! 4 ！·È :' + ′ . ：:----… -----∞****** * * * * *， *）（.* * * * * ******* • ** • * ， ， ºwº-wę zº º -ae （） - º -，： -， -， -， *， *s*… º- ·|-----·~• • . \， ，******* 2. *******************-pº gº: sr.， , ， ， * o *********** ********** . ： “si se :： ` ~ ~~~~）; -r - -~ - s -- :)• -،··o º ty， spºº :： ， ， , ）.*ae · ·,·，≤F ÍY. [* ! # zºº \; |||||||||||||||||||| ###|IIIT Tſº ||Nº|| t º-r Nº. : r gº- º '. º Nººn º::.- º. ', º . . . ‘,C#3 . ºll! . . . . . . ... I - - - fºrº...sº a w ...] *-i-g §§§º. tºj H. É º tº: - ºljºš jää 3|E|| ºill —d §º ºx: #hiſmiſſimºiſmiſſiſſiſſ; | Cº-º-º-º- Fºllº ºº::. ††† |I Uſ | !!!\!\!\!\; | Fº TIII ( ; IFT OF | : C C º º f C º C C f º ſ ſº C C º C Cº. i T C C f C f l |- º C J º i i L ſ i C º º C C ſ C C i ſ f f i QC. 3.35T G 77, /837 ABSOLUTE MEASUREMENTS IN ELECTRICITY AND MAGNET ISM. EY THE SAME AUTHOR. THE THEORY AND PRACTICE OF ABSOLUTE MEASUREMENTS IN ELECTRICITY AND MAGNETISM. Vol. I., Crown 8vo. I2s. 6d. Vol. II. (In the press). ABSOLUTE MEASUREMENTS ELECTRICITY AND MAGNETISM BY ANDREW GRAY, M.A., F.R.S.E. PROFESSOR OF PHYSICS IN THE UN IVLRSITY COLLEGE OF NORTH WALES SAE COMWD / AD/7”/OAV A’A. WZSA./D AAV/O GA’A.-1 / / V AºAV/LAA’GA2/) ºftonbc.m. M A C M I L L A N A N D C O. AND NEW YORK 1889 ZThe Right of Translation and Rezzoduction is Reserved : : : TICHARD CLAY AND SONs, LIMITED, LONDON AND BUNGAY. PREFACE. THE following statement was prefixed to the first edition of the present work, published in 1884 — “This little book was originally intended to be mainly a reprint of some papers on the Measurement of Electric Currents and Potentials in Absolute Measure contributed to Maſure during the winter of 1882-3 ; but as these were being reprinted, many alterations and additions suggested themselves, which it was thought would render the book more generally useful. Most of the additional matter is mentioned in the introductory chapter, but I may here refer to a sketch of the theory of alternating machines, and of methods of measurement available in such cases, contained in Chapter X., and to Chapter XII. on the Dimensions of Units, which I have thought it desirable to introduce.* “The work has of course no pretensions to being a complete treatise on Electrical and Magnetic Measure- * Chapters IX. and XI. of the present edition. ~) --, -, -, -, - A * , * ~... --> y ... * ~ * - `-- * ~!-- ~~~~~~~ vi PREFACE. ments, but is rather designed to give as far as is possible within moderate limits a clear account of the system of absolute units of measurement now adopted, and of some methods and instruments by which the system can be applied in both theoretical and practical work. - * - “I am under great obligations to Sir WILLIAM THOMSON and to Mr. J. T. BOTTOMLEy, who have kindly examined some of the proofs, and favoured me with valuable suggestions.” About a year ago the first volume of what is designed to be a more comprehensive treatise on this subject was published by Messrs. Macmillan and Co., under the title of The Theory and Practice of Absolute Measurements in Electricity and Magnetism, and the second volume is now in progress. It was intended that that work should supersede the original small book, which had been long out of print. It has, however, been represented to me that the smaller work has been found useful by students, and that a desire exists for a second edition on the original plan. I have therefore prepared the present volume, incor- porating in it a few parts of the larger work, in which the original matter had been amplified and brought more nearly to date. I have besides revised the PREFACE. vii whole, and made several additions which I hope may be found to add to its interest and usefulness. Among these are a fuller account of the Determination of H, a description of Sir William Thomson's Standard Electrical Instruments, a more complete treatment of the Graduation of Instruments, espe- cially by Electrolysis, an extension of the Theory of Alternating Machines so as to include Dr. Hopkin- son's Theory of the Working of Alternators in Series and in Parallel, additional information regarding the Measurement of Activity, &c., in the circuits of Alternators, and what I hope is an improved Chapter on the Theory of Dimensions of Physical Quantities. In the last mentioned part I have adopted Professor Rücker's suggestion that the dimensions of the electric and magnetic inductive capacities should be left undetermined, and regarded as so related as to render the dimensions of every physical quantity the same in the electrostatic as in the electromagnetic system of units. Except in the part relating to Sir WILLIAM THOMSON's instruments, which has been revised by Sir WILLIAM himself and his assistant Mr. J. RENNIE, I have had little or no assistance in reading the proof sheets; and it is probable therefore that many slips and viii PREFACE. inaccuracies have escaped my notice. I shall feel greatly obliged if any reader who may detect errors will kindly communicate them either to Messrs. Macmillan or to myself. In conclusion I wish here to acknowledge with great respect the many and deep obligations I am under to Sir WILLIAM THOMSON, at whose suggestion I undertook the original work, and to whose encourage- ment and kindly help is due what little progress in scientific studies I have been able to make. A. GRAY. THE UNIVERSITY COLLEGE OF NORT (I WALES, BANGOR. October, 1889. CONTENTS, CHIAPTER I. I N T R O D U C T O R Y. CHAPTER II. DETERMINATION OF THE HORIZONTAI, COMPON ENT OF THE EARTH's MAGNETIC FIELD. PAGE Gauss's Method of determining H' 6 Bottomley’s Magnetometer . . . . . . 6 Tempering and Magnetization of Magnets . 8 Setting-up of Instruments . . . . . . 9 Observation of Deflections of Needle . - 9 Convention as to Blue and Red Magnetism . . . . . . IO Observation of Oscillation-Periods of Magnets . . . . . ſ3 Calculation of Moments of Magnets and of H from Result I5 Corrections . . . s e e e º e s ∈ a e I6 Correction for Magnetic Induction . • to e º e º 'º 17 Account of an Actual Determination of H . . . . . I9 et seq. Determination of Induction Correction . . . . . . . . 29 Relation between Ratio of Length to Diameter and In- duction Coefficient . . . . . . . . . . . . . . . 33 Effect of Hardness on the Induction Coefficient . . . . 35 Magnetic Distribution. . . . . . . . . . . . . . . . 36 X CONTENTS, CHAPTER III. ABSOLUTE UNITS OF MAGNETIC POLE, MAGNETIC FIELD, AND ELECTRIC CURRENT. Electromagnetic System of Units . . . Unit Magnetic Pole . . . . . . . . . Uniform Magnetization . . . . . . e tº 3 s a s tº Dynamical Basis of Definition of Unit Pole . . . . . . Rinetic Measure of Force . . . . • C.G.S. System of Units. • * * * * @ e & Dyne or C. G.S. Unit of Force . . . . . . . . . . Unit Magnetic Pole in C. G. S. System Magnetic Moment . . . . . . • tº Magnetic Field . . . . . . Magnetic Field of Unit Intensity defined by Unit Pole. . Magnetic Field due to a Single Pole Ampère's Theory of Electromagnetic Action . . . . . . Definitions of Unit Current derived from Unit Pole . Equivalence of Current-Circuit and Magnetic Shell . C.G. S. Unit of Current . . CHAPTER IV. PAGE 4O 4. I 4 I 4 I 42 42 42 42 42 43 43 43 44 44 44 47 MEASUREMENT OF A CURRENT IN ABSOLUTE UNITS, AND PRACTICAL CONSTRUCTION OF A STANDARD GALVANO- METER. Theory of Tangent Galvanometer with Short Needle . Correction for Dimensions of Coil Correction for Torsion s & e º s e & Method of Kohlrausch for determining A/. Sir William Thomson's Method Helmholtz's Double-Coil Galvanometer . g Construction of Standard Tangent Galvanometer . Winding of Coil. tº e º e s Adjustment of Needle, Scale, &c. . 49 50 53 54 55 56 57 58 58 CONTENTS. xi CHAPTER V. DEFINITION OF ABSOLUTE UNITS OF POTENTIAL AND RESIST- ANCE, AND DERIVATION OF PRACTICAL UNITS-VOLT, OHM, AMPERE, COULOMB. PAGE Difference of Potential . . . . . . . . . . . . . . . 65 Absolute Electrometer . . . . . . . . . . . . . . . . 66 Electric Resistance . . . . . . . . . . . . . . . . . 66 Ohm's Law . . . . . . . . . . . . . . . . . . . . 66 Unit of Potential based on Unit Magnetic Field, and de- fined by means of Rails and Slider . . . . . . . . 67 Electromotive Force . . . . . . . . . . . . . . . . 68 Derivation of Volt, or Practical Unit of Difference of Potential . . . . . . . . . . . . . . . . . . . 69 Absolute unit of Resistance based on Unit of Difference of Potential and Unit of Current . . . . . . . . . . 69 Derivation of Ohm, or Practical Unit of Resistance . . . 69 A Resistance in Electromagnetic Units has the Dimensions'' of a Velocity s a tº & 4 tº tº º e º w tº e - e º 7o Dimensions of Units . . . . . . . . . . . . . . . . 7 Realization of the Ohm . . . . . . . . . . . . . . . 71 Derivation of Practical Units of Current and of Quantity of Electricity—Ampere and Coulom/ . . . . . . . 72 T]efinition of Work . . . . . . . . . . . . . . . . . 73 Foot-Pound . . . . . . . . . . . . . . . . . . . . 73 Horse Power . . . . . . . . . . . . . . . . . . . 73 Erg, or C.G. S. Unit of Work . . . . . . . . . . . . 74 Definition of Activity. 74 Proof by means of Rails and Slider of the Expression for the Total Electrical Activity in any Circuit. . . . . 74 Joule's Law of the Generation of Heat by a Current . . 75 }oule, or Unit of Electrical Work . . . . . . . . . . 76 Watt, or Unit of Electrical Activity . . . . . . . . 77 xii CONTENTS. CHAPTER VI. POTENTIALS AND CURRENTS IN DERIVED CIRCUITS. Ohm’s Law true in any Homogeneous Part of the Circuit of a Battery . . . . . . . . . Electromotive Force . * * * * * * is e Electromotive Force in a Heterogeneous Circuit Electromotive Force of a Cell . . . . tº º a tº tº g Relation of Electromotive Force to Difference of Potential at Extremities of External Resistance . Activity in Voltaic Circuit. . . . . . Arrangement of Cells in a Battery 6 & ſº tº Arrangement in Series and in Multiple Arc . . . . . Arrangement for Greatest Current and for Greatest Economy . . . . . . . . . . . . . . . Principle of Continuity in Flow of Electricity . * Kirchhoff’s Theorems for Currents in a Network of Con- ductors . * & tº # 6 tº $ tº e º ſº º e s tº gº Resistance and Conductance of a System of Conductors arranged in Multiple Arc . . . . . . . . . . . . Resistance of Incandescence Lamps arranged in Parallel Circuit . & & © tº e º ſº e & Applications of Kirchhoff's Theorems Theory of Wheatstone's Bridge . $ tº dº tº g º & Resistance of Network of Five Conductors in Wheatstone's Bridge Useful Results . * * * * * * * * * Effect of Adding Single Wire to Network . Theory of Potential Galvanometer . PAGE 78 79 8O 8I 82 83 84 85 86 88 88 88 89 90 9I 92 92 94. CONTENTS. xiii CHAPTER VII. STANDARD ELECTRICAL MEASURING INSTRUMENTS AND THEIR GRADUATION. Sir W. Thomson's Standard Electrical Balances Principle of their Action . . . . Standard Centi-ampere Balance Standard Deka-ampere Balance Standard Kilo-ampere Balance . . * * * Use of Centi-ampere Balance as a Voltmeter Standard Composite Balance Use of, as Voltmeter . ,, , , Wattmeter ,, , , Current-meter Ranges of Standard Balances Magnetostatic Current-meter. Electrostatic Instruments • * * * * Sir W. Thomson's Quadrant Electrometer . Details of Construction . Idiostatic Gauge . Replenisher . . Adjustments Tests for Insulation Theory of . * * * Grades of Sensibility . • * : * ~ * * * Vertical Electrostatic Voltmeter (Sir W. Thomson's) . Graduation of Vertical Electrostatic Voltmeter . Multicellular Voltmeter (Sir W. Thomson's). Graduation of Multicellular Voltmeter. Graduation of Instruments in general . e - 6 - Convenience of a Galvanometer of Great Resistance of Measurements of Differences of Potential Graduation of a Potential Galvanometer. e - e. Methods by Comparison with Standard Galvanometer. PAGE 95 96 xiv CONTENTS. PAGE Verification of Graduation by means of Standard Cell . . 151 Standard Daniell's Cell . . s tº & © a e . . . . I5 I Clark's Standard Cell . . . . . . . . . . . . . . . . I5 I Comparison of a Large Difference of Potential with Electromotive Force of a Standard Cell . . . . . I 56 Graduation of a Current Galvanometer or Current Balance 157 Methods for Evaluating the Horizontal Force at Needles given by Field Magnet of Galvanometer. . . . . . I58 Graduation of a Current Galvanometer or Balance by Electrolysis . . . . . . . . . . . . . . . . . . I62 Forms of Electrolytic Cells and Precautions in Working (for Silver and Copper). . . . . . . . . . . I63–172 Electro-chemical Equivalents of Silver and Copper . . . I73 Effect of Size of Plates on Electro-chemical Equivalent of Copper . . . . . . . . . . . . . . . . . . . . I 73 Effect of Change of Temperature . . . . . . . . . . . I 73 Graduation of Sir W. Thomson's Current-Balances by Electrolysis . . . . . . . . . . . . . . . . . . 174 CHAPTER VIII. COM PARISON OF RESISTANCES. Reflecting Galvanometer . . . . . . . . . . . . . . ISI Dead-beat Form . . . . . . . . . . . . . . . . 183 Adjustment of Field ſor Sensibility . . . . . . . . 185 Resistance Coils . . . . . . . . . . . . . . . . . . 185 Arrangement in Resistance Boxes. . . . . . . . IS5, 188 Material and Mode of Winding. . . . . . . . . . I88 Ohm, or Practical Unit of Resistance . . . . . . . . . . I 89 Resistance Boxes. . . . . . . . . . . • ? • * * 189 Arrangement of Coils in Geometrical Progression. . I 91 Dial Form . . . . . . . . . . . . . . . . . . I92 Bridge Arrangement . . . . . . & e º a 9 º' º I94. Resistance Slides. . . . . . . . . . . . . . . . . . I94 CONTENTS. XV PAGE Conductance, or Conductivity Boxes . . . . . . I95 Effect of Rise of Temperature on Resistance of Coils . . I97 Determination of Temperature of Interior of Box by an Auxiliary Copper Coil . . . . . . . . . . . . . I98 Process of Testing a Resistance Box . . . . . . . . . I99 - Care of Resistance Boxes . . . . . . . . . . . . . . I99 Rheostat . . . . . . . . . . . . . . . . . . . . . 200 - Wheatstone's Bridge . . . . . . . . . . . . . . . . . 202 Arrangement for Greatest Sensibility in Practical Case of Invariable Battery and Galvanometer . . 203 v- Practical Rule for Sensibility when Full Adjustment not needed . . . . . . . . . . . . . . . . . 204 Arrangement cf Keys . . . . . . . . . . . . . . . 205 Mode of Operating . . . . . . . . . . . . . 205, 207 Effect of Self-Induction . . . . . . . . . . . . . 206 Avoidance of Thermo-Electric Disturbance . . . . 207 Kirchhoff's Slide-Wire Bridge . . . . . . . . . . . . 209 Mercury Cups for Connections . . . . . . . . . . . . 2 IO Calibration of a Wire . . . . . . . . . . . . . . . . 212 Matthiessen and Hockin’s Method . . . . . . . . 212 Carey Foster's Method . . . . . . . . . . . . . 213 T. Gray's Method . . . . . . . . . . 217 Precautions against Thermoelectric Disturbance . . 219 Comparison of Coils by Slide-Wire Bridge . . . . . . 2 IQ Method of finding Resistance of Slide-Wire . . . . . . 22O Comparison of two Nearly Equal Resistances . . . . . 222 Comparison of Very Low Resistances . . . . . . . . . 224 Sir W. Thomson's Bridge with Secondary Conduc- tors. Practical Apparatus . . . . . . . . . . . 224 Matthiessen and Hockin's Method . . . . . . . . 229 Method of Comparison of Differences of Potential . 231 Tait's Method by Differential Galvanometer . . . . 236 T. Gray's Potential Method with Double Arc . . . 236 Measurements of Specific Resistance . . . . . . . . . 238 Thin Wire . . . . . . . . . . . . . . . . . . 239 xvi CONTIENTS. PAGS. Corrections in Weighing . . . 239 Thick Wire or Rod . . 24O Specific Resistance of Mercury. 242 Forms of the Standard Ohm . . 243 Measurement of High Resistances 246 Method by Galvanometer . 246 Specific Resistance of Glass . © s tº e 247 Method by Leakage . . . . . . . . . . . . 249–256 Measurement of Battery Resistance. 256 Mance's Method . . • . . . . . . . . . . 259 Measurement of the Resistance of a Galvanometer, Thom- son's Method . . . . . . . . . . . . . . . . . 26o CIHAPTER IX. MEASUREMENT OF ENERGY IN ELECTRIC CIRCUITS. Activity in Circuit of Generator and Motor 262 Counter Electromotive Force of Motor . . . . 263 Electrical Efficiency in Circuit of Generator and Motor . 264 Maximum Activity . . . . . . . . . . . . . 264 Arrangement for Maximum Activity not that of Maximum Efficiency . • e º e º a • * 264 Calculation of Electrical Efficiency . . . . 265 Relation of Electrical Efficiency to Speed . 265 Relation of Electrical Efficiency to Electromotive Force . 265 Efficiency of a Motor . 267 Friction Ergometer . . . . . . . . . . 268 Determination of Efficiency of a Motor . e - 269 Work Spent in Charging a Secondary Battery . . . . . 269 Determination of the Efficiency of a Secondary Battery. 272 Electro-Dynamometers 273 Sir William Thomson's Electric Balances 273 CONTENTS. xvii Weber's Electro-Dynamometer . . . . Siemens’ Electro Dynamometer • e º 'º e Activity Meters . . . . . . . . . . . . . . . e e Q Law of Variation of Current in Alternating Machines . . Measurement of Mean Square of Current by Electro- Dynamometer . . . . . . . . tº e º 'º - © e º 'º Measurement of the Mean Square of a Difference of Potentials by Electrometer . . . . . . . . . . . Determination of Electrical Activity in any part of a Circuit where Self-induction is negligible . . . . . Theory of Alternating Machines—Effects of Self-Induction Theory of Single Machine . . . . . • * * * * e “e Two Equal Machines in Series . . . 3 2 3 y * 3 in Parallel . tº e - Machine and Motor . . . . . . . . . . . . . . . Measurement of Activity in Inductive Circuit . . . . . Relation of True Activity to Apparent Activity as shown by a Wattmeter . . . . . . . . . . . . . . . . Measurement of Activity by Electrometer . tº tº 40 tº o CHAPTER X. MEASUREMENT OF INTENSE MAGNETIC FIELDS. Method by Determination of Electromagnetic Force on a Wire carrying a Known Current . . . . . . . . . Application to Field Magnets of Ordinary Dynamos . . . Application to Field Magnets of Siphon Recorders . . Method by Determination of Critical Resistance for Non- Oscillatory Subsidence of Coil Deflected in Field . Rallistic Method . . º e º 'º º $ tº ſº * * * * * * Use of Earth Inductor for Reduction of Ballistic Results to Absolute Measure e e o 'º Theory of Ballistic Galvanometer . . . . . . . . . . . Calculation of Field Intensity from Experimental Results. Ö PAGE 274 275 275 276 277 279 28 I 282 283 285 288 29O 296 299 3O2 3O7 3IO 3I2 316 317 3I9 32O xviii CONTENTS. PAGE Sir William Thomson's Method for Reduction of Ballistic Results to Absolute Measure . . . . . . . . . . . 320 Determination of Magnetic Distribution by the Ballistic Method . . . . . . . . . . . . . . . . . . . . 321 CHAPTER XI. DIMENSIONS OF THE UNITS OF PHYSICAL QUANTITIES. AWeemeric . . . . . . . . 323 Expression of Physical Quantities Numerically . . . . . 323 Change Ratio . . . . . . . . . . . . . . . . . • 324 Dimensional Formulas . . . . . . . . . . . . . . . 324 Fundamental Units . . . . . . . . . . . . . . . . 325 Derived Units . . . . . . . . . . . . . . . . . . , 333 Dimensions of Derived Units . . . . . . . . . . . . 333 Dimensions of Dynamical Units . . . . . . . . . . . 336 Examples of Problems on Units . . . . . . . . . . . .342 Dimensions of Electrical and Magnetic Units in the Elec- trostatic System . . . . . . . . . . . . . . . . 343 Dimensions of a Conductivity in Electrostatic Units . . .347 Sir William Thomson's Illustration . . . . . . . 348 Dimensions of Electrical and Magnetic Units in the Electro- magnetic System . . . . . . . . . . . . . . . . 348 Magnetic Permeability . . . . . . . . . . . 350 Attraction of a Material Disk on a Particle in its Axis (note) 351 Dimensions of a Resistance in Electromagnetic Units . . .352 Sir William Thomson's Illustration . . . . . . . . .353 Tables of Dimensions of Units . . . . . . . . . . 355, 356 Units adopted in Practice . . . . . . . . . . . . . . 357 Practical Units considered as an Absolute System . . . . 358 Relation of Practical Units to C.G. S. Units . . . . . . 358 Relation between Electrostatic and Electromagnetic Units 360 Evaluation of v . . . . . . . . . . . . . . . . . . .361 CONTENTS. xix APPENDIX. Resolutions of Paris Congress (1884) . . . . . . . . . Resolutions of Paris Congress and British Association Com- mittee (August and September, 1889) . . . . . . . Table I.-Resistances, Conductivities, and Weights of Wires according to the New Wire Gauge Table II.-Resistances, Conductivities, and Weights of Wires according to B. W.G. . . . . . . . Table III.-Conductivities of Pure Metals at tº C . . . Table IV.-Conductivity and Resistance of Pure Copper from of C to 40° C . . Table V.--Specific Resistances in Legal Ohms of Wires of Different Metals and Alloys . . . . . . Table VI. —For Reduction of Period of Oscillation for Finite Range to Period for Infinitely Small Range . . . . . . . . . Table VII.--Units of Work or Energy . . . . . . . . Table VIII.-Units of Activity or Rate of Working . . Table IX. —Doubled Square Roots for Sir W. Thomson's Standard Electric Balances . . . . . . . Table X. —French and English Units . . . . . . & & Table XI.-Numerical Data for Calculating Resistance of Conductors carrying Alternating Cºrrents Table XII.-Virtual Resistance of Conductors carrying Alternating Currents . g * * g & PAGE 364 366 368 37O 37 I 372 373 374. 375 375 376 378 379 38O ERRATA. P. I35, lines 5, II, and 12, for “40” read “ , 6.” ABSOLUTE MEASUREMENTS IN ELECTRICITY AND MAGNETISM. CHAPTER I. I N T R O D U C T O R Y. WENTY-FIVE years ago the experimental sciences of electricity and magnetism were in great measure mere collections of qualitative results, and, in a less degree, of results quantitatively estimated by means of units which were altogether arbitrary. These units, de- pending as they did on constants of instruments and conditions of experimenting which could never be made fully known to the scientific public, were a source of much perplexity and labour to every investigator, and to a great extent prevented the results which they expressed from bearing fruit to the furtherance of scientific progress. Now happily all this has been changed. The absolute system of units introduced by Gauss and Weber, and rendered a practical reality in this Country by the labours of the British Association Com- mittee on Electrical Standards, has changed experimental electricity and magnetism into sciences of which the very 3E B | | 2 INTRODUCTORY. essence is the most delicate and exact measurement, and enables their results to be expressed in units which are altogether independent of the instruments, the surround- ings, and the locality of the investigator. The record of the determinations of units made by members of the Committee, for the most part by methods and instruments which they themselves invented, forms one of the most interesting and instructive books” in the literature of electricity, and when the history of electrical discovery is written the story of their work will form one of its most innportant chapters. But besides placing on a sure foundation the system of absolute units they conferred a hardly less important benefit on elec- tricians by giving them a convenient nomenclature for electrical quantities. The great utility of the practical units and nomenclature, which the Committee recom- mended, soon became manifest to every one who had to perform electrical measurements, and has led within the last few years to their adoption, with only slight al- terations, by nearly all civilized nations. Although it is only five years since the recommendations f of the Paris Congress of Electricians were issued; they have been almost universally adopted and appreciated by those en- gaged in electrical work, and have thus begun to yield excellent fruit by rendering immediately available for comparison and as a basis for further research the results of experimenters in all parts of the world. But in order that the full benefit of the conclusions of the Paris Congress may be obtained it is essential in the first place that convenient instruments should be * Reports of the British Association Committee on Electrical Standards, edited by Prof. Jenkin, F.R.S. - # See Appendix below. INTRODUCTORY. 3 used, adapted to give directly, or by an easy reduction from their indications, the number of amperes of current flowing in a particular circuit, and the number of volts of difference of potential between any two points in that circuit. To be generally useful in practice these instruments should be easily portable, and should have a very large range of sensibility; so that, for example, the instrument, which suffices to measure the full potential produced by a large dynamo-electric machine, may be also avail- able for testing, if need be, the resistances of the various parts of the armature and magnets by the readiest and most Satisfactory method; namely, by comparing by means of a galvanometer of high resistance the difference of potential between the two ends of the unknown resistance with that between the ends of a known resistance joined up in the same electrical circuit. In like manner the ampere measurer should be one that could be introduced without sensible disturbance into a circuit of low resist- ance to measure either a small fraction of an ampere, or the whole current flowing through a circuit containing a large number of electric lamps. These conditions are more or less fulfilled by a large variety of practical instruments recently patented by different inventors. Among these is a very complete set of potential and current measurers, due to Sir William Thomson, and adapted for all kinds of work. My main purpose in the present work is to give an account, both theoretical and practical, of the measurement of currents and potentials in absolute measure ; and to apply this to the graduation of instruments for use in practical electrical work. Of Such instruments I shall take some of Sir William Thomson's as examples, and after describing them, show how they may be graduated, or their graduation tested, B 2 4 INTRODUCTORY. by the experimenter himself. It will be convenient to introduce definitions of absolute units of measure- ment of magnetic and electric quantities when they are required ; and in so doing I shall endeavour to give a clear account of the foundation of the absolute system of electrical measurement, and to show how from the fundamental units are derived the practical units of current, quantity, potential, and resistance. I shall then give and explain a few rules for the calculation of currents and resistances in derived circuits: and describe among others some methods of determining resistances which are useful in many important practical cases, but which so far as I know are not treated of in the ordinary text-books of electricity. The remainder of the work will contain a brief account of the measurement of energy spent in the circuits of generators transmitting power to electric lamps or motors, or in the charging circuit of a secondary battery; a chapter on the practical determination in absolute units of the intensities of powerful magnetic fields, such as those of the field magnets of dynamo machines, and another on the dimensions of units; and, lastly, a few tables of useful electrical data. CHAPTER II. DIETERMINATION OF THE HORIZONTAL COMPONENT OF THE EARTH'S MAGNETIC FIELD. LL the methods by which galvanometers may be graduated so as to measure currents and potentials in absolute units, involve, directly or indirectly, a compari- son of the indications of the instrument to be graduated with those of a standard instrument, of which the constants are fully known for the place at which the comparison is made. There are various forms of such standard instru- ments, as, for example, the tangent galvanometer which Joule made, consisting of a single coil of large radius and a small needle hung at its centre, or the Helmholtz modification of the same instrument with two large equal coils placed side by side at a distance apart equal to the radius of either ; or some form of “dynamometer,” or instrument which instead of the needle of the galvano- meter has a movable coil, in which the whole or a known fraction of the current in the fixed coil flows. The measure- ment consists essentially in determining the couple which must be exerted by the earth’s magnetic force on the needle or suspended coil, in order to equilibrate that exerted by the current. But the former depends on the value, usually denoted by H, of the horizontal component of the earth’s magnetic force, and it is necessary there- fore, except when some such method as that of Kohlrausch, 6 DETERMINATION OF H. referred to below, is used, to know the value of that quantity in absolute units. * The value of H may be determined in various ways, and I shall here content myself with describing the method which is most convenient in practice. It con- sists in finding (I) the angle through which the needle of a magnetometer is deflected by a magnet placed in a given position at a given distance, (2) the period of vibration of the magnet when suspended horizontally in the earth’s field, so as to be free to turn round a vertical axis. The first operation gives an equation involving the ratio of the magnetic moment of the magnet to the hori- zontal component H of the terrestrial magnetic force, the Second an equation involving the product of the same two quantities. I shall describe this method, which was given by Gauss,” somewhat in detail. A very convenient form of magnetometer is that devised by Mr. J. T. Bottomley, and made by hanging within a closed chamber, by a silk fibre from 6 to Io cms, long, one of the little mirrors with attached magnets used in Thomson's reflecting galvanometers. The fibre is care- fully attached to the back of the mirror, so that the magnets hang horizontally and the front of the mirror is vertical. The closed chamber for the fibre and mirror is very readily made by cutting a narrow groove to within a short distance of each end, along a piece of mahogany about IO cms. long. This groove is widened at one end to a circular space a little greater in diameter than the diameter of the mirror. The piece of wood is then fixed with that end down in a horizontal base-piece of wood furnished with three levelling screws. The groove is thus * “Intensitas vis magneticae ad mensuram absolutam revocata.”—Coaſt- 7nent. Soc. Zºeg. Gotting. 1833. DETERMINATION OF H. 7 placed vertical ; and the fibre carrying the mirror is sus- pended within it by passing the free end of the fibre through a small hole at the upper end of the groove, adjusting the length so that the mirror hangs within the circular space at the bottom, and fixing the fibre at the top with wax. When this has been done, the chamber is closed by covering the face of the piece of wood with a strip of glass, which may be kept in its place either by cement, or by proper fastenings which hold it tightly against the wood. By making the distance between the back and front of the circular space small, and its diameter very little greater than that of the mirror, the instrument can be made very nearly “dead beat,”—that is to say, the needle when deflected through any angle comes to rest at once, almost without oscillation about its position of equilibrium. A magnetometer can be thus constructed at a trifling cost, and it is much more accurate and con- venient than the magnetometers furnished with long magnets frequently used for the determination of H; and as the poles of the needle may always in practice be taken at the centre of the mirror, the calculations of results are much simplified. The instrument is set up with its glass front in the magnetic meridian, and levelled so that the mirror hangs freely inside its chamber. The foot of one of the levelling Screws should rest in a small conical hollow cut in the table or platform, of another in a V-groove the axis of which is in line with the hollow, and the third on the plane surface of the table or platform. When thus set up the instrument is perfectly steady, and if disturbed can in an instant be replaced in exactly the same position. A beam of light passes through a slit, in which a thin vertical cross-wire is fixed, from a lamp placed in front of * 8 DETERMINATION OF H. 4 the magnetometer, and is reflected, as in Thomson's reflecting galvanometer, from the mirror to a scale attached to the lamp-stand, and facing the mirror. The lamp and scale are moved nearer to or farther from the mirror, until the position at which the image of the cross- wire of the slit is most distinct is obtained. It is convenient to make the horizontal distance of the mirror from the Scale for this position if possible one metre. The lamp- stand should also have three levelling screws, for which the arrangement of conical hollow, V-groove, and plane should be adopted. The scale should be straight, and placed with its length in the magnetic north and south' line ; and the lamp should be so placed that the incident and reflected rays of light are in an east and west vertical plane, and that the spot of light falls near the middle of the scale. To avoid errors due to variations of length in the scale, it should be glued to the wooden backing which carries it, not simply fastened with drawing pins as is often the case. The magnetometer having been thus set up, four or five magnets, each about IO cms, long and ‘I cm. thick, and tempered glass-hard, are made from steel wire. This is best done as follows. From ten to twenty pieces of steel wire, each perfectly straight and having its ends carefully filed so that they are at right angles to its length, are prepared. These are tied tightly into a bundle with a binding of iron wire and heated to redness in a bright fire. The bundle is then quickly removed from the fire, and plunged with its length vertical into cold water. The wires are thus tempered glass-hard without being seriously warped. They are then magnetized to saturation in a helix by a strong current of electricity. A horizontal east and west line passing through the mirror is now DETERMINATION OF H. 9 laid down on a convenient platform (made of wood put together without iron and extending on both sides of the magnetometer) by drawing a line through that point at right angles to the direction in which a long thin magnet hung by a single silk fibre there places itself (See also p. 23). One of the magnets is placed, as shown in Fig. I, with its length in that line, and at such a distance that a convenient deflection of the needle is produced. This deflection is noted and the deflecting magnet turned end for end, and the deflection again noted. In the same way a pair of observations are made with the magnet at the same distance on the opposite side of the magnetometer ; and the mean of all the observations is taken. These deflections from zero ought to be as nearly as may be H FIG. I. the same, and if the magnet is properly placed, they will exactly agree; but the effect of a slight error in placing the magnet will be nearly eliminated by taking the mean deflection. The distance in cms. between the two positions of the centre of the magnet is also noted, and is taken as twice the distance of the centre of the magnet from that of the needle. The same opera- tion is gone through for each of the magnets, which are carefully kept apart from one another during the experiments. The results of each of these experiments give an equation involving the ratio of the magnetic moment of the magnet to the value of H. Thus if IO DETERMINATION OF H. M denote the magnetic moment of the magnet, M the magnetic moment of the needle, r the distance of the centre of the magnet from the centre of the needle, 2X the distance between the “poles” (p. 36) of the magnet which, for a nearly uniformly magnetized magnet of the dimen- sions stated above, is nearly equal to its length, and 2X the distance between the poles of the needle, r, A, and X’ being all measured in cms, we have for the repulsive force (denoted by F in Fig. 1) exerted on the blue * pole of the needle by the blue pole of the magnet, supposed nearest to A/ /l/7 I 2X 2x (~x); since the value of X is small compared with A. Similarly for the attraction exerted on the same pole of the needle by the red pole of the magnet, we have the expression A/ //7 I 2X 2X' ' (r-EX)? by the magnet on the blue pole of the needle is A///’ I I A/7 7- ----- i – or AZ * . —. ... [wº-ºoºººº-ºº: Proceeding in a precisely similar manner, we find that the magnet of moment M exerts an attractive force equal to A/ g º wº on the red pole of the magnet. The needle is therefore acted on by a “couple’’ which tends to turn it round the Suspending fibre as an axis, and the amount of this couple, when the angle of deflection is 6, is plainly O 7” e -- e. - e equal to MM’ 2r , cos 6. But for equilibrium this (r” – A) * The convention according to which magnetic polarity of the same kind as that of the earth’s northern regions is called blue, and magnetic polarity of the same kind as that of the earth’s southern regions is called red, is here adopted. The letters B, R, 6, 7, in the diagrams denote blue and red. the needle, as in Fig. I, the value Hence the total repulsive force exerted DETERMINATION OF H. I I couple must be balanced by M"H sin 6; hence we have the equation— AZ (* ~ * tang . . . . (I) Aſ 27- If the arrangement of magnetometer and straight scale described above is adopted, the value of tan 6 is easily obtained, for the number of divisions of the scale which measures the deflection, divided by the number of such divisions in the distance of the scale from the mirror, is then equal to tan 26. Instead of in the east and west horizontal line through the centre of the needle, the magnet may be placed, as represented in Fig. 2, with its length east and west, and its centre in the horizontal north and South line through the centre of the needle. If we take M, M', \, X', and r to have the same meaning as before, we have, for the distance of either pole of the magnet from the needle, the expression Wrº-FX”. Let us consider the force acting on one pole, say the red pole of the needle. The red pole of the magnet exerts on it a repulsive force, and the blue pole an attractive force. Each of these forces has the value AM //7 I 2X 2X 2 + x2' equivalent to a single force, F, in a line parallel to the magnet, tending to pull the red pole of the needle towards the left. The magnitude of this resultant force is plainly 2 M. M. X . Or A//// 2\ | 2N (2 + x2) 2X(r” + X): it can be shown that the action of the magnet on the red pole of the needle is a force of the same amount tending to pull the blue pole of the needle towards the right. The needle is, therefore, subject to no force tending to But the diagram shows that they are In the same way I 2 DETERMINATION OF H. produce motion of translation, but simply to a “couple’” tending to produce rotation. The magnitude of this couple when the needle has been turned through an angle . MM' 2\! cos 6 M//’ 6, is Tax (ZEW); Or (2-EX3); cos 6. If there be B - R FIG. 2. equilibrium for the deflection 6, this couple must be balanced by that due to the earth's horizontal force, which, as before, has the value M'A sin 6. Hence equating these two couples we have— # = (a+xº an a . . . (2) DETERMINATION OF /7. I 3 Still another position of the deflecting magnet relatively to the needle may be found a convenient one to adopt. The magnet may be placed still in the east and west line, but with its centre vertically above the centre of the needle. The couple in this case also is given by the formula just found, in which the symbols have the same meaning as before. The greatest care should be taken in all these experi- ments, as well as in those which follow, to make sure that there is no movable iron in the vicinity, and the instru- ments and magnets should be kept at a distance from any iron nails or bolts there may be in the tables on which they are placed. We come now to the second operation, the determina- tion of the period of oscillation of the deflecting magnet when under the influence of the earth’s horizontal force alone. The magnet is hung in a horizontal position in a double loop formed at the lower end of a single fibre of unspun silk, attached by its upper end to the roof of a closed chamber. A box about 30 cms. high and 15 cms. wide, having one pair of opposite sides, the bottom, and the roof made of wood, and the remaining two sides made of plates of glass, one of which can be slided out to give access to the inside of the chamber, answers very well. The fibre may be attached at the top to a horizontal axis which can be turned round from the outside so as to wind up or let down the fibre when necessary. The suspension-fibre is so placed that two vertical scratches, made along the glass sides of the box, are in the same plane with the magnet when the magnet is placed in its sling, and the box is turned round until the magnet is at right angles to the glass sides. A paper screen with a small hole in it is then set up at a little distance in such a position that the I 4 DETERMINATION OF AZ. hole is in line with the magnet, and therefore in the same plane as the scratches. The magnetometer should be removed from its stand and this box and suspended needle put in its place. If the magnet be now deflected from its position of equilibrium and then allowed to vibrate round a vertical axis, it will be seen through the small hole to pass and re-pass the nearer scratch, and an observer keep- ing his eye in the same plane as the scratches can easily tell without sensible error the instant when the magnet passes through the position of equilibrium. Or, a line 1may be drawn across the bottom of the box so as to join the two scratches, and the observer keeping his eye above the magnet and in the plane of the scratches may note the instant when the magnet, going in the proper direc- tion, is just parallel to the horizontal line. The operator should deflect the magnet by bringing a small magnet near to it, taking care to keep this small deflecting magnet always as nearly as may be with its length in an east and west line passing through the centre of the suspended magnet. If this precaution be neglected the magnet may acquire a pendulum motion about the point of Suspension, which will interfere with the vibratory motion in the horizontal plane. When the magnet has been properly deflected and left to itself, its range of motion should be allowed to diminish to about 3° on either side of the posi- tion of equilibrium before observation of its period is begun. When the amplitude has become sufficiently small, the person observing the magnet says sharply the word “Now,” when the nearer pole of the magnet is seen to pass the plane of the scratches in either direction, and another observer notes the time on a watch having a seconds hand. With a good watch having a centre Seconds hand moving round a dial divided into quarter-Seconds, DETERMINATION OF A/. I 5 the instant of time can be determined with greater accuracy in this way than by means of any of the usual appliances for starting and stopping watches, or for regis- tering on a dial the position of a seconds hand when a spring is pressed by the observer. The person observing the magnet again calls out “Now * when the magnet has just made ten complete to and fro vibrations, again after twenty complete vibrations, and, if the amplitude of vibra- tion has not become too small, again after thirty; and the other observer at each instant notes the time by the watch. By a complete vibration is here meant the motion of the magnet from the instant when it passes through the position of equilibrium in either direction, until it next passes through the position of equilibrium going in the Same direction. The observers then change places and repeat the same operations. In this way a very near approach to the true period is obtained by taking the mean of the results of a sufficient number of observations, and from this the value of the product of me and H can be calculated. For a small angular deflection 6 of the vibrating magnet from the position of equilibrium the equation of mo- tion is - d/26) , MAſ zºº + -- where p is the moment of inertia of the vibrating magnet round an axis through its centre at right angles to its length. The solution of this equation is e - a sin(x/* - A £1. and therefore for the period of oscillation 7 we have 6 = O, *-i-x/#, I6 DETERMINATION OF H. Hence we have ºr — 47°u A/// = 75 Now, since the thickness of the magnet is small compared with its length, if W be the mass of the magnet and 2/ its actual length, p. is VV/*/3, and therefore 47%.” W Al/// = (3) Combining this with the equation (1) already found we get for the arrangement shown in Fig. 1, 7. / ) 2 T2(2” – X2)2/2 W/ tan 6 Al/ 2 :- 3 a -- º ---- (4) and . 2/2 A/2 == 8 74/22 1// (5) T 3 72(? – X3), tan 6 If either of the other two arrangements be chosen we have from equations (2) and (3) g 4 trº /* // —--- (a-2 2\? C Ul I. U → • 3 T2 r” + A*)3 W tan 6 (6) and 4 T2/3 W/ H* = . . --—"... Tº----. . . . (7) 3 (24 + \?)? Tº tan 6 Various corrections which are not here made are of course necessary in a very exact determination of H. The virtual length of the magnet—that is, the distance between its poles or “centres of ‘gravity’ of magnetic polarity” as they have been called *—should be determined by experi- ment. This may be done by observing the deflections 6 * Strictly speaking no such points exist. See p. 36 below. DETERMINATION OF H. 17 and 6' of the magnetometer needle produced by the magnet when placed in the position shown in Fig. I at distances r and r", both great in comparison with /, from the centre of the needle. We have the equations M (,” – X?)? 7- \ + = Y → - tan 6 = \{------" " Aſ 2 7° 27-' and therefore, 7' 2 N/7 tan 6 – 2 N/7tan 6 X2 - -- - - - - - ~~ Wr tan 6 – W7tan 9 (8) Preferable methods of taking account of the magnetic distribution will be found at pp. 28, 29 below. Allow- ances should be made for the magnitude of the arc of vibration ; the torsional rigidity of the suspension fibre of the magnetometer in the deflection experiments (pp. 9–12), and of the suspension fibre of the magnet in the oscillation experiments; the frictional resistance of the air to the motion of the magnet; the virtual increase of inertia of the magnet due to motion of the air in the chamber ; and the effect of induction and, if necessary, of changes of temperature in producing temporary changes in the moment of the magnet. The correction for an arc of Oscillation of 6° is a diminution of the observed value of 7" of only tºo per cent., and for an arc of IO’ of 35 per Cent. Of the other corrections that for induction is no doubt the most important; but its amount for a magnet of glass-hard steel, nearly saturated with magnetism, and in a field so feeble as that of the earth, may, if only a roughly accurate result is required, be neglected. This correction arises from the fact that the magnet in the deflection experiments is placed in the magnetic east and west line, whereas in the oscillation experiments it is C I8 DETERMINATION OF H. placed north and south, and is therefore subject in the latter case to an increase of longitudinal magnetization from the action of terrestrial magnetic force. The in- crease of magnetic moment may be determined by the following method, which is due to Mr. Thomas Gray. Place the magnet within, and near the centre of, a helix, considerably longer than the magnet and made of insu- lated copper wire. Place the helix and magnet in position either as shown in Fig. 2, or as in Fig. 3, for giving a deflection of the magnetometer needle, and read the deflection. Then pass such a current through the wire of the helix as will give by electromagnetic induction a magnetic field within the helix nearly equal to the horizontal component of the earth's field, and again observe the deflection of the magnetometer needle. The intensity of the field thus produced within the helix at points not near the ends is given in C.G.S. units (p. 43) by the formula 4trn C, where n is the number of turns per centimetre of length of the helix, and C is the strength of the current in C.G.S. units (pp. 44–48). Experiments may be made with different strengths of current, and the results put down in a short curve, from which the correc- tion can be at once read off when the approximate field has been determined by the method of deflection and oscillation described above. Care must of course be taken in experimenting to eliminate the deflection of the magnetometer needle caused by the current in the coil. This is easily done by observing the deflection produced by the current when the magnet is not inside the coil and subtracting this from the previous deflection, or by arranging a compensating coil through which the same current passes. This plan has several advantages (see p. 31 below). The change of magnetic moment produced DETERMINATION OF H. I9 in hard steel bars, the length of which is 12 cms, and diameter 2 cm., and previously magnetized to Saturation, is, according to Mr. Thomas Gray's experiments, about gº per cent. (For particulars of actual experiments, see pp. 30–36 below.) The deflection experiments are, as stated above, to be performed with several magnets, and when the period of oscillation of each of these has been determined, the magnetometer should be replaced on its stand, and the deflection experiments repeated, to make sure that the magnets have not changed in strength in the meantime. The length of each magnet is then to be accurately deter- mined in centimetres, and its weight in grammes; and from these data and the results of the experiments the values of M and of H can be found for each magnet by the formulas investigated above. Equation (5) is to be used in the calculation of H when the arrangement of magnetometer and deflecting magnet, shown in Fig. I, is adopted, equation (7), when that shown in Fig. 2 is adopted. The object of performing the experiments with several magnets, is to eliminate as far as possible errors in the determination of weight and length. The mean of the values of H, found for the several magnets, is to be taken as the value of H at the place of the magnetometer. We shall show in the next chapter how to apply this value to the measurement of currents. The following is an account of a determination of H made by this method, with several improvements in the practical carrying of it out, by Mr. Thomas Gray in the Physical Laboratory of the University of Glasgow, during the summer of 1885. The apparatus and its arrangement is shown in Fig. 3. T represents a table which supports C 2 2O DETERMINATION OF H. · 8 · 31 J ozīS 11n4 % ºppos | - - -; «- «-»suo- 634 ----> DETERMINATION OF H. 2 I the magnetometer M, two stands A and B for the deflecting magnets, and a lamp and Scale S. The magnetometer consisted, as described above, of a light mirror about ‘8 cm. in diameter, suspended by a single silk fibre within a recess in a block of wood, and carrying on its back two magnets each I cm. long and ‘O8 cm. in diameter. Two holes cut in the wood at right angles to one another (and plugged when not in use), permitted the position of the mirror and magnets to be seen and adjusted. [A prefer- able form of magnetometer since adopted consists of a mirror and attached magnet Suspended within a glass-tube from a brass mounting at the upper end which allows the fibre to be wound up or down. For definiteness and ease of determination of the magnetic centre of the needle, a Single Small cylindrical magnet is used, carried at the lower end by a short strip of aluminium to which the mirror is attached. At the lower end half the tube is cut away over a length of two or three cms, and the part remaining closes the back of the chamber in which the mirror hangs. The sides of the chamber are of wood attached to the base piece, and the front, or side toward the lamp, is closed by a sliding panel of glass which can be placed in either of two slots so as to give a greater or less space for the mirror to swing in..] The sole plate P, made of mahogany, is supported on three brass feet, which rest in a hole, slot, and plane arrangement cut as described above in a horizontal plate of glass cemented to the table. The deflector stands A, B rest each on a base plate P, of mahogany, supported, according to the hole, slot, and plane device, in precisely the same way as the magneto- meter, on plates of glass ſº, Z, cemented to the table T. Each stand consists of a horizontal carriage for the deflec- 22 DETERMINATION OF H. tor magnet, and is constructed as follows: A strip of hard wood, about I 3 cms. long and 4 cms, broad, has a V-shaped groove run along its whole length in the middle of one side. One end is faced with a plate of brass in which a brass screw works, and the piece is cemented with the groove upwards to a plate of glass g. This plate is sup- ported on three feet of hard wood, resting on the mahogany sole plate P, and is free to turn in azimuth round a closely fitting centre pivot c fixed in the sole plate. The apparatus is so adjusted that the bottom of the V-groove is just over the pivot c. The magnet when placed in the carriage lies along the groove, and the screw s serves to give a fine adjustment of the position of one end which abuts against it. Over each carriage a wire of brass or copper bent into a semi-circle serves as a support for a suspension fibre with double loop, by which the deflector can be suspended for purposes of adjustment or for the oscillation experiments. A glass shade can be placed on the plate P to prevent currents of air from disturbing the magnet in the oscillation experiments, In Fig. 3 the deflecting magnets d, d, are shown in positions at equal distances east and west of the magneto- meter, at a distance of 70 cms, between their centres. Four plates of glass are fixed to the table in two end on positions and in two side on positions, each pair of posi- tions being at equal distances from the magnetometer needle, and on opposite sides of it. The Scale S, shown at a distance from the mirror of I29 cms., is a millimetre scale carefully divided on transparent glass So that the spot of light may be observed either from the front or the back. The first adjustment, made in Setting up the apparatus, was to place the table so that the line joining the centres DETERMINATION OF AZ. 23 of A and B should be exactly at right angles to the magnetic meridian. This was done by one or other of the following two methods according as (a) the end-on, or (5) the side-on position was required. (a) After the adjustment had been first roughly made, a plane circuit was formed by stretching a thin wire along the line join- ing the centres of A, B under the magnetometer needle, and then carrying the wire back, either above the magneto- meter, or below it, at a greater distance, in a vertical plane. An electric current was then sent through the wire, and the table T, with the apparatus turned until the current pro- duced no deflection of the needle. (b) One of the deflecting magnets was placed in its carriage, either South or north of the needle, and lifted out of the V-groove by the sus- pension fibre, and the table turned until the suspended magnet produced no deflection of the magnetometer needle. The magnet and needle were then in one line, and if the needle was in its proper position this line produced through the centre of the needle passed through the position of the deflector on the other side. The deflector was placed on the opposite side of the needle, and the table T, turned until no deflection was obtained. The position of the needle was then altered, if necessary, by the levelling screws until the positions of the table for no deflection with the magnet first on one side then on the other of the magnetometer were coincident. If this could not be done the plates ? were not placed with Sufficient accuracy, and their position had to be changed. This process gave the direction of the magnetic meridian with accuracy and ensured that the plates / in the north and South line were properly placed on the table. The two methods taken together ensured that all four plates ? were properly placed. 24. DETERMINATION OF H. Deflectors of different relative lengths and thicknesses, and of different degrees of hardness, were used. These were originally magnetized by placing them between the poles of a large Ruhmkorff magnet excited by a consider- able current, and afterwards by the same magnet excited by a much stronger current. The relative strengths of the magnets were unchanged by the Second magnetization, and their absolute strengths only very slightly. The dimensions are given in the table of results, p. 35 below. The method of observing the deflections was as follows: According to a suggestion of Sir William Thomson two deflectors were used at the same time, one on each side of the magnetometer. This arrangement was more sym- metrical than that of a single deflector, and, what was of very great importance, it enabled a readable deflection to be obtained with the magnets at a much greater distance from the needle, thus diminishing error due to uncertainty as to the actual magnetic distribution. As each magnet was transferred on its carriage from One glass plate to another the magnet was not handled during the experi- ments. One deflector A was placed east another B west of the magnetometer, and the plate g turned for each until their lengths were accurately in the east and west line, and their poles so turned that each magnet gave a deflec- tion of the needle to the same side of zero; and the deflection was then noted. The plates g were then turned through 180°, and the deflection on the opposite side of zero read off. The carriages were then turned back to the first position and the deflection again read. The difference between the mean of the first and third readings and the second reading gave twice the deflection for the position of the magnets. The same operation was then repeated with the deflector in inter-changed positions. Two similar DETERMINATION OF H. 25 series of observations were next made with the magnets in the north and south line through the magnetometer and at equal distances on opposite sides of the needle. The Imean deflection for the east and west positions and that for the north and south positions were calculated, and the results were used in the calculation of H in the manner described below. Scaſe % Full Size FIG. 4. After the deflection observations for a particular magnet had been completed, the magnetometer was removed and the deflector stand put in its place. The magnet was suspended from the brass bow & Over its carriage by a length of single cocoon fibre, in a double stirrup formed by twice doubling the lower end of the fibre and knotting. The suspension thus obtained was sufficiently fine to be practically devoid of inertia, and long enough to give a negligible moment of torsion. The magnet was deflected in the manner already described (p. 14 above), and then 26 DETERMINATION OF H. left to oscillate. The period was observed in some cases by noting the times of the successive transits of the needle across the vertical cross wire of an observation telescope; but the method finally adopted was to attach to the stirrup as shown in Fig. 4 a light silvered mirror /e (.3 cm. in diameter and ‘OI gramme in mass), and to use the same lamp and scale as in the deflection experiments. This latter arrangement enabled the amplitude of oscillation to be reduced to less than a degree and so reduced to zero the correction necessary for arc. The moment of inertia of the mirror was only about roºoo of that of the deflector, and its neglect therefore introduced an error of only rºw per cent. • Time was observed in these experiments by means of a very accurate watch provided with a centre Seconds hand moving round a dial divided into quarter seconds. When two observers were available, one counted the oscillations and called sharply “Now” at the end of every four or five periods, while the other observed the time at each call. When only one observer counted the oscillations he used a chronometer beating half seconds. Having read time, he counted the beats until he could observe a transit. He then counted the beats until he observed another transit. From the result he estimated the number of periods in one minute, and therefore observed the time of the first transit after each minute so long as there was sufficient amplitude. The fractions of half seconds were estimated from the positions of the magnet at the beat next before and the beat next after the transit. With the mirror and scale arrangement these observations could be made with great accuracy. The observations were combined in the following manner so as to give the most probable value of the DETERMINATION OF H. 27 period. Supposing the number of observations to have been even, 27, say. The interval between the 77th obser- vation and the (n + 1)th, three times that between the (n − 1)th and the (n + 2)th, five times that between the (n − 2)th and the (n + 3)th, and so on to that between the 1st and the 2nth were added together, the sum divided by the sum of the series 1* + 3” + 5° + . . . -- (27 – 1)*, and the result by the number of periods (which was the same in each case) between each successive pair of observations. This gave the average period to a high degree of approximation. If an odd number of obser- vations (2n + 1) was taken, the interval between the 77th and the (n + 2)th, twice that between the (?? – 1)th and the (n + 3)th, three times that between the (n − 2)th and the (n + 4)th, and so on to the Ist and (27. -- I)th, were added together, the sum divided by twice the sum of the series 1* + 2* + 3 + ... + m”, and the result divided by the number of periods in each interval gave the average period. The period adopted was always the mean of those given by two closely agreeing sets of observa- tions. Assuming (see p. 36) that the magnet has two definite poles, that is (in this connection) points at which the whole of the free magnetism in each half of the magnet may be Supposed concentrated in considering the external action of the magnet (an assumption not seriously erroneous in the case of the thin magnets and the distances used); the distance between them can be calculated from the results of deflection experiments in the side-on and end-on positions obtained as described above, since the effect of the distribution is opposite in the two cases. For if r be the distance for the end-on position, 6 the deflection, and r", 6" the distance and deflection for the side-on 28 DETERMINATION OF H. position, we have by equating the values of M/H given by equations (I) and (2): (* = N*)* tan 0. 27 (r. 2 + \?); tan 6' Expanding the numerator and denominator of each side and neglecting terms smaller than those of the second Order we get: 2.36 — 27-’36' ~ ārāTº . . . . . (9) X2 By this equation the value of A used in the calculation of Aſ and M was found. The results for magnets of different lengths and diameters are interesting in themselves. The moment of inertia of the bar was found by weighing the bar and carefully measuring its length and cross- Section, and calculating for a vertical axis through the centre of the magnet supposed hung horizontally. The axis of Suspension of the magnet in any case was not, however, that vertical, but another near it owing to the compensation for the tendency of the magnet to dip in the earth's field. The distance between these two axes can be found approximately for each magnet from the magnetic moment, mass, and length as given in the table below, and is so Small that any error caused by Supposing the magnet simply to vibrate round the former vertical is well within the possible limit of accuracy. For a cylindrical magnet of mass W, actual length 2/ and diameter d, the moment of inertia is W (Z%|3 + dº/64). Hence (3) becomes: 4 * (* + ºr d") W. . . (IO) 3 72 Hence for a single deflector we get instead of equations A/A/ = DETERMINATION OF H. 29 (4), (5), (6), (7) equations obtained from these by substi- tuting instead of /*, 2* + 3d%64. If two deflectors be used, each of the actual length 2/, and diameter d, but of masses W1, W., periods 7, 7, and nearly equal effective lengths which give a mean A, we get from (1) and (2) instead of (5) and (6) for the end-on and side-on positions respectively : 8 Tºr (ſº + š, d”) (7 °W. -- 7% W.) Aſ2 = II 3 (,” – X?)? 7,27.2 tan 9 (II) Aſ2 = 4. Tº (/* + š d”) (7.2 W, -- T.” W.) (I2) 3 (r” + A*); 737.* tan 6 In these formulas 6 and 6 are the angular deflections found from the mean readings taken as described above (p. 24). There are two corrections for alteration of moment of the magnet, produced (I) by variation of temperature, (2) by induction when the magnet is in or near the magnetic meridian when oscillating. The first correction was found by placing the magnet within a bath, in one of two principal positions at such a distance from the magneto- meter needle that a deflection of I,000 divisions was obtained, and then raising the temperature through about 40°C. It was found that such a rise of temperature pro- duced a change of deflection of only about two divisions. Thus the magnets changed in magnetic moment by only gººd per cent, for a change of temperature of 1° C. Hence as the variation of temperature in the experiments never exceeded 2°C. or 3° C. this correction was neglected. The correction for induction was found by immersing the deflecting magnet in an artificially produced magnetic field of known strength, and ascertaining the alteration of 3O DETERMINATION OF H. magnetic moment which resulted. The field was produced by surrounding the magnet with a magnetizing coil, and its intensity calculated from the number of turns of wire per unit of length of the coil and the current-strength, which was measured. The coil was sufficiently long to project beyond the magnet at each end some distance, so that the magnetic field was uniform, and equal to 4tnc, where m = number of turns per cm. of length, and C the current strength in C.G.S. units (see below, Chap. III.). Fro, 5. Fig. 5 shows the arrangement of apparatus for these experiments; m is the magnetometer needle, C, C" are coils each consisting of silk-covered copper wire wound on glass tubes 5 cm. in external diameter, S is the lamp scale, ſº a box of resistance coils, G the current galvano- meter, K a reversing key, and B a battery. DE represents a horizontal line through the needle and in the magnetic meridian, and AF a horizontal line at right angles to OE, and also passing through the centre of the needle. As DETERMINATION OF H. 3I shown in the diagram the coil C was placed with its axis parallel to AF and its centre on the line D.E. C.’ had its axis in the line AF, and the relative distances of the coils from the magnetometer needle were so adjusted that the magnetic effect of the current passing through the coils was zero at the needle, although the current flowing was made many times greater than that used in the experiments. The magnet for which the induction correction was to be determined was then placed in one of the coils and the deflection read while as yet no current flowed. A field of about 1% of a C.G.S. unit was then produced by passing a current, and the deflection was once more read. The current was then reversed, and the deflection again noted. The same operations were then repeated with greater and greater currents until a field of from I to 2 units had been reached. The magnet was then transferred to the other coil, and a similar series of observations made. It was found that a field of considerably greater intensity than the highest thus used is required to produce any permanent change of the magnetic moment of hard-tempered magnets. The increase of magnetic moment being plotted as ordinate of a curve, the corresponding abscissa of which was the field intensity, enabled the change produced by the earth's field to be obtained by interpolation as described above (p. 18). A comparison of the results obtained with the two coils showed that the percentage change of deflection produced by the field was smaller for the coil C than for the coil Cº. This was undoubtedly due to change of magnetic dis- tribution, the effect of which on the deflection is opposite in the two cases. Assuming that the magnet has an effective half-length \, the deflection in the first case is 32 DETERMINATION OF H. given by (I) and in the other by (2). Thus by using the coils in the two positions as described, the change of distribution as well as the change of moment can be approximately estimated. The plan of two coils has also the advantage of allowing the change of magnetic moment to be obtained free from any error caused by want of exact compensation between the two coils of their direct effect upon the needle. The results of the experiment showed that to make the effect of induction Small the magnet should be hard tempered, and its length should be at least 40 times its diameter. The results are shown in the table on p. 35 below. The effects of variations in the intensity and direction of the earth’s magnetic field were quite marked. The latter showed itself by changes of the magnetometer zero, which were eliminated by reading the zero before and after each deflection, and by reversing the magnets. The effect of change of intensity was allowed for by observing the period of a permanent magnet kept sus- pended for the purpose. This period was observed at the beginning of the experiment, after the deflection experi- ment, and again after the oscillation experiment. The necessary correction was estimated from the results and applied. It will be observed that the effect of diurnal variation is quite perceptible. The results in the table on p. 34 are tabulated in the order in which they were obtained, and it will be noticed that the earlier results of each day are generally the smaller. On some occasions on account of magnetic storms it was found impossible to obtain results at all. This was notably the case on Sept. 1, 1885. The results of this determination are shown in the fol- DETERMINATION OF H. 33 lowing two tables. The variation of the effect of induction on the magnetic moment with different ratios of the length of the deflecting magnet to its diameter is shown in the curve of Fig. 6. Curve illustrating the effect of Ratio of Length to Diameter on the Inductive Coefficient. i : Ratio of length to diameter FIG. 6. It will be observed that the effect of induction diminishes, rapidly at first, then more and more slowly, towards a constant value of about 4 per cent. for unit field for glass-hard magnets of the kind of steel experi- mented on. - #29 ‘A/ JO NOIJ.VNIWRIGILGICI i |- *— B- S. S. : : ; 5 - 3 5 § 3 ; ; QQ ; GO --' Sº-,--> § § º § - Diameter of deſlector, Ol Oſt 111 Centlmetres. N N tº N. N. N. ‘. . . Jº; ‘. . .'; Weight of deflector, in § 3 ; ; ; ; g\! Sº S $ $ 3.3 3,5 grammeS. ON CO ON CO CN CO ONS ON On tw) ON ~. On OJ -º- Distance of centre of Cy) U.J. Oſt (JJ O) deflector from the ‘. . ; ; ; ; ; ; ; ; ; ; . magnetometer-needle, 8 8 O O ON in cms. (East and west positions.) Distance of centre of (j) (J.) Cy) Cy) {v} N) deflector from the ... . . ; ; ; ; ; ; ; ; ; ; magnetometer-needle, 3 3 8 $º Č. Oſl in cms. (North and t - e. South positions.) H - H |- Distance of the scale & & & 3 g : ; ; ; ; ; 2 & from the magneto- VC c • ^l meter mlrror. \O CO WO Cy) tº C.) tº 92 o' S oys o 4 e * Tº ſº H C - ' co N 03 H Cy) - \O Eſſective length of the -R CO -R- tw) UJ to to H H O H deflector. ^ *-* CNS. CNSI CNS, On CN Cn U. U.J. O.J. Cn On Oſt Magnetic moment per H. O. N. H. N. H. = + v. H. v., on tº + c2+. gramme of the co o S o S o on 93 co S S N 03 - Qi o deflector & § § § § & § § º § § º § Moriº º 111 3, ^l CN S & S *Q -- -R- UN H -- O . Y. Ulſ]ltS. H. : : : : * : : : : : © . ; § ; ; § ; ; § ; ; ; Mean tº set of à i ; & : : -; ; ; ; ; ; ; lºeSUlitS. Double Remarks. deflector method. DETERMINATION OF H. 35 TABLE II.-Showing the effect of Length and of Hard- ness on the Induction-Coefficient of Magnets. Unit field. §: 5 9 - 5 e --> .# | 8 || ### | ##3 | #j | # H 5 ºf E. E. S. # 5 # , C ſº E 45 Q: 35 b) . 8 O 90 . 8 O ſ: E º 6 E † : ## | #53.3 #5.5 || 3 || | # 5 o E º # ă's; #5; ; J # .9 ; Remarks. *5 E O & # 9 ºr 3 E 9°F 3 o E 3 503 Sº $ 3.5 2. § 3 tº P. 5.3 5 § 5 ö º § 3 5 § { 5 § 3 of C4 H £4 à:3 !- £3 !—t : C > 4. I6 o'67 o'70 o'69 35 5 y 6 2O O'51 o'67 O'59 36 » y 7 3I O'5I o'58 O'54 3 y y 8 32 O'5I o°58 O'S4 54 y y 8 32 O 5 I o'58 O'54 52 x 3 IO 34 9°46 o'56 O'5I 4O x 3 IO 44 O'4O o'56 o'48 43 j } 7 47 o°46 o'5I O'49 57 3 x IO 5O O'44 o' 58 O'51 67 3 y IO 5O o'48 O'54 O'5I 6o y 3 IO 5O o°46 O'55 O'5I 53 y 3 IO 5O O'46 O'52 O'49 71 > y IO 5O o'46 o'56 O'5I 6o 2 3 - IO 67 O'4I O '5I o'46 65 3 × 7 73 O'4I O'50 O'47 64 x 5 IO IO5 O'42 O'45 O'43 66 5 y IO 34 O'47 O'53 O'50 41 5 || Glass hard. IO || 34 o'63 o'67 o°65 44'5 || Yellow. IO 34 o'84 o'98 O'91 54"I | Blue. IO 48 o'32 O'4O o'36 45 Glass hard. IO 48 O'43 O'55 O'49 46 Yellow. IO 48 O'S3 o'67 o'60 71 Blue. 36 DETERMINATION OF H. The method given above (p. 28) for the determination of the correction for the non-uniform magnetization of the deflecting magnet, gives of course only a first approxima- tion to the true correction, but under the condition that the length of the bar is sufficiently small in comparison with the distance r, say from 4 to #5 of r, and on the sup- position that the magnet is reversed at the position on either side of the needle (Fig. 1), it is generally sufficient. In general no such point as a “centre of ‘gravity” of magnetic polarity,” that is, a point at which the whole of the free magnetism in the actual distribution in each half of the magnet may be supposed to be concentrated, so as to have the same action on the needle as the actual dis- tribution has, in strictness exists; but, in dealing with the equilibrium or motion of a magnet in a uniform field, and therefore acted on by a magnetic couple, or system of Couples, we may regard the Centre of inertia of a material system following the same law of distribution as that of the magnetic matter of either half of the magnet as the “centre of mass” of that magnetic matter. This point or “ pole” is the centre of the horizontal parallel magnetic forces acting on the particles of ideal free magnetic matter of either half of the magnet when it is hung horizontally in the earth's field, as in the vibration experiments, and is deflected from the magnetic meridian, or of those acting on each magnetism of the needle in the deflection experi- ments. This is the most proper use of the term “pole,” and not that in which it is regarded as a “centre of ‘gravity’ of magnetic polarity” in the sense just explained. A method of observation and reduction which amounts to determining the distribution of ideal magnetic matter over the surface of the magnet which has the same external magnetic action as the actual distribution, is given in DETERMINATION OF H. 37 Maxwell's Electricity and Magnetism, vol. ii., chap. vii., where also will be found full information as to corrections necessary in an exact determination of Åſ. Let two deflections be taken (Fig. 1) by reversing the deflecting magnet at a distance r1 on the west side of the needle, and similarly two deflections at the same distance on the east side, and let Di be the mean of the tangents of these four deflections. I.et this process be repeated at a second distance rº, and let D, be the mean tangent for that distance. It is easy to prove that, approximately : 2/4 - ??”/2 - ??"/22 (I3) Aſ r,” – r.” It has been shown (Maxwell’s Electricity and Magnetism, vol. ii., p. IOI ; or Airy's Magnetism, p. 68) that if approxi- mately r = 1:32 rº, the effect of errors in the observed deflections on the value of A//// will be a minimum for these distances. If long thin bars are used in the determination of H, their magnetic distribution could first be determined very accurately by Rowland's method (Chap. X. below), and the proper corrections applied. On the other hand, short thick bars of hard steel have the advantage of giving greater magnetic moment for a given length, and they can there- fore be placed at a greater Comparative distance from the needle, so that the correction for the distribution becomes of less importance. So far, then, as the deflection experi- ments are concerned it is better to use thick short magnets of the hardest steel, and to place them at such a distance from the needle that the error caused by neglecting A becomes vanishingly small. On the other hand, the magnets must be sufficiently long and thin to render it possible to determine with accuracy their moments of 38 DETERMINATION OF H. inertia, and therefore to reduce correctly the results of the vibration experiments. When X is Small in comparison with r, we may use the approximate formula 23 M = ′ Aſtan 0 . . . . (14) 2 for the position shown in Fig. 1 ; or M = r" // tan 6 . . . . (15) for the position shown in Fig. 2. A magnetic Survey of horizontal force in the neighbour- hood of a place for which H has been determined may very readily be made with one of the magnets used in the deflection experiments, by simply observing its period of vibration at the various places for which a knowledge of H is desired. The magnetic moment /l/ of the magnet being of course known from the previous experiments, H can be found by equation (5) or (7) of Chap. II. above. By keeping a magnetometer set up with lamp and scale in readiness, the magnetic moments of large magnets can be found with considerable accuracy by placing them in a marked position, at a considerable distance from the needle, and observing the deflection produced. By having a graduated series of distances, for each of which the constant r" H or r" //, as the case may be, by which tan 6 must be multiplied to give //, has been calculated, the magnetic moments can be very quickly read off. The magnetic moments of large magnets of hard steel well magnetized can be determined very conveniently with considerable accuracy by hanging them horizontally in the earth's field, and determining the period of a small oscilla- tion about the equilibrium position. They should be hung DETERMINATION OF H. 39 by a bundle of as few fibres of unspun silk as possible, at least six feet long, so that the effect of torsion may be neglected. The suspension thread should carry a Small cradle or double loop of copper wire, on which the magnet may be laid to give it stability, and to allow of its being readily placed in position or removed. Two vertical marks are fixed in the meridian plane containing the suspension thread, and the observer placing his eye in their plane can easily tell very exactly when the magnet is passing through the equilibrium position, and so determine the period. Or a north and south line may be drawn on the floor or table under the magnet, and the instant at which the magnet is parallel to this line observed by the experimenter by stand- ing opposite one end of the magnet and looking from above. The value of M is given in terms of // by equation (3) above, Care must of course be taken to avoid undue disturbance from currents of air, and to prevent the magnet, when being deflected from the meridian, from acquiring any pendulum Swing under the action of gravity. The deflec- tion from the meridian should be made with another magnet brought, with its length along the east and west line through the centre of the suspended magnet, near enough to produce the requisite deflection, and then withdrawn in the same manner. CHAPTER III. ABSOLUTE UNITS OF MAGNETIC POLE, MAGNETIC FIELD, AND ELECTRIC CURRENT, N the preceding investigation nothing has been said as to the units in which the quantities M and H are measured. It will be convenient, before proceeding further, to state how the absolute unit of current is defined in the absolute electromagnetic system now generally adopted for most electrical measurements. This definition involves those of the absolute units of magnetic pole and magnetic field, which therefore must be considered first. In the electromagnetic syste/ſt of measurement, all magnetic and electrical quantities are expressed in units which are derived from a magnetic pole chosen as the pole of unit strength. This unit pole might be defined in many ways: but in order to avoid the fluctuations to which most arbitrary standards would be subject, and to give a convenient system in which work done in the dis- placements of magnets or conductors, relatively to mag- nets or to conductors carrying currents, may be estimated without the introduction of arbitrary and inconvenient numerical factors, it is connected by definition with the absolute unit of force. It is defined as a ſole which, if UNIT MAGNETIC POLE. 4. I Ż/aced at unit distance /rom an equal and similar Žole would be refelled with unit force.* The poles referred to in this definition are purely ideal, for we cannot separate one pole of a magnet from the opposite pole of the same magnet : but we can by proper arrangements obtain an approximate realization of the definition. Suppose we have two long, very thin, straight,steel bars, which are uniformly and longitudinally magnetized ; their poles may be taken as at their extremities; in fact, the distribution of magnetism in them is such that the magnetic effect of either bar, at all points external to its own substance, would be perfectly represented by a certain quantity of one kind of imagin- ary magnetic matter placed at one extremity of the bar, and an equal quantity of the opposite kind of matter placed at the other extremity. We may imagine, then, these two bars placed with their lengths in one line, and their blue poles turned towards one another, and at unit distance apart. If their lengths be very great compared with this unit distance, say IOO or IOOO times as great, their red poles will have no effect on the blue poles comparable with the repulsive action of these on one another. But there will be an inductive action between the two magnets which will tend to diminish the mutual repulsive force of the two blue poles, and this we cannot in practice get rid of. The magnitude of this inductive effect is, however, less for hard steel than for soft steel, and we may therefore imagine the steel of the magnets so hard that the action of one on the other does not appreciably affect the distribution of magnetism in either. If, then, two equal blue poles repel one another with unit force, each according to the definition has unit strength. * The medium between the poles is supposed to be air. 42 UNIT MAGNETIC FIELD. The magnitude of unit pole is by the above definition made to depend on unit force. Now unit force is defined, according to the system of measurement of forces founded on Newton's Second Law of Motion (the most convenient system), as that force which, acting for unit of time on unit of mass, will give to that mass unit of velocity. The unit pole is thus based on the three fundamental units of length, mass, and time. According to the recom- mendations of the B.A. Committee, and the resolutions of the Paris Congress, it has been resolved to adopt gener- ally the three units already in very extended use for the expression of dynamical, electrical, and magnetic quanti- ties, namely, the centimetre as unit of length, the gramme as unit of mass, and the second as unit of time ; and these units are designated by the letters C.G.S. With these units, therefore, unit force is that force which, acting for one Second on a gramme of matter, generates a velocity of one centimetre per second. This unit of force has been called a dyne. The unit magnetic pole, therefore, in the C.G.S. system of units is that pole which, placed at a distance of one centimetre from an equal and similar pole, is repelled with a force of one dyne. Each of the poles of the long thin magnets of our example above is therefore a pole of strength equal to one C. G.S. unit, if the mutual force between the poles is one dyne. The magnetic moment M of any one of the deflecting magnets is equal to the strength of either pole multiplied into the distance between the poles, which for magnets of such great length in comparison with their thickness is nearly enough the actual length of the magnet. There- fore either pole has a strength of M/2A units. If r and A are measured in centimetres, and Win granmes, the Strengths ABSOLUTE UNIT OF CURRENT, 43 of the magnetic poles deduced from equation (4) or (6) will be in C.G.S. units. A magnetic field is the space surrounding a magnet or a system of magnets, or a system of Conductors carrying currents, at any point of which, if a magnetic pole were placed, it would be acted on by magnetic force. The intensity of the magnetic field due to any system of magnets or of conductors carrying currents, or any Combination of such systems, is measured at every point by the force which a unit magnetic pole would there experience, and the direction, and magnitude of this force Can, theoreti- cally, be calculated if the magnetic distribution is given, Hence from the definition of unit magnetic pole we get at once a definition of magnetic field of unit intensity. A magnetic ſe/d of 7//// ?//ensity, or, as we may call it, unit magneſic /ie/d, is f/a/ /ie/d in which unit magnetic/o/e is acted on by unit ſorce, and in the C.G.S. system, therefore, it is that field in which unit magnetic pole is acted on by a force of one dyne. In the theory of the determination of H, given above, the horizontal force on either pole of the magnetometer needle due to the horizontal component of the earth's field is taken as M' /// 2\", and again the horizontal force on cither pole of the deflecting mag- net as M. H/2 N. H. therefore measures in units of mag- netic field intensity the horizontal component of the earth's field. 13y formula (5) or (7), when r and X are taken in centimetres, and VV in grammes, // is given in dynes ; that is, it is the number of dynes with which a unit red pole would be pulled horizontally towards the north, and a unit blue pole towards the south if acted on only by the earth's magnetic field. In the magnetic field due to a single magnetic pole, the direction of the result- ant magnetic force at any point is in the straight line 44 ABSOLUTE UNIT OF CURRENT. joining the point with the pole producing the field, and the magnitude of this force, which measures the intensity of the field at the point, is the force which a unit pole would there experience, and is measured by the strength of the pole producing the field, divided by the square of its distance from the point in question. According to the theory of electromagnetic action given by Ampère,” a plane closed circuit carrying a cur- rent in a magnetic field, and of very small dimensions in Comparison with its distance from any part of the mag- netic system producing the field, is equivalent, as regards magnetic action, to a small magnet, having a magnetic moment directly proportional to the strength of the current and to the area of the circuit, placed with its length at right angles to the plane of the circuit at some point with- in it, and so turned that to an observer towards whom the red pole is pointing, the current circulates in the direction opposite to the motion of the hands of a watch. If we measure the current in such units that the magnetic moment of the equivalent magnet is equal numerically to the strength of the current multiplied by the area of the circuit, we define unit current (I) as that currezzz zwhich ſlowing 777 a sma// ?/azze circześć, Žs effzzzz/a/ezzi,j Žn mag- 7tetic action to a sma/Z magnet of 7/zoº/ae/af Zetzmerically equa/ ſo the area of the circuzzi. From this result for a small plane closed circuit it follows that the mutual action between a finite closed circuit of any form carrying a current in a magnetic field and the magnetic system producing the field is the same * See for a fuller treatment of this subject the author’s treatise on 7/ke Theory and Practice of Absolute Measurements in Złectricity and /l/ag- zze??s/, vol. ii. # A circuit and magnet equivalent in one medium are not necessarily equivalent in another medium. It is assumed here as elsewhere that the medium is air. ABSOLUTE UNIT OF CURRENT. 45 f as that which would exist if the circuit were replaced by any magnetic shell (that is a thin material Surface magnetized so as to have, on one side, blue magnetism, and on the other side red magnetism), whose bounding edge occupies the position of the circuit, provided the shell does not be- tween any two positions include any part of the magnetic System, and is magnetized in the proper direction, and So that the magnetic moment of every small part is equal to the strength of the current multiplied into the area of that part. To see this it is only necessary to conceive the circuit as the bounding edge of a net formed by two sets of crossing conductors and to suppose round every mesh an indepen- dent current to circulate of the same strength and in the same direction as the current in the circuit. Each mesh may be taken so small that it may be regarded as plane, and it is therefore equivalent in magnetic action to a Small magnet specified as above. By taking the meshes infinitely small we get by the equivalent small magnets the magnetic shell just described. But it is plain that in every conductor which is common to two meshes, there will be two equal and opposite currents, which cancel oue another, and that this will hold for all parts of the net- work except those included in the bounding edge. Hence the shell may have any geometrical disposition provided its bounding edge coincide with the circuit, and it do not include between two possible positions any part of the magnetic system the mutual action between which and the shell is considered. By considering the work (p. 72) done in any small dis- placement of an element of the boundary of an equivalent shell, it can be shown that every element of the circuit is acted on directly by a force, at right angles to the 46 ABSOLUTE UNIT OF CURRENT. plane through the element and the direction of the resultant magnetic force at its centre, the magnitude of which is obtained by multiplying together the length of the projection of the element on a plane at right angles to the direction of the resultant magnetic force, the magni- tude of this force, and the strength of the current. Thus if (Fig. 7) ds denote the length of the element, / the resultant magnetic force, or, which is the same, the intensity of the field at the element, 6 the angle between the length of the element and the direction A B of the magnetic force, C the FIG. 7. strength of the current, and d F the force on the element, we have dR = C / d’s sin 6. If we take 6 = 90°, and I = 1, we get d'A' = Cds, and hence the definition :-(2) Uni'ſ current is ſhe current ſlowing in an element of a conduc/or which placed at 7:ght angles to the direction of the resu//ant force in a ſield of unit intensity at the element, is acted on by an electromagnetic force, the amount of which Žer unit of Zemg//, of the element is equal to unity. If an observer be supposed immersed in the current so that it flows from his feet to his head, and to look along the line of the resultant magnetic force in ABSOLUTIE UNIT OF CURRENT. 47 the direction in which it would move a blue magnetic pole, the element if free to move would do so towards his right hand. If we suppose the magnetic system to consist of a single magnetic pole, and put 7 for the distance of the pole from the element, and u for the strength of the pole, I becomes pſ”, and we have d F = Cu. d.s. sin 6/7”. Considering now the opposite aspect of the stress in this case, the action of each element of the circuit on the pole is, by Newton's law, that action and reaction are equal and opposite, a force equal to d F reversed, and therefore acting in the same line. But this is equivalent to an equal and similar parallel force acting in a line through the pole ; together with a couple, the moment of which is r dF, acting in the plane of r and d F. For a closed circuit the resultant moment of the couples for all the elements is zero, and the total action is the same as if the forces alone acted on the pole. From these considerations we may define unit current as (3) that current which, ſlowing in a thin wire forming a circle of unit radius, acás on a zºnić wagnetic fo/e/laced at the centre with unit force per unit of length of the circum- Jerezice, that is, with a total force of 2 T units; or which produces at the centre of the circle a magnetic field of 2 tr units of intensity. Thus in the C. G.S. system unit current is that current which flowing in the circle of unit radius acts on a unit magnetic pole at the centre with a force of 2 T dynes. This force acts along the axis of symmetry towards one side or the other of the plane of the circle, according to the nature of the pole and the direction of the current, and its direction may be found for any case by remember- ing that the earth may be imagined to be a magnet turned 48 MEASUREMENT OF A CURRENT. into position by the action of a current flowing round the magnetic equator in the direction of the sun's apparent motion. These three definitions of unit current are equivalent, and in the applications which follow we shall use that which is most convenient in any particular case. CHAPTER IV. MEASUREMENT OF A CURRENT IN ABSOLUTE UNITS AND PRACTICAL CONSTRUCTION OF A STANDARD GALVANOMETER. IF we take next the simple case of a single wire bent round into a circle and fixed in the magnetic meridian, with a magnet, whose dimensions are very small in comparison with the radius of the circle, hung by a torsionless fibre so as to rest horizontally with its centre at the centre of the circle, we may suppose that each pole of the magnet is at the same distance from all the elements of the wire. A current flowing in the wire acts, by Ampère's theory, with a force on one pole of the needle towards one side of the plane of the circle, and on the other pole with an equal force towards the other side of that plane. The needle is thus acted on by a couple tending to turn it round, and it is deflected from its position of equilibrium until this couple is balanced by the return couple due to H. Let us suppose the strength of each pole of the needle to be m units, r the radius of the circle, and C the strength of the current in it. Then by Ampère's law we have for the whole force without regard to sign, exerted on either pole of the needle by the current, the value 27tr. Cm/?” or 2TmC/r. If Z be the distance between the poles the couple is 2TCm//r before any E 5O MEASUREMENT OF A CURRENT. deflection has taken place. After the needle has been deflected through the angle 6 the arm / of the couple has become / cos 6, and therefore the couple 2 T C m / cos 6/r: and the return couple due to H is me H / sin 6. Hence we have equilibrium when C 772 ar / COS 6 = 7/7 H / sin 6 and therefore , H 7- C = ** – tan 6 . . . . . (I) 2 TT if 6 be the observed angle at which the needle rests in equilibrium when deflected as described from the mag- netic meridian. If instead of a single circular turn of wire we had AV turns occupying an annular space of mean radius r, and of dimensions of cross-section small com- pared with r, we should have C = ** tan a . . . . . (2) 2 TT AW In practice the turns of wire of the tangent galvano- meter may not be all contained within such an annular space. It is necessary then to allow for the dimen- sions of the space occupied by the wire. For a coil made of wire of Small section we may suppose that the actual current flowing across a unit of area is every- where the same. Hence if C be the current strength in each turn, and 77 the number of turns in unit area, we have for the current crossing the area A of an element E the value 77 CA. Taking a section of the coil through its centre, at right angles to its plane, and Supposing the annular space of rectangular section, let B C (Fig. 8) be a radius drawn from the centre C in the plane cutting the coil MEASUREMENT OF A CURRENT. 5 I into two equal and similar coils, and taking C P (-3) and D E ( = 4) at right angles to one another, we have A = d ardy and CAE * = a + y”. Hence the force exerted on a unit magnetic pole at the centre C by an element Of area A, and of length d s, supposed at right angles to - - ** ~ *... * * - * * - - **~- * * ** *- - * * * the plane of the paper, is n C da dy/(a” + y”) in the direction at right angles to C E and in the plane of the paper. The component of this force at right angles to B C is n Cydar dyſ(* + y”)}. Hence if we call d F the component in this direction of the whole ring at right lº, 2 52 MEASUREMENT OF A CURRENT. angles to the plane of the paper of which this element is the Cross-section we have 2 T 7: Cy” da dy (** + y”); Hence for the whole force at right angles to B C we have aſ A" = + c y”dardy & 2. F-2 ºncſ | #. 22 2); –& , - . (* + y”); where r is the mean radius of the coil, 2 b its breadth, and 2 c its depth in the plane of the circle. Integrating, and putting AW for the whole number of turns 4 7, & c, we get Jº’ = a wellogº tº: Mr-Fºº-Fº . (3) If 6 be the angle at which the deflecting couple is equili- brated by the return couple due to H, we have as before the equation Jº = H tan 6. Hence, substituting the above value for F and solving for C, we have finally Aſ tan 6 r —- c + V(r)+2)2+ 2 r – c + x/(r. 2)2 + 32 C = . (4) a Włlog When the value of r is great in comparison with 0 and c this reduces to the equation A/7" tan 6 C = −. 2TAV (5) which we found before by assuming all the turns to be contained in a small annular space of radius 7". In MEASUREMENT OF A CURRENT. 53 practice, in galvanometers used as standards for absolute measurements, generally neither & nor c is so great as 'a of r, and the needle cannot be made infinitely short ; hence in these cases the difference between the values given by equations (4) and (5) is well within the limits of experi- mental error, and the correction need not be made. The value of C given by (5) is then to be used. When the dimensions of the coil are measured in centi- metres, and H is taken in C.G.S units the value of C is given by (4) or (5) in C.G.S units of current strength. In this investigation the suspension fibre has been sup- posed torsionless. If a single fibre of unspun silk is used as described below for this purpose, its torsion may for most practical purposes be safely neglected. The error produced by it may however be easily determined and allowed for by turning the needle, supposed initially in the magnetic meridian, once or more times completely round, and noting its deviation from the magnetic meri- dian in its new position of equilibrium. The amount of this deviation, if any, may be easily observed by means of the attached index and divided circle, or scale and reflected beam of light used as described below, to measure the deflections of the needle. From the result of this experiment the effect of torsion for any deflection may be calculated in the following manner. Let a be the angular deflection, in radian * measure, of the magnet from the magnetic meridian produced by turning the magnet once round, then the angle through which the thread has been twisted is 2 T – a. The couple produced by this torsion has for moment H / ?/z sin a. * A 7adian is the angle subtended at the centre of a circle by an arc equal in length to the radius. It has generally been called in books on trigonometry hitherto by the ambiguous name unit angle in circular measure, 54 MEASUREMENT OF A CURRENT. Hence, by Coulomb's law of the proportionality of the force of torsion to the twist given, we have for the couple corresponding to a deflection 6 the value 6 º — Aſ 772 / Sin a. 27T —- Ct If then under the action of a current in the coil the deflec- tion of the needle is 6, the equation of equilibrium is Cm # / cos 0 = m /// (in 6 — 6 sin a) 7- 27ſ — Cl and therefore instead of (5) we have C = ( + 6 sin ..) A/7. tan 6 . . (6) 27ſ — a sin 6 / 27t/V If a be an angle of say I*, and 6 be 45°, 6/(2T – a) is very nearly 1/8 and sin aſsin 6 is 1/(57'3 × 707) or 1/40'5. Hence ::) Hr 324/27/V The error therefore is somewhat less than ; per cent. The determination of H and the measurement of a Current in absolute units, can be effected simul- taneously by the method devised by Kohlrausch, and described in the Philosophical Magazine, vol. xxxix. 1870. This method consists essentially in sending the current to be measured through two coils, of which all the con- stants are accurately known. One of these is the coil of a standard galvanometer, the other is a coil hung by a bifilar suspension, the wires of which convey the current into the coil. The latter coil rests in equilibrium when no current is passing through it, with its plane in the magnetic meridian. When a current is sent through it, c= (1+ MEASUREMENT OF A CURRENT. 55 it is acted on by a couple due to electromagnetic action between the current and the horizontal component of the earth's force, which tends to set it with its plane at right angles to the magnetic meridian ; and this couple is resisted by the action of the bifilar. The coil comes to rest, making a certain angle with the magnetic meridian, and, as the couple exerted by the bifilar suspension for any angle is supposed to have been determined by experi- ment, a relation between the value of H and the value of the current is obtained. But, as the same current is sent through the coil of the standard galvanometer, the ob- served deflection of the needle of that instrument gives another relation between H and C. From the two equa- tions expressing these relations the values of Aſ and C can be found. Full details of the construction of Kohl- rausch's apparatus and of the calculation of its constants will be found in the paper above referred to. In this method it is assumed that the value of H is the Same at both instruments, an assumption which for rooms not specially constructed for magnetic experiments cannot safely be made. An instrument which is not liable to this objection has been suggested by Sir William Thomson. A short account of this instrument and its theory will be found in Maxwell's Electricity and Magnetism, vol. ii. p. 328, and in the author’s Treatise on the Theory and Practice of Absolute Measurements in Electricity and Magnetism, vol. ii. The galvanometer may with advantage in some par- ticulars be composed of two equal coils of which the radii are great in Comparison with the transverse dimensions, placed parallel to one another so that their line of centres is normal to their mean planes. A short needle is sus- pended symmetrically with respect to the coils on the 56 CONSTRUCTION OF A line of centres, and the current is sent through the coils arranged in series so that the magnetic action of each tends to turn the needle round in the same direction. The wire passing from one coil to the other is placed close to that passing back to the electrodes of the instrument, which are preferably two well-insulated wires twisted to- gether, and so long that no loop of the circuit, caused by binding Screws or any other contact-making arrangement, is near the needle. Thus no magnetic action on the needle is produced except that due to the current in the coils. Let AV be the number of turns of wire in each coil, r the mean radius of each, 2a their mean distance apart, and C the current producing an angular deflection 6, then C = (*-ī-a”)*H tan 6 . . . . . (7) 47%.N. Helmholtz has shown that for an instrument of this construction the mathematical theory º (for a needle not infinitely short) is greatly simplified if r = 2a, and he has accordingly advised that the instrument should be so constructed. In ordinary practice however a single coil (or, as described below, p. 61, a long coil having only one layer) with needle at the centre gives results which are sufficiently accurate, and it is much more easy to con- Struct. It is evident from the equation that the greater a is made the smaller is the sensibility. By making the coils movable therefore along a graduated bar to different dis- tances apart, an instrument (though not conveniently a * See Z'heory and Practice of Absolute Measurements in AElectricity and Magnetism, vol. ii. STANDARD GALVANOMETER. 57 Standard instrument) having a Sensibility variable at will through a considerable range can be constructed. The accuracy of the measurements of currents, made according to the method of which the elementary theory is given above, of course altogether depends on the careful adjustment of the standard galvanometer, and the care and skill of the observer. The standard galvanometer should be of such a form that the values of its indications can be easily calculated from the dimensions and number of turns of wire in the coil. Such a galvanometer can be made by any one who can turn, or can get turned, a wooden or brass (see p. 62) ring with a rectangular groove round its outer edge to re- ceive the wire. It is indeed to be preferred that the experi- menter should at least perform the winding of the coil and the adjustments of the needle, &c., himself, to make sure that errors in counting the number of turns or in determin- ing the length of the wire, or in placing the needle at the centre of the coil, are not made. If there are to be several layers of wire, the breadth and depth of this groove ought to be small in comparison with its radius, and each should not be greater than 15 of the mean radius of the coil, which should be at least I 5 cms. The gauge of the wire with which the coil is to be wound must depend of course on the purposes to which the in- strument is to be applied, but it should be good well insulated copper wire of high conductivity, and not so thin as to run any risk of being injured by the strongest currents likely to be sent through the instrument. For the exact graduation of current as well as potential galvanometers directly by means of the standard instru- ment, it is convenient to make it have two coils—one of comparatively high, the other of low resistance. The 58 CONSTRUCTION OF A latter may conveniently in Some cases in which great accuracy is not required, be a simple hoop of Say I 5 cms. radius, made of copper strip I cm. broad and I mm. thick. As however the distribution of the current in a massive Conductor is uncertain in consequence of want of homo- geneity in the material, and it is besides difficult to allow exactly for any irregularity that may exist where the ends are led out, so that it is impossible to determine by calculation the exact constant of such a coil, it is better to use instead several turns of thick wire. Each spire of the coil may then be regarded as explained above, as a circular conductor coinciding with its circular axis. To form electrodes to which wires can be attached the ends of the strip are brought out side by side in the plane of the ring with a piece of thin vulcanite or paper between for insulator. Insulated wires are soldered to the ends of the circle thus arranged, and are twisted together for a sufficient distance to prevent any direct effect on the needle from being produced by a current flowing in them. In constructing the fine wire coil the operator should first subject the wire to a moderate stretching force, and then carefully measure its electrical resistance and its length. He should then wind it on a moderately large bobbin, and again measure its resistance. If the second measurement differs materially from the first, the wire is faulty and should be carefully examined. If no evident fault can be found, on the removal of which the discrep- ance disappears, the wire must be laid aside and another substituted. When the two measurements are found to agree the wire may then be wound on the coil. For this purpose the ring may either be turned slowly round in a lathe or on a spindle, so as to draw off the wire from the STANDARD GALVANOMETER. 59 bobbin, also mounted So as to be free to turn round. The wire must be laid on evenly in layers in the groove (which may be done with the greatest uniformity by using a self-feeding lathe), and the winding ended with the completion of a layer. Great care must be taken to count accurately the number of turns laid on. The resistance should now be again tested, and if it agrees nearly with the former measurements the coil may be relied on. The ring carrying the coil thus made should then be fixed to a convenient stand in such a manner that if necessary it can be easily removed. The stand ought to be fitted with levelling screws, so that the plane of the coil may be made accurately vertical. A shallow hori- zontal box with a glass cover and a mirror bottom should be carried by the stand at the level of its centre. Within this the needle and attached mirror or index are to be suspended. Or, what is more convenient in many cases, a platform should be arranged below the level of the centre a sufficient distance to allow the magnetometer described in Chap. II. above, to be placed with the centre of its needle at the level of the centre of the coil. The needle should be a single small magnet about a centimetre long, hung by a single fibre of unspun silk about ten cms. long from the top of a tube fixed to the cover of the shallow box, so that the centre of the needle when the coil is vertical is exactly the centre of the coil. To allow of the exact adjustment of the height of the needle, the fibre should be attached to the lower end of a Small Screw Spindle, made so as to be raised or lowered, without being turned round, by a nut working round it above the cap of the tube. If the instrument is to be used with scale and pointer 60 CONSTRUCTION OF A (or, as is desirable in some cases, is to be furnished with Scale and pointer as well as mirror) the pointer may be made by drawing out a bit of thin glass tube at the blow- pipe into a thread, so thick as to remain nearly straight under its own weight when suspended by its centre. In order that the zero position of the pointer may not be under the coil, the pointer ought to be fixed horizontally with its length at right angles to the needle, so as to project to an equal distance on both sides of it. To test that this adjustment is accurately made, draw a Couple of lines accurately at right angles to one another on a sheet of paper. Then suspend a long thin straight magnet Over the paper, and bring one of the lines into accurate parallelism with it. Remove then the magnet and put in its place the little needle and attached index. If the index is parallel to the other line the adjustment has been carefully made. The needle may then be suspended in position and the box within which it hangs closed to prevent disturbance from currents of air. A circular scale graduated to degrees, with its centre just below the centre of the coil and its plane horizontal, is placed with its zero point on a line drawn on the mirror bottom of the box at right angles to the plane of the coil, so that when the needle and coil are in the magnetic meridian the index may point to zero. The accuracy of the adjustment of the zero point is to be tested by finding whether the same current reversed produces equal de- flections on the two sides of zero. To test whether the centre of this divided circle is accur- ately under the centre of the needle supposed at the centre of the coil, draw from the point immediately under the centre of the needle two radial lines on the mirror bottom, one on each side of the zero point and 45° from STANDARD GALVANOMETER. 6 I it, and turn the needle round without giving it any motion of translation. If the index lies along these two radial lines when its point is at the corresponding division on the circle the adjustment is correct. When taking readings the observer places his eye so as to see the index just cover its image in the mirror bottom of the box, and reads off the number of divisions and frac- tion of a division, indicated on the scale by the position of the index. Error from parallax is thus avoided. A mirror with attached magnets may be used, as in the magnetometer, instead of the needle and index. Very conveniently a magnetometer with long fibre may be used placed on a platform fixed within the bobbin. A hole slot and plane arrangement cut in the platform for the adjusted position will enable the magnetometer to be taken away, and replaced at pleasure. The adjustment of scale, &c., is the same as that described in Chap. II. above. When a mirror is employed the coil is in the magnetic meridian when equal deflections of the spot of light on the scale on the two sides of zero are produced by re- versing any current. The scales used, as has been already remarked, should, if of paper, always be carefully glued to a wooden piece, instead of being, as they frequently are, fixed with drawing pins. Preferably however they should be ruled on glass by any one of the simple methods now available for copying an accurately engraved standard. The author has had constructed lately a standard gal- vanometer which seems to have Several advantages. It consists of a cylindrical bobbin of about 50 centimetres in diameter and 25 centimetres in length, wound with a single layer of fine wire. The needle (1 cm. long) is suspended at the centre of the bobbin, and the magnetic field produced by a current flowing in the wire is in this 62 CONSTRUCTION OF A arrangement practically invariable over a distance in any direction at the centre considerably exceeding the length of the needle. Very accurate placing of the needle is thus not necessary, as a displacement of so much as half its length from the central position (an error of adjust- ment which is practically impossible with the slightest care) produces a quite imperceptible effect in the deflec- tion with any given current. The distribution of the wire, since there is only one layer, is known with perfect certainty, and hence the constant of the instrument can be calculated with great exactness. At each end of the bobbin is wound one of two equal coils of Small transverse dimensions in com- parison with their radii. These are of thick copper wire arranged so as to form a Helmholtz double-coil galvano- meter of the kind described above (p. 56), available for strong currents. When the instrument was being designed it was thought desirable to have the bobbin made of some material which could not contain magnetic substances in sufficient quantity to affect the accuracy of measurements of currents flowing in the wire. The fear then felt by the author that the bobbins of brass ordinarily employed for standard galvanometers might very probably contain iron in sufficient quantity to cause disturbance through its inductive magnetization, has since been found by Prof. Thomas Gray to be in part at least justified. The measure- ments of currents made by a new standard galvano- meter were found by him to be so much disturbed by the effect of magnetic substances contained in a brass box surrounding the needle as to be practically useless. It was resolved therefore to construct a bobbin of wood in such a manner as to avoid all risk of alteration of STANDARD GALVANOMETER. 63 figure by warping, or of dimensions through variation in the amount of moisture contained in the wood. Accord- ingly a sufficient number of pieces of mahogany were cut from a dry well-seasoned board about $ inch thick. Each piece was about 4 cms, broad, 2O Cms, in length, and was cut so as to form a segment of a ring the outside diameter of which was about 50 cms. and the inner dia- meter about 8 cms. less. Four of these cut so that the grain of the wood ran in different directions in adjoining pieces and placed end to end, gave a complete circular ring, or rather cylinder, $ inch in length. Above that was placed a similar ring with the grain of the wood in the pieces crossing that in the pieces below, and the pieces themselves overlapping the end joints in the preceding ring. Above that was placed another ring, and so on until the whole bobbin, rather more than 25 cms, in length, had been built up. The whole was then put together under considerable pressure, with glue between the joints. The cylinder thus roughly formed was then turned carefully down to a true cylindrical figure of the size desired, and the pores all over the surface, inside and outside, filled with spirit varnish to prevent the absorption of moisture. Two edges of wood, projecting slightly beyond the out- side cylindrical surface, were fixed on the ends to keep the wire in its place. The coil was then carefully wound, the turns counted, and the wire covered with American cloth to preserve it from injury. The two ends of the thin wire coil were brought out together at one end of the coil to be connected to two electrodes closely twisted together, and several yards in length, by which the in- strument could be joined to any circuit in which it might be required. The end of the wire which had to be 64 CONSTRUCTION OF A STANDARD GALVANOMETER. carried from the further extremity of the coil was (sup- posing the coil set up in position) brought along horizont- ally in a vertical plane through the axis of the coil until it met the other extremity at the termination of the last spire of the coil. The current in this part of the wire of course just compensates by its effect on the needle the effect of the component of current in each element of the spires in the direction of the axis. The constant of the instrument was calculated by putting in equation (3) or (4) above c = 0. Thus if M be the total number of turns in the single layer of the coil, we have , (r” + /2)}// tan 6 C = \' ' ' '... “ . . . . . (8) 2 TT AV For the thick wire coils the equation connecting the current with the deflection is given above (7), p. 56. We have here described the tangent galvanometer as it is the instrument most easily constructed ; but what are called sºme ga/vanometers are sometimes employed. In these the coil can be turned round a vertical axis to allow its medial plane to be placed parallel to the needle in the deflected position of the latter. Then if 0 be the angle of deflection of the needle (which can be measured by observing the angle through which the coil has been turned from the meridian) we have for the equations of equilibrium (I), (2), &c. above, with faſt 6 replaced by sºn 6. Particulars of the construction of such galvanometers will be found in the author's Theory and Practice of Absolute Measure/tents in Z/ectricity and Magnetism, vol. ii. CHAPTER V. DEFINITION OF ABSOLUTE UNITS OF DIFFERENCE OF POTENTIAL AND RESISTANCE, AND DERIVATION OF PRACTICAL UNITS--VOLT, OHM, AMPERE, COULOMB. RATE OF WORKING IN AN ELECTRIC CIRCUIT. WO conductors of the same material, are at different potentials if on their being put in conducting con- tact electricity tends to pass from one to the other. A difference of potential may also exist between two con- ductors of different materials in contact, or may be maintained between two parts of the same conductor by electrical forces. In all cases a difference of potential is measured by the work (p. 73) which would be done by electric forces on unit quantity of positive electricity if, while the potentials remained the same, it were carried from the place of higher to the place of lower potential. Two bodies at different potentials attract one another ; and therefore, if one be connected with an electrically insulated plate carried at one end of a delicate balance, and the other with a second insulated plate fixed facing the first at a very short distance from it, these two plates will, if they have been previously put for an instant in conducting contact and balance obtained, attract one another, and the force of attraction may be weighed by F 66 ABSOLUTE UNIT OF DIFFERENCE OF POTENTIAL. restoring balance. With certain arrangements necessary to ensure accuracy, a balance may be constructed by means of which the difference of potential between two conductors can be measured. Such an instrument has been made by Sir William Thomson, and called by him an Absolute Electrometer. It is found experimentally by measuring with a delicate electrometer that, between any two cross-sections A and P of a homogeneous wire, in which a uniform current of electricity is kept flowing by any means, there exists a difference of potential, and that if the wire be of uniform section throughout, the difference of potential is in direct proportion to the length of wire between the cross-sections. It is found further, that if the difference of potential between A and B is kept constant, and the length of wire between them is altered, the strength of the current varies inversely as the length of the wire. The strength of the current is thus diminished when the length of the wire is increased, and hence the wire is said to oppose resistance to the current ; and the resistance between any two cross- sections is proportional to the length of wire connecting them. If the length of wire and the difference of poten- tial between A and B be kept the same, while the cross- sectional area of the wire is increased or diminished, the current is increased or diminished in the same ratio ; and therefore the resistance of a wire is said to be inversely as its cross-sectional area. Again, if for any particular wire, measurements of the current strength in it be made for various measured differences of potential between its two ends, the current strengths are found to depend only on, and to be in simple proportion to, the differences of potential so long as there is no sensible heating of the wire. Hence we have Ohm’s law, by which to find the ABSOLUTE UNIT OF DIFFERENCE OF POTENTIAL. 67 strength C of the current flowing in a wire of resistance R, between the two ends of which a difference of potential W is maintained, namely I/ Y C = . . . . . . . . . () In this equation the units in which any one of the three quantities is expressed depend on those chosen for the other two. We have defined unit current, and have seen how to measure currents in absolute units ; and we have now to show how the absolute units of V and ſº are to be defined, and from them and the absolute unit of current to derive the practical units—volt, ampere, coulomb, and ohm. We shall define the absolute units of potential and resistance by a reference to the action of a very simple but ideal magneto-electric machine, of which, however, the modern dynamo is merely a practical realization. First of all let us imagine a uniform magnetic field of unit intensity. The lines of force in that field are every- where parallel to one another: to fix the ideas let them be vertical. Now imagine two straight horizontal metallic rails running parallel to one another, and connected to- gether by a sliding bar, which can be carried along with its two ends in contact with them. Also let the rails be connected by means of a wire so that a complete con- ducting circuit is formed. Suppose the rails, slider, and wire to be all made of the same material, and the length and cross-sectional area of the wire to be such that its resistance is very great in comparison with that of the rest of the circuit, so that, when the slider is moved with any given velocity, the resistance in the circuit remains practically constant. When the slider is moved along the rails it cuts across the lines of force, and so long as it moves F 2 68 DERIVATION OF VOLT AND OHM. with uniform velocity a constant difference of potential will be maintained between its two ends by induction, and a uniform current will flow in the wire from the rail which is at the higher potential to that which is at the lower. If the direction of the lines of force be the same as the direction of the vertical component of the earth's magnetic force in the northern hemisphere, so that a red or north- tending pole placed in the field would be moved down- wards, and if the rails run South and north, the current when the slider is moved northwards will flow from the east rail to the west through the slider, and from the west rail to the east through the wire. If the velocity of the slider be increased the difference of potential between the rails, or, as it is otherwise called, the electro- motive force producing the current, will be increased in the same ratio ; and therefore by Ohm’s law so also will the current. Generally for a slider arranged as we have imagined, and made to move across the lines of force of a magnetic field, the difference of potential produced would be directly as the field intensity, as the length of the slider, and as the velocity with which the slider cuts across the lines of force. The difference of potential produced therefore varies as the product of these three quanti- ties; and when each of these is unity, the difference of potential is taken as unity also. We may write therefore V= / Z v, where Z is the field intensity, Ł the length of the slider, and v its velocity. Hence if the intensity of the ficlal we have imagined be I C.G.S. unit, the distance between the rails I cm., and the velocity of the slider I cm. per second, the difference of potential produced will be I C.G.S. unit. This difference of potential is so Small as to be incon- THEORY OF DIMENSIONS. 69 venient for use as a practical unit, and instead of it the difference of potential which would be produced if, every- thing else remaining the same, the slider had a velocity of IOO,OOO,OOO cms. per second, is taken as the practical unit of electromotive force, and is called one volt. It is a little less than the difference of potential which exists between the two insulated poles of a Daniell's cell. We have imagined the rails to be connected by a wire of very great resistance in comparison with that of the rest of the circuit, and have supposed the length of this wire to have remained constant. But from what we have seen above, the effect of increasing the length of the wire, the speed of the slider remaining the same, would be to diminish the current in the ratio in which the resistance is increased, and a correspondingly greater Speed of the slider would be necessary to maintain the current at the same strength. We may therefore take the speed of the slider as measuring the resistance of the wire. Now Suppose that when the slider I cm. long was moving at the rate of I cm. per second, the current in the wire was I C.G.S. unit ; the resistance of the wire was then I C.G.S. unit of resistance. Unit resistance therefore corresponds to a velocity of I cm. per second. This resistance, how- ever, is too small to be practically useful, and a resistance I,OOO,OOO,OOO times as great, that is the resistance of a wire, to maintain I C.G.S. unit of current in which it would be necessary that the slider should move with a velocity of 1,000,000,000 cms. (approximately the length of a quadrant of the earth from the equator to either pole) per Second, is taken as the practical unit of resistance, and called one oſt/z. In reducing the numerical expressions of physical quan- tities from a system involving one set of fundamental units 7o DIMENSIONS OF A RESISTANCE. to a system involving another set, as for instance from the British foot-grain-second system, formerly in use for the expression of magnetic quantities, to the C.G.S. system, it is necessary to determine, according to the theory of “dimensions” (first given by Fourier, and extended to electrical and magnetic quantities by Maxwell), for each a Certain reducing factor, by Substituting in the dimensional formula which States the relation of the fundamental units to one another in the expression of the quantity, the value of the units we are reducing from in terms of those we are reducing to. For example, in reducing a velocity, say from miles per hour to centimetres per second, we have to multiply the number expressing the velocity in the former units by the number of centimetres in a mile, and divide the product by the number of seconds in an hour; that is, we have to multiply by the ratio of the number of centimetres in a mile to the number of seconds in an hour. The multiplier therefore, or change-radio as it has been called by Professor James Thomson, is for velocity simply the number of the new units of velocity equivalent to one of the old units, and may be expressed by the formula Z/7, where Z is the numerical expression of the old unit of length in terms of the new contained in one of the old, and T is the corresponding expression for the old unit of time. In the same way the change ratio for rate of change of velocity or acceleration is L/7”; and the change-ratio of any other physical quantity may be found by determining from its definition the manner in which its unit involves the fundamental units of mass, length, and time. Now the theory of the dimensions of electrical and magnetic quantities, in the electromagnetic system of units, shows (Ch. XI.) that the dimensional formula DETERMINATIONS OF THE OHM. 7 I for resistance is the same as that for velocity ; that in fact a resistance in electromagnetic measure is ex- pressible as a velocity; and hence we may with propriety speak of a resistance of one ohm as a velocity of IO" centimetres per Second. The first experiments for the realization of the ohm were made by the British Association Committee, and later determinations have been made with varying results by experimenters in different parts of the world. The method used by the committee was one suggested by Sir W. Thomson, in which the current induced in a coil of wire revolving round a vertical diameter in a magnetic field was measured by the deflection of a magnetic needle hung at the Centre of the coil. The principle of the method is exactly the same as that of the ideal method, with slider and bars, sketched above. According to the results of the committee the ohm is represented approximately by the resistance of a column of pure mercury IO4'8 centimetres long, one square millimetre in section, at the temperature O’ C. Coils of an alloy of two parts of silver to one of platinum, which had a resistance of one ohm at a certain temperature, were issued by the committee as standards to experi- menterS. Concordant experiments by Prof. Rowland in America, by Lord Rayleigh and by Mr. Glazebrook at Cambridge, and by Messrs. Mascart, de Nerville, and Benoit in Paris, have shown that the B.A. unit is probably about 1.3 per cent. too small. A very careful determination by Lord Rayleigh and Mrs. H. Sidgwick gives '98677 earth-quad- rant per second as its value.* Lord Rayleigh and Mrs. Sidgwick also measured the specific resistance of mercury * Phil. Traws. A. S. 1883. 72 DERIVATION OF THE AMPERE AND COULOMB. in terms of the B.A. unit.* Their result, together with their value of the B.A. unit gave IO6’21 cms. as the length of a column of pure mercury of Section I sq. mm. and temp. O’C. which has a resistance of I ohm. A determination of the ohm in terms of the B.A. unit has recently been made by Messrs. L. Duncan, G. Wilkes, and C. T. Hutchinson at the Johns Hopkins University, Baltimore, with apparatus designed by Prof. H. A. Row- land. Their result is ‘9863 earth-quadrant per second = I B.A. unit, and this taken with a redetermination of the specific resistance of mercury made by the last-named two gentlemen, gives IO6'34 cms, as the length of the mercury column specified as above. It has been agreed by the Congress of Electricians held at Paris in 1884, to adopt temporarily as the Legal Ohm, a resistance equal to that of a Column of pure mercury IO6 cms. in length and one Square millimetre in section, at the temperature of O’C. When a sufficient number of exact determinations of the ohm have been obtained the legal value will no doubt be brought into as close as possible accord with the true value. It is obvious from equation (I) that if V and A', each initially one unit, be increased in the same ratio, C will remain one unit of current ; but that if V be, for ex- ample, Io" C.G.S. units of potential, or one volt, and A’ be a resistance of IO" cms. per Second, or one ohm, C will be one-tenth of one C.G.S. unit of current. A current of this strength—that is, the current flowing in a wire of resistance one ohm, between the two ends of which a diſſerence of potential of one volt is maintained,—has been adopted as the practical unit of current, and called * Phil. Trans. A', S', 1883. See also below, p. 242. f /’/ºil. Mag. Aug. 1889. See also below, p. 242. | Zöld. July, 1889. WORK AND ACTIVITY. 73 one ampere. Hence it is to be remembered one ampere is one-tenth of one C.G.S. unit of Current. The amount of electricity conveyed in One Second by a current of one ampere is called one cou/omb. This unit, although not quite so frequently required as the others, is very useful, as, for instance, for expressing the quantities of electricity which a secondary cell is capable of yielding in various circumstances. For example, in comparing different cells with one another their capacities, or the total quantities of electricity they are capable of yielding when fully charged, are very conveniently reckoned in coulombs per square centimetre of the area across which the electrolytic action in each takes place. The magneto-electric machine we have imagined gives us a very simple proof of the relation between the work done in maintaining a current, the strength of the current, and the electromotive force producing it. In dynamics work is said to be done by a force when its place of appli- cation has a component motion in Zhe direction 27, which the ſorce acts, and the work done by it is equal to the product of the force and the distance through which the place of application of the force has moved in that direction. The rate at which work is done by a force at any instant is therefore equal to the product of the force and the component of velocity in the direction of the force at that instant. The work done in overcoming a resistance through a certain distance is equal by this definition to the product of the resistance and the distance through which it is overcome. Among engineers in this country the unit of work generally used, is one ſoof-found, that is, the work done in lifting a pound vertically against gravity through a distance of one foot, and the unit rate of working is one /orse-power, that is 33,000 foot-pounds per minute. The 74 ACTIVITY IN A VOLTAIC CIRCUIT. weight of a pound of matter being generally different at different places, this unit of work is a variable one, and is not used in theoretical dynamics. In the absolute C.G.S. system of units, the unit of work is the work done in over- coming a force of one dyne through a distance of one centimetre, and is called one centimetre-dyne or one erg. The single word Activity has been used by Sir Wm. Thom- Son as equivalent in meaning to “rate of doing work,” or the rate per unit of time at which energy is given out by a working system ; and to avoid circumlocutions in what follows we shall frequently use the term in that sense. We have seen above (p. 46) that every element of a conductor, carrying a current in a magnetic field, is acted on by a force tending to move it in a direction at right angles to the plane through the element, and the direction of the resultant magnetic force at the element, and have derived from the expression for the magnitude of the force a definition ( (2) p. 46) of unit current in the electro- magnetic system. From these considerations it follows that a conductor in a uniform magnetic field, and Carrying a unit current which flows at right angles to the lines of force, is acted on at every point by a force tending to move it in a direction at right angles to its length, and the magnitude of this force for unit length of conductor, and unit field, is by the definition of unit current equal to unity. Applying this to our slider in which we may suppose a current of strength C to be kept flowing, Say, from a battery in the circuit, let L be the length of the slider, v its velocity, and I the intensity of the field ; we have for the force on the moving conductor the value / Z. C. Hence the rate at which work is done by the electro- magnetic action between the current and the field is I Z. Cdr/dt or // C v, and this must be equal to the rate at ACTIVITY IN A VOLTAIC CIRCUIT. 75 which work would be done in generating by motion of the slider a current of strength C. But as we have seen above, J L zy is the electromotive force produced by the motion of the slider. Calling this now E, the symbol usually em- ployed to denote electromotive force, we have E C as the electrical activity, that is, the total rate at which electrical energy is given out in all forms in the circuit. By Ohm’s law this value for the electrical activity may, when the work done is wholly spent in producing heat, be put into either of the two other forms, namely, E*//ć, or C*R. In the latter of these forms the law was dis- covered by Joule, who measured the amount of heat generated in wires of different resistances by currents flowing through them. This law holds for every electric circuit whether of dynamo, battery, or thermoelectric arrangement. We have, in what has gone before, supposed the slider to have no resistance comparable with the whole resistance in the circuit. If it have a resistance 7", and ſº be the remainder of the resistance in circuit, the actual difference of potential between its two ends will not be / Z v or Æ, but E. /ē/(R – ?) (p. 83). The rate per unit of time at which work is given out in the circuit is however still E C, of which the part E. C. r.ſ (R-H 7) is given out in the slider, and the remainder, E. C. A/(A + 2), in the remainder of the circuit. In short, if V be the actual difference of potential, as measured by an electrometer, between two points in a wire Connecting the terminals of a battery or dynamo, and C be the current flowing in the wire, the rate at which energy is given out is VC, or if ſº be the resistance of the wire between the two points, C*R. The activity in the part of the circuit considered is 76 ELECTRICAI, UNITS OF ACTIVITY AND WORK. always VC, but this may be greater than C*/'; in which case work is done otherwise than in heating the con- ductor. C*/ú is then the part of the activity employed in generating heat. One of the great advantages of the system of units briefly sketched here, is that it states the value of the rate at which work is given out in the circuit, without its being necessary to introduce any coefficient such as would have been necessary if the units had been arbitrarily chosen. When the quantities are measured in C.G.S. units, the value of /2 C is given in centimetre- dyncs, or in ergs, per Second. Results thus cxpressed may be reduced to horse-ſhower by dividing by the number 7'46X IO"; or if /, is measured in volts, and C in amperes, / C may be reduced to horse-power by dividing by 746. Thus, if 90 volts be maintained betwccn the terminals of a pair of incandescent lamps joined in Series, and a current of 13 ampere ſlows through these lamps, the rate at which energy is given out in the lanps is approximatcly ‘I 57 horse-power. If the rate at which work is done in 1main- taining a current of one annpere through a resistance of one ohm were taken as the practical unit of rate of work- ing, or ac//7///y, and / reckoned in volts and C in amperes, the rate at which electrical energy is given out in the circuit would be measured simply by / C ; and calcula- tions of clectrical work would be much simplified. This was proposed by Sir William Sicnn.cns (Brit. Assoc. Address, 1882), who suggested that the name 7Ua// should be given to this unit rate of working. The rate at which cncrgy is given out in the lamps of the above example is 90 × 1-3 - 1 17 waſ/s. A waſ/is therefore cquivalent to I o' crgs per Scoond. The Elcctrical Congress held at Paris in August 1889 ELECTRICAL UNIT OF WORK. 77 has adopted the watt as the practical electrical unit of work, and the term ///owałł, proposcq by Mr. W. H. Preece to designate an activity of IOOO watts or Io" ergs per second. It is intended that the latter unit should be used instead of the horse-power. An activity given in kilowatts can be reduced to horse-power by dividing by ‘746 or roughly by multiplying by 4 and dividing by 3. Sir William Sicmens also proposed to call the work done in one second, when the rate of working is onc watt, one joule. Thus the work done in one second in main- taining a current of one ampere through one ohm, or the work obtaincol by letting down one coulomb of electricity through a difference of potential of one volt, is one joule. A jouſe is therefore equivalent to IO' ergs, and the work done in one second in the above cxamplc is I 17 fore/es. The Electrical Congress of August 1889 has also adopted the joule as the practical unit of electrical work. CHAPTER VI. POTENTIALS AND CURRENTS IN DERIVED CIRCUITS. N the preceding chapter we have given a short explan- ation of Ohm’s law ; in the present chapter we shall consider the consequences of this law a little more at length, and show how to derive from it rules for the calculation of the equivalent resistance of any arrange- ment of derived circuits, and the strength of the current in any part of that arrangement. Suppose that we have an electric generator E, the two terminals of which are joined by a single copper wire. Then if C be put for the strength of the current flowing in a wire of resistance ſº between two cross-sections at potentials V, V, respectively, the results stated at p. 66 above are all expressed, and unit resistance is defined, by the equation V1 – V. A’ (I) C = Ohm used the expression “Gefälle der Elektricität’ for a quantity which, in the earlier works which appeared after the publication of his essay, was called “Difference of Tensions,” but which is now recognized as proportional to V1 – V., ; and it is still usual to give a special name to difference of potential when considered in connection OHM's LAW IN HETEROGENEOUS CIRCUITs. 79 with the flow of electricity. Thus the name Electromotive Force is frequently given to the difference of potential between two points or two equipotential surfaces in a homogeneous conductor, when thus considered with refer- ence to flow of electricity from one to the other, and in accordance with custom and authority the term may be thus employed. A somewhat more general sense in which the term is used will presently be explained. It is to be carefully remembered, however, that electromotive force is not a force : the two words must be taken together as a single term having the meaning assigned to it in its definition. A constant difference of potential may be maintained between the extremities of a homogeneous conductor, and therefore also a current maintained in the conductor, in several different ways: for example, by a voltaic battery, a thermo-electric pile, or a dynamo-electric or magneto- electric machine. Particulars regarding different forms of voltaic batteries, and the practical construction of other electric generators, are given in various treatises; at present we deal only with principles which are generally applicable, reserving for consideration later their applica- tions in particular cases. Equation (I) is not fulfilled in general by a conductor made up of different homogeneous portions, put end to end, or by a conductor moving across the lines of force of a magnetic field. For such cases V1 – V. y = −. A. + + . . . . . (2 A’ (2) where l’i, V., denote as before the potentials at the extremities of the conductor, and R the sum of the resist- ances of the homogeneous portions of the conductor in 8O ELECTROMOTIVE FORCE IN A VOLTAIC CIRCUIT. the former case, or the actual resistance of the conductor in the latter. The conductor in such cases is said to contain, or to be the seat of, an electromotive force E, or (as frequently in what follows) an electromotive force AE is said to be 272 the conductor. The total electromotive force producing a current in the conductor is now V1 – 1/2 + A. Since in a heterogeneous conductor (I) applies in the first case to every part, except any, however Small, which includes a surface of discontinuity, the electromotive force is said to have its seat at the surface or surfaces of discontinuity. In the other case electromotive force has its seat in every part of the con- ductor moving in the field, according to a law which we shall afterwards discuss. In a circuit composed of different homogeneous conductors not moving across lines of magnetic force let adjacent points be taken on opposite sides of each surface of continuity, and let the difference of potential between the pair of points in each conductor be measured ; the sum of these differences taken in order round the circuit is equal to the sum of the parts of Æ contributed by the discontinuities. For going round in the direction of the current from a point (not in a surface of discontinuity) to the same point again we have by (2) V = V2 and y = . . . . . . . . (3) But denoting the successive homogeneous conductors in their order round the circuit by the suffixes I, 2 . . . . 72, and the differences of potential between their ex- tremities by - V – V’l, V2 - V', . . . . Wu - V m, ELECTROMOTIVE FORCE OF A VOLTAIC CELL. 8 I the corresponding resistances by Æ1, Æ2, . . . . ſºn, and the total resistance ſºi -- R, + . . . -- A', by ſº, we have – W. " – Kºi º – ... ', " - *(***) () y A’l A's R, A2 Hence y >(V – V’) = E . . . . . . (5) A is called the total electromotive force in the circuit, or simply the electromotive force of the circuit. We will consider here as an example of the principle just stated its application to the case of a simple voltaic cell composed of two plates of dissimilar metals connected by a liquid, for example, copper and zinc immersed in hydrochloric acid, and connected externally by a copper wire. Let c and 2 denote the copper and zinc plates, / the liquid between them. By the theory of the voltaic cell now generally adopted, there is a certain finite difference of potential on the two sides of the junction of the dis- similar metals, and on the two sides of each junction of a metal with the liquid. We may suppose for simplicity the plates to be such that they add no sensible resistance to the circuit, and that therefore the potential may be taken as the same at every point of each. Let V., denote the potential of the copper plate ; V, the potential of the copper wire close to its junction with the zinc plate; Via the potential of the stratum of the liquid close to the zinc plate ; and Vic the potential of the stratum of the liquid close to the copper plate. The difference of potential between two points in the copper conductor near its ends is therefore V, - Vº, and that between the two sides of the liquid is Via – V. Both of these differ- ences are positive, and the current flows from the copper * * 82 ELECTROMOTIVE FORCE OF A VOLTAIC CELL. plate to the zinc plate through the wire, and from the zinc plate to the copper through the liquid. Further it is an experimental fact, as we shall see later, that the current across any cross-section is the same at every part of the circuit. Calling ſº the resistance of the copper conductor joining the plates, and r the resistance of the liquid of the cell, we have by (3) y = Ka – º – Vie – Vic A’ 7- 2 and therefore also - 1/2 - l//. —H. Viz — l/ſ, y A -- 7" - But V, - Vic is the finite difference between the potential of the copper plate and that of the liquid in contact with it, and Vic – V. is similarly the difference between the potential of the liquid in contact with the zinc plate and that of the extremity of the copper wire adjacent to the zinc plate, and the sum of these two differences con- stitutes what is called the Electromotive Force of the cell. Calling this E, we have y - #2 (6) R –H 7- Any other case, however more complicated, might be treated in a similar manner. If V be the difference of potential between any two points in the Copper wire, ſº the resistance of the wire between these two points, and r the remainder of the resistance in circuit, we have from the equations VOLTAIC BATTERY. 83 the result A& W = E —-. A -- r (7) The activity in the wire is by (2) V2 A = Viv = - . . . . . . (8 y = }; (8) and for the whole circuit A = Ex = € ) = ey= + (9 By (8) the activity in any wire not containing an electromotive force can always be found, whatever be the arrangement of which it forms part. The activities in the different parts of more complicated circuits Con- taining Electromotive forces of different kinds will be considered in the chapter on the Measurement of Electric Energy. - If instead of a single cell we have a battery of several cells, its electromotive force is found in exactly the same manner by summing all the finite differences of potential at the surfaces of separation of dissimilar substances in the circuit. Hence if there be 7, cells in the battery joined in series, that is to say the zinc plate of the first cell joined to the copper plate of the second cell, the zinc plate of the second to the copper plate of the third, and so on to the last cell, the total electromotive force of the arrange- ment, if the cells have each the electromotive force E, is ſt B. If the copper plate of the first cell and the zinc plate of the last be joined by a wire, and R denote as before its resistance, r the internal resistance of each cell, a current of strength y given by the equation 22 A. 7 * F.I., (IO) 84 ARRANGEMENT OF A BATTERY. will flow in the wire. This equation may be written y = A. — — …, 7 –– R 72 which shows that when ºn is So great that ſº/?? is small in comparison with 7", little is added to the value of y by further increasing the number of cells in the battery. - The method of joining single cells in series is advan- tageous when R is large, but when Æ is comparatively small it fails as shown above, and it is necessary then to diminish r as much as possible. The value of r is, for cells in which, as is generally the case, each plate nearly covers the cross-section of the liquid, nearly in the inverse ratio of the area of the plates, and directly as the distance between them. Hence, by increasing the area of the plates and placing them as close together as possible, the resistance may be diminished. One large cell of small resistance may be formed of several small cells by putting all the copper plates into metallic connection with one another, and similarly all the zinc plates. Several Com- pound cells of large surface thus made may be joined in series. The electromotive force of each compound cell will be E as in a simple cell, but if ſ/z cells be joined so as to form one compound cell its resistance will be r|m. If an of these compound cells be joined in Series, we have, calling the total external resistance R, 72 E 777 72 E y = 7. Tº m ſº-E my ' ' ' (II) R+m. - If R be not too great, and we have a proper number of MOST ECONOMICAL ARRANGEMENT. 85 cells, it is possible to arrange the battery so that y may have a maximum value. There being ºn 27 cells in the battery the numerator of the above value of y does not change when the arrangement of cells is varied, and there- fore, in order that y may have its greatest possible value, m ſº -i- m r must be made as Small as possible. But identically, 772 R + m r = (V7m A. ----- W77)? + 2 Wºm m /ºr. As the last term on the right-hand side does not vary with the arrangement of the battery, it is plain that 777 R + 777 will have its smallest value when W mR – W mr. vanishes, that is when m^+ nr or Å = mr/m, or, in words, when the total external resistance of the circuit is equal to the internal resistance of the battery. It may not be possible in practice so to join a given battery as to fulfil this condition, but if the strongest possible current is required it should be fulfilled as nearly as possible. This method of arranging the battery is called joining it in multiple arc. It is to be carefully observed that this theorem applies only to the case in which we have a given battery and have to arrange it so as to produce the greatest current through a given external resistance R ; and the fallacy is to be avoided of supposing that of two batteries of equal electromotive force, but one having a high, the other a low, resistance, the former is better adapted for working through a high external resistance. Nor is this method of using the battery to be confounded with the most economical method. By this arrangement the greatest rate of working in the external part of the circuit is obtained ; for by (9) the total rate of working is n Ey, and the part of this which belongs to the external con- 86 PRINCIPLE OF CONTINUITY. ductor is muzu Ey K/(m A' + nr), which is a maximum under the same conditions as y. As much energy is thus given out in the battery itself as in the external resistance, and it is plain that for economy as little as possible of the energy of the battery must be spent in the battery itself, and as much as possible in the working part of the circuit. Hence for economical working the internal resistance of the battery and the resistance of the wires connecting the battery with the working part of the circuit must be made as Small as possible. We shall return to this question in Chapter IX. FIG. 9. We shall now consider shortly the theory of a system of linear conductors (homogeneous wires of uniform section) in which steady currents are flowing. It has been stated above that when a steady current is kept flowing across any section of a conductor, the current strength is the same across every other section of the conductor; or, in other words, that the flow of electricity at any instant into any portion of the conductor is equal to the flow out of the same portion. This is what is called the frºncºle of continuity as applied to the case of a steady flow of electricity. By the same principle we have, in the case in which steady currents are maintained in the various parts of a network of conductors, the theo- rem that the total flow of electricity towards the point at RESISTANCE AND CONDUCTANCE IN MULTIPLE ARCS. 87 which several wires meet is equal to the total flow from that point. Thus the current arriving at A (Fig. 9) by the main conductor is equal to the sum of the currents which flow from A by the arcs which connect it with B. By Ohm's law, if two points A and B, between which a difference of potential V is maintained, be connected by two wires of resistances r, and r2, the current in that of resistance ri will be V/71 and in the other V/r2. But if y be the whole current flowing in the circuit we have by the principle of continuity IV V V y = - + — = 7.9 where R is the resistance of a wire which might be sub- stituted for the double arc between A and B without altering the current in the circuit. Hence, I I I 71 r, ſº and . . . . . (I2) 2^l 7". *-*. 71 - ?, - The reciprocal of the resistance A of a wire, that is, I/R, is called its conductance.* Equation (II) therefore affirms that the conductance of a wire, the substitution of which for 71 and r2 between A and B, would not affect the current in the circuit, is equal to the sum of the conductances of the wires r, and r3. From equation (12) we see that the resistance R of this equivalent wire is equal to the product of the resistances of the two wires divided by their sum. * Formerly called its conductivity. 88 KIRCHHoFF's THEOREM If for rº, we were to substitute two wires having an equivalent resistance, so that A and B should be con- nected, as in Fig. 9, by three separate wires of resistances 71, r2, r3, we should have in the same manner for the Current in r, V/r, ; in rº, V/r, ; in 73, V/rs, and I I I I % = a + 2 + 2. a zºº” wre- (13) A’ = 7.17 2 73 71 re -- ?, ?’s + 737, Generally, if two points A and B are connected by a multiple arc consisting of n Separate wires, the conductance of the wire equivalent to the multiple arc connection is equal to the sum of the conductances of the n connecting wires ; and its resistance is equal to the product of the 7. resistances divided by the sum of all the different products which can be formed from the 7z resistances by taking 72 – I at a time. As a simple example, we may take the case of a number 77 of incandescence lamps joined in multiple arc. If the resistance of each lamp and its connections be r, the equivalent resistance between the main conductors, the resistance due to the latter being neglected, is by (II) riſm . .” - 1 = rſ7e. Thus if r be 60 ohms when the lamp is incandescent, and there be twenty lamps, their resistance to the current will be 3 ohms. By the considerations stated above, we at once deduce from Ohm’s law the following important theorem.* In any closed circuit of conductors forming part of any linear system, the sum of the products obtained by mul- * This theorem and the application of the principle of continuity were first stated explicitly by Kirchhoff, Pogg. Ann. Bd. 72, 1847, also Ges. Aëhand., D. 22. REGARDING A CLOSED CIRCUIT. 89 tiplying the current in each part, taken in order round the circuit by its resistance, is equal to the sum of the electromotive forces in the circuit. This follows at once by an application of Ohm’s law to each part of the circuit, exactly as in the investigation in p. 81 above of the electromotive force of the circuit composed of a cell and a single conductor. As an example of a circuit containing no electro- motive forces, consider the circuit formed by the two wires (Fig. 9) of resistances r1, r, joining AB. We FIG. Io. have, for the current flowing from A to B through ri, the value V/r1; the product of this by r is V; for the current flowing from B to A through r, we have – V/ra, and the product of this by r, is – V: the sum is V – V or zero. As another example, consider the diagram, Fig. IO, of resistances r1, r2, rs, rº, rs, between the two points A and B. By what we have seen, if Va, V, V., Va., be the potentials at A, B, C, D, respectively, the current from A to C is (V. - V.)/ri, from C to D (V – V.)/r, and from D to A (W2 – V.)/r, 90 NETwork of WHEATSTONE's BRIDGE, Hence, taking the sum of the products of these current strengths by the corresponding resistances for the circuit A CDA, we get - W. – V. -- V. – V., + V - V. = O . (I4) To illustrate the use of the principles which have been established, we may apply them to find the current strength in 73 (Fig. IO) when re contains a battery of electromotive force E. Let 7'o be the resistance of the battery and the wires connecting it with A and B, and let yi, y2 &c., be the strengths of the currents flowing in the resistances r1, r2, &c., respectively, in the directions indicated by the arrows. By the principle of continuity we get - 'y: = yi — ya Y4 = y2 + y; . . . . . (15) 'yo = y + y, -: Applying the second principle to the circuits BA CB, A CDA, CB DC, and using equation (14), we obtain the three equations, 'yi (?-, + r + 78) + y, zºo — yº 73 = E 'yl 71 – yº?? -- ys 7, = 0 }. (16) 'yl 73 - — y2 ra – yº (7.3 + 7.4 + 2*) = O Eliminating yi and yº, we find E (?, 73 – ?, ??) ºys = 2 7, 1 / 4 e tº tº tº (17) where D = ?, ?, (2) + 7, -ī- rs + r.) + 7.3 (r. -- 7:3) (r) + r.) + zo (r. -- rº) (rs + r.) + riſ's (rº + r.) + 737, (? -- ??) . (18) RESULTS OF FOREGOING PRINCIPLES. 9I By substituting for y, in the second and third of equa- tions (16) its value yo — yi, we get, yì (ri + r.) + yºr; - yd?? ...) {} (19) Yi ("s-H ra) – yº (r. -- ", + r.) – yº” - O From these we obtain by eliminating yi, -: - 'yº (r., ra -r, r.) 'y; = 73 (?? -- ?? -- rs + r.) + (ri + r.) (rs + r.) ' (20) By means of equations (17) and (20) we can very easily Solve the problem of finding the equivalent resistance of the system of five resistances r1, r2, &c., between A and B. For let R be this equivalent resistance, since ye is the current flowing through the battery, we have 'ye = E/(ro + A'). Substituting this value of ye in (20), equating the values of ys given by (17) and (20), and solving for ſº, we get A’ = 7;(ri-H 7.3) (re-H ri)+ r rs (r3+??)+?,74(ri-H 73) (21) 7's (ri + r., + 73 + r.) + (ri + 72) (rs + r.) It follows from Ohm’s law, and the theorems which have been deduced from it, that any two states of a system of conductors may be superimposed ; that is, the resulting potential at any point is the sum of the potentials at the point, the current in any conductor the sum of the currents in the conductor, and the electromotive force in any circuit the sum of the electromotive forces in the circuit, in the two states of the system. The following result, which is a direct inference from the foregoing principles, and can be verified by experiment, will be of use in what immediately follows.” * For an elementary proof of certain general theorems regarding a network of conductors see the author's Theory and Practice of Aðsolute Measure- 77tents in Electricity and Magnetism, vol. i. p. 166. 92 RESULTS OF FOREGOING PRINCIPLES. Any two points in a linear circuit which are at different potentials may be joined by a wire without altering in any way the state of the system, provided the wire contains an electromotive force equal and opposite to the difference of potential between the two points. For the wire before being joined will in consequence of the electromotive force have the same difference of potential between its extremities as there is between the two points, and if the end of the wire which is at the lower potential be joined to the point of lower potential it will have the potential of that point, and no change will take place in the system. The other end will then be at the potential of the other point, and may be supposed coincident with that point, without change in the state of the system. The new System thus obtained plainly satisfies the principle of Continuity, and the theorem of p. 88 above, and is there- fore possible ; and it can be proved that it is the only possible arrangement under the condition that the state of the original system shall remain unaltered. As a particular case of this result any two points in a linear circuit which are at the same potential may be con- nected, either directly or by a wire of any resistance, without altering the state of the system. The following result is easily proved, and is frequently useful. If the potentials at two points A, B, of a linear system of conductors containing any electromotive forces, be V, V respectively, and ſº be the equivalent resistance of the system between these two points, then if a wire of resistance, r, be added, joining AB, the current in the wire will be (V – V')/(A + r.). In other words the linear system, so far as the production of a current in the added wire is concerned, may be regarded as a single conductor of resistance ſº connecting the points A B and containing DERIVED CIRCUIT OF POTENTIAL GALVANOMETER. 93 an electromotive force of amount V – V". For let the points A and B be connected by a wire of resistance r, containing an electromotive force of amount V – V' opposed to the difference of potential between A and B, no current will be produced in the wire, and no change will take place in the system of conductors. Now imagine another state of this latter system of conductors in which an equal and opposite electromotive force acts in the wire between A and B, and there is no electromotive force in any other part of the system. A current of amount (V– V')/(R -- ?) will flow in the wire. Now let this state be superimposed on the former state, the two electromotive forces in the wire will annul one another, and the current will be unchanged. The potentials at different points, and the currents in different parts, of the system, will be the sum of the corresponding potentials and currents in the two states, and will therefore, in general, differ from those which existed before the addition of the wire. As an example consider a circuit between two points of which there is a difference of potential V, and let r, r" be the resistances of the two parts of the circuit between the two points. (These two parts may of course be any two networks of conductors joining the points.) Then the equivalent resistance is rº/(r–H r"); and if another con- ductor of resistance R, and not containing any electro- motive force, be connected between the two points, the new difference of potential V' will be given by V ( = r—“––. 77° A’ --- + 7" -- 7" since it has been shown above that V'/A' is the current in the conductor. 94 DERIVED CIRCUIT OF POTENTIAL GALVANOMETER. Hence in order that V’ may be approximately equal to V, R must be great in comparison with r"/(r + 1"). But "r'/(r + r") can be written in either of the forms rſ(I + r(r), 7'/(1 + r"/r), which shows that r and r" are each greater than rr'/(r + r"). Hence if R be great in com- parison with either of the two resistances r, r, V' will be approximately equal to V, no matter how the electromotive force may be situated in the circuit. This result is useful in connection with the measurement of the difference of potential between two points of a circuit by a galvano- meter, as it is only necessary to make the resistance in the circuit of the instrument great in comparison with that of either part of the circuit lying between the two points, to be sure that the difference of potential is practically unaltered by the application of the instrument. CHAPTER VII. STANDARD ELECTRICAL MEASURING INSTRUMENTS AND THEIR GRADUATION. NE of the most important operations now performed in an electrical laboratory is the standardizing of instruments for the absolute measurement of currents and differences of potential. The process consists in so graduating the scale or indicating arrangement of the apparatus that currents or differences of potential are either read off directly in absolute units, or are so obtained with only a very simple reduction from the reading. One object of this book is to describe methods of standardizing such instruments, and to do this to most advantage it is desirable to first describe an actual instrument of each class, and to use it as an example in describing methods of graduation. The methods will lose nothing in gener- ality by being described in this connection, and will be easily applicable to other instruments. The instruments adopted for description are Sir William Thomson's well-known Standard Electric Balances, which are now constructed so as to give a very long range of sensibility in the measurement of currents and potentials, his Magnetostatic Current-Meter, and his Quadrant Electrometer and Electrostatic Volt- Inleters. 96 CURRENT BALANCES. The first-named instruments are based on the principle, set forth in Chap. III. above, of the mutual action between the fixed and movable portions of a circuit carrying a current. Each of the mutually influencing portions consists in most of the instruments of one or more com- plete turns or spires of the conductor, but in some cases consists of only half or part of a turn. In all cases in what follows we shall, following Sir W. Thomson, call each portion a ring. - In each of the balances, except that for very strong Currents (the kilo-ampere balance), the movable portion of the conductor consists of two rings, carried with their planes horizontal at the extremities of a balance beam free to turn in the ordinary way round a horizontal axis. Above and below each ring on the beam is a fixed ring with its plane parallel to that of the movable ring. The rings are (except in the Composite Balance, p. IO7 below) all joined in series, and the current to be measured is sent through them so that the mutual action between the movable ring at one end and each of the two fixed rings there is to raise that movable ring, while the mutual action of the other group of three rings is to depress the cor- responding movable ring. The action is therefore to turn the beam round the horizontal axis on which it is pivoted, with for any given position a couple varying as the square of the current flowing. - Fig. I I shows diagrammatically the rings and the course of the current through them. a, e, , f are the two pairs of fixed rings, c, d the movable rings. The current en- tering by the terminal T passes round all the rings in series, in the two movable rings in opposite directions, and returns to the terminal T. Since each movable ring is in general in a magnetic field, terrestrial or CURRENT BALANCES. 97 artificial, which has a horizontal component, it tends to set itself so that the greatest number of horizontal lines of force may pass through it (Chap. III. above) and therefore is acted on by a couple which tends to turn the beam round its axis. But since the current passes round the movable coils in opposite directions, and these are very approximately equal, the two couples are nearly equal and opposite, and the instrument is practically free from disturbance by horizontal magnetic force. | T T. FIG. II. The turning couple produced by the mutual action of the fixed and movable rings is balanced for the hori- zontal or “sighted position ” of the beam by an equal and opposite couple produced (in the manner more particularly described below) by a stationary weight at the end of the beam, and a sliding weight placed, steelyard fashion, at a suitable point on a graduated bar attached to the beam. The amount of the current flowing in the rings is de- H 98 CURRENT BALANCES. duced from the amount of the equilibrating couple thus applied, or rather from a number proportional to it, by the end of a table of reckoning, as described below. In the kilo-ampere Balance (Figs. I2 and I 5) there are only two rings, each nearly rectangular in shape, and hori- zontal in position. These, since they have to bear very Strong currents, are made of massive copper strip. The ſº § : º & C § s i i movable ring is mounted as shown above the fixed ring, and is free to turn round a horizontal axis halfway be- tween its two ends. In the diagram (Fig. I2) a plan of the fixed and movable rings (A, B, respectively), with the course of the current in them, is shown. The current enters the movable ring by the terminal T, and divides, passing round its two halves in opposite directions. At CENTI-AMPERE BALANCE. 99 the middle of the other side the two parts of the current meet and pass to the fixed ring, then completely round that ring once or more times, and across to the terminal T. One half of the movable ring thus tends to rise, the other to fall, and hence the ring tends to turn on its axis. The turning couple is resisted by a return couple due to weights precisely as in the former case. Having thus stated the general principle of these in- struments, we now consider their construction and mode of action with greater minuteness. Most of the con- structive details will be made out from Fig. 13 which shows what is called the Standard Centi-ampere Balance, and illustrates the arrangement of the beam, the graduation, and the mode of applying the equilibrating couple, for all the instruments. . The beam in all the instruments is hung on two trunnions, each supported by a flat elastic ligament made of fine copper wires, through which the current passes to and from the movable rings. The horizontal or sighted position of the beam is that in which the pointers on the extreme right and left are at the middle divisions of their scales. This position, in all the instruments in which a movable ring is acted on by two fixed rings between which it is placed, is not that midway between these two rings, as that would be a position of minimum force and therefore of instability. For stability it is so chosen that the movable ring is nearer to the repelling fixed ring than to the attracting ring by such an amount as to give about ; per cent, more than the minimum force. Fixed to the beam and parallel to it is a finely gradu- ated bar, and above this is a horizontal fixed scale, called the Inspectional Scale, less finely divided. Both gradua- H 2 IOO CENTI-AMPERE BALANCE. - s |#. \ *S* \ |\ $25 ſiliſillº * ſ f N. Y il. . - | \ r \ N º = - - ſº | Wºº. | & W \,\, \! |S}#: ſ\Vijff's ſº *E É == |º | §Sº ſº/| Nl* \s Nº-: G §NN º\s s§ º CENTI-AMPERE BALANCE. IOI tions begin from zero on the extreme left and have numbers increasing towards the right. A carriage is moved along the graduated bar to any required position by a sliding piece controlled by a cord which can be pulled from either end, and this carriage, by itself or with an additional weight, forms the movable weight referred to above. The position of the carriage is indicated by a pointer which moves along the lower scale. Each additional weight has in it a small hole and slot which pass over conical pins in the carriage. This ensures that the weight is always placed in a definite position. The balancing weight is moved along the beam by means of a self-releasing pendant carried by the sliding piece above referred to. To this pendant is attached a vertical arm (seen in the figure) which passes up through the recess in the front of the weight and carriage and so enables the carriage to be moved with the sliding piece. The stationary weight is placed in the trough shown at the right-hand end of the instrument. The trough is V shaped, and the weight cylindrical, with a cross pin which passes through a hole in the bottom of the trough. The weight is thus placed in a perfectly definite position and always has the same leverage. It is so chosen as just to keep the beam in the sighted position when the sliding weight is at the zero of the scale. Since the mutual action of the rings is to bring the beam towards the sighted position when displaced by the weights, and the equilibrating couple is that due to the displace- ment of the sliding weight from zero, the latter couple in- Creases as the current increases, and hence motion of the sliding weight towards the right corresponds to increasing Currents. The use of the stationary weight gives a scale of double the length which would be obtained without it. I O2 CENTI-AMPERE BALANCE. In the top of the lower or finely graduated scale are notches which correspond to the exact integral divisions in the upper fixed scale. Thus the reading in the fixed scale is got when the pointer is at a notch, without error from parallax due to the position of the eye. The read- ing when the pointer is between two notches is easily obtained by inspection and estimation with sufficient accuracy for most practical purposes. When however the greatest accuracy is required, the reading is taken on the lower scale, with the aid of a lens, and the current strength calculated from the table of doubled square roots given in the Appendix below. Four pairs of weights are given with each instrument. Of these one set is for the sliding platform, the other set are the corresponding Counterpoises. The weights of each set are in the ratios I : 4: 16: 64, and are so adjusted that, when the carriage is placed with its index at a divi- sion of the inspectional Scale, the instrument shows a current of an integral number of amperes, half-amperes, or quarter-amperes, or some decimal subdivision or multiple of one of these units of current. The accurate adjustment of the zero is effected by a small metal flag as in a chemical balance. This flag is set in any required position by means of a fork moved by a handle beneath and outside the case of the instrument. The sliding weight is brought to zero with the correspond- ing counterpoise in the trough, and then the flag is turned to one side or the other until the pointer of the beam (seen on the extreme right and left in Fig. 13) is just at ZCrO. When necessary for transit or otherwise, the beam in the centi-ampere and deci-ampere balances is lifted off its supporting ligament by turning an eccentric by a DEK A-AMPERE BALANCE IO3 |iſ|||||| *GION vº Ivy g. 3 NGA INv-vxº GT CIXI vCINYLS –** I· O I Į ~~~~）;/ №z， ~ ſºſ |---- |||||||||||||||||||| 。· № Ī||||||||| ſae ||§§§ §S S™Ē № E=E=<！--- ſì （)§ №š Ē:№_º = ſ IO4 KILO-AMPERE BALANCE. shaft under the sole-plate of the instrument. In the other balances the beam is fixed for carriage by placing distance pieces between the upper and lower parts of the trunnions and screwing them together by milled headed screws kept always in position for the purpose. Fig. I4 shows the standard deka-ampere balance for the measurement of currents ranging from I to IOO annperes. The only essential difference in construction in this instrument consists in the use of a small number of turns of thick copper-wire for the rings, more massive connections to carry the current to and from the movable rings, and Special electrodes, so that the instru- ment can be placed in an electric-light current without perceptibly increasing its resistance. In this balance (and in the similar hekto-ampere balance adapted for currents from 6 to 600 amperes), when made so as to measure alternating as well as con- tinuous currents, the current is carried by a twisted rope of copper wires, each of which is insulated from its neigh- bours along its length. The object of this arrangement is to prevent the distribution of an alternating current over the cross-section of the conductor from being affected by inductive action. Fig. I 5 shows the kilo-ampere balance, for currents varying from 25 to 2,500 amperes. Of this we have already described the main parts. The details are similar to those of the other balances, and will be easily made out from the cut. The rings are massive, and the lower fixed ring, shown in the figure as having several turns, is made of bare copper strip, the different turns of which are kept apart by slips of mica. The centi-ampere balance may be used as a voltmeter KILO-AMPERE BALANCF. JOS 106 USE OF CURRENT BALANCE AS VoltMETER. by connecting it to two points in a circuit between which the difference of potential is to be ascertained. (On account of heating effects it is not desirable to use in this application of the instrument currents which are greater than can be measured with the lightest weight on the beam, viz. the carriage itself.) According to the difference of potential to be measured, one or more of four platinoid resistances, anti-inductively wound (p. 188 below) provided with the instrument, is to be included in series with the instrument between the two points in question. One of these with the resistance of the coils of the instrument just makes up 500 ohms at a certain specified temperature; each of the others has a resistance of 500 ohms at that tem- perature. The smallest of these resistances is used when the smallest differences of potential are being measured, and in other cases one or more of the others as may be required to bring the current passing through the instru- ment down to one which can be measured with the smallest weight on the beam. Of course deviation of the temperatures of the coils and the resistance employed, produced by the current or otherwise, from the temperatures at which they are correct, must be determined and allowed for in this application of the instrument. The correction is, for the copper coils of the balance 38 per cent. per 1* C., and for the platinoid resistance coils about ‘O24 per cent. per I* C. The temperatures within the coils and the resistances are found by inserting the bulb of a thermometer sent out with the instrument into orifices provided for the purpose. The difference of potential in volts, V, between the points at which the terminals of the derived circuit, formed by the balance and coils, are applied, is obtained COMPOSITE BALANCE. Io'7 from the current in amperes, C, and the resistance in ohms, R, by the equation V = C A'. The value of R is, as stated above, either 500, I,OOO, 1,500, or 2,000, and hence the reduction from the value of C can be made at once mentally without risk of error. If R be great in comparison with the resistance of the part of the circuit included between the terminals, this value of V may of course be taken as that of the dif- ference of potential existing between these points when the instrument is not applied. At present this application may be considered as made only for continuous currents. The subject of alternating currents will be considered in Chap. IX. A form of balance called a Standard Composite Balance (Fig. 16) has been constructed for the measurement of the current flowing in a circuit, or the difference of poten- tial between two points, or the product V C of these two quantities expressed in amperes and volts respectively, that is the activity in watts (see p. 76 above) between the two points considered. The details of the beam, Scales, &c., are similar to those of the balances already described, but the arrangement of coils is different. This is shown in the diagrammatic sketch (Fig. 17). Each coil of the fixed pair at one end of the beam is made of thick wire represented by the heavy lines on the right-hand side. These coils have a separate pair of electrodes, E, E. The coils of the pair at the other end, and the coils on the beam, are of fine wire, as in the centi-ampere balance, and can be joined in the manner described below with a pair of electrodes T. T. The switch in the diagram enables the fine wire coils IO8 COMPOSITE T3ALANCE. “Go Nwºvg. 3）LISOåIKOO–’91 ºbIJ <%！！！！！ Íſlae áāſ ¿5ms，|FE=EET-}، ،ºſ į\\№. șğ№r=|-ſaephae| || №ĒģēŤ"， "\\>'\,§ ：（… Ķē （; ， !!! }}!·Jį.- |||||！، }•№！© ¡iº：№Ēș>}}|ſ|ſſſſſ：S #ffffffffffſīſ'， （ ?，?=@$.*; #S №ŹŹŽŽ; COMPOSITE BALANCE USED AS WATTMETER. Io9 on the beam to be joined in series with the fixed fine wire coils and connected with 7. Tº or to be connected with the terminals T, T1, by themselves while the fixed fine wire coils are left out of circuit. The former arrangement has place when the switch is turned to watts, the latter when the Switch is turned to volts. The actual switch with these directing words will be readily seen in Fig. 16. IFIG. 17. When the switch is turned to zvažs the instrument is joined for use as an activity-meter. The thick wire coils are included in the circuit, so that the main current flows through these, while the movable coils are placed by the terminals T, T, so as to connect in series with a suitable anti-inductive resistance, sent out with the instru- ment, two points of the circuit through which the current C in the thick wire coils is flowing. The action between I IO COMPOSITE BALANCE USED AS WATTMETER. the fixed and movable coils is then directly as the pro- duct of V and C. These quantities are to be found as follows, and V C calculated from their values. To find V the instrument is arranged for use as a volt- meter or centi-ampere meter by connecting the switch to volts, and connecting the instrument through a suitable resistance to the two required points. The fixed fine wire coils and the movable coils at one end thus act mutually on one another. One or other of three special weights marked V. W1, W. W., W. W., is used, and the current calculated from the constant given in the certi- ficate sent with the instrument. The difference of potential is then found in volts by multiplying this by the resistance in the circuit of the instrument taken in ohms. The instrument may also be used as a hekto-ampere meter and the value of C found. The switch is turned to watts, and the thick wire coils so joined by their electrodes in the circuit that the movable coil at that end is repelled up. A measured current is then sent through the movable coils. This current, as used in the gradua- tion of the instrument, is 25 ampere, but any other current which is convenient and not too great may be used. Either the carriage alone or the special weight marked W. W is to be used in bringing the beam to the sighted position. If the measured current in the fine wire coils be c the current C in the thick wire coils will be obtained in amperes by multiplying the constant given in the certificate for the reading by 4c. The current in the thin wire coils may be measured by the fixed and movable thin wire coils arranged as for the measurement of volts. The switch is turned to z/o/ts, and the terminals, including the instrument and a suitable anti-inductive resistance, are connected with STANDARD BALANCES. I I I those of an electric installation or battery and the current measured ; then the switch is turned to watts and a re- sistance added to the circuit equal to that of the fixed fine wire coils which has been withdrawn. The measure- ment is then made as described above. If Vbe the potential in volts, ſº the resistance included in ohms, c the measured current in amperes sent through the fine wire coils, A the activity in watts, we have A = VC = c('A'. The following is a list of the Standard Balances with their ranges, and a table showing for each instrument as numbered, the current per division of the inspectional scale corresponding to each of the four pairs of weights. Name of Balance. Range. I. Centi-ampere Balance. I to roo Centi-amperes. II. Deci-ampere Balance. I to Ioo Deci-amperes. III. Deka-ampere Balance. I to Ioo Amperes. IV. Hekto-ampere Balance. 6 to 600 Amperes. V. Kilo-ampere Balance. 25 to 2,500 Amperes. VI. Composit º { 'o2 to 3oo Amperes. } mposite Balance 1oo to 25, coo Watts (at 100 volts.) Current for each division of inspectional scale for the different balances. 3 : cy) QU) - - # 5 | # 5 a 5 a 5 a 5 §§ ## ##|s #3 £: ; : S | – E E | – 3 S | S. g. • ?"; CU .2- e -4 C. : * *4 Q-, -, -jā Fää E #3 F #3 *i; 5 § | 3 || | < | | || ArtRAcTING É PLATE |- § 2 j- INDUCTION PLATE s rº ſº ELECTRODE O to LAMP AND seaſººr---- * * *- :- - - MP AND scalºº::=#|--|Miñºn CIRCULAR F LEVEL w END OF nEEDLE COVER § Hinduction J| PLATE ADRANTS ; | ! | ; TIN FOIL FIG. To the extremities of a small cross-bar at the top of the aluminium wire are attached the lower threads of a bifilar made of two single silk-fibres. The upper I 2C) THE QUADRANT ELECTROMETER. ends of these fibres are wound in opposite directions round the pins c, d, each of which has, in its outer end, a square hole to receive a small key, by which it can be turned round in its socket so as to wind up or let down the fibre. By this means the fibres can be ad- justed so as to be as nearly as may be the same length ; and as the whole Supported mass of needle, &c., is then symmetrical about the line midway between the fibres, each bears half the whole weight. The pins c, d, DETAILS OF CONSTRUCTION. I 2 I are carried by the upper ends e, /, of two Spring pieces which form the continuations of a lower plate Screwed firmly to the supporting piece. Through e, /, and work- ing in them, pass two screws a and 0, the points of which bear on the brass supporting plate behind. Dy the screw a the end e of the plate e /, can be moved forward or back through a certain range, and thus the pin c carried forward or back relatively to d'; similarly d can be moved by the screw 5. Thus the position of the needle in azimuth can be adjusted. The distance of the fibres apart can be changed by Screwing out or in a conical plug shown between the springs e,f, The aluminium wire carries between its upper end and the needle a small concave mirror of silvered glass, to be used with a lamp and scale to show the position of the needle. The mirror is guarded against external electric influence by two projecting brass pieces, which form nearly a complete cylinder round it. The part of the wire just above the needle is protected by the tube shown at the bottom of Fig. 22. This tube extends down below the needle a little distance, and is cut away at each side to allow the needle free play to turn round. The interior coating of the Leyden jar is formed by a quantity of Sulphuric acid which it contains, and which also serves to preserve a dry atmosphere within the jar, the exterior Coating by Strips of tinfoil pasted on its outer sur- face. The acid has been boiled with sulphate of ammonia to free it from volatile impurities which might attack the metal parts of the instrument. The jar itself is enclosed within a strong metal case of octagonal form, Supported on three feet, with levelling screws. The line joining two of these feet (which are in front) is, when level, parallel to the axis of the needle if the latter is properly adjusted. Í 22 THE QUADRANT ELECTROMETER. The needle is connected with the inner coating of the jar by a thin platinum wire kept stretched by a platinum weight at its lower end, which hangs in the acid. The wire is protected from electrical influence by a guard-tube forming a continuation of the narrower guard-tube, (partly shown in Fig. 22) and therefore extending from below the quadrants to a short distance above the acid, and con- nected also by a platinum wire with the acid. The supporting plate in Fig. 22 carries the disc of an idiostatic * gauge of the kind described in p. 123 below. The height of the disc is adjustable by means of a fine Screw and jam-nut below it. The supporting plate, with the Suspension and disc of the gauge, &c., is enclosed within an upper brass case, called the lantern, which closes tightly the central opening of the cover. The top of the lantern is the guard-plate of the gauge, and carries the aluminium trap-door and lever with sighting plate and lens as already described. A glass window in the lantern allows light to pass to the mirror, and the suspension to be seen. A small opening in the glass, closed when not in use by a Screw- plug of vulcanite, enables the operator to adjust the suspension without removing the lantern. The principal electrodes of the quadrants are brass rods cased in vulcanite, and are arranged so as to be movable vertically. Each is terminated above in a small brass binding screw, and is connected below by a light spiral spring of platinum with a platinized contact piece, which rests by its own weight on a part of the upper surface of the quadrant, also platinized to ensure good contact. * An electrometer in which an auxiliary electrification, independent of that being tested, is employed (as in the ordinary mode of using the quadrant electrometer) is said by Sir William Thomson to be heterostatic ; if the electrification being tested is alone employed it is said to be idiostatic. THE IDIOSTATIC GAUGE. I 23 They are placed one on each side and in front of the mirror. One is in contact with the quadrant connected below to the micrometer quadrant, the other to the quadrant connected to that below the induction plate. An insulated charging-rod descends through the lan- tern, and carries at its lower end a projecting Spring of brass. When the rod is not in use the Spring is not in contact with anything ; but when the jar is to be charged the rod is turned round until the spring is brought into contact with the supporting-plate, which, as stated above, is in contact with the acid of the jar. The difference of potential between the inner and outer coatings of the jar is tested by an auxiliary attracted disc electrometer used idiostatically. This electrometer, which is called the gauge, is contained within the cylin- drical box / on the cover of the jar. The arrangement is shown in detail in Fig. 23. The disc is a square piece of aluminium forming a continuation of a lever /, of the same metal. This lever is forked and the prongs joined by a black opaque hair which moves in front of a white enamelled plate on which are two black dots in a vertical line. The position of the hair is seen through the plano-convex lens /, which is carried by a sliding platform attached to the guard-ring G. Torsion given to the platinum wire /, to which the lever is attached in the manner shown in Fig. 23, and round which the lever turns as a fulcrum, forces the disc upwards. This upward force is balanced when the hair is just between the dots by electric attrac- tion between the disc and a parallel plate below it, which is in contact with the interior coating of the jar, while the guard-ring and disc are in contact with the exterior coat- ing. The attracting plate below is mounted on a fine Screw, by which its distance from the disc and therefore I 24 THE QUADRANT ELECTROMETER. the sensibility of the gauge can be varied at pleasure within certain limits. The sensibility of the gauge varies with any alteration in the elasticity of the torsional springſ. This however is of little consequence as the variations are not sudden, and it is never necessary to know the actual potential of the jar. Between each end of the wire f and the supporting block is interposed a U shaped spring (not shown in Fig. 23) made of light brass. The end of the wire is attached to the extremity of one limb of the U, a pin passing through the supporting block to the extremity of the other limb. The two pins, the extremities of the springs, and the wire are in line. The springs can be turned round the pins as axes, so as to give any initial torsional couple to the wire which may be required, and by their elastic reaction cause the wire to be stretched with a nearly constant force. The mode of attachment of the wire to the lever /, deserves notice. The wire is threaded through two holes in the broader part of the lever, near the square disc, so that the part between the holes is above the lever. Half- way between the holes it passes over a small ridge piece of aluminium, which prevents the lever from turning round without twisting the wire. THE REPLENISHER. I 25 The difference of potential between the coatings of the jar is kept nearly constant by means of a small induction machine R, called by Sir William Thomson the Replenisher. The construction and action of this part will be easily understood from Figs. 24 and 25; * Fig. 24 shows the mechanism full-size in perspective elevation ; Fig. 25, the Same in plan. * These cuts are taken with permission from Prof. Ayrton's Practica/ Flectricity. (Cassell & Co., London.) I 26 THE QUADRANT ELECTROMETER, Two similar metal carriers C, D, each part of a cylin- drical Surface, are fixed on a cross-bar of paraffined ebonite so as to be slightly noncoaxial but inclined at the same angle to the Cross-bar. Through the cross-bar and rigidly fixed to it, passes a cylinder of ebonite having at FIG. 25. its ends metal pieces which form the extremities of an axle. The carriers turn round this axle within the circular cylinder marked out by the cylindrical metallic pieces A, B, which are insulated from one another and act as inductors. A receiving spring, s or sº, projects obliquely inwards through a hole in each inductor, with which it is THE REPLENISHER. 127 also connected at the back, and is bent so that the carriers touch the springs on their convex sides, and pass on but little impeded by the friction. Two contact springs S, S', connected by a metallic arc, project slightly inwards be- yond the inductors so that the carriers, while opposite the inductors, come into contact with these two springs at the same time, and are therefore put into conducting contact. One of the inductors, A, is connected to the inner coating of the jar, the other, B, is attached by the Supporting plate of brass to the cover of the instrument and therefore to the outer coating. A milled head attached to ſº pro- jects above the cover and forms a handle by which the carriers are twirled round. The electrical action is easily understood. An initial charge has been given to the jar, so that a difference of potential exists between the coatings, the interior for example being positive. When the carriers come into contact with the springs S, S', they are brought to the same potential, and, since they are under the influence of the inductors, one carrier becomes charged positively, the other negatively. Then, turning in the direction of the arrow, they come into contact with the receiving springs, and being each (electrically) well under cover of the corresponding inductor, they give up the greater part of their charges, thus increasing the difference of potential between the inductors. If the carriers are turned in the opposite direction the action is of course reversed, and the difference of potential is diminished. A spring catch keeps the knob of the replenisher, which is on the upper side of the cover, in such a position when not in use that the carriers are not in contact with any of the Springs. I 28 THE QUADRANT ELECTROMETER. On the upper side of the cover of the jar are screws, three in number, Securing the cover to a tightly fitting flat ring-collar below it, to which the jar is cemented, and to which the case is screwed ; two screws, one on each side, which fix the lantern in its place ; a cap covering an orifice communicating with the interior of the jar ; two binding screws by which wires can be connected to the case; and a knob similar to that of the replenisher, which, when turned against a stop marked “contact,” connects by an interior spring the quadrant below the induction plate with the case, and when turned in the opposite direction to an adjoining stop marked “no contact,” in- sulates that quadrant from the case. Two keys, for turning the pins a, b, c, &c., are kept let down outside the case through holes in the projecting edge of the cover. The cover also carries a small circular level, set so as to have its bubble at the centre when the cover is levelled by an ordinary level. When this has been done the accuracy of construction of the quadrants ensures that they are also level. The level has a slightly convex bottom, and is screwed down with three screws, so that when the instrument is set up for use, a final adjustment, to show horizontality of the quadrants, can easily be made by turning the Screws. Full instructions for setting up and adjusting the quad- rant electrometer are sent out with each instrument by the maker, and are therefore available, if kept, as they ought to be, beside it in the case. We shall suppose therefore that the detached parts have been put into their places, the acid poured into the jar, and the instrument set up and levelled ; but as a quadrant electrometer is now part of every well-equipped physical laboratory, and is used over a wide range of electrical work, we shall describe here the principal adjustments. SETTING UP OF INSTRUMENT. I 29 The two front quadrants are pulled out as far as possible, to allow the operator to observe the position of the needle, which should rest with its plane horizontal and midway between the upper and under surfaces of the quadrants. If it requires to be raised or lowered, the operator winds or unwinds the fibres by turning the pins c, d, to which they are attached. The suspension wire of the needle should pass through the centre of the circular orifice formed in the upper surface of the quadrants, when these are symmetrically arranged. If the wire is not in this position the pins a, à, are turned so as to carry the point of suspension forward or back until the wire is adjusted, and then one pin is carried forward and the other back, without altering the position of the wire, until the black line along the needle is parallel to the transverse slit separating the quadrants. The scale is placed at the proper distance to give a distinct image of the wire across the line of divisions in front of the lamp flame, then levelled and adjusted so that, when the image is at rest in the centre, the extremities of the scale are at equal distances from the needle. - - When the best relative positions of the instrument and the stand for the lamp and scale have been ascertained, these are fixed by the “hole, slot, and plane’ arrangement (p. 7) adopted by Sir William Thomson, to allow any instru- ment supported on three feet or levelling screws to be re- moved at pleasure, and replaced without readjustment in its original position. A conical hollow, or better, a hole shaped like an inverted triangular pyramid, is cut in the table so as to receive the point (which should be well rounded) of one of the levelling screws, without allowing it to touch the bottom. A V-groove, with its axis in line K I 3O THE QUADRANT ELECTROMETER. with the hollow, is cut for the rounded point of another levelling screw, and the third rests on the plane surface of the table. If it is desired to insulate the electrometer case it is supported on three blocks of vulcanite cemented to the table ; and in one of these the hollow is cut, in another the V-groove. When the jar is being charged, the main electrodes, the induction plate electrode, and one of the binding screws on the cover, are kept connected by a piece of fine brass or copper wire. The charging electrode is turned round So as to bring the spring at its lower end into contact with the Supporting brass piece, and a positive charge is given to the jar by means of the small electrophorus which accompanies the instrument. The cover of the jar is tapped during the process to release the balance lever from the stop, to which it may be adhering. When the lever rises the charging rod is turned so as to disconnect the spring, and the charge is then adjusted to the normal amount (determined by the distance of the attracting disc from the trap door) by the replenisher. The spot of light may in the process of charging have moved from its position for no electrification, and must be brought back by moving out or in the quadrant carried by the micrometer screw. In ordinary circumstances the leakage of the jar will cause the hair to fall down in twenty-four hours about half the breadth of the lower black spot. This loss of charge from the jar is made good by the replenisher ; but if the leakage is considerably greater, the main stem should be washed by means of a piece of hard silk ribbon (to avoid shreds) with soap and water, then with clean water, and finally carefully dried. Shreds and dust on the needle and quadrants may tend to discharge the jar, TESTING OF INSULATION. I 3 I and anything of this kind should be removed by carefully and lightly dusting the needle and quadrants with a clean camel's-hair brush. The jar is selected for its high in- sulating power, but if the acid has in careless handling of the instrument been splashed over the interior surface there may be considerable leakage over the surface of the jar to the case. This can be remedied by removing the acid and carefully washing the jar. The replenisher may also cause leakage of the jar through a deterioration of insulating power of the vulcanite sole-plate which connects the inductors. Such a deterioration with lapse of time is not uncommon in ebonite, and is a consequence of slow chemical action at the surface. A nearly complete cure can be effected by removing the piece and washing it carefully by prolonged immersion in boiling water, and then re-covering its surface with a film of paraffin. The insulation of the quadrants is now tested. One pair of quadrants is connected to the case and a charge producing a difference of potential exceeding the greatest to be used in the experiments is given to the insulated pair by means of a battery, one electrode of which is con- nected to the electrometer case, while the other is connected for an instant to the electrode of the insulated quadrants; and the deflection of the spot of light is read off. The percentage fall of potential produced in thirty minutes or an hour is obtained merely by taking the ratio of the diminution of deflection which has taken place in the interval to the original deflection. If this is inappreciable the quadrants insulate satisfactorily. In any case, for satisfactory working the rate of loss of potential shown by the instrument should not be greater than that of the body tested. If the insulation is imperfect the glass stems supporting K. 2 I32 THE QUADRANT ELECTROMETER. the quadrants should be washed by passing round them a piece of hard silk ribbon well moistened and Soaped, then with clean water to remove the soap, and dried by a clean piece of silk ribbon well dried and warmed. If this does not succeed, the fault probably lies in the vulcanite insulators of the electrodes, which should be well steeped in boiling water, then re-covered with clean paraffin and replaced. Care must be taken when this is done not to bend the electrodes. The final adjustment of the tension of the threads to equality is now made. One pair of quadrants is connected to the case, and the other pair insulated. The poles of a single Daniell's cell are then connected to the electrodes, and the extreme range of deflection produced by reversing the battery, either by hand or by a convenient reversing key, is observed. One side of the instrument is then raised by screwing up that side by one or two turns of one of the front pair of levelling screws, and the range of deflec- tion again noted. If the range is greater the fibre on that side is too short, if the range is smaller the fibre is too long ; and the length must be corrected by turning one or other of the pins to which the fibres are suspended. The pins can be reached by the aperture in the window of the lantern ordinarily closed by the vulcanite plug ; and to prevent discharge of the jar the key with vulcanite’ handle should be used to turn them. The black line on the needle will require readjustment by the screws after each alteration of the suspension. The ordinary method of using the quadrant electro- meter is heterostatic, since the jar is kept at a constant potential, generally much higher than any potential which the instrument is used to measure. The shape of the needle is such that for most practical purposes equation THEORY OF THE INSTRUMENT. I 33 (I) (p. I 34) may be regarded as giving accurately the couple deflecting the needle, when the quadrants are symmetrical about the needle and close. The following is the elementary theory of the instru- ment. Consider three conductors maintained at different potentials, and fulfilling the following conditions:—One of the conductors (A) (in the quadrant electrometer the needle) is symmetrically placed with reference to the other two (B and C) (in the electrometer the two pairs of quadrants), and is so formed that one of its two ends or bounding edges is well under cover of B, and the other end or edge under cover of C, so that the electric dis- tribution near each end or edge is uninfluenced except by [… A T FIG. 26. the near conductor. One such simple symmetrical arrange- ment is shown in the figure. Let the potentials of A, B, C be respectively V, V, V, ; and let A be slightly displaced from B towards C. This displacement may be angular or linear, according to the arrangement adopted ; in the quadrant electrometer it is measured by the angle through which the needle is turned. Let 6 denote the displacement ànd ſº the electrostatic capacity of A per unit of 6 at places not near the ends or bounding edge of A, and well under cover of B and C. Then the quantity of electricity lost by A in consequence of its displacement relatively to B is Æð (V – V.), and the quantity lost by B is Æ6 (V – V). Similarly the quan- tities gained by A and C in consequence of the motion I 34 THE QUADRANT ELECTROMETER. of A towards C are respectively £6 (V – V.) and Żó (V2 - V). Multiplying the first and second of these quantities by V and V1 respectively, the third and fourth similarly by V and V, subtracting the sum of the two first products from the sum of the second two, and dividing by 2, we get for the work done by electrical forces in the displacement the value But this must be equal to the average couple multi- plied into the displacement if the latter is angular, or the average force into the displacement if the latter is linear. We have therefore, denoting the force or couple by F. F = 4 (n − iº (P - ***) . . (I) In an arrangement of this kind when the displacement is small the couple or force acting on A is nearly the same over the whole displacement, and thus is nearly equal to the equilibrating couple or force due to the torsion wire, or bifilar, or other arrangement producing equilibrium. But for small displacements this will generally be proportional to the displacement, and therefore also to the deflection D on the scale of the instrument, and thus p = me – cov, - tº (V – " ") (2) where c is a constant depending on the instrument and the mode of reckoning of D. If V be, as it usually is, great in comparison with V, or l’, then V1 – V. = C(ſ) . . . . . (3) DEGREES OF SENSIBILITY. I 35 where Iſ C" is the now practically constant value of c{V - (V, + V)/2}. If the angle of deflection 6 of the ray of light is not a very small angle, the couple given by the bifilar, it is to be remembered, is proportional to sin #6. Hence if D be the distance in divisions on the scale (supposed straight and at right angles to the zero direction of the ray) through which the spot of light is deflected, and R the horizontal distance of the scale from the mirror in the same divisions, we have tan 6 = D/R, from which 6 can be found and hence #6. We have then Z, A sin #6 = (V1 – V.) (w -sº * : *) . (4) where A is a constant. Equation (I) would be more nearly satisfied if the central portions of the needle to well within the quadrants were as much as possible cut away, leaving only a framework opposite the orifice at the centre of the quadrants to support the needle. The electrometer, when used heterostatically, admits of a number of different grades of sensibility. These are shown in the two following tables, where Z denotes the electrode of the pair of quadrants, one of which is below the induction plate, ſº the electrode of the other pair of quadrants, I the electrode of the induction-plate, O an electrode of the case of the instrument, and C the electrode of the conductor to be tested. LC denotes that L is connected to C, RO that R is connected to O, A/CC that A'ſ, and C are connected together, and so on, (L) that the quadrants connected with L are insulated by raising L, (R) that the quadrants connected with R are similarly insulated, (RL) that both L and R are I36 THE QUADRANT ELECTROMETER. raised. The disinsulator mentioned above (p. 128) is used to free the quadrants connected with Z from the induced charge which they generally receive when Z is raised. GRADES OF SENSITIVEN ESS. A. B. Inductor connected with quad- Inductor connected as indi- rant beneath it. cated below. FULL POWER. FULL POWER. Inductor Insulated. A.C A' C J. C r A C T A’O O I A.O A’O O A.O IDIMINISHED POWER. GRADES OF DIMINISHED POWER. A’ JC T A' ſ (2) [*3] or (A) | *% |[# | #5 (L) [*2] or (Å) [*g | JC IC | A’O LO M (RAE) [3] Either of these grades of sensibility may of course also be varied by increasing the distance of the fibres apart. The quadrant electrometer can be made to give results in absolute measure by determining the con- stant C of equation (3), by which the deflection must be multiplied to give the difference V - V. This can be done by observing the deflection produced by a battery of electromotive force of convenient amount, determined by direct measurement with an absolute electrometer or by connecting the electrodes of the instrument to two points of a circuit the difference of potential between which is at the same time measured by a standard balance used as a voltmeter (see p. IO4). Dif- ferent such electromotive forces may be employed to give deflection of different amounts and thus give a kind of IDIOSTATIC USE. I 37 calibration of the scale to avoid error from non-fulfilment of condition of proportionality of deflection to difference of potential. The quadrant electrometer may also be used idio- statically for the measurement of differences of potential of not less than about 30 volts (p. 69). When it is so used the jar is left uncharged, the charging-rod is brought into contact with the inner coating of the jar, and joined by a wire with one of the main electrodes, so as to connect the needle to one pair of quadrants. The other pair of quadrants is either insulated or connected to the case of the instrument. The instrument thus becomes a con- denser, one plate of which is movable, and by its change of position alters the electrostatic capacity of the con- denser. The two main electrodes are connected with the conductors, the difference of potential between which it is desired to measure. A lower grade of sensibility can be obtained by connecting the needle through the charging-rod to the electrode R, and using the induction-plate instead of the pair of quadrants connected with J., which are insulated by raising their electrode. When the instrument is thus used idiostatically V in equation (2) above becomes equal to V1, and instead of (2) we have C * \s) P = 3 (V – * . . . . . (4) that is, the deflection is proportional to the square of the difference of potential and therefore independent of the sign of that difference. It is to the left or right according to the electrode connected to the needle. This independence of sign in the deflection renders the I 38 VERTICAL ELECTROSTATIC VOLTMETER. instrument thus used applicable to the determination of difference of potentials in the circuits of alternating dynamo- or magneto-electric generators. (See Chap. IX.) The quadrant electrometer has been modified by different makers. In a form made in Paris for M. Mascart, the needle is kept at a constant potential by being connected to the positive pole of a dry pile, the negative pole of which is connected to the case, and the replenisher is dispensed with. In another form devised by Prof. Edelmann of Munich, and suitable for some purposes as a lecture- room instrument, the quadrants are longitudinal seg- ments of a somewhat long vertical cylinder, and the needle consists of two coaxial cylindric bars connected by a cross-frame, and suspended by means of a bifilar. A glass vessel below contains strong sulphuric acid in which dips a vane carried by a platinum wire attached to the needle. For practical work Sir William Thomson has lately constructed a form of electrometer to be used idio- statically, and has called it an electrostatic voltmeter. It is represented in Fig. 27, and may be described as an air condenser, one plate of which, corresponding to the needle of the quadrant electrometer, is pivoted on a horizontal knife-edge working on the bottoms of rounded V-grooves cut in the supporting pieces. This plate by its motion alters the electrostatic capacity of the condenser. The fixed plate consists of two brass plates in metallic connection, each of the form of a double sector of a circle, which are placed accurately parallel to one another, with the movable plate between them as shown in the figure. The upper end of the movable plate is prolonged by a fine pointer which == º== . s s---º º -sº===#=F I4O VERTICAL ELECTROSTATIC VOLTMETER. moves along a circular scale, the centre of which is in the axis. The fixed plates are insulated from the case of the instrument ; the needle is uninsulated. Contact is made with the plates by insulated terminals fixed outside the case. The two shown on the left- hand side in the figure belong to the fixed plate, and a similar pair on the right-hand side are in connection with the movable plate through the supporting V- groove and knife-edge. The terminals of each pair are connected by a safety arc of fine copper wire con- tained within a U-shaped glass tube suspended from them and the terminals in front in the diagram which are separated from the plates by the arcs of wire are alone used for connecting to the conductors, or two points of an electric circuit, the difference of potential between which is to be measured, and are therefore called the working terminals. The instrument is enclosed in a wooden case to diminish the risk of injury to incautious or uninstructed persons who might chance to touch its terminals. When a difference of potential is established between the fixed and movable plates the latter moves so as to increase the electrostatic capacity of the condenser, and the couple acting on the movable plate in any given position is, as in the quadrant electrometer when used idiostatically, proportional to the square of the differ- ence of potential. This couple is balanced by that due to a small weight hung on the knife-edge at the lower end of the movable plate. The scale is graduated from O’ to 60° so that the successive divisions represent equal differences of poten- tial. Three different weights, 32°5, 97-5, 390 milli- grammes respectively, are sent with the instrument to GRADUATION. I4 I provide for three different grades of sensibility. Thus the sensibility with the smallest weight on the knife- edge is a deflection of one division per 50 volts, with the two smaller weights, that is four times the smallest, one division per Ioo volts, with all three weights or sixteen times the smallest weight, one division per 200 volts. The electrostatic voltmeter is graduated as follows. A known difference of potential is obtained by means of a battery of from 50 to IOO cells with a high standard resistance in its circuit. An absolute galvanometer or current balance measures the current in the circuit, and the product of the numerics of the current and the resistance gives that of the difference of potential between the terminals of the latter (see pp. IO4, IOC). These terminals are connected to the working terminals of the voltmeter, and the deflections noted with the smaller weights on the knife-edge. For the higher potentials a number of condensers of good insulation are joined in series, and charged by an application of the wires from the terminals of the re- sistance coil to each condenser in succession from one end of the series to the other. This is done so as to charge each condenser in the series in the same direction, and as the same difference of potential, V say, is produced between the plates of each condenser, the total difference between the extreme plates is n V, if there be m con- densers. A convenient large difference of potential can thus be obtained with sufficient accuracy, and being applied to the working terminals of the voltmeter is made to give divisions for a series of different weights hung on the knife-edge. These divisions correspond of course to deflections for known differences of potential with one of the weights on the knife-edge. I 42 MULTICELLULAR VoITMETER. The divisions thus obtained are then checked by using three instruments which have been dealt with in this way. They are joined in Series and a difference of potential established between the extreme terminals, which is observed also by the third joined across the other two. Thus by a process of successive halving and doubling the scale is filled up. For smaller differences of potential, ranging from 4o to 800 volts, Sir William Thomson has constructed a multi- cellular voltmeter (Fig. 28) on the same principle. The indicator consists of a number of equal vanes w, v, ..., in shape similar to the needle of the quadrant electrometer, attached horizontally at equal distances apart on a vertical spindle. The spindle passes at its upper end through a small hole at the centre of a shallow circular box, in the bottom of which is the scale of the instrument. This is covered with a glass plate to guard the indicator from currents of air and keep the scale and other interior parts free from dust. A vertical brass tube f, carries a torsion head h, from which the indicator is suspended by a fine platinum wire w. Between the upper end of the spindle and the wire is interposed a small coach-spring which having sufficient resilience to allow the spindle to touch a guard-stop, saves the suspension from being broken down by acci- dental shaking or jolting of the instrument. By turning the torsion head the needle can be adjusted so as to point to zero when no difference of potential is upon its terminals. Each vane is within a cell similar to that surrounding the vane of the vertical voltmeter but placed in a hori- zontal position, so that the stationary part of the instru- ment is a pile of cells or condenser plates e, e, ... arranged MULTICELLULAR VOLTMETER. I 43 FIG, 28. I44. MULTICELLULAR VOLTMETER. vertically above one another. These are all in metallic connection, and between them and the vanes which form the other plate of the condenser the difference of potential to be measured is produced. * Two vertical brass repelling plates d, d are attached to the sole-plate of the instrument, and prevent the indicator from turning too far. The lower end of the spindle passes through a hole in a guide-plate carried by these vertical plates, and is prevented from passing back by a little brass head attached to it below. The spindle hangs free by the suspending wire when the instrument is level, and the vanes are then horizontal, each in a plane halfway between the top and bottom of its cell. The index, i, is of aluminium and is attached to the top of the spindle so as to indicate on the scale the difference of potential between the indicator and the fixed plates. The instrument is graduated so as to give volts directly by its readings. The scale can be read in an engine-room from a distance by means of a mirror placed above the instrument at an angle of 45° with the plane of the scale, giving thus a vertical scale by reflection. When the instrument is to be carried from one place to another, a thumb-screw r, provided below the Sole-plate is screwed up so as to take the weight of the spindle off the suspension wire, and clamp the head above referred to against the guide plate. The instrument is made of four different ranges be- tween the extreme limits of 40 and 800 volts. The graduation is performed in the following manner. (1) For differences of potential under 140 volts :-A circuit composed of a standard resistance, a standard centiampere balance and a variable resistance joined in GRADUATION. I 45 series, is arranged. The multicellular electrometer, properly adjusted, has its terminals joined to the extremi- ties of the standard resistance, and the deflection of its indicator and the current flowing through the balance are both observed. The difference of potential is of course at once got by multiplying the observed current and the standard resistance together. Its value is altered to give different points on the scale by properly changing the variable resistance. Of course the part of the circuit composing the standard resistance and the centiampere balance must be so well insulated as to ensure that all the current which flows through the standard resistance also passes through the centiampere balance. (2) For differences of potential above 140 volts:—A voltaic pile is used attached by one terminal to one ex- tremity of the standard resistance of the above arrange- ment, so that the terminals of the multicellular voltmeter attached one to a point in the pile, and the other at the further extremity of the standard resistance, have upon them a difference of potential equal to that between the terminals of the standard resistance plus a constant amount obtained by including the requisite number of couples in the voltaic pile. A previously graduated Standard multicellular of the proper grade of sensibility enables this latter difference of potential to be kept Constant during the progress of the standardization, while the different points, on the scale of the instrument being Standardized, are obtained by varying the difference of potential on the terminals of the standard resistance by means of the variable resistance. Repeating this process the necessary number of times, using as “standard multicellular” first one of the first grade of sensibility (to graduate another of the second grade), then a second L I46 GRADUATION OF INSTRUMENTS. grade, then a first and Second grade joined in series, and so on, with greater and greater differences of potential from the pile, we obtain a step-by-step graduation up to the limit of the range of this type of instrument. Of course when a complete set of standard instruments of this type is available, the Ordinary method of graduating by comparison, division for division, may be used ; care being always taken to re-test the “standards” from time . to time, and to guard them from usage that might tend to produce errors in their measurement of difference of potential. * The pile referred to above is composed of copper-zinc Couples, each about an inch Square, Separated by a layer of blotting-paper moistened with water. The plates are Supported on edge, with a diagonal of each plate vertical, on a slotted support, the edges of which are two strips of vulcanite. They are pressed together between two end- pieces of vulcanite by a screw, and are thus kept in position. The papers are wetted by letting the lower edge of the pile dip for a short time into a bath which can be placed below for the purpose. GRADUATION OF INSTRUMENTS. We have already considered the graduation of electro- static voltmeters: we shall now treat, very briefly, the gradu- ation of other instruments for measuring volts and amperes in practical work, and shall take as our examples partly the balances above described and partly other instruments such as ordinary non-absolute galvanometers. The graduation of these instruments may be effected in various ways ; for example, by a direct comparison of their indications with those of a standard galvanometer, such as that described POTENTIAL GALVANOMETER OR VOLTMETER. I.47 in Chap. IV., or those of an absolute dynamometer, or by such a comparison effected indirectly by keeping a con- stant current flowing through the instrument and measur- ing its amount by the electrolytic decomposition which the current produces in a given time in a voltmeter included in the current. We shall consider first the graduation by the direct comparison method of a potential galvanometer, or gal- vanometer the resistance of which is so high that the attachment of its terminals to two points in a conductor carrying a current does not perceptibly change the differ- ence of potential formerly existing between these points. Of course every absolute galvanometer measures dif- ferences of potential, for, if its resistance is known, the difference of potential between its terminals can be calculated from Ohm’s law ; but the convenience of a galvanometer especially made with a high resistance coil is that its terminals may be applied at any two points in a working circuit, and the difference of potential, thus calculated as existing between these two points while the terminals are in contact, may, in most cases, be taken as the actual difference of potential which exists between the same points when nothing but the ordinary conductor connects them. For, let V be this actual difference of potential in volts, let r ohms be the equivalent resistance of the whole circuit between the two points and R ohms the resistance of the galvanometer. Then (p. 92) by the application of R, V is diminished in the ratio of R to A + r, and therefore the difference of potential between the ends of the coil is now VA/(R+ r). Hence by Ohm's law we have for the current through the galvanometer the value V/A (1 + 7/A). If r be only a small fraction of Å, r/K is inappreciable, and the difference of potential L 2 I48 GRADUATION OF INSTRUMENTS. calculated from the equation C= V/R will be nearly enough the true value. So far, it is to be observed, r is the equivalent resistance between the two points, and the result stated holds, however the electromotive force may have its seat in the circuit, if only ſº be great in compari- Son with r. If, however, either of the two parts of the circuit between the two points in question have a resist- ance r, then as shown at p. 94 above, if only ſº be great in comparison with r the value of the difference of potential between the terminals of r is practically un- changed by the addition of R as a derived circuit. The instrument to be graduated is first tested as to the adjustment of its coil, needle, &c. We shall suppose in the first place that the current through its coil is proportional to the tangent of the deflection, according to the Conditions stated above (Chap. IV.), as necessary for a tangent galvanometer. The standard galvano- meter and it are then properly set up with their needles pointing to zero, and their coils in the magnetic meridian, in positions near which there is no iron, and at which the values of H have been determined. The high re- sistance coil of the standard galvanometer and the coil of the potential instrument are joined in series with a constant battery of as many Daniell’s cells as gives a deflection of about 45° on the standard galvanometer, or a conveniently readable deflection if a mirror and scale are used, and a deflection of the needle also of nearly 45° on the potential instrument. If, however, the instrument, as often happens, is very much more sensitive than the Standard, the method next described (p. 149) may be employed. The current actually flowing in the circuit is calculated by equation (II) or (I2) from the reading obtained on the standard, and reduced to amperes by VOLTMETER. I49 multiplying the result by I.O. Care of course must be taken that there is no leakage in the circuit which might cause the current flowing through the standard galvano- meter to be different from that flowing through the other instrument. The difference of potential between the two ends of the coil of the potential galvanometer is found in volts by multiplying the number of amperes thus found by the resistance of the coil in ohms. From this is found the number of divisions of deflection which cor- responds to one volt between the terminals when the needle is in any other field of known intensity. We can then obtain, by an obvious calculation, the number of divi- sions of deflection which corresponds to one volt between the two ends of the coil, and thence from the value of H the number of divisions which would correspond to one volt if the intensity of the field were one C.G.S. unit. For example, let the deflection be 40 divisions for 20 volts, and let the value of H at the instrument which is being graduated be 17. The number of divisions of deflection which would correspond to one volt for that position, if the field were of unit intensity, would be 48 × 17 = 34. Hence if the instrument be placed in a field where the horizontal intensity is ‘I85 the number of divisions of deflection which would correspond to one volt would be '34/ 185 = I 838. Another method sometimes convenient is as follows. The standard instrument, a few good Daniell's cells, and a resistance which makes the deflection about 45° on the standard, are joined in series, and the galvanometer to be graduated is applied at two points in the circuit which include between them such a portion of the resistance as gives a deflection of about the same amount. Let ſº ohms be the portion of the resistance included between I 50 GRADUATION OF INSTRUMENTS. the terminals of the galvanometer, and let G ohms be the resistance of the galvanometer coil. Let the current cal- culated from the deflection on the standard be Camperes, then if V be the difference of potential in volts between the terminals of the potential instrument, we have by Ohm’s law— where R' is the resistance equivalent to the divided circuit of R and G. But (p. 87) R = RG/(A + G), and therefore c p & H (7. /ø G Hence, A G V = C –- . . . . . . R -- G (5) This last equation gives the number of volts indicated by the deflection on the potential instrument, for the field in which its needle is placed ; and from this, in precisely the same manner as described above, the number of volts for any other field or position of the needle may be found. If the scale of the instrument does not follow the tan- gent law, it is necessary to determine by direct experi- ment the electromotive force corresponding to different deflections and thus, so to speak, calibrate the instrument. To do this the most convenient plan is to divide the scale accurately into equal divisions and to number these from zero at the position of equilibrium with no current. Then the current measured by the standard galvanometer is varied conveniently by introducing resistance into the STANDARD DANIELL. I 5 I current by a rheostat, and the deflection observed for several different values. The corresponding differences of potential are then plotted on Squared paper as ordinates for which the numbers of divisions of the deflections are the corresponding abscissae. A curve is then carefully drawn through the extremities of these ordinates and the ordinate of this curve drawn for any chosen abscissa will be the difference of potential for that deflection. The reason for using a deflection of 45° in these experi- ments is this, that a given absolute error in reading the deflection gives at that deflection a minimum percentage error in the estimation of the current. For verifying the accuracy of the graduation of the potential instruments when performed by either of these methods, a standard Daniell’s cell of the form proposed by Sir William Thomson at the Southampton meeting of the British Association, or a standard Clark's cell of the form, and prepared with the precautions, recommended by Lord Rayleigh (see p. I 53 below), may be used. The Daniell’s cell is represented in the annexed cut (Fig. 29). It consists of a zinc plate at the bottom of the vessel resting in a stratum of Saturated zinc sulphate, on which has been poured, so gently as to give a clear sur- face of separation, a stratum of half saturated sulphate of copper solution, in which is immersed a horizontal plate of copper. The copper-sulphate solution is introduced by means of the glass tube shown in the diagram dipping down into the liquid, and terminating in a fine point, which is bent into a horizontal direction so as to deliver the liquid with as little disturbance as possible. This tube is connected by a piece of india-rubber tubing with a funnel, by the raising or lowering of which the sulphate of copper can be run into or run out of the cell. By this means the I 52 4 ºf GRADUATION OF INSTRUMENTS. sulphate of copper is run in when the cell is to be used, and at once removed when the cell is no longer wanted. Daniell's cells show considerable variation of electro- motive force with different solutions, and therefore the liquids should not be twice used. The solutions should be carefully prepared, and kept in stock bottles till wanted. In all cases careful determinations of the ==...:# --- --- º-- º - |. .- #-- - i-hº ſº T ſº ;: =#|É ##! º §§ { º º * le-sº ---. F. : electromotive force should be made from time to time. The electromotive force of this cell has been determined very carefully and found, according to Lord Rayleigh's latest determination of the ohm, to be I'O72 volt at ordin- ary temperatures. The temperature at which it has been employed has usually been about 12°C. The cell is so used (p. 155) that a feeble current flows through it, and a feeble current should be passed for a little time just before CLARK’s STANDARD CELL. I 53 to freshly coat the copper plate. The variation of electro- motive force with temperature has not been determined, but has been found not to cause any perceptible error for the small changes of temperature to which the cell was exposed. The direct application of this cell to the galvanometer gives at Once a check on the graduation. If the resistance of the galvanometer is always over, say, 6,000 ohms, there is practically no polarization. The behaviour of Clark's standard cell has been studied with great care by Lord Rayleigh. It may be made in a reliable and handy form in the following way, which in- cludes the precautions that Lord Rayleigh's experience has shown to be necessary. The vessel is a weighing tube, or for small sizes merely a test-tube, with a platinum wire sealed through the bottom, and rests on a suitable stand as shown (Fig. 30). This wire makes contact with mercury, which occupies the bottom of the cell and forms one of the plates. The mercury must be pure, and it is desirable to ensure its being so by redistilling in vacuo good clean commercial mercury. On the mercury rests a paste made by adding to I 50 grammes of mercurous sul- phate 5 grammes of zinc carbonate, and Sufficient Saturated zinc sulphate solution to give a stiff pasty consistency. The zinc sulphate solution should be made from pure zinc sulphate dissolved under gentle heat in distilled water so as to make a Saturated Solution, and, after having been allowed to stand for some time to precipitate any iron which may have been present in the Sulphate, filtered in a warm place into a stock bottle. When required the solution is gently warmed, and drawn off by a siphon from just above the crystals at the bottom. The paste is made by placing the mercurous Sulphate and zinc carbonate in a mortar and rubbing in the zinc sulphate at I 54 GRADUATION OF INSTRUMENTS, intervals during two or three days, to give time for all Carbonic acid to pass off. - A rod of what is called “redistilled zinc” resting in the paste, and held upright in the vessel by a notched ring of cork, forms the other plate. The zinc is cleaned before putting it in position by dipping it in Sulphuric acid and then washing it in distilled water. Connection with it is made by a gutta-percha-covered copper wire soldered to it, and passed up through a cork which closes the cell and nearly fills the upper part of it so that very little air is included. The cork is flush with the top of the tube, USE OF STANDARD CELL. I 55. and the edges of the tube and the whole upper surface of the cork is covered with marine glue to Seal up the cell. A cell thus made, if used with only the very feeblest currents, never short-circuited, nor exposed to great variations of temperature, will have a constant electro- motive force E in true volts at temperature & C., given according to Lord Rayleigh's determination by the equation A. = I-435 (I – ‘OOO77 (£ — I 5). The standard Daniell’s cell is very conveniently used along with a Daniell's battery in the manner represented in the diagram, Fig. 31. C is the Standard cell, and B a battery of from 30 to 40 Small gravity Daniells.” A circuit is formed of a resistance box, the galvanometer G to be graduated, and the battery B joined in series with the standard cell C. A sensitive galvanometer Z), which may be a reflecting galvanometer, or any very sensitive galvanometer of low resistance, has one terminal attached at a point M between the battery and the standard cell, and the other terminal through the key K to an interme- diate terminal Z of the resistance box. The resistances in the box, on the two sides of Z, are adjusted until no current flows through the galvanometer D, when the key K is depressed. Let R be the resistance in the box to the right of Z, r * These can be very easily made by using large preserve-pots as containing vessels, and placing at the bottom of each a copper disc of from three to three and a half inches in diameter, in a stratum of saturated copper sulphate solution, and a grating or plate of zinc a little below the mouth of the vessel immersed in a solution of zinc sulphate, of density 1-2. The copper sulphate may he kept saturated by crystals dropped into a glass tube passing down through a hole in the zinc plate to the copper. A copper wire well covered with gutta percha should be used as the electrode of the copper plate. I 56 GRADUATION OF INSTRUMENTS. the resistance of the cell C, and G the resistance of the galvanometer. Then if V be the difference of potential, in volts, between the terminals of the galvanometer, (6) In practice a resistance of from 300 to 400 ohms is gener- ally required for ſº. The electromotive force of the stand- ard cell is taken as I'O72, as, notwithstanding the large C |||||--|-- FIG, 31. battery in the circuit, the total resistance is so great that there is very little polarization. This method in fact is peculiarly well adapted for the Daniell's cell, as the slight current flowing through serves to keep its plates in a constant and fresh State. The difference of potential, the magnitude of which is thus obtained, is chosen such as to give a convenient de- flection on the instrument to be graduated. Another method of using the standard cell for determin- ing the difference of potential given at the terminals of the galvanometer G by the battery /3, and one better adapted for a standard like Clark's through which it is not desir- CURRENT-METERS. I 57 able to send even a slight current for any length of time, is obtained by placing R between Z and M in Fig. 31, K and D in the position occupied by Æ, and reversing the cell C. A.' is adjusted until no current flows through D when the key A is tapped down for an instant. When this is the case the electromotive force of C is balanced by the difference of potential at the two ends of ſº pro- duced by B. Hence the difference of potential in volts then existing between the terminals of G is given (for a Clark's cell at 15° C.) by the equation, G V = I ‘435 – . . . . . . . 1435 ; (7) By this method, which is an application of Poggen- dorff's method of comparing the electromotive forces of batteries, balance is obtained when no current is flowing through the standard cell, and disturbance from polariza- tion is altogether avoided. It has been found very easy and convenient in practice. The methods first described (pp. 147–151) are of course applicable to the graduation of current galvanometers or balances. The standard galvanometer, of which in this case the low resistance coil is used, and the current gal- vanometer to be graduated are joined in series with a battery, which with some resistance in circuit is sufficient to give a deflection in each of about 45°, if the indicators are pointers, or as large as possible a deflection in each case as can be conveniently obtained within the ranges of the instruments if their utmost range is less than 45°. The current flowing in amperes is given by the standard, and this, of course, is the number of amperes which is indicated by the deflection of the current instrument. By an obvious calculation from the value of H at the current I 58 GRADUATION OF INSTRUMENTS. instrument, precisely similar to that above described, the number of divisions of deflection corresponding to a current of one ampere for a field of unit strength is found, and from this the deflection corresponding to one ampere with any given field. In some instruments the field at the needle of the gal- vanometer is varied in order to obtain greater or less sen- sibility by superimposing on the earth's field a horizontal field due to a permanent magnet of hard steel (as in the magnetostatic current-meter described at p. 112 above), or to an electromagnet. The value of the field intensity given by this magnet at the needle of the galvanometer when in position, may be determined in the following manner. A battery of about 30 of Sir William Thomson's Tray Daniells is joined in series with a resistance, R, of about 7,000 ohms. The electrodes of a potential galvano- Imeter,” in which the needle is acted on only by the earth's field, H, are attached at two such points in this resistance that the deflection of the needle produced is from 30 to 4o divisions on the Scale. The current through the galvanometer is now stopped and the magnet placed in position, so that the index of the galvanometer is again at zero. The electrodes are now placed so as to include a resistance which makes the deflection nearly what it was in the former case. Let Z be the electromotive force, and B the resistance of the battery, I the total horizontal intensity of the magnetic field at the needle when the magnet is in position ; R1, R2 the resistance included between the electrodes of the galvanometer in * This, according to convenience, may be the instrument being tested, or another which allows the magnet to be placed in the same position relatively to the needle. - * This and the other methods described below are applicable to the compari- son of the sensibilities of the magnetostatic current-meter for any two positions of its field magnets. - ARTIFICIAL FIELD AT NEEDLE. I 59 the first and second cases respectively; V, and l’, the potential difference in volts on the instrument in the same two cases; D1 and D2 the corresponding deflections, and G the resistance of the galvanometer. By Ohm’s law E R, G P = (ATA-ºw. Hay Tºa - "A Pl, A. Zºo G l’, 2 = ??? / D2, (B-ER – R.) (W,+G)=F78, G where m is a constant. Therefore we have D, R, {(B+ R – R.) (R, + G) + R, G} D, R, {(B+R – R.) (R,+ G)+ R, G} M = H (8) If the resistance B of the battery be small in comparison with G, or if the galvanometer be sensitive enough to allow B/G to be made sufficiently small by resistance added to G, AE may be neglected ; and it is generally possible, by properly choosing ſº, ſº, R2, to simplify very much this formula. The number / thus found, diminished by H, is the number of C.G.S. units which measures the horizontal intensity of the magnetic field produced at the needle by the action of the magnet alone. The value of Z– H obtained should be carefully verified from time to time, and the value after such determination marked on the magnet. - The known difference of potential obtained by com- paring a Daniell's battery with the standard cell by the methods described on pp. I 55—I 57, may be used to give a deflection with the magnet in position, and this deflection compared with that obtained at the same position of the galvanometer, with the earth’s force alone, and the Standard cell directly applied to the instrument. From this the ratio of / to H can be at once obtained. I6o GRADUATION OF INSTRUMENTS. The value of Z may be found at any place and for any magnet by means of a potential galvanometer, as follows.” Set up the instrument and apply a difference of potential of V volts (measured by comparison with a standard cell as described in p. 156) to the terminals of the coil. Let the deflection produced be D Scale-divisions, and let n be the number of divisions (de- termined as already explained) which would be given by one volt if the field were of unit strength ; then, Y= m V/D. Ga G6 FIG. 32. Denoting the electromotive force of the standard cell in volts by E, we have as in p. 157, V- AEG/AE, and therefore = ?? FG//DA’. The intensity of an artificial magnetic field at the needle of a galvanometer may also be determined as follows. A circuit is arranged containing as shown in Fig. 32 a battery of about 30 Daniell’s elements and * This method is very convenient for testing the constancy of the field magnets from time to time. Such tests must of course be made with a reliable standard cell, and at a place remote from magnets and iron. ARTIFICIAL FIELD AT NEEDLE. I6 I two resistances /e, A’ of about IOO ohms and 600 ohms respectively. Two potential galvanometers, Ga., Gº, are connected to the points A, A' of the circuit; of these Ga has the artificial field at the needle which it is proposed to measure, the other has a magnetic field of convenient intensity, and we shall suppose that with the arrangement made a deflection of from 30 to 4o divisions of the Scale is obtained on each instrument. Calling Da the deflection of the galvanometer Ga, and D6 that of Gº, then as the two instruments measure the same difference of potential V, V = A/Da = BD, . . . . . (9) in which Y is the field intensity at the needle of Ga, and A, B are constants. The artificial field is removed from Ga., the index of Ga brought to zero, and its terminals placed on the Smaller resistance A', while G3 is left as it was. The deflections now produced, ZX2, Z)', are read off. If now V' is the difference of potential between the ter- minals of Gº, that between the terminals of Ga has the value 17, AG 17. —“tºº--—º-a – AHD., (to R+*- 1 +...(1 +.) At —– G A \* (; where G represents the resistance of the galvanometer Ga. This gives us the relation V’ = A H D'a | I + #( + £)} = B'D' (II) M I 62 GRADUATION OF INSTRUMENTS Equations (9) and (II) give Pa Po ſ, L &(, , Æ) = AA T * * * a- - - * * = , ( I 2 J. ## 1 + ( t ) (12) I – H gives in C.G.S. units the intensity of the artificial magnetic field at the needle. When the instrument has been graduated for a field of intensity equal to I C. G.S. unit, and the intensity of the field given by the magnet at the needle has been deter- mined, the graduation is complete. In the practical use of the instrument with the magnet in position, the number of volts, or the number of amperes (according as the instrument used is a potential or a current galvanometer) corresponding to a deflection of any number of divisions is found by the following rule :- Mu////y the number of divisions in the deſ/ec/ion by the 27atensity of ſ/he magnet’s ſield increased by the horizonta/ intensity of the earth's field,” and divide by the number of divisions in the de/lection which corresponds to one volt 27, a ſield of one C.G.S. unit ºntensity. When the magnet is not used the rule is the same as the above, except that the divisor to be used is the value of H for the place of the galvanometer. - The graduation of the current galvanometer may also be performed by means of electrolysis. The electro- chemical equivalents of a large number of the metals have been determined, and it is only necessary therefore, in order to graduate the instrument, to join it in series with a proper electrolytic cell and a constant battery, and to compare the amount of metal deposited on the negative plate of the cell with the total quantity of electricity which flows through the circuit in a certain time. * The mean value of H for Great Britain may, when the magnet is used, be taken with sufficient accuracy as '17 C.G.S. IBY ELECTROLYSIS. 163 The two best electrolytes for this purpose are solutions, of proper strength, of nitrate of silver and of sulphate of copper. With the former solution electrodes of silver, with the latter solution electrodes of copper are used. When proper precautions are taken very reliable and accurate results can be obtained with either electrolyte. These precautions relate to the preparation of the Solu- tions, the adaptation of the size of the plates to the strength of the current passing through the voltameter, and the washing and drying of the plates before being weighed.* • A form of cell very convenient for use with both solutions when the current strength is not greater than Io amperes, is shown in Fig. 33. It consists of three * parallel plates of pure silver, or pure copper, suspended from spring clips in a glass vessel containing the proper solution. This form of cell has the advantages of giving light plates, which facilitate the accurate weighing of the amount of loss or gain of metal, and of allowing, when silver is used, and the size of the plates is properly pro- portioned, the loss from the anode to be used as a check in estimating the gain on the kathode. There is of course the objection which attends the use of vertical plates, that the solution becomes less dense near the kathode, but the only practical effect due to this has been found to be a slightly greater thickness of deposit in the lower part of the plate due to the greater density there. Lord Rayleigh has used as voltameter a platinum bowl as kathode, and a silver plate as anode. This cell, though it has several advantages, is more difficult to * See a paper on the ‘Electrolysis of Silver and Copper, T. Gray, Phil. AZag. Oct. 1886, from which the details here given are mostly taken. See also a paper by A. W. Meikle, Electrica! Engineer, Mar. 23, 1888. M 2 I64 GRADUATION OF INSTRUMENTS manipulate than that described above. The anode must be protected to prevent silver from falling from it to the kathode below. The amount of silver deposited may be determined in two ways, both of which should be used so that one may be a check on the other: (1) the platinum bowl is previously carefully weighed, and again with the deposit adhering, (2) the silver is dissolved from the FIG. 33. platinum bowl by nitric acid, and its amount afterwards estimated. The form of clip, as illustrated in Fig. 34, almost explains itself. It is made of stiff platinoid or brass wire. A piece is taken of the proper length, bent into a close loop at the middle, then each half wound two or three times round a rod of metal to form 'springs as shown, and the two ends bent round to meet side by side, BY ELECTROLYSIS. I65 and there soldered to a stiff back-piece of brass. The springs when soldered in position should cause the loop to press firmly against the back-piece so as to form a firm clip. The stems of the two outer clips when in position are connected by a cross-piece a of copper. Both are insulated from the inner clip by a block of vulcanite through which its stem passes. This whole arrangement E. # FIG. 34. of Cross-piece and insulating block is fixed on the top & of the wooden framing shown in Fig. 33. The two plates attached to the outer clips form the anode of the electrolytic cell, and the plate between them the kathode. The kathode thus gains on both sides, and as it is safer to use the gain than the loss of metal in estimating the current, the weight of the plate itself is thus made as small as possible in comparison with the alteration in weight to be determined. I66 GRADUATION OF INSTRUMENTS Since the form of cell shown in Fig. 33 was arrived at it has been improved by the substitution for the cover & of a rectangle of wood, well soaked in paraffin or var- nished, which carries on one side the clips for the anode, and at the middle of the opposite side the single clip for the kathode. g When currents of over IO amperes are to be used the form of cell shown in Figs. 35, 36 is preferable. An insu- lating rim rests on the top of the Cell, which for the larger \;==sº FIG. 35. FIG. 36. sizes is conveniently made of earthenware and of rect- angular shape. A groove in the rim fits the top of the cell loosely so that the rim with its attachments can be easily removed and cleaned. To the rim are fixed on opposite sides two sets of spring clips, each made as shown in Fig. 37, by soldering flat strips of springy metal to a stiff base-piece which can be screwed to the in- sulating rim of the cell. To make the effective area of the plates as great as possible in comparison with the ineffective part, the part above the liquid is cut away to BY ELECTROLYSIS. 167 two narrow strips connecting the lower part to an upper cross bar c, d. One end c of this cross-bar rests in a clip, the other in a notch in the insulating rim. Anode plates and kathode plates alternate with one another, and there is one more of anodes than of kathodes, SO that each kathodc is between two anodes. In large cells where the plates are close and liable to touch, they are kept apart by two Ú-shaped glass tubes hung over each alternate plate. With regard to the size and preparation of plates experience has shown that in the cases of both silver and copper there is a certain density of current (current strength per unit of area of plate) which gives the most adherent and in the case of silver most finely crystalline deposit. When silver is used there is a tendency, if the plate be too large or two small, for the Crystals of deposited silver to grow out branch-like from one plate to the other, an effect which is most marked where there is a sharp edge or corner. Hence the plates must have their edges and corners rounded off to prevent the formation of these “trees,” which cause great risk of loss of silver from the plate in its treatment before being weighed. The best deposit has been found to be obtained with a Solution made with five parts by weight of nitrate of silver to 95 of water, and a kathode plate giving not more than 600 sq. cms. nor less than 200 sq., cms. of active face to the ampere of current. If a stronger solution be used, the density of current may be somewhat increased, but the strength should not be less than 4 per cent. nor greater than Io per cent. The anode plates should be considerably greater in 168 GRADUATION OF INSTRUMENTS area than the kathode plates if their surface is to remain bright and moderately hard so as to admit of the plates being weighed if necessary. The density of the current for them should be less than one ampere to 400 sq. Cms. If the anodes are of rolled sheet silver the surface skin should be polished off with fine silver sand, and the plate washed in distilled water before being used ; as otherwise the silver would be dissolved away from under the skin, which would hang as a loose sheet ready to break away when the plate was moved. A plate of silve: becomes soft and inelastic by repeated use as an anode, owing to solvent action going on below the . Surface, and to remedy this should be heated after being used each time to a red heat in the flame of a spirit lamp. The following mode of treating silver plates has been found very successful. The plate cut from the new sheet has its corners first rounded and smoothed, then is polished with fine silver sand in water, rubbed on with a soft clean pad of cloth, so as to remove the skin above referred to, and still leave a smooth surface. A gentle stream of clean water is then run over the surface from a tap to remove the sand, next the plate is washed, first with clean soap and water, then with water alone, then immersed for a few minutes in a boiling solution of cyanide of potassium, and finally washed thoroughly in a stream of clean water. The plate is dried in a current of hot air, for example before a clear fire; and great care must be taken in handling it after it has been cleaned not to touch it with the fingers, otherwise the parts which have been in contact with the skin will re- ceive no deposit. Of course the plate must be allowed to cool before it is weighed to obviate risk of disturbance from air currents in the balance case. BY ELECTROLYSIS. I69 When the silver deposit is to be washed and weighed, the plates are gently removed by easing the Springs to prevent risk of rubbing off metal by the friction of the clips, then dipped gently in clean, recently distilled water contained in a glass vessel, so that any Small crystals which may fall from the plate may be detected. The adherent nitrate Solution is thus to a great extent re- moved ; and the plates are then laid in the bottom of a shallow glass tray containing clean distilled water, and washed by gently tilting one side then the other of the tray so as to make the water flow gently over their surfaces. Then they are washed in a second tray in the same way, and allowed to Soak for a quarter of an hour before being dried. To dry the plates one corner is laid on a pad of blotting-paper and the greater part of the water drained off. The plate is then dried by holding the upper end in a spirit flame. The electrolysis of copper sulphate with copper anode and kathode gives results which for very high accuracy in standardizing are but little if any inferior to those obtained with silver : for most practical purposes results quite accurate enough can be obtained with much less experimental skill on the part of the operator. Repeated experiments made in the Physical Laboratory of the University of Glasgow, in connection with the graduation of Sir W. Thomson's standard instruments,” have shown that under certain easily fulfilled conditions the method of standardizing by the electrolysis of copper sulphate is accurate and trustworthy. * See the Ref. p. 163 above. The remarkable concordance of standard- izings made at different times is illustrated by results quoted in Mr. Meikle's paper, - I7o GRADUATION OF INSTRUMENTS The size of plates is not of so great importance as in the case of silver, but the kathode plate for the best results in long-Continued electrolyses should have about 50 cms. of active surface or upwards per ampere. When the current is of Small density deposits are obtained which are much more solid and adherent than those of silver, and there- fore much more easily dealt with. As in the case of silver the anode should be of much greater area than the opposed surface of the kathode. With a density of current of upwards of 35 of an ampere per sq. cm. the resistance at the anode becomes variable and very con- siderable, sometimes almost stopping the current, which after a little, with evolution of gas at the anode, regains nearly its former strength. It was found by Prof. T. Gray in the experiments above referred to that satisfactory and concordant re- sults could be obtained with a solution of any ordinary pure commercial copper Sulphate made with pure water, provided the density did not fall below I'os, and the solutions were made slightly more acid than in their normal state. An addition for example of Hº per cent. of sulphuric acid to different solutions, which gave results differing among themselves, brought them into complete accordance. The loss of weight which is well known to take place when a copper plate is left Standing in a copper sulphate solution, was also carefully in- vestigated. This loss it was found seldom exceeds gººd of a milligramme per sq. cm. per hour, or about sºon of that which would be deposited by a current of one ampere per 50 sq. cms. When the current density is smaller than this the loss is nearly the same as when no current flows. The effect seemed to have a minimum for a density of solution between I'Io and I’I 5, and seemed for BY ELECTROLYSIS. 171 this density to be rather retarded than the reverse by the addition of a small percentage of free acid. The kathode plate having been cut and rounded at the corners is polished with silver sand in the same manner as the silver plate. It is then placed in the cell and a thin coating of copper deposited over it, while the current (if a large current is to be used), is adjusted to its proper strength by placing resistance in the circuit. The plate is then removed, washed in clean water and dried before a clear fire without being sensibly heated. Any defect in the first cleaning will be shown by the deposit, and if no Such defect is shown, the plate is weighed and replaced in the cell for the continuation of the electrolysis. If feeble Currents are to be used this preliminary adjustment is hardly necessary, as it is preferable then to use a larger number of cells than are absolutely necessary to produce the current, and bring down the current to the necessary Strength by adding an amount of resistance which can be easily enough estimated. After the electrolysis the plates are carefully removed and at once dipped in ordinary (not necessarily distilled) Clean water, Containing two or three drops of sulphuric acid per litre, then washed in a tray like the silver plates, The plates are then rinsed in clean water without acid, and dried first in a clean pad of white blotting paper, and then before a fire or over a spirit lamp. If this is carefully done and the deposit be fairly good no copper will be lost and there will be no gain of weight by oxidation. The plates may be weighed after having been allowed to Cool down to the ordinary temperature. The anode plates are treated in a similar manner (except as regards the drying in a blotting-pad, which might cause loss of silver) without loss of copper, or gain 172 GRADUATION OF INSTRUMENTS by oxidation, but owing to loss of weight in the solution, &c., they give much less satisfactory results than do the kathode plates. # É Éà • O ; ; ; The arrangement of the circuit for electrolytic experi- ments consists of a battery of tray Daniells, or other BY ELECTROLYSIS, I73 constant cells, joined in series with the electrolytic cells to be used, a sensitive galvanometer, and a rheostat (or other readily variable resistance) by which the current is to be regulated. The current is adjusted so that a con- venient deflection is obtained, which is restored by slightly turning the rheostat in the proper direction, if any alteration takes place. The conduct of an experiment will be understood from the description of the process of Standardizing given below. Lord Rayleigh has carried out the electrolysis of silver nitrate with very great care, and measured the electro- chemical equivalent of the metal, that is the amount of silver deposited per absolute unit of electricity passed through the voltameter. He found that a coulomb deposits very approximately ‘OOI I I 79 gramme of silver.” From this result Professor T. Gray has experimentally determined by comparison the electro-chemical equivalent of copper, and found it to be very approximately ‘OOO3287 (or for practical purposes ‘OOO3290) at ordinary tempera- tures, and with a current density of one ampere per 50 Sq. CIms, of active surface of kathode. This number can be corrected for other current densities by the dotted curve given in Fig. 38. The results from which this curve has been plotted are given in the Table on the following page. The effect of variation of temperature f on the amount of copper deposited has been found by Mr. A. W. Meikle to be very slight at Ordinary temperatures; for a change from 12°C, to 28°C. it is a diminution for a given size of plate of only a per cent. At temperatures rising above 20° C. the effect of varia- tion of size of plate becomes more and more important. * Phil. Trams. Pt. II. 1884. f See Ref. p. 163 above. I 74 GRADUATION OF INSTRUMENTS AMOUNTS OF COPPER DEPOSITED BY THE SAME QUAN- TITY OF ELECTRICITY ON KATHODE PLATES OF DIFFERENT AREAS. Area of plate Amount of deposit Amount of deposit * l II]. r ln Sq. Cms. (firſt tºn). Gecºmen) 3 '3534 '3533 5 '353O '3529 II '3528 ‘353O 18.5 '3526 '3527 36 3524. '352 I 73 I 35O3 *35O2 The application of electrolysis to the standardizing of instruments will now be illustrated by a short account of its application to the determination of the proper weights for use in the current balances described above. The arrangement of apparatus is shown in Fig. 39 which may be taken as a plan of the standardizing table with instruments in position. C C C C C C are six of the Electric Power Storage Co's secondary cells, shown joined in series, by being connected to a series of mer- cury cups m, m, . . . which are connected across by thick copper rods as indicated by the full and dotted lines. (These cups are on a vulcanite base, and have bottoms of thick copper to ensure contact.) When however currents of great strength are required for the graduation of low resistance instruments, these cups are joined in parallel by two rods of copper which have teeth at the proper distance apart to fit into the cups, so as to join all in each row together. The battery fully charged and thus joined in parallel will maintain a current of 200 amperes for IO hours, BY ELECTROLYSIS. I75 The terminal cups of the commutating board are shown joined to a distributing board provided with cups, 1, 2, ... 12, by which the battery is put in series with a rheostat R, in parallel arc with a set of conductance bars D, a galvanometer G, a pair of large electrolytic cells P ; ; } }}} \}} ºn Yū M ; ºr E. C n fºom º *** E] C F- C *-ºs- { || || joined by a movable cup M, and finally the balance B to be standardized. The conductance bars are con- structed as shown in Fig. 40. Rods of platinoid of thick- mess according to the conductance required are bent into U-shape as shown, and the limbs held at proper distances 176 GRADUATION OF INSTRUMENTS apart by wooden blocks at intervals, or by a strip of wood running along their whole length, according as the rods are thick or thin. The length of rod in each U is about 4 metres, and the thickness is chosen such that one or two volts difference of potential produces very little heating of the wire. The troughs, tt (Fig. 39), are made with bottoms of thick copper and contain mercury in which the ends of the rods (or thick Copper pieces soldered to the wires if thin) rest pressed down by their own weight. The different Us beginning from one side are graduated so as to have conductances nearly in the ratios I : I : 2 : 4, &c. so that the total conductance in the set may be increased at will by a step equal to the lowest conduct- ance (since each conductance is that amount greater than the sum of all that precede it in the series). When BY ELECTROLYSIS. 177 any bar is not in use its lower ends are lifted out of the troughs as shown in the Figure. The rheostat, which has a least conductance rather less than that of the smallest bar, furnishes an auxiliary variable bar by which the conductance can be gradually altered. Its wire is of Stranded copper and can carry Io annperes without damage. The current balance has previously had its scale graduated and attached as described above, and it re- mains only to show how the constant of the instrument is determined, or in other words the weight which placed on the beam will enable the current to be obtained from its indications in the manner already described (p. 102). A chosen arbitrary counterpoise weight is placed in the trough, and another, which then just brings the beam to the sighted position without current when at the zero of the scale, is placed on the beam with the index at some division near the right-hand end so that a current of, say, Io amperes (more or less according to the instrument) is required to bring the beam to the sighted position. The electrolytic cells are then arranged to give about 500 sq., cms, of kathode surface, and are joined up with a conductance sufficient to give nearly the required current. The balance will come nearly to zero, and is brought to zero exactly by adjusting the current by means of the rheostat. These adjustments having been made, the kathode plates are removed, washed, weighed, and replaced. At an instant observed on an accurate time- keeper the circuit is closed, and any deviation of the current corrected by means of the rheostat. The current is brought to its correct value in from five to ten seconds, and hence in an electrolysis of say an hour (the usual duration of an experiment) the error due to its deviation N 178 GRADUATION OF INSTRUMENTS from the final constant value for this short variable period is quite imperceptible. Any variations of the current strength are observed on the instrument itself, or if (which rarely happens) that is not sensitive enough, on a more Sensitive galvanometer G (Fig. 39), which is introduced when required, and kept out of circuit at other times. Any Sufficiently sensitive instrument which can have its (not necessarily known) constant changed by any required amount by varying the field at the needle, or by using an instrument provided with two parallel coils with the needle midway between them, and arranged to permit the distance of the coils apart to be altered at pleasure, is convenient for this purpose. The electrolysis having thus been carried on and com- pleted, the circuit is broken, and the plates washed and weighed. The current is calculated from the result by dividing the gain of copper on the kathode expressed in grammes, by the electro-chemical equivalent of Copper (OOO3287, or, as explained above, the proper value for the density of current), and the result by the number of seconds during which the electrolysis has lasted. Let C be the current for the position of the weight on the beam as given by the table of doubled Square roots, 701, wg, the corresponding counterpoise and beam weights respectively, C' the current given by the electrolysis, wi, wº, the counterpoise weight and beam weight applied, then we have - cº -: wid, +- wed, C’” w'idi + z0/2d, where dº, d, are constants. But wiſwa = 'w'1/w's ; hence this equation gives -- BY ELECTROLYSIS. I79 Thus wi, w, are found by multiplying the ratio C*/C’” by w', we respectively, and the determination is complete. When a very strong or a very weak current is required, as in the graduation of a hektoampere or a centiampere. balance, it is desirable in the former case to allow the whole current to flow through the instrument, and only a convenient part through the electrolytic cell, and in the latter case to use a considerably greater current through the electrolytic cell than through the instrument. The current must therefore be divided in both these cases into two parts whose ratio is accurately known, and this may FIG. 41. be done by the conductance bridge shown in Fig. 4I. A set of parallel straight wires of platinoid are each soldered at one end to a thick terminal bar of copper Ö, and have soldered to them at the other ends thick terminal pieces of copper by which they can be connected in two groups by means of mercury troughs 51, 59. In the Figure they are shown in two groups of IO and I respectively. The wires are adjusted so that when they are in position they have all precisely the same resistance. (The method by which this equality is ensured is described in Chap.VIII. below.) Between the troughs 51, 52 a sensitive reflecting galvanometer (see p. 182 below) gis joined which indicates N 2 I 8o GRADUATION By ELECTROLysis. no current when Ö1, Ö, are at the same potential. The electrolytic cells E, AE', and the instrument G to be standardized, are placed as shown in the figure when the standardizing current must be greater than that which the cells can carry, and the positions shown are inter- changed when the reverse is the case. The currents are adjusted to balance in both cases by the rheostats R, A". The currents are of course in the ratio of the conduct- ances of the groups r, r' of the wires of the bridge, CHAPTER VIII. THE COMPARISON OF RESISTANCES. E give here some account of methods for the com- parison of the resistances of conductors in which steady currents are kept flowing. In most cases the con- ductor to be compared is arranged in a particular way in connection with other conductors, which are then ad- justed so as to render the current through a certain conductor of the system zero. From the known relation of the resistances of the other conductors the required comparison is deduced. tº The form of galvanometer generally employed in the measurement of resistances is the reflecting galvanometer invented by Sir William Thomson, one arrangement of which is shown in Fig. 42. For most purposes the ordin- ary form of the instrument can be used. In this a mirror of silvered glass to which the needle-magnets are cemented at the back is hung within a cylindrical cell about half a centimetre in diameter. The ends of the cylinder are closed by glass plates from four to five millimetres apart, held in brass rings which can be screwed out or in so as to increase or diminish the length of the cell. The mirror is hung by a piece of a single silk fibre passed through a small hole in the cylindrical I 82 COMPARISON OF RESISTANCES. surface of the chamber and fixed there with a little shellac. The mirror is only of slightly smaller diameter than the cylinder in which it hangs, so that in this rºſ º . i. * : : # º º º | - tº º: ||||||| | arrangement the fibre is very short, rendering it necessary in cases in which deflections have to be read off to allow for the effects of torsion. The cylindrical chamber is º ——é-ºtº s-ºl. MIRROR GALVANOMETER. 183 screwed into one end of a cylinder of slightly greater diameter which fits the hollow core of the coil, and is called the galvanometer-plug. When the plug is in posi- tion the mirror hangs freely within its cell, with therefore the point of suspension on the highest generating line of the cylinder. Deflections of the needle are observed either by the Poggendorff telescope method, or, and much more generally, by the ordinary projection method described on pp. 7 and 8 above. The weight of the needle and mirror is under one grain, and hence the period of free vibration of the sus- pended system about any position of equilibrium is short. The needle is also made to come quickly to rest by the Smallness of the chamber in which it hangs. Since the mirror nearly fills the whole cross-section of the cell, the air damps the motion of the mirror to a very great ex- tent even when the cell has its largest volume. The mirror may be made quite “dead-beat,” that is, to come to rest without oscillation, by screwing in the front and back of the cell until the space is sufficiently limited. In instruments in which it is desirable to avoid effects of torsion the galvanometer coil is made in two lengths which are fixed end to end, with a narrow space between them to receive the suspension piece. This piece forms a chamber in which the needle hangs between the two halves of the coil, and gives a length of fibre which at shortest is equal to the radius of the outer case of the coil, and which can obviously be made as long as is desired. The part of the hollow core at the needle is closed in front and at back by glass plates carried by brass rings. These can be screwed in or out by a key from without So as to diminish or increase the size of the chamber, and I84 COMPARISON OF RESISTANCES. thus render the needle system more or less nearly “dead- beat.” w We shall suppose the galvanometer set up so that the deflections are read by the ordinary deflection method. It is only necessary to arrange that the needles when no current is flowing in the wires shall hang parallel to the plane of the coils. This is done as follows. A straight thin knitting wire of steel is magnetized and hung by a single silk fibre of a foot or so in length. This can easily be done by taking a sufficiently long single fibre of silk and forming a double loop on one end by doubling twice and knotting. In this double loop, made widely divergent, the steel wire is laid horizontally, and the single end of the fibre is attached to a support carried by a convenient stand, which is then placed so that the wire takes up a position in the direction of the horizontal component of the magnetic field where the needle is to be placed. A line can now be drawn parallel to the wire on the table beneath it. All that is necessary then is to place the galvanometer so that the front and back planes of the coil are vertical and parallel to this line, and adjust the lamp and scale as described above. i It is sufficient for our present purpose to state that if the needles be so small as in the Thomson reflecting gal- vanometer, and torsion can be neglected, the current in the coil may be taken as proportional to the tangent of the deflection angle, and therefore if that angle be not greater than three or four degrees the current may, with an error not greater than } per cent, be taken as propor- tional to the deflection simply. The galvanometer should be made as sensitive as pos- sible by diminishing the directive force on the needle as RESISTANCE COILS. 185 far as is practicable without rendering the needle unstable. This is easily done by placing magnets near the Coil SO that the needle hangs, when the current in the coil is zero, in a very weak magnetic field. That the field has been weakened by any change in disposition of the mag- nets, made in the course of the adjustment, will be shown by a lengthening of the period of free vibration of the needle when deflected for an instant by a magnet and allowed to return to zero. The limit of instability has been reached when the position of the spot of light for zero current changes from place to place on the Scale, and the intensity of the field must then be slightly raised to make the zero position of the needle one of stable equilibrium. Although not absolutely essential, except when accurate readings of deflections are required, it is always well when the field is produced by magnets, to arrange them so that the field at the needle is nearly uniform. It may therefore be produced by two or more long magnets placed parallel to one another at a little distance apart symmetrically with respect to the centre of the needle above or below it, and with their like poles turned in the same directions; or a long magnet placed horizontally with its centre over the needle, and mounted on a vertical rod so that it can be slided up or down to give the required sensibility, may be used. Sensibility is sometimes obtained by the use of astatic galvanometers, but these are rarely necessary and are more troublesome to use than the ordinary non-astatic instrument. - For the comparison of the resistances of conductors other resistances the relations of which are known are employed. These are generally coils of insulated wire I 86 COMPARISON OF RESISTANCES. wound on bobbins which are arranged so that the coils can be used conveniently in any desired combination. }| ºE. \ * 3. S$ iºf |º -§*=: sº-- w # º § º . %: : wº § º º-3; º i º M * §Wº: : ºk=| º - -g | * §ºf t§º§ #iš :}Ö i ſ N ;y; º| ltºº- Such an arrangement of coils is called a resistance box. Figs. 43 and 44 show resistance boxes of different forms. RESISTANCE COILS. 187 In a resistance box each coil has a separate core, which ought to be a brass or copper cylinder split longitu- dinally to prevent induction currents, and covered with thin rubber or varnished paper for insulation. These cores are shown in Fig. 45. The metallic core facilitates the cooling of the coil if an appreciable rise of temperature is produced by the passage of a current through it. After each layer of the coil has been wound it is dipped Gºº.º.) . ‘. NS - im:#3 § | | º ſ Nº N: Zºº; sº º }_fth | | || Nºl, imº sº º liºl||Nº||m|| Wilmillièſ'ſ |Sº |º sº §º i º[IIIlliſD ºum ; ſº sº ſº III | à º * ! º Eº | sº- liſ FIG. 44. in melted paraffin, so as to fix the spires relatively to one another, preserve them from damp, and insure better insulation. It is of great importance to use perfectly pure paraffin, and especially to make sure that no sulphuric acid is present in it. Unless this precaution is observed trouble may be caused not only by the action of the acid on the metal of the conductor, but by the polarization effects due to electrolytic action in the acid paraffin. I 88 COMPARISON OF RESISTANCES. Paraffin which is at all doubtful should be well shaken up when melted with hot water to remove the acid. The wire chosen for the higher resistances is generally an alloy of one part platinum to two parts silver. This has a high Specific resistance (p. 238 below) combined with a small variation of resistance with temperature. For the lower resistances wire of greater thickness is employed on account of its greater conductivity, which enables a greater length of wire to be used and thus facilitates accurate adjustment. Coils are now sometimes made of “platinoid,” a species of German silver which does not tarnish seriously with exposure to the air and has a low variation of resistance with temperature (see Table V.). When a coil of given resistance is to be wound, a length of well-insulated wire of slightly greater resistance (deter- mined by comparison at ordinary temperature by one of the processes to be described) is cut, doubled on itself at its middle point, and wound thus double on its core. This is done to avoid the effects of induction (see p. 206) when the current is in a state of variation, as when start- ing or stopping. After the coil has been wound its resist- ance is again measured, and if good insulation has been obtained, it ought now to show a slightly increased resistance, on account of the change produced in the wire by bending. The coil is fixed in position by two long brass or copper Screws d, d, Fig. 45 passing through ebonite discs in the ends of its core, which fasten it to the cover of the box. These should be sufficiently massive to give no appreciable resistance. These Screws are attached to two adjacent brass pieces, a, a, on the outside of the cover, and have the ends of the wire of the coil soldered to them so that the coil bridges across the gap RESISTANCE COILS. 189 shown in the figure between every adjacent pair of brass pieces. The coil is now brought to the temperature at which it is to be accurate and finally adjusted so that its resistance taken between the brass pieces is the required resistance. Coils are made in multiples of the “Ohm’’ or practical unit of resistance. The ohm is defined absolutely in jº I r—I t I § I º | = | | I º || 2 || 2 || 2 || || 2 || ~ * o! d | FIG. 45. Chap. V. above: and it is there stated that the Legal O/// as adopted by the International Congress of Electricians held at Paris in 1884 is equal to the resistance of a uniform column of pure mercury IO6 centimetres long and one square millimetre in cross-section, at the tem- perature o' C. Different forms in which copies of such a standard are made are described below, p. 243. A series of coils are arranged in a resistance box in ,, . " I90 COMPARISON OF RESISTANCES. Some convenient order either in series or in multiple arc. Fig. 46 shows a series arrangement suitable for many purposes. The numbers indicate the number of ohms in the corresponding coils. The space between each pair of blocks is narrow above and widens out below, as shown in Fig. 45, to increase the effective A. N à-#–#* e = <= 2O (O (3 O /OO 200 * . O ſooo 2.Ooo 300a 4000/000 & a () a. - #: F distance along the vulcanite from block to block. In the adjacent ends of the brass pieces, between which is the narrow gap, are cut two narrow opposite grooves, so as to form a slightly conical vertical Socket. This fits a slightly conical plug, f in Fig. 45, which when inserted bridges over the gap by making direct contact between RESISTANCE COILS. I91 the blocks, and when not thus in use is held in a hole drilled in the middle of the upper surface of the block. The coil is short-circuited when the plug is inserted, that is a current sent from one block to the other passes almost entirely across the plug on account of the much greater resistance of the coil. The handle, f, of the plug is generally made of ebonite. The plan of arranging a series resistance box which is most economical of coils is a geometrical progression with common ratio 2. In such a box two units are generally provided to enable the box to be conveniently tested. The inconvenience of the arrangement is in the reduction of any resistance which it is proposed to unplug in the box to its expression in the binary scale of notation. For example if the resistance 370 is to be found on the box, this is expressed as 28 + 29 + 2* +2* +2 or IOI I IOOIO, and the corresponding plugs inserted, namely the first, fourth, fifth, sixth, seventh, and eighth plugs beyond the units. The process of reduction is performed as follows by dividing successively by 2, and writing the remainders as successive figures of the number from right to left in the order in which they are obtained, ending with the last quotient, which is of course I. 2 |37O 185 92 46 23 I I º I92 COMPARISON OF RESISTANCES. Hence 370 = IOI I Iooſſo in the binary scale. It is not however always necessary to go through this process. Practice with a box on this principle leads soon to readiness in deciding what coils are to be unplugged, or what is the resistance of any set of coils which may be unplugged. It is well to remember that any coil of the series is greater by unity than the sum of all the pre- ceding coils of the series. The coils form a geometrical series from I to 4096 with a common ratio 2. The unit is duplicated for the reason stated above. The “Dial" form of series resistance box shown in Fig. 43 above, is preferable to the ordinary forms for many purposes. It contains three or four or more sets of equal coils, each nine in number. One set consists of nine units, the next of nine tens, the next of nine hundreds, and so on. Besides these the box sometimes contains a set of nine coils each a tenth of a unit. Fig. 47 is a plan of a five-dial box. The sets of coils are arranged along the box in order of magnitude. Each set is arranged in series, and the blocks to which the extremities of the coils are attached are arranged in circular order round a central block, which can be connected to any one of the ten blocks of the set surrounding it, by inserting a plug in a socket provided for the purpose. Each central block, except the first and last, is connected by a thick copper bar inside to the initial block of the succeeding series of nine coils, as shown in Fig. 47 by the dotted lines. The ten blocks of each set of coils are numbered O, I, 2, . . . 9, as shown. Thus a current passing to one of the central blocks passes across through the bar to the next series of coils, then through the coils until it reaches a block con- nected to the central piece by a plug, when it passes across DIAL RESISTANCE BOX. I93 to the centre and then to the next series of coils. If no coil of a series is to be put in circuit, the plug joins the central block to the coil marked zero. In a five-dial box the cen- tral blocks are marked re- spectively TENTHS, UNITS, TENS, HUNDREDS, THOU- SANDS, and the resistances are read off decimally at once. Thus supposing the centre in the first dial to be connected to the block marked 5, in the second dial to the block marked 7, in the third to that marked 6, the resistance put in circuit is 67.5 units. The advantage of the arrangement consists in the fact that only one plug is re- quired in each dial whatever the resistance may be, and since the plugs when no coils are included complete the circuit through the zeros, there is always the same number of plug contacts in circuit, instead of a variable number as in the ordinary arrangement. I94 COMPARISON OF RESISTANCES. Besides the dial resistances there is generally in each box a set of resistances arranged in the Ordinary way, and comprising two tens, two hundreds, two thousands, and sometimes two ten-thousands, fitted with terminals to allow the box to be conveniently used as a Wheatstone Bridge, as described below. The extremities of this series of resistances can be connected by means of thick copper straps with the series of dial resistances. Each pair of equal coils are sometimes wound on one bobbin to ensure equality of temperature. It is sometimes desirable to have a ready means of varying the ratio of two resistances, or of increasing a single resistance by steps of any required amount. For this purpose a resistance slide is a convenient arrange- ment. A form devised by Sir William Thomson is shown at C/D in Fig. 48. Along a metallic bar r in front of a series of equal resistance coils slides a contact piece s by which r is put in conducting contact with any one of the series of brass or copper blocks by which the coils are connected. The figure shows a combination of two slides used by Sir William Thomson and Mr. C. F. Varley for cable testing. Each resistance in AB is five times that of each coil in CD, and there is the same number in each, so that the whole resistance of C/D is twice that of each coil in A.B. The slider, S, of AB consists of two contact pieces insulated from one another on the slider, and at such a distance apart as to embrace two coils. The terminals of C/D are connected to CC as shown in the figure, and therefore in whatever ratio the resistance CZ) is divided by the contact piece s, in that ratio is the joint resistance of the two coils CC divided. CO thus forms a vernier for A.B. In the arrangement figured the resist- ance CD is divided into the two parts I2 and 8, and CONDUCTANCE BOXES. I95 therefore the sixth and seventh coils of 4/3 which are between the terminals of S are divided into two similarly situated parts 12 and 8. Hence the whole resistance between A and B is divided into the two parts 56 and 44. Fig. 49 shows a dial form of the double resistance slide. The main coils are on the left, the vernier coils on sEEEEEEº-EEE JC ſº %. 42, A, 4%. A 22, #. A 42 ſº 4. Q 2 iſºft.jºſ.iºffiti Q (Đ cTE---re B the right. Each slide may be detached and used inde- pendently if required. Boxes in which the coils in circuit are in multiple arc were first made at the suggestion of Sir William Thomson, and called Conductivity Boxes, because the conductance % in circuit is obtained by adding the Con- . ." The word “Conductance” as stated above (p; 87) is now widely used instead of “Conductivity,” and is also adopted in this book. O 2 I96 COMPARISON OF RESISTANCES. ductances of the coils. Fig. 50 shows the arrangement. Each coil is a resistance coil wound on a bobbin as |:ſºf ſ -| # |§º W j # | |\\|| - w | - ||| s: º | * . sº . º | | described above and has one extremity connected to a massive bar a, the other to a brass block c, outside the box, which can be connected by a plug to the massive TEMPERATURE VARIATION OF COILS. 197 bar &. The resistance in circuit is obtained at once by adding the conductances of the coils thus in circuit, and taking the reciprocal of their sum. The conductances of the coils are marked on the corresponding blocks outside the cover, This arrangement is very convenient for the measure- ment of low resistances such as one ohm and under, as it gives a long gradation of fractions by combination of the coils. Sir William Thomson has proposed to call a box arranged thus a Mho-Box, where “Mho” is the word “Ohm’ read backwards to indicate that the box gives conductances, that is reciprocals of resistances. - The resistance of almost all wires increases with rise of temperature, and the box is generally adjusted to be correct at a convenient mean temperature which is marked on the cover. The value of the resistance shown by the box at any other temperature is obtained when the change of temperature can be ascertained from the 198 COMPARISON OF RESISTANCES. known variation of resistance with temperature. A table of the variation of the resistances of different substances with temperature is given at the end of this volume. The general internal temperature can be observed by means of a thermometer passed through one of the orifices which should be left in the side of the box to allow free circulation of air. Local changes of temperature may sometimes be produced in the coils without affecting ap- preciably the general internal temperature. These changes cannot be accounted for, as it is impossible to observe them with any accuracy, but can be avoided by using only the very feeblest currents, and continuing these for the shortest possible time. The general internal temperature can also be measured by means of an auxiliary coil provided for the purpose. This is constructed of thick copper wire wound on ebon- ite, and extends along the whole length of the box. Since the variation of resistance of copper relatively to that of the wire of which the coils are constructed is known, we can by measuring the resistance of this auxiliary unit by the box itself obtain a closely approximate estimate of the internal temperature. The temperature variation may be made for all the coils the same as the highest variation for any one, by introducing into each a piece of copper (conveniently at the bight after the coil is wound) just sufficient for the purpose. In every case the blocks to which the coils are attached should be pierced with a socket for special plugs with binding terminals attached, by means of which any coil in the box may be brought into circuit itself. This is necessary for the testing of the box, which is done as CARE OF A RESISTANCE BOX. I99 follows. In the case of the ordinary arrangement of coils Figs. 44, 46, each of the units is compared with a standard unit, then the two units together are tested against each of the 2s, then the 2s and a I are attested against the 5 and so on, until the IOOS are reached. All the preceding coils put together give IOO, which can be attested against each of the IOOs, and this process is continued until the box is completely tested. The process can be checked by other possible combinations, and the whole of the results, if necessary, put together by the ordinary methods Of combination. If a dial box is to be tested the auxiliary unit, if it has one, Suffices ſor the comparison of each of the units, then the nine units and the auxiliary unit give IO for the com- parison of each of the nine tens. These when compared give with the ten units IOO for the Comparison of each of the hundreds, and so on. In the case of a box arranged in geometrical progres- sion with common ratio 2, and first term I, the unit is duplicated for the sake of comparison. Each unit having been Compared with a standard, they give together a Comparison of the next coil, which is 2, then that with the two units give 4, with which the coil of 4 units can be Compared, and So on. The actual methods of comparing coils are described below (p. 219 eſſ seq.). It is to be remembered that in the comparison of the coils of low resistance the connecting wires (which should be in all cases short and thick) must be taken into account. In the use of a set of resistance coils it is important that the plugs be kept clean, and the ebonite top of the box, especially between the blocks of brass, kept free from dust and dirt. The ebonite may be freed from 2OO COMPARISON OF RESISTANCES. grease by washing it with benzole applied sparingly by means of a brush, and a film of paraffin oil should then be spread over its surface. The plugs and their sockets may also be freed from adhering greasy films by washing in the same way with benzole or very dilute caustic potash. The latter should not however be allowed to wet the ebonite surface. If necessary the sockets may be scraped with a round-pointed scraper. On no account should the plugs or sockets be cleaned with emery or sand paper. It is frequently necessary to adjust a current to a con- venient strength by varying the amount of resistance in circuit. When the amount of resistance in circuit need not be known, this can be done most readily by means of a rheostat, or resistance coils in series with a rheostat, an arrangement which has the advantage of giving a con- tinuous variation of the resistance. A form of rheostat constructed by Sir William Thomson is shown in Fig. 51. Two metal cylinders are mounted side by side on parallel axes and are geared so as to be driven at the same rate in the same direction by a third shaft turned by a crank. Along this shaft from end to end of the cylinder is worked a screw, which when turned moves a nut along a graduated Scale at the top of the instrument. One of the cylinders is covered with well varnished paper, the other has a clean metal Surface. A bare wire of platinoid or other material is wound partly on one cylin- der partly on the other, and in passing from one cylinder to the other threads through a hole in the nut. Thus if the cylinder be turned the relative amounts of wire on the two cylinders can be varied at pleasure, and the wire is laid on them helically in a regular manner. The toothed wheel by which one of the cylinders is turned is con- RHEOSTAT. 2O I. nected with the axle by means of a spring previously wound up so as to give a couple tending to wind the wire ū ſº §§ | ==His šič sº | == fº-ES *- ------- S-E- . * # = É- :sº # s es *|- #: -- dº on the cylinder. The wire is thus kept taut and the spires prevented from shifting on the cylinder. The course of the current is along the wire on the . 2C2 COMPARISON OF RESISTANCES. paper covered cylinder, then to the bare cylinder. Thus the resistance in circuit is regulated by the amount of the wire on the former cylinder. A comparative esti- mate of this is given by the scale along which the nut In OVéS. The arrangement of screw and nut for guiding the wire above described seems to have been first used in a rheostat constructed by Mr. Jolin of Bristol. A simpler form of rheostat, first used by Jacobi, consists of a single cylinder of insulating material round which the wire is wound in a helical groove. A screw of the same pitch as the groove is cut in the axle of the cylinder, and works in a nut in one of the supports. The cylinder, when turned, moves parallel to itself, so that the wire is kept in contact with a fixed rubbing terminal. Another rubbing terminal rests on the axle, to which one end of the wire is attached. The method of comparing resistances of most general use is that known as Wheatstone's Bridge. The arrange- ment of conductors used is that shown in Fig. 52, with a battery, generally a single Daniell’s or Menotti's cell, included in 76, and a galvanometer in 7's. A much higher battery power is however sometimes required, especially in cable and other testing. The three conductors whose resistances are r1, r2, r3 are coils of a resistance box pro- vided with terminals so arranged that connections can be made at the proper places to form the bridge, for example as in Fig. 46, which shows a resistance box fitted up as a Wheatstone Bridge. It will be easy to make out in Fig. 52 the terminals corresponding to A, B, C, D respectively of Fig. IO. Fig. 44 above shows a so-called “Post-Office Resistance Box * in which the battery and galvanometer keys are mounted on the cover, WHEATSTONE'S BRIDGE. 2O3 and permanently connected to the proper points inside the box ; and Fig. 48 a Wheatstone Bridge arrangement of resistance slides. The resistance to be compared is placed in the position BD (Fig. 52), and convenient values of 71 and 7% are chosen, while 7's is varied until no current flows through the galvanometer. The value of 7, is then found by (17) of Chapter VI., which since ya is zero, may be written 2 * = .. 73 . . . . . . . (I) FIG. 52. If r1 and 7% are equal, r, is equal to rº, and is read off at once from the resistance box. In the practical use of Wheatstone's Bridge we have generally to employ a certain battery and a certain galvanometer for the measurement of a wide range of resistances; and it is possible if great accuracy is required So to choose the resistances of the bridge as to make the arrangement have maximum sensibility. An approxi- mate determination is first made of the resistance to be 2O4. COMPARISON OF RESISTANCES. measured. Call this ra. It has been shown independ- ently by Mr. Oliver Heaviside,” and by Mr. Thomas Gray, that if the battery and galvanometer are invari- able we should make 2^4 –H 7° 77.1 —H 7’s 7.1 = Wrr, n-vº. 4 + 5, r = Mrrºº6. 2.4 + 7.6 71 -H 7.5 If the resistances of the battery and galvanometer are at the disposal of the experimenter, then on the Sup- position that the resistance of the galvanometer may be taken equal to rº, the most sensitive arrangement is that in which each of the resistances is equal to 74. Unless in particular cases in which great accuracy is necessary, any convenient values of r1, r2 will give results sufficiently accurate for all practical purposes ; but in arranging the bridge with these the following rule should be observed : of the resistances 7-5, 7% of the galvanometer and battery respectively, connect the greater so as to join the junction of the two greatest of the four other resist- ances to the junction of the two least. This rule follows easily from (17) of Chap. VI. For interchanging 7's and 7"s we alter only the value of D, and calling the new value D' we get D’ – D = (rs — re) (? – ??) (rs – ?) , (2) The expression on the right will be negative if re-ra and 71, r3 be the two greatest or the two least of the other resistances. Hence on this supposition the value of D has been diminished, and therefore the current through the galvanometer for any small value of 7,73–?,”, increased by making re join the junction of r1, rs to that of 72, r3. * Phil. Mag. vol. xlv. (1873), p. 114. f Ibid. vol.xii. (1881), p. 283. For a proof of these results see the author's larger treatise, vol. i. WHEATSTONE'S BRIDGE. 2O5 In cases in which the resistances in the bridge are large, a galvanometer of high resistance should also be used. In the practical use of the method the electrodes of the battery should be carried to the terminals of a revers- ing key, so that the testing current may be sent in opposite directions if desired through the resistances of the bridge. Also a single Spring contact-key, which makes contact only when depressed, should be placed in 7%. These keys are convenient when arranged side by side, so that the Operator placing a finger on each can depress one after the other. A convenient form of wire rocker with mercury cups, combining the two keys, may be easily made by the operator. When the bridge has been set up and a test is about to be made, the single key in rs is first depressed to test whether any deflection of the galvanometer needle is produced without closing the battery circuit. If there is a deflection, this must be due either to thermoelectric action in the galvanometer cir- cuit, or to leakage from the battery to the galvanometer wires. The procedure in this case will be stated presently, If there is no deflection, the operator then opens the galvanometer circuit, depresses the key which completes the battery circuit, and immediately after, while the former key is kept down, depresses also the galvanometer key. After the circuits have been completed just long enough to enable the operator to see whether there is any deflec- tion of the needle, the keys are released so as to break the contact in the reverse order to that in which they were made. This order of opening the circuits enables him to make a second observation of deflection without its being necessary to again send a current. It is easy to imagine and construct a form of contact-making key, which being depressed a certain distance completes the 2O6 COMPARISON OF RESISTANCES. battery circuit, and on being depressed a little further completes the galvanometer circuit, and therefore on being released interrupts these circuits in the reverse order. This form of key is of use in the testing of resistance coils in which there is considerable self- induction. For general work, however, it is inconvenient, as the reverse order of making the contacts may have to be adopted. Again, in many practical operations, such as cable testing, &c., the contacts have to be made after different intervals of time in different cases. The object of thus completing and interrupting the battery circuit before that of the galvanometer is partly to avoid error from the effects of self-induction. When a current in a conducting wire is being increased or diminished, an electromotive force, the amount of which depends on the arrangement of the conductor, is called into play, so as to oppose the increase or diminution of the current. The effect of this electromotive force is to produce, therefore, a weakening of the electromotive force of a battery for a very short time after the circuit is completed, and a strengthening during the very short interval in which the current falls from its actual value to zero at the interruption of the circuit. Its value is small when the wire is doubled on itself so that the two parts lie along side by side, the current flowing out in one and back in the other; but is very considerable if the wire is wound in a helix, and still greater if the helix contains an iron core. The electromotive force of self-induction is directly proportional to the rate of variation of the current in the circuit, and thus is explained the bright spark seen when the circuit of a powerful electro-magnet is &ro/teſt. If, then, one or more of the coils of a Wheatstone WHEATSTONE'S BRIDGE. 2O7 Bridge arrangement were wound so as to have self-in- duction, the electromotive force thus called into play would, if the galvanometer circuit were completed before that of the battery, produce a sudden deflection of the galvanometer needle when the battery circuit is closed. All properly constructed resistance coils are, as has been Stated, made of wires which have been first doubled on themselves and then wound double on their bobbins, and have therefore no self-induction. The wire tested, however, and the connections of the bridge have gene- rally more or less self-induction, the effect of which, unless the contacts were made as described above, might be mistaken for those of unbalanced resistance. This mode of winding the coils also avoids direct electro- magnetic effects on the coils on the galvanometer needle when the coils are placed near it. If on depressing the galvanometer key at first as described above a current is found to be produced by thermoelectric or leakage disturbance, and the spot of light is therefore displaced, the operator keeping down the galvanometer key depresses the battery key, and observes if there is any permanent deflection of the spot of light from its displaced position during the time that the battery key is kept down. This is easily distinguished from the sudden deflection due to self-induction, as that immediately dies away to zero as the current rises to its permanent value. If the sudden deflection of the galvanometer, as it may be in the case of a dynamo or electro-magnet, is too violent and long continued, the reversing key of the battery should be used, the battery contact made first in each case, and the mean of the results taken. When comparing a resistance the operator first observes 2O8 COMPARISON OF RESISTANCES. the direction in which the mirror or needle is deflected when a value of 73 obviously too great is used, and again when a much smaller value of ra is used. If the deflections are in opposite directions, the value of r3, which would produce no deflection of the needle, lies be- tween these two values, and the operator simply narrows the limits of rs, until on depressing the galvanometer key no motion, or only a very small motion, of the needle is produced. It may happen, however, that the value of the resistance which is being compared may be between two resistances which have the smallest difference which the |, *~ l'E (l, C. al, b ſº Cºrs). /~~A A Trºy © G 3) & 3) & 3) & G) & G) 9 c e f 0 h. (SN # º 3. 3% £=#. # Sf =#–G =# §* # 3. box allows. Thus with a resistance box by which with equal values of 71 and r2 he cannot measure less than '5 of an ohm, he may either by making the ratio of r1 to 2, IO to I, or IOO to I, obtain the values of 74 to one or two places of decimals. Any inaccuracy in the relation of the arms of the bridge may be eliminated by reversing the arrangement, that is, interchanging 71 and 72, and 7"s and 74, and taking the mean of the results. Whatever be the ratio of ri to rap if he can read the deflections when first one and then the other value of ra (between which r, lies, and which differ by only to of an SLIDE-WIRE BRIDGE. 209 ohm) is used, he can find r, to another place of decimals by interpolation by proportional parts. For example, let the value I2O6 of ra produce a deflection of the spot of light of 6 divisions to the left, and I2O-5 a deflection of 14 divisions to the right: the value of ra which would produce balance is equal to I2O’5 + ‘I X 14 / (14 + 6) = I2O'57. The most accurate form of Wheatstone's Bridge is that introduced by Kirchhoff. In this an exact balance is obtained by moving a sliding contact-piece along a gradu- ated wire which joins the two resistances 7-1, 7% of Fig. 52. A diagrammatic sketch of the arrangement is shown in Fig. 53. S is the sliding-piece, A, B the wire along which it slides. A, B is stretched in front of a scale a metre in length graduated to half-millimetres and doubly numbered, from left to right and from right to left. The coils a, c, d, Ó of the diagram have the respective resist- ances 71, 73, 7:3, 7.4. Fig. 54 shows a form of the instrument manufactured by Messrs Elliott Bros. Fig. 55 shows an easily-made and cheap form of wire bridge devised by Prof. T. Gray. w, w is the wire, made of platinoid or German silver, which is stretched above, but not in contact with, a base-board, passes round the insulating and supporting vulcanite block B from the P 2 IO COMPARISON OF RESISTANCES. mercury Cup ci to the other cs. A vulcanite crossbar A clamps the wire in position near the cups. If the wire be long Several such crossbars may be used. Each end of the wire is soldered to a large mass of copper, bent, as shown in Fig. 56, so as to dip into a mercury cup with- out any risk of contact of the mercury with the soldered junction. The cups should be of copper, and may conveniently be made of the form shown in the figure, and fixed in holes in the wooden or ebonite sup- porting-block. The ends of the copper pieces dipping into them should be carefully squared and bear against the Copper bottoms. / Osei-H • {} * ºš i º) The wire is divided into parts of equal resistance by a process of calibration (p. 212 et seg., below) and marks indicating these parts are made on a rule attached to the base-board, along which the contact-piece slides. A movable scale is used to subdivide the space between two divisions. On a plate of ebonite or well-paraffined hard wood are fixed mercury cups cº, ca, ći, made as just described. The auxiliary resistances r1, 7% of the bridge when required are placed between ci and cº, ca and ca, while the wires to be compared connect cº and ca, cs and ca. Since the wire w, w can be made long, the auxiliary resistances are not frequently required. When they can be dispensed with, SLIDE-WIRE BRIDGE. 2 II c., and c, are removed and cº placed in the Socket h, and the wires to be compared are then placed between ca and 61, C3 and 65. A form of slide-wire bridge was used by Matthiessen and Hockin in the very careful comparisons of resist- ance made by them in their work as members of the British Association Committee on Electrical Standards ; and it was found by these experimenters that an alloy of 85 parts of platinum with 15 parts of iridium formed an excellent material for the graduated wire. This alloy, they found, did not readily become oxidized. Platinum- silver alloy is however frequently employed. m % S&N The contact piece is generally a well-rounded edge of Steel with a slight notch to receive the wire. The knob pressed by the operator bends a spring which presses the Contact piece with just sufficient pressure against the wire. A turning bar can be put into position to keep down the contact when desired. The sliding piece carries a vernier which enables fractions of a division to be read on the scale. The method of testing by this instrument is precisely the same as by the ordinary Wheatstone Bridge, except P 2 2 I 2 COMPARISON OF RESISTANCES. that when balance has been nearly obtained in the usual way, by varying the relation of the resistances r1, 2:2, 7.3, for a particular position of the sliding piece, an exact balance is obtained by shifting the sliding piece in the proper direction along the wire. Supposing that the resistance of the wire per unit of length has been deter- mined for different parts of the wire, and that the resist- ances of contacts have been determined (p. 223) and allowed for, the value of ra is at once found by taking into account the resistances of the segments of the wire AB on the two sides of the point, contact at which gives zero deflection. The wire A B may be “calibrated ” by one of the following methods. The first is that which was employed by Matthiessen and Hockin.* Let r and r, (a, 6 in Fig. 53) be such resistances that balance is obtained at some point P in A/3, with two coils 22, 2’, (c, d in Fig. 53) differ- ing in resistance by say #3 per cent. Let r + a be the total resistance, including contacts, between C and P, and * -- 3 that between D and P. Now alter 71 by inserting a short piece of wire. This will shift the zero point along the wire through a certain distance to the left. Balance So as to find this point, which call Pi ; then interchange 72 and 73, and balance again, and call the Second point thus found P2, Let 2 denote the resistance between P and Pi, 2' the resistance between P and P2, a the resistance of the short piece of wire added to r, and Z the length of wire be- tween Aºi and P2. We have, neglecting connections of r2,73, * + a + 3 − 2 r + 3 + 2 73 73 (3) rl -- a + a - 2' ra. -- 3 + 2 73 ?? * Reports on Electrical Standards, p. 119. CALIBRATION OF A WIRE. 2 I 3 from which we obtain for the resistance per unit of length between P1 and P2, r. ey/ - 7T T /(r. Hz) The value of a is easily obtained with sufficient accuracy from either of equations (3), as 2 is approxi- mately known from the known resistance of the whole wire. In this way the resistance per unit of length at different parts of the wire can be easily found, and, if necessary, a table of corrections formed for the different divisions of the scale. Professor Carey Foster has given the following method for the calibration of the bridge wire. The arrangement is shown diagrammatically in Fig. 57. The battery shown in Fig. 53 is removed, and two equal copper bars are attached at C, D (Fig. 53), at right angles to the bars of the bridge at those points. Between the extremities of these is stretched a second slide wire. Or the slide wire of a second bridge, from which all other connections have been removed, may be connected to C and D by wires from the end bars to which it is attached. In place of the coils c, d of Fig. 53, and the middle bar of the bridge is substituted a single Daniell’s or other cell. One terminal of the galvanometer is connected to a sliding piece on the wire W, the other to a sliding piece on the other wire, W. In place of r, and r, are Substituted two small resistances, one simply a piece of thick wire c, the other a resistance g, equal to that of a convenient portion say from 80 to IOO millimetres of the bridge wire. The former of these has been called the connector, the latter the gauge. They are connected to the bridge by mercury cups in the manner described ſ\x. (ri + r + a + 3 + 4) . (4) 2I4 COMPARISON OF RESISTANCES. on p. 2 II above, and some form of Switchboard is usually employed to effect the interchanges described below. - Supposing the gauge placed first on the left and the connector on the right, the slide on W is moved close up to the extremity B, and balance is obtained by placing the slider on W’ at some point near D. The gauge and connector are then interchanged, and balance is again obtained by shifting the slider on W’ towards the left to Some point à. - The gauge and connector are again interchanged, and balance obtained by shifting the slide on W to the left, E — H C W' g e b D N º, g º, ; \, C A W `s, B f d a FIG. 57. and so on until both wires have been traversed almost completely from end to end. The distance through which the slider is moved at each interchange of the resistance is read off, and gives, as we shall now show, a determination of the average resistance per unit of length over that portion of the wire. Let P and P' be points of contact on W and VV' when balance is obtained, let the permanent resistances included with W, W’ at the left- hand ends be denoted by a, b, and at the other ends by a', 6' respectively, the resistance of the connector by c, of the gauge by g, of the wire from A to P by 2, of the whole wire by w, of the wire W’ from C to P' by 2", and CALIBRATION OF A WIRE. 2 I 5 of the whole wire by w. If the connector be on the left and the gauge on the right, we have c + n + 3 g + 3 + w – 2 (5) at' -- 2' W-E w – 2' ' ' ' and if the gauge and connector be interchanged so that 2 receives a new value 21, 3 + 4 + 81 6 -H & H w – 31 (6) a' + 2' ô' + w( — 2' From these equations we get at Once g – c = 21 – g . . . . . . (7) that is, the steps along W have each a total resistance equal to g- c, a result evident without calculation at all. Again, Supposing the gauge at first on the left, and next on the right, the slider on W is shifted, and we get the equations a' + 2' Ö' + w' – 2 g-Ha -- 2 & H. c-Hay – 2 a' + 2', º' -- w' – 2', 3-Ha -E2 T 3 + g + zy Ta’ These give — 2/- — - a' + 5' + zºy 31 2' = (g ^a Hºi. Hºw . . (8) The quantities on the right-hand side are all con- stants, and therefore the wire VV' is thus divided into parts of equal resistance. From the known resistance of the whole wire, which can be found as shown on p. 354 below, the resistance of each part can be found. The steps on each wire are thus steps of equal resist- &ll] CC. 2I 6 COMPARISON OF RIESISTANCES. The following are the actual results obtained in the calibration of the slide-wire of a bridge performed in the Physical Laboratory of the University College of North Wales by the method just described. I’arts of the wire of cqual - Resistances of the parts .. resistance (= 7). included between the corresponding readings. &cadings (zero take e - I () /* *:::::: ſº I engths /. Readings. Resistance = / o... IO'59 IO'50 O... IO '944.29 7" 9'79... 20'35 1 o'56 IO ... 2C '94697 , , 1970.30'26 Io'56 2O... 3O • ‘94(107 , , 29'84.40'41 Io'57 3O. . . 4Q ‘946O7 , , 39'69... 50'22 Io'53 A.O... 50 ‘94907 , , 49'71 ...GO’27 Io'56 5o... 6o '94697 , 59'8o...70'35 IO’55 Óo... 7o ‘94787 , , 69'82...8o'32 I O'50 7o... 8o '95238 , , 79°86...90'38 Io'52 8o... 9o '95057 , , 89'41 ... 99'97 Io'56 90... IOO ‘94697 , , O . . . I OO 9°47873 ): 'The numbers in the right-hand column are taken from tables. The results are of course not correct to the number of decimals given. It will be noticed that the scoond reading in any line of the first column is not exactly the same as the first reading in the next line. This was caused through its being difficult to balance by adjusting the contact on the auxiliary wire. Balance was therefore obtained after a step was taken along the auxiliary wire by moving the slider through a short distance on the wire which was being calibrated. The value of r found as described below, p. 22O, was CALIBRATION OF A WIRE. 217 ‘OA52 ohm. From this the resistance of the part of the wire between two readings of the scale is found as shown in the table. A modification of this method which works well in practice and avoids some difficulties has been made by Prof. T. Gray. The two wires W. W.", are arranged parallel to one another as in I’ig. 58, and are connected at the ends A, C and /3, D by two equal small resistances of suitable amount g. The equality of these resistances can be tested with great case and delicacy by connecting the battery at A, B, and balancing with the galvanometer between a point on VV and another on PV", then trans- ferring the battery contacts to C, Z) and observing if the balance is disturbed. If it is not the resistances are equal. When the resistances have been adjusted to equality, the battery is brought into contact at A and D and balance is obtained by placing onc galvanometer terminal close to B on W, and the other at 6 on W. The battery contacts are then transferred to /3 and C, and balance is obtained by shifting the terminal of the galvanometer on W to some point a while that on 2 I 8 COMPARISON OF RESISTANCES. W' is kept at 5. The battery contact is then transferred to A, D, and balance obtained by moving the ter- minal on W’ So that the points of contact are a, e, and SO OI). The readings on the graduated scales are taken for the Successive points of contact, and divide each wire, as will be shown presently, into steps each of resistance g. The contact of the battery at A, D or B, C can be made by means of two simple rockers K, K, working between mercury cups or ordinary metal contacts, or by means of any simple key. This renders unnecessary any mercury Cup switchboard arrangement for transferring coils. Thus the method has the great advantage that the contacts are all permanent except those of the battery and the sliders, no one of which of course introduces any €IrOr. Let contact be made by the battery at A and ZD, and balance be obtained with the galvanometer at points a and e on the wires W and W", then calling as before 2, 2' the resistances of the wires between A and a, C and e respectively, and w, w the resistances of the whole wires, we have, neglecting (which will not affect the result) constant resistances of connecting bars, &c., f 70) — 2. O" 70’ — 3. * + 3 - ". . . . . . . . (9) a T 2 - g Let the battery now be transferred to B and C and balance be obtained at d and e. Denoting the resistance between A and d by 21, we again have ſ? (?? - ' — ' + g’ * : * ~ * ~ * T & . . . . (10) 21 + g’ 2. COMPARISON OF COILS WITH A STANDARD. 219 Equations (9) and (IO) give ey - 70 –– g 3 – 31 = g(1+,+). . . . (II) or the steps along W are steps of equal resistance. The same can of course be proved for VV". To avoid thermoelectric effects in such processes, the mean of the two positions of balance for opposite currents should always be adopted as the true position. The slide-wire bridge may be used for the accurate comparison of resistance coils with a standard, say for the adjustment of single ohms with a standard ohm. Fig. 53 (p. 208 above) shows the arrangement adopted. 7-1 and 7% are the resistances of the coils a, b to be com- pared, and are nearly equal. 2, and 73 are the resistances of the two coils c, d, and are each nearly equal to 71 or 74. The connections are made by mercury cups as already described. Balance is obtained with the contact-piece somewhere near the middle of the slide-wire. The coils 7.1, 74, are then interchanged and balance again obtained. By (7) above we have 71 - 7'4 - 31 - 32 & * * g in (I2) where 21, 22 are the resistances of the wire from A to the point of contact in the two cases. If p be the resistance per unit of length for the whole wire, si, so the distances (reduced, if necessary, by calibration, as shown above, to distances along a wire of uniform resistance p per unit of length) measured along the wire from A, we have 7-1 — 74 – p(S, - s2) & o º e s (I3) These results are evidently free from any uncertainty as to the resistance of the junctions of the slide-wire to the 2 2O COMPARISON OF RESISTANCES. copper bars at its ends, and from any error due to want of correspondence between the index mark on the sliding- piece and the point of contact.* If a separate experiment be made with a coil of accu- rately known resistance ri, just a very little less than that of the whole wire, and a second conductor of resistance r. So Small that it may be neglected, the value of p may be obtained from the equation 71 -- . . . . . . (14) p = If the coils compared are too unequal to allow balance to be made on the wire, a series of intermediary coils may be obtained, so as to give a gradual descent from one coil to the other. The resistance of the wire between any two readings may also be determined by the following method, which is due to Mr. D. M. Lewis. The total resistance of the wire is approximately found by measuring it with an ordinary bridge consisting of a post-office set of coils or other available form of a resistance box. Two coils are then made, the resistance of each of which is less than unity by a quantity which is nearly equal to, but not greater than, the total resistance of the wire. These can be also made by means of an ordinary resistance box. Let /Pi, A', be the as yet not accurately known resistances of these coils. Each is tested as follows in the slide wire bridge against a unit coil, a standard ohm for example. The unit coil is first placed in the position a of Fig. 53 and one of the two resistances, A'i say, is placed in the * The resistance of a coil may be accurately adjusted to any required value by first making it slightly too great, and then joining it in multiple arc with a thin wire cut so as to give as nearly as possible the required correction, If the observed resistance be 74, and that required ri, the resistance of the correcting wire is r1,r1/(71 – 7:1). RESISTANCE OF THE SLIDE WIRE. 22 I position à. The connections should be made by mercury cups as already described. In the positions marked c, d are placed permanently two coils of nearly equal resist- ance. The magnitudes of these need not be known, but should not be greater than one or two units. Balance is obtained with the slide S at a point near the end B of the slide wire, and the reading on the slide scale is taken. The coil R1 and the unit are then interchanged, and bal- ance obtained with the slide near A. The difference of the two readings gives the length of wire intercepted between them, and this must be equal in resistance to I — A'i. The other coil Ā’, is now substituted for R1 and two readings for which balance is obtained taken in the same way. These give a length of the wire the resistance of which is I – R2. The two resistances are now put together in series and tested against the unit in precisely the same way, and give between the two readings taken a length of wire of resistance R1+A2 – 1. Now from a previously made calibration of the wire the resistances of the three portions of the wire thus observed can be obtained in terms of the resistance of the cali- bration-step, and three equations are thus available for the determination of the three unknown quantities R1, Æ, and r, the resistance of the step used in calibration, as in p. 216 above. The following table gives the results of this process applied to the slide wire the calibration of which is given above. 222 COMPARISON OF RESISTANCES. Positions of the Resistances. . Resistances between these readings in tºº terms of 2. Obtained from Table, p. 216 º: above. Leſt. Right. W* R1 I I '4O 9 I31 I JPl 97'72 | [9'47875 – ‘I 3220 – 21590 = 9' 13065?) R2 I O'14 9°368 I R2 98’97 | [9'47873 – or 322 – ‘og754 = 9°367977.) R1 + R2 I 69'70 3' 625 I A’1 + R2 || 31'45 | [379058 – ‘o2843 – 13717 = 3-62498?] Here I – R1 = 9. 13 Ir, I – R2 = 9-3682: and therefore A'. -- R2 — I = 3-625?" , - " - A - 1 - Zºº A'. -H A2 – I 9' I 3 I 9-368 3:625 22° I 24 Substituting this value of 7 in the first two equations we find R1 and R2. This can be used, as shown at p. 216 above, to find the resistance of the portion of the wire between any two readings of the scale. An accurate comparison of two nearly equal resistances, for example a unit with its copy, can be obtained by mak- ing rº and 73 to be compared occupy the positions c, d, of Fig. 53. Balance is first obtained with r, and 73 in one pair of positions, then they are interchanged and balance again obtained. Assuming that the permanent resistances are included in ri, r, r2, r3, and giving 3r, 29 the same meanings as at p. 219 above, we have ^2 _ _ " + °l *, + " - º 73 74 + 29 – 2, 7-1 + 22 7" -- r + w -- 21 – 22 7", + r + w – (21 – 22) = ‘O452. COMPARISON OF COILS. 223 and therefore ^2 '3 = 2(2) - 22) tº e (I 5) 73 71 + 7.4 + 70 – (21 – 22) ^ Hence the greater ri + 7.4 the greater 21 – 22. Thus by choosing a pair of resistances as nearly equal as possible, and sufficiently great, 72 and 7.3 may be compared to any needful degree of accuracy. The permanent resistances, a, 8 Say, corresponding to the coils a, b of Fig. 53, may be estimated by the following method, by which two low resistances can be measured when the ratio of two others is accurately known. Let the resistances 7-2, r3 of c, d in Fig. 53 have the known ratio p.. We shall suppose 71 and r, to be so low resistances that, with a value of p differing considerably from unity, balance can be found on the wire. Balance is obtained with the coils in the positions c, d, shown in Fig. 53 ; then r2 and 7's are interchanged, and balance is again obtained. We have 7" -- 21 – ’a t w = **. 74 + 70 – 21 71 + 22 From these equations we obtain 21 — L2 2.1 — 27, _ _ 31 - |132 - |131 2 7-1 = **—ſ:#, r1 = — w —H tº º pu - I p — I (16) If thick copper pieces be substituted for the coils a, b of Fig. 53, their resistances, if the connections as is under- stood are made with proper mercury cups, may be taken as zero, and a and 8 are approximately given by (16). The values of a, 8 thus obtained may be used for the cor- rection of the values of r1, r, found as just described. This correction will not be appreciably affected by the 224 COMPARISON OF RESISTANCES. unknown permanent resistances corresponding to the coils c, d, if r2, r3 are taken moderately large So that the actual ratio may be taken as equal to their known ratio. Neither of the arrangements of Wheatstone's Bridge described above is at all suitable for the comparison of the resistances of short pieces of thick wire or rod, for example, specimens of the main conductors of a low resist- ance electric light installation, the resistances of which are so small as to be comparable with, if not less than, the resistances of the contacts of the different wires by which they are joined for measurement. To obtain an accurate result in such a case, we must compare, directly or in- directly, the difference of potential between two cross- sections in the rod which is being tested, with the difference of potential between two cross-sections in a standard rod, while the same current flows in both rods, in a direc- tion parallel to the axis at and everywhere between each pair of cross-sections. - Sir William Thomson has so modified Wheatstone's Bridge, by adding to it what he has called secondary conductors, as to enable it to be used with all the con- venience of the ordinary arrangement, for the accurate comparison of the resistance of a foot or two of thick copper conductor with that between two cross-sections in a standard rod. The arrangement is shown in Fig. 58. CZ) are two cross-sections, at a little distance from the ends of the conductor to be tested, and A B are two simi- lar cross-sections of the standard conductor. These rods are connected by a thick piece of metal, so that the re- sistance between B and C is very small, and the terminals of a battery of low resistance are applied at the other extremities of the rods as shown. The sections B, C are connected also by a wire BLC, and the sections A, D by a LOW RESISTANCES. 225 wire AMD, in each case by as good metallic contacts as possible. BZC and AMD may very conveniently be wires, along which sliding contact-pieces Z and M can be moved, with resistances R, R, A', ſº of half an ohm or an ohm each, inserted as shown in the figure. The sections A, D are so far from the ends of the rods, and the wires AMD, BZC are made of so great resistance (one or two ohms is enough in most cases), that the current through- out the portions of the conductors compared is parallel to 1, Batter'ſ 2– Tim A B Hº- 0. JD | | L I ITI l — | I, Galv?" `s _^ M FIG. 59. the axis, and the effect of any small resistance of contact there may be at A, B, C, D is simply to increase the effective resistance of BZC and AMD by a small fraction of the actual resistance of the wire in each case. The terminals of the galvanometer G are applied at L and //, and the circuits of the galvanometer and battery are Com- pleted through a double key as in the ordinary bridge. A reversing key is inserted in the battery circuit as in other cases, to enable the comparison to be made with both directions of current. - Q 226 COMPARISON OF RESISTANCES. Let the resistances AM, DM be denoted by r, ra; BL, CZ by a, b ; AB, CD by rº, r, ; and BC by s. Suppose r. § § Hill|ºil iſ P T :-I \-/ TV / V-2 FIG. 60. and r, to be varied by moving the sliding piece at AZ till no current flows through the galvanometer. To find the LOW RESISTANCES. - 227 relation which must hold among the resistances when this is the case, we may suppose the point Z. Connected by a bar of zero resistance, with the cross-section of E which is at the same potential as L. Call this cross-section K. The resistance of the portion of BC to the left of K is as/(a + 5), and the portion of to the right &s/(a + 5). The resistance between B and KL is therefore {a}s/(a + AE)/ {a + asſ(a + Ö)}, or asſ(a + b + s), and similarly that between C and KZ is Ós/(a + 6 + s). Hence by (I) we have (M.S. Ös ~(~ +++++) = r(r. 4 ºr ), Or (ars – bri) . . (17) rº, - ºr -, P. H. Now s has been supposed very Small in comparison with a + š, and a and 6 can be easily chosen so as to make ars — bri approximately equal to zero. Hence equation (4) reduces to 7.1 = ’87, * * * * * * (18) 7.1 the formula found above for the ordinary Wheatstone Bridge. - The apparatus illustrated in Fig. 60 is convenient for the carrying out of this method in practice. On a massive sole plate of iron, P, are mounted two vertical guide-rods of copper, A, A, and parallel to these the rods to be com- pared, viz., a standard rod C, and the rod to be tested C. C., C are supported with their lower ends in two mercury cups cut in a single block of copper. This block corresponds to the piece E in Fig. 60. The upper ends Q 2 228 COMPARISON OF RESISTANCES. of C, Ci are fixed in screw blocks of copper, t, f, to which also are attached the terminals of a constant battery B of low resistance. A reversing key K is interposed between Ž, ä and the battery. A scale D graduated along its two edges nearly fills the space between the rods C, C1, A pair of resistance coils, 7, r, are fixed to the sole plate, and have one terminal of each connected by a strip of copper, which also carries the terminal screw 7. The other terminals of these coils are fixed to two copper slides, S2, S3, which move along, but are insulated from, the guide-rods, and carry contact pieces c, c, each of which is bevelled off to a knife-edge on a level with its upper side. This knife-edge is pressed against the Corresponding rod by springs s, s, which are insulated so as not to touch the rods. The coils r, r are attached directly to the contact pieces c, c. Thus S., r 7 r Sº, corresponds to the partial circuit BAZAC of Fig. 60. Near the upper ends of C, C, is a similar arrangement of sliders, S, S, with spring contacts and attached coils, A’, A’. These coils are connected by a copper strip which carries the terminal T. The coils R, ſº are attached to the upper ends of the guide-rods A, A, and through these to the sliders S, S1. The guide rods are so thick that no appreciable change is made in the ratio of the resistances of the parts of the partial circuit SATA'S, on the two sides of T by varying the positions of the sliders. This partial circuit corresponds to AAMAZD of Fig. 60. Each pair of coils, 7", r and ſº, ſº, may be wound on a single bobbin with advantage. The arrangement is thereby rendered more compact, and there is less risk LOW RESISTANCES. 229 of error from difference of temperature between the bobbins, or of thermoelectric disturbance between their terminals. Between T and T is placed the galvanometer G, which is provided with a simple key A, placed for convenience in the actual arrangement beside the reversing key A. In the use of the instrument the rods to be compared are placed in position, and the sliders on the rod of lower resistance are placed so that their upper edges, and therefore their knife-edges, are opposite the lowest and uppermost divisions of the scale. The lower contact piece on the other rod is placed with its upper edge opposite the lowest division of the scale on that side. The upper contact piece on the same rod is then shifted until no current flows through the galvanometer. Balance is obtained for both directions of the current, and the mean position of the slider taken, to eliminate error from thermoelectric disturbance. A number of standard rods of different thicknesses are provided with the instrument in order that nearly equal ratios may be obtained over a wide range of low resistances. The following method was used for the same purpose by Messrs. Matthiessen and Hockin in their researches on alloys. A B, CD, Fig. 61, are the two rods to be compared. They are connected in circuit with two coils of resistances r, s, which have between them a graduated wire WW’’, as in Kirchhoff's bridge. SS" are two sharp knife-edges, the distance of which apart can be accurately measured, fixed in a piece of dry hard wood or vulcanite, and connected with mercury cups on its upper side. This arrangement is placed on the conductor AB, so that the knife-edges making contact include between them a length 23O COMPARISON OF RESISTANCES. SS' of the rod. T7 is a precisely similar arrangement placed on CD. One terminal of the galvanometer is applied at S, and the resistances r, s adjusted so that a point P on the wire which gives balance is found for the other terminal. The terminal of the galvanometer is shifted to S', and a second point P' found by varying the resistances of the coils from ri, s, to r", s', in such a manner as to keep the sum r+s constant. Similarly Battery | – L *= S’ T T' iſ H | I- H == | 'A | Bl—C ID S " * ºw | P º Galv Ž' FIG. 61. balance is found for TT with values rº, so, r'2, s', for the resistances of the coils, fulfilling the condition that the sum r + s is the same as in the former case. Let a, b, c, d, AE denote the resistances between Z and S, Z and S', L and T, L and T′, Z and M respectively ; a, 8, y, 6 the resistance between Wand P in the four cases, k the resist- ance of the whole wire IVIM’. We have by (1) LOW RESISTANCES. 23 I and therefore (! -: 71 + (t - - I /* A (19) where /* = 2 + 3 + k, Similarly & 71-H 3 Ž A Therefore & – a – ?' – 21 H 8 – a (20) /* - A’ º In the same way we get d = c r? – ?? -H 8 - Y . . . (2 I) /. A’ and combining the last two equations we get for the ratio of the resistances of the conductors between the pairs of knife edges— & – a r" – r + 8 – a . . . (22) d = c r', - r + 6 – y The following method of Comparing resistances is in principle the same as Thomson's Bridge with secondary conductors, and Matthiessen and Hockin's method described above, as, like them, it consists in comparing the difference of potential between two cross-sections near the ends of the conductor to be tested with the difference of potential between two cross-sections in a standard conductor, when the same uniform current is flowing in both. It is, however, more readily applicable in practice, and is very useful for a great many purposes, as for example, in the testing of the armatures or magnet coils of machines, in the estimation of the resistances of Contacts, and in the determination of the specific con- 232 COMPARISON OF RESISTANCES. ductivities of thick copper wires or rods. All that is required is a small battery, a suitable galvanometer of sufficient sensibility, and two or three resistance coils of from , ohm to I ohm. These coils may very conveniently for many purposes be made of galvanized or tinned iron wire of No. 14 or 16 B.W.G., wound round a piece of wood ; inch thick, from 8 to IO inches broad, and from 12 to 18 inches long, with notches cut in its sides, at intervals of a quarter of an inch, to keep the wire in position. To avoid any electromagnetic effect which may be produced by the coils if they happen, when carrying currents, to be placed near the galvanometer, the wire should be doubled on itself at its middle point, the bight put round a pin fixed near one end of the board, and the wire then wound double on the board, the two parts being kept far enough apart to insure insulation. Resistance coils made in this way are exceedingly useful for electric- lighting experiments, as the thickness of the wire and its exposure everywhere to the air prevent undue heating by strong currents, or, if there is much heating, obviate the risk of damage. For the battery a single cell, as for example a gravity-Daniell, or, if the battery is to be carried from place to place, two hermetically sealed chloride of silver cells, which may be joined in Series or in multiple arc as required, may very conveniently be used. As instrument of comparison Sir William Thom- son's centiampere balance used as voltmeter or his old form of graded potential galvanometer* is convenient for many practical purposes; but when very great * That is a high resistance galvanometer in which the needle system, or magnetometer, can be placed with its centre at different distances from the centre of the coil to give different degrees of sensibility, and further provided with one or more magnets to intensify the magnetic field at the needles when required. See Theory and P7 actice of Aðs. Meas. 77, ZA, and AZag., Vol. ii. LOW RESISTANCES. 233 accuracy is aimed at, as when the method is used for the measurement of the (specific) conductivity of short lengths of thick metallic wires by comparison with a standard, a sensitive reflecting galvanometer of resist- ance great in comparison with that of the conductor between the points at which the terminals are applied should be employed, and the battery should be of as low internal resistance as possible. The galvanometer is first set up and made of the requisite sensibility either by adjusting, as described in p. 185 above, the intensity of the field in which it is placed, or, if it is a graded galvanometer, by placing the magneto- meter at the position nearest the coil, and dispensing with the field-magnet. The conductor whose resistance is to be compared, and one of the coils whose resistance is known, are joined in series with the battery. It is advisable to have this circuit at a distance of a few yards from the galvanometer, so that accidental motions of the wires carrying the current may not have any sensible effect on the needle. One operator then holds the electrodes of the galvanometer so as to include between them, say, first the wire which is being tested, then the known resistance, then once more the wire being tested, in every case taking care not to include any binding screw connection, or other contact of the conductors. The known resistance should, when great accuracy is required, be so chosen that the readings obtained in these two operations are as nearly as may be equal. Let the mean of the readings for the first and third operations be V scale divisions, for the second V’; let r denote the known resistance, and a the resistance to be found. 234 COMPARISON OF RESISTANCES. Since by Ohm's law the difference of potential be- tween any two points in a homogeneous wire, forming part of a circuit in which a uniform current is flowing, is proportional to the resistance between those two points, we have V * = ºr (23) The resistance of a contact of two wires whether or not of the same metal may be found in the same manner, by placing the galvanometer electrodes so as to include the contact between them, and comparing the difference of potential on its two sides with that between the two ends of a known resistance in the same circuit. Care must however be taken in all experiments made by this method, especially when the galvanometer circuit includes conductors of different metals, to make sure that no error is caused by thermal electromotive forces. To eliminate such errors the observations should be made with the current flowing first in one direction and then in the other in the battery circuit. - The following results of some measurements of the resistance of a Siemens S D, dynamo machine, made on May 4, 1883, in the Physical Laboratory of the University of Glasgow, may serve to illustrate this method. An iron wire coil, of half an ohm resistance, was joined to one of the terminals of a standard Daniell, and short wires attached to the other terminal of the cell and the free end of the coil were made to complete the circuit through the armature, by being pressed on two diametri- cally opposite commutator bars, from which the brushes and the magnet connections had been removed. The LOW RESISTANCES. 235 electrodes of the galvanometer, which was one of Sir William Thomson’s dead-beat reflecting galvanometers of high resistance, were applied alternately to the same commutator bars, and to the chds of the half ohm, and the readings recorded. The following are the results, extracted from the Laboratory Records, of three con- secutive experiments: ExPERIMENT I. n H Reading on T)eflection of Operation. Scale. Spot of Light. Galv. zero read 2I4 Electrodes on 3 ohm 857 643 } } ,, armature 597 383 ExPERIMENT II. Galv. zero read 2I4 Electrodes on armature 607 393 5 y ) > } ohm 874 660 } } ,, armature 607 393 ExPERIMENT III. Galv. zero read 2 IA Electrodes on 3 ohm 874 66o y v ,, armature 607 393 } } ,, 3 ohm 872 658 The first experiment gives for a the value, 383 × 5/643, or ‘298 ohm. The other two experiments, although their numbers are different, give very nearly the same result, which agrees closely with a measurement made about eight months before, by the same method, with a graded potential galvanometer. In the ordinary testing of the armatures of machines by this method, the circuit of the battery may be com- pleted through the brushes ; but if the machine has been wound on the shunt System, care must be taken to previously disconnect the magnet coils. In every case 236 COMPARISON OF RESISTANCES. the galvanometer electrodes must be placed on the com- mutator bars directly. Prof. Tait * has used a differential galvanometer (see below, p. 238) for this method of determining low resist- ances. The conductors to be compared were arranged in series, so that the same current flowed through both. The terminals of one coil were then placed at two points On One Conductor, the terminals of the other coil at two points on the other, such that the galvanometer deflection was zero. The difference of potential between the points T FIG. G2. of each pair was therefore the same in the two cases. Hence the lengths of portions of the two conductors of equal resistance were obtained. The following zero method, due to Prof. T. Gray, is founded on the same principle. The arrangement of apparatus is shown in Fig. 62. One terminal of a battery of one or two low resistance cells is attached to a stud on a thick copper bar P, the other terminal to a metallic axis round which the copper bar / turns. The bar / makes contact at its outer end with a bare wire and a bare rod bent round into concentric circles with centre at the axis of the bar, and having a pair of remote extremities con- nected with mercury cups or binding terminals, and the * Trans. R. S. A., vol. xxviii. 1877–8. LOW RESISTANCES. 237 other pair of extremities free as shown. To one of these terminals is connected one end of the bar to be tested, to the other one end of the standard bar. The other end of one of these bars, say the standard, is connected to a mercury cup S, which is in line with, but is insulated from, a row of mercury cups or a mercury trough cut in a copper bar placed parallel to P. Between this bar and the trough are stretched a series of parallel wires all of the same material and length and as nearly as possible of the same resistance ; and a single wire, of the same resistance, material, and length, connects the bar P and the cup S with which the standard bar is in contact. These wires may be conveniently straight rods of platinoid, an eighth of an inch in diameter, and six feet long, soldered at one end to the bar P, and at the other to stout well-amalgamated copper terminals dipping into the mercury cups or trough. The wires may be made of the same resistance by means of a slide wire bridge, or by the method described below. The cup S and the terminals T are now brought to one potential by turning the bar / round on the circular wire until a sensitive galvanometer, ſ, joining them shows no deflection. This galvanometer is then left connected, and by means of a second sensitive galvanometer, g, two pairs of points a, d and c, d are found between which in each case no current flows when they are connected by a wire. Each pair of points are therefore at the same potential. Hence if we denote by r, the resistance of the standard hetween 5 and d, by r, that of the other rod between a and c, and by 7, the number of wires joining P and T, we have * = + . . . . . . . (24) 238 COMPARISON OF RESISTANCES. A differential galvanometer* with two independent pairs of terminals may be employed for this method. One coil may be made to join a, b, the other c, d, or one Coil may be made to join b, d, and the other a, c. In the former case either the effect on each coil must be made zero, or care must be taken to connect the terminals to a, Ö and c, d so that the magnetic effects of the two coils at the needle may be opposed. The resistance of the galvano- meter coils, except when the current in each coil is made zero, must be so great as not to cause any sensible alteration of the potentials at the points at which the terminals are applied. The wires joining P to S and T may be tested for equality as follows. Two nearly equal wires are made to join P to S and P to T, and h is placed so that the galva- nometer f shows zero current. The wire joining P to T is then removed and another put in its place. If the current in f still remain zero for the same position of /, the latter wire and the former are of the same resistance. If not the necessary correction is made (see footnote p. 22O) and the comparison repeated. • In order that the conducting powers of different sub- stances may be compared with one another, it is neces- sary to determine their specific resistances, that is, the resistance in each case of a wire of a certain specified length and cross-sectional area. We shall here define the specific resistance of any substance at any given temperature as the resistance between two opposite faces of a centimetre cube of the material at that temperature. Î * For an account of this instrument and its use in the measurement of resistance see Maxwell's Electricity and Magnetism, Vol. i., or the author's Theory and Practice of Absolute //easurements, &c., Vol. ii. f The reciprocal of this (called below the specific conductivity) may be advantageously called the electric conductivity of the substance, if the word conductivity be set free by the general adoption of the word conductance for the reciprocal of the resistance of a given conductor. SPECIFIC RESISTANCE. . 239 This resistance has been very carefully determined for several different substances at ordinary temperatures by various experimenters, and a table of results is given below (Table V.). To measure the specific resistance of a piece of thin wire, we have simply to determine the resistance of a sufficiently long piece of the wire by the ordinary Wheat- stone Bridge method described above, and from the result to calculate the specific resistance. Let the length of the wire be / cms., its cross-section s Square Cns., and its resistance Æ ohms. Then the specific resistance of the material would be Rs/Z ohms. The length / is to be carefully determined by an accurately graduated measur- ing-rod ; and the area s may be found with sufficient accuracy in most cases by direct measurement, by means of a decimal wire gauge measuring to a hundredth of a millimetre. If, however, the wire be very thin, the cross- Section may, if the density is known, be accurately obtained in square cms. by finding the weight in grammes of a sufficiently long piece of the wire (from which the insulating covering, if any, has been carefully removed), and dividing the weight by the product of the length and the density. Very thin wires are generally covered with silk or cotton, which may very easily be removed, without injury to the wire, by making the wire into a coil, and gently heating it in a dilute solution of caustic soda or potash. The coating must not in any case be removed by Scraping. If the density is not known, it may be found by weigh- ing the wire in air and in water by the methods described in books on hydrostatics. All the weights, from I gramme upwards, ordinarily used in weighing are made of brass, and hence when conductors of nearly the same specific 24O COMPARISON OF RESISTANCES. gravity as brass are weighed in air, the correction for buoyancy may be neglected. The weighing in water however must be corrected both for expansion of water with rise of temperature and for the weight of air dis- placed by the weights. For a temperature of 13°C. these corrections are as follows:—for expansion of water an increase of loss of weight in water of ‘oj9 per cent.; for buoyancy of air a diminution of apparent weight in water of about OI43 per cent. Care should be taken in weighing to prevent air bubbles from adhering to the sides of the specimen ; and the water used for weighing should first have been carefully boiled to expel the air contained in it. All error of this kind may be avoided by boiling the water with the specimen in it, and then allowing both to cool together. • If the wire be thick, and a sufficient length of it to render possible an accurate measurement of its resistance by the ordinary bridge method is not conveniently available, one of the methods of comparing small resistances described above (p. 224 et seq.) is to be used. The most convenient in many cases is that described in p. 231–236, in which the resistance between two cross-sections of the bar to be tested is compared with that between two cross-sections of a standard rod of pure copper. The cross-sections should, if the distance between them be not thereby made too small, be chosen so as to make the two resistances nearly equal. If we put V for the number of divisions of deflection on the Scale of the potential galvanometer, when the electrodes of the galvanometer are applied to the standard rod, at cross-sections / cms. apart; V' that when they are applied to the rod being tested, at cross-sections / cms, apart, then we have for the ratio of the resistance of unit length of the wire tested SPECIFIC RESISTANCE. 24 I to the resistance of unit length of the Standard at the temperature at which the comparison is made, the value V’//V/. If s and sº be the respective cross-sectional areas, which in this case are easily determinable by measurement with a screw-gauge, and we assume that the temperature at which the measurements of resistance are made is o' C., we get for the ratio of the specific resistances at o' C. the value V’ls'/V/s, and therefore also for the ratio of their specific conductivities V/s/ V’ls. This last ratio multiplied by IOO gives the percentage conductivity at of C. of the substance as compared with that of pure copper. If, as will generally be the case, the temperature at which the experiments are made be above the freezing-point, the value of IOO V/s/V'/s', may be taken as the percentage of the specific conductivity of pure copper at the observed temperature possessed by the substance, and this, if the wire tested is a specimen of nearly pure copper, will be nearly enough the same at all ordinary temperatures. If in experiments by this method the electrodes are applied by hand to the conductors, the operator should, especially if the electrodes and the conductors tested are of different materials, be careful not to handle the wires, but should hold them by two pieces of wood in strips of paper passed several times round the wires, or by some other substance which conducts heat badly, so that no thermal electromotive force may be introduced into the circuit of the galvanometer (see above, p. 205). He may conveniently make the galvanometer contacts by means of two knife edges fixed in a piece of wood which can be lifted from one conductor to the other without its being necessary to handle the galvanometer wires in any way. This will besides render any measurement of the length R 242 COMPARISON OF RESISTANCES. of the conductor intercepted between the galvanometer electrodes unnecessary, as Z is equal to l'. We have then for the percentage specific conductivity of the substance the value Ioo Vs/V's'. As an example of this method we may take the follow- ing results of a measurement (made in the Physical Laboratory of the University of Glasgow) of the specific conductivity of a short piece of thick copper strip. The specimen was joined in Series with a piece of copper wire of No. o B.W.G. of very high conductivity, in the circuit of a Daniell’s cell of low resistance. The electrodes of a high resistance reflecting galvanometer applied at two points 700 cms. apart in the copper wire gave a deflection of I 53°5 divisions, when applied at two points 500 cms. apart in the strip 270 divisions. The weight of the wire per metre was 443 grammes, of the strip per metre I 86.3 grammes. Hence the specific conductivity of the copper strip was 96.6 per cent. of that of the wire against which it was tested. The accurate realization of a standard ohm, as defined on p. 72 above, involves the determination of the specific resistance of mercury, an operation which requires great care and considerable experimental skill. This determi- nation has been made by several experimenters, among others by Lord Rayleigh and Mrs. Sidgwick and by Messrs. Glazebrook and Fitzpatrick at Cambridge, and by Messrs. Hutchinson and Wilkes at Baltimore.* The value obtained by Lord Rayleigh and Mrs. Sidg- wick for the resistance at O’ of a column of mercury * An account of Lord Rayleigh and Mrs. H. Sidgwick's determination will be found in Phil. Trans. R.S., Part I., 1883, or the author's Theory and Practice, &c., vol. i. For the later determinations of Messrs. Glazebrook and Fitzpatrick and Messrs. Hutchinson and Wilkes see Phil. Trans. R.S., 1888, and Phil. Mag. July, 1889. CONSTRUCTION OF STANDARD OHMS, 243 I metre long and I square millimetre in Cross-section was '954.12 B.A. unit (p. 71 above). Messrs. Glazebrook and Fitzpatrick's value for the same resistance is 95.352 B.A. unit, Messrs. Hutchinson and Wilkes found it to be '95341 B.A. unit. Previous measurements made by Werner Siemens and Matthiessen gave '9536 B.A. unit and '96 I9 B.A. unit respectively for this resistance. It will be noticed that the mean value for this resistance given by the three later measurements quoted lies between these, but much nearer to the former. Messrs. Siemens Brothers for a long time used the resistance of a column of mercury specified as above as the unit of resistance, and standard units were issued by them to experimenters. One of these examined by Lord Rayleigh gave '95365 B.A. unit for its resistance at the temperature 16.7° at which it was stated to be correct. Standard ohms have been made in mercury, by using tubes bent so that the requisite length is obtained in a compact form, but they are not very convenient in use, and are of course liable to breakage. A copy of the standard ohm can however be easily made when the resistance of a column of mercury of definite cross-section and length has been accurately found. Figs. 63 and 64 show such copies. Fig. 63 is the usual form of the standard. It is made of platinum-silver wire, wound within the lower cylinder. The space within up to the top of the wider cylinder is filled with paraffin wax. The ends of the coil are attached to two thick electrodes of copper rod, bent as shown and kept in position by a vulcanite clamp. The ends of these when the coil is used are placed in mercury cups in the manner already explained, and should always, before the coil is placed in position, be freshly amalgamated with mercury. The lower cylinder R 2 244 COMPARISON OF RESISTANCES. up to the shoulder is placed in water when the coil is in use, and the temperature of the water is ascertained by means of a thermometer in the hollow core of the cylinder. The variation of the resistance of the coil with temperature is known, and hence its resistance at any observed tem- perature can be obtained. Of course care must be taken not to expose the standard to too strong currents, and to keep the temperature as near as possible to the || || &# # m |||}== º In º it. | | | | | | Hº!'ſ ſ normal temperature at which the standard is given as correct. Fig. 64 shows a form of the standard constructed by Messrs. Elliott and Co. according to a suggestion made by Professor Chrystal. A thermoelectric couple, of which one junction is within and close to the coil, and the other outside the case, is used to determine the temperature of the coil. In the form in which the instrument is now made the external junction is not brought out through FIG. G3. CONSTRUCTION OF STANDARD OHMS. 245 the bottom of the case as shown, but the wire is brought out at the top of the case, and then joined to a wire of the other metal which is entirely outside and attached to one of the binding screws. The external junction is of course placed in water the temperature of which is measured, and the thermal current is observed by means of a galvanometer connected to the terminals. This gives the difference of temperatures between the junctions and therefore the temperature of the coil. tiº ºis wº i. i. º Hº-º: º: | | º | FIG. 64. On account of the uncertainty of the temperature of the coil, and its liability to loss of insulation by deposition of moisture on the upper surface of the cylinder, Prof. J. A. Fleming * has constructed a standard in which the case containing the coil is a hollow circular ring of brass made by screwing together by projecting flanges two square sectioned circular troughs. The electrodes (rods arranged as in Fig. 63) proceed to the ring through two upright brass tubes from 5 to 6 inches in length, from which they are insulated by vulcanite collars at the bottom, and at the top by two vulcanite funnels corrugated on the outside, * Phil. Mag. Jan. 1889. 246 COMPARISON OF RESISTANCES. and projecting above the tubes. Paraffin oil placed in these vulcanite funnels prevents loss of insulation by condensation of moisture on the insulating pieces. The measurement of a very high resistance such as that of a piece of insulating material cannot be effected by means of Wheatstone's Bridge, and recourse must be had in most cases to electrostatic methods in which the required resistance is deduced from the rate of loss of charge of a condenser, the plates of which are connected by the substance in question. If, however, the resistance of the material be not too great, and a large well-insulated battery of from IOO to 200 cells, and a very sensitive high resistance galvanometer are available, the following method is the most convenient. First join the galvano- meter, also well insulated, and the resistance to be measured (prepared as described below, p. 25, to prevent leakage) in series with as many cells as gives a readable deflection, which call D. Now join the battery in series with the galvanometer alone, and reduce the sensibility of the instrument to a suitable degree by joining its ter- minals by a wire of known resistance, and, to keep the total resistance in circuit great in comparison with the resistance of the battery, insert resistance in the circuit. Let E and B denote respectively the electromotive force and resistance of the whole battery, G the resistance of the galvanometer, S the resistance joining its terminals in the second case, R the resistance introduced into the circuit of the galvanometer in that case, and X the resist- ance to be found ; we have for the difference of potential between the terminals of the galvanometer in the first case the value, A. G. Giºr y = ** . . . . (25) |HIGH RESISTANCES. 247 where m is the factor by which the indications of the galvanometer must be multiplied to reduce them to volts. In the second case the resistance between the galvano- meter terminals is SG/(S+ G), and therefore the difference of potential between them is, SG S+ G A.S.G = 7/2/) (26 2.I.I.S.T (5 + 0 (s- G) + so m D (26) S –– G Hence combining equations (25) and (26) So as to eliminate E and in, and solving for X, we get, D (B + R)G\ x - #(a + k + G+***) (B + G) . (27) If X be great in comparison with the remainder of the resistance in circuit the term (B + G) may be neglected. This method was used by Mr. T. Gray and the author for the determination of the specific resistances of different kinds of glass. The specimens of glass were in the form of thin, nearly spherical flasks about 7 cms. in diameter, with long narrow and thick walled necks. The thin walls of the flask were brought into circuit by filling it up to the neck with mercury, and sinking it to the same level in a bath of mercury, then joining one terminal of the battery to the external mercury by a wire passed down the long neck, and the other to the mercury in the bath without. This mercury bath was an iron vessel contained in a sand- bath which could be heated to any required temperature. A well-insulated galvanometer (constructed by aid of a grant from the Government Research Fund to a special 248 COMPARISON OF RESISTANCES. design º of high resistance and great sensitiveness was included in the current. A battery of over Ioo Daniell’s cells was used, and after a reading of the galvanometer in one direction had been taken and recorded, with the Corresponding temperature of the glass, the coatings of the flask were connected together until the next reading was about to be taken. For this the current was reversed, and the deflection taken as before, and so on. The “electric absorption ” was thus reversed between every pair of readings, and lasted in most cases about three minutes. The resistances were therefore those existing after three minutes' electrification. The result for the glass of highest insulation tested, which was lead glass of density 3: I4, was a specific resistance at IOO’C. of about 84OO × Io'0 ohms. The resistance was halved for each 8'5° or 9° rise of temperature. A modification of this method for which a potential galvanometer or voltmeter is very suitable, may be used for the determination of the insulation resistance of the conductors in an electric light installation. The conductors are disconnected from the generator and both ends from one another. They are then joined at one end by the potential galvanometer in series with a battery of as many cells as gives a readable deflec- tion. The number of divisions corresponding to this deflection is read off, and the number of divisions which the battery gives when applied to the galvanometer alone is then observed. Call the latter number V and the former V’; and let E divisions be the total electro- motive force of the battery. Let the resistance of the battery which may be determined by the method described below (p. 258) be B ohms, the resistance of the galvano- * Proc. A.S. vol. xxxvi (1884). HIGH RESISTANCES. 249 meter G ohms, and the insulation resistance to be found A’ ohms; we have plainly, = ~ * : W = –“ . A + G’ A -- G -- /ć Therefore, - R = (A + 6)(# - i). . . . (28) If B be small in comparison with G we may put V – V/ A = G – . . . . . Iz' (29) A shunt-wound generating machine giving sufficient electromotive force may be used instead of the battery, and in this case R is found by equation (29). The insulation resistance for unit of length is found from this result by multifflying by the length of either of the conductors. This method is applicable to the measurement of the insulation resistance of cables or telegraph lines, but for details the reader is referred to the manuals of testing in connection with these special subjects. In the case of insulating substances the method just described requires the use of so powerful a battery that it is quite inapplicable except when the specimen, the resistance of which is to be measured, can be made to have a large surface perpendicular to the direction of the current through it, and of very Small dimensions in that direction. Such a case is that of the insulating covering of a submarine cable in which the current by which the insulation-resistance is measured flows across the covering between the copper conductor and the salt water in which the cable is immersed. 25O COMPARISON OF RESISTANCES. In general, therefore, in the determination of the insulating qualities of Substances which are given in com- paratively small specimens it is necessary to have recourse to the electrometer method mentioned above (p. 246), of which we shall give here a short account. The most convenient instrument for this purpose is Sir William Thomson's Quadrant Electrometer. For a full description of this instrument, and a detailed account of the mode of using it, see Chap. V. above. The electro- meter, having been carefully set up according to the most Sensitive arrangement, and found to be otherwise in good working order, is tested for insulation. One pair of quadrants is connected to the case according to the instructions for the use of the instrument, and a charge producing a difference of potential exceeding. the greatest to be used in the experiments is given to the insulated pair by means of a battery, one electrode of which is connected for an instant to the electrometer-case, the other at the same time to the electrode of the insulated quadrants, and the percentage fall of potential produced in thirty minutes or an hour by leakage in the instrument is observed. If this is inappreciable, the instrument is in perfect order. For practical purposes the insulation is sufficiently good when the same battery being applied to charge the electrometer alone as is applied to charge the cable, or condenser formed as described below, there is not a more rapid fall of potential without the cable or specimen than with it; for there can then be no error due to leakage. An air condenser, well insulated by glass stems var- nished and kept dry by pumice moistened with strong sulphuric acid, is adjusted to have a considerable capacity, and its insulated plate is connected to the insulated quad- HIGH RESISTANCES. 25 I rants of the electrometer, and the other to the electrometer- case, to which the other pair of quadrants is also connected. A charge producing as great a difference of potential as before is given to the condenser and electrometer thus arranged, and the fall of potential observed by means of the electrometer. If the loss in a considerable time be also inappreciable, the condenser insulates properly, and its resistance may be taken as infinite.* The specimen of material to be tested is now placed so as to connect the plates of the condenser. The manner in which this is to be done of course depends on the form of the specimen. If it is a flat sheet, it may be covered on each side, with the exception of a wide margin all round, with tinfoil, and thus made to form itself a Small condenser which is to be joined by thin wires in multiple arc with the large condenser. The edges and margins of the sides of the specimen should be carefully cleaned and dried, and covered with a thin coating of paraffin to prevent conduction along the surface between the two tinfoil coatings, when the condenser is charged. It is advisable, when possible, to coat the whole surface in- cluding the tinfoil with paraffin, and to make the con- tacts with the tinfoil plates by means of thin wires also coated with paraffin for some distance along their length from the tinfoil. The plate condenser thus formed should be supported in a horizontal position on a block at the middle of the lower surface. The upper coating is made the insulated plate. If the specimen be cup-shaped, as, for example, if it be * A condenser of any other kind, such as those used in cable testing, the insulating material between the plates of which is generally paper soaked in paraffin, may be used instead of an air condenser, but as the resistance of the latter may, if the glass stems be well varnished and kept dry, be taken as infinite, and there is besides no disturbance from the phenomenon called electric absorption, it is preferable to use an air condenser if possible. 252 COMPARISON OF RESISTANCES. in the usual form of an insulator for telegraph or other wires, the hollow may be partially filled with mercury, and the cup immersed in an outer vessel containing mercury, so that the mercury stands at nearly the same level outside and inside. The lip of the cup down to the mercury on both sides is to be cleaned and coated with paraffin, as before, to prevent leakage across the surface. A thin wire connected with the insulated plate of the condenser is made to dip into the mercury in the cup, and a similar wire connected with the other plate of the condenser dips into the mercury in the outer vessel. Strong sulphuric acid may, on account of its drying properties, be used with advantage instead of mercury as here described, when the substance is not porous and is not attacked by the acid. - In every case in which, as in these, the insulating sub- stance and the conductors making contact with it form a condenser of unknown capacity, the condenser used in the experiment must be arranged to have a capacity so great that the unknown capacity thus added to it, together with the capacity of the electrometer, may be neglected in the calculations. The condenser, if it has been disconnected, is again connected as before to the electrometer. One electrode of a battery of from six to ten small Daniell’s cells in good order, is also connected with the electrometer case, and the other electrode is brought for a short time, thirty seconds say, or one minute, into contact with the insulated plate of the condenser at any convenient point, such for example as the electrode of the electrometer connected with the insulated pair of quadrants. The battery elec- trode is then removed, and the condenser and electro- meter left to themselves. LEAKAGE METHOD, 253 The condenser has thus been charged to the potential of the battery, which will be indicated by the reading on the electrometer scale at the instant when the battery is removed. The deflection of the electrometer needle will now fall, more or less slowly according to the insulation resistance of the condenser with its plates connected by the material being tested. Readings of the position of the spot of light on the electrometer scale are taken at equal intervals of time, and recorded, and this is continued until the condenser has lost a considerable portion, say half, of its potential. The resistance of the insulating material is easily calculated from the results in the following manner. Let V be the difference of potential between the plates of the condenser at any instant, Q the charge of the condenser at that instant, which may be taken as proportional to the deflection on the electrometer scale, and C its capacity (Chap. xi.). We have Q = CV, and therefore d'O/dt = Cal V/dt. But—-d'O/dt is the rate of loss of charge, that is, the current flowing from one plate to the other, and this is plainly equal by Ohm’s law to V/A’. Hence – d'Q/dt– V/A’ and therefore Z V . V. C — = # + 4 = o Integrating we get, f l IV --- - º * & e * og V + C/Q (28) where A is a constant. If V be the difference of potential seconds after it was V, we get by putting # = o in (28) A = log V2. Hence (28) becomes f l/, – = log “ CA’ S 17 254 COMPARISON OF RESISTANCES. and f I /* -: 7. -p: - ‘k. º & t g t (29) log O |2 If V = # V., we have R = ?/C log 2. If the Condenser have a resistance so low as to add materially to the rate of discharge, an additional experi- ment must be made in the same way to determine the resistance of the condenser alone, with its plates con- nected only by its own dielectric. Let R. denote the resistance of the condenser, determined by equation (29) from the results of the latter experiment, and ſº, the resistance of the specimen; by equation (12) (p. 87) I/A = I/A'; + 1//P., and therefore Jºſe. ^ = z*z, . . . . . (30) If C has been obtained in C.G.S. electrostatic units of capacity, it may be reduced to electromagnetic units by dividing by the square of the number of electrostatic units of capacity equivalent to the electromagnetic unit, that is (see Chap. XI.) by 9 × Io” nearly. When an air condenser is used, its capacity can gene- rally be obtained approximately by calculation from the dimensions and area of the plates. For example, if two parallel plates of metal, placed at a distance d apart, very Small in comparison with any dimension of either surface, have a difference of potential V, and there be no other conductor or electrified body near, it can easily be shown that the capacity on a portion of area. A near the centre * It is to be remembered that the logarithms to be here used are Naperian logarithms. The Naperian logarithm of any number is equal to the ordinary or Briggs' logarithm multiplied by 2' 302585. * LEAKAGE METHOD, 255 of either plate is A/4tra. Hence in the example below, we have for the capacity of the disk of area A the value A/4tral, if we neglect the non-uniformity of the electrif al distribution near the edge. If C has been taken in absolute C.G.S. electromagnetic units of capacity (see Chap. XI.), we obtain A' from (29) in cms. per second,” which may be reduced to ohms by dividing by Io". When a condenser such as one of those used in Sub- marine telegraph work is used, the capacity C, of which is known in microfarads,f then since a microfarad is I/I o” C.G.S. electromagnetic units of capacity, we have for ſº in ohms the formula A – of * . . . . . (31) log V The following are results actually obtained in tests of a specimen of insulating material made in the form of an ordinary telegraph insulator. An air condenser consisting of two horizontal brass disks, the distance of which apart could be regulated by means of a micrometer screw, was joined with the insulator made into a small condenser with mercury inside and outside, as described above. The lower disc was of considerably greater diameter than the upper, which had a diameter of 12'54 cms., and the distance between them was adjusted to be 1 cm. The upper disk was connected to the insulated pair of quad- rants, and the lower to the electrometer case. Calling A the area of the upper plate, and d the distance between them, we have, neglecting the effect of the edges of the * In the electromagnetic system of units a resistance has the dimensions of velocity. See Chap. Xl f See Chap. XI, below. 256 COMPARISON OF RESISTANCES. upper disk, for the capacity of this condenser the value A/4tral in C.G.S. electrostatic units. Hence in the actual case C = 9:828. The interior surface of the insulator covered by the mercury was so small, and the thickness of the material so great, that, even allowing the material to have a high specific inductive capacity, the capacity of the condenser which it formed was small in comparison with that of the air condenser. The experi- ment gave, when the condenser and insulator were joined as described, V., - 25 I, W = IOO, f = 564O seconds. Hence, A’ = 5640 - 25 I e * • l - 9:828 × 2.303 × 9810i. = 623, in seconds per centimetre (C.G.S. electrostatic units of resistance). As the condenser was not insulating per- fectly, a separate test was made for it alone, with the results V, - 239, V1 = 182, f = 612O, Hence A'e = ——------------ J239 - 2286, 9.828 × 2'3O3 × logio 182 and therefore by (30) A'. - 623 x 2286 857, * T 2286 T623 T in seconds per centimetre. Multiplying this result by 9 × Io” (the approximate value of wº, see Chap. XI.), to reduce to electromagnetic units, we get for the resistance of the insulator 771.2 × Io”0 cms. per second, or 771 × 10% ohms. We shall now consider very briefly the measurement of the resistance of a battery. This term is not perfectly BATTERY RESISTANCE. 257 definite in meaning, as there is reason to believe that the resistance as well as the electromotive force of a battery depends to some extent on the current flowing through the battery, and further the resistance and the electro- motive force, and possibly also the polarization of the battery are affected by differences of temperature. But the information which in practice we generally require from the test, is really what available difference of potential can be obtained with a certain working resistance in the external circuit. This could be obtained at once by connecting the terminals of the battery by this resistance, and measuring the difference of potential by means of a quadrant electrometer or a potential galvanometer. If we call this difference of potential V, and the electromotive force of the battery when on open circuit E, then putting R for the external resistance we may write A. IV •) 7. Tº = x ~ y . . . . (31) where r is a quantity the definition of which is simply that it satisfies this equation. If the battery had the same electromotive force E, when generating the current 'y, as when on open circuit, then r would be the effective resistance of the battery; but, although this is not the case, we may without being led into error still speak of it as the resistance of the battery for the current y. In fact, the value of r, thus found for a particular value of R, does actually enable us to calculate from the known electromotive force for open circuit, with a moderate degree of approximation in the case of a constant battery, and also, but less Surely, in the case of a secondary battery, what available difference of potential S 258 COMPARISON OF RESISTANCES. will exist between the terminals of the battery when connected by other and somewhat widely differing values of Æ, and therefore also to find what arrange- ment of a battery it will be best to adopt in any given circumstances. So far as this practical result is con- cerned, the numerous methods which have been devised for the determination of the resistance of a battery before any sensible polarization (which requires time to develop) has been set up are, though interesting in them- Selves, of no practical value, and we shall not here describe any of them.* From equation (31) we have E – V ––P- To determine r therefore we have simply to measure with a potential galvanometer the difference of potential which exists between the terminals of the battery when on open circuit, or connected only by the galvanometer coil, the resistance of which we suppose to be very great in comparison with ?, and again to measure in the same way the difference of potential when the terminals are connected by a resistance R, also small in comparison with that of the galvanometer.f If the galvanometer scale be graduated so that readings are proportional to the tangents of the corresponding angles, we have, if D be the deflection in the first case, and O' the deflection in the second case, the equation = *z, *R. . . . . . (33) * Some account of the Measurement of Electrolytic Resistance will be found in the author's larger treatise, vol. i. t If the battery consist of a large number of cells, it may be divided into sections and so tested, or each cell may have its resistance measured separately. 7- A'. . . . . . (32) 7- BATTERY RESISTANCE. 259 Instead of a potential galvanometer a quadrant electro- meter may be employed if the battery is not too large, and the same formula applies when D and D are taken proportional to the sines of the angles through which the mirror is turned. A resistance coil, which may be of German silver wire, constructed as described in p. 232, should be used for the resistance connecting the terminals, and if the current passing through it be considerable its resistance should be determined when the current is flowing. This may be done by including in its circuit a current-galvanometer, and determining the current y through the wire in amperes, when V is read off in volts on the potential instrument. The resistance of the wire with that of the current-galvanometer is in ohms V/y, and this is to be used as the value of ſº in equation (33). If a galvanometer of high resistance be not available, an approximate test can be made by means of a sensitive galvanometer of low resistance. The battery and gal- vanometer are joined in series with a resistance Æ, and again with a resistance A". Let D and D' be the deflec- tions, which must have a difference comparable with either. Then, supposing E and r to be the same in both cases, and putting G for the resistance of the galvanometer we have D = 7/4 A. A. ==. Z2' = ??? — R + G + r.' *WIGI2 where m is a constant. Therefore we find A)/A’ – 7)/Q - —— – G . . . . D — O' (34) 7 = Mance has shown how to determine the resistance of S 2 26o . COMPARISON OF RESISTANCEs. a battery by means of Wheatstone's Bridge. The battery is placed in the position BD of Fig. 52 above, and a key is connected between A and B. The resistances r1, r2, ra are adjusted until the depression of the key produces no alteration in the galvanometer deflection. The galvan- ometer and the key, with their respective connecting wires, are then conjugate conductors; * and it is easy to show that the resistance of the battery is then rºrs/r1. The needle of the galvanometer is kept nearly at zero by means of a Small magnet during the adjustment of the resistances, so that it is as -sensitive as possible to any alteration of current produced by depressing the key. This method is so troublesome as to be practically useless, chiefly on account of the variation of the effective electromotive force of the cell produced by alteration of the current through the cell which takes place when the key is depressed. Prof. O. J. Lodge i has discussed the method, and shown how it may be improved by inserting a condenser in series with the galvanometer between C and D. Still it is inconvenient and gives no informa- tion which may not be obtained more easily in another way, and we shall therefore not enter into further detail regarding it. Sir William Thomsoni has however shown how the same mode of operating may be made to give the resistance of a galvanometer when there is no other galvanometer available. The arrangement of Fig. 52 is varied by placing the galvanometer in the position BZ), and a key in the position there shown as * That is an electromotive force in either produces no current in the other. See the author's larger treatise, vol. i., p. 159. f Phil. Mag, 1877, p. 515. | Proc. R. S. Vol. xix, (Jan. 1871). BATTERY RESISTANCE. 26 I occupied by the galvanometer. The deflection of the galvanometer produced by depressing the battery key is nearly annulled by means of a magnet, and the resist- ances r1, r2, r3 are adjusted until no alteration of the galvanometer deflection takes place when the key in CD is depressed. When this is the case C and D are at the same potential, since the addition of the conductor CD does not disturb the current distribution in the network; and we have for the resistance 7% of the galvanometer - CHAPTER IX. THE MEASUREMENT OF ENERGY IN ELECTRIC CIRCUITS. HEN a circuit in which a current of electricity is flowing contains a motor, or machine by which work is done in virtue of electromagnetic action, the whole electrical work done in the circuit consists, as was first shown by Joule, of two parts, work spent in heat in the generator and motor and in the conductors connect- ing them, and work done in moving the motor against external resistance. The total rate at which electrical energy is given out in the circuit is, as we have seen, E C watts, where E is the total electromotive force of the generator in volts, and C is the number of amperes of current flowing. The rate at which work is spent in heat is in watts, by Joule's law, C*R, where R is the total resistance in circuit in ohms ; hence, if we call W the rate at which work is done in the motor,” we have, FC = C*R + W. . . . . . (1) We may write this equation in the form, E - || C = . . . . . (2 A2 (2) * We consider here a system in which C is constant, and neglect loss of energy due to local currents, &c., in the motor. For fuller information regarding motors and their action see a paper by Profs. Ayrton and Perry, Proc. Soc. Tel. Engs., 1883, republished in the electrical journals, also Prof. S. P. Thompson's Dynamo-Electric Machinery. ELECTRICAL EFFICIENCY. 263 which shows that the current flowing is equal to that which would flow in the circuit if, the resistance remain- ing the same, the motor were held at rest, and the electro- motive force diminished by an amount equal to W/C. This is what is called the back electromotive force of the motor, and is due to the action of the motor in Setting up when driven an electromotive force tending to send a current through the circuit in the opposite direction to that of the current by which the motor is driven. We shall denote the back electromotive force by E1. Hence equation (2) becomes, C = Z - Zºl A2 (3) and the rate at which work is spent in driving the motor is El C. To determine E we have simply to measure with a potential galvanometer or voltmeter, the difference of potential between the two terminals of the generator. Calling this V, and R1 the effective resistance of the generator, we have plainly, F = V + CA’. . . . . . . (4) Again, since C and also the total resistance ſº in the circuit can be found by measurement, we find by (3) E = E – CA . . . . . . (5) where all the quantities on the right-hand side are known. The ratio of EIC, the electrical energy spent per unit of time in the circuit otherwise than in heating the con- ductors, to the whole electrical energy EC spent in the circuit per unit of time, that is the ratio of E, to E, we may call the electrical efficiency of the arrangement. 264 CIRCUIT OF GENERATOR AND MOTOR. Denoting this efficiency by e, we find, by equation (4), = ** – – º – – 4 ~ * e = z = i # = 1 —z- . . (6) Hence the Smaller C is made, that is, the slower the energy is given out, the value of the efficiency of the arrangement is the more nearly equal to unity, the value of the efficiency of an arrangement in which the energy in the motor done against external resistance is exactly equal to the whole electrical energy given out in the circuit. When however energy is spent at the maximum rate in working the motor, El C has its greatest value. But by (5) AE, C = E C – C*R = W. This equation may be written, C” R — EC + W = o, a quadratic equation of which the solution is, C = 2 /ø g Now in order that these values of C may be real, 4 R W cannot be greater than AE*. Hence the greatest value, W can have is E*/4/8. When W has this maximum value, C is equal to E/2/8, and therefore E1 equal to E/2. Hence the electrical efficiency is #. It is to be very care- fully observed that although in this case the arrangement is that of greatest electrical activity, it is not that of greatest electrical efficiency, as it has only about one-half the efficiency of one in which energy is given out at a very slow rate. The case is exactly analogous to that re- ferred to in p. 85, of a battery arranged so as to give maximum current through a given external resistance. ELECTRICAL EFFICIENCY. 265 All that has been stated above is applicable to the case of a motor fed by any kind of generator whatever. The generator employed however is generally some form of dynamo- or magneto-electric machine driven by an ex- ternal motor, such as a steam- or gas-engine or a water- wheel, and a few of the results obtained below apply only to such cases, which will be indicated as they occur. When the generator and motor are exactly similar machines, and the same current passes through both, the ratio of Æ, to Æ will be that of 7. A f(C) to n' A f(C); where n and n' are the speeds of the machines, A a constant depending on the form and disposition of the magnets, and f (C) a function of the current. Hence in this case the efficiency is measured simply by the ratio of the speed of the motor to that of the dynamo. The more nearly therefore the speed of the motor approaches to that of the generator, the greater is the efficiency. It is to be observed however that two machines identically alike will not in practice be perfectly similar in their action, even with the same currents flowing in their armatures and field-magnet coils. The armature currents tend to weaken the field in the generator, and to strengthen the field in the motor. In general, the higher the speed at which the motor is run, the greater is the electrical efficiency of any arrange- ment, for it is obvious that the higher the speed the more nearly does E1 approach to E, and therefore the value of AEI/F, the measure of efficiency, to unity. For a constant difference E – E1, the ratio of the energy spent in heating the conductors by the current to the whole energy expended in the circuit, may be reduced by increasing the total electromotive force E of the circuit. The energy spent in heat is C*R, or (E-AE)?/R, and the 266 CIRCUIT OF GENERATOR AND MOTOR. ratio of this to E C is CR/E. But C A' is equal to the constant difference E – E1, hence the ratio is (E - E)/F, and this becomes smaller as E is increased. A greater efficiency is therefore obtained by using high potentials than by using low potentials. Hence a greater electrical efficiency is realized, with a given magneto- or dynamo- electric machine used as generator and a given motor, when both generator and motor are run at higher speeds. Consequently the generator should be run as fast as possible, and the motor loaded lightly, or the speed with which the working resistance is overcome reduced by gearing between it and the motor. When high potentials are obtained by the use of machines wound with fine wire, or by using as generator a battery of a large number of cells joined in series to drive a high potential motor, the gain of electromotive force is accompanied by an increase of resistance in the circuit. But if we suppose the speed of the motor to be so regulated that the difference between the total electro- motive force in the circuit and the back electromotive force of the motor remains the same in the different cases, it is easy to show that the electrical efficiency of the arrangement is greater for high electromotive forces than for low. If, as supposed, E – E1 remains constant, while F is changed to 7. A., we have for the total activity of the motor m EC – (E – Ei) C. Dividing this by nEC we get for the electrical efficiency, 72 — I I E1 62 - 72 m F g e ſº tº & (7) As n is made greater and greater, the first term on the right becomes more and more nearly equal to unity, and the last term to zero. Hence, on the supposition made, EFFICIENCY OF MOTOR. 267 the efficiency is increased by increasing the working electromotive forces. Taking as a particular case ?? = 2, we see that the efficiency is ; together with one-half of the former efficiency; if n = 4, the efficiency is # together with one-fourth of the former efficiency, and so on for other values of 72. This result holds for any case what- ever in which the condition that E– E, should remain constant is fulfilled ; and hence it is independent of any change that may have been made in the resistances of the generator or motor in order to obtain the higher electro- motive force at E. For example, it is plain that no Sensible change in the actual rate of loss by heating of the conductors by the current will be produced by increasing the resistances of the generator and motor, if these be very small in comparison with the remainder of the resist- ance in circuit ; as, since E – E1 remains constant and the resistance is practically the same as before, the current strength will not be perceptibly altered. The ratio, how- ever, of the activity wasted in heating to the total activity will be only I/mth of what it was before. In the opposite extreme case, in which the generator and motor have practically all the resistance in circuit, the current, C, (= (E – E1)/A) is diminished in the ratio in which the resistance is increased ; and the actual rate of loss by heat according to Joule’s law, (E-AE)*//e, is diminished in the same ratio, so that, as in the former case, its ratio to the total activity n EC is 1/77th of what it was for the electromotive force E. We see, therefore, that here also the efficiency must be the same in both cases. We have called E1/E the electrical efficiency of the arrangement, but this is not to be confounded with the efficiency of the motor itself. The activity El C includes the wasted activity or rate at which work is done against 268 EFFICIENCY OF A MOTOR. frictional resistances in the motor itself, and in the gearing which connects it with its load, as well as the useful activity or rate at which it performs useful work. Hence, although the electrical efficiency of the arrangement be very great, only a small amount comparatively of the energy given to the motor may be usefully expended, and vice versé, and we define therefore the efficiency of a motor at any given speed as the ratio of the useful activity to the whole activity, taking as the latter the total rate at which electrical energy is expended in the motor ; that is, E1C + C*R1, or, which is the same, V C, where V is the difference of potential between the terminals of the motor. Accordingly, if A be the useful activity, we have for the efficiency of the motor the ratio A/VC. To determine this ratio in any particular case the motor is run at the required speed, V is measured with a potential galvanometer, and C with a current galvano- meter, and their product taken, or VC is determined with some form of electrical activity-meter, while A is deter- mined by means of a suitable ergometer. A very con- venient and accurate friction ergometer may be formed by passing a cord once completely round the pulley of the motor, and hanging a weight on the downward end, while the other is made to pull on a spiral spring fixed at its upper end and provided with an index to show its extension. The weight is adjusted so that the motor runs at the required speed, while wasting all its work in overcoming the friction of the cord, and the extension of the spring is noted, and the corresponding pull found in the same units of force as those used in estimating the downward pull due to the weight. Let the weight used in any experiment be taken in grammes, and be denoted by w, and let w/ be the number of grammes required to } CHARGING CIRCUIT OF SECONDARY BATTERY. 269 stretch the spring by gravity to the same amount, then the total force overcome is in dynes (w = w”) g, where g is the acceleration, in centimetres per second per second, produced by gravity at the place of experiment (at London g = 981-71 nearly). If n be the number of revolutions per second, and c the circumference in cms. of the pulley at the part touched by the rope, the velocity with which this force is overcome is m c, and therefore the activity in ergs per second is n c (w – w!) g. If A is reckoned in watts, we have the equation, A = ºn c(w = wº)g . . . . (8) IO/ If w — wº be taken in pounds, and c in feet, and n be the number of revolutions per minute, the activity in horse-power is given by – ºne(w = wo . . . . (9) and in watts approximately by A = ‘O2267, c (w — wº) . . . . (9) We have now considered cases in which electrical energy is transformed into mechanical work by means of motors working by electromagnetic action, and have seen that the whole electrical activity EC in the circuit is equal to the useful activity of the motor together with the unavailable part spent in heating the conductors in cir- cuit, and in overcoming the frictional resistances opposing the motion of the motor. Part of the electrical energy developed by a generator may however bespent in affecting chemical decompositions in electrolytic cells placed in the circuit, as, for example, in charging a secondary battery or “accumulator.” Each cell in which electrolytic action 27o CIRCUIT OF SECONDARY BATTERY. takes place, so that the result is chemical separation at the plates of the constituents of the solution acted on, opposes a counter electromotive force to that causing the current, and the work done per second in each cell over and above that spent in heat according to Joule's law (p. 75), is equal to the product of this counter electromotive force into the strength of the current. In most cases the counter electromotive force exceeds the electromotive force required to effect the chemical decompositions, and the energy corresponding to the difference of electro- motive force appears in the form of what has been called local heat in the electrolytic cells. In the case of a secondary battery charged by the current from an electrical generator, which is the only case we shall here consider, the activity spent in the battery while it is being charged is equal to the product of the difference of potential existing between the terminals of the battery while the current is flowing, multiplied by the strength of the current. Let V be this difference of potential in volts, and C the current strength in amperes, then VC joules is the whole work per unit of time spent in the battery. The whole activity spent in the circuit is EC, or VC + C*R, where E is the total electromotive force of the generator, and R is the resistance of the generator and the wires connecting it with the secondary. Again if A^i volts be the electro- motive force of the secondary battery, which may be measured by removing the charging battery for an instant and applying a potential galvanometer to the terminals of the secondary, the activity actually spent in charging the battery may be taken as E1C. Hence the ratio of the activity spent in charging the battery to the whole activity in the circuit is E1/(V+RC) or E1/E, and the ENERGY SPENT IN CHARGING. 27 I activity wasted in heating the conductors in circuit is (E – FI)C. This ratio Eſ Eis the same as that found above in the case of a generator and a motor, and may be called as before the electrical efficiency of the arrangement. Hence, in order that as nearly as possible the whole electrical energy given out in the circuit may be spent in charging the battery, as many cells should be placed in circuit as suffice to nearly balance the electromotive force E of the generator, that is, the charging should be made to proceed as showly as possible. In practice, however, a very slow rate of charging is not economical, as the work spent in driving the generator, if a dynamo- or magneto- electric machine, against frictional resistances would be greater than the useful work done in the circuit; and if the speed of the generator slackened for a little the battery would tend to discharge through it. As in the case of the motor (p. 265), the electrical efficiency of the arrangement can be increased by increasing Æ and F1, so that E – E is maintained constant. E may, in the present case, be increased by running the generator faster, or by using a machine adapted to give higher potentials. As before, if E be increased to m E, while E, is changed to E', so that n E – E = E – E1, the electrical efficiency becomes (n − 1)/n + E1/n E as in (7) above. The electromotive force of a Faure or storage cell is about 2.2 volts when fully charged, but is considerably less when nearly discharged. When the cell is placed in the charging circuit, the counter electromotive force which it opposes rises quickly to a little less than this value, and thereafter gradually increases, while the charging current falls in strength. In order to measure, therefore, the whole energy spent in charging a secondary battery, we must either use some form of integrating energy-meter which 272 ACTIVITY IN ELECTRIC LIGHT CIRCUITS. gives accurate results, or measure, at short intervals of time, V with a potential galvanometer, and C with a Current galvanometer placed permanently in the circuit. After the battery has been charged, the total number of joules spent is obtained by multiplying each value of VC by the number of seconds between the instant at which the Corresponding readings were taken and that at which the next pair of readings were taken, and adding all the results. Or, more exactly, values V and C are obtained for each interval by finding the arithmetic means of the values of V and of C at the beginning and end of each interval, and taking the product of these two means as the value of the activity for that interval. Each product is multiplied by the number of seconds in the corresponding interval, and the sum of the products is the whole energy spent. The integral work in joules having been thus estimated, the efficiency of the battery may be obtained by finding in the same manner the total number of joules given out in the external working circuit while the battery is discharging. The ratio of the useful work thus obtained to the whole work spent in charging is the efficiency of the battery. In discharging in an electric light circuit, the greatest economy is obtained when the resistance of the working part of the circuit is very great in comparison with that of the battery and main conductors. Neglecting the latter part of the resistance, we see that, if a large number of lamps are arranged in multiple arc, a large number of cells should also be joined in multiple arc, so that, while the requisite difference of potential is obtained the resistance of the battery is still Small in comparison with that of the external circuit. As regards the measurement of energy spent in electric light circuits, in which continuous currents are flowing, ELECTRO-DYNAMOMETERS 273 we have already sufficiently indicated above (Chap. VII.) how this may be done. To find the activity, or work spent per unit of time, in any part of a circuit, we have only to find the dffference of potential, V, in volts between its extremities with a potential galvanometer, and the current, C, in amperes flowing through it with a current instrument. If the activity be constant, we have simply to multiply VC by the number of seconds in any interval of time, to find the number of joules spent in that time in the part of the circuit in question. If the activity is variable, the whole energy spent in any time may be estimated by finding V C at short intervals of time, and calculating the integral as explained above (p. 272). So far we have been considering only measurements made in the circuits of batteries or of continuous current generators. Alternating machines in which the direction of the current is reversed two or three hundred times a second are, however, frequently em- ployed, especially in electric light circuits, and it is necessary therefore to consider the methods of electrical measurement available in such cases. We shall consider briefly, first, a class of instruments some of which can be used in a circuit of either kind, and we shall deal in the first place with their application in continuous current circuits. These are instruments the fundamental principle of which is the mutual electromagnetic action between two circuits, to be calculated conveniently in most cases in which this can be done, by replacing each circuit according to Ampère's theory, by an equivalent magnetic shell (p. 42), and considering the mutual action of the systems. Sir William Thomson's Electric Balances, described in Chap. VII., are instruments which act on this principle, and can T 274 ELECTRO-DYNAMOMETERS. be used both in alternating and in continuous current circuits. Weber's well-known electrodynamometer% is another instrument of this class, but arranged to be used as an absolute or standard current-meter. It may be considered as Constructed by replacing the needle of the standard tangent galvanometer (Chap. IV.) with a coil of radius small in Comparison with that of the galvanometer coil, and suspended by a torsion wire or bifilar, so as to hang in equilibrium when no current is passing through it, with its plane at right angles to that of the large coil. Hence when a current C is made to flow through the fixed coil, and a current C through the suspended coil, a couple is exerted on the latter, tending to set it with its plane parallel to the large coil, and this tendency is resisted by the action of the torsion or bifilar suspen- sion, so that there is equilibrium for a deflection 6, the magnitude of which plainly depends on the product CC’ of the two current strengths. To avoid disturb- ance from the action of the local horizontal magnetic force, the large coil may be placed parallel to the direction of that force, and the small coil brought back when deflected by the current to the initial position at right angles to the large coil, by turning the upper end of the suspension wire or wires through a measured angle. By Ampère's theorem (p. 42) the suspended coil is equivalent to a small needle of moment zu A Cº, where 7, is the number of turns of wire in the coil, A their mean area. Hence if M be the number of turns in the large coil, r its mean radius, we have as in p. 47 for the electromagnetic couple on the suspended coil, when brought back to the * See Maxwell's Electricity and Magnetism, vol. ii. p. 330. ACTIVITY-METERS. 275 initial position, the value 27t/r. W.C. m. A C or 2T/r. 7, AVA CC’. This couple is balanced by the opposite couple given by the supension, the magnitude of which for all angles within a certain range is supposed known from experiment. Calling this latter couple /, we have / A 7- CC’ = — . . . . . (IO) 2 TT 22 AVA If the two coils be joined in series so that the same current ſlows through both, we have C = C', and therefore JL 7- C* = ––––– 2 TT 27 A/A (II) With such an instrument therefore an absolute measure- ment of a current can be made without its being necessary first to determine H.” In practical work the instruments on this principle usually employed are such as require to have their constants determined by comparison with standard instruments, such as a standard tangent galvano- meter, or a standard dynamometer, and are dealt with in Chap. VII. We may here mention, however, Siemens' electro-dynamometer, in which a suspended coil is acted on by a fixed coil, and the strength of the current deduced, by means of a table of values for different angles, from the torsion which must be given to a spiral spring to bring the coil back to the zero position. When an instrument on this principle is arranged for use as an activity-meter, one set of coils, the fixed or the movable, is made of thick wire so as to carry the whole current in the circuit, while the other set is made of high resistance and is connected to the two ends of the part of * For fuller particulars regarding absolute instruments of this class see Theory and J’ractice of Aðsolute Measurements in Electricity and Magnetism, vol. ii. T 2 276 ALTERNATING CURRENTS. the circuit in which the electrical activity is to be measured. In this case the force or couple required to restore the movable coils to the zero position is proportional to the product V C of the difference of potential and current, that is to the activity, for that part of the circuit ; and if the instrument has been properly graduated this can be at once read off in watts, or in any other chosen units of activity. Instruments of this kind have been made by Professors Ayrton and Perry, Sir William Thomson, and Sir William Siemens. Sir William Thomson's form is described above (Chap. VII.). We shall now consider the measurement of currents and differences of potential, and therefore also of electrical energy in the circuits of alternating machines or of trans- formers. In all such circuits the march of the current in each complete alternation may be stated roughly as a rise from zero to maximum in one direction, then a diminution to zero, then a change of sign and a rise to maximum in the opposite direction, followed by a diminution again to zero. The law according to which these changes take place is more or less complex in the various cases, and the complete mathematical representation of the current strength at any time would require an application of Fourier's method of representing any arbitrary periodic function, by means of an infinite series of simple harmonic terms of the form A; sin (in #– e.), where n is 2T divided by the period 7 of a complete alternation, A.; and eſſ constants and Ž any integer. It has been found experimentally by M. Joubert that the variation of electromotive force in a Siemens' alternating machine can be expressed by the single harmonic term E sin m t, where we reckon f from the instant at which the electromotive force was zero when changing from the direction reckoned as negative MEASUREMENT OF CURRENTS. 277 to that reckoned as positive. There is good reason to believe that this law is not very seriously in error for the majority of alternating machines, and we shall assume its truth in what follows. The current strength is affected by the action of self-induction (p. 206) to a greater or less extent in all such machines independently of the disposition of the external circuit, especially if the revolving armature contains iron ; but, as shown below, it follows, with a difference in phrase, the same law as does the electro- motive force. The effect of variations in the field magnets produced by the rotating armature has also in a rigorous theory to be taken into account, but this effect, in well- designed machines without iron in their armatures is not great, and where experiments have been made to detect it, has been found to be slight, and we shall therefore neglect it. Writing then C for the current, at a time t, less than 7/2, reckoned from the instant at which the current was zero, we have C = A sin m/ . . . . . . (I2) The whole quantity of electricity generated in a half period 7/2 is therefore T; 7/2 . A 7T Ca' ſ = Aſ sin a t d f = ′ . . (I 3) 62 O T Hence if C denote the mean current in that time, we have Cm -: 2A º - te g o e (14) T Now if an electro-dynamometer be placed in the circuit So that the same current passes through both its fixed and movable coils, the current in both will be reversed at the 278 ALTERNATING MACHINES. same instant, and their mutual action will be the same for the same current strength, and will be proportional to C* that is to A* sin”f. If the period of the alternation be small in comparison with the period of free oscillation of the movable coil system of the dynamometer, the mutual action of the fixed and movable coils will be the same as if a continuous current C’ given by the equation / 2 I 7. Q A2 7. * - 2 C = }ſ. C2 aſ # = ºſsin 77 f d # . . (I 5) were kept flowing through them. But by integration C’” = . . . . . . (16) and substituting from (14) in this equation, we get c. – ºc – ‘90O3 Cº. . . . (17) T In order therefore to find the actual mean current strength in the circuit of an alternating machine from the value of C' given by a current dynamometer we must multiply the latter by '9; in other words the mean current strength is 9/10 of the strength of the continuous current which would give the same deflection. The product, if C has been taken in amperes, multiplied by the number of seconds in any interval of time during which the machine has been working uniformly on the same circuit, will give the number of coulombs of electricity which has flowed through the circuit in that time. The measurement of potentials is however attended with more difficulty on account of the effect of the self- induction of any electromagnetic instrument which can be applied to the circuit for this purpose. The following MEASUREMENTS BY ELECTROMETER. 279 method of employing Sir William Thomson’s quadrant electrometer for this purpose has been used by M. Joubert.* The needle of the instrument is left uncharged, and the charging rod connected with it and used as a third electrode. If then we suppose the needle connected to one point in the circuit at which the potential is V, one pair of quadrants at a point at which the potential is V1, and the other pair at a third point where the potential is V, then if D be the deflection of the spot of light cor- responding to the angle (supposed small) through which the needle has been turned against the bifilar suspension, then as we have seen above (p. 134) we have D = AE (V – * (v-Ai º) . . (I 8) where Æ is a constant. If the needle be connected to the pair of quadrants whose potential is V, then we have as in (4) of Chap. VII. / * \ . P = (V – Vº)”. . . . . (19) The constant AE is determined by a process of calibra- tion with known differences of potential similar to that described at p. 136 above. The multicellular or the vertical voltmeter described in Chap. VII. may (preferably) be used instead of the quad- rant electrometer, except when three points at different potentials are to be connected to the electrometer at the same time, as in the measurement described below, p. 300. Any doubt as to the applicability of the expression on the right of (19), with Å a Constant, is avoided, for in these * Com/tes Rendus, July, 1880. Annales de Chimie et de Physique, May, 1883. 28O ALTERNATING CURRENTS. instruments the values of different deflections on the scale have been fixed by experiment. - If the terminals of the electrometer employed be con- nected to any two points in the circuit of a machine in which the period of alternation is short in comparison with the free period of the needle, the couple acting on the needle will be at each instant proportional to the second power of the difference V1 – V., of potential exist- ing between these two points at that instant, and this of course is independent of the sign of the difference. Also as in the similar case of the dynamometer above, the deflection of the needle will be the same as that which would be produced by a constant difference of potential V given by the equation Q I 7. * - #ſ (n − iºn, If we denote the actual mean difference of potential by V, then since the difference of potential follows the same law of variation as the current we get also Vºn = 9003 V . . . . . . (20) If we know the resistance in the part of the external circuit between the points at which the electrometer electrodes are applied, then calling this resistance R, and supposing that this part of the circuit contains no motor or other arrangement giving a back electromotive force, and that the ratio of its self-induction to the period of alternation is zero or negligible in comparison with R, we have for the mean value of the current Vºnſ R, and thus by means of an electrometer alone we can measure not MEASUREMENT OF ACTIVITY. 28I only the difference of potential between the ends of, but also the current in, that portion of the circuit. It has been pointed out by Lord Rayleigh (P/º/, //ag. May, 1886) that the resistance offered by a conductor to the passage of a current through it is greater the Smaller the period of alternation. This variation is due to the fact that as the alternation increases in rapidity the current is more and more confined by inductive action to the outer strata of the conductor which is therefore virtually reduced in section. This is not to be confounded with the fictitious increase of resistance seen in the expression V/º-H 747." (see p. 293 below) which arises directly from the electro- motive force of self-induction, but is a real increase of the value of R for the current in question. A very useful table of the resistances of conductors at different periods of alternation drawn up by Mr. W. M. Mordey and given in his paper on Alternate Current Working (ſoc. cit. p. 282) will be found in the Appendix. The theoretical data from which the table has been calculated were given by Sir William Thomson as an addendum to his presidential address (1889) to the Institution of Electrical Engineers. An abstract of it is printed with the table. Denoting by Am the mean value of the electrical activity in this part of the circuit, still supposing the self-induction of this part to be negligible, we have plainly V2 I / 7" O Aze = Aſ Gº - V.)” dz - (21) In the same way, since the value of the electrical activity at any instant is C*A*, we have from the results of experi- ments made by an electro-dynamometer, A / T An - ºf Cºd f = C'2 A. . . . (22) 282 THEORY OF ALTERNATING MACHINES. From these two results we get Am = PC’ e º - - e º (23) that is, the mean value of the electrical activity is equal to the product of the square root of the mean square of the difference of potential, by the square root of the mean Square of the current strength. It can therefore be deter- mined by means of an electrometer and an electro- dynamometer of negligible self-induction without its being necessary to know the resistance. We shall now consider the case in which the self- induction cannot be neglected. Let R be the total resistance in the circuit, C the current flowing in it at the time t, E the total electromotive force of the machine, and Z the coefficient of self-induction, or the “inductance” (as, following a usage introduced by Mr. Oliver Heaviside, we shall henceforth call it) for the whole circuit, that is, the number which multiplied into do|df gives the electromotive force opposing the increase or diminution of the current. We shall suppose / a constant, although there can be no doubt that in some alternating machines its value is different in different positions of the armature. The iron cores of the field magnets act to a greater or less extent as cores for the armature coils, and as the magnetic susceptibility of iron is a function of the strength of the magnetizing current, L, which is the mag- netic induction through the armature produced per unit of its own current, must vary accordingly. Still for certain alternators which have no iron in their armatures the variation of L with the position of the armature is slight.” It will also be assumed that there * See the discussion on Mr. W. M. Mordey's paper on “Alternating Current Working” Inst. of Elect. Eng. May, 1889 (The AE/ectrician, May 24, 31, June 7, Aug. 2, 1889). THEORY OF ALTERNATING MACHINES. 283 are no masses of metal in which local currents can be generated moving in the field. On these assumptions the equation of the current is aſ C R C = E – Z 77 . . . . . . (24) But by the law which we have assumed for the machine, E = me sin n f = E0 sin m #. . . . . (25) where e is a constant such that Æo is the maximum value of E for the given speed. Substituting in (24) we get d C -- = F., si as e - 26. Z 274 R C = E0 sin a . (26) which integrated becomes A’ * c-4'-º'-ºasin (nº-o: (27) where ---—#– --—“– (28) S11) e = WAZ-E 7272' COS e = w/A3+ 7,272 * \ -º The term A e-#" is only important immediately after the circuit is closed, and will therefore be neglected. We may remark that if Z were equal to zero (27) would reduce to C = Eo/A’. Sin n f, which corresponds to (I2) above. From (27) we get for the mean current Co. F 2.Eo ſ e/n-H T/2 2 Ao Triºſ. sin(*- )* = F(xi+;75,99) s' Also for the mean square of the current strength as 284 THEORY OF ALTERNATING MACHINES. given directly by an electro-dynamometer we have by (27) the equation - w A. 2 7. ’2 — 0 - *- zººſ. sin " (zz à — e) d ? E.” — . . . . . O A” + 77* /2 (30) I 2 and we have therefore as before, the relation Cm F ‘90O3 C'. From (27) we see that the effect of self-induction is to diminish every value of the current in the ratio of Eo/(ſº + 77° Z”)} to Eo/R, and to produce a retardation of phase which measured in time is eſn seconds; that is, the resistance is virtually increased in the ratio (A*-ī-77°/2)}/A', and the current in following the law of sines passes through any value e/77 seconds after it would have passed through the corresponding value if there had been no self-induction. It is plain also that, for any finite resistance /ć, by diminishing 7, that is, by increasing the speed of the machine, the current can, by (25), be made to approach the limiting value c = }sin (a, - .) . . . . (31) which is independent of the resistance, and has a retarda- tion of phase of 7/4 seconds, a quarter period of a complete alternation. Hence integrating over a half period from zero current to zero current again, and dividing by T/2 we get for the maximum mean current 2 € Cm - 7. / º º & g º g (32) TWO EQUAL MACHINES IN SERIES. 285 To find the mean value Amt of the total electrical activity in the circuit, we have by (25), and (27) I ſT Zoº T 4m - – A cº-ºſ. sin (n & - c) sin 72 Zaſz I ƺ R - T 2 A3-E 7,273 (33) Hence by (30) Am - C'2 A’ e - s - - - (34) that is, the true mean value of the total electrical activity is equal to the mean Square of the current strength multi- plied by the total resistance in circuit. It may be shown, from (33), by the method used in p. 85 above that the total activity in the circuit is greatest when ſº-n/, that is, for a given speed and a given value of Z, the activity is a maximum when A = 77/. It must be observed however that for a given resistance ſº the activity is greater the Smaller the value of Z, that is, the greater the speed. When A' has the value 77/L we have, by (27) e = T/4; that is, the retardation of phase is then one-eighth of the whole period.* If in the circuit there be two sources of electromotive force of the same period T but of different phases; for example, two machines driven so as to have the same period of alternation, the solution here given applies. For the two electromotive forces combine to give a single electromotive force of the same period as the components * The conclusions as to maximum work and retardation of phase, as well as most of the theoretical results stated above as to the action of alternating machines, were first we believe given by M. Joubert, Com/tes A'ezidzes, 1880. The problems of alternating machines joined in series, or in parallel, or as motors, were considered by Dr. J. Hopkinson in a lecture to the Inst. of Civil Engs. 1883, and in a paper “On the Theory of Alternating Currents,” Soc. Tel. Engs. and Els., Nov. 1884. The principal results of this latter paper are reproduced below. See also Mr. Mordey's paper loc. cit. 286 THEORY OF ALTERNATING MACHINES. but differing in phase from either ; so that, to use the solution it is only necessary to take this resultant electro- motive force as Eosin ſet, reckoning the time from an instant at which sin mt is zero and increasing. If the difference of phases be 2q, reckoned in angle, the interval between the successive instants at which a component is increasing through zero is 2q3/71. Hence taking the zero of reckoning of time midway between these two instants we may denote the two components by E1 sin (nº 4 q), AE, sin (nf – p). Calling their resultant Eo sin (nt – lº), we have Eosin (n : – ;) = E1 sin (n : + ºp) + Ey sin (n : - 'b) (35) By elementary trigonometry we get E3 = E.” + Ey + 2 E. E. cos 24 and E., - E . . (36) tan / = **——tº tan (p A 1 + £2 When q = o, y = 0, and Æo = AE1 + Z, as is evident without calculation, since the machines are then in the same phase. If E1 = E2, that is if the machines are equal, the resultant is in phase halfway between its com- ponents. When this is the case we have also Fo = 2 AE1 Cos @ . . . . . (37) which when q = 0 gives, as it ought, Æo = 2E1. Considering still two unequal machines and remember- ing that when the value of the resultant electromotive force is increasing through zero, the value of the current is given by (27), that then the electromotive force of the leading machine is E1 sin (7tt -H (p + \lº), and that of the following machine AE2 sin (mt – b –– Nº), we have for the mean activity A10, of the leading machine. Two EQUAL MACHINES IN SERIES. 287 r, ſ?" 41m -- #ſ Jº Caſſ 7 Jo E0E1 ſºn (m f – e)sin (n : + q + \,) d f 7(A&E ºzº), I Fo E € -- #(ºyº” (q – ſ; + e) Aº Aº & = };}}A(8 cos(p+ 0 + n, sin (p + b (38) To find the mean activity of the following machine we have only to change the sign of p in this expression. We get I Æ, ź - tº 42m = 2 #2 #7. (k cos (q – Nº) -- 7: L sin (q) — v)} (39) If the machines be equal E1 = E2, and W = O, so that A.,” cos º A 12m = #% (R cos p – 71 / sin q}) . (40) A.2 g 4.3m = #(º cos q + 7. Z sin ºp) . (11) Since p is less than T/2, both cos q and sin q are positive, and therefore the following machine does more work than the leading machine. Hence, unless each is completely controlled by the prime-mover, the leading machine will increase its lead, and this will go on until 245 = tr, when the two machines will be in exactly oppo- site phases, and will exactly neutralize one another. This tendency to assume opposition of phase depends on the difference Agº. – 41,,,, and this having the factor n//(R*-H m”/A), has a maximum value, for a given resist- ance and a given period of alternation, when 77 L = R. 288 THEORY OF ALTERNATING MACHINES. The machines thus arrange themselves so that no current passes in the wires joining their terminals, and these wires alternate in relative potential with the period of the machines, and each is at any instant very approxi- mately at one potential throughout. It might therefore be inferred that if a working circuit be joined from one wire to the other, a current will pass through that circuit, and that the two machines will control one another so as to keep in the same phase in supplying it. We shall consider this case as a further example of the theory. Let 2 q be the difference of phase with reference to the external circuit, so that at time t, E sin (m/+@), Esin (mt – q}) are the electromotive forces of the two machines, C1, C, the currents, L the coefficient (supposed constant) of self. induction for each, r the resistance of each machine from one point of attachment to the other point, and R the re- sistance of the external circuit. We shall suppose that the external circuit has no sensible Self-induction, and that the whole work there developed is spent in overcoming resistance, for example, in lighting glow lamps. By considering the circuit through each machine and the external resistance,” remembering that the current in the latter is C. -- C, and therefore the difference of potential between the terminals R(C + C), we find the equation Zºº; + , c. 4 K (c) + C) = E sin (a t + i) d? (42) Zº + r C., + R (C + C) = E sin (n : - (b) * According to Kirchhoff's rule, p. 88, above, taking into account the electromotive force of self-induction in each circuit. TWO EQUAL MACHINES PARALLEL. 289 Adding and subtracting we get 1%(c. 4 c) + (2 k + r (C + cj = 2 E cos q} . Sin 77 & . . . . (43) Ž(c. – Cº) + r(C1 – C.) = 2 E sin ºp. Cos má (44) Solving these we find as in (27) 2 AE cos q) {(2 R + rj + nº Zºº 2 E sin q (* + m2 Z2)} Ci + C = sin (n : - ) (45) C – C = . Cos (?? & – e') . (46) where 7. /. 2 A -- 7° tan e tan & = **. . . (47) 7- Hence if Aiºn be the mean activity of the leading machine 7. 41m = #ſ (C + C3 -H C – C.) sin (n : + qi) d ? _ E°ſ cos (p ſtar + r.)” –F *L* ſ in (mf – e) sin (m.t-H (f)dź e 7. - + zººſ cos (mt – e') sin (mě -- ºp)d: 2 - ; (2R-F ; + #128 +r)cos”q — m/sing cos q} I A.2 + 2. aizzº sin ºp + mL sin q cos (b) . . (48) 29O THEORY OF ALTERNATING MACHINES. The mean activity Agn of the following machine may be got from A1, by altering the sign of q throughout the expression on the right. Hence 2 4.3m = . axiºleºn cos^4 + m/sing cosºp) + I £* (rsin” — m/ sin ºp cos (b) . . (49) 2 rº + m2/.” Alºn – 42m is positive, that is more work is done by the leading than by the following machine. The lead will therefore tend to zero, and the machines to settle down into coincidence of phase with reference to the external circuit, that is, into opposite phases with refer- ence to their own circuit, which agrees with the result already obtained. We shall consider only one more case of this theory, that of an alternating motor connected by its terminals to two conductors upon which an alternating difference of potential is impressed by other machines. Let the motor be started so as to have the same period of alternation. Then denoting by ſº the resistance of the motor-armature and the leads up to the point at which the difference of potential is impressed, by Z the co-efficient of self-induc- tion for the same part of the circuit by E1 sin (nf – p), the impressed difference of potential at time t, by A., sin (nt – q), the back electromotive force of the motor at the same instant, we have for the equation of the current 4% + K C = E1 sin (n : + qi) — Z, sin (?! # – p) (50) This equation differs only in the sign of E, from that from which (38) and (39) above are deduced. Hence ALTERNATING GENERATOR AND MOTOR. 29I taking the value of Azº, in (41) we have for the mean electric activity received by the motor A. - 42m =#24&cos º – Nº) + n L sin (ºp – V)} (51) where Eo = (E1% + E.” – 2 AE1 E. CoS 2 (b)} l tan / = - ###tan q) ſ . (52) The second equation of (52) gives cos \| = (E1–E) cos q}/Eo, sin \, = – (E1 + Z2) sin (b//º, and these values substituted in (5 I) yield I E., Aom = --→ 2 2 /ć” –- 77°/.” {E,(Rcos 24 + m1 sin 25)–E.A.) (53) Now 2 q being the difference of phase, cannot be numerically greater than TT, and therefore the work zeceived by the motor is less when 2 p is negative than when it is positive, that is, is less when the motor is leading than when it is following. Hence the motor will tend to run slower when leading and faster when following; or the difference of phase will tend towards zero. Also as long as 2 q is not far from zero Agy, is less the greater the lead, and greater the greater the lag, and in nearly the same proportion. Hence when the machines are once in phase any small deviation is opposed by a proportional corrective tendency. This depends almost entirely on the term involving the factor n//(R* + 77°/.”) in the value of Aºm given in (53), and therefore for a given resistance R, and period of alternation 7, has its greatest value when n/2 = R, or, Z//ē = Tl2T. U 2 292 THEORY OF ALTERNATING MACHINES. Writing in (53) sin 2q3" = –*– cos 2q3" = –4– (54) (A* + m2 /2)}’ (R*-H 75%/2)} we get I A. • a rº- f 4, -ºº-ººsinº-ºo-ºº! Gs) which is obviously a maximum when q + p = T/4. We have then 42m - As ſ I O t * ; zºº (ºr nº- E,F} (56) This value of Agº, is positive if A A2 * E., T (Rº-Eºſ2). which may be the case even if Z, - AEI. Hence we have the Curious result that an alternating machine may act as a motor even if its electromotive force be greater than the impressed or driving electromotive force. The theory just given of the working of alternating machines on the same circuit is due to Dr. J. Hopkinson, F.R.S. (Soc. Tel, Engs. and Els., Nov. 1884). Its con- clusions were verified by him in 1884 in experiments made with two De Meriten's machines made for the lighthouse at Tino. Some very striking experiments are described in Mr. Mordey's paper referred to above (p. 281), and it contains moreover much interesting practical information On this subject. Difference of opinion at present exists as to whether Mr. Mordey's results are in accordance with the mathematical theory. It is to be remembered how- ever that the theory does not take account of the action of the armature currents of the field magnets nor of the variation of self-induction. The whole subject requires further investigation. MEASURE OF ACTIVITY IN INDUCTIVE CIRCUIT. 293 We may apply (as we have already done in the problem of the alternating motor just treated) the formulas given above for the whole circuit to a part, taking for E the impressed electromotive force on the part of the circuit considered, and for R and Z the proper values for that part only. We find that the effect of self-induction on the current is virtually to increase the resistance from ſº to v ŽIºE, and to produce a difference of phase between the current and E given also by (27) and (28). But the resistance of a conductor is (p. 75) the activity spent in it by unit current in producing heat; hence by (22) and (34) the resistance in this sense is not increased. The quantity V /* + m” L*, or one analogous if the simple harmonic law is not followed, is very important, and Mr. Oliver Heaviside has proposed for it the very appropriate name impedance.* The impedance of a current electro-dynamometer or current balance, through both coil-systems of which flows the whole current in the main circuit, cannot if it be low (as it generally is) in comparison with that of the rest of the circuit, affect appreciably the strength of the current by its introduction ; and since the whole current passes through both sets of coils, the instrument will give the mean Square of the current passing. It may be otherwise, however, with a fine wire instru- ment used as a shunt to measure the difference of potential between two points of the circuit. The induct- ance of such an instrument may be considerable, and if it be used alone its impedance will seriously affect the result. * The Electrical Congress held at Paris in August last (1889) voted in favour of calling this quantity the “résistance apparente” of the part of the circuit considered. The term impedance has however been approved by the British Association Committee on Electrical Standards in its Report to the Newcastle meeting held in Sept. last (1889), For its definition see Note in Appendix. 29.4 ALTERNATING MACHINES. Since the value of the impedance depends on the period of alternation, it will have different values when connected to circuits in which the periods are different. To obviate the uncertainty and inconvenience arising from this cause, the instrument is made sensitive enough to allow a con- siderable non-inductive resistance to be joined in series with its own coils. This makes the value of R/V Rº-F772 approximately unity. Some calculations made by Prof. T. Gray * for Sir William Thomson’s vertical scale volt- meter, give for this ratio with only the resistance of the instrument (640 ohms) included, and a period of alternation of Tºro of a second, the value '9976, which is within + per cent. of unity. Plainly the error caused by the impedance in this case is small with any period commonly employed, and can be made still Smaller by the introduction of non- inductive resistance. The difference of phase between the currents through the coils of the instrument, and the difference of potential (given by (28) above) is therefore small. This difference of phase, it is to be remembered, does not affect the value of the mean square of the differ- ence of potential, provided the amplitude be corrected for the effect of inductance. It is however of importance in the action of a wattmeter, of which one coil is placed in the main circuit, and the other as a shunt between the extremities of the portion of the circuit in which the activity is to be estimated. For let the circuit divide into two parts, each forming a derived circuit on the other, and Z1, L3, Æ1, A2, C1, C2 be the inductances, the resistances, and the maximum currents in the two parts, C the maxi- * A lectrical Review, Jan. 20, 1888. The instrument here referred to is not described above, but is similar in principle and construction to the instrument shown on p. 99. It has however only two fixed and two movable coils and reads with a pointer on a vertical scale. WATTMETER ON ALTERNATE CURRENT CIRCUIT. 295 mum total current in the circuit, and ei, e, the differences of phase between C and C1, C2 respectively, a similar analysis to that given above shows that if there be no mutual induction between the two parts of the circuit - C.? - C.? - C2 R,”--m”Zºº Riº+7°/21° - (R, + A*)*-H77°(Z1+/2)* tan e1 = 72 (Z1 Æ2 - Z2 A’ſ.) (57) (R, + A*2) A'. -- 77° (/1 + Z.) Z2 with a similar formula for es: If either Z1, Z2 be both Small, or Z1//2 = R1//ē2, the difference of phase between the two currents C1, C., will be insensible. If the first condition is fulfilled both parts of the circuit will have currents agreeing in phase with the difference of potential between the terminals, and, on the usually allowable supposition of negligible mutual induc- tion, a wattmeter whose coils are included in them will measure accurately the power expended. It will, on the same supposition, also measure accurately the power expended while the wattmeter is on circuit if the ratio R/V Rº + 7.” Z” be approximately unity for the fine wire circuit, since the main current passes through the other coil, and it can be shown that the deflection will be the same as would be produced by a constant activity A,” given by the equation I ſT A m = }| V Czyż e º º tº º (58) - O Where V, C, are the values of the difference of potential and the current at time t. If also N' Rº-E mº/2 for the thick wire coil be small in comparison with the same quantity for the part of the main circuit in which the activity is being measured, the inclusion of the wattmeter 296 WATTMETER ON ALTERNATE CURRENT CIRCUIT. will not affect the circuit and the activity shown by the instrument may be taken as that existing when it is not applied. - But it is important to notice that the second condition may be fulfilled without an accurate measurement of power by the wattmeter. The currents through the two coils will have the same phase, but this may not be the phase of the difference of potential between the terminals if between these there be sensible self-induction in the main circuit; and the wattmeter would give too high as result. It is essential for accuracy that the current through the fine wire coil-system of the wattmeter should have the phase of the difference of potential, while that in the thick wire coil should have the phase of the main current, The general problem of finding the ratio of the apparent activity as shown by the wattmeter to the true activity can be solved with great ease by aid of the theory given above. For let A, B be the points at which the terminals of the fine wire coil-system are attached to the main circuit; let Aºi, A', ſ.1, Z2 be the resistances and inductances of the fine wire and thick wire circuits between A, B, and C1, C2 the currents in them, then by (27) if the difference of potential between the terminals A, B is Æo sin ºut. A. º C = —t"—a sin (n : - 1. (R,” + 7.” Z1”)} ( e1) C2 = , ; ; Æo sin (?! # – e.) (R,” – 774 L.”)} (59) where "I A's The current through the fine wire is therefore the same as if the resistance in its circuit between the points A, B RELATION OF APPARENT TO TRUE ACTIVITY. 297 were without inductance, and the difference of potential E between A, B had the value Fo R1/(R,” + 77* L1%)} . sin (n f – ei). Hence if A'm be the apparent activity in the main circuit between A, B I ſ?" A'm -> }ſ F.C., dź Z O = + Foº, 'sinou- ei)sin(???– e.g)d: T(Riº+?,*Z)}(R,”--m°/...”)} | . 1 2 __ I Fo” R1 cos (el – e.) | 8 2 (A’,” + n” Z1”) (AE,” + 77* Lºº) . [by (38)] - Fo” A'. (R1 A's + A1 Z2) - - (60) I T 2 (R,” + m” Z1”) (R.” + 7:27.2) since by (59) sin e1 = n Z1/(R,” + 77° W.1%)}, cos et = R1/(R,” + 7.” Z1”)} with similar values for sin e2, cos ea. But if A, be the true mean activity then in the same way I Enº Cos e” 7. A z = —t —tº-º-º- & - - 7/ T(R.” + m” L.”)}ſ O sin 77 t sin (n : – eg) dº - I. AE,” COS e - I. Eº A's - (61) 2 (R.3-F 7:37.); T 2 F.3-E 7:27. ' Hence 4m - A’z R.? + m? L12 - - I + 7:2 riº (62) A', R. R. R, + Za Z, 1 + m2.T.T., * where T1, T2 are written for Zi/A'i, Z2/A’, respectively, the so- called “time constants” of the two parts of the circuit, 298 CASE OF NoN-INDUCTIVE CIRCUIT. Now in general T1 > r2, hence as a rule the wattmeter will give too high a result. If the time constants and the period of alternation be known, then Am can be calculated from the apparent activity by this equation.* In any case in which a wattmeter is inapplicable, if the actual resistance of the portion of the circuit considered is known, and the mean square of the current can be measured with accuracy, the product of the two will, as shown above (p. 285), be the true mean value of the activity. This of course will be given in watts, if the resistance is taken in ohms and the current in amperes. As we have seen above, the proper mean value of the current, and of the difference of potential, and therefore also of the activity, can be found for any part of a circuit in the case of negligible self-induction, either by means of an electro-dynamometer, or by means of an electrometer, when the resistance of the part of the circuit is known. When the resistance is unknown or uncertain, as for example in the case of incandescence lamps, the current and difference of potential may be measured for the lamp circuit in the following manner. A coil of german silver wire, having a resistance considerably greater than that of the lamps as arranged, constructed as described above (p. 188), so as to have no self-induction, is connected in series with a current-meter between the terminals of the machine so as to be a shunt on the lamps. The lamps are brought to their normal brilliancy, and the mean square C* of the current through the german silver wire measured. If R be the resistance of this wire, including, * This result was given without proof by Prof. Ayrton in his remarks on “Testing the Efficiency of Transformers,” Proc. Soc. Tel. Æzig, and Æð. Feb. 1888. For methods of determining time-constants, see the author's Theory and Practice, &c., vol. ii. APPLICATION OF ELECTROMETER. 299 if appreciable, the resistances of the current-meter and its connections, and ſº be great in comparison with the coefficient of self-induction of the current-meter divided by 7, we have for the mean square V* of the difference of potential between the terminals of the lamp system, the value C* ſ”. The current-meter is now employed to measure the whole current flowing to the lamps while their brilliancy is kept the same. Denoting the mean square of this current by C*, we have for the value A, of the mean activity spent in the lamp system, the equation A, - VC = C'C' /č . . . . (63) An electrometer may be used in the following manner to give the mean square of the current, and of the difference of potential for any part of a circuit whether containing motors, or arc lamps, or any arrangement with or without counter-electromotive force or self-induction. A coil of thick german silver wire (or to prevent sensible heating a set of two or more equal coils arranged in multiple arc) having no self-induction is included in the part of the circuit considered, so that the current to be measured also flows through the wire. The mean square of the differ- ence of potential between the ends of this resistance is measured as described above (p. 279) by connecting one pair of quadrants of the electrometer to one end, and the needle and the other pair of quadrants to the other end, and the mean square C" of the current found by dividing by the square of the resistance of the wire. The mean Square of the difference of potential between the terminals of the part of the circuit considered, is then found in the Same manner. The product is not generally to be taken as the mean Square of the activity in the part of the circuit 3OO MEASUREMENT OF TRUE ACTIVITY considered, for it is evident that in this case what is obtained is the value of }ſ, 172 d: X ſcº, O where V and C are the difference of potential and the current at any instant. The square root of this quantity is not generally the same thing as I ſ?" +, V Co. # }ſ. the true mean value of the activity. This is, however, given directly by the following method.* - Let the two ends of the resistance coil of zero self- induction and known resistance ſº be called A and B, and let the extremities of the portion of the circuit for which the measurements are to be made be called C and D. One of the pairs of quadrants is connected to A, the other pair to B, and the needle to C, and the reading d say taken. The quadrants remaining as they were, the needle is connected to D, and the reading d" taken. Now if at any instant V be the potential of A, V., of B, VA’ of C, and V, of D, we get by (18) above Å; ſ 7 f + 1/. d = #ſor – Vº (V, — }*) ... A ſº... ... (…, P + P. º d" = #ſº - 19 (P. — 1.- '. *).) * This method is described by A. Potier, Jozernal de Physiºnee, t. ix. p. 227, 1881, but was independently invented also by Prof. W. E. Ayrton, and Prof. G. F. Fitzgerald (see Prof. Ayrton on “Testing the Power and Efficiency of Transformers,” Proc. Soc. Tel. Engs. and EZs., Feb. 1888), 2 (64) 2 BY ELECTROMETER. 3OI and by subtraction and division by AE /* d — d" I ſT / *...* - ſº ſº. — Po (P - rod. (6) But we have seen that the expression on the right hand side of (65) is the true mean value of the activity required. If V1' – V. be great in comparison with Vi – V., and A, Say, be connected with the case of the instrument, the first of (64) becomes A ſ T f a- #ſor, − V.) V, dź . . . (66) 0 If A and D coincide V, - V1, and the activity in the part of the circuit between C and D is, by (61), given by (62) alone when put in the form d I ſ? y \ r > / A. - %ſ, ºr - Foºd, & This observation is due to Mr. Sayers, a pupil of Professor Ayrton. It is thus possible in the case supposed to use an electrometer as a direct reading wattmeter. If a quadrant electrometer is used as here explained, care must be taken to see that the equation (17) holds for the instrument, and if it does not hold what the deviation from fulfilment of this relation is. Dr. Hopkinson found (Phil. Mag. Aff. I885), that the indications of his instru- ment were very exactly expressed by the equation /3 V -- i. D = — v, - V. (V – Kii º Hº (% 2) 2 ) where m is a small constant. Hence for high values of V it was necessary to know and use this second constant. 3O2 ADJUSTMENT OF ELECTROMETER. The deviation from fulfilment of the ordinary equation here shown was found to be in great part due to down- ward electrical force on the needle caused by its hanging a little too low in the quadrants. If the needle hangs at its proper level and is otherwise properly adjusted, and the quadrants are close, the equation may be taken as accurate enough for practical purposes, but as a precaution it should be tested by a battery which can be made to give different relatively known differences of potential. CHAPTER X. MEASUREMENT OF INTENSE MAGNETIC FIELDS. J E have seen above (p. 45) that every element of a conductor carrying a current in a magnetic field is acted on by a force tending to move it in a direction at right angles to its length and to the direction of the resultant magnetic force at the element, and have stated how the magnitude of the force may be calculated in terms of the intensity of the field and the strength of the current. Hence if we know the strength of the current flowing in a conductor placed in a magnetic field, and measure the force exerted in virtue of electromagnetic action on any element of the conductor, we can calculate the intensity of the field at the element. On this principle are founded the following simple methods, mainly suggested by Sir William Thomson, of determining in absolute measure the intensity of magnetic fields in dynamo machines or other electromagnetic apparatus. We shall take first the case of two long straight pole faces oppositely magnetized and placed at a short distance apart, facing one another, with their lengths vertical. In the middle of the space between the poles a wire, w (Fig. 65), somewhat longer than the poles, so as to extend a little above and below them, is hung vertically, by a cord 3O4 MEASUREMENT OF INTENSE MAGNETIC FIELDS. of four or five feet in length, attached near its upper end from a fixed peg above, and is stretched by the weight, W, attached near its lower end. Two pendulums, made A. S1 Aſ2 H--- *—E 2. TTTTTTI, O |TTTTTTTTi O Tº łºmmeº Tº TTTTTTTTTTTTTITLE: sQ - | * - QL) I t g l i !--- P2 FIG. 65. of weights, P1, P2 carried by fine threads, are hung from two sliding pieces which can be moved along a graduated cross-bar S, above, so placed that the ends are as nearly as possible in a plane parallel to the pole faces MEASUREMENT OF INTENSE MAGNETIC FIELDS. 305 and passing through the middle of the space between them. The two pendulum threads and the wire, w, are thus nearly in one plane. One of these pendulums is made so long as to have its bob below the level of the lowest part of the pole faces, while the other has its bob a little below the level of the top of the pole faces, and the former is placed at the greater distance from the suspended wire. A thin thread attached to the upper end of the suspended wire is carried out horizontally and made fast at its other end to the suspension thread of the nearer pendulum. A similar thread is attached at one end to a point of the wire near the bottom of the pole faces, and carried out Similarly and made fast at the other end to a point nearly on the same level in the suspension thread of the further pendulum. The upper and lower ends of the wire, ºv, are placed, as shown, in mercury cups, to which are also connected the electrodes of a battery, by means of which a current can be sent through the wire w, and measured by means of a galvanometer in the circuit. A scale, S2, is placed a little behind the plane of the threads, so that the position of a point in each, on the same level near their lower ends, can be easily read off. When an experiment is made, the sliding pieces, Žiž», are moved towards the left until the threads, #1 *, are quite slack, and the positions of each thread on the upper and lower scales are read off and noted. The position of the wire, w, when #1, #2 are quite slack is also marked at the upper and lower ends of the pole faces or elsewhere. A current is then sent through the wire, w, in such a direction that the electromagnetic force acting on it moves it to- wards the left. The sliding pieces, #1, #2 are then moved towards the right so as to cause the pendulums to pull the wire by means of the threads, #1, #2, back again to its initial X 306 MEASUREMENT OF INTENSE MAGNETIC FIELDS. position. When the upper and lower ends have come back to their former positions, the electromagnetic force on the wire is balanced by the pulls exerted by the pendu- lums. The positions of the pendulum threads are again read off on the upper and lower scales and noted with the strength of the current flowing in w. From these results we can easily calculate the average intensity of the field at the place occupied by the wire, zw. For let W be the mass of each of the pendulum bobs in grammes, d the distance through which the top of the pendulum thread has been carried by £1 to the right of the point of the thread opposite to the lower scale S2, d, the correspond- ing distance for the other pendulum, Z the vertical distance between the levels of the tops of the pendulum threads and the lower scale measured in the same units as di, d2, Z the length of the opposed pole faces, and C the strength of the current in C.G.S. units (one-tenth the number of amperes). The downward force in dynes on each of the masses in W.g., where g is the acceleration in centimetres per second per second (=981'4 in latitude of Glasgow) produced by gravity in a falling body at the place of experiment. The total pull to the right exerted by the threads on the wire is therefore Wg (d. + dā)//, and this is equal to the pull towards the left on the wire produced by the electromagnetic action. If I be the average inten- sity of the field along the wire in C.G.S. units, we have for this pull in dynes I L C. Hence we get the equation I LC - Wºjº. and therefore, I = }; -º-º: . . . . . (I) MEASUREMENT OF INTENSE MAGNETIC FIELDS. 307 In an experiment made on September 16, 1882, with a similar arrangement, W was IOO grammes, / IOO centi- metres, C. "I 88 in C.G.S. units of current, Z. 30 centimetres, and di -- d. 25'84 centimetres. Hence, y 100 × 98 I 4 25'84 e - 6. 30 × 188 I OO 449 The wire, w, should not be so flexible as to bend per- ceptibly under the influence of the forces to which it is subjected, so that the value of I found may be nearly enough the average value of the intensity along a straight line in the space between the pole faces. In cases in which, as in many dynamo machines, the opposite pole faces of the electromagnets are at a con- siderable distance apart, with or without pieces of soft iron in the intermediate space, it is practically useful to find simultaneously the magnetic field intensity along two lines in the same plane, one in the vicinity of each pole face. This may be done by so placing the electromagnets that the two lines along which the field is measured are in a horizontal plane, and using, instead of the single wire carrying the current, a rectangle of copper wire, or strip, of which the opposite sides are in these lines, supported on knife-edges in the bisecting line parallel to the pole face so that it can turn round that line as axis. The frame should be weighted symmetrically on the two sides of the line of knife-edges, so that it rests with just enough of stability in the horizontal position. The ends of the wire or strip forming the rectangle are brought out one above the other at one of the knife-edges with a piece of insulating material between them, and bent over so that the end of each dips into a mercury-cup in line with the knife-edges. The electrodes of a battery are connected X 2 308 MEASUREMENT OF INTENSE MAGNETIC FIELDS. to the mercury-cups and a measured current is sent round the rectangle. Since the poles have opposite magnetisms, the electromagnetic action causes one side of the rectangle to move upwards, the other side to move downwards, and thus turns the rectangle round the knife edges. The moment of the electromagnetic forces is balanced by the action of weights, which may be riders of known weight made of wire, placed on the sides of the rectangle, which is thus brought back to its initial position. If we call / the average intensity of the fields along the two sides of the rectangle in the equilibrium position, and C the current strength, both as before measured in C.G.S. units, L the length of each side, and d the distance be- tween them in centimetres, the moment of the electro- magnetic forces round the knife-edges is / C / d. The opposite moment resisting the motion is, if only one weight of PV grammes at a distance of d' crims. from the line of knife-edges is used, Wg d". Hence, equating these moments, we get Wg d" CZ 7 ° from which / can be calculated. If more than one weight, W, is used, each must be multiplied by its distance from the line of knife-edges, and the sum of the products multiplied by g for the equilibrating moment. In some cases it may be convenient to use more than one turn of wire in the rectangle. If there be 77 turns, each of length L, 72 / is to be used instead of Z in the formula above. An obvious modification of this arrangement, which may be useful in some cases, is a rectangle suspended in a vertical plane, and kept in equilibrium in the proper MEASUREMENT OF INTENSE MAGNETIC FIELDS. 309 position when no current is flowing through it, by means of a bifilar suspension, or a single thread or thin wire under torsion. When a current is sent through the frame, it is deflected round a vertical axis by the electromagnetic action, and is brought back to the initial position of equi- librium by means of two pendulums, the points of sus- pension of which are on sliding pieces which can be moved along horizontal parallel bars fixed above at right angles to the plane of the rectangle when in the equili- brium position, and in the same vertical planes as its sides. Each pendulum cord has attached to it a thread which pulls horizontally at the middle of one side of the rectangle. When the rectangle is deflected, the sliding pieces are moved in opposite directions, so that, in con- sequence of the opposite inclinations of the pendulums to the vertical, forces restoring equilibrium are applied to the rectangle. As before, we have for the electromagnetic couple / C / d. Supposing the two points of suspension of the pendulums to be on one level, and the points of attachment of the pulling threads to the pendulum cords to be on a level lower by a distance of / cms., the distances of the verticals through the points of suspension from the Corresponding verticals through the attachments of the threads to the pendulum cords to be dº, d, cms. for the respective pendulums, and W grammes the mass of each bob, we have, for the moment of the equilibrating forces, the value Wg. *# * Hence, equating moments, we get y = **.*, *, *. . . . . . (3) C /. / If I L is the same for both sides of the rectangle, d, and 3 Jo MEASUREMENT OF INTENSE MAGNETIC FIELDS. d, will be equal; but in general there will be a small difference between the two values. In some important practical cases the pole faces are of small area and are at only a small distance apart. If there is room, a small rectangular coil, similar to that of a siphon recorder (see Fig. 66), but of comparatively few turns of wire, and without an iron core, may be hung, as described above, between the poles, with its plane parallel to the lines of force, by a bifilar or a torsion thread or wire, and a measured current sent through it. A rigid projecting arm fixed to the coil at the middle of its upper end and at right angles to the plane of the coil, has rest- ing against it the Suspension thread of a pendulum, attached at its upper end to a sliding piece movable along a horizontal bar carrying a millimetre scale, above and at right angles to the projecting arm ; and by this means the coil is brought back to the initial position. When no current is flowing through the coil, the thread is allowed to hang vertically just touching the bar and the reading on the scale above noted. Let the difference between this reading and that obtained when the pendulum is deflected be d, and let / be the vertical height of the point of suspension above the projecting arm. The horizontal force exerted by the pendulum is Wg.d//, and the moment of this round the vertical axis about which the coil turns Wgr.d//, where r is the distance of the pendulum thread from the central plane of the coil. If ze be the number of turns in the coil, 6 cms. its mean breadth, and Z cms, the mean length of each side, the moment of the electro- magnetic forces is n & / /. C. We have, therefore, * - jº, . . . . . . (4) This method has frequently been used for the determina- MEASUREMENT OF INTENSE MAGNETIC FIELDS. 311 tion of the magnetic field-in- tensity of the magnets of siphon recorders. The coil hanging in its place was used as the measuring coil, and when no current was flowing through it, was kept hanging vertically in stable equilibrium with its plane parallel to the lines of force by the bifilar threads attached beneath it. These threads were kept taut and bearing against the bridge /3 by the weights W, resting on a plane slightly inclined to the vertical. A current from one or two cells was then sent through the coil, and the difference of potential between the terminals of the coil measured by means of a potential galvanometer. The thread of the pendulum was made to pull against the project- ing aluminium arm to which the siphon is attached as shown in the figure, so as to bring the coil back to the initial position. The value of d was then read off, and that of C deduced from the fºr 66. known resistance of the coil and the result of the measurement with the galvanometer, and being substituted with those of the other quantities W., 77, Ö, &c. in (4), gave the value of Z. 312 MEASUREMENT OF INTENSE MAGNETIC FIELDS. The field intensities of siphon recorders have sometimes been determined by the following method, which is in- teresting theoretically. Advantage is taken of the signal-coil, which consists of a rectangular coil a little more than 5 cms. long and 2 cms. broad, made of thin wire and supported by a silk thread above, so as to hang in a vertical plane round a rectangular core of iron, which nearly fills, but nowhere touches, the coil. To the lower end of the coil two silk threads are attached, as shown in Fig. 66, and are stretched against a bridge B by two weights resting on the inclined plane W. This bifilar arrangement gives a directive force, tending to bring the plane of the coil into parallel- ism with that of the bifilar threads ; so that when the coil is disturbed from that position; which is one of stable equilibrium, and then left to itself, it will, if the circuit be not closed, vibrate about the position of equilibrium with a determinate period of oscillation, with slowly diminish- ing range, until at last it comes to rest. But if the cir- cuit be closed through a high resistance, the coil will come more rapidly to rest ; and if we gradually diminish this resistance, deflecting the coil through the same angle and noting its subsidence at each diminution, we shall find it come more and more quickly to rest, until a resistance is obtained with which in circuit it just returns to the position of equilibrium without passing that position. When this resistance has been determined, the strength of the field can be calculated. Let 6 be the deflection of the coil from the position of equilibrium at time t, and Tits period of oscillation when the circuit is not closed. We have then, neglecting the resistance of the air and other disturbances, for the equation of motion, MEASUREMENT OF INTENSE MAGNETIC FIELDS. 313 d°6 2 # + 40 = o . . . . . (5) Let now the circuit of the coil be closed ; a retarding force due partly to air-resistance, but in the main to the current induced in the wire, and, if the effect of self- induction be neglected, proportional to the angular velocity, will act on the coil ; and the equation of motion or this case will be of the form 2 2 # + 4 + 4.6 = o . . . . (6) For let I be the mean intensity of the magnetic field over the space occupied by the coil at time t, Z the inductance of the circuit for that position of the coil, ſº the total resistance in the circuit, p, the moment of in- ertia of the coil round a vertical axis passing through its centre, Z the effective length of wire in the coil (that is, the length of wire in its two vertical sides), and & the mean half-breadth of the coil. If we call AW the number of lines of force which pass through the coil at time #, and y the strength of the induced current in the coil at that instant, we have plainly AW = b / / sin 6 — Zy. The rate at which Wincreases per unit of time is therefore aſ W d/6 . . d I d * : * = / / / 6 * * / 6 * * – (/Ly): a f cos dº, + . . sin 6%, - #(º); and if 6 be small, and / be therefore supposed the same for every position of the coil, we have approximately aſ AW d 6 d * : * = & J / - – - (/, v). d : dź ...' y) But dA/dt is the electromotive force due to the inductive 314 MEASUREMENT OF INTENSE MAGNETIC FIELDs. action ; hence the current y is by Ohm's law given by the equation /, / / aſ 6 I d = . . . . . — (L y). # , ~ ; ; (+3) It was assumed that the second term of this expression for y would prove negligible in comparison with the first ; and this assumption was so far justified by the results of the experiments, which agreed fairly well with results obtained, for other instruments of the same pattern, by a modification of the second method described below. The couple due to the action of the field on the current is Ö//y; and therefore, on the supposition of negligible self-induction, the retardation of the angular velocity of the coil at time # is * d6 32 /2 22 dé dź a R 77° Hence (2) becomes d2 0 , 52 /2 /3 d6 4 trº dº? T Tº A ºf #6 – o . . . (7) The motion represented by this differential equation will be oscillatory or non-oscillatory, according as the roots of the auxiliary quadratic are imaginary or real– that is, according as 4T/T ~ or < &*A*/*/u R. Hence, if A be the critical resistance at which the motion just ceases to be oscillatory, we have _ 4 ºr /ć p. /* = }; . . . . . . (8) When Zand & are expressed in centimetres, p. in grammes and centimetres, T in seconds, and R in cms, per second, I is given by this equation in absolute C.G.S. units of magnetic field intensity. MEASUREMENT OF INTENSE MAGNETIC FIELDS. 31.5 The method of experimenting consisted in first finding the value of T, the free period of vibration of the coil with its circuit uncompleted, then finding the resistance which, being placed in circuit with the coil, just brought the needle to rest without oscillation. This resistance was con- veniently obtained by means of a resistance-box included in the circuit, and therefore added no self-induction to that in the coil. An aluminium arm attached to the coil, and carrying the siphon, served as an index to render the motions of the coil visible. The resistance A2 was first made much too great, so as to give a slow subsidence, then gradually diminished until the value which just prevented oscillation was reached; and it was found that this value could be determined easily within 50 ohms, and with great care, to 20 ohms. As the experiments on the recorders had to be made somewhat hurriedly, and on account of the disturbances neglected, and, further, as p. was taken as equal to Wö%, where W is the mass of the Coil, the results could not be taken as giving more than a rough approximation to 1 : but those for two instruments are given below in illustration of the method. For both instruments the values of W, Z, and 5 were the same, and were respectively taken as 3.343 grammes, 3338 cms., and ’95 cm. Each coil had a mean vertical length of 5' 3 cm., a mean breadth of I-9 cm., and contained 45'72 metres of fine wire arranged in 290 turns, and had a resistance of about 500 ohms. 7. A’. A. (I) 465 sec. 3330 × Io” Crns. per Sec. 5150 C. G. S. (2) 500 , 3530 × Io" , 3, 5 I2O >, This method is obviously applicable in any case in which a coil can be suspended by a torsion wire, or bifilar, 316 MEASUREMENT OF INTENSE MAGNETIC FIELDS. or other arrangement so as to have a measurable free period of vibration.* The following method, which has been frequently used in the Physical Laboratory of the University of Glasgow, is very convenient and useful in many cases. It consists in exploring the magnetic field by means of the induced current in a wire moved quickly across the lines of force over a definite area in the field. The wire is in circuit with a reflecting “ballistic" galvanometer— that is, a galvanometer the system of needles of which has so great a moment of inertia that the whole induced current due to the motion of the wire has passed through the coil before the needle has been sensibly deflected. The deflection thus obtained is noted, and compared with the deflection obtained when, with the same or a smaller resistance in circuit, a portion of the conductor is made to sweep across the lines of force over a definite area of a uniform field of known intensity, such as that of the earth or its horizontal or vertical component. In performing the experiments, it is necessary to take precautions to prevent any action except that between the definite area of the field selected and the wire cutting its lines of force. For this purpose the conducting-wire, which is covered with insulating material, is bent so as to form three sides of a rectangle, the middle one of which is of the length of the portion of field to be swept over. This side is placed along one side of the space over which it is about to be moved so that the connecting wires lie along the ends of the space ; and the open rectangle is then moved in the direction of its two sides * The method just described gives (theoretically) a means of determining the ohm. For suppose the coil hung in a sufficiently intense and uniform field, the intensity of which has been measured by another method, and the decrement of the oscillatory motion produced by the induction observed. Then the resistance could be calculated. MEASUREMENT OF INTENSE MAGNETIC FIELDS. 317 until the opposite side of the space is reached. The con- necting wires thus do not cut the lines of force, and the induced current is wholly due to the closed end of the rectangle. Instead of a single wire cutting the lines of force, a coil of proper dimensions (for many purposes conveniently of rectangular shape), the mean area of which is exactly known, may be suspended in the field with its plane parallel to the lines of force, and turned quickly round through a measured angle of convenient amount not exceeding 90°; or it may be suspended with its plane at right angles to the lines of force and turned through an angle of 180°. If n be the number of turns, A their mean area, and / the mean intensity of the field over the area swept over in each case, then, in the first case, if 6 be the angle turned through, the area swept over is 7. A sin 6 and the number of lines cut is 72 M A sin 6 ; in the Second, the area is 2 m A, and the number of lines cut is 2 m / A. In order that with the feeble intensity of the earth's field a sufficiently great deflection for comparison may be obtained, it is necessary that a relatively large area of the field should be swept over by the conductor. One con- venient way is to mount on trunnions a coil of moderately fine wire of a considerable number of turns wound round a ring of large radius, like the coil of a standard tangent galvanometer, and arranged with stops so that it can be turned quickly round a horizontal axis through an exact half-turn, from a position in which its plane is exactly at right angles to the dip. This coil, if the ballistic galvano- meter is sensitive enough, may always remain in the circuit. The change in the number of lines of force passing through the coil in the same direction relatively to the coil, pro- duced by the half-turn, is plainly equal to twice as many 318 MEASUREMENT OF INTENSE MAGNETIC FIELDS. times the area of the turn of mean area as there are turns in the coil (the effective area swept over) multiplied by the total intensity of the earth's magnetic force at the place of experiment. Or, and preferably when the horizontal component of the earth's magnetic force has been determined by experiment, the coil may be placed in an east and west (magnetic) vertical plane, and turned through an exact half-turn. The magnetic field intensity by which the effective area is to be multiplied is in this case the value of H.3% A sufficiently large area of the earth's field for compari- Son may, in Some cases, be obtained very readily by carry- ing the wire along a rod of wood, say two or three metres long, and suspending this rod in a horizontal position by the continuations of the conductor at its ends from two fixed supports in a horizontal line at a distance apart equal to the length of the rod, and securing the remaining wires in circuit so that they may not cause disturbance by their accidental motion. The rod will thus be free to Swing like a pendulum by the two suspending wires. The pendulum thus made is slowly deflected from the vertical until it rests against stops arranged to limit its motion. When the needle is at zero, the rod is quickly thrown to the other side against similar stops there, and caught. The straight conductor thus sweeps over an area of the vertical component of the earth's field equal to the product of the length of the rod into the horizontal distance between the two positions of the conductor at the ex- tremities of its swing. The rod may be placed at any azimuth, as the suspending portions of the conductor in * The method of reducing results of observations to absolute measure by means of an earth inductor was used by Professor H. A. Rowland in his experiments on the magnetic permeability of iron, steel, and nickel.—Phil. Mag, vol. 46, 1873. MEASUREMENT OF INTENSE MAGNETIC FIELDS. 319 circuit, moving in vertical planes, can cut only the horizon- tal lines of force ; and the induced currents thus produced have opposite directions and neutralize one another. The calculation of the results is very simple. By the theory of the ballistic galvanometer* (the same mutatis mutandi's as that of the ballistic pendulum), if q be the whole quantity of electricity which passes through the circuit, and if 6 be the angle through which the needle has been deflected, or the “throw,” we have, neglecting air resistance, &c., 2 . Aſ . 6 w , - . Vºn: . . . . . (9) 772 2 where p is the moment of inertia of the needle and attach- ments, m the magnetic moment of the needle, H the earth’s horizontal magnetic force, and G the constant of the galvanometer. If 6 be small, as it generally has been in these experiments, we have - Mºff. Ø G m . . . . . (IO) and the quantities of electricity produced by Sweeping over two areas, A and A', are directly as the deflections. Let A be the total area swept over in the field or portion of field the mean intensity / of which is being measured, A' and I' the same quantities for the known field, R, K2' the respective total resistances in circuit, 7, 7' the quantities of electricity generated in the two cases, 6, * For further particulars see the author's Theory and Practice of Absolute Measzerements in Elect2ticity and Magnetisma, vol. ii. 320 MEASUREMENT OF INTENSE MAGNETIC FIELDS. 6" the corresponding deflections supposed both small; we have , - 42 – 1,797/9 º A T G. 7/7 ’ g' +: 4' Z. - I Mººg A” G. 772 and therefore A' A' 6 , , = j. Z' . . . . . . ( ) If convenient, 6 and 6' may be taken as proportional to the number of divisions of the scale traversed by the spot of light in the two cases. The error caused by neglecting the effect of air resist- ance, &c., in diminishing the deflection will be nearly eliminated if A and R' be chosen so that 6 and 6 are nearly equal. The following new method of reducing ballistic obser- vations to absolute measure has been given by Sir William Thomson. A short induction coil wound round the centre of an ordinary magnetizing helix, whose length is great com- pared with its diameter, is kept in circuit with the galvano- meter. A measured current is sent through the wire of the helix, and when the needle is at rest the circuit of the helix is broken, and the galvanometer deflection read off. If AV be the number of turns of wire per cm. on the helix, C the current in electromagnetic C.G.S. units, the magnetic force within it is 4 T n C parallel to the axis; and if A' be the proper mean area of the cross-section of the helix, and 7' the number of turns in the induction coil, the number of lines (unit tubes) of force passing out of the galvanometer circuit when the current is stopped is 4T AW n'A' C.” * See the author's Theory and Practice of Absolute Measurements in Electricity and Magnetisma, vol. ii. IDEAL SURFACE DISTRIBUTION OF MAGNETISM. 321 Hence, R' denoting the total resistance in circuit, the total quantity q' of electricity generated is 4 T AV n'A' C/A" and instead of (9) we get A' A' 6 Z = 47 Wm. Czºg * * * * (I2) The ballistic method of investigation was also used by Prof. H. A. Rowland (Phil. Mag. vol. i., 1875) for the deter- mination of the ideal surface distribution of magnetism on magnets. A thin ring of wire was made just large enough to pass round the magnet experimented on, and was placed in circuit with a ballistic galvanometer. It was then, while encircling the magnet and held with its plane at right angles to the axis of the magnet, slided quickly along the magnet through equal short distances, and the deflection of the needle noted for each motion. The deflections thus obtained gave for thin magnets an approximate Com- parative estimate of the density at different points along the magnet of the surface distribution of ideal magnetic matter by which the action of the magnet could be produced, and the results were reduced to absolute measure by means of an earth-inductor. This method, used along with Sir William Thomson's method of reduction to absolute measure, gives a very ready means of estimating with much exactness the total quantity of imaginary magnetic matter in one pole or one end of a magnet, whether of bar, horse-shoe, or other shape. The ring, which for the present purpose may be larger, and thick enough to contain any convenient number of turns, is placed at the centre or nearly neutral region of the magnet, and then quickly pulled off and away from the magnet, and the galvanometer deflection (6) noted. A measured current is then sent through the helix, and the deflection (6') produced by suddenly opening the circuit of Y 322 IDEAL SURFACE DISTRIBUTION OF MAGNETISM. the helix also observed. Let n be the number of turns in the ring of wire, and p the total quantity in C.G.S. units of imaginary magnetic matter on the portion of the magnet swept over, then the number of lines of force cut through by each turn of wire in the ring is 4 trop, and if R be the total resistance in circuit, the total quantity (g) of electricity generated is 4 trn p/R. We have therefore 4 tº 4 – 2 J/uſ/... 6 T A G N 772 sing and for the helix we get from the calculation above *. , 4 ºr Nzz' A' C 2 p A/. 6' ſ/ -: —º- := G V m S1 Il 2. By division we get * 6 zz' A' Sin 2 q = WA’ C — . . . . (13) / ... • 6 zz AC" sin 2. and if the deflections are small angles, q) – watcº . . . . (14) This equation is of course also applicable to the reduc- tion to absolute measure of the results of determinations of magnetic distribution made by the ballistic method. The value of p deduced for each deflection divided by the area of the corresponding small portion of the magnet is approximately the surface density of the ideal distribution, the distribution on the end faces being of course included in the end deflections. CHAPTER XI. THEORY OF THE DIMENSIONS OF THE UNITS OF PHYSICAL QUANTITIES. E have, in p. 70 above, explained the term change- ratio of a physical quantity: in order to show clearly the relations of the various absolute units of electrical and magnetic measurement to the units on which they are based, we shall here investigate for each of the principal quantities the formula of dimensions from which the numerical value of the change ratio is to be found in any particular case. A physical quantity is expressed numerically in terms of some convenient magnitude of the same kind taken as unit and compared with it. The expression of the quantity consists essentially of two factors, a numeric,” and the wnit with which the quantity measured is compared ; and * The term nazmeric has been introduced by Prof. James Thomson (Thomson’s “Arithmetic,” Ed. LXXII., p. 4). It denotes a number, or a proper fraction, or an improper fraction, or an incommensurable ratio. We shall find it convenient to employ it here where we wish to lay stress on the fact that we are dealing with what are essentially numerical expressions. Of course what is actually meant by the conveniently brief expressions “a length, L,” “a mass, M,” “a force, F,” and the like, is simply that L., M., F, &c., denote the numerics which express the respective quantities in terms of the units chosen, that is, are, as we shall say below, the 7zzzzzzerics /or the quantities in terms of those units. Further in such phrases as “the product of mass and velocity,” or “the product of charge and potential,” and so on, the product (or whatever other function is specified) of the numerics is of course what is intended. If all such expressions were made verbally un- exceptionable, the resulting prolixity would be intolerable. 3. Y 2 324 DIMENSIONS OF UNITS. the numeric is the ratio of the quantity measured to the Quantity chosen as unit. Thus when a certain distance is said to be 25 yards, what is meant is that the distance has by some process been compared with the length, under Specified conditions, of a certain standard rod (which length is defined as a yard) and the ratio of the former to the latter found to be 25. The unit of measurement is of course itself capable of being expressed numerically in terms of any unit of the same kind, and in the same way therefore its full expression consists of a numeric and the new unit. Hence if M be the numeric for any physical quantity in terms of any unit, W’ in terms of another unit, and 7. the numeric for the first unit in terms of the second, we have M’ = n . AW . . . . . . (1) In order therefore to find the numerical expression AV’ of the quantity in terms of the second unit from its numerical expression AV in terms of the first we have to multiply by 7, the ratio of the first unit to the second. This numeric has been appropriately called the change- zatio for the change from the first unit to the second. The change from M to AV' cannot be made unless the Change-ratio ºn, is known. Each unit may have been arbitrarily chosen without reference to any other unit, and 77 determined by some process of measurement ; or the units may have been derived from certain chosen fundamental units, and the ratio deduced from the relation of one system of fundamental units to the other. In the measurements described in this work the units employed are entirely of the second kind here referred tC). FUNDAMENTAL UNITS. 325 The task before us is to determine the manner in which the various derived units involve the fundamental units, that is, we have to determine for each quantity (p. 180 above) the change-ratio m in terms of the funda- mental units. The formula which expresses ze for a unit of measurement of any quantity we shall call the Formula of Dimensions or the Dimensional Formula of the quantity. To prevent the necessity for the con- stant repetition of these terms we shall denote the dimensional formula of any quantity, of which the nu- merical expression in terms of some chosen unit is de- noted by any particular symbol (see p. 326, footnote), by the same symbol inclosed in Square brackets. Thus we denote the dimensional formula of the quantity (2 by the symbol [Q]. Examples of the values of [Q] will be found in dealing with the various units, to which we now proceed. We shall first consider the definitions and relations of the fundamental units in common use and the derivation from them of the units of other physical quantities. In doing so we shall find the dimensional formula in each case and its numerical values for certain changes of units. For brevity we shall sometimes in what follows call the numerical part of the complete expression of any quantity a numerical guanſ?!y of that Áind, as this will at once serve to indicate that we are dealing with a numeric, and refer to the quantity with which it is connected. Thus we shall frequently use the phrases, a numerical length, a 7tt/merical velocity, and the like to denote the numerics for the quantities in terms of the respective units, what- ever these may be, chosen for their expression. In this way L, which denotes the numeric for a length in terms 326 DIMENSIONS OF UNITS. of Some chosen unit, will be called a 7tumerical length, and so on for other quantities.* FUNDAMENTAL UNITS. (I) / ength. The standard unit of length in Great Britain is defined by Act of Parliament in the following terms: ‘f “The straight line or distance between the centres of the transverse lines in the two gold plugs in the bronze bar deposited in the Office of the Exchequer shall be the genuine standard of length at 62° F., and if lost it shall be replaced by means of its copies.” Authorized copies are preserved at the Royal Mint, the Royal Society of London, the Royal Observatory at Greenwich, and the New Palace of Westminster. The comparison of the length of the standard with the lengths of its copies has been effected with the utmost Scientific accuracy, and formed a most elaborate and important scientific investigation. The length of a simple pendulum which beats Seconds has been determined for several places by means of very careful observations, and repeated pendulum experiments at these places would in the event of the destruction of the standard and all its copies give a means of accurately renewing them. * In the former edition of this work L was called ‘‘the numeric of a length,’’ and so for other quantities; but it has been pointed out to me by Prof. James Thomson that as we may say for example that 25 yards is 25 of the unit distance of the yard, so the phrase “the numeric of a length" might be supposed to mean that length multiplied by the numeric, and thus lead to confusion. He prefers to say “the numeric of the unit’’ which expresses the quantity referred to. It was stated in the former edition that the term was used as an abbreviation of “numerical expression "; and therefore the phrase “the numeric of a quantity” meant the numerical expression of that quantity in terms of some chosen unit, but lest there should be any ambiguity I have adopted the phraseology in the text. That the quantity itse//, and not merely its numerical expression in terms of some unit was meant, Prof. Thomson would indicate by the adjective intrinsic, as in an intrinsic length, an intrinsic enezºy. f 18 and 19 Vict. c. 72, July 30, 1855. FUNDAMENTAL UNITS. 327 In France and in most Continental countries the standard of length is the Metre. This is defined as the distance between the extremities of a certain platinum bar when the whole is at the temperature o’ of the Centigrade scale. This rod was made of platinum by Borda, and is preserved in the national archives of France. As in the case of the yard, authorized copies whose lengths have been carefully compared with the standard are preserved in various places. The metre was constructed in accordance with a decree of the French Republic passed in 1795,” which enacted, on the recommendation of a Committee of the French Academy of Sciences, consisting of Laplace, Delambre, Borda, and others, that the unit of length should be one ten-millionth part of the distance, measured along the meridian passing through Paris, from the Equator to the North Pole. The arc of that meridian extending between Dunkirk and Barcelona was measured by Delambre and Méchain, and from their results the standard metre was realized in platinum by Borda. The metre, it is to be observed, is not now defined in relation to the earth's dimensions, and later and more accurate results of geodesy have therefore not affected the length of the metre, but are themselves expressed in terms of the length which Borda’s rod has at O’ C. In the French system the decimal mode of reckoning has been adopted for multiples and sub-multiples of all the units. Thus the metre is divided into ten equal parts each called a decimetre, the decimetre into ten parts each called a centimetre, and the centimetre into ten parts each called a millimetre. Again, a length of * Loi du 18 germinal, an iii. 328 DIMENSIONS OF UNITS. ten metres is called a decametre, of one hundred metres a hectometre, and one thousand metres a kilometre. Of these, in accordance with the prevailing practice of scientific experimenters who adopted the suggestions of the B. A. Committee, the centimetre has been very generally chosen as the unit of length for the expression of scientific results, and on it as unit of length the electric and magnetic units approved by the Inter- national Congress of Electricians held at Paris in 1882 have been founded. The reason for this choice will appear when we consider the unit of mass. We shall denote a numerical length by Z. The dimensional formula is therefore [/]. For example, if we wish to find from the numeric for a length in terms of the yard as unit the numeric for the same length in terms of the metre as unit, we have [Z] = '91439, the ratio of one yard to one metre ; and this of course is equal to 36/39°3704, or 91°439/IOO, &c., the ratios of the numerics for the two units directly obtained according as the inch, or the centimetre, &c., is taken as unit of comparison. Similarly the value of [Z] for a change from the foot as unit to the centimetre as unit is 3O'47945. (2) Mass. The legal standard of mass in Great Britain is the Imperial standard pound avoirdupois, a piece of platinum marked “P. S. 1844, 1 lb.,” preserved in the Exchequer Office. In the Act of Parliament (the Act already referred to) which gives authority to the standard, it is called the “legal and genuine standard of weight; ” and the Act provides that if the standard is lost or destroyed it may be replaced by means of authorized copies, which are kept in the same national repositories as the copies of the standard of length. FUNDAMENTAL UNITS. 329 It is to be noted that the word “weight” used in the Act, is one which is constantly used in two senses : (1) as here, to signify the quantity of matter in a body : (2) in its proper sense, to signify the downward force of gravity on the body. It is evident that these two senses are distinct. The quantity of matter in a body is invariable; the force of gravity upon the body depends on the situation of the body, and may even be zero. At a given place the forces of gravity on different bodies are, as was proved by Newton by pendulum experiments, pro- portional to their masses, and thus a comparison of the weights of different bodies gives a direct comparison of their masses. º The pound has been generally used in Great Britain as the unit of mass for the expression of dynamical results, but in engineering and the arts, larger units, for example, the ton, or mass of 2240 lbs., and the hundred-weight, or mass of I I2 lbs., are frequently employed. The French standard of mass is a piece of platinum called the Ki/ogramme des Archives, made also by Borda in accordance with the decree of the Republic mentioned above. It was connected with the standard of length by being made a mass as nearly as possible equal to that contained in a cubic decimetre of distilled water at the temperature of maximum density, 4°C. The comparison was of course made by weighing, and so far as this process was concerned it was possible to obtain great accuracy, but the density of water is somewhat difficult to determine with exactness, and is still in a Small degree uncertain. The relation between the standards is, however, so nearly that stated above that, for practical purposes, the error may be neglected. But on account of this uncertainty it 33O DIMENSIONS OF UNITS. is important to remember that the standard is defined as the kilogramme made by Borda, and not as the mass of a cubic decimetre of distilled water at 4° C., which it approximately equals. - A Comparison between the French and British standards of mass made by Professor W. H. Miller gave the mass of the Kilogramme des Archives as 15432:34874 grains. The gramme, defined as I/IOOO of the mass of the Kilogramme des Archives, and approximately equal to the mass of one cubic centimetre of water at 4°C., was recommended by the B. A. Commitee as the unit mass for the expression of experimental results generally, and this choice has now been ratified by the general practice of scientific men. The convenience of the adoption of this unit of mass lies in the fact that it is approximately the mass of unit volume of the substance (water at its temperature of maximum density), usually taken as standard of comparison in the estimation of specific gravi- ties of bodies, which therefore become in this case the same numbers as the densities of the bodies. The multiples and sub-multiples of the gramme proceed decimally, and are distinguished by the same prefixes as those of the metre. We shall denote a numerical mass by M, and hence its dimensional formula by [M]. The value of [M] for a reduction from the pound as unit to the gramme as unit is 453'593, for reduction from the grain as unit to the gramme as unit I/I 5'432. (3) Time. The definition of equal intervals of time belongs to dynamics and cannot here be entered on, but according to it the times in which the earth turns through equal angles about its axis are to a very high degree of FUNDAMENTAL UNITS. 33I approximation equal. These intervals correspond to equal intervals of time shown by a correct clock, and if the clock just goes twenty-four hours in the time of an exact revolution of the earth about its axis (or, which is the same, the interval between two successive passages of a fixed star in the same direction across the meridian of any place), showing Oh. Om. OS. each time a certain point of the heavens called the First Point of Aries crosses the meridian in the same direction, it is said to show sidereal time. Though sidereal time is used in astronomical obser- vatories, it is more convenient in ordinary civil affairs to use solar time ; but as the actual Solar day, the interval between two successive transits of the sun across the meridian of any place, varies in length during the year, the standard interval is the average of such intervals, and is called a mean solar day. On account of the orbital motion of the earth the mean Solar day is about 3m. 55-9s. longer than the sidereal day. The mean solar second, defined as I/864OO part of the mean Solar day, is taken as the unit of time for the expression of all scientific results. We have seen that the choice of the fundamental units is entirely arbitrary, and there is nothing in their nature which entitles them in any just sense to the term “absolute.” The unit of time, which is based on the period of rotation of the earth, is, we have reason to believe, subject to a slow progressive lengthening, due to tidal retardation of the earth's rotation, and possibly also to frictional resistance of the surrounding medium ; and as a matter of definition, without reference in all cases to realization, it would be easy to find many much more Satisfactory standards. Thus it has been suggested by 332 DIMENSIONS OF UNITS. Sir William Thomson* that the period of vibration of a metallic Spring, kept in a hermetically sealed exhausted chamber at a constant temperature, or the period of a particular mode of vibration of a quartz crystal (or other Crystal of definite composition) of a specified size and shape and at a given temperature, would be theoretically preferable to the mean solar second, as fulfilling with a much nearer approach to perfection the condition of Constancy. Clerk Maxwell i has also suggested as units of time the period of vibration of a gaseous atom of a widely diffused substance easily procurable in a pure form ; and the period of revolution of an infinitesimal Satellite close to the surface of a globe of matter of the Standard density, which may be any density determined by a definite physical condition of any substance, for example the maximum density of water. This period is independent of the size of the globe £ and it has been pointed out that this advantage would also be obtained by founding the unit of time on the period of a small simple harmonic vibration of a globe of standard density. The mass of a gaseous atom, for example that of a Sodium or hydrogen atom, and the wave-length of a definite line in the spectrum of an easily obtainable sub- stance, for example one of the O lines in the spectrum of Sodium, might be chosen as the units of mass and length. These units would be quite definite since, according to the kinetic theory of gases, the atoms of any one substance are undistinguishable from one another by any physical teSt. * Electricity and Magnetism, vol. i. p. 3; Thomson and Tait, Nat. Phil, vol. i. part i. p. 227 (second edition). + /öza. f For water at the temperature of maximum density, it is approximately Ioh. 3m. DERIVED UNITS. 333 We shall denote a numerical time-interval by 7. The dimensional formula of time will therefore be [T]. DERIVED UNITS. Let us suppose that the numerical expression of a physical quantity is given by the equation, N = C. L. M.” 7". Z,” M. T.' . &c., . . (2) where L1, Z2, &c., M1, M2, &c., Zi, T., &c., are numerics for different lengths, masses, and times in terms of a cer- tain chosen unit for each, and C is a numerical multiplier (generally equal to unity) which does not depend on the units adopted. Now let other units of length, mass, and time be chosen and let AV be the numerical expression of the same quantity in terms of these units, and Zi, L'2, &c., M'i, M'2, &c., T'i, Z’2, &c., those of the lengths, masses, and times. Then we have N' = C. L. M',” Z’,”. Zºº M'. 7". &c. . (3) But by equation (I) Z” = Z.” [/]", M” = M”[M]”, and so on. Hence (3) becomes, N = C. L. M.” T". Lº My 7,” . &c. [Z]***** º [M]” + 2 + &c. e [7] Frt &c. © & © (4) By equation (I) therefore the dimensional formula [W] of the quantity is [Z]**** [M]***** [7]***** In accordance with the notation [AV], we shall denote this in future by the more convenient expression [złºrse. Mºn-F7+&c. zºr-sº The numerics / -- ?) + &c., &c., correspond to what 334 DIMENSIONS OF UNITS. Fourier * called exposants des dimensions of the quantities which entered into his analysis, and it is these numerics, not the dimensional formulas, which are properly the “dimensions” of the units. It was pointed out by Fourier that in equations involving the numerics for physical quan- tities every term must be of the same dimensions in each unit, otherwise some error must have been made in the analysis. This consideration affords in physical mathe- matics a valuable check on the accuracy of algebraic work, It is obvious from equation (I) or (4) that the dimen- sional formula of the product of any number of numerics M, N, for different physical quantities is the product [AV1. W. &c.] of their dimensional formulas, and more generally that the dimensional formula of the product [N,”. AW.”.] &c., of any powers whatever of these ex- pressions, is the product of the same powers of the corresponding dimensional formulas. We are now prepared to find the dimensional formulas of the various derived units. The process will consist in finding for each quantity the formula corresponding to the right-hand side of (2), and thence deriving according to (4) the proper formula of dimensions. We shall consider first the units of Area, Volume, and Density; then the various dynamical units which are involved in those of electrical and magnetic quantities. Area. The general formula for the area of any surface can be put in the form C/Lº, where L is a numeric expressing a length, and C is a numeric which does not change with the units. Hence by (4) the formula of dimensions for area is [LP]. Wo/ie/le. Similarly the formula for a numerical volume * Théorie Analytique de la Chalezer, Chap. II. Sect. IX. DERIVED UNITS. 335 can be written CL8, and the formula of dimensions is [L3). Density. The Density of a body is numerically expressed by the numerical mass per unit of volume. We shall denote it by the symbol D. If the body be of uniform density, the density is obtained by finding the mass contained in any given volume of the body: the ratio of the numerical mass to the numerical volume is the density. If the body be of varying density, the density at any point is the limit towards which the ratio of the numerical mass contained in an element of volume including the point, to the numerical volume, approaches as the element is taken smaller and smaller. Thus if 8 V be an element of volume including a point at which the density is D, and & M be the mass of the element, we have . . . § Al/ aſ /l/ AD = Limit § V - aſ 17 In either case we have for the numerical expression of the volume taken CZ", and for that of the mass contained in it M. Hence [D] = [M Z-3]. The Specific Gravity of a body is the ratio of the density of the body to the density of the standard substance, and is therefore a numeric independent of the system of units adopted, that is, its dimensional formula is I. If G denote the specific gravity of a body whose density is D, and IDs be the density of the standard Substance, AD = G. Ds. In the French system of units Ds is taken as unity and we have D = G. This is one great convenience of 336 DIMENSIONS OF UNITS. the French units of length and mass; but it is to be remembered that Density and Specific Gravity are essentially different ideas, and only coincide in numerical value when Z), - I. l)YNAMICAL UNITS. Velocity. The velocity of a body is measured by the numerical length described per unit of time. Its specifica- tion involves direction as well as magnitude ; but in dealing with the dimensions of velocity we are only concerned with the latter element. If the velocity is uniform, its numerical expression is the ratio of the numerical distance / traversed to the numerical time-interval T in which it is described. If the velocity is variable, its numerical expression zy at any instant is the value towards which the ratio of the numerical distance 6/, traversed in an interval of time including the instant, to the numerical time-interval & 7, approaches as the interval is taken smaller and smaller. Hence d/, * = 77. where / denotes in Newton's fluxional notation the time- rate of variation of Z. We shall use this symbol for velocity. We see that the numerical expression of a velocity is the ratio of a numerical length to the numerical time- interval, and therefore we have [/] = [/] T-1]. As multiplier for a change from mile-minute units to centimetre-second units we have = }, n = 5280 × 30:4797/60 = 2682'2136. DYNAMICAL UNITS. 337 In statements of amounts of velocities there ought clearly to be a distinct reference to the unit of time: thus the expressions one mile per minute, 88 feet per second, 2682-2136 centimetres per Second, are per- fectly definite, and express the same velocity, while such a statement as a “velocity of 88 feet” is devoid of meaning. Acceleration. The acceleration of a body is the rate of change of velocity per unit of time. - Like velocity, acceleration involves in its signification the idea of direction as well as of magnitude; and it is through a want of clear apprehension of this fact that difficulty is found by students in the theory of curvi- linear motion. Let 67 be the velocity given in direction and magni- tude which compounded with the velocity J. which a particle possesses at the beginning of an interval of time 67 would give the velocity in direction and mag- nitude at the end of that interval, then 8//87 is the average acceleration during that interval, and the limit towards which this ratio approaches as 67 is made smaller and smaller is the true value of the acceleration at the beginning of the interval. That is, we have & d’A. tº ge Acceleration = iſ 7 = Z. where 7 denotes in the fluxional notation the time-rate of variation of Ł, that is, of velocity. We shall use this symbol for acceleration, and the two dots above the L will serve to recall the double reference to time which is plainly involved in the notion of acceleration. This should be clearly expressed in Z 338 DIMENSIONS OF UNITS. Statements of amounts of acceleration. Thus such a statement as an acceleration of 981 centimetres per second per second, or 32 feet per second per second, is perfectly definite, while such phrases as an “accele- ration of 981 centimetres” or “an acceleration of 32 feet per second,” which are often used, are meaningless. The dimensional formula is [...] = [Z T-ºl. For a change from mile-minute units to centimetre- second units, 7t = 5280 × 30-4797/60° = 447-O356. Momentum. Taking for simplicity the case of a rigid body moving without rotation, that is, so that each particle of the body has the same velocity at the same instant, the momentum of the body is expressed as the product of the numerics for the mass of the body and its velocity. Hence it is expressed symbolically by Mſ. The dimensional formula is therefore [M].] = [M L T || Time-Rate of Change of Momentum. If the momentum of the body be not constant, then, since we suppose the mass constant, we must have for the time-rate of variation the expression MŽ, that is, the product of the numerics for the mass and the acceleration. The dimensional formula is therefore [iſ?] = [M L 7-4). Force (F). A force acting on a body is proportional to the time-rate of change of momentum. Hence the dimensional formula just found is that of force. DYNAMICAL UNITS. 339 According to the system suggested by Gauss, a force is measured by the time-rate of change of momentum, that is, the constant, C, of equation (3) is in this case, as in the other cases we have considered, taken equal to unity. Unit force is therefore that force which acting for unit of time on unit mass produces unit change of velocity, or simply that which produces unit acceleration in unit mass. When the unit of mass is one pound, the unit of length one foot, and the unit of time one second, then unit force is that force which acting for one second on a pound of matter generates a velocity of one foot per second. This unit force has been called a Zozenda/. The unit force in the C.G.S. system, is that force which, acting for one second on one gramme of matter, generates a velocity of one centimetre per second. To this unit force the name dyne has been given, This method (sometimes called the Äänetic method) of measuring forces has now superseded, for scientific purposes, the gravitation system formerly in use. In that system the unit of force is the force of gravity on the unit of mass, and has, therefore, different values at different places on the earth's surface, and at different vertical distances from the mean surface level. This substitution of an invariable unit of force, depending only on the standards adopted for length, mass, and time, instead of the former variable unit, is at the foundation of the system of units of measure- ment established by Gauss. It is to express this fact of invariability with locality and other circumstances that, as already explained, the term “absolute" is used for the unit of force and other derived units in this System. 34O DIMENSIONS OF UNITs. Work (PV). In dynamics work is said to be done by a force when the place of application of the force receives a component displacement in the direction in which the force acts, and the work done by it is equal to the product of the force and the distance through which the place of application of the force has moved in that direction. The time-rate at which work is done by a force at any instant is therefore equal to the product of the force and the component of velocity in the direction of the force at that instant. The work done in overcoming a resistance through a certain distance is equal by this definition to the product of the resist- ance and the distance through which it is overcome. Among engineers in this country the unit of work gene- rally used is one ſoof-float/td, that is, an amount of work equal to that done in lifting a pound vertically against gravity through a distance of one foot. The weight of a pound of matter being generally different at different places, this unit of work is a variable one, and is not used in theoretical dynamics. In the absolute C.G.S. system of units, the unit of work is the work done in over- coming a force of one dyne through a distance of one centimetre, and is called one centimetre-dyne or one erg. In practical electricity IO' ergs is frequently used as unit of work, and is called a /ozzle. If F denote a numerical force and Z the numerical space- interval through which it has acted, the numeric for work done is FZ. Hence we have [W] = [FZ] = [M L*T*. Activity (A). The single word Activity has been used by Sir William Thomson as equivalent in meaning to “time-rate of doing work,” or the rate per unit of time DYNAMICAL UNITS. 34 I at which energy is given out by a working system ; and to avoid circumlocutions in what follows we shall frequently use the term in that sense. Among engineers in this country the unit rate of working is one /orse- £ower, that is 33,000 foot pounds per minute. Unit Activity in the C.G.S. system is one erg per second. In practical electricity an activity of Io ergs per Second is frequently employed as unit. This unit has been called a Wałż. Since an Activity is measured by the numerical work done per unit of time, its dimensional formula is given by [A] = [M /2 Tºl. Fnergy (E). When a material system in virtue of stresses between its own parts and those of bodies external to it does work or has work done upon it, in passing from one state to another, it is said to give out or to gain energy. The energy given out or gained is measured by the work So done. If the change be a change of motion, then, according as energy is given out or gained by the system, it is said to lose or gain Kinetic energy. If the change be of any other kind, it must be classed under change of con- figuration, and, according as the System gives out or gains energy, it is usually said to lose or gain Żołenſia/ eview gy. When we consider the work done by mutual forces be- tween different parts of the same system, a loss of kinetic energy in the system is accompanied by an equal gain of potential energy, and vice versä, So that the total energy of the system remains unchanged in amount. This is the principle called the Conservation of Ænergy. Energy is measured by the same units as work, and 342 DIMENSIONS OF UNITS. its dimensional formula is the same as that of work, that is - [E] = [M Z2 T-ºl. We shall here, for the sake of illustration, give three examples of the application of dimensional formulas to the solution of problems regarding units. The problems are taken from Professor Everett's Units and Physica! Constants. - Ex. I. If the unit of time be the second, the unit density 162 lbs. per cubic foot, and the unit of force the weight of an ounce at a place where the change of velocity g produced by gravity in one second is 32 feet per second, what is the unit of length 2 * Here the change-ratio by which we must multiply the density of a body in the system of units proposed, to find its density in terms of the pound as unit of mass, and the foot as unit of length, is 162. We have therefore, omitting the brackets in the dimensional formulas, AZ Z-8 == 162, where M is the number of pounds equivalent to the unit of mass, and Z, the number of feet equivalent to the unit of length. Also, it is plain that the unit of force in the proposed system is two foot-pound-second units. Hence we have also, since T = I, M L T * = Aſ L = 2. By division therefore we get Zº- 1/81 or Z = 1/3. The unit of length is therefore 4 inches. Ex. 2. The number of seconds in the unit of time is equal to the number of feet in the unit of length, the unit of force is 750 lb. weight (g being 32), and a cubic foot of ELECTRICAL UNITS. 343 the substance of unit density contains I 3,500 ounces. Find the unit of time. Using M and Z as in the last problem, and putting 7. for the numerical expression of the unit of time in Seconds, we have plainly M L * = I 3500 Tº I6 and M L T * = 750 × 32. Therefore by dividing, and remembering that Z = 7, we get 7.2 32 × 750 × I6 16°. I 3500 3° That is the unit of time is 5% seconds. Ex. 3. When an inch is the unit of length and T seconds the unit of time, the numeric of a certain acceler- ation is a ; when 5 feet and I minute are units of length and time respectively, the numeric for the same accelera- tion is Iow. Find 7. The change-ratio or value of ZT-” for reduction to foot- second units is plainly in the first case T-2/I 12, in the second 5/3600. We get therefore * ~– 5 + TTº a = I 2 36OO X IOd, Or T = W6. DERIVED ELECTRICAL UNITS. I.—ELECTROSTATIC SYSTEM. Quantity of Electricity [7]. In the electrostatic system of units which is convenient when electrostatic results, independently of their bearing on electromagnetic pheno- 344 DIMENSIONS OF UNITS. mena, are required, the units of all the other quantities are founded on the following definition of unit quantity of electricity. Unić quantity of electricity is that quantify which, concen/rated at a point at 7/7, it distance from an equal and similar Quantity, also concentrated at a point, is 7 afte//ed with unit force when the medium across which f/he e/ectric action 2's transmitted 2s a certain standard 771su/a/7/19 weedium. An ideal vacuum is sometimes taken as standard, but we shall suppose at present that the medium is air at temp. O’C. and at standard atmospheric pressure. We shall call this simply air. This definition is precisely similar to the definition (p. 40 above) of unit magnetic pole which forms the basis of another system of units called the electromagnetic system, of much wider and more important application than the electrostatic. Hence by Coulomb's law that (the numerical values of) electric attractions and repulsions are directly as the products of the (numerics for the) attract- ing and repelling quantities, and inversely as the second power of the (numeric for the) distance between them, if a quantity of positive electricity expressed by 7 be placed at a point distant / units from an equal quantity of electricity, then the medium being air, the numeric F for the force between them is 7°//*. If the medium across which the electric action is transmitted be some other medium than air, the force between the charges is numerically gº/KZ” where K is the numerical measure of a quantity called the electric Ž/16//c//7/e ca/ac2/y, or usually the specific inductive capacity of the medium. This quantity is precisely analogous to the conductivity of a substance for heat # and to magnetic * See Théozy azua Practice of Absol//fe Al/cas/7′emzewits in E/ectricity and /l/agnet/s/t, vol. i. chap i. sect. v. ELECTRICAL UNITS. 345 permeability (see p. 350 below). In the ordinary electro- static system of units this quantity is defined (as at p. 347) so as to have a dimensional formula I, that is to be a mere numeric. But we might proceed otherwise and regard K as a quantity of undetermined dimensions as regards the fundamental units, but such that 7”/ATſ.” has the dimensions of a force. We may then, in the absence of special reasons for preferring one dimensional formula for A to another, assign its dimensions according to any convenient hypothesis. One such hypothesis is that which forms the basis of the ordinary electrostatic system, namely, that A is, as regards the fundamental units, of zero dimensions, that is has a dimensional formula [1]. But in the ordinary electromagnetic system of units, which has quite a different derivation from the electrostatic, the dimensional formula of K is [Z-27°] and the numerical value of K depends on the choice made of fundamental units. We shall in what follows suppose the dimensions of Á undetermined, and therefore allow the symbol Å express- ing it to appear in the dimensional formulas of the other quantities. We shall thus obtain a more general electro- static system in which the absolute dimensions of the quantities are not settled. From this the ordinary electro- static system is obtained by simply deleting K.” The dimensional formula of quantity of electricity is accordingly [Fº Z. Kººl or [Mé Zi Ti" Kºl. Electric Surface Density [or]. The density of an elec- tric charge on a surface is measured by the quantity of electricity per unit of area. Therefore [a] is [g LT") or [M L-4 7'-' Kºl * This method of proceeding is advocated and its advantages pointed out by Prof. A. W. Rücker, F.R.S. in a paper on the “Suppressed Dimensions of Physical Quantities,” Phil. Mag., Feb. 1889. - 346 DIMENSIONS OF UNITS. Electric Force and Intensity of Electric Field [f]. The electric force at any point in an electric field, or, the intensity of the field at that point, is the force with which a unit of positive electricity would be acted on if placed at the point. Hence if the numeric for the quantity of electricity at a point P be g and that of the electric force at that point be ſ, the numeric F for the force on the electricity is g/, and we have the equations f = F2-1. Therefore [f] is [F2-1] or [M Z-3 7 - K-3]. Flectric Potential [v]. The difference of electric potential between two points is measured by the work which would be done if a unit of positive electricity were placed at the point of higher potential and made to pass by electric force to the point of lower potential. Hence in transferring g units of electricity through a difference of potential expressed numerically by v, an amount of work is done for which the numeric W is equal to gy. We have therefore y = W g-1, and hence [v] is [Wſ, -1] or [M Zł z - K -}]. * Capacity of a Conductor [c]. The capacity of an insulated conductor is the quantity of electricity required to charge the conductor to unit potential, all other con- ductors in the field being supposed at zero potential. Hence denoting the numeric for the capacity of a given conductor by c, those for its charge and potential by 7 and v, we have c=qv-1, and for [c] therefore [qv-1] is [ZK]. The unit of capacity has therefore the same dimensions as the unit of length provided [K]= I ; and the capacity of a conductor is then properly expressed as so many centimetres. The electrostatic capacity of a conducting sphere is in ordinary electrostatic units numerically equal to the radius of the sphere. A conducting sphere of I Cm. radius has therefore I C.G.S. unit of capacity. ELECTRICAL UNITS. 347 Specific Inductive Capacity [A]. The specific inductive capacity of a dielectric has already been virtually defined above, but it is usual to define it as the ratio of the capacity of a condenser, the space between the plates of which is filled with the dielectric, to the capacity of a precisely similar condenser with air as dielectric ; or, according to Maxwell’s “ Theory of Electric Displace- ment, it is defined as the ratio of the electric displacement produced in the dielectric to the electric displacement produced in air by the same electric force. Thus in the ordinary electrostatic system of units its dimen- Sions are taken as zero, that is, it is simply a numerical coefficient which does not change with the units. Hence in the ordinary electrostatic system [A] = I. Electric Current [y]. An electric current in a con- ducting wire is measured by the quantity which passes across a given cross-section per unit of time. If q be the numeric for the quantity which has passed in a time for which the numeric is 7, then denoting the numeric for the current by y, we have y = 7/7, and [y] is [z T-1] or [M # /* T-” K*. A'esistance [r]. By Ohm's law the resistance of a conductor is expressed by the ratio of the numeric v for the difference of potential between its extremities to the numeric y for the current flowing through it. We have therefore r = v/y, and [r] is [vy-l] or [LT' TK 7"). Conductance (see p. 87 above). The dimensional formula of conductance is plainly [LTT' A |. Hence in the ordinary electrostatic System its dimensional formula is [LTT'] which is that of velocity. Hence a conductance in electrostatic C.G.S. units is properly expressed in centi- metres per second. * El, and Mag. vol. i. 2nd edition, p. 154. 348 DIMENSIONS OF UNITS. The following illustration of this result has been given by Sir William Thomson. Suppose a spherical Conductor charged to a potential v to be connected to the earth by a long thin wire, of which the Capacity may be neglected ; and let r be the resistance of this wire in electrostatic measure. The current in the wire at the instant of contact is v/r. Now let the sphere diminish in radius at such a constant rate that the potential remains v. The current remains v/r, and the quantity of electricity which flows out in f seconds will be viſ?". If the radius be initially ar, and in f seconds has diminished to ar', the diminution of capacity is a -a- (p. 346). Hence the loss of charge is v(r – a '), and we get v//r = v(a — ar), or 1/r = (+ – ar')/f. But (a – aſ )/# is the velocity with which the radius of the sphere diminishes. The conductivity 1/r of the wire is therefore measured numerically by the velocity with which the surface of the sphere must approach the centre, in order that its potential may remain constant when the surface is connected to the earth through the wire. II.-ELECTROMAGNETIC SYSTEM. Magnetic Pole or Quanſity of Magnetism [m], Surface density of Magnetism [a']; Magnetic Force or Magnetic field intensity [/]; Magnetic potenţia/ [V]. The electromagnetic system of units is based on the unit magnetic pole as defined above (p. 40). This definition is exactly the same as that of unit quantity of electricity on which the electrostatic system is founded ; and therefore the purely magnetic quantities here men- tioned, which bear the same relations to the unit quantity of magnetism that the corresponding electric quantities MAGNETIC UNITS. 349 bear to the chosen unit quantity of electricity, have, with the substitution of the magnetic analogue to K, in the electromagnetic system the same dimensional formulas as those just found for the latter quantities in the electrostatic system. Observations precisely similar to those made above regarding specific inductive capacity apply here regarding its analogue, magnetic inductive capacity, or, as it is frequently called, magnetic permeability. The force between two poles each of strength m at distance Z in a medium of magnetic inductive capacity p is numerically mºla Lº, and hence [m] = [Mº Lº T' pºl. In the ordinary electromagnetic system p is defined (see p. 350 below). So as to be a mere numeric. We shall not here make this assumption, but allow p to appear in the formulas, and its dimensions may be afterwards assigned. By simple deletion of pi from the dimensional formulas they become those for the ordinary electromagnetic system in which [p] = I. Magnetic Moment [m]. The numeric ni, the magnetic moment of a uniformly magnetized bar-magnet, is the product of the numerics for the strength of either pole and the length of the magnet, Hence we have [m] = [M Lº T -1]. [L] = [Mk Lº Tºpºl. Intensity of Magnetization [v]. The intensity of magnetization of any portion of a magnet is measured by - the magnetic moment of that portion per unit of volume. Hence, if v denote the numeric for the intensity of magneti- zation of a uniformly magnetized magnet, the numerics for the magnetic moment and volume of which are m and A Z”, we have 1/ = #: and [v] = [Mk I.- T-1 pº). 35O DIMENSIONS OF UNITS. It is plain that the intensity of magnetization of a uniformly and longitudinally magnetized bar is equal to the surface density of the magnetic distribution over the ends of the bar, and therefore intensity of magnetization has the same dimensional formula as magnetic surface density. Magnetic Permeability% [p]. The magnetic per- meability of an inductively magnetized substance, or its magnetic inductive capacity, is, as has already been stated, the analogue in magnetism of specific inductive capacity of a dielectric in electricity, and of the conductivity of a body for heat in heat conduction. It is usual to define it as measured at any point by the ratio of the force which a unit pole would there experience if placed in a narrow crevasse, cut in the substance so that its walls are at right angles to the direction of magnetization, to the force which it would experience if placed in a narrow crevasse, the walls of which are parallel to the direction of magnetization. In the first case we must suppose, in consequence of the formation of the crevasse, a distribution of blue magnetism over one of its walls, and a similar distribution of red magnetism over the opposite wall. These magnetic distributions are exactly equal and opposite to the distributions on the sides of the portion of the substance removed to form the crevasse. If we suppose the substance uniformly magnetized, the density of this distribution will be uniform on both walls, and if we denote the density in one by or', that on the other will be denoted by – o'. A unit blue magnetic pole placed in the crevasse would be repelled from the blue side and attracted towards the red with a force in each case equal * See Reprint of Papers on Electrostatics and Magnetism/, by Sir W. Thomson, p. 484. MAGNETIC UNITS. 35 I to 2 To'; % and therefore would be acted on towards the red side by a total force of 4 T o'. Besides this force due to the magnetic distribution in the crevasse, the pole is acted on by the resultant of the forces due to the other magnetic distributions of the system. If we suppose the substance isotropic as to magnetic quality, that is, to have the same magnetic quality in different directions, this latter force will be in the same direction as the former. Calling it ſ, we have for the total force acting on the pole towards the red side of the crevasse the expression f + 4 T o'. If, on the other hand, the pole were placed in the same position but in a narrow crevasse cut with its walls parallel to the direction of magnetization of the substance, the force acting on it would be simply f. Hence, if we call p the magnetic permeability of the substance, we have p = 1 + 4 tro'ſſ. It is clear that this mode of defining p makes it in the ordinary electromagnetic system a mere numeric, that is its dimensional formula is [u] I. Magnetic Susceptibility. The ratio o'/f is called the * Imagine a material particle of unit mass (such a particle being here defined as a particle which placed at unit distance from an equal particle would be attracted with unit force) situated at a point on the axis of a thin uniform material disk of radius 7". Let the mass of the disk per unit of area be or, and the distance of the particle from the disk measured along the axis a. The attraction on the particle of a small element of the disk of area d'S at a distance v from the centre is ords/(a” + x2) and the component resolved * & S g - - & * along the axis is a o d'S /(a” + a 9)*. Hence the attraction in this direction due to a narrow ring of radius at and breadth d x is 2 to a a dir/(a 2 + x2)*, and that of the whole disk therefore |. ard r ( (Q. ) 2 Tr Cr & T. T.N.: = 2 T or ( 1 — Tz-H, ). J o (a 2 + x2); vaº Tzº If the distance a of the particle from the disk be very small in comparison with the radius r of the disk, this expression for the total attraction becomes simply 2 tro. We have only to substitute a unit magnetic pole for the unit particle and a distribution of imaginary magnetic matter of surface density o' for the material disk, and we have the result used in the text. 352 DIMENSIONS OF UNITS. magnetic Susceptibility of the substance. Its dimen- Sional formula is therefore also I in the ordinary electro- magnetic system, that is magnetic susceptibility is in that system a mere numeric. Current Strength [T]. By the theory of electromagnetic action stated above in p. 44, and the definition of unit current (3), p. 47, we have, for any actual case of a mag- netic pole placed at the centre of a circle of wire carrying a current, the equation T = F Z/2 iſ m, where F, L, m, and T are the numerics respectively for the force acting on the pole, the radius of the circle, the strength of the pole, and the strength of the current. Hence [T] = [F/m-*) = [Mk LA 7"-"pºl. Quantity of Electricity [Q]. The numeric Q for the quantity of electricity conveyed in T seconds by a current the numeric for the strength of which is C is equal to C T. Hence [Q] = [CT) = [M Lºu Tºlº Electric Potential, or Electromotive Force [V]. As above (p. 346), but using in this case the symbol V for a numerical difference of potential, we get W = WQ. Thus we have [V] = [M* Lº Tºp}}. Electrostatic Caffacify [C]. Using for a numerical capacity in electromagnetic units the symbol C, we find, by the same process as in p. 346, the equation C = Q/ V, [C] = [ZT* Tºp"). A'esistance [ſº]. Using here A* to denote a numerical * We might pass in the electrostatic system from the dimensional formula of unit current to that of unit quantity of magnetism, precisely as we pass here in the electromagnetic system from the dimensional formula of unit quantity of magnetism to that of unit current, and we should find for the dimensional formula sought that here obtained [M Z} R-3) as might be inferred at once. . From this the formulas in the electrostatic system for all the other magnetic quantities might be found. DIMENSIONS OF RESISTANCE. 353 resistance, we get as formerly /ē = V/C, and therefore [R] = [Z Tºp). Thus if [u] = 1, the dimensional formula for resistance is the same as that for velocity, and therefore a resistance in ordinary electromagnetic units is properly expressed as a velocity, and accordingly, in C.G.S. units, as so many centimetres per second. This fact is directly shown by the following illustration, due to Sir William Thomson. Let the rails of the ideal machine, described in p. 67, be supposed to run horizontally at right angles to the magnetic meridian, and let their plane be vertical. Let a tangent galvanometer be included in the wire con- necting the rails. The slider when moved along the rails will cut the lines of the earth’s horizontal force, the inten- sity of which in electromagnetic measure we have denoted by H. If the slider have a length / and be moved with a velocity v, the electromotive force developed will be Aſ Z v. If ſº be the total resistance in circuit, C the current flowing, r the mean radius of the galvanometer coil, and // the length of wire in the coil, we have by (2), p.28, C = H rº//'tan 6. But by Ohm's law C = H L z//°. Hence H L v/A = Aſ rº/// tan 6, or A’ = Z/ 7, 7-4 tan 6 Now we may suppose the radius r of the coil so taken that r* = Z.Z.', and that the slider is moved at such a speed, v, that the deflection of the needle is 45°. Under these conditions we get /ē = v. The resistance ſº of the circuit is therefore measured in electromagnetic units by the velocity with which the slider must be moved, so that the deflection of the needle of the tangent galvanometer may be 45°. A A 354 DIMENSIONS OF UNITS. Coefficient of Self-Induction (or Inductance). Denoting by / (instead of L to avoid confusion) the inductance of a circuit, the current in which is T, we have (p. 282) /dT/dt for the electromotive force of self-induction. Hence /dT/dt has the same dimensional formula as electro- motive force, that is [/T TT") = [M Zº T-ºp-ºl. and therefore [/] = [/] pil. Coefficient of Mutual Induction. If A be the co-efficient of mutual induction between two circuits, T the current in one of them, then the electromotive force in the other circuit due to mutual induction is AdI/dt. Hence by the same process as before we get [ſº] = [Zp]. A co-efficient of self- or of mutual induction is therefore in ordinary electromagnetic measure in dimensions simply a length, and in C.G.S. units is properly expressed as so many centimetres. -- * But if resistance is taken in terms of the true ohm which is Io9 cms. (or nearly one earth-quadrant) per second, the corresponding unit of induction is IO" cms. If the legal or any other ohm is used, the unit of induc- tion is that length which replaces Io9 cms. in the definition of the ohm. For the unit of induction defined by any ohm Profs. Ayrton and Perry have proposed the name secohm. The Paris Congress has however adopted the name guadraná. Plainly this can only in Strictness be applied in connection with the true ohm. - We have now investigated the dimensional formulas of the absolute units of all the principal electric and mag- netic quantities in the electrostatic system, or in the electro-magnetic system, according as each quantity is generally measured in practice. Each may, however, be expressed either in electrostatic or in electromagnetic TABLE OF DIMENSIONS. 355 units, and we give the following table of dimensional formulas for all the quantities in both systems. In Tables II. and III. A and p, have been introduced into the formulas as stated above, pp. 345, 349. The ordinary electrostatic and electromagnetic Systems are obtained by Supposing A and p each unity. One advantage of thus exhibiting the dimensions is that it enables electrostatic and electromagnetic quan- tities to be regarded as of the same absolute dimensions, since A and p, not being fixed as to dimensions, Can, un- less restricted by definition, have dimensions assigned to them which fulfil this condition. For example, as sug- gested by Professor G. F. Fitzgerald,” each may be taken as having the dimensions [7 ZT*]. Another advantage is that problems in which passage from one set of units to the other is involved are solved with greater ease from first principles (See Professor Rücker's paper loc cit.) FUNDAMENTAL UNITS. Quantity. Dimensional Formula. Length [/] Mass [M] Time [T] DERIVED UNITS. Z. Dynamical Unzás. Velocity [Z T- | Acceleration [Z 7-2 Force [M Z 7-2 Work - S) Energy [M Zº 7–21 * Phil. Mag., April 1889. A A 2 356 DIMENSIONS OF UNITS. NOTE.--Col. A below with K deleted gives the ordinary electro- Static formulas, Col. B with p deleted gives the ordinary electro- magnetic formulas. A/. F/ectric U772/s. Quantity of Electricity Surface density of Electricity Electric Displacement Electric Force, or Intensity of Electric Field Electric Potential } Electromotive Force Electrostatic Capacity Specific Inductive Capacity Current Strength Resistance J//. Quantity of Magnetism, or ! Magnetic Pole. Surface density of Magnetism [J/ #1-3 k-4 Magnetic Moment Intensity of Magnetization Magnetic Force or } Intensity of Magnetic Field Magnetic Potential Magnetic Permeability Magnetic Susceptibility Coefficient of Self-Induction Coefficient of Mutual Induction [Arizº T-2A.) A - In terms of In terms of L., M., T, A. L., M., T, M. [M*A* I -1K. [M*L*u-?] {[Mºz-z-Kº Lºz-in- [Mºz-37 - K-3) [Mºz, T-2,3] [Mºz T-1A-#1 (M*117-??] [K) [Z-27°,-] [ZK) J .[L-47 a 'j [M}/47-4A-3) [Mºzł z - wº [Z-17-kº [LT-ºp-3 Magnetic Units. [Mºzł 7-º] [MłZ-47-pº] [Mk/#7-1,2] [M*/37-'º [Mł Z-37-10-3] [A/3/37-'gi-3) [u] [Z pil [J/k/A-3] [A/3/4 K-3] [A/º L-3 K-4] [Aziz T-3K/ [Z-272K-1] }(z-z-z- PRACTICAL UNITS. 357 As an example of the use of dimensional formulas we may find the multiplier for the reduction of numerics for magnetic field intensities given in terms of British foot- grain-second units to the corresponding numerics in terms of C.G.S. electromagnetic units. Let H be the numerical intensity in terms of British units, H' the numerical intensity in C.G.S. units. We have, by equation (4), p. 333, H = H [H] = H [M3 LT # 7–1). Since I gramme = I 5'43235 grains, and I centimetre = 1/3O'47945 foot, we have % I M3 L- # 7 - 1] = I = ——- - -— [ J (sº X sº) 2 I-688° The earth's horizontal force is given as 3'92 in British units at Greenwich for 1883. We get therefore Pł’ = 3.92 * = 18075, in C.G.S. units. Onits adopted 77, Practice. In practical work the resistances and electromotive forces occurring to be measured are usually so great that if the absolute electromagnetic C.G.S. units were used, the resulting numerics would be inconveniently large ; while, on the other hand, capacities are generally so small that their numerics in C.G.S. units would be only very Small fractions. Accordingly certain multiples of the C.G.S. units of resistance and electromotive force, and a Submultiple of that capacity have been chosen for use in practice. The first two, the ohm and the volt, together with the practical units of current and quantity, the ampere and the Coulomb, have been explained above 358 DIMENSIONS OF UNITS. (Chap. V.). The practical unit of electrostatic capacity is called the farad, and may be defined as the capacity of a condenser which, when charged by an electromotive force of one volt, has a charge of one coulomb. If C be the numerical capacity of such a condenser in C.G.S. electro- magnetic units of capacity, we have C = Io-I/Io9 = Io-9; or one farad is equivalent to IO-9 C.G.S. In some cases, when the quantities to be expressed are very large, units one million times greater than the chosen practical units are employed. These are denoted by the names of the corresponding practical units with mega (great) prefixed. On the other hand, for the expression of very Small quantities, units one million times smaller than the practical units are sometimes used, and are denoted by the corresponding names of the practical units with micro (small) prefixed. º Such units are however rarely employed, with the ex- ception of the 77tego/im, used for expressions of resistances of insulating substances, and the microfarad, which is really the most convenient unit for expressions of capaci- ties. A megohm is a velocity of IO” centimetres per second ; one C.G.S. unit of capacity is equivalent to IO” microfarads. The practical units which have been adopted may be considered as belonging to an absolute system, based on a unit of length equivalent to one thousand million (IO") centimetres (approximately the length of one quadrant of the earth's polar circumference), a unit of mass equivalent to one one-hundred-millionth of a milligramme, or Io-ll gramme, and the second as unit of time. The verification of this in the different cases will furnish examples of the use of dimensional formulas. For example, let us find what the expressions of RELATIONS OF PRACTICAL TO C.G.S. UNITs. 359 resistances and electromotive forces in C.G.S. units become when these new units of length and mass are substituted for the centimetre and the gramme. Let A’ be the numeric of a resistance in C.G.S. units, and ſº its numeric in terms of the new units. We have, by (4) p. 333 above, R = R [Z 7-1) = R × -- IO) since the unit of time remains unchanged. One ohm is therefore equivalent to Io9 C.G.S. units of resistance, that is 10° centimetres per second. Again let V be the expression of an electromotive force in C.G.S. electromagnetic units, E' its expression in terms of the new units. The dimensional formula for electro- motive force is [M Lº 7–1). We have therefore E = E [Mł Zł T-1). We have only to consider what [M* L*] becomes. This is plainly (1/10−")} x (1/10)” or 1/10°. Hence E = E x → IOS that is, one volt is equivalent to IO* C. G.S. units of electromotive force. The following table gives the numerics for the various practical units in terms of C. G.S. units:— Name of Practical Equivalent in Quantity. Unit. C.G.S. Units. Resistance Ohm Io9 Electromotive Force Volt IOS Current Strength Ampere IO-l Quantity of Electricity Coulomb IO – 1 Farad IO – 9 Electrostatic Capacity Microfarad IO - 15 360 RELATIONS OF We have seen above that if Q, Q’ be the numerics of two quantities the dimensional formula of Q'/Q is [Q]/[Q],and this of course applies to the expressions of the same quan- tity in two different systems of units. Thus if g denote the numerical expression of a quantity of electricity in electrostatic units, and Q that of the same quantity in electromagnetic units, the same fundamental units being employed in both cases, the dimensional formula of g/Q is [g]/[Q]. But from the table (p. 357) we have [g]=[M * Z. T'J and [Q]=[M Z}]. The dimensional formula of 7/Q is thus the same as that of velocity, that is to say 7/Q is equal to a certain definite velocity, the numerical expression of which depends on the funda- mental units of length and time employed. In other words the number of elec/rostaſic units of electricity which is equivalent to one electromagnetic unit is numerically equal to this velocity. The same velocity is derivable from the ratios of the numerical values of any of the other electrical or magnetic quantities in the two systems of units. For instance, if e be the numerical value of an electromotive force in electro- static units, and E that of the same electromotive force in electromagnetic units, we have for the dimensional formula of eſ E the value [e]/[E]=[M*A*]/[M * Z, T']=[ZT*T]. The ratio eſ AE has thus the dimensional formula of the reciprocal of a velocity, and therefore E/e, or which is the same, the number of electromagnetic units equivalent to one electrostatic unit of electromotive force, is properly expressed as a certain definite velocity. It is easy to see that this velocity is identical with the former. For if g and Q be the numerical values in the two systems of a certain quantity of electricity, then since e and E denote the same electromotive force, the work e º must be ELECTROSTATIC AND ELECTROMAGNETIC UNITs. 361 numerically equal to the work A. Q. We get therefore A/e = 7/Q, that is, the two velocities are the same. By taking the more general dimensional formulas given in the table (p. 357) we find that !!! = [K-,-] [Q] when [7], [Q] refers to the ordinary systems. Hence the product K Tºp-4 has the dimensions of a velocity. It is in fact the velocity g/Q above referred toº Denoting this velocity by v, f we get for the various quantities the following relations. The numerator of the ratio on the left of each equation denotes the numeric of the quantity in electrostatic units, the denominator the numeric of the same quantity in electromagnetic units. A given Quantity of Electricity % = Zy ,, Current 2 = v T -* g 62 I , Electromotive Force Ž . & g C ,, Electrostatic Capacity z = 7/2 Resi 7- I 22 GeSiStan Ce # = 2 Therefore if g and Q, e and E, or the numerics for any other given quantity, be determined in the two systems of units, the value of v can be at once obtained. Experiments of this kind have been made by Maxwell, Sir W. Thomson, * See Maxwell, Ed. and Mag. chap. xx, or the author’s larger treatise, vol. ii. & f For illustrations of the physical meaning of v see Maxwell's Electricity and Magnetisme, vol. ii. chap. xix. 362 DETERMINATIONS OF v. Weber, Ayrton and Perry, J. J. Thomson, H. A. Rowland, E. B. Rosa, and others,” with the result that w = 3 × 1 o'0 centimetres per second approximately, or very nearly the velocity of light in air as deduced from experiments made by the methods of Foucault and Fizeau. According to Maxwell’s Electromagnetic Theory of Light (Electricity and Magnetºszyz, vol. ii., chap. xx.) this relation should hold, and thus the theory is so far confirmed. Full information regarding experiments for the deter- mination of v will be found in Maxwell's Electricity and Magnetism, vol. ii., chap. xix., and an account of the principal determinations made down to the present time will be found in the author’s larger treatise, i (vol. ii.), which contains a much more detailed and complete account of electric and magnetic measurements than it is the object of this book to give. But to Maxwell’s work also we refer the reader who wishes to obtain a complete account of the mathematical theory of electricity and magnetism measurements. He may consult also portions of Sir William Thomson’s ſe/rint of Paſers on Electrostatics and Magnetism, Mascart and Joubert's Leçons sur / Elec- tricité et le Magnétisme: and Prof. Chrystal’s articles on Electricity and Magnetism,' which contain an admirable digest of the whole mathematical theory together with much valuable information as to experimental results. * An account of the principal determinations of this quantity with the results will be found in the author's Theory and Practice of Absolute Measurements in Electricity and Magnetism, vol. ii. f The Theory and Practice of Absolute Measurements in Electricity and Magnetism, vols. i. and ii. Macmillan & Co. - f An English translation of this work by Prof. Atkinson has lately been published. § Encyclopaedia Britannica, New Edition. APPENDIX. APPENDIX. NOTE. A'ecommendations of the Paris Congress and the British Association as to Practical Electrical Units. At the meeting of the Electrical Congress held in Paris in 1884, it was decided to adopt for the present, as practical unit of resistance, a resistance equal to that of a uniform column of mercury one square millimetre in Section, IO6 centimetres in length, and throughout at the temperature O’ C. The mercury column thus specified expressed approximately and in round numbers the value of the olim according to the latest and most accurate experiments. It was resolved to give this unit the name Mega/ O//z. - The Congress also arrived at certain conclusions re- garding the practical units of current, electromotive force, quantity of electricity, and electrostatic capacity, as follows:– (1) That the unit of current should be called the Ampere, and be defined as 'o of a C.G.S. electro- magnetic unit of current. (2) That the Volt or practical unit of electromotive force, or difference of potentials, should be defined as the electromotive force required to maintain a current of one ampere through a resistance of one ohm. APPENDIX. 365 (3) That the unit quantity of electricity should be called the Coulomó, and be defined as equal to the quantity of electricity transferred by a current of one ampere in one second. (4) That the Farad or practical unit of capacity should be the capacity of a conductor which is charged to a potential of one volt by one coulomb of electricity. The British Association at its meeting in 1886 agreed that the Committee on Electrical Standards should re- commend to Her Majesty’s Government :— (1) “To adopt for a term of ten years the Legal Ohm of the Paris Congress as a legalized Standard sufficiently near to the absolute Ohm for commercial purposes. (2) “That at the end of the ten years' period the Legal Ohm should be defined to a closer approximation to the absolute Ohm. (3) “That the resolutions of the Paris Congress with respect to the Ampere, the Volt, the Coulomb, and the Farad, be adopted. (4) “That the Resistance Standards belonging to the Committee of the British Association on Electrical Standards, now deposited at the Cavendish Laboratory at Cambridge, be ac- cepted as the English Legal Standards con- formable to the accepted definition of the Paris Congress.” At the meeting of the Electrical Congress held at Paris in August, 1889, resolutions were carried of which the following is an English equivalent adopted by the British Association Committee on Electrical Standards (Report, Newcastle Meeting, September, 1889):— 366 APPENDIX. (I) The name of the practical unit of work shall be the joule. The joule is equivalent to Io' C.G.S. units of work. It is the energy expended during I second by a current of I ampere when traversing a resistance of I ohm. [See above, p. 77.] (2) The name of the practical unit of power' shall be the zwałł. The watt is the rate of working of a machine per- forming I joule per second. The power of a machine could naturally be expressed in kilowatts instead of in horse-power. [See above, p. 78.] (3) The name of the practical unit of light intensity shall be the cand/e. The candle is equal to the twentieth part of the absolute standard of light as defined by the International Con- ference of 1884.2% (4) The name of the practical unit of induction shall be the “quadrant.” I quadrant is equal to Io9 cms. (5) The Žeriod of an alternating current is the duration of a complete oscillation. (6) The frequency of an alternating current is the number of complete oscillations per second. (7) The mean current through a circuit is the time- average of the current. [It is defined by I T/2 Imean Current = #ſ. C d f, C being the current at each instant of the time 7/2 (T = period of the current) which is reckoned from an instant at which the current is increasing through zero. * This standard is what was defined by that Conference as the Aractical zenit of white light, viz, the total quantity of light emitted by a square centimetre of melted platinum at the temperature of solidification. APPENDIX. 367 The mean current is thus Cº., of p. 283 above, there also called the “mean current.”] (8) The effective current is the square root of the time- average of the square of the current. Thus I ſ?” % effective current = | }| cºal) º O [The effective current is therefore C of p. 284 above.] (9) The effective electromotive force is the square root of the time-average of the square of the electromotive force. Thus e e ſ I ſ? % effective electromotive force = l #| J22 & *} - () A. being the actual electromotive force at each instant of the time 7. (IO) The Zºnfedance is the factor by which the effective current must be multiplied to give the effective electro- motive force. Thus, in the case of a circuit of A' ohms and Self-induction / quadrants in which a simple harmonic electromotive force of frequency m/27 is acting, impedance = WA” + 7.” M.”. (II) In an accumulator the positive pole is that which is connected with the positive pole of the machine when charging, and from which the current passes into the external circuit when discharging. 368 TABLE I. CROSS-SECTION OF ROUND WIRES, witH RESISTANCE, CONDUCTIVITY, AND WEIGHT OF HARD-DRAWN PURE COPPER WIRES, ACCORDING TO THE NEW STANDARD WIRE GAUGE LEGALIZED BY ORDER IN COUNCIL, AUGUST 23, 1883. Temperature 15° Cent. Area of Cross-section. Con- Weight. Resistance. ductivity. | (Density=8'95) No. Diameter. Legal | Legal ||Yards |Metres L bs. Grms. * - Sq. Ohms Ohms per | per Ins. Cms. | Sq. ins. CIY1S. per per | Legal Legal yº M. Yard. | Metre. Ohm. Ohm. &ll Cl. etre ooooooo '500 || 27o ‘IgG3 I'267 ooor 25|'ooor 36.8055 7365 2'285 II 34 ooooool'464 || '179 | 1699 I ‘og1 |"Ooor 44|’oool 5716937 || 6343 | I '970 |376'3 ooooo, ‘432 |I ‘og7 || 1466 ‘946 ooor 66|"Ooo 1826or 3 || 5498 I '706 |846'3 oooo. 4 Co | I or 6 || 1257 '811 |"Ooo 194|'OOO213,5054 || 4714 | I 463 725-6 ooo. 372 '945 || Io97 ‘70I 'ooo225|"Ooo245|4459 | 4077 | I 265 627:6 ooj'348 '884 |’og51 "614 || “ooo.256 ‘ooo28o39ot 3568 I ‘Io/ 549'6 o|'324 ‘823 o824 '532 |"Oooz96|ºooo323|3384 || 3093 ‘56o 476°r 1|'3oo 762 |'o'707 ‘456 ‘ooo3.45|"Ooo377|2899 || 2652 ‘823 408"I 2| 276 || 701 |'oS98 "386 |'ooo.408|’ooo446|2454 2244 '696 |345°4 3|“252 '640 |'O499 *322 |-ooo.489 ooos36|2O46 1871 ‘581 288°o 4|232 '589 |'O423 ’273 |'ooos'77|"Oooo.31|1734 I586 '492 244 I 5|’212 '538 |’o353 ‘228 |'oooë91|’OOO756|I45 t I 324 '4II 203 °8 6|' 192 || 488 |'ozgo ‘187 ſºooo.842|-ooog21|1197 | Io96 '337 [66°8 7| 176 447 "O243 "1 57 |-ooroo 'oor Io 988 912 ‘283 I 4O 5 8}*16o '406 |'ozor '130 °oor 22 |'oor 35 | 824 748 "234 t 16' I 9 : 144 || 366 |'oró3 ‘105 |’oor 49 |ooI64 | 669 6 II 190 94 "O Io "I 28 325 °o 129 ‘o83o |'oor 90 °oo208 || 528 482 "I50 74 3 I Il 116 || 2 25 °orcó 'o682 'ooz3o 'ooz 52 || 434 396 "I 23 ÓI ‘o 12.194 ‘264 | co849 ‘os48 |'ooz87 |"Oo314 || 348 318 'og89 || 49'o 13°og2 234 |'ooG65 ‘c 429 |'oo367 'oozło2 273 250 'o'774 || 38'4 14 "o8o 203 |'oosog ‘og24 |"ooq85 |’ooS30 206 I88 'o685 29°o I5'oZ2 | "I'83 | "OoAOZ 'oz63 |’oos99 |'ooč57 | 167 I53 "O474 || 23’5 36.264 ’163 |'oog22 ‘o208 |'oo752 |'oo839 || 132 I 2G "O374 I8°6 17|'osó | "142 |'oozá6 ‘ol 59 |'oog.9 |’oro8 I O I 91°5] 'o.287 I4 2 18|'o.48 || '122 |'oor 81 “or 17 or 35 or 47 74°2' 67"8| ‘o21 I Io'4 19 'o.4o 'Io2 'oo 126 ‘oo811 or 94 'O2 I2 51 '6| 47 'I or 46 7°26 2O ‘O36 og I4 |'oor.O2 ‘ooô57|'o239 |'O262 41 ‘8 38-2 or 18 5'88 21|'o62 | "o813 |'ooo804 || "Oo319|'o3O4 'o63I 32 '9 3o'I] ‘oog36 4'64 22|'oz8 || 'oZ II |'oooé16 || “oo397|'o396 |'O433 25'3| 23'o "oo'ſ 17 | 3’56 23|'oz4 'o61o |'ooo452 | "oozg2|'o639 °o589 18'5| 17 to ‘oos26 || 2 61 24 oz.2 ‘os 59 |'oco 38o ‘oo245|'o642 |'o'70I I5'6 I4'3| "OO443 2 "Ig 25}'ozo 'o608 |'ooo314 | co203|'o'778 |'o849 I2 °8 II '8| ‘oo366 I '8o TABLE I. (continued.) 369 CROSS-SECTION OF ROUND WIRES, CONDUCTIVITY, AND WEIGHT OF HARD-DRAWN PURE WITH RESISTANCE, COPPER WIREs, ACCORDING TO THE NEW STANDARD WIRE GAUGE LEGALIZED BY ORDER AUGUST 23, 1883. Temperature I 5° Cent. IN COUNCIL, No. Diameter. cºin. Resistance. Conductivity. (pº '95) Legal Legal Yards Metres & 6. Ohms per per Lbs. Grms. Ins. Cms. | Sq. ins. Sq. cms. per per per per | Legal Legal Yard. | Met Yard. | Metre. Ohm. Ohim. a ſū. €tre. 26 'ol 8 |'oq57 °ooo.254 ‘oor 64 'og58; ‘Io5 |Io'4 9'54 , "ooz96 I '47 27 or 64; 'OA 17 | "Ooo2II "Oor 36 ‘II6 '123 || 8'65 || 7 '93 |'ooz46 I "22 28 or 48°og76 |“ooor 72 'OO III "I4I "I 55 7'o'7 6'45 |'OO2Oo ‘893 29 |'or 36, 'O?45 |'ooo145 ‘OOO937 "I68 ‘I33 || 5'95 5'45 °oor 69 ‘839 3o 'oï24, 'O315 'OOO12 I ‘OOO779 "2O2 221 || 486 || 4'53 |’oor 41 '697 31 or 16 oz.95 ‘oooroč 'oooč82 ’230 252 || 4’34 3'96 |"Oor 23 ‘6 Io 32 |'oro8'o274 oooog 16 "Oooš91 ’266 ‘291 3’75 3'44 |’OOIO7 ‘529 33 |’o too, ‘O254 oooo785 ooošo'7 "3 II “339 3'22 || 2 '94 |'OOO914 453 34 |'oog2 O234 |"Oooo&65 'ooo.429 '367 ‘402 || 2 |73 || 2'50 |’OOO774 '384 35 °oo84|’O213 | "Oooos 54 |"Ooo358 '44o ‘431 || 2 (27 2°03 | "Ooof,45 '32O 36 °oo76 or 93 | "Oooo.454 |"Ooo293 '54O '587 || 1 '86 I '70 'Ooos48 "262 37 |'ooë8|’or 73 | "oooo363 | "Ooo234 '672 ‘736 I'49 I '36 ooo.423 2 IO 38 °oofio or 52 oooo.283 | "Ooo 182 ‘862 ‘944 || 1 || 6 I'OA 'OOO329 ‘163 39 ‘OOS2 'OI 32 | "OOOO2 I2 ‘OOOI37 I ‘I 5 I 26 ’870 ‘796 |'ooo.247 'I23 4o "ooq8 or 22 | "ooool 81 | "Ooor 17 I 32 I 47 '759 '679 ooo.211 IO4 41 |"ooq 4 or 12 oooor 52 'oooog81 || 1 '60 I ‘75 ‘624 '570 ooo 177 ‘oS78 42 |"ooqo, 'oro2 | "Ooool 26 |"Oooo&II | I '94 2 13 '516 '47 I |"Ooo 146 'o'726 43 |oo36, ’oog14|-oooo.192 ſooooë57 || 2:39 2'62 '418 '382 |'ooo 118 oš88 44 |"Oo32, 'Oo313|"Ooooo&O4 |"Oooos 19 || 3'O4 || 3'32 "33O '301 || Oooog36 oA64 45 °oo28, ‘oo/II] 'Oooooºr 6 |"Ooºo397 || 3'96 || 4’33 253 ’230 "Oooo717 ‘og56 46 |'oozá'ooé19 ooooo.452 |"Oooo.292 || 539 5'90 ‘185 ‘I 7o oooos 27 | ‘o.261 47 'oozo, oosog ooooo.314 |'oooo.203 776 | 8:49 ‘I 28 ‘I 18 oooo 366 or 81 48 ooi6, ooloº ‘OOOOO2OI ‘OOOO130 |I2'2O |I3'3 'o824' 'oZ54|’oooo.234 of 16 49 |ool 2"oo305°oooool 13 Fooooo?30|2I 6 23'5 ‘O464, 'o:25|'oooo 132 ‘ooé53 5O colºs. "Oooooo785!'OOOOO507|31 "I 33'9 'o622, 'oz'9, 'Ooooog I4 OoA53 Not E. –The resistances and conductivities in Tables I. and II. are calculated by taking I-624 × Io-6 Legal Ohm as the res.stance at o' C., between the ends of a hard-drawn copper wire I cm. long and I sq. cm. in cross-section. Matthiessen and Hockin’s result (B. A. Rep., 1864, and Phil. Mag., vol. xxix., 1865) of 1469 B.A. unit as the resistance at o' C. of a wire one metre long weighing one gramme, if the density, 895, of cast specimens of their copper be taken as approxi- mately the density of the wires experimented on, which was not determined. This agrees with To reduce the numbers to accord with the B.A. unit add 1'12 per cent. to the resistances and subtract 1 12 per cent, from the conductivities, B B a- 37O TARLE I [. CROSS-SECTION OF ROUND WIREs, witH RESISTANCE, CONDUC- TIVITY, AND WEIGHT OF PURE COPPER WIREs, According To THE BIRMINGHAM WIRE GAUGE. (See Wote to Table I.) Temperature 15° Cent. B. W in ºn of a - Area of cross- s: .*.*... Weight. (density G. Diameter. Section. Resistance. Conductivity. = 8'95). 5. 5. "..." "...” Its per |Gins. *- * : -- ~ - S - - * * * - Ins. Cms. | Sq. ins. Sq. cms. per per | Legal Legal Yard. Mºre Yard. | Metre. ()hm. Ohm. - oooo ‘454 I I S3 | "162 I "O444 ‘ooor 50 ooor 65 | 664.o 6072 |I '884 934 7 oCo 425 || 'oZ9 142 '915 ‘ooor 72 ‘ooom 88 || 5819 || 5321 |I'651 819. I oo 38o ‘965 1 13 ‘732 ‘ooo.215 ‘ooo.235 | 4653 || 4254 |I'329 654'8 Ó '34o '864 ‘ogo3 '586 ‘ooo269 ‘ooo.294 || 3735 | 3415 || OSO 524'2 I 3oo 762 'oZo.7 '456 ‘ooo?45 || “ooo378 2899 || 2652 '822 408" I 2 284 || 721 'o633 "4O9 ‘OOO385 ‘ooo.420 2599 || 2377 '73 365'8 3 250 '658 ‘os 27 '34O ‘ooo.463 | "ooo;06 || 2162 | 1996 ‘613 3O4'2 4 238 ‘Gos of 45 287 ‘oooš48 ‘ooo;99 || 1825 | 1669 '51 256'9 5 22O '559 'ogº'o "245 "Ooo(42 ‘ood7or 1561 || 1427 442 219' 5 6 203 '516 ‘og2 | ‘209 ‘ooo754 ‘ooo324 || 1328 || 1214 '377 1869 7 18o 457 'oz 54 164 ‘OOOO58 ‘ooros IO44 IOO4 ‘296 I46'9 8 165 '419 || 'oz 4 138 ‘OOI 14 ‘oo I 25 877 | 802 ‘249 123'5 9 148 376 ‘ol 72 "I I I ‘OOI 41 OO [ 55 706 || 645 2CO 99'3 I O | I 34 || 3-ſo | ‘OI. I ‘og IO ‘OOI73 oo 189 578 529 || '164 81 “4 I 1 || '12o '305 or 13 ‘o'73o ‘oo216 'ooz 35 463 || 424 | "I 32 65'5 I 2 | "Io9 277 || "Ooo 33 'o6O2 'ooz61 ‘oo286 382 350 IO9 53'9 J 3 | "O!)5 ‘241 'OO 709 "O457 "OO344 oo:376 291 266 o82 40'9 14 ‘OS3 || 2 || 1 || ‘oos. I 'O349 ‘OO4S I OO492 22 I 2O3 o630 31 2 15 ‘O72 183 ‘oo.407 || 'oz'63 "OO599 OO655 167 | 153 | "O474 23'5 16 'o65 165 ‘oo331 ‘o214 ‘oo735 oo8o.; 136 | 124 ogS6 19'2 17 | ‘os3 147 'oozó4 or 7o ‘oo923 O J O I IoS 98.7 O3O7 I5'3 18 'o.49 || '124 'oo 189 || 'ol 22 'or 3o O 14 I 77° 3 || 7o'7 O2. Q I O'9 I 9 'o.;2 | " I O'7 || “oo I 39 ‘oo894 ‘o 176 ‘ol 94 568 52 'o or 61 8 ‘oo 20 ‘oj5 ‘o889 'oooſ)02 ‘ooG21 ‘O253 'oz77 39°4 || 36°r O 1 2 3 5'56 2 I ox2 | "o813 ooo&o.4 ‘oos 19 || 'ogo.4 'O331 32 to 30' I ooq36 4 64 22 || “oz8 of 1 1 ‘oooſ, 16 || ‘oo 397 "O395 "O433 25'3 23' I oo716 3.35 23 'oz 5 'o635| "Ooo.10 I ‘oo 317 'o.496 "O543 20'2 18°4 OOS7 I 2.83 24 ‘O22 | "OS59| "Ooo 38o 'ooz.45 'o642 'o'701 15 6 || 14 3 OO.[42 2.9 2 'ozo ‘oso8| ‘ooo314 'oozog ‘o?78 ‘o849 12 ‘8 || 1 || 7 Oo367 I '82 26 to 18 ‘o.157| ‘ooo.254 ‘ool 64 'O').59 Ios 1 o’2 0'53 oozoá 47 2 'or 6 || 'odo6 ‘oooºo 1 || 'OO 130 "I 22 I 33 8'25 || 7 54 OO234 . 6 28 or 4 || 'o656 ‘oool 54 ‘ooooo.3 | "158 173 6' 31 577 OOI79 889 29 or 3 | "o 330 °ooo 133 |: ‘ooo856 | 18.4 *2O I 5' 4 1 || 4 98 OOI 54 766 3o or 2 | ‘osos oool 13 | "Ooo732 || 2 | 6 “235 4'64 4'24 OOI 32 653 31 || 'oro 'oz 54| ‘oooo.70s oooso/ | 3 | 1 330 3'23 || 2 '95 ooooº; 454 32 | "ooq) 'oz 29| “oooc.636 ‘OOO. Io 384 4 IQ 2 5 I 2'30 ooo746 367 33 ‘oo8 o203| 'o-ooso | 'ooo;24 || 486 '53O 2 'o6 || 1 '88 oces85 299 34 ‘oo7 | "o 178 oooo.385 'coo248 '634 '693 I '58 I 45 OOO. 42 229 35 | ‘oos on 27 oooo 196] 'Cool 27 in 25 I '35 '806 || 736 "Oooz29 II 3 36 ‘oot 'oro.2 oooo.126 'oooo.81 II '94 2 'I 3 '516 || 471 ‘ooor 46 'o'726 37 I TABLE III. CONDUCTIVITIES OF PURE METALS AT * C.* Conductivity at o' = 1. Metal. Conductivity at tº C. Silver 1 – ‘oo38278? -- 'ooooog848.1% Copper 1 – ’oog87orf + ooooooooot!” old 1 – ‘oo36745: + ‘ooooo&443?” Zinc t – ‘oo37047? -- 'ooooo3274tº Cadmium 1 – ‘oo?6871 t + ‘ooooo7575?? ^ Tin 1 – ‘oo36029t + ‘oooooë136tº Lead 1 – ’oog8756t + ‘oooooorá6t? Arsenic 1 – ’oo:38996; + ‘ooooo&879t? Antimony 1 – ‘oo3O826t + ‘ooooto364?” Bismuth 1 – ’oog5216# - ‘ooooos'728tº Iron 1 – ‘oos, 182t + ‘oooor 2916#2 * From the results of Matthiessen's experiments; and to be used only for temperatures between o’C. and 100°C. The formulas, excluding that for iron, agree closely, and give the mean formula I – ‘oo37647t + 'oooo.834ot?, 372 TABLE IV. CONDUCTIVITY AND RESISTANCE OF PURE COPPER AT TEMPERATURES FROM o' C. TO 40° C. Calculated by the formula for the Conductivity of Copper in Table III. Temp. Conductivity. Resistance. Temp. Conductivity. Resistance. o° I 'OOOO I "OOOO 219 o'9227 I'o838 I O'9961 I too388 22 O'9192 I'o679 2 O'9923 1°oo776 23 o 9158 I ‘og2O 3 o 9885 I or 16 24 O'91:23 I ‘og61 4. C’9847 I or 56 25 o'9089 I IOO3 5 O'9809 I ‘O I95 26 || o'9054 I IO44 6 o’977 I I O234 27 o'902O I '8o35 7 O'9734 I ‘O274 28 o'8987 I ‘II 27 8 o'9696 I'O3I3 29 o’8953 I 1169 9 O'9559 I ‘O353 3O o'8920 I "1 2 II IO O 9622 I ob93 3 I o'8887 I I253 I I o 9585 I'O433 32 o'8854 I I295 I 2 O 9549 I'O473 33 o'8821 I "I 337 I 3 O'95I2 I OSI 3 34 o:8788 I ‘I379 I4 o 9476 I ‘OS53 35 o 8756 I 142 I I 5 O'944o I of 93 36 o'8723 I ‘I464 I6 O 94O4. I'o634 37 o'8691 I 1506 17 o'9368 I 'o675 38 o 8659 I ‘I 548 18 O'9333 I'oZ IS 39 o'8528 I ‘I 591 19 O'9297 I 'oZ56 40 o'8596 I 1633 2O o’9262 I'oZ.97 TABLE V. 373 SPECIFIC RESISTANCES IN LEGAL OHMS OF WIRES OF DIFFERENT METALS AND ALLOYS.* * - Q2 (d . . ." ..., 35 | Sº : | . . . ; 3 L 3 o 5 ; # 95 |% ##|% ºf ‘ā Śā |: 5 U 3.5 ; F ºf . , ; c. E . , ; ; ; ; , ; 5 #5 | ? ( ; ; ºp; 9.2 :, 9. § |9.g. || 9 || || 3: ; 33; 33 g|% ##|33 || 3 #5 |###g Substance. § 3.5 cd “º o : ā ā ;3.5 § 3 ; ‘. .33°g 3 sº 3 * b.) | 3 p 3 § 3–5 | # 3 à ºf gº, . º 9 E § 3.5 | E 5: 90 5 § 3 || 5 #:- || 3 3 S : § E 3 ; : ; ; #| 3 || 3 || 3 ##|# 5 # 3 ° # 8 Tb) || 3 p 3 || 5 §2 Ž j = § 3 ; "aſ do Cº ‘5 's ‘35 || “...: 'bſ, 'º'; ' ' 'º, tº *2.4 3 §§ " | # * #|###|}* | | ###|### : > H ~ 5 3 - 9 Silver, annealed I'504 × 10–0 o'org16 o'I527 | 9'049 “2190 o'377 Silver, hard drawn I '534 } } o'o.208o o' 1661 | 9'825 ’2388 ...so Copper, annealed I '598 3 y o'o.2034 o' 1424 || 9'609 ‘2041 o'383 Copper, hard drawn I '634 3 × o ozo&I o' 1453 9'829 ‘2083 g Gold, annealed 2°os8 2 3 o'o.2621 o'4035 | 12'38 '5784 o' 365 Gold, hard drawn 2 'OO5 3 y o'o2667 o'4 IO4 1260 '5883 Aluminium, annealed *}. > 3 . o'o'749 I7 3. ..ºz. 6 H Zinc, pressed 5'0 I4 ,, . 9'O7I63 o'4022 || 33’84 ‘5766 o'365 Platinum, annealed 9'O55 3 y O'I I53 I '94 54'47 2.779 tº & ºn Iron, annealed 9°715 * x o' 1237 o'7570 58'44 I ‘o85 º Nickel, annealed 12'46 3 y o’ I 586 I ‘os9 || 74°93 I '518 tº e e Tin, pressed I3'21 3 * o' 1682 o'9629 || 79°46 I'381 o'365 Lead, pressed I9'63 * x o’2498 || 2:232 | T 18°os | 3:200 O'387 Antimony, pressed 35' 50 * 3 o'4520 || 2 384 214 3'418 c'389 Bismuth, #. I31 °2 : 5 I 670 12 '89 || 789 1844 O'354 Mercury, liquid (see * * - & e - y )5 "I I 3 * I 21 12 12'92 || 572 I 18'51 o'oZ2 Note) 9 Platinum - Silver - Alºhard Or 24’39 3 y o'31os 2'926 | 1.46"69 || 4 195 o'os I annea.I.G German SilverAlloy, º * $ * £º º ºr * (*). hard or annealed 2099 , , o'2665 | 1 83 125'89 || 2 623 o'o.44 Gold-Slver Alloy, I e * T rº g * , , * ~ * * ‘,-6 - hard or annealed Io'87 3 y o' 1384 || 1 '65o 65°36 2'365 O OS 5 Dlatinoid * 32-8 3 y tº $ tº • * * tº E & g º is o2 I* Hºangan) 68° about , , e ‘º e tº º º tº e ºs tº gº tº ‘I 2.2% * Reduced (with the exception of platinoid and manganese steel) from a table given by Professor Jenkin as expressing the results of Matthiessen’s experiments. The numbers for platinoid and Hadfield's manganese steel are taken from a paper by Professor J. A. Fleming (Electrician, March 9, 1888). The percentage variation of resistance for these two substances is the average for the range between o' C. and roo” C. t Two parts platinum, one part silver, by weight. † Two parts gold, one part silver, by weight. Mote.—According to the determinations of the specific resistance of mercury referred to above (p. 243), the value given in this table is about 8 per cent. too high. A column of pure mercury therefore one sq. millimetre in section, which at o°C. has a resistance of one ohm, has, according to the B.A. determination of the ohm, a length of Ioq8 cms. nearly, and according to Lord Rayleigh and Mrs. Sidgwick's experiments, ro5 21 cms. According to the legal ohm the length is, as stated above, p. 72, 106 cms. The value given in Col. I. for hard-drawn copper has evidently been calculated from the corresponding observed result in Col. III., by using the density 8-89 for copper, and is therefore higher than that on which Tables I, and II. are founded. (See AVožet S Table I.) 374 TABLE VI. For reduction of Period of Oscillation observed for finite amplitude to Period for infinitely small amplitude. If T be the observed period and I — k the reducing factor, so that AET is to be subtracted, the values of k are as follows:– Amplitude. Ž Amplitude. /* O "OOOOO I I • "OO23o I "OOOO2. I2 ‘OO274 2 ‘oooo& I3 "Oo322 3 "Ooo 17 I4 "Oo373 4. "Ooo3O. I5 'oozł28 5 'ooo.48 I6 ‘oo487 6 ‘ooo.69 17 ‘OOS5o 7 ‘OOOO3 I8 ‘ooôI6 8 "OO I22 I9 "ooô86 9 "OOI54 2O ‘oo76r IO ‘oo L90 375 TABLE VII. UNITS OF WORK OR ENERGY. 2 y - - - - - - - • . . . . . 1 centimetre-gramme at Paris . . x 5 3 y 2 y at London . 3 y ..?? 3 x 33 . . » * I metre-kilogramme at Paris . . ,, at London 2 3 3 y I foot-pound at London . . . . . I./ottle . . TABLE VIII. UNITS OF ACTIVITY OR RATE OF WORKING. 2.374 × ro-6 foot-poundal. 7.375 × 10-8 foot-pound at London. 981 ergs. 981' 17 ergs. 2: 329 × Io-3 foot-poundal. 981 × 105 ergs. 981' 17 × 105 ergs. 7'236 foot-pounds. 42I 390 ergs. 13° 56 × 100 ergs. IoW ergs. I erg per second 1 horse-power . . . . . . . . . 3 y ,, at London . . . . I force-de-cheval at Paris . . . . I Waff • 0 & 0 tº q = ~ * * * 1'34 × 10T10 horse power at London. 33ooo foot-pounds per minute. 7'46 × 109 ergs per second. 7'36 × 109 ergs per second. Io? ergs per second. IOOO Watts, 7 6 TABLE IX. —DOUBLED SQUARE ROOTS FOR - I i O. KYOO 2 °C OO 2 °828 3°464 4 'OOO 4'472 4 '899 5’292 5 (657 6'ooo 6' 325 2CO 3OO 4OO 500 6oo 8oo 9do 28 °28 28 3 28°43 28'50 28'57 28° 64 34°64 34°7o 34 76 34°81 34'87 34'93 4o'oo 4O'oS 4o Io 4O I 5 4O'2O 4O 25 44'72 44 (7.7 44 81 44'86 44'90 44'94 48'99 49'O3 49'O7 49 II 49 IS 49 I9 56'57 56-60 56'64 56-67 56'7 I 56'75 6o'oo 6o'o 3 6o'o'7 6o 1 o 6o 13 6o 17 * 28°7 I 6'633 6 928 7' 21 I 7'483 7 '746 34 '99 35 OA. 35° ſo 35' 16 35’21 35 27 35' 33 35'38 35' 44 35 5O 4O'3O 4O'35 4 o' 40 40' 45 4O SO 4O'55 4O'60 4o 64 4O 69 4O 74 44'9) 45 O3 45°o8 45' I2 A 5 I 7 45 2 I 45' 25 45' & O 45'34 45' 39 49 °23 49'27 49 32 49°36 49 4O 56-78 56'82 56'85 56'89 56'92 49 44 49 48 49'52 49 56 49' 6o 56'96 56'99 57 og 57 'o6 57 Io 6o 20 6o 23 6o'27 Óo'3o Čo 33 6o'37 I 8'ooo 8'246 8’485 8’718 8'944 35' 55 35'61 35'67 35'72 35'78 40'79 45' 43 45°48 45' 52 45'56 45 61 49'64 49'68 49 72 49'76 49°8o 9' 165 9° 381 9'592 9'798 I ( ) "OOO 35'83 35'89 35 94 36 ‘OO 36 oé 3o 27 3O 33 45.65 45'69 45'74 45°78 4583 49.84 49'88 49'92 49'96 5o'OO 45'87 45'91 45 96 46°oo 46'04 5O'O4. 5o'o6 5o I2 5o 16 go'2O 3O'4O 3O'46 3O'53 3O 59 3o'66 46'09 46’13 46' 17 46’22 46’26 50'24. 5o 28 5O'32 5o'36 5O'4O 3O'72 3o'79 3o '85 3O'Q2 3O 98 3 I'os 3I I I 31 18 3 I 24 3I 3O 3I '37 31'56 31 ‘62 46.30 46'35 46'39 46'43 46′48 CO 44 5o 48 5O'52 5o 56 5o Go 46'52 46'56 46'60 46 65 46"69 5o'64 5o 68 5o'71 5O'75 50'79 46'73 4678 46 82 46-86 46'90 5o '83 5o'87 50'QI 5O 95 5o'99 | I OO 2O"OO 2O" IO 2O "2O 20 'go 2O'Ao 2O'49 2O '59 2d 69 2 I 45 2 I's 4 21 63 2 I '73 2 I '82 2 I '91 57 'I 3 57 17 57 20 57 24 57 27 22 OO 22 'o) 22 18 22'27 22°36 | Io' 198 IO'392 I O '583 Io 770 IO’954 22 45 22 '54 22 '63 22 72 22 '8o 36 II 36 17 36°22 36:28 36°33 57'31 57'34. 57 '38 57° 4 I 57 ° 45 57°48 57'52 57' 55 57°58 57-62 1 I’t 36 II 3 I4 Y I'489 1 ſ 662 1 r 832 22°89 22 Q8 23'O? 23 15 23'24 36°39 36'44 36'50 36'55 36'61 57-65 57' 69 57'72 57'76 57'79 J 2 OOO I 2 I 66 12'329 12' 490 I2 O49 23'32 23 4 I 23' 49 23'58 23 '66 I 2 '806 12 '961 I 3’ I I 5 r 3'266 13°416 23°75 23 °83 23'92 24'oo 24°o8 13’565 I 3 '71 I 13°856 14 "OOO I4 I42 24, 17 24 25 24 33 24 4 I 24 49 36'66 36.72 36 77 36 82 36'88 57.83 s? '86 57 '90 57 '93 57'97 36'93 36 Q9 37 'O4 37'O3 37’ I 5 58 ‘oo 58'03 58'ſ 7 58'Io 58'14 37'2O 37'26 37'3 I 37 36 37 '42 58 17 58'21 58°24 58:28 58°31 SIR. W. THOMSON'S STANDARD ELECTRIC BALANCES. O ICO 2CO 3OO 4OO 5OO 6oo 7oo | 8oo | 900 I4'283 I4” 422 I4'56o I4-697 I4'832 24'58 24'66 24'74 24'82 24'90 31'69 3I’75 31°81 31'87 3I'04 37' 47 37'52 37' 58 37:63 37°68 42'47 42'52 42'57 42'61 42-66 46'95 46'99 47 O3 47'O7 47' I 2 5I ‘O3 51'07 5.I.' I I 5.I.' I 5 51 ‘I 9 54'81 54'85 54'88 54'92 54'95 58'34 58°38 58° 41 58°45 58° 48 61 '68 61'71 61 '74 61-77 61 ‘8 I I4'967 I5' Ioo I 5’232 I 5'362 I5'492 24'98 25°C6 25' I 4 25'22 25'30 32°oo 32-c6 32' 12 32' IQ 32'25 37'74 37'79 37°84 37°89 37'95 42'71 42'76 42'8o 42'85 42'90 47° 16 47'2O 47'24 47' 29 47'33 51'22 51'26 5I'30 51'38 54'99 55'O3 55 c6 55' lo 55. I4 I 5'62o I 5’748 15'875 I6'ooo I6' 125 25'38 25'46 25" 53 25'61 25 69 32'3 I 32°37 32' 43 32°50 32°56 38°oo 38°o 5 38'11 38'16 38'21 42'94 42'99 43'O3 43'c8 43' I 3 47'37 47'4 I 47°46 47' 50 47'54 I6'248 16' 37 I I6'492 16-613 16'733 25'77 25'85 25'92 26'oo 26'o8 32’62 32’68 32 74 32°8o 32'86 38°26 38°31 38-37 38°42 38°47 43'I 7 43'22 43'27 43'3 I 43'36 47'58 47°62 47°67 47 '71 47° 75 5I 42 51'46 51 5O 5 I'54 51°58 55' I 7 55’2 I 55’24 55’28 55'32 5 I'61 5I'65 51'69 5I '73 51'77 55' 35 55' 39 55 43 55'46 55. So 58'51 $8'55 58'58 58-62 58-65 58-69 58-72 58°75 58'79 58-82 58-86 58-89 58'92 58'96 58'99 61-84 61-87 61 ‘90 6I '94 61.97 62'oo 62°o3 62'o6 62. Io 62. 13 62. I6 62' 19 62°23 62'26 62'29 16'852 I6'971 17°o88 17 205 17'321 26’ I 5 26'23 26'31 26:38 26'46 32'Q2 32°98 33 O5 33 II 33'I 7 38°52 38°57 38-63 38.68 38°73 43'4I 43'45 43’50 43’54 43’59 47'79 47°83 47-87 47'92 47'06 51 ‘81 51 '85 51-88 5I '92 51'96 5.5" 53 55' 57 55-61 55'64 55'68 59'O3 5 9'o6 59 OO 59. I 3 59' 16 62'32 62'35 62°39 62'42 62'45 17'436 I7'55o 17°664 17776 17:889 26'53 26-6 I 26'68 26'76 26'83 33'23 33'29 33'35 33. 4 I 33' 47 43'63 43'68 43'73 43 77 43'82 48°oo 48°o4 48°o8 48° 12 48° 17 52'oo 52'O4 52'o8 52' 12 52’ I 5 55'71 55’75 55'79 55'82 55'86 59' 19 59°23 59'26 59'3O 59' 33 62'48 18°ooo I8’ III 18°221 1833o 18°439 33'53 33'59 33°65 33 '70 33°76 43'86 43'91 43 95 44 'OO 4.4"o 5 48'21 48'25 48'29 48°33 48°37 52'IQ 52'23 52'27 52'3 I 52' 35 55'89 55'93 55'96 56°oo 56°o 4 59'36 59' 4o 59' 43 59'46 59'50 18'547 18°655 18°762 I 8' 868 18'974 33'82 35'88 33'94 34 Co 34'06 I9'C79 19 183 19°287 I9'391 I9 494 34 I2 34° 18 34 23 34'29 34'35 44'O9 44' I 4 44° 18 44'23 44'27 48°41 48°46 48'50 48 54 48°58 52°38 52'42 52'46 52 '50 52 '54 56'o7 56 II 56° 14 56°18 56°21 59'53 59'57 59'60 59'63 59'67 44, 32 44'36 44' 41 44'45 44'50 48°62 48°66 48°7o 48°74 48°79 52 57 52'61 52 65 52'69 52 '73 56'25 56'28 56'32 56:36 56'39 59'70 59'73 19'596 19'698 I9'799 I9'90o 20" OOO 26'91 26'98 27'o6 27 13 27:20 27:28 27°35 27 '42 27'50 27°57 27 '64 27 71 27-78 27.86 27 93 28°oo 28'o'7 28'14 28°21 28°28 34°4 I 34'47 34' 53 34°58 34 64 44'54 44'59 44'63 44 68 44 72 48°83 48.87 48 91 48'95 48'99 52.76 52'8o 52 '84 52 88 52 '92 56'43 56'46 56'5o 56'53 56'57 378 TABLE X. FRENCH AND ENGLISH UNITS. Metre in inches . . . . . . . . . . . . . . . . . . 39'37O43 Foot in centimetres. . . . . . . . . . . . . . . . . 3O'4797 Mile in kilometres . . . . . . . . . * * * * * * * * I'6093 Kilometre in miles . . . . . . . . . . . . . * * ‘621.37 Gramme in grains . . . . . . . . . . . . . . . . . I5' 43235 Grain in milligrammes . . . . . . . . . . . . . . . 64'799 Pound in grammes . . . . . . . . . . . . . . * * - 453'593 Kilogramme in pounds . . . . . . * * * * * * * * * 2°2O46 British ton in French tons of Iooo kilos . . . . . . . I or 6 Litre in cubic inches . . . . . . . . . . . . . . . . 61 oz.53 Cubic inch in cubic centimetres . . . . . . . . . . - 16:3862 Cubic foot in cubic centimetres . . . . . . . . . . . 2831.5°3 VIRTUAL RESISTANCE OF CONDUCTORS CARRYING ALTERNATING CURRENTS. Sir William Thomson has given (Presidential Address Inst. Elect. Engs. Jan. 9, 1889) the following formulae for calculating the resistance of wires carrying alternating currents, Let a denote the specific resistance of the wire in C.G.S. units (see above, . 2 2 & ). a ,, the radius of the wire. R(S) ,, the value of ord/tra?, that is the resistance in cms, per second of a length of 2 cms. of the wire with a steady current flowing through it. R(N) ,, the effective ohmic * resistance of the same length, 2, with an alternate current of N periods per second flowing through 1t. c(N) ,, the current density at distance r from the axis at time t. C(N) ,, the current density at the axis at time t. then c(N) = C(N) (berg cos 6 - beig sin 6). . . . . . (1) where q = 2 T 2: S/2 A/or, 6 = 2 TAV t, and ber and bei denote two functions defined as follows 4 8 be a = 1 - # 4 gºas – &c. 2 6 — ? g * beiz = }; - gºat & Also if A denote the value of g with r = a, & X) = , a ber 4, beiº - bei?ber 2 g . (2) R (S) - (ber' p)2 + (bei’A)? where the accents denote differential coefficients. For copper at o' C. a = 1610 sq. cms. per sec. Hence with W = 80, Q = 27 nearly. * Not including any impedance effect of self-induction. 379 Thus for copper with W = 80, the column below headed 7 may be taken as containing the diameters of the wires and in respect to the distribution of the current through the wire expressed by the formula in (1) above, 7 may be taken as the diameter of the cylindric shell in which the current density is to be calculated. TABLE (XI.) OF NUMERICAL VALUES. (Calculated for Sir W. Thomson by Mr. M. Maclean.) t beit ber – ber bei 1, bei' ber - ber' bei Q ber 2 + be.’ ” - ber 2 + bei' 2 O'o CO I 'OOOO O'S 4'OOOO I OOOO I 'O 2 "OOOI 4 I 'OOOI I 5 I 3678 I'oz 58 2 "O I 'o605 I ‘o805 2 '5 '9398 - I 1747 3'O ’8787 I 318o 3'5 ‘8526 I '492O 4 'O ‘8389 I 6778 4 5 ‘8279 I 8628 5'o ‘817.2 2'O43O 5 5 ‘8069 2*219o 6'o ‘7979 2 3937 8 '7739 3’ og S6 IO ‘7588 3 794O I5 '743 I 5'5732 2O '7325 7'325O From the data here given and some further data supplied by Sir William Thomson, Mr. W. M. Mordey has calculated the following table :- (See preceding page.) TABLE XII. VIRTUAL RESISTANCE OF CONDUCTORS CARRYING ALERNATING CURRENTS. |From Mr. W. M. Mordey's paper on Alternate Current Working, Inst. El. Eng., May 23, 1889, E/ectrician, May 31. - Diameter. Area. Percentage Number Increase of of Complete Milli- S.- : Ordinary Periods metres. Inches. Sq. mm. Sq. in. Resistance. per Second. IO '3937 78'54 * I 22 Less than rºo I5 '5905 1767 ‘274 2' 5 2O ‘7874 || 314 16 '487 8 25 '9842 490°8 ‘76o I7' 5 8o 4O I'575 | 1,256 I '95 68 IOO 3'937 7,854 12 17 38 times IOOO 39'39 785,400 I, 217 35 times 9 3543 63-62 ‘og8 Less than rêo I3'4 ‘528o I4I 3 “218 2' 5 IOO I8 ‘7086 || 254'4 394 8 22'4 ‘8826 394 *6II 17' 5 7°75 '3OI3 47'2 ‘o? I Less than rºw II ‘61 “457O Ioô ‘I 64 2' 5 U , I33 I9'36 ‘7622 294 '456 17' 5 I5'5 ‘61oz 189 .292 8 | INDEX. Acceleration, definition of 337 dimensions of, 338 Activity, definition of, 74 dimensions of, 341 Alternating machines, measurement of currents, 277, 283 measurement of differences of potential, 278, 298 measurement of activity, 281– 3OI - - theory of, single machine, 283 two machines in Series, 285 two machines in parallel arc, 288 generator and motor, 290 AMPERE, A. M., theory of electro- magnetic action, 44 Ampere, the, or practical unit of current, 72 Arrangement of cells in a battery, 4 . * - Attraction of circular disc on particle in axis, 351 AYRTON, W. E., 262, 276, 298, 30o, 354 Balances,Sir W. Thomson's standard . . electric, 195 et seq. Ballistic galvanometer, theory of, 3I9 method for measuring magnetic field intensity, 316–319 method for measuring density of ideal surface distribution of magnetism, 321 Battery, arrangement of, 84 resistance, 256—261 BOTTOMLEY, J. T., 6 Calibration of a wire, Matthiessen and Hockin's method, 2 I2 Carey Foster's method, 213 T. Gray’s method, 217 Capacity, electrostatic, 346 dimensions of 346, 352 Change-ratio, 70 CHRYSTAL, G., 244, 362 CLARK, LATIMER, standard cell, I53 Condenser, capacity of two-plate, rx use of, in measurement of great resistances, 250 Conductance (or Conductivity), defi- nition of, 87 of multiple arc, 87 a velocity in ordinary electro- static units, 348 Continuity, principle of 86 Currents, measurement of, 49–64, 95-II.4, I46 et seq., 273 Current-balances, Sir W. Thomson’s 95–111 graduation of, 174–179 of magnetic needle, measurement of, 9 Derived circuits, 87 units, 65 definitions and dimensions of, 336–362 Determination of H, by modification of Gauss's method, 5–39 by method of Kohlrausch, 54 Difference of potential, measurement of 93, IoA-107, 113,133– 146, 155, 162, 276, 298 Deflection 382 INDEX. T)imensions of units, 323–362 tables of, 355, 356 DUNCAN, HUTCHINSON, and WILKIts, Messrs. , 72 Dynamical units, dimensions of, 336 JDyne, or C.G.S. unit of force, 42 TCarth-inductor, 317 Iºfficiency, electrical, in circuit of generator and motor, 263 — 267 of motor, 267 of secondary battery, 271 Flectric current, absolute unit of, 44–47 practical unit of, 72 dimensions of 347, 352, 356 (see also Currents) displacement, 347 light circuits, measurements in, 76, 27.2 et scº. motors, measurements in circuits of, 263-269, 290, 299 activity in circuits of, 262— 265, 291, 3OO potential, difference of, defini- tion of, 346 resistance, absolute unit of, 69 meaning of, 66 practical unit of, 69–72 velocity in electromagnetic Ineasure, 70, 353 resistances, comparison of 181 —261 Electrodynamometers, 95 ct seq., 273 —275 Electrolysis, graduation of instru- ments by, 162–180 Electromagnetic action, Ampère's theory of, 44 system of units, 40, 348 Ekectrometer, definition of an, 113 absolute, 66, 114 ir W. Thomson’s divided ring, II.4 quadrant, I 14–138 replenisher for, 125 theory of symmetrical, 133 measurement of differences of potential and of activity in alternate-current cir- cuits by, 278, 299 Electromotive force, meaning of, 78 of a cell or battery, 81 of a machine, 67 of standard cells, 152, 155 Electrostatic capacity of a conductor, 346 - absolute unit of, 346 dimensions of 346, 352 System of units, 343–348 voltmeter, Sir W. Thomson's vertical, 138 multicellular, 142 ELLIOTT BROTHERS, Messrs., 209, - 244 Energy, given out in electric circuits, measurement of, 262—302 spent in charging secondary bat? tery, 271 given out by secondary battery, 272 Farad, or unit of capacity, 358 Field magnets of galvanometers, graduation of, 158—162 FITZGERALD, G. F., 3oo mote, 355 FIZEAU, H. L., 362 FLEMING, J. A., 245 Force, kinetic unit of, 42 Formulae of dimensions, 325 tables of, 355, 356 Foster, G. CAREY, 213 Fouca ULT, L., 362 Friction ergometer, 268 Fundamental units, definitions of, 326 Galvanometer, mirror, 181 potential, 94, 147 standard tangent, 49–64 resistance of, 26o - GAUss, C. F., 1, 6, 339 GLAZEBrook, R. T., 71 and FITZPATRIck, Messrs., 242 Graduation of galvanometers, 147 et Seº. current balances, 174—180 GRAY, A. and T., 247 T., 18, 19, 163, 203, 217, 236,294 Great resistances, comparison of by galvanometer, 246 by electrometer, 250–256 HEAVISIDE, O., 203, 293 HOPKINSON, J., 285, 292 Horizontal component of terrestrial magnetic intensity, deter- mination of, 5–39 Horse-power, 76 HUTCHINson and WILKEs, Messrs., 242 INDEX. 383 Induction, correction for, in deter- mination of ſ/, 29 electromagnetic, 67, 206, 282 coefficients of, 354 units of, 354, Wote in Appendix Insulation resistance, measurement of 246, 250—256 * Intensities of magnetic fields, deter- mination of, 303–322 Intensity of magnetization, definition and dimensions of, 349 JOUBERT, J., 285, 362 Joule, J. P., law of generation of heat by current, 75 Jozele, the, or practical unit of elec- trical work, 77 Kilowatt, 77 KIRCHHoFF, G., theorems regarding flow of electricity, 88 examples of, 87, 89 LEwis, D. M., 220 LODGE, O. J., 26o Magnetic field intensity, 43 dimensions of, 348 determination of, 5–39, 300 –320 distribution, determination of surface density of, 321 moment, 42 determination of, 5–39 dimensional formula of 349 permeability, 35o dimensions of, 351, 356 susceptibility, definition of 351 dimensions of, 351, 356 Magnetization, uniform, definition of, 4t Magnetometer, 6 Magnetostatic current-meter, Sir W. Thomson’s, I 12 MASCART, E., 362 MASCART, DE NERVILLE, and BENort, Messrs., 243 MATTHIEssen and HockIN, Messrs., 2 I 2, 23 I, 243 MAXWELL, J. CLERK, 55 Megohm, 358 Microfarad, 358 MoRDEY, W. M., 281, 282 mote Multiple arc, arrangement of cells 11), 85 resistance and conductance of system of wires arranged in, 87, 88 Mutual induction, 354 dimensions of coefficient,354,356 Numeric, meaning of, 323 OHM, G. S., law of, 66, 78 O/ime, the, or practical unit of resist- ance, 69 evaluated as a velocity in elec- tromagnetic units, 70, 353 JCegal, defined by column of mercury, 72, 242 Ohms, standard, construction of, ... . 243 º Oscillations of magnetic needle, observation, 13, 25 Period of oscillation of a magnet, determination of, 13–25 PERRY, J., 262 note, 276, 354 Polarization of a cell, 257 Potential, diſſerence of, (see Difference of potential, measurement O POTIER, A., 3oo note Practical units, 68, 357 considered as an - system, 358 PREECE, W. H., 77 absolute Quadrant electrometer, theory of, I r 4 — I 36 application of, to alternating - circuits, 278, 299 Quantity of electricity, unit of, 343 dimensional formula of 343, 352, 356 RAYLEIGH, I, ORD, 71, 153, 242, 281 Resistance (see Electric resistances) Resistance coils, winding of, 187 construction of, 187 - arrangement of, in resistance box, 189 arrangement of, in resistance slide, 194 . Resistances, comparison of, by Wheatstone's bridge, 202 et seq). methods of comparing two nearly equal, 219 et seq. comparison of low, 224 of high, 246—256 specific determination of 238 standard, construction of, 242 384 INDEX. Resistances of batteries, measure- ment of, 256 of pure copper wires, Tables I., II., 368–370 of copper at different tempera- tures, Tables III., IV., 37 I, 372 of metals and alloys, Table V.,373 of conductors carrying alternat- ing currents, Table XII., 38o Rheostat, 200 ROSA, G. B., 362 ROWLAND, H. A., 318 note, 321, 362 RUCKER, A. W., 345, 355 Secondary batteries, efficiency of, 27o reckoning of energy given out or spent in circuits of, 270 —272 Self-induction, or inductance, 206 avoidance of effect of, in Wheat- stone bridge, 207 in alternating machines, 282 et Seº. dimensional formula of coeffi- cient of 354 SIDGWICK, Mrs. H., 71, 242 SIEMENS BROTHERS, Messrs., 243 SIEMENs, Sir W., 76 SIEMENS, WERNER, 243 Small resistances, measurement of, 224 ; by bridge methods, 224–231 ; by comparison of potentials, 231 Specific resistance, 238 measurement of, 239–243 of mercury, 242 Speed of motor, relation of, to efficiency, 265 Standard cells, 151–157 use of, in graduation of galvano- meters, I 56, I 59 TAIT, P. G., 236 THOMPso N, S. P., 262 THOMSON, J., 70, 323, 326 : Jº. 302 Sir W., current balances, 95 magnetostatic current-meter, 112 quadrant electrometer, 115– I3 electrostatic voltmeters, 138– 146 bridge with secondary conduc- tors, 224 resistance of a galvanometer, 2OO reduction of ballistic observa- tions, 320 illustrations of dimensions of units, 348, 353 determination of z., 361 resistance of conductors carrying alternating currents, 378 Torsion of silk fibre, determination of, 54 Units of physical quantities, C. G.S. System of, 42, 67 et seq. Units of physical quantities, dimen- sions of, 323–362 Volt, the, or practical unit of elec- tromotive force, 68, 359 Watt, the, or unit of electrical acti- vity, 76, AVote in Appendix WEBER, W., I Work, definition of, 73 units of 73 dimensional formula of 340, 355 Wheatstone's bridge, theory of 90 use of 202–231 Kirchhoff's modification of, 209 Matthiessen and Hockin’s ar- rangement of, 229 Thomson's arrangement of, with secondary con- ductors, 224 iMAR i 2 1918 RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAY. MESSRS, MAGMILLAN & CO.'S PUBLICATIONS. A Y 7A/A2 SAMAZ A U 7THZOA’. ABSOLUTE MEASUREMENTS IN ELECTRICITY AND MAGNETISM, THE THEORY AND PRACTICE OF. Second Edition. In Two Volumes. Crown 8vo. Vol. I., 12s. 6d. Iron says:—“This first volume includes the apparatus for measurement of angular deflection oscillations, couples, potential, resistance and specific inductive capacity. It is all very correct and exact. 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