tid.u. Kg. 12. L,J\I ij u 1 m LJ a i o rv i LOVVKK. LIHKS Pi. \l. EXPERIMENTS UPON WE ''a a a a a a , > AT THF, f,0\VF,R LOCKS ' KKI'F.IUMLNTS I'l'llN WF.I Scale for Figures '3,3 and 4' So.ilo tor Fio . 1 . Pan AT TIIK I.OWKIl LOCKS .xm. VN i r. \> ui Inr, I, UVVr.lv MM l\> m. 4.. I.Q I I j' ' > I) A r Hiram ?. .1,11, ,M B E Fid. H. Kid. Exp. 7 Ixp . 12$ ._ EXP. 6 Exp.g-J. Tlie ABSC1 The. ORD1I The frt/tn of Hie tu j.l. hn a Exp. ;)(>. S .of the /mints rn,ti //,;/ o represent III,' me, in ,//.r/,if/e,:r in Ji-et ,if Me ,ret>er,ii ti/be.r. Jhu t/ic /e/i '.S ,-/' til,' [>,iin/.r tn,i, /,;;/ n-finvt'til ///, t>et,-Hi/' //!< .rein-rill. Ilil',:- in _ /',;>! per set-tin,! . i>/' t/ie irregularly firrt>rer .reeiitiil in f.n-t.r ,>/' th<' n>iru/inat.r /!wn n>/neh the nieiiii ne/tieiiie.f have bi'fii determined . 10 G. Lxp.g/. /t> /.i .172 Exp. ing. Kxp. u< 3.0 3,5 Exp. * ' 22 23 21 2S 27 2S 29 30 HI 32 S3 3k 3S 36 31 3ft 39 Ifff EXPERIMENTS ON THE FLOW OF WATER TflROU SUBMERGED ORIFICES AND DIVERGING TUBES. PL XX. EXPERIMENTS ON THF. FLOW OF WATER . TIIKOn Fio.ii. F Scale except for Figures 8 and SPBMKKOKI) OR1FICKS AND DIVKKGINC. Tl II KS TURBINE WATER - WHEK: >F 700 HORSE POWF.R PI. xxn. Srale for liourc I. Srah- la I' fidllivs _> ft 7) . TURBM WATER- WEE1 700 HORSF, I'OWKK \ UNIVKRKITY OP CALIFORNIA LIBRARY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW OCT 13 , y?ODISCMARu6'90 957 30m-l,'15 U.C. BERKELEY LIBRARIES fflBRfflJHHBHHnBB $${$$m(l$w Illilllli