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PRACTICAL ELECTRICITY 
 
 AND 
 
 MAGNETISM 
 
PHYSICAL AND ELECTRICAL ENGINEERING 
 LABORATORY MANUALS.—Vol. 1. 
 
 ELEMENTARY PHYSICS. By John Render- 
 SON, B.Sc. (Edin.), A.I.E.E., Lecturer in 
 Physics, Manchester Municipal Technical School. 
 Crown 8vo, 2j. 6d. 
 
 LONGMANS, GREEN, AND CO. 
 
 LONDON, NEW YORK AND BOMBAY 
 
PHYSICAL AND ELECTRICAL ENGINEERING 
 LABORATORY MANUALS 
 
 VOL. II. 
 
 PRACTICAL ELECTRICITY 
 
 AND 
 
 MAGNETISM 
 
 JOHN HENDERSON, B.Sc. (Edin.), A.I.EE. 
 
 LECTURER IN PHYSICS MUNICIPAL TECHNICAL SCHOOL, MANCHESTER 
 
 LONGMANS, GREEN, AND CO. 
 
 39 PATERNOSTER ROW, LONDON 
 
 NEW YORK AND BOMBAY 
 
 1898 
 
 All rights reserved 
 
PREFACE TO THE SERIES 
 
 In bringing before the public these laboratory manuals, it has 
 been the object of the authors to provide a course of instruc- 
 tion for carrying out a progressive series of experiments in 
 Physics and Electrical Engineering, arranged so that the usual 
 apparatus at the disposal of a laboratory, though not especially 
 designed for any particular experiment, may nevertheless be 
 used with advantage in a variety of ways. 
 
 Able courses of instruction ' in . experimental work have 
 already appeared, and have done a vast amount of good; 
 but as these usually require expensive apparatus made and 
 arranged for each experiment, they have not become so 
 generally useful as otherwise might have been the case, 
 especially in such instances where the scope of the work 
 undertaken, precludes all possibility of separate and special 
 apparatus being provided for each independent experiment! 
 In technical work this is more particularly the case, seeing that 
 in actual practice a set of instruments must be put to very 
 divergent uses, in order that results may be obtained quickly 
 and with sufficient accuracy for commercial work. This use 
 of apparatus for ends not specially intended, is in itself a 
 training of considerable importance, to any student who will 
 afterwards, in his daily life, have to so adapt for different 
 purposes, such instruments as may be available at the time. 
 
vi Preface to the Series. 
 
 It has not, however, been forgotten that most apparatus thus 
 used is too often placed under circumstances inconsistent with 
 accurate work, and to this end very careful instructions will 
 be found in the more advanced volumes, for guarding against 
 such disturbing influences as time, situation, temperature, and 
 magnetic forces ; this being too often neglected in general labo- 
 ratory and commercial work, it being frequently forgotten that 
 a set of apparatus arranged for a particular test, is sometimes 
 not only a centre of disturbance itself, but is Uable to disturb- 
 ance from other apparatus in use in its neighbourhood. The 
 precautions thus indicated are of especial importance in 
 technical work, where the disturbing influences are of such a 
 powerful nature as may be found in engine-rooms and. dynamo- 
 houses, where high and varying temperatures and - leakage 
 magnetic lines are very prevalent. 
 
 Another way in which an alteration has been effected is to, 
 as far as possible, arrange experiments where a student working 
 alone, may be able to obtain satisfactory results. In, a large 
 proportion of the existing laboratory manuals, groups of 
 students are expected to work together; but a number of 
 years of practical experience with students of all kinds has 
 convinced the authors, that habits of individual accuracy and 
 self-reliance can only be acquired by separate and unaided 
 work. Of course, in advanced work it is often necessary, that 
 two or more students should work together, in order that 
 simultaneous observations should be taken; but it is most 
 desirable that students so combined, should have had con- 
 siderable individual training and experience. 
 
 It is particularly desirable that every experiment should be 
 repeated until a set of consistent results have been obtained. 
 In this way only, can experience and accuracy be acquired. 
 
 This series is divided into two courses, to which Vol. I. is 
 a general introduction. The future volumes will contain the 
 
Preface to the Series. vii 
 
 advanced work in both Physics and Electrical Engineering, 
 and it is intended, that students should be able to take either 
 the one or the other, thus specializing in Physics or Electrical 
 Engineering; but they may combine the two courses with 
 advantage, where time will admit. 
 
 J. HENDERSON. 
 
 S. JOYCE. 
 Manchester, 1895. 
 
PREFACE 
 
 The present volume, which forms the second of the series of 
 Physical and Electrical Engineering Laboratory Manuals, has 
 been entirely devoted to practical work in electricity and 
 magnetism, as it was found that the importance of this de 
 partment of physical work demanded a volume to itself. 
 
 The arrangement and treatment of the subject-matter in 
 the present volume is essentially diflferent from that adopted 
 in Vol. I., since it was found that, although it was extremely 
 desirable to present instructions to elementary students in as 
 brief and concise a manner as possible, it was impossible to 
 follow the same method with more advanced work. A con- 
 siderable amovmt of explanatory matter has therefore been 
 added to the descriptions of the various measurements. In 
 very few cases have proofs of formulae assumed been given, 
 as it was found that this would lead to the book ass uming 
 unwieldly dimensions; besides, mathematical investigations 
 do not come within the scope of a practical manual, but of a 
 theoretical treatise. In order, however, to assist the student 
 as much as possible in this direction, references are given to 
 sources where the required proofe may be obtained. 
 
 One of the principal features of the book, and one which 
 the author hopes wiU commend itself to others engaged in writing 
 scientific works, is the list, given at the end of each chapter, of 
 references to the more important original papers published in 
 
X Preface. 
 
 the various scientific periodicals, which bear on the subject- 
 matter of the chapter. So much valuable matter is buried in 
 the "Proceedings" and "Transactions" of various learned 
 societies, and in monthly and weekly scientific publications, 
 which in many cases are only indexed for each volume, there 
 being no general index, that it becomes a very formidable 
 task, and one involving a great expenditure of time, to ransack 
 perhaps many volumes of back numbers in order to get a 
 single paper. 
 
 Feeling the want of such references himself, and knowing 
 that others were necessarily in the same position, the author, 
 after considerable labour, has compiled lists of references to 
 the more important papers on subjects treated of in the 
 present volume which have appeared in — ; 
 
 (i) The Philosophical Transactions of the Royal Society of 
 London. 
 
 (2) The Proceedings of the Royal Society of London. 
 
 (3) The Transactions of the Royal Society of Edinburgh. 
 
 (4) The Proceedings of the Royal Society of Edinburgh. 
 
 (5) The Philosophical Magazine. 
 
 (6) The Electrician. 
 
 Several references have also been made to " Nature," 
 the " B.A. Reports," the " Journal of the Institution of Electrical 
 Engineers," and to a few foreign scientific periodicals. In 
 many cases, where the same paper appears in several of the 
 above publications, more than one reference is given, as the 
 student may not have access to all the above-mentioned 
 books. 
 
 These lists are not to be looked upon as giving all the refer- 
 ences to the subjects with which they deal, but they will 
 supply the student with a ground-work to start upon, and if 
 he consults the papers mentioned he will in them find the 
 further references which he requires. It is also hoped that 
 
Preface. x i 
 
 these lists will be found useful by students and others engaged 
 in original research. . 
 
 As regards the arrangement of the material in the present 
 volume, the measurement of resistance is dealt with first, on 
 account of its great importance. At the end of this chapter 
 there is a very brief account of some of the methods of 
 measuring resistance in absolute units ; this is not given with 
 the hope of the student making highly accurate determinations 
 of the length of the mercury column which represents the 
 practical unit of resistance, but simply to familiarize him 
 with the methods of making such determinations, and also as 
 an exercise in absolute measurement, albeit his results may 
 not be correct to one or even ten parts in ten thousand. 
 
 In choosing methods for the various measurements it has 
 been the author's aim to take only the best suited for the 
 purpose, and the book does not pretend to be a dictionary of 
 possible methods, or, as is sometimes the case with practical 
 handbooks on physics, to show how many different experiments 
 could be performed with any given piece of apparatus. 
 
 Figures and diagrams of representative pieces of apparatus 
 have been included, and in a few cases they are taken from 
 photographs of apparatus in the author's own laboratory. 
 
 The last chapter has been devoted to a description of 
 experiments with electro-magnetic waves, this the author 
 believes is also new to practical books on electricity, but the 
 rapid growth of this section of the subject, and the many 
 possible practical applications of electro-magnetic phenomena 
 in the near future, is sufficient justification for its inclusion. 
 
 Tables of logarithms, etc., and of physical constants have 
 been added for the convenience of the student. 
 
 The author takes this opportunity of acknowledging his 
 indebtedness to Messrs. Elliott Bros., The Davies Motor Co., 
 The Royal Society, and the Editor of the Philosophical 
 
xii Preface. 
 
 Magazine for permission to reproduce illustrations ; also to Mr. 
 Jas. White, Glasgow, the constructor of Lord Kelvin's current 
 balances, for illustrations and for special permission to use a 
 large part of his descriptive pamphlet on that instrument. 
 
 The author will be obliged to any of his readers who will 
 point out errors in the text which have escaped correction in 
 the proof-reading. 
 
 JOHN HENDERSON. 
 
 Physical Laboratorv, 
 
 Municipal Technical School, Manchester, 
 November, 1897. 
 
CONTENTS 
 
 I. — Measurement of Resistance. 
 
 PAGE 
 
 Adjustment and care of apparatus . 2 
 
 Contacts, keys, batteries, and resistances 10 
 
 Calibration of a sensitive galvanometer . ig 
 
 Measurement of resistance . 
 The metre bridge ... . . 
 
 The Carey Foster bridge . . 
 
 Measurement of very low resistance 
 „ of very high resistance 
 
 ,, of liquid resistance 
 
 23 
 2S 
 
 S3 
 57 
 65 
 80 
 
 ,, of battery resistance . . . . . 84 
 
 ,, of galvanometer resistance . . . -87 
 
 ,, of specific resistance . ... . . 90 
 
 ,, of temperature variation of resistance . . 95 
 
 Fault testing 102 
 
 Absolute measurement of resistance . . . 106 
 
 Standards of resistance . -115 
 
 References to original papers .... .120 
 
 II. — Measurement of Current. 
 
 Standard galvanometers . . 127 
 
 „ electro-dynamometer 140 
 
 „ current balance ... .... . . 143 
 
 Voltameters 151 
 
 Lord Kelvin's current balances . ... i6z 
 
 Siemens' electro-dynamometer . . . . 169 
 
 References to original papers . . . . . 173 
 
xiv Contents. 
 
 III. — Measurement of Electro-Motive Force. 
 
 PAGE 
 
 Standards of E.M.F . . . -.175 
 
 Standard cells . . . . 176 
 
 Comparisons of E.M.F. . . .... 185 
 
 Temperature coefficient of a battery ... . 187 
 
 Test of a primary battery .... ... 190 
 
 Electrometers ... ... . . 192 
 
 Absolute electrometer ... . . 193 
 
 Quadrant electrometer ... . . 195 
 
 Potentiometer . . . 200 
 
 References to original papers ... . . 209 
 
 IV. — Measurement of Quantity of Electricity. 
 
 Theory of ballistic galvanometer ... .... ...211 
 
 Measurement of logarithmic decrement . .... 215 
 
 Calibration of ballistic galvanometer . ... 217 
 
 V.^Measurement of Capacity. 
 
 Absolute measurement of capacity . ... . ... 232 
 
 Comparison of capacities .... . 235 
 
 Measurement of S.I.C. . . ... . . 241 
 
 References to original papers . . .' . ■ . . 252 
 
 VI.— Magnetism. 
 
 Determination of H . . . ... . 254 
 
 „ of the angle of dip .... . . 26& 
 
 Measurement of the magnetic qualities of iron and steel ... . 272 
 
 „ of permeability . . . 273 
 
 Magnetic hysteresis . . . . 294 
 
 Effect of temperature on magnets . . 296 
 
 Self and mutual induction . ... . . 300 
 
 Methods of measuring the coefficient of self-induction .... 312 
 
 Secohmmeter .... . 313 
 
 Standards of self-induction .... . . . . . 31 j 
 
 Measurement cf the coefficient of mutual induction . .' . .316 
 
 Comparison of capacity and mutual induction ... .... 322 
 
 References to original papers .... . . . . 324 
 
Contents. xv 
 
 VII. — Electro-Magnetic Waves. 
 
 PAGE 
 
 Description of phenomena . . . . 328 
 
 Velocity of propagation .... 329 
 
 Indexof refraction and S.I.C. ... .... 331 
 
 Apparatus for producing electro-magnetic waves . . 332 
 
 „ for detecting electro-magnetic waves .... 337 
 
 Transmission, reflection, and absorption . . ... 341 
 
 Refraction ... . . 342 
 
 Measurement of wave-length 344 
 
 Polarization ... .... . . 346 
 
 References to original papers . . .... ... 349 
 
 APPENDIX. 
 
 Tangent positions of Gauss .... ... . . 351 
 
 Tables of logarithms . . .... 356 
 
 of antilogarithms 3S8 
 
 of squares ... ..... 360 
 
 of reciprocals .... . .' " . . . . 362 
 
 of tangents .... ... 364 
 
 of sines . . . . . .... . 366 
 
 Physical constants . . . . 368 
 
 Index .... . . . 379 
 
REFERENCES TO SCIENTIFIC PAPERS 
 
 The following abbreviations are used in connection with the abmie references 
 
 The Philosophical Transactions of the Royal Society of 
 
 London (Phil. Trans.). 
 
 The Proceedings of the Royal Society of London . . (Pro. Roy. Soc). 
 
 The Transactions of the Royal Society of Edinburgh . (Trans. R.S.E.). 
 
 The Proceedings of the Royal Society of Edinburgh . (Pro. R.S.E.). 
 
 The Philosophical Magazine . . (Phil. Mag.). 
 
 The Electrician (Elect.). 
 
 The Reports of the British Association . .... (B. A. Report). 
 
 The Journal of the Institution of Electrical Engineers . (J.E.E.). 
 
 Po^endorff's Annalen der Physik und der Chemie . (Pogg. Ann.). 
 
 Wiedemann's Annalen der Physik und der Chemie . . (Wied. Ann.). 
 
 Comptes rendus de 1' Academies des Sciences . . . (Comp. Rend.). 
 
 Journal de Physique . . Qour. de Phys.). 
 
 Wiener Berichte ... . . (Wien. Ber.). 
 
PRACTICAL 
 
 ELECTRICITY AND MAGNETISM 
 
 INTRODUCTION. 
 
 I. In all phjrsical measurements, with any pretence to accuracy, 
 and which are intended to be of any permanent value, the 
 student must grudge no amount of time and trouble in making 
 them. He must never be in a hurry. A week spent in 
 discovering and overcoming some source of error will be 
 well-spent time, and may be of more educational value than 
 the performance of the original experiment itself. Above all 
 things, however, the experimenter must be methodical; all 
 results and measurements must be recorded exactly as observed. 
 No corrections, however simple, should be made mentally at 
 the time of observation, and the results must be recorded 
 immediately after they have been obtained — the memory ought 
 never to be trusted. 
 
 In order to impress the student with the necessity of being 
 methodical in his work, a few hints are given respecting the 
 method of recording results. 
 
 The observation of general conditions which may affect the 
 exjieriment — 
 
 {a) The date, time, and place where the experiment is 
 performed should invariably be recorded. 
 
 {b) The temperature of the room during the experiment, 
 and the barometric pressure, should be noted in cases where 
 a variation of either of these quantities might affect the results. 
 
 {c) There should also be a complete description of all the 
 apparatus employed, with the reference numbers of the various 
 
 B 
 
2 Practical Electricity and Magnetism. 
 
 instruments, and a diagram showing the disposition of the 
 apparatus, and giving some idea of the relative distances of 
 various pieces of apparatus from one another, more especially 
 when there is reason to suspect that one instrument may exert 
 an effect on others near it. The great importance of these 
 precautions is very evident. Suppose, for instance, that after 
 some experiment it was found that a resistance coil in a box 
 was faulty, then unless the exact part played by that particular 
 coil in the experiment was known, a correction for the error 
 so introduced would be impossible. Too great elaboration 
 cannot be given to this somewhat tedious and uninteresting 
 part of experimental work, and the author would advise every 
 student who intends to devote his time to experimental physics, 
 to first of all read Faraday's " Experimental Researches in 
 Electricity,'' not only for the sake of the valuable information 
 contained in them, but also to learn how to record results of 
 experiments, nothing being too unimportant to record to that 
 prince of experimenters, even to the lengths of the connecting 
 wires employed. 
 
 2. Before commencing any experiment, it is assumed that 
 the student is perfectly familiar with its object. He should 
 also read as much of the literature bearing on the subject as 
 he can lay his hands on. Should any irregularities occur, the 
 causes of them must be hunted down and proz'ed — they should 
 never be shelved as being merely "experimental errors." 
 Nothing must ever be taken for granted, everything must be 
 cotwlusively proved. No deductions should be made from 
 results at the time of recording, as it is better that the mind 
 should be entirely given up to accurate observation ; then after- 
 wards, on reading over and studying his results, the student 
 will be better able to make deductions, and to devise further 
 experiments to test uncertain points, or to be crucial tests of 
 any theory he may have proposed. 
 
 Adjustment and Care of Apparatus. 
 
 3. Galvanometers. — The number of forms which galvano- 
 meters may take is legion, but there are a number of remarks 
 on their adjustment and care which apply equally to all forms . 
 
Adjustment of Galvanometers. 3 
 
 The table or stand on which the galvanometer is placed should 
 be separate from that on which the rest of the apparatus is 
 arranged, otherwise it is liable to be shaken when adjusting the 
 other instruments. The table or stand must be arranged to be 
 as free from vibration as possible. In some laboratories the room 
 in which the sensitive galvanometers are kept is in the base- 
 ment, the stands being built up specially from the foundation, 
 in order to secure freedom from vibration. This cannot always 
 be managed, and very often small brackets are attached to the 
 walls, on which the galvanometers may be placed; provided 
 the walls are thick, and there is no heavy running machinery 
 near them, this makes a very satisfactory stand. All large 
 masses of iron which would influence the galvanometers must 
 be removed from the neighbourhood of the testing-room. 
 Stationary masses of iron, such as iron pillars, are not so 
 important, as the effect due to them will be constant, unless the 
 magnetic field around them alters in value. Also movable 
 magnets, or anything of a magnetic nature, such as knives, keys, 
 watches, etc., must be placed where they will exert no influence 
 on the instruments. 
 
 Great care must be taken to prevent the magnetic effect 
 of the current in the other pieces of apparatus affecting the 
 galvanometer. In order to ascertain whether or not there is 
 any such action, it is advisable to send a current through the 
 apparatus, having previously disconnected the galvanometer. 
 Any deflection obtained under these circumstances points to 
 the magnetic effect of some part of the apparatus on the 
 galvanometer, and the apparatus must be rearranged until no 
 such effect is observed. For this reason it is advisable in 
 some cases to twist the wires leading to the galvanometer 
 together for some distance from it. 
 
 4. In some experiments, as for instance the measurement 
 of insulation resistance, the insulation of the galvanometer 
 becomes very important, and the instrument must be tested 
 for leakage. This can best be done by connecting up one 
 terminal of the galvanometer to one pole of a battery of 
 several cells, the other pole of the battery being earthed, whilst 
 the other galvanometer terminal is insulated. A deflection of 
 
4 Practical Electricity and Magnetism. 
 
 the needle under these circumstances points to a leakage to 
 earth from some part of the galvanometer coil. This test 
 should be made to each temiinal of the galvanometer 
 separately, since the leak might be close to one, when of 
 course no deflection would be obtained on the battery being 
 connected to that terminal. Should a small leakage be foimd 
 which would interfere with the accuracy of the measurements, 
 the galvanometer should be placed on blocks of freshly 
 scraped paraffin wax, one under each levelling screw ; this will 
 completely insulate the instrument from the earth. 
 
 5. The galvanometer must be set up so that the needle 
 oscillates freely when disturbed from its zero position, the instru- 
 ment being levelled so as to bring the needle into the centre of 
 the field. In the case of a reflecting galvanometer, any friction 
 between the needle and the coil can be detected by observing 
 the motion of the spot of light on the scale when the needle is 
 set oscillating, an irregular, jerky motion denoting friction 
 between the two. 
 
 It is also advisable, whenever possible, to so arrange the 
 galvanometer that the needle will point to the zero on the scale 
 when under the influence of the earth's field alone. This is 
 not essential, as the needle can always be brought to zero by 
 means of a bar magnet, but it is often desirable. 
 
 6. Most galvanometers are supplied with a movable directing 
 magnet, attached to the frame of the instrument, and by means 
 of which the needle may be adjusted to zero by turning the 
 magnet on its axis, or the controlling force may be altered by 
 altering the distance between the needle and controlling magnet. 
 Such an arrangement is shown in Fig. i. Here the 'directing 
 magnet is attached to the top of the case of the galvanometer, 
 and the rotation is given to the magnet by means of a worm 
 and worm-wheel, whilst the controUing force is altered by slid- 
 ing the magnet up or down the vertical brass rod. The great 
 objection to this arrangement is that it is almost impossible 
 to alter the position of the magnet -nithout seriously shaking 
 the whole instrument, thus making it very difficult to rapidly 
 vary the controlling field so as to give a long period of swing 
 to the needle. This difficulty may be overcome by mounting 
 
Control of Galvanometers. 
 
 5 
 
 the directing magnet on a separate stand. A simple form 
 employed by the author is shown in Fig. 2. A represents 
 a sliding board, moving in the frame F, between the guide 
 bars G, G; to it is fixed the vertical rod P, on which the 
 magnet slides, so as to adjust it to the height of the gal- 
 
 FlG. , 
 
 vanometer needle. The rod P is free to rotate in its socket, 
 and is turned by means of a small band passing round 
 the pulley S and the pulley T, to which a milled head is 
 attached. The to-and-fro motion is given by means of a 
 band, B, attached to A, and passing over two rollers, R, R, 
 to one of which a milled head, M, is attached. The whole 
 
6 Practical Eieciricity and Magnetism. 
 
 apparatus may be made in the laboratory workshop. In use it 
 is placed behind the galvanometer, and the magnet adjusted on 
 the rod P until it is on a level with the needle. The milled 
 head M is then turned so as to pull the magnet nearer the 
 galvanometer until the required sensitiveness is obtained. 
 
 7. Suspension of the Magnetic System. — The suspension 
 of the needle of a galvanometer is a very important matter. 
 The function of the suspending fibre may be twofold — either 
 simply to suspend the needle, or, in addition, to supply the 
 controlUng force — and of necessity the nature of the suspen- 
 sion varies with the above conditions. If a suspension only 
 is required, the fibre must be as torsionless as possible; if 
 
 G 
 
 1 
 
 r-1 
 
 3M 
 
 Fig. z. 
 
 a controlling force is required, that must be supplied by the 
 torsion of the fibre. In the first case it is necessary to get some 
 substance which, when of sufficiently small diameter, is practically 
 free from torsion, and is at the same time strong enough to sup- 
 port the weight of the suspended system. One such substance 
 is unspun cocoon silk. This is produced by the silkworm as a 
 double thread, each part having a diameter of about 0*0005 
 inch. The threads are separated from each other by wash- 
 ing them with warm water, painted on from a camel's-hair brush, 
 in order to dissolve the gum by which they are fastened together. 
 One such thread has been found to support a weight of about 5 
 grammes before breaking.' A number of such fibres should be 
 kept ready for use, suspended inside a glass tube, with small 
 
 ' See Gray's " Absolute Measurements in Electricity and Magnetism," 
 vol. i. p. 241. 
 
Galvanometer Suspensions. 7 
 
 weights attached to them to keep them stretched. The torsion 
 in silk fibres is very small, but the fibre is affected by both 
 temperature and moisture, so that the zero of instruments in 
 which such suspensions are employed is liable to alteration. 
 
 8. A much better material for a suspension is a thread from 
 a garden spider's web ; this, although possessing the properties 
 of an almost ideal suspension, seems to be very little used. It 
 is very strong, considering its size. Joule found that it could 
 support a weight of 2 grammes without breaking.' The 
 greatest point in its favour is that experiments go to show that 
 it is almost absolutely torsionless.^ Experiments by Bottomley 
 and Tanakadate go to show that a spider line capable of carry- 
 ing a mirror and magnet weighing o'2 gramme has a torsional 
 rigidity yj^ that of a single cocoon silk fibre. Its extension 
 with temperature is also very small, a fibre 23 inches long, 
 loaded with a weight of i gramme, not altering more than about 
 07 % in length for a rise of over 50° C, a lengthening of about 
 2 % taking place when dry air was changed for moist. This, 
 however, need be no drawback, since, on account of its great 
 freedom from torsion, very short fibres may be employed. 
 
 9. Another material for galvanometer needle suspension is 
 quartz fibre, originally prepared by Professor C. V. Boys, F.R.S.' 
 Such fibres can be obtained of any desired degree of fineness. 
 As regards strength. Professor Boys has shown that, for a fibre 
 nearly ^qq^q of an inch in diameter, the breaking stress was 
 5 1 7 "tons per square inch. In addition to its other properties, 
 quartz fibre is an almost perfect insulator even in a damp 
 atmosphere, which makes it invaluable as a means of suspend- 
 ing charged bodies, such as the needle of an electrometer. 
 The following hints regarding the method of suspending sub- 
 stances by quartz fibres have been taken from Professor Boys' 
 Cantor Lectures on " Instruments for the Measuring of Radiant 
 Heat":— 
 
 " Having chosen a fibre of the right diameter, and longer 
 
 ' Joule's " Scientific Papers," vol. i. p. 479. 
 
 ' Cantor Lecture on " Instruments for the Measuring of Radiant 
 Heat," Boys, p. 11 ; also Bennet, P/iil. Tratis., 1792. 
 5 See Pro. Roy. Soc, vol. 46, p. 253 ; also Phil. Mag., Aug., 1S89. 
 
8 Practical Electricity and Magnetism. 
 
 than is ultimately required, the first thing to do is to fasten 
 a small fragment of gummed stamp-paper to one end. This 
 acts as a weight and makes the following processes more 
 easy. The upper or fixed support must next be fastened to 
 the free end of the fibre. I prefer a common blanket-pin 
 passing through a cork to any of the more elaborate contriv- 
 ances in common use ; however, whatever is going to be used 
 for the fixed support should be pointed at the lower end. If 
 the needle or other thing to be supported is very light, i.e. 
 nowhere approaching the breaking weight of the fibre, shellac 
 varnish is the best thing to use as the cement. Just moisten 
 the last five millimetres above the pin with this varnish — hold- 
 ing the fibre near its free end in one hand, and the pin in the 
 other, with the little finger of one hand resting against <the little 
 finger of the other for the sake of steadiness — immediately 
 apply the fibre to the varnished surface, to which it will stick. 
 Then pull it endways through a distance of half a millimetre 
 about, to make sure that when all is dry there will be no sudden 
 bend in the fibre. A hot piece of wire, or knife, or pair of 
 pliers must then be applied to the pin above the varnish, so 
 that the heat may be conducted down slowly to drive ofi" the 
 remaining alcohol and melt the shellac After this the fibre is 
 securely held at that end. If the thing to be supported is very 
 heavy, varnish is not so good as melted shellac, but this is 
 much more difficult to apply. The pin must be warmed and 
 smeared near the joint, and while hot the fibre must be 
 applied and slightly pulled. Assuming one end of the fibre is 
 properly fastened, the next thing is to determine the exact 
 length required. For this purpose I always make a drawing 
 on a perfectly clean and smooth board, showing the point of the 
 pin at one end, the extreme end of the thing to be supported, 
 and the position of the mirror or whatever else determines the 
 length of the fibre. The holding pin is then raised up until the 
 paper weight is hanging in the air. This is then allowed to rest 
 on the board, and slowly dragged along until the point of the 
 pin is exactiy over the corresponding mark on the board, 
 and the paper wafer is in the line of, but beyond the other 
 mark. . 
 
Galvanometer Suspensions. g 
 
 " The fibre is then straight, and must pass over the mark which 
 indicates the upper end of the apparatus that is to be suspended, 
 though of course nothing can be seen. A knife is then drawn 
 across the board, say 5 milHmetres beyond this mark. By 
 this means the fibre is cut, and the 5 milHmetres are left for 
 the purpose of attachment. The needle or suspended thing is 
 then fastened the same way as the pin." 
 
 10. Metallic suspensions are employed in galvanometers 
 built on the D'Arsonval principle, in which case they have not 
 only to supply the controlling force, but also to make electrical 
 contact between the suspended coil and the terminals. Flat 
 strips of phosphor bronze have been found very satisfactory for 
 suspensions of this kind, but it has been shown that a better 
 material would be platinum silver for both coil and sus- 
 pension, since the strip is liable to have its torsional rigidity 
 altered by change of temperature, and in the case of platinum 
 silver the percentage diminution of torsional rigidity is almost 
 equal to the percentage increase . of electrical resistance ; so 
 that the same potential difference at the terminals of a 
 galvanometer possessing such a suspension would always pro- 
 duce the same deflection throughout a considerable range 
 of temperature, the one variation being made to compensate 
 for the other. 
 
 11. In making connection to the galvanometer, fine double 
 silk-covered wire (about No. 26) should be used. If thick 
 wires are employed, mechanical vibrations may be transmitted 
 along them from the rest of the apparatus, which will keep 
 the galvanometer needle continually shaking; also the wires 
 leading to the galvanometer should not be coiled into spirals, 
 as these form electro-magnets which produce a disturbing effect 
 on the galvanometer needle, but should rather be twisted 
 together, or, at least, run parallel to one another close together, 
 so that their magnetic effects will tend to neutralize. 
 
 12. In some cases, especially where a busy street runs near 
 the laboratory, it may be found impossible to get a support for 
 the galvanometer which is free from vibration. In the event of 
 this, it is sometimes the practice to place the galvanometer on a 
 board suspended from the roof by long indiarubber ropes. 
 
10 Practical Electricity and Magnetism. 
 
 which absorb the greater part of the vibration. A new trouble 
 is, however, sometimes introduced in the shape of a pendulum 
 swing of the whole apparatus. 
 
 Contacts, Connecting Wires, etc. 
 
 13. All connections should be made with well-insulated 
 copper wire ; and it is not a bad plan to record the lengths and 
 diameters of the various connecting wires employed in any experi- 
 ment, as some point might turn up afterwards in which a small 
 correction had to be applied for the resistance of the leads, 
 and if this was not measured at the time, it may be estimated 
 afterwards from the record of the nature and distribution of the 
 connecting wires. 
 
 In cases where the resistances of leads, connecting wires, etc., 
 become important, however, it must be borne in mind that to 
 have the resistances of the connecting wires small is not enough, 
 since the resistance introduced at the contacts between the 
 wires at terminals and binding screws, often far exceeds the 
 resistance of the leads, and especial care has to be exercised in 
 order to make all such contacts as perfect as possible. Two 
 methods of connecting wires together are available : (i) Mer- 
 cury contacts, (2) metallic contacts. 
 
 14. Mercury Contacts. — In mercury contacts a cup of some 
 insulating material is required to hold the mercury. Such a 
 cup may be cut out of a solid, homogeneous block of paraffin 
 wax, this block being afterwards fixed to a wooden stand ; at 
 the bottom of the cup is placed^a plate of well-cleaned copper, 
 which has been amalgamated by placing in a vessel containing 
 dilute nitric acid and mercury, and rubbing the mercury over 
 the surface of the plate. On the top of this plate is poured 
 pure mercury, into which the ends of the wires to be connected 
 together, and which have also been amalgamated, dip, pressing 
 on the copper plate. A cover should be placed over the cup 
 to keep dust and dirt from getting at the mercury. This form 
 of mercury cup will probably give as good a contact as it is 
 possible to obtain, so long as the mercury and the wires remain 
 clean. It is, however, so difficult to manage this, and mercury 
 
Contacts. 1 1 
 
 is so difficult to purif}', that this form of connection is not re- 
 commended for ordinary laboratory work. 
 
 15. Metallic Contacts. — In order to get a good metallic 
 contact, the surfaces in contact should be large and perfectly 
 clean. Such a contact may be obtained between two copper 
 plates, which have had their surfaces filed and scraped true, 
 and which are then bolted together with a strong bolt having a 
 large nut, the under surface of which has also been filed and 
 scraped true, so that it makes contact all over. Such surfaces 
 may be easily cleaned from time to time, but after cleaning 
 should never be touched by the hand, as a thin film of oil 
 is liable to get on the contact surface, thus increasing its 
 resistance. Metallic contacts such as this will in general be 
 found much more satisfactory than mercury ones. 
 
 16. Keys. — In all circuits which have to be made and 
 broken, suitable keys must be employed. 
 
 I. Simple Make-and-Break Key. — This may take the form 
 of either a plug key (see Fig. 3), which may be arranged as a 
 one-way or multiple-way key, or a tapping key (see Fig. 4). 
 
 Fig. 3. Fig. 4. 
 
 In the first form, the metal part consists of brass blocks fixed 
 to an ebonite base, and insulated from one another. They 
 may be connected together by means of a plug, in the same 
 way as the brass blocks in a resistance box are. The tapping 
 key consists of a brass spring attached to an ebonite base; to 
 the lower side of the spring a plantinum contact piece is 
 attached, which, when the key is depressed, makes contact 
 with a similar platinum contact piece on the base, which is 
 in connection with the other terminal of the key. The contact 
 points should be as large as possible, and arranged so that 
 
12 
 
 Practical Electricity and Magnetism. 
 
 they may be easily accessible, in order to clean and adjust 
 them. The brass spring has an ebonite button attached to it 
 for the finger to press on. 
 
 17. II. Reversing Keys. — One form of reversing key is shown 
 
 in Fig. 5, and is similar to 
 the tapping key, only there 
 are two tappers instead 
 of one. Normally, both 
 springs press on the upper 
 )ntacts, which are at- 
 ched to the brass bridge 
 over the key. When the 
 keys are depressed they 
 make contact with the 
 lower contacts, which are connected to the terminal on the far 
 side of the key. In using this key, the battery must not be 
 connected to the two tappers, otherwise, when the key is not 
 depressed, it will be short circuited, but to the terminals con- 
 nected to the upper and lower contacts respectively, the circuit 
 
 Fig. 5. 
 
 in which the current is to be reversed being connected to the 
 terminals attached to the tappers. A very highly insulated 
 form of reversing key is also shown in Fig. 6, which is supplied 
 with cams to hold the tapping springs in any position. 
 
Keys. 
 
 13 
 
 AnotJier very useful form of reversing key is Pohl's mercury 
 commutator (see Fig. 7). This key may not only be used as a 
 reversing key, but also for connecting either of two circuits to a 
 third, which is very useful, for instance, in comparing the fall 
 of potential in an unknown resistance 
 with that in a standard resistance. The 
 following diagrams show the connections 
 in the tw^o cases : — 
 
 (i) Asanordinaryreversingkey(Fig.8). 
 
 In one position of the rocker, i is con- 
 nected to 3 and 2 to 4 ; in the other posi- ^ ••=• ?■ 
 tion, I is cormected to 5 and 2 to 6 ; contacts 3 and 6, also 4 
 and 5, being permanently connected by copper straps. 
 
 /•** Circuit 
 
 if 
 
 f2 
 
 \. J 
 
 2"^ Circuit 
 3"^ Circuit 
 
 Fig. 9. 
 
 (2) For connecting either of two circuits to a third, the two 
 copper straps are removed (Fig. 9). 
 
 ^^'hen I is connected to 3 and 2 to 4, the first and third 
 circuits are connected together ; also where i is connected to 5 
 and 2 to 6, the second and third circuits are connected. 
 
 18. High-lnsulaHon Keys. — In cable and condenser work, 
 keys of special construction and of high insulation are re- 
 quired. One form largely used is shown in Fig. 10. This 
 consists of a movable brass tongue attached to a long ebonite 
 pillar. The tongue moves between an upper and lower con- 
 tact, both of which are highly insulated from the base ; there 
 
14 Practical Electricity and Magnetism. 
 
 are also two detents attached to levers marked " discharge " 
 and " insulate.'' When the tongue is depressed, it is held on 
 the lower contact by the detent attached to the "insulate" 
 lever. If this lever is depressed, the tongue springs up until 
 it is caught by the other detent, which holds it midway 
 between the two contacts. When the lever marked " dis- 
 charge " is depressed, the tongue springs up to the upper 
 contact. The ebonite insulating pillars should be corrugated, 
 in order to offer a greater resistance to surface leakage ; and 
 the screws by which they are attached to the base should only 
 enter the ebonite a short distance, otherwise the advantage of 
 all the insulation is lost. Such keys, and in fact all keys, 
 
 Fig. 
 
 should be carefully cleaned before being used. The ebonite 
 should occasionally be well washed with soap and distilled 
 water and dried before a good fire ; drying by rubbing is apt 
 to electrify the ebonite, and this in some cases would introduce 
 errors into the experiments. Such instruments also should 
 always be lifted by the metal part, as the hand is apt to leave 
 a film of moisture on the ebonite which might spoil its in- 
 sulating properties. 
 
 19. Batteries. — In many experiments one of the necessary 
 requirements is a battery of constant E.M.F. No ordinary 
 primary battery will fulfil this condition, unless the current 
 taken from it is very small and is only required for a short 
 interval of time. When large currents are required, or a 
 constant E.M.F., for any length of time, a secondary battery 
 
Batteries. 1 5 
 
 must be employed. Small-size secondary batteries of about 
 14 ampere hours' capacity will be found very useful for general 
 laboratory work, and they may be relied upon to maintain a 
 constant potential difference at their terminals, provided they 
 are not used immediately after being charged, but after having 
 been partially discharged. Great care must be taken not to 
 accidentally short circuit such cells, since the internal resist- 
 ance is very small, and a large current might not only damage 
 the cell but also the external resistance. It will sometimes be 
 found a good plan to make up sets of three such cells in boxes, 
 inside which is a resistance in series with them, sufficient to 
 protect the cells from a dangerous short circuit. 
 
 Dry batteries and Leclanchds will be found very useful for 
 ordinary testing and bridge work, where the constancy of the 
 current is not of such vital importance. The latter cells will 
 be much improved if potassium permanganate crystals are 
 mixed up with the manganese doxide in the porous pot ; and 
 if this cannot be done, the cells may be " revived " from time 
 to time by pouring a solution of potassium permanganate into 
 the porous pot through one of the air tubes. Other forms of 
 battery, such as Daniell's and Latimer Clark's cells, will be dealt 
 with later on in coimection with standard cells. 
 
 20. In man)' cases an E.M.F. very much smaller than that 
 given by any of the above batteries, but of known value, is 
 
 )e-(J) WWWVVAAA/WWWVV^A/WWWVAA 
 
 II 
 
 Fig. II. 
 
 required ; this may be obtained as follows. A current from an 
 ordinary cell is sent through a very high resistance, say 10,000 
 ohms, in series Tvith a low-resistance sensitive galvanometer 
 (see AB in Fig. 11). The terminals A, C supply a potential 
 difference which may be any desired fraction of AB, by simply 
 
1 6 Practical Electricity and Magnetism. 
 
 moving C nearer to or farther from B. The object of the 
 galvanometer is to enable the potential between A and C to 
 be kept constant when the resistance across AC varies ; as 
 the external circuit attached to AC diminishes in resistance, the 
 potential diiference between A and C will fall, and this will be 
 indicated by the galvanometer deflection falling off. The 
 point B must now be adjusted so as to diminish the total 
 resistance between A and B until the galvainometer deflection 
 is restored to its original value, the P.D. between A and C 
 then being the same as before, and the same fraction of the 
 original value of AB, the fall of potential down which may 
 be taken as equal to the E.M.F. of the cell. 
 
 2 1 . Resistances. — In addition to the ordinary resistance boxes, 
 post-office bridges, etc., which are accurately adjusted, but meant 
 only to carry small currents (in no case more than o'ooi ampere 
 should be allowed to pass through them), resistance coils are 
 often required, which, while still being accurate, are capable of 
 
 Fig. I 
 
 carrying much stronger currents without heating to such an 
 extent as to alter their resistance appreciably. Such coils may be 
 made of manganin, wound in non-inductive spirals and stretched 
 on a large wooden frame, so as to expose a large cooling 
 surface, the ends of the coils being soldered to thick copper 
 wires ; a very convenient form used by the author being that 
 
Resistances. 17 
 
 shown in Fig. 12, one of the copper end pieces being shown 
 in Fig. 13, and consisting of a piece of thick copper wire first 
 doubled in two, the end of the resistance coil being soldered into 
 
 the bend, and then one , 
 
 limb being bent at right X [^;) j 
 
 angles at the middle point, '" '. . ^ 
 
 B. The object of this 
 being to enable the cur- 
 rent to be sent in at A, _B 
 and B to be used as a 
 potential terminal when 
 it is required to measure A 
 the fall of potential down I 
 the coil. F'°- '3- 
 
 The following table gives the sizes of wires that may be 
 employed for the various resistance coils : — 
 
 Size in B.W.G 
 
 1-5 ohms 1 20 
 
 10-50 „ I 26 
 
 IO0-2CO „ : 30 
 
 500-1000 „ \ 34 
 
 1000-5000 ,, 36 
 
 10,000 ,, 40 
 
 The wires in all cases are double silk covered. Also, in 
 cutting the lengths of wire for the coils, it is advisable to test 
 each gauge for specific resistance, as various specimens of 
 manganin are found to vary among themselves. The coil, before 
 being finally adjusted, should be artificially aged by heating to 
 a temperature of 140° C. in an air-bath for 5 hours ; some 
 authorities also recommend coating the wire with shellac var- 
 nish, to protect it from the oxidizing action of the air. The 
 advantage of using manganin is its high specific resistance and 
 negligible temperature coefficient. 
 
 22. In addition to such resistances as the above, variable un- 
 known resistances are often required to regulate the current in 
 a circuit. For purposes of this k ind, Varley's carbon rheostat (see 
 
 C 
 
1 8 Practical Electricity and Magnetism. 
 
 Fig. 14) will be found very useful. This consists of m number 
 of discs of carbonized cloth, through which the current passes, 
 and by compressing the discs the resistance can be diminished. 
 Ranges from 500 ohms to -^ ohm may be obtained with such 
 instruments. 
 
 Flc. 14. 
 
 Liquid resistances will also be found \ery useful. A simple 
 form which can be easily constructed in any laboratorj' is shown 
 in Fig. 15. A is a long glass specimen jar, fitted with an india- 
 rubber cork, B, through which a hole has been bored and fitted 
 with a brass connector, C, the lower screw of which has been 
 removed ; through this passes a copper rod, R,to the end of which 
 is attached a copper plate, P, forming one electrode ; the other 
 electrode, P', is connected to a wire, ^^', which is well insulated 
 down to the point where it is attached to P'. A solution of 
 copper sulphate is employed. To vary the resistance, the rod 
 R is pushed in or pulled out, so causing the plate P to approach 
 or recede from P', the upper screw of the connector being 
 employed to clamp the rod R in any position. Such a resist- 
 ance is capable of considerable variation, not only by altering 
 the distance apart of the plates, but also by employing solutions 
 of various densities. 
 
Galvanometer Calibration. 
 
 19 
 
 ^ 
 
 7^ 
 
 B 
 
 P' 
 
 r 
 
 Tne Calibration of a Sensitive Galvanometer. 
 
 23. In the majority of experiments in which the indications 
 of a galvanometer, reflecting or other- 
 wise, are employed to denote the 
 relative values of the currents passing 
 through it, the assumption is fre- 
 quently made (except in the case of a 
 standard galvanometer as a tangent 
 or sine) that the currents are directly 
 proportional to the. deflections. This 
 may be true for the majority of re- 
 flecting galvanometers for small de- 
 flections on either side of the zero, 
 but, as the deflections increase, the 
 needle of the galvanometer may 
 move out of the uniform magnetic 
 field, and then this assumption is no 
 longer correct. In all experiments 
 of an accurate nature it is desirable 
 to calibrate the galvanometer em- 
 ployed, and to deduce the relation- 
 ship between the currents and the 
 deflections from the calibration curve. 
 The shape of this curve depends on 
 the winding of the galvanometer, tlie 
 relative positions of the fixed and 
 movable magnetic systems, and the 
 nature and strength of the controlling 
 force. Consequently, if any of these 
 are altered, the calibration curve is 
 also liable to alter. It is therefore 
 necessary to carefully specify the 
 nature of these conditions at the time 
 of calibration, so that it will be 
 possible to readjust them to their original values should they 
 at any time be altered. 
 
 In the following method of calibration, advantage is taken 
 
 Fig. 15. 
 
20 Practical Electricity and Magnetism. 
 
 of the relation that the current in a simple circuit varies inversely 
 as the total resistance of the circuit, when the E.M.F. acting 
 in it remains constant. Therefore, by comparing various 
 
 deflections with the values of j^ corresponding to them, where 
 
 R represents the total resistance of the circuit, the calibration 
 of the instrument may be effected. . 
 
 Since the galvanometer itself is part of the circuit traversed 
 by the current, it will be necessary to determine its resistance. 
 The most satisfactory method of doing this being, to connect it 
 up to a Wheatstone bridge and measure its resistance in the 
 ordinary way, taking the precaution in the case of a suspended- 
 
 kI^^Z^)^ 
 
 R 
 A ; f— ^)-^WW\lWWW\/WWWWW\/W\AA/WV^^ 
 
 Fig. i6. 
 
 needle galvanometer of removing the needle before making 
 the measurement, and in the case of a D'Arsonval galvanometer 
 of fixing the movable coil, so as to prevent the suspension from 
 being strained by the relatively large testing current employed. 
 Having obtained the galvanometer resistance and the tempera- 
 ture at which the measurement was made, the calibration is 
 next proceeded with. Since the galvanometer is a sensitive 
 one, it will be impossible to connect it up in series with a 
 battery, unless there are available some very high and accurately 
 adjusted resistances. It is therefore more convenient to use 
 the method described in par. 20, of employing a small 
 fraction of the total E.M.F. of the battery. Fig. 16 shows a 
 diagram of the connections. S represents a secondary battery 
 connected to the resistance AB, which should not be less than 
 
Galvanometer Calibralioti. 21 
 
 10,000 ohms. In series with this resistance, and forming part 
 of it, is the low-resistance galvanometer g, to show variations in 
 the current strength ; and once the experiment is started, the 
 deflection of ^ must be kept constant by altering, if necessary, 
 the value of AB, i.e. by bringing the point B nearer A. G is the 
 galvanometer to be calibrated, and is connected, through the 
 reversing key K and resistance R, to A and C ; AC representing 
 a small fraction of the total resistance in the battery circuit AB. 
 The galvanometer g may be dispensed with if the resistance 
 of the galvanometer G is not less than a hundred times AC, 
 as in this case the percentage variation of P.D. across AC, with 
 a variation of resistance in the galvanometer circuit, is ex- 
 ceedingly small. The galvanometer G having been adjusted 
 properly, R is increased until a small deflection is obtained 
 on G ; this is recorded. The reversing K is then rocked over, 
 and the deflection on the opposite side of the zero obtained ; 
 this is also recorded, along with the value of R. R is then 
 varied so as to increase the galvanometer deflection by a few 
 degrees, and deflections to right and left of zero are again 
 obtained, and so on, until the deflection is as large as can be 
 measured. The temperature of the coils R and of the galva- 
 nometer must be determined, also the time of swing of the 
 galvanometer needle must be taken. The needle is set swing- 
 ing, the galvanometer having been first disconnected from the 
 rest of the apparatus, and a stop-watch is started just as the 
 spot of light or galvanometer needle is crossing the zero on 
 •the scale ; the time that elapses until the next transit is made 
 in the same direction is the periodic time of swing. By taking 
 ten or twenty such swings, and dividing the total time by the 
 number of complete swings, a more accurate determination 
 will be got. The periodic time specifies the relative value of 
 the controlling force acting on the needle or movable magnetic 
 system. The distance of the mirror from the scale must be 
 measured, in the case of a reflecting galvanometer, and the 
 length of a scale-division, in centirhetres. The readings may 
 then be tabulated thus — 
 
22 
 
 Practical Electricity and Magnetism. 
 
 Resistance R. 
 corrected for 
 
 temperature. 
 
 Deflection to 
 right. 
 
 Deflection to 
 left. 
 
 Galvanometer 
 resistance 
 
 corrected for 
 temp, g' 
 
 Total 
 resistance 
 
 Values of 
 
 ^+s- 
 
 Periodic time of swing = 
 Length of a scale division = 
 Distance of mirror from scale = 
 Temperature during test = 
 
 The calibration curves are obtained by plotting the deflec- 
 tions to right and left respectively against the corresponding 
 
 values of , values of ~ — being taken as ordinates 
 
 •!<.+,?■ R + .?" 
 
 and the deflections as abscissae. Two curves will thus be 
 
 obtained ; but if the deflections on either side of the zero are 
 
 identical, they will coincide. 
 
 ISO 
 
 c 
 
 ? 
 
 O 
 
 ■oooas 
 
 ■OOOIS OOOZ OOOZS 0003 oooss oow 
 
 Values of -rTj 
 
 Fig. 
 
 24. The following data were obtained in calibrating a reflect- 
 ing galvanometer : — 
 
 The galvanometer resistance = 1609 ohms at 14° C. The 
 
Measurement of Resistance. 
 
 23 
 
 temperature during the calibration was 9° C. Using as tem- 
 perature coefficient of copper 0-388 % per 1° C, the galvanometer 
 resistance corrected for temperature was 1578 ohms. Since the 
 box of known resistances was adjusted for a temperature near 
 that of the room, no temperature correction was made on 
 them, as the galvanometer was not sensitive to that amount. 
 
 Resistance 
 
 Deflection to 
 
 Deflection to 
 
 Galv. resist. 
 
 Total resist. 
 
 1 
 
 R. 
 
 right. 
 
 left. 
 
 g- 
 
 R-t-f. 
 
 R+f 
 
 13330 
 
 23-5 
 
 24-0 
 
 1578 
 
 I4gc8 
 
 0-000 67 
 
 11000 
 
 28-0 
 
 29-0 
 
 1578 
 
 12578 
 
 0-000080 
 
 1 0000 
 
 30-5 
 
 31-0 
 
 1578 
 
 11578 
 
 0-000087 
 
 <;ooo 
 
 33-5 
 
 ?3-5 - 
 
 1578 
 
 10578 
 
 0-000095 
 
 8000 
 
 37-5 
 
 370 
 
 1578 
 
 9578 
 
 000104 
 
 7000 
 
 41-5 
 
 41-5 
 
 1578 
 
 8578 
 
 0-0001 [6 
 
 60C0 
 
 47-5 
 
 46-5 
 
 1578 
 
 7578 
 
 0-000132 
 
 S50O 
 
 50-5 
 
 50-0 
 
 1578 
 
 7078 
 
 0-000141 
 
 50C0 
 
 55-0 
 
 540 
 
 1578 
 
 6578 
 
 0-000152 
 
 45CO 
 
 59-0 
 
 58-5 
 
 •578 
 
 6078 
 
 o'ocoi65 
 
 4000 
 
 645 
 
 630 
 
 1578 
 
 5578 
 
 o-cooi8o 
 
 3SOO 
 
 710 
 
 69-5 
 
 1578 
 
 5078 
 
 0-000197 
 
 3C00 
 
 79-0 
 
 770 
 
 1578 
 
 4578 
 
 0-000219 
 
 2600 
 
 860 
 
 845 
 
 '578 
 
 4.78 
 
 0-CC0240 
 
 2400 
 
 91-0 
 
 88 
 
 1578 
 
 3978 
 
 0-000252 
 
 2200 
 
 95 "5 
 
 925 
 
 1578 
 
 3778 
 
 0-C00266 
 
 2CO0 
 
 101-5 
 
 97 5 
 
 1578 
 
 3578 
 
 0-C00280 
 
 1900 
 
 104-0 
 
 100-5 
 
 1578 
 
 3478 
 
 0-000289 
 
 1700 
 
 11 1-0 
 
 107-0 
 
 1578 
 
 3278 
 
 0-000306 
 
 i6oo 
 
 1140 
 
 1 100 
 
 1578 
 
 3'78 
 
 0-000316 
 
 1500 
 
 1:0 
 
 112-5 
 
 1578 
 
 3078 
 
 0-000326 
 
 14CO 
 
 123-5 
 
 116-5 
 
 1578 
 
 2978 
 
 0-000337 
 
 1300 
 
 127-5 
 
 119-5 
 
 1578 
 
 2878 
 
 0-000349 
 
 1200 
 
 i33'o 
 
 124-0 
 
 1578 
 
 2778 
 
 0-000362 
 
 IICO 
 
 >38-5 
 
 128-5 
 
 1578 
 
 2678 
 
 0-000375 
 
 1000 
 
 1430 
 
 133-5 
 
 1578 
 
 2578 
 
 0-000390 
 
 Periodic time of swing = 6-5 seconds. 
 Distance of mirror from scale = i metre. 
 Length of one scale-division = i mm. 
 Temperature during test = 9° C. 
 
 Measurement of Resistance. 
 
 25. One of the most important of all electrical measure- 
 ments, and perhaps that most frequently made, is the measure- 
 ment of the resistance of a conductor. In 1827 Ohm first 
 
24 Practical Electricity and Magnetism. 
 
 stated the relationship, which is now known as Ohm's law, that 
 for any given material, provided its physical conditions remain 
 unaltered, there is a constant ratio between the potential 
 difference at its ends, and the steady current flowing through it 
 due to that potential difference. This ratio is defined as the 
 resistance of the conductor. 
 
 The methods employed in determining this ratio in absolute 
 measure will be given later ; at present we are only concerned 
 with the methods of comparing different resistances with one 
 another, and of expressing the resistance of a conductor in 
 terms of a suitably chosen standard, this latter process being 
 meant when we speak of measuring a resistance. 
 
 26. The following measurements will be dealt with in this 
 chapter : — 
 
 (a) The measurement of resistances of ordinary value. 
 {b) The measurement of very low resistances. 
 
 (c) The measurement of very high resistances. 
 
 (d) The measurement of liquid resistance. 
 {e) The measurement of battery resistance. 
 
 (f) The measurement of specific resistance. 
 
 (g) The variation of specific resistance with temperature. 
 
 (h) The variation of specific resistance with molecular 
 change. 
 
 (y) Fault testing. 
 
 {k) The absolute determination of resistance, and construc- 
 tion of standards. 
 
 27. Two of the methods employed in the measurement of 
 resistances of ordinary value, and in the comparison of resist- 
 ances which are nearly equal to one another, have already 
 been mentioned in Vol. I. (p. 47). The first of these, the 
 substitution method, is not susceptible of the same accuracy as 
 the Wheatstone bridge method, and we will therefore pass it over 
 in favour of the latter, the proof of which we will now proceed to 
 give. Four coils. A, B, C, D, are connected together, as shown in 
 Fig. 18. A galvanometer, G, is connected across one diagonal 
 of the diamond-shaped figure from c to b, and a battery across 
 the other from a to d. If now the resistances are so arranged 
 that when the battery circuit is closed and then the galvano- 
 
Wheatstone Bridge. 
 
 25 
 
 meter circuit, there is no deflection in the galvanometer, the 
 points c and b must be at the same potential. Hence the 
 current in A will be the same as that in C, and that in B 
 the same as that in D. Let c^ be the current flowing in A and 
 
 Fig. i8. 
 
 C, and c, that flowing in C and D. Also let a, b, c, d represent 
 the potentials at the corners of the figure, and A, B, C, D the 
 resistances from a X.o c,a to b, c to d, and b to d respectively. 
 Then by Ohm's law — 
 
 A = '1^", C = '-- '^, B = "— ^ D = *-- '^, 
 A c, c. 
 
 '■I 
 
 J a — c c — d ■, 
 
 and f, = = — „ — ; also c„ 
 
 A C 
 
 a — b _ b — d 
 "B W 
 
 But since there is no galvanometer deflection, the potential at 
 c is equal to that at b, and a — c = a — b = Yi, while c — d = 
 b- d=7„. 
 
 Therefore ^' = 
 A 
 
 ^"'^l^^ 
 
 Hence Pj = 
 
 AP, BP, 
 C D 
 
 A 
 
 B 
 D 
 
 It will be observed here that in order to calculate the resist- 
 ance of any one coil, we do not need to know the resistances 
 
26 
 
 Practical Electricity and Magnetism. 
 
 of the other three, but only that of one of them, provided we 
 know the ratio of the resistances of the other two. 
 
 28. An apparatus for measuring resistances according to this 
 method, and known as the post-office resistance box, is shown 
 in Fig. 19, and consists of three rows of resistance coils attached 
 
 Fig. 19. 
 
 to brass blocks fixed on an ebonite plate, and so arranged that 
 by placing a brass plug in between two blocks the coil between 
 them is short circuited. The values of the coils and the 
 method of connection are shown in Fig. 20, the lettering being 
 
 ^1000 I r ^00 
 Fig. zd. 
 
 the same as that employed in Fig. 18. The coil to be measured 
 is connected from b to d, the galvanometer from b to c, and the 
 battery from a to d. The coils from a io c and c to d repre- 
 sent coils A and c in the other figure, and are known as the 
 ratio coils or the proportional arms. The maximum possible 
 
Post- Office Bridge. 27 
 
 limits to which the bridge could measure would be, for high 
 
 resistances, when the arms were in ratio, i^ = - ^— , and B 
 
 C 10,000 
 
 had all the plugs withdrawn, giving- 11,110 ohms ; then — 
 10 _ 11,110 
 10,000 I) 
 
 or n = 11,1 10,000 
 
 = 11 megohms pi-aetically ; 
 
 and lor the inferior limit, ' = io<°°o . ^j^j^,^ — 
 C 10 
 
 10,000 _ I 
 
 10 ~ D 
 
 or D = o'ooi ohm. 
 
 The above measurements could not, however, be made with 
 any pretence lo accuracy, the ]iractical limit of working of the 
 above bridge being from about o'l ohm to a megohm. 
 
 It will be observed, in Fig. 1 9, that a hole is drilled in the 
 centre of each block to admit of a plug being inserted. This 
 is exceedingly useful, as it allows of sets of coils being com- 
 )>ared against one another, say the 1, 2, 3, 4 against the 10, 
 etc., and also it admits of a fall of potential down a known 
 resistance being taken from the box. 
 
 The wires of which the coils are wound consist, in the more 
 expensixe boxes, of platinum-silver, and in the cheaper forms, 
 of platinoid, tierman siher, or manganin. They are adjusted 
 so as to be correct at the temperature stated on tlie box. A 
 hole is provided in tlie ebonite top for the insertion of a 
 tbormometer to measure the temperature of tlie coils. 
 
 :*9. When making a measurement of resistance, if an exact 
 balance ainnot be obtained, the galvanometer deHections for 
 resistances above and below the true value are taken, and the 
 exact ^•alue of the resistance obtained by interpolation, as 
 exi>lained in \'ol, I. 
 
 In using a post-office box of coils, great care must be exer- 
 cised in seeing that the plugs are fitting tightly into the holes, 
 and make good contact. For this reason the metal part of the 
 plugs should never be held in the hand, since a film of oil is 
 
28 
 
 Practical Electricity and Magnetism. 
 
 apt to get over the surface and affect the contact between the 
 plug and the brass blocks. 
 
 30. In order to minimize the effect of the plugs and reduce 
 their number as much as possible, the form of bridge shown in 
 Fig. 21 is sometimes employed ; this is known as the dial bridge. 
 The coils in this form are arranged in four sets — units, tens, 
 hundreds, thousands. The ten coils attached to each dial are 
 all of the same value, and are in series with one another ; so 
 that by means of one plug, one coil, or any number up to ten 
 in series, can be inserted into the circuit. This arrangement 
 
 is convenient for the inter-comparison of the coils with one 
 another, since the ten coils in series in one dial should equal 
 the first in the next above it. 
 
 The Metre Bridge. 
 
 31. If it is desired to compare the resistances of two coils, 
 or to determine the value of an unknown resistance in terms 
 of that of a standard coil, the most satisfactory and accurate 
 method to adopt is to compare the resistances on the wire or 
 metre bridge. 
 
 In this apparatus, so called because the wire employed is 
 usually I metre in length, the ratio of the resistances required 
 is obtained in terms of the ratio of two parts of the bridge wire. 
 
Metre Bridiic. 
 
 29 
 
 'riic theory of llie inslrumciU is tlic same as lliiU of the ^Vheat- 
 slone hi'iitgo just descriheil, the wnv taking the plaee of the 
 proportional arms in that arrangement. .V rough outline of 
 the method of using the wire bridge has ahvacly been given 
 in Vol, I.,' which contains a description of an apparatus 
 
 KiG, aa. 
 
 capable of giving results sufficiently accurate for most com- 
 mercial purposes; for more accurate work, however, such as 
 the standardization of coils, etc., a more carefully made. and 
 elaborate apparatus is required. The following is the descrip- 
 tion of such a bridge, which has been designeil by the autlior 
 
 ^ 
 
 W 
 
 "at 
 
 a"-ifiV-a"-t* s'-hi-a'-i«ciy-fli 
 
 ' B gh o (ia c 
 
 a->l«\i'->i»- -reii' 
 
 ¥' 
 
 >;i)'' J' ■1 
 
 Kill. jj. 
 
 for his own laboratory, and in whidi there is a special com- 
 n\utating device for interchanging die coils witliout having to 
 eiuploy mercury contacts, ilesigned by the audior's colleague, 
 Mr. S. Joyce. 
 
 iMg. >j gives a general \iew of tlie bridge, wiiile Figs. 23, 
 J4, and 25 give dimensioned sketches of various parts. 
 
 The base of the instrument is of teak, and tlie copper con- 
 ductors are fastened to it by screws which pass tlirough 
 ebonite washers, thus conipletely insulating diem from it, while 
 
 ' Sec vol. i. p. 47. 
 
30 
 
 Practical Electricity and Magnetism. 
 
 allowing the whole of the bridge to be taken to pieces in a 
 very short time for cleaning. The scale, which is of boxwood, 
 is also insulated from the base by pieces of ebonite. 
 
 There are in all four gaps in the bridge for the insertion 
 of coils, these are shown in A, B, C, D (Fig. 23), the coils, the 
 ratio of whose resistances is required, being placed in the gaps 
 
 Fig. 24. 
 
 B and C ; then, when the copper connecting arm Aj is turned so 
 as to connect terminals (i) and (3), and the arm A^ connects 
 (4) and (6), the resistance in the gap B is next to that in gap A, 
 and the resistance in gap C is next to that in gap D. But by 
 
 KiG. 25. 
 
 simply swinging the arms round so as to connect (2) and (3), 
 and (4) and (5), the coil in C is next A, and that in B is next D, 
 or the coils B and C have been interchanged with respect to 
 the other conductors in the bridge. The rotating arms A, and 
 A2 make contact with large terminal screws, so that the contact 
 resistance is small, also the time occupied in making the change 
 over is very small. The galvanometer is connected to the 
 blocks to which terminals (3) and (4) are attached ; one of the 
 battery terminals is connected to the terminal on the central 
 
Wire Calibration. 31 
 
 small block between B and C and marked /;, the other being 
 attached to the sliding contact. 
 
 The bridge wii;e is not soldered at its ends to the copper 
 bars, but clamped by rtieans of small copper plates screwed 
 down on the copper bars. 
 
 The details of the sliding contact are shown in Figs. 24 and 
 25. The tapping contact slides along a brass rod, | inch ex- 
 ternal diameter, and can be clamped tightly to it by means of 
 the small set screw S, the final adjustment being made by 
 means of the micrometer screw M, which moves the whole rod. 
 
 A wooden cover encloses the whole bridge when not in use, 
 thus keeping it clean and protecting it from damage. Also on 
 the inside of the cover is pasted the calibration curve of the 
 stretched wire. 
 
 32. Calibration of a Bridge Wire. — Before describing the 
 methods of comparison of resistances on a wire bridge, we 
 must first consider the various sources of error likely to affect 
 such a measurement. 
 
 The method of comparison is essentially the determination 
 of the ratio of the resistances of the two coils in terms of the 
 resistances of two portions of a stretched uniform wire. If the 
 wire is uniform, and has the same physical properties through- 
 out its length, then the resistances of the portions will be 
 directly proportional to their lengths. This is the first assump- 
 tion, and a set of experiments made with a view to discover 
 whether or not it is warranted, is usually termed the calibration 
 of the bi'idge wire. 
 
 33. The first method of calibration to be described is a 
 modification of that due to Mattheissen and Hockin.^ Two 
 coils, A and B (see Fig. 26), of equal resistance, which should be 
 about twenty times that of the wire, are placed in the outer gaps 
 of the bridge. Two other coils of resistances, Rj and Rj, 
 differing from each other by about -~ per cent., are placed across 
 the inner gaps, the galvanorheter and battery being connected 
 up in the usual way, a balance is obtained at some point, Bj, 
 on the scale. A small known resistance, ;-, is then inserted 
 
 ' See Report of Electrical Standards Cimimittee of British Ass., l86/), 
 Appendix C. 
 
32 Practical Electricity and Magnetism. 
 
 in series with A, and the balance is altered to some point, B^. 
 A + ;■ and B are then interchanged, and a third balancing 
 point, B.„ is obtained. Then, if the resistance of the bridge 
 wire and end contacts is represented'by w, and the length of 
 
 Bj B3 B, 
 
 Fig. 26. 
 
 the bridge wire between B^ and B3 in millimetres by /, it can 
 be shown that the resistance of the part B2B3 of the wire is 
 equal to — 
 
 |^;(A + B + . + .), 
 
 and the resistance per millimetre of this part of the wire equals — 
 
 $S)(^ + ^ + - + '■>• 
 In order to proceed with the calibration of the wire, the value 
 of ;• is gradually increased, and so the balance is obtained at 
 various parts of the wire. To get the value of (A+B -{-w+r), 
 the coils Ri and Rj should be disconnected, also the battery 
 and galvanometer, and the resistance between P and Pi 
 measured on a post-office bridge in the usual way. A table 
 giving resistance per millimetre, at various parts of the wire, 
 may be made, and a calibration curve plotted from it with 
 resistance in ohms per millimetre for ordinates, and scale- 
 divisions for abscissae. 
 
 34. Calibration of a Wire by fall of Potejitial. — The second 
 method of calibrating a bridge, or any other wire, is a fall of 
 potential method. A steady current from a secondary battery, 
 with resistance in series with it, is sent through the wire to be 
 calibrated, the current being kept small in order to prevent any 
 
Wire Calibration. 
 
 33 
 
 serious heating in it, which might alter its resistance. The fall 
 of potential along the wire is then proportional to its resistance ; 
 so that by comparing the P.D.'s per centimetre run of the wire, 
 we get a series of numbers proportional to the resistance 
 per centimetre of the wire. 
 
 In order to tap the wire at points i cm. or other convenient 
 distance apart, the following arrangement is adopted. Two 
 small thin brass or copper plates are attached to two separate 
 small rectangular blocks of ebonite or wood well boiled in 
 paraffin (see Fig. 27). The brass springs are attached so as 
 to project over the ends of the ebonite blocks. To the under 
 surface of the brass plates are soldered short straight pieces of 
 stout platinoid wire, which have been filed to form knife-edges. 
 These small blocks are laid on the bridge in such a way that 
 
 Fig. 27. 
 
 the platinoid knife-edges press on the wire, the pressure being 
 maintained by the springs to which they are soldered, and the 
 ends of the knife-edges projecting on to the scale indicate 
 their position on the wire. The two blocks are then arranged 
 at a suitable distance apart on the wire, and are clamped in 
 positiop by means of a small wooden clamp. The galvanometer 
 is connected to the terminals attached to the brass springs, and 
 should have a high resistance compared with that of the length 
 of wire experimented upon, so as not to sensibly affect the 
 potential difference by the current which it takes ; the galva- 
 nometer must have been previously calibrated, since the values 
 of the P.D.'s are to be deduced from its deflections. 
 
 The apparatus is connected up as shown in Fig. 28. The 
 secondary battery, B, is connected to the ends of the bridge 
 
 D 
 
34 Practical Electricity and Magnetism. 
 
 wire through a regulating resistance, R, and a key, K. Starting 
 at one end with the tapping contacts, deflections are obtained 
 for every centimetre length of the wire. The first reading should 
 be repeated frequently, to test for constancy in the current flowing 
 through the wire. The readings obtained in this way give us 
 a means of calibrating the wire relatively to one of the centi- 
 metre lengths. If the calibration is required to be absolute, 
 
 B R K 
 
 1 1 — \A/WWWVWWW •/ 
 
 ■■>:--<T) 
 
 that is, to determine the ohmic resistance per centimetre of 
 the wire, a standard resistance of ^o, y^, or yoVo o'^'^ "^^isX 
 be included in series with the wire during the test, and the fall 
 of potential between its ends measured on the galvanometer, 
 thus enabling the ohmic resistances of the sections of the 
 wire to be calculated from the deflections produced by them. 
 The results should be tabulated thus— 
 
 Part of wire under test. Galvanometer deflection. 
 
 The sum of all the deflections represents the deflection that 
 would have been obtained if the galvanometer had been con- 
 nected to the ends of the bridge wire. 
 
 35. The calibration curve is obtained by plotting the lengths of 
 
Wire Calibration. 
 
 55 
 
 the bridge wire as ordinates against the sum of the deflections 
 obtained from the commencement to the end of these given 
 lengths as abscissae. Thus the ratio of the length of the 
 abscissa at any part of the curve to the total length of the 
 horizontal scale minus that length, will be the ratio of the re- 
 sistances of the two portions into which the wire has been 
 divided at that point. Thus let Fig. 29 represent such a curve, 
 and let tlie sum of all the deflections come to 865. Then, if 
 the balance on tlie wire came at 500 in a metre bridge, the 
 ratio of the other resistances is not -f^, but f^, on account of 
 the inequality of the wire, 400 being the length of the abscissa 
 
 JOOO 
 
 865 
 
 ections 
 
 Fig, 39. 
 
 at the 500 point, and 465 is the total length of the horizontal 
 scale minus the abscissa, or 465 = (865 - 400). 
 
 Such a curve having been drawn for a bridge wire, it may be 
 pasted on a wooden board and kept beside the bridge, and 
 when a measurement is made on the bridge, the effective lengths 
 of the two portions of the wire on either side of the balance 
 point are obtained from the curve, instead of taking the actual 
 lengths as determined from the scale over which the wire is 
 stretched. In all experiments like the above the hand must 
 not be brought near the contacts, as thermo-electric currents 
 will be set up which will interfere with the results. To make 
 quite sure, that no such effect is taking place, the battery circuit 
 should be broken at K (see Fig. 28), when, if there is no 
 disturbing effect in the tapping circuit, the galvanometer needle 
 should remain at zero. 
 
36 
 
 Practical Electricity and Magnetism. 
 
 36. The following are the data obtained in the calibration 
 of the wire of a metre bridge, the sensitiveness of the galvano- 
 meter being such that a variation of the distance between the 
 tapping-points of i mm. represented a variation of about six 
 scale-divisions on the galvanometer scale : — 
 
 Tapping-points. 
 
 Galvanometer 
 deflection. 
 
 Sum of deflections. 
 
 (centimetres). 
 
 
 
 0-6 
 
 418 
 
 418 
 
 6-1 1 
 
 336 
 
 754 
 
 11-16 
 
 337 
 
 I091 
 
 16-21 
 
 336 
 
 1427 
 
 21-26 
 
 335 
 
 1762 
 
 26-31 
 
 330 
 
 2092 
 
 31-36 
 
 330 
 
 2422 
 
 36-41 
 
 330 
 
 2752 
 
 41-46 
 
 338 
 
 3090 
 
 46-51 
 
 335 
 
 3425 
 
 51-56 
 
 336 
 
 3761 
 
 56-61 
 
 336 
 
 4097 
 
 61-66 
 
 337 
 
 4434 
 
 66-71 
 
 337 
 
 4771 
 
 71-76 
 
 337 
 
 5108 
 
 76-81 
 
 325 
 
 5433 
 
 81-86 
 
 334 
 
 5767 
 
 86-91 
 
 330 
 
 6097 
 
 91-96 
 
 335 
 
 6432 
 
 96-100 
 
 251 
 
 6683 
 
 Fig. 30 represents the curve drawn from the above data. 
 
 In this case it will be seen that if balance was obtained at 
 the middle point on the wire, instead of the resistances being 
 equal they would be in the ratio of — 
 
 3360 3360 _ roi 
 
 6683 - 3360 ~ 3323 ~ 1 
 
 In the following modification of the last method of calibrating 
 the wire, which, if not so accurate, has the advantage of being 
 simpler and of not requiring any special tapper, the sliding 
 contact of the bridge is itself used as a movable contact-piece, 
 the other contact being fixed permanently to one end of the 
 bridge wire. Fig. 31 shows the connections of the apparatus. 
 The battery B, with a large resistance, R, in series with it, is 
 
\ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 iC 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \, 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 Q Q Q ^ ^ 
 
38 
 
 Practical Electricity and Magnetism. 
 
 connected to the ends of the wire XY through the key K. One 
 terminal of the galvanometer is connected to X, while the other 
 goes to the movable contact-piece P of the bridge. Readings 
 are taken with the contact-piece P at various positions between 
 X and Y, say every centimetre, until the middle point of the wire 
 is reached, when the wire from the galvanometer is disconnected 
 from X and connected to Y, the readings being then continued 
 imtil P reaches Y. The object of changing the galvanometer 
 terminal from X to Y when half the wire has been calibrated is 
 
 BR K 
 
 1 1 — VwVWWWWV /^ 
 
 J 
 
 ^-^SV 
 
 Fig. 31. 
 
 in order to allow larger deflections to be obtained on the 
 galvanometer, since the maximum P.D. at the terminals of the 
 tapping circuit will then be half the total fall down the wire, 
 whereas, if the change had not been made, the total P.D. at the 
 tapper and down the wire would have been the same. The 
 change of course alters the direction of the current in the 
 galvanometer, unless a reversing key is placed in its circuit. 
 The results are tabulated thus — 
 
 Distance, XP. 
 
 Galvanometer deflection. 
 
Wire Calibration. 
 
 39 
 
 From this table the deflections corresponding to equal 
 lengths of the wire may be calculated, and the result plotted 
 graphically, as in the last case. The remarks on thermo-electric 
 effects at the tapping-point apply equally to this method, and 
 the test for such an effect should be applied between each pair 
 of readings. If such an effect persists it must be allowed for 
 by adding or subtracting the deflection produced by it from 
 the deflection when the current is flowing in the wire, according 
 as it acts against, or with, the main effect. 
 
 The galvanometer deflections are supposed to be propor- 
 tional to the currents flowing through it ; the instrument must 
 be calibrated, however, and, if necessary, corrections applied 
 from its calibration curve, should it not be a straight line. 
 
 37. The following numbers were obtained in calibrating a 
 wire according to the above method. On account of a deflec- 
 tion due to thermal currents in the conductors a correction had 
 to be applied to each reading. 
 
 Distance along 
 
 Deflection due 
 
 Deflection on 
 
 Corrected 
 
 Deflection per 
 
 XY 
 
 to thermal effect. 
 
 tapping. 
 
 deflection. 
 
 10 cm. 
 
 0-10 
 
 — ro 
 
 + 34-5 
 
 35-S 
 
 35-5 
 
 0-20 
 
 -i-S 
 
 +68-5 
 
 70-0 
 
 35-5 
 
 0-30 
 
 — 2-0 
 
 -H03-S 
 
 105-5 
 
 34-5 
 
 0-40 
 
 — 2-0 
 
 + 139-5 
 
 I4I-5 
 
 360 
 
 0-50 
 
 -3"o 
 
 + «7S'o 
 
 178-0 
 
 36-5 
 
 100-90 
 
 -5'S 
 
 +29-5 
 
 35-0 
 
 35-0 
 
 100-80 
 
 -4-5 
 
 +64'S 
 
 69-0 
 
 34-0 
 
 ICO-70 
 
 -4-5 
 
 +99-0 
 
 103-5 
 
 34-5 
 
 100-60 
 
 -4-5 
 
 + 134-5 
 
 139-0 
 
 36-5 
 
 100-50 
 
 -3-5 
 
 + 171-5 
 
 175-0 
 
 36-0 
 
 From these numbers the following table can be calculated 
 from which the curve is plotted (see Fig. 32), a previous 
 experiment having shown that the galvanometer deflections 
 were proportional to current : — 
 
40 
 
 Practical Electricity and Magnetism. 
 
 Part of wire. 
 
 Deflection. 
 
 O-IO 
 
 35-5 
 
 0-20 
 
 700 
 
 0-30 
 
 105-5 
 
 0-40 
 
 141-5 
 
 0-50 
 
 1780 
 
 0-60 
 
 2t4-o 
 
 0-70 
 
 250-s 
 
 0-80 
 
 285-0 
 
 0-90 
 
 3190 
 
 0-100 
 
 354-0 
 
 loo 
 
 
 
 
 
 
 
 
 
 ■0-3 
 
 TH 60 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 / 
 
 ^ 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 mo 150 zoo 250 300 
 
 Galvanometer Deflections 
 Fig. 32. 
 
 38. Correction for t/te Contacts at tlie Ends of the Wire and tlie 
 End-Pieces of the Bridge. — In bridges where the wire is soldered 
 at X and Y, or as in Fig. 23, where it is clamped by copper 
 clamps, if the contact is not good a small resistance is intro- 
 duced at each end of the wire in addition to the resistance of 
 the copper end-pieces, which practically lengthens it. The 
 value of the resistance thus introduced is best determined in 
 terms of a length of the wire, and the correction is applied as 
 such. One method of determining this is to insert known 
 
 resistances Rj and Rj in the gaps; the ratio of ^ should be 
 
 R2 
 
Correction for End-Contacts. 
 
 41 
 
 large. Let the balancing-point be Bi, and let the distance 
 XBi = -Vi, also let the length XY = /, 0-1 = resistance at the 
 end, X and (t.^ = resistance at the end Y, expressed in terms 
 of a length of bridge wire. Having got a balance, R, and 
 
 Ra are interchanged, and a second balancing-point, B^, is 
 obtained, let XB., = x.. Then — 
 
 
 ^1 4- o-i 
 
 and- 
 
 R2 (/ — Xi) + 0-2 
 
 Ri (/-^,)+<r. 
 
 ^ + 0-1 
 
 R, 
 
 Calhng the ratio ^ = K, we get — 
 R2 
 
 K(JC2 + 0-1) — I — .To + <T., 
 j^(a'i + o-i) = / - ^1 -f o-a 
 
 Whence subtracting we get — 
 
 _ Xi -f K(jCi — .V,) — K"x, 
 '"' l^^i " 
 
 In a similar manner we get — 
 
 _ K(x, — atj) — /(i — K^) + X, — K^Vj 
 I — K- 
 
 \Ve thus have the eifects of the end resistances expressed in 
 terms of the effective lengthening which they produce on the 
 bridge wire. In making the above measurement it is assumed 
 
42 Practical Electricity and Magnetism. 
 
 that when taking the bridge wire readings corresponding to the 
 ratio of the resistances Rj and R2, allowance is made for any 
 irregularity in the bridge wire by the aid of a previously 
 determined calibration curve. 
 
 39. The following data were obtained in order to determine 
 the resistances of the side-pieces and end-contacts in a metre 
 bridge : — 
 
 Ri was a standard coil of 100 ohms resistance, 
 Ra was a standard coil of 10 ohms resistance. 
 
 The temperature at which the measurement was made was 
 
 1 5 '5° C, and was that at which the coils were correct. 
 
 The first balance on the wire was obtained at 91 "15 cm. 
 
 The second balance, after interchanging Rj and Rj, was at 
 
 8'9o cm. 
 
 91'iS -I- 10(91*15 -r 8-90) — 100 X 8*90 
 Hence 0-1= ^ _ ^^2 
 
 = 0*23 cm. 
 
 io(8'9o — 91'is) — 100(1 — io^)-|- 8*90— 10^ X 9i'iS 
 and <r, = 1 _ lo'' 
 
 = o'28 cm. 
 
 40. Fall of Potential Method of finding the End Resistances on 
 a Metre Bridge. — The following method may be employed 
 instead of the last, in order to determine the resistances of the 
 
 Fig. 34. 
 
 ends of the bridge. The bridge wire is connected in series 
 with a battery b of constant E.M.F., and a large resistance R 
 (see Fig. 34). The galvanometer G has one of its terminals 
 
Correction for End-Contacts. 43 
 
 connected to one of the bridge side-pieces and the other to the 
 movable tapping-contact, and its sensitiveness is so adjusted 
 that when the tapping-point Bj is about r cm. from the end of 
 the wire a large deflection is obtained. Galvanometer readings 
 are taken with the tapping-point at Bj, B., etc., points about 
 o-i cm. apart. The first reading should from time to time be 
 repeated in order to test the constancy of the current in the 
 wire. A similar set of readings are taken with the galvano- 
 meter, at tlie other end of the wire. 
 
 If curves are now plotted with the galvanometer deflections 
 for ordinates, and the tapping-points as abscissas, they will be 
 found to run as shown in Fig. 35. The curve obtained does 
 
 Deflections 
 
 P> 
 
 TappinS Points 
 
 Fig. 35. 
 
 not run to zero, but to some point, a, on the vertical scale ; this 
 is on account of the resistance of the end-contact. If we 
 project tlie curve backwards till it cuts the horizontal axis at j8, 
 tlien the length o^, measured on the same scale as tlie hori- 
 zontal axis to the right of die zero, represents the length of 
 bridge wire which would have the same resistance as that of 
 the end-contact. 
 
 This method will be found very convenient, and its accuracy 
 depends on the number of readings taken at each end of the 
 wire. 
 
 41. To determine the resistances at the ends of a bridge wire, 
 the following data were obtained from the calibration of a 
 centimetre of wire at each end of the bridge by means of the 
 fall of potential method : — 
 
44 Practical Electricity and Magnetism. 
 
 I. Left-hand end of wire — 
 
 Division of wire. 
 
 Galvanometer 
 
 deflection. 
 
 centimetre. 
 
 
 o-o-i 
 
 86 
 
 0-0-9 
 
 78 
 
 0-0-8 
 
 72 
 
 0-07 
 
 65 
 
 0-06 
 
 56 
 
 o-o-s 
 
 50 
 
 o-o'4 
 
 42 
 
 0-03 
 
 36 
 
 0-0'2 
 
 29 
 
 O-O-I 
 
 22 
 
 II. Right-hand ehd of wire — 
 
 Division of wire. 
 
 Galvanometer 
 deflection. 
 
 centimetres. 
 
 
 99-0-100 
 
 92 
 
 99-1-100 
 
 82 
 
 99-2-100 
 
 74 
 
 99-3-100 
 
 65 
 
 99-4-100 
 
 5S 
 
 99-5-100 
 
 47 
 
 99'6-ioo 
 
 38 
 
 997-100 
 
 30 
 
 99-8-100 
 
 22 
 
 99-9-100 
 
 I.S 
 
 99-95-100 
 
 8 
 
 The curves drawn from these data are shown in Figs. 36 
 and 37. 
 
 42. The Tapping Error. — A source of error is sometimes 
 introduced into bridge measurements on account of the pointer 
 which indicates the position of the tapping knife-edge on the 
 wire, not being situated immediately above the knife-edge, but 
 a little to one side or other of it ; this is known as the tapping 
 error, and may be eliminated by obtaining two balances with 
 the coils interchanged. Thus let R and x in Fig. 38 repre- 
 sent the two coils whose resistances are to be compared. Also 
 
The Tapping Error. 
 
 45 
 
 let Bi represent the balancing-point, as indicated by the pointer 
 attached to the tapping-key, and let the true point of contact of 
 
 100 
 
 t 
 
 
 
 
 
 
 
 y 
 
 ■S so 
 
 
 
 
 / 
 
 y 
 
 
 
 
 / 
 
 
 
 E « 
 
 I 
 
 4 
 
 X 
 
 ^ 
 
 
 
 
 
 
 IcAiatanc 
 
 : - -21 Cn 
 
 , of brid 
 
 e wire 
 
 o 
 
 too 
 
 
 
 
 / 
 
 Q 
 
 80 
 
 
 
 V 
 
 /^ 
 
 
 60 
 
 
 J 
 
 
 
 
 
 
 / 
 
 
 
 
 90 
 
 / 
 
 
 
 
 -5 
 
 ta / 
 
 
 
 
 
 O 
 
 5 
 
 £cai&tance - *05 cm oF brid e Vfire 
 
 Divisions on Bridge Wire (left) 
 Fig. 36. 
 
 _^ M-6 »•* S9Z 
 
 S Divisicns on Bridtfe Wire (rifht) 
 Fig. 37. 
 
 the knife-edge be at a distance t on the Y side of Bi. Then — 
 R XBi -irr_ 
 
 is the length of the bridge wire XY. 
 
 On interchanging the coils R and x, a second balancing- 
 point, B.>, will be obtained, and — 
 
 R /-(XBa-l-T ) 
 
 .V ~ XB + T 
 
46 Practical Electricity and Magtietism. 
 
 By adding together the numerators and denominators of these 
 two equations, we get — 
 
 R XBi + Z-X B, 
 a: ~ / - XBi + XB2 
 
 An expression which, it will be noticed, is free from t. So, by 
 taking the mean value of the resistance of the coil x in the two 
 positions, the tapping error, if any exists, will be eliminated. 
 
 43. The Thermo-Electric Effect. — ^When a metre bridge is 
 connected up to measure the resistance of a coil, the circuit 
 may include several junctions of dissimilar metals where thermo- 
 electric effects might occur. There are always the contacts 
 between the stretched wire and the copper end-pieces ; and if 
 the wire to be measured is not of copper, it will also introduce 
 two such junctions ; besides these, there is the junction between 
 the wire and the tapping knife-edge. Should these junctions 
 be at different temperatures, thermo-electric currents will be set 
 up in the network which will disturb the balancing of the 
 resistances. The tapping-point is one of the most likely places 
 for such an effect to occur, since it is liable to get heated by 
 radiation from the hand ; it is therefore advisable to work the 
 tapper by means of a long ebonite rod, so as to keep the hand 
 as far as possible from the point of contact. The whole of the 
 bridge should be kept as far as possible at the same tempera- 
 ture, and thus minimize as far as possible the' thermo-electric 
 effects at the other junctions. It must, however, be borne in 
 mind that the passage of the testing current itself will tend to 
 produce differences of temperature at the various junctions on 
 account of the reversible Peltier effect ; it should therefore be 
 kept small, and only allowed to flow for as short a time as 
 possible. The presence of thermo-electric effects may easily 
 be observed by opening and closing the galvanometer circuit, 
 the battery being disconnected ; if any deflection is obtained it 
 must be due to thermo-electric currents. If these effects are 
 present they may be allowed for by taking the deflection due to 
 the thermo-electric currents as the true zero on the scale, and 
 balancing so as to obtain that deflection. A more satisfactory 
 method is to reverse the battery current, and take two readings 
 
The Thermo-Electric Effect. 47 
 
 for each position of the coils, since in one case the thermo- 
 electric effects will act with, and in the other against, the battery 
 current, so that the mean of the two results will be free from 
 this error. In order to diminish as far as possible any thermo- 
 electric effect at the tapping-point, the galvanometer may be 
 replaced by the battery in the tapping-circuit ; any such effect 
 then will only increase or decrease the battery current, and the 
 balancing-point is independent of the current. An objection 
 sometimes urged against this method is, that local heating might 
 occur at the point of contact between the tapping knife-edge 
 and the wire, and so injure the latter ; this, however, is very 
 unlikely, on account of the very small value of the bridge 
 currents employed. There is, however, one point that must be 
 borne in mind if this method is adopted, namely, that if there 
 is self-induction in the circuit whose resistance is being 
 measured, the galvanometer connection must not be made 
 until after the battery circuit has been closed, otherwise a 
 swing will be obtained on the galvanometer, due to the back 
 E.M.F. of self-induction, which might be mistaken for a want 
 of balance. This effect is only instantaneous, and occurs at 
 the moment of making or breaking the battery circuit. 
 
 44. Sensitiveness of the Wheatstone Bridge. — In Vol. I. the 
 student, when making measurements of resistance with the post- 
 office bridge, would find out the best arrangement for the arms 
 of the bridge, experimentally. It can, however, be shown 
 mathematically '^ that the bridge is most sensitive when all four 
 arms are of equal resistance, and the battery and galvanometer 
 resistance each equal to that of one of the arms. This arrange- 
 ment is, of course, not always practicable, nor can we always 
 manage to have the battery and galvanometer resistance 
 variable. Generally speaking, the galvanometer will be wound 
 with two coils, one of high resistance and the other of low 
 resistance ; the unknown resistance may, however, have a value 
 halfway between these. When this is the case, it can be 
 shown ^ that the best arrangement for the bridge is to connect 
 the battery or galvanometer — whichever has the higher 
 
 ' See Gray's "Absolute Measurements," vol. i. p. 331. 
 2 See Kemp's " Handbook of Electrical Testing," p. 195. 
 
48 Practical Electricity and jSIagnetism. 
 
 resistance — across from the junction of the two higher resistances 
 in the bridge arms to the junction of the two lower resistances. 
 
 In the wire bridge, for maximum sensitiveness, the wire 
 should have twice the resistance of the coil to be measured. 
 In general, the wire has a low resistance, but this can be 
 increased by placing equal resistances in R3 and Rj, thus 
 virtually lengthening the wire, the resistances R3 and R4 being 
 expressed either in ohms or in terms of so: many centimetres of 
 bridge wire. By employing an arrangement of this kind, the 
 sensitiveness of the bridge is greatly increased, but, at the same 
 time, the range of resistances capable of being compared is 
 very much smaller, since the arrangement is equivalent to a 
 bridge wire, perhaps ten or a hundred times the length of the 
 bridge, but of which only a small portion in the middle is 
 available for tapping on. 
 
 The resistances R3 and R4 are generally wound on the same 
 bobbin, so that they are both subject to the same temperature 
 variations, and therefore the ratio of their resistances will 
 remain constant. 
 
 It may be found convenient to make a set of coils of this 
 kind for use with the bridge, one pair of 10 ohms each, one of 
 100 ohms, and one of 1000 ohms, the resistances being carefully 
 marked on them in terms of centimetres of bridge wire. 
 
 45. In making a measurement of resistance, after obtaining a 
 balance, the tapping-key should be displaced until a deflection 
 on the galvanometer is just noted ; this gives a practical test of 
 the sensitiveness of the arrangements, and may be expressed as 
 so many scale-divisions deflection per millimetre of bridge wire. 
 
 46. Measurenient of Temperature. — One of the most im- 
 portant precautions necessary in an accurate comparison of 
 resistances is that required in the measurement of temperature. 
 The temperature of each of the coils employed must be 
 accurately known ; this is liable to alter, both on account of 
 the coil being above or below the temperature of the surround- 
 ing space, and on account of the heat generated in it by the 
 testing current. This latter cause of heating may be reduced 
 to a minimum by keeping the testing current small, and by 
 only allowing it to flow for very short intervals of time. 
 
Measurement of Temperature. 49 
 
 In cases where the testing current might cause an alteration 
 of temperature of the coil, an estimate of the amount may be 
 made if we know the value of the current employed. Calling 
 C the current in ampbres flowing in the coil, and R its resistance 
 in ohms, the energy expended in the coil per second which 
 causes heating equals C^R ; this produces a rise of temperature 
 of the wire, which continues until the rate of loss of heat by 
 cooling equals the rate of production. The heat lost per 
 second by cooling being represented by (t° — /i°)SK, t° 
 and t° are the temperatures of the space outside the wire and 
 the wire itself respectively, S the surface from which the 
 radiation takes place, and K a constant depending on the 
 radiating surface, the construction, and surroundings of the coil, 
 and represents the number of units of heat dissipated per square 
 centimetre per second per degree excess temperature. 
 
 47. It is almost impossible to give data for K, since it 
 depends so largely on the construction of the coil, and should 
 be determined experimentally from a cooling curve. The 
 following numbers are given by Dr. St. Lindeck' for a 
 manganin standard coil immersed in paraffin oil, similar to 
 that described in par. 125 : — 
 
 Excess temperature 
 of wire. 
 
 1-1° c. 
 
 67° 
 
 25-0° 
 
 44-0° 
 
 00022 
 0-0035 
 0'0O4S 
 O-0OS5 
 
 In order to insure constancy of temperature of the space 
 round the coil, it should be immersed in a vessel of paraffin oil, 
 which latter may be jacketed by a much larger vessel of water, 
 the oil being kept constantly stirred, and a thermometer placed 
 in it to register the temperature. 
 
 In cases where the coil has to be kept at 0° C., it is usually 
 arranged so that it may be placed in a vessel, and packed round 
 
 ' See Electrician, vol. xxxvi. p. 509. 
 
so Practical Electricity and Magnetism. 
 
 with melting ice. Even this, Iiowever, is not always sufficient 
 to insure a temperature of o° C, since heat is liable to be 
 conducted to the coil from the outside along the heavy copper 
 connecting wires. 
 
 This imcertainty in measuring the temperature is a strong 
 argument in favour of adjusting standard coils to a temperature 
 near or a little above that of the average temperature of a 
 room, in which case the standard temperature can almost 
 always be reproduced. 
 
 48. In cases where the temperature of a coil requires to be 
 adjusted and kept constant for some time at different values, 
 the most convenient method is to employ 
 some form of automatic gas regulator or 
 I "thermostat." This may take various 
 
 TcBumtn fjj^jjjg^ jj^j jjj^j devised by Ostwald, which 
 
 the author uses in his laboratory, is per- 
 haps the most satisfactory. In this instru- 
 ment the expansion of some liquid is 
 employed to regulate the gas-supply to the 
 Bunsen burner. 
 
 The liquid is contained in a long thin- 
 walled glass vessel (a test-tube answers 
 very well), which is immersed in the water 
 in the outer jacket of the heating vessel ; 
 i/wAiwy ttL- this is connected to the side tube of a 
 
 I U-tube by means of a long small-bore glass 
 
 j.j^ tube (see Fig. 39). In the bottom of 
 
 the U-tube there is placed a little clean 
 mercury. The gas-supply enters by a glass tube, which slides 
 through a cork in the other limb of the U-tube, the end of 
 the glass tube being cut off at right angles to the axis of the 
 tube, the gas to the burner escaping through the side tube. 
 
 To set up the apparatus, the test-tube and connecting tube 
 are filled with petroleum, and connected to the U-tube by 
 means of a small piece of rubber tubing ; the cork c is then 
 removed, the space above the. mercury filled with petroleum, 
 and the cork replaced, care being taken not to admit any air 
 into the tube. As the temperature rises the petroleum expands, 
 
The Thermostat. 51 
 
 and forces the mercury up to the open end of the gas inlet 
 tube, thus diminishing the gas-supply, and lowering the Bunsen 
 flame. The temperature at which this regulation takes place 
 may be altered by sliding the gas-supply tube further in or out 
 of the U-tube. When working properly, the temperature may 
 be regulated to a fraction of a degree, and maintained at that 
 temperature for many hours. 
 
 The sensitiveness depends, to a certain extent, on the 
 diameter of the U-tube, and Ostwaldt gives 3 mm. as the best 
 size. The author, using a tube of half-inch bore, has 
 succeeded in maintaining a temperature constant to half a 
 degree Centigrade. The temperatures with which this regulator 
 may be employed cannot be very high, on account of the low 
 boiling-point of petroleum ; if a high temperature is required, 
 the petroleum may be replaced by a ten per cent, solution of 
 calcium chloride. 
 
 In addition to the regulator the Bunsen burner should be 
 provided with a by-pass, so as to relight the gas in the event of 
 its being shut off at any time by the regulator ; the flame of the 
 Bunsen should also be protected from draughts and air-currents 
 by a small metal screen. 
 
 49. Air thermometers employed in the measurement of the 
 temperatures should be graduated to -j-V" C., should be com- 
 pared with a standard thermometer, and, if necessary, correc- 
 tions applied to their readings. 
 
 50. When the final tests are being made, no two successive 
 tests should be made with a smaller interval of time than a 
 quarter of an hour between them, in order to allow the coils to 
 settle down to a steady temperature. 
 
 51. Resume. — In making an accurate measurement of the 
 resistance of a coil by means of a wire bridge, a rough measure- 
 ment is first made to determine approximately what its value is. 
 Then, if necessary, the resistances R3 and R4 of the proper 
 magnitude are placed in position, and a standard resistance 
 chosen, which is, as nearly as possible, equal in resistance to 
 the unknown coil. The galvanometer coil most suitable is 
 chosen, and connected to the bridge in the manner described 
 previously. A balance is then obtained, the battery current is 
 
52 Practical Electricity and Magnetism. 
 
 reversed, and a second balance got, thus correcting for any 
 thermo-electric effect. The standard coil and the unknown are 
 then interchanged, and a third balance obtained, this reversal 
 eliminating the tapping error if it exists, a reversal of the 
 battery current, and a fourth balance again eliminates thermo- 
 electric effects. The resistance in each case should be calcu- 
 lated, the effective lengths of bridge wire, as taken from the 
 calibration curve, being used, and the correction made for the 
 resistances of the contacts at the ends of the wire. The mean 
 of these determinations is taken as the true resistance of the coil. 
 The resistances of the standard and R3 and R4 are, of course, 
 corrected for temperature before making the calculation, and 
 the temperature at which the measurement is made is carefully 
 taken. 
 
 52. The following data were obtained in the accurate 
 measurement of a resistance. 
 
 A preliminary determination of resistance proved that the 
 resistance to be measured was approximately 100 ohms, and 
 the bridge wire having a resistance of i'738 ohms, coils 
 of resistance, approximately 100 ohms each, were introduced 
 at either end of the bridge wire. These were as follows : 
 R3 = 1 01 7 6 ohms = 585 5 "GO cm. of bridge wire, and R4 = 
 ioi'56 ohms = 5843*50 cm. of bridge wire; these being of 
 manganin, as well as the bridge wire, they had no appreciable 
 temperature correction. A previous measurement had shown 
 the end resistances of the bridge to be respectively a = 0*23 
 cm. of bridge wire, and /8 = 0*28 cm. of bridge wire. The 
 bridge wire was uniform. The standard resistance of 100 
 ohms balanced against the coil was correct for a temperature 
 of 1 5 '5° C, aud was of platinoid. 
 
 Temperature of coil during experiment = 2 2'o° C. 
 Therefore the true resistance of standard = ioo*i69 ohms. 
 
 First balancing-point i9'6o cm. 
 
 Balancing-point with current reversed ig'6o ,, 
 
 Interchanging coil and standard, balance ... 70*30 ,, 
 
 Reversing current, balancing-point 70*30 ,, 
 
The Carey -Foster Bridge. 
 
 53 
 
 Hence (i) x = £^°:i69ls843-So + 0-28 + 80-40) 
 (s855'oo + °"23 + i9'6o) 
 = lofoi ohms @ 22° C. 
 
 (2) X = i°°'i69(58 55'°o + o'23 + 29-70) 
 (S843"S0 + 0-28 + 70-30) 
 = 101-04 ohms @ 22° C. 
 Therefore the mean value is a; = 101-02 ohms @ 22° C. 
 
 The Carey Foster Bridge. 
 
 53. The method of using the wire bridge due to Professor 
 Carey Foster is specially applicable to the measurement of 
 low resistance, since the resistance to be measured is ex- 
 pressed in terms of a certain length of the bridge wire. One 
 advantage of the method is that it is independent of the 
 contact resistances at the ends of the wire; it, however, 
 assumes that the bridge wire has been carefully calibrated. 
 
 The connections for this method of using the bridge are 
 shown in Fig. 40. In the two middle gaps of the bridge are 
 
 xb-^- 
 
 '/• 
 
 B 
 
 3 
 
 B2 
 
 Fig. 40. 
 
 B, 
 
 placed two coils, A and B, of known resistance, nearly equal 
 
 to one another, and such that the ratio of ^ does not differ 
 
 from unity more than does the ratio of x to the resistance of 
 the bridge wire. A and B may with advantage be wound on 
 the same bobbin, so that they are both subject to the same 
 temperature changes. The coil x represents the resistance to 
 be measured, while D is a thick copper strap of negligible 
 resistance. The connections are made as shown, and a balance 
 is obtained at some point, B^, on the wire. The coil ^ a,nd the 
 
54 Practical Electricity- and Magnetism. 
 
 strap D are then interchanged, and a balance is again got at 
 some other point, Bo. Then we have — 
 
 / V A _ X + XB. 
 ^ ' B ~ / - XBi 
 
 where / is the length of XY ; 
 
 (^) %= ^^^ 
 
 B ~ « + (/ - XB2) 
 From (i) and (2) we get — 
 
 X = XBi - XBj 
 
 or is equal in resistance to that part of the wire between the 
 two balancing-points, the value of which may be got from a 
 measurement of the resistance of the whole wire, and from the 
 calibration curve ; the resistance of XBi — XB^ bearing the 
 same ratio to resistance of XY as the deflection produced by 
 the fall of potential between Bj and Bj does to the sum of all 
 
 Fig 41. 
 
 the deflections obtained between X and Y in the calibration 
 (see par. 34). 
 
 54. The above method of using the wire bridge is specially 
 useful in comparing standard coils, or in standardizing resistance 
 
Standardizing Resistance Coils. 
 
 55 
 
 coils, since the difference in the resistances is then very small. 
 A special form of bridge for comparing coils is shown in Fig. 
 41, and was described by Mr. F. H. Nalder in the Physical 
 Society, 1893.^ It consists of an ebonite base, on which are 
 fixed thick copper bars with mercury cups at their ends. The 
 connections being as shown in Fig. 42, AA' and BB' are 
 
 f(o ®N o. ' 
 
 ' 6 n — 
 
 Fig. 42. 
 
 connected to two coils of equal resistance wound on the same 
 bobbin. The standard coil is connected to IL, and the coil 
 to be compared with it to JJ'. The bridge wire, which is very 
 short, is mounted on an ebonite plate, and is detachable from 
 the rest of the apparatus, so that various wires of different 
 resistances may be used as occasion demands ; it is connected 
 
 ' See Electrician, vol. xxxi. p. 241. 
 
5 6 Practical Electricity and Magnetism. 
 
 to HHi. The commutator for interchanging R and X consists 
 of a circular ebonite plate, rotating about a vertical axis, to 
 which the copper connectors, which dip into the mercury cups, 
 are attached, a spring keeping them pressed down into the 
 cups. When it is desired to interchange the coils, the plate 
 is raised, then rotated through i8o° and lowered. The whole 
 instrument is very compact, and the short bridge wires greatly 
 reduce the labour of calibration. 
 
 55. Standardization of a Coil by tlte Carey Foster Bridge. — 
 In order to standardize a coil of resistance, say i ohm, by 
 means of this method, two coils of approximately equal re- 
 sistance, which should be about i ohm, are attached to AA' 
 and BB', a standard 'i-ohm coil is connected to IL, and 
 the coil to be standardized to connected to j J' ; a bridge wire 
 of low resistance, the value of which is known, and which has 
 been carefully calibrated, is used. The battery and galvano- 
 meter connections are made, and a balance is obtained at some 
 point on the wire. Should it be found impossible to get a 
 balance on the wire, then the resistance of the unknown coil 
 must differ from that of the standard by an amount greater 
 than the resistance of the bridge wire, and the coil must be 
 removed and roughly readjusted until a balance can be obtained 
 on the wire. The position of the standard and unknown coil 
 is then interchanged, and a fresh balancing-point obtained on 
 the wire. The resistance of the bridge wire between the two 
 balancing-points represents the difference in resistance between 
 the standard and unknown coil, and can, as previously stated, 
 be determined from the resistance of the whole length of the 
 wire and the calibration curve. 
 
 In making such a comparison, all the corrections employed 
 in making a measurement with the meter bridge equally apply. 
 It will be noticed that it is not necessary to know the re- 
 sistances of the two middle coils, or even their ratio, pro- 
 vided the latter keeps constant during the measurement, 
 and for this reason it is usual to wind them together on a 
 single bobbin, so that temperature changes will affect both 
 equally. 
 
 56. The following experiment was made to determine the 
 
Measurement of Very Low Resistance. 57 
 
 resistance of a piece of copper wire 64 cm. long and o'ii4 cm. 
 diameter. 
 
 The coils A and B were standard i-ohm coils of platinoid, 
 both at the same temperature, which was also the temperature 
 of the rest of the apparatus and of the copper wire, viz. 21-5° C. 
 D consisted of a very thick strap of copper, the resistance of 
 which was negligible. 
 
 First balance 50"i8 cm. 
 
 Balance with current reversed ... .. S0'I4 ,, 
 
 Resistances interchanged balance ... 49 '60 ,, 
 
 Balances with current reversed 49'59 i> 
 
 Mean balance in first case 50' 1 6 cm. 
 
 Mean balance with coils interchanged 49"S9 >> 
 
 Difference 0'57 „ 
 
 The resistance of the wire was therefore equivalent to the 
 resistance of o"57 cm. of the bridge wire. This resistance had 
 previously been determined to be i"738 ohms, and the wire 
 being of manganin, the temperature variation of resistance is 
 negligible. The ratio of the resistance of the part of the bridge 
 wire between 49*59 and 50*16 to the resistance of the whole 
 wire, was found from the calibration curve to be the ratio of 
 36 to 6683. Hence the resistance of the copper wire is — 
 
 _ 1*738 + 36 
 - 6683 
 
 = 0*00936 ohm @ 2i'5° C. 
 
 Measurement of Very Low Resistance. 
 
 57. In measuring accurately resistances below y^ ohm, it is 
 advisable to employ either the wire bridge, after the method of 
 Carey Foster, or one of the fall of potential methods to be 
 described. The first of these involves the reading of the de- 
 flections of a galvanometer, the otlier is a zero method. 
 
 In the first method, the resistance to be measured is con- 
 nected in series with a standard resistance, which should have 
 a value as nearly as possible equal to it (the method of con- 
 structing such low-standard resistances being described in 
 
58 
 
 Practical Electricity and Magnetism. 
 
 pars. 61-63), a current from a battery, which has a regu- 
 lating resistance, r, in series with it, being sent through the 
 two resistances (see Fig. 43). Wires from the ends of the two 
 resistances X and R are taken to a Pohl's commutator, K, to 
 which the galvanometer G is also connected, so that it may 
 either be placed across the ends of X or R, according as the 
 rocking lever of the key is turned to one side or the other. 
 The galvanometer G must be made very sensitive, so that 
 when connected across the ends of the smaller of the two 
 resistances a considerable deflection may be obtained, whilst 
 the current flowing through the coils must not be allowed to be 
 so large as to alter their resistance by heating. The galvano- 
 
 ■ «i/VWWVVwWW> ' 
 
 Fig. 43. 
 
 meter is connected across the ends of X, and the deflection 
 recorded. The key is then quickly turned over so as to con- 
 nect the galvanometer across the ends of R, and the deflection 
 obtained is again recorded. To insure that in the mean time 
 the current from the battery has remained constant, the reading 
 across the ends of X is repeated ; if it is still the same, the 
 current flowing may be assumed to have remained the same. 
 If the galvanometer deflections are directly proportional to the 
 currents passing through it, then the two resistances are directly 
 proportional to the deflections ; if not, the relative values of the 
 two potential differences may be obtained from the calibration 
 curve of the galvanometer. 
 
Measurement of Very Low Resistance. $9 
 
 58. The following experiment was performed in order to 
 measure the resistance of a manganin resistance coil. 
 
 The coil to be measured was placed in series with a standard 
 coil of manganin, whose resistance at 15" C. was o'ooggo ohm, 
 an additional resistance, and a secondary battery and key. The 
 galvanometer employed was a sensitive D'Arsonval instrument. 
 On completing the battery circuit, and connecting the galvano- 
 meter across the terminals of the standard coil, a deflection of 
 185 scale-divisions was obtained; while when placed across the 
 terminals of the coil to be measured, it gave 190 scale-divisions. 
 On repeating the first experiment, the deflection was found still 
 to be 185, thus showing that there had been no alteration of 
 resistance due to heating. .On consulting the calibration curve 
 of the galvanometer, it was found that deflections in the ratio 
 
 IQO , , ... ^ 0'00020I 
 
 — „- corresponded to currents m the ratio 01 -": , 
 
 185 o oooigo 
 
 X 0'00020I 
 
 Hence = -: ;— > 
 
 000999 0000196 
 
 and x= o'oioi9 ohm 
 
 The temperature during the experiment was 15 '6° C, so that 
 no correction was required for the resistance of the manganin 
 standard. 
 
 59. The other fall of potential method consists in cornparing 
 the fall of potential down the unknown resistance and standard 
 with that down a calibrated wire, the ratio of the resistances 
 being the ratio of the lengths of calibrated wire down which 
 there is the same fall of potential. A calibrated Wheatstone 
 bridge wire may be employed as follows. A battery is con- 
 nected up so as to send a constant current through a bridge 
 wire (Fig. 44), the gaps in the bridge being connected over 
 with thick copper straps, except one where the battery is in- 
 serted, and one where a regulating resistance, r^, is included. 
 The resistance to be measured and the standard resistance are 
 connected in series with a battery and regulating resistance, r^. 
 Wires from the ends of these are brought to a Pohl's commu- 
 tator, K, whilst the other terminals of the commutator are con- 
 nected, one with the end of the bridge wire at A, and the other 
 
6o 
 
 Practical Electricity and Magnetism. 
 
 through a sensitive galvanometer, G, with a tapping contact, B. 
 The currents in the two circuits must be arranged so that the 
 fall of potential down the bridge wire is greater than the fall 
 down the larger of the two resistances, X and R ; also so that 
 the value of the current in either circuit is not likely to produce 
 heat sufficient to alter the value of the resistance, and that the 
 currents in the two are in such a direction that a balance is 
 
 Fig. 44. 
 
 obtainable. The commutator K is then arranged so that the 
 ends of the resistance X are connected to A and B, and a 
 balance is obtained by adjusting the position of the contact B 
 until there is no deflection in the galvanometer; the tapping- 
 point is noted and recorded. The key is then altered so that 
 R is connected to A and B, and a second balancing-point 
 obtained. To insure that the currents have remained constant 
 during the change, the first reading is repeated. If this is the 
 same, then the resistances of X and R are directly as the 
 lengths of wire AB on which a balance was obtained for each 
 respectively, assuming the wire uniform ; if not uniform, the 
 relative lengths are obtained from the calibration curve of 
 the wire. It must be remembered in this measurement that 
 
Standards of Low Resistance. 6i 
 
 the contact resistance at the end of the bridge wire is included 
 in addition to the resistance of the portion of wire AB. 
 
 60. The following comparison of resistances was made by 
 this method. 
 
 A platinoid standard coil of o'oioo ohm resistance at iS'S" 
 C, was placed in series with the coil to be measured, the 
 rest of the connections being as shown in the above diagram. 
 A balance was obtained for the standard coil when the 
 tapper of the bridge was placed at 85 "35 cm,, and for the 
 unknown coil at 86"oi cm. The temperature of both coils 
 was i5'6° C. No temperature correction was therefore 
 judged necessary for the platinoid standard. The left-hand 
 end contact of the bridge had a resistance equivalent to o'zz 
 cm. of bridge wire. The calibration curve of the wire showed 
 that the lengths of wire, 8S'35 and 86'oi, were proportioned to 
 resistances in the ratio of 4262 and 4298 respectively. So that 
 
 X 4262 + 0*22 
 
 O'OIOO ~ 4298 + 0'22 
 
 X = o'oo99i ohm @ i5"6° C. 
 
 Ci. Standards of Low Resistance. — Standard coils of low 
 resistance for use in measurements, such as are described in 
 pars. 57-60, should be supplied with two sets of terminals, one 
 for the current entering and leaving, and the other pair— the 
 potential terminals — the resistance between which is the value 
 marked on the coil. Such resistances may be made as follows. 
 
 AC D B 
 
 C3-1 ^— E=D 
 
 Fio. 45. 
 
 A piece of manganin wire is chosen of such a diameter that 
 the largest current likely to be sent through it will not heat it 
 dangerously. From a measurement of resistance of a consider- 
 able length of the wire, the approximate length required for 
 the standard is calculated, and a piece slightly longer is cut off. 
 The ends are bared and soldered to copper plates, A and B 
 (see Fig. 45), to which the current terminals are attached. 
 
62 
 
 Practical Electricity and Magnetism. 
 
 The insulation is removed for a few centimetres at either end 
 of the wire, and a copper or manganin wire is soldered to the 
 wire AB at a point, C, near one end. Some other point, D, 
 near the other end must now be found, such that the resistance 
 between C and D is exactly the value required for the 
 resistance. To find this the calibrated wire of a Wheatstone 
 bridge is employed. The wire AB is fastened down to a 
 board, and the end A is connected to one end, X, of the bridge 
 wire (see Fig. 46), whilst the other, B, is connected to the end Y. 
 
 Fig. 46. 
 
 A current is sent through the two wires in parallel from the 
 battery b, regulated by means of the small resistance r. A 
 Pohl's commutator, K, is arranged so that the galvanometer G 
 can be placed across from C to E, the contact being adjusted 
 till there is no deflection on the galvanometer, C and E then 
 being at the same potential. From the calibration curve of 
 the wire XY, and its known resistance, a point F can be found 
 such that the resistance betw^een E and F is the value required 
 between C and D. The contact at E is then placed at F, and 
 the key K is rocked over so as to connect F with a wire which 
 makes contact with the wire AB near the end B. By shifting 
 this contact-piece about, a point, D, is found such that when 
 the galvanometer is placed across FD no deflection is obtained ; 
 the point D on the wire is marked, and the second potential 
 terminal wire is soldered to it. 
 
 62. In using such a standard, it is connected in series with 
 the resistance to be measured and the battery, etc., by the 
 
Standards of Low Resistance. 
 
 63 
 
 terminals A and B (see Fig. 47), and the fall of potential 
 between Tj and Ta is compared with that down the unknown 
 resistance; the fall of potential between Tj and T2 being 
 proportional to the resistance between C and D. 
 
 © 
 T, 
 
 © 
 
 Fig. 47. 
 
 The advantage of such a standard is that it is quite inde- 
 pendent of the resistance of connecting wires, etc. 
 
 When made of manganin it is advisable to varnish it over 
 with shellac varnish, to prevent oxidation from the air. 
 
 63. Another form of low resistance which is very easy to 
 make and adjust, and which will carry considerably stronger 
 currents than the last form, can be constructed out of a sheet 
 of manganin as follows. 
 
 A rectangular sheet of 
 manganin of proper thick- 
 ness has a series of saw- 
 cuts made in it, as shown 
 in Fig. 48, thus forming a 
 zigzag strip of manganin, 
 as shown by the shaded 
 portions. Large termi- 
 nals, A and B, are attached to the ends of the strip to convey 
 the current to it, while small potential terminals, Tj, T,, are 
 bolted to the plate at some distance from them. The resistance 
 of the strip is taken between Tj and T3. The end strips of the 
 resistance are at least twice the breadth of the others, and the 
 holes into which Tj and T2 are bolted are oblong, so as to admit 
 of a small adjustment. A preliminary adjustment is made by 
 altering the positions of Tj, T2 in the rectangular holes, the 
 final adjustment being obtained by filing the saw-cuts deeper. 
 The end strips are screwed to ebonite blocks raised above 
 the surface of the base-board, so as to allow the air to circulate 
 
 Fig. 48. 
 
64 Practical Electricity and Magnetism. 
 
 about the resistance, which may be coated over with lamp- 
 black, in order to make it a better radiator of heat. 
 
 64. Making and adjusting an Ordinaiy Resistance Coil. — 
 In making a resistance coil of an ohm or a few ohms in 
 value, a piece of well-insulated, double silk-covered manganin 
 wire is selected of a length considerably more than will be 
 required. This must first be artificially aged by heating in an 
 air bath to 140° C. for several hours; its resistance is then 
 measured roughly, and a piece cut off having a resistance 
 slightly above that required. The wire is then doubled on 
 itself and wound on a bobbin or frame of ebonite ; the ends, 
 and the part of the wire at the bend, where it is doubled on 
 itself, are left free. The ends may now be soldered to two 
 thick copper wires or terminals, and the whole wire varnished 
 over with shellac varnish and again heated in the air bath at 
 140° C. for several hours. If it is required to be very accurate, 
 the coil should be reheated several times, extending over a few 
 weeks, in order to allow it to recover from strains that may 
 have been set up in winding, and to come to a permanent state. 
 This latter condition may be found by measuring the resistance 
 of the coil from time to time against a standard coil, it always 
 being at the same temperature during the measurement, until it 
 shows no permanent alteration in resistance after heating and 
 cooling. In order to finally adjust it to the correct value we 
 may proceed in two ways. 
 
 (i) The coil should be connected up to a Metre or Carey 
 Foster bridge and placed against a standard resistance coil, 
 both coils being placed in thermostats adjusted to temperatures, 
 which in the case of the standard is that at which it is correct, 
 and in the case of the coil to be measured should be about the 
 temperature of the room {i.e. 15°-! 6° C). The tapping-contact 
 of the bridge is adjusted to the reading at which it should be 
 when the coil is exact, allowance being made for inequalities 
 in the bridge wire and for end contact resistances, etc. The 
 current is then momentarily sent through the bridge; if the 
 galvanometer deflects the coil is not exact. The free end at 
 the bend is now bared of its insulation, and the wires twisted 
 together by means of a pair of pliers till a balance is obtained. 
 
Measurement of Very High Resistance. 65 
 
 A drop of solder is then run into the twisted wire to keep it in 
 position, and the bared part carefully insulated and varnished 
 over with shellac. Should the drop of solder make the 
 resistance slightly too low, it may be readjusted by scraping 
 the wire a little, either with a knife or with sandpaper, until 
 exact balance is obtained. 
 
 65. (2) The other method of adjusting, although simpler to 
 carry out, is apt to ma,ke the coil bulky and, consequently, slow 
 to alter in temperature. 
 
 The coil is placed in a thermostat as before, regulated to the 
 temperature at which it is desired that it should bs exact. Its 
 resistance is then accurately measured on a bridge, and its 
 excess over the desired value calculated. A simple calculation 
 is now made to find the resistance of a wire which, if placed as 
 a shunt across the coil, will make it the exact value. If S is 
 the resistance of the shunt coil required, r the resistance of the 
 coil as measured above, and R the correct value required, then — 
 
 s= '-^ 
 
 r-R 
 
 The wire for the shunt coil should be of much smaller gauge 
 than the wire of the coil, and should be of manganih which 
 has been artificially aged by heating. The required length is 
 soldered on to the main terminals and wound non-inductively 
 round the bobbin. This method of shunting cannot well be 
 applied to coils above 10 ohms resistance. 
 
 Measurement of Very High Resistance. 
 
 66. In the case of resistances too high to be measured 
 accurately on the Wheatstone bridge, such, for instance, as those 
 over a megohm, special methods have to be employed and 
 special precautions taken. Also in the measurement of the 
 dielectric resistance of cables, the E.M.F. applied in the test 
 must always be slightly in excess of the E.M.F. likely to be 
 employed with the cable ; thus electric light leads intended to 
 carry current at a pressure of 100 volts say, must be tested for 
 insulation with an E.M.F. of 100 or zoo volts at least, and not 
 of I or 2 volts, since the insulation might be sufficient for a 
 
 F 
 
66 
 
 Practical Electricity and Magnetism. 
 
 low voltage, whereas it would be broken down by the greater 
 strain of the high. 
 
 The first method to be described, which is also the simplest, 
 and applicable for measuring all except exceedingly high resist- 
 ances, is that known as the direct deflection method, the con- 
 nections being as follows (see Fig. 49). A battery, B, of 
 
 suitable E.M.F., is connected to the ends of a very high re- 
 sistance, R, which should be about 10,000 or 100,000 ohms, and 
 may consist of two or more resistance boxes in series. To the 
 ends of R is connected the resistance X to be measured, in 
 series with a high-resistance galvanometer, G, and high-insula- 
 tion key, K, The deflection obtained on the galvanometer is 
 
 Fig. so. 
 
 noted. Should this not be large, the sensitiveness of the 
 galvanometer, or else the value of the resistance R, must be in- 
 creased until a good deflection is obtained : call this deflection 
 Sj. The resistance X is then removed, and a resistance box put 
 in its place, the connections being altered as in Fig. 50. The 
 
Measurement of Very High Resistance. 67 
 
 galvanometer and resistance R3 being placed across only a 
 small part, R3, of the large resistance R, by altering R3 and R^ 
 a convenient deflection may be obtained on the galvanometer ; 
 this may be arranged to be the same or nearly the same as Sj : 
 call it 82. Then, assuming the currents in the galvanometer to 
 be proportional to the deflections (the galvanometer must be pre- 
 viously calibrated, and, if necessary, the relative currents obtained 
 from the calibration curve), and calling E the E.M.F. of the 
 battery, in the second experiment the E.M.F. at the ends of 
 
 R. 
 
 the galvanometer circuit = ^^E, and hence we have, calling 
 
 R 
 
 .V the unknown resistance, and g the galvanometer resistance — 
 
 (i) 8, « —5 
 
 (2) s,«4i 
 
 R.E 
 
 ^z+g 
 
 S (R,+A -) Ri 
 
 Hence ^ = —, — ; — c-^ 
 
 S2 {x -\- g) Ro 
 
 R 8 
 and^ = J^^g•^(R,4^i^)-^iJ■ 
 It is here assumed that the E.M.F. of the battery B remains 
 constant during the change, and also that R3 + ^ is great in 
 comparison with Rj. 
 
 67. Precmitiojis . — In all high-resistance measurement there 
 are several practical points that must be attended to, or else the 
 measurements will be of little value. First the experimenter 
 must be sure that it is the dielectric resistance of the specimen 
 under test that he is measuring, and not that of the apparatus 
 employed ; consequently all keys, wires, and connections from 
 which leakage might occur, must be very carefully insulated. 
 Should wires have to stretch long distances, they must be sup- 
 ported on glass or ebonite rods, or, better still, suspended by 
 silk fibres, and in no case must the insulation of the wire alone 
 be depended upon. Keys must be carefully cleaned and dried 
 before use. The galvanometer should be well insulated from 
 earth, as should also the battery and resistance R. The 
 
68 Practical Electricity and Magnetism. 
 
 apparatus should be arranged so that the resistance X is 
 between the key and galvanometer, and the high-potential 
 terminal of the battery, as then the tendency to leakage at the 
 key and galvanometer will not be so great. 
 
 68. If the resistance to be measured is the dielectric re- 
 sistance of a long cable, it should be coiled up and placed 
 into a large metal tank filled with water, completely covering it, 
 except only the two ends, which are left outside. These ends 
 require very carefully insulating, in order to prevent surface 
 leakage. 
 
 One method of preparing the ends is to strip off 3 or 4 inches 
 of insulation from each end, and to scrape and clean the insula- 
 tion for at least 6 inches beyond that point The whole 9 or 
 10 inches is then immersed repeatedly in a bath of melted 
 paraffin wax until it is completely encased in a thick layer of 
 insulation. The wax is then partly cut off, so as to bare the 
 core at one end to make contact to the battery with ; the other 
 terminal of the resistance being the metal vessel which, through 
 the medium of the water, is in contact with every part of the 
 surface of the insulation. 
 
 6g. A much simpler and better method of preventing surface 
 leakage is that due to Mr. W. A. Price,'^ and is known as the 
 guard-ring method. Leakage can only take place between two 
 points at different potentials ; if, therefore, the outer surface of 
 
 the insulation near the ends 
 Insulation is raised to a potential 
 
 ^f'^ equal to that of the inner 
 
 coating there will be no 
 
 QuanlKina tendency for leakage to take 
 
 j.,^. jj •' place along the end surfaces. 
 
 This method of preventing 
 
 leakage in the case of a cable consists in baring the ends of 
 
 the cable for 2 or 3 inches, and then over the insulation, and 
 
 close to the end, is placed the guard-ring, consisting of a few 
 
 turns of fine bare copper wire (see Fig. 51). The connections 
 
 m the direct-deflection method of measuring the resistance are 
 
 xlii.,^A«p^r5r "'■ "'""'• P- ^°^' '^^S; also Phil. Mas., vol. 
 
 t 
 
Guard- Ring for prevaitivg Leakage. 
 
 69 
 
 then as shown in Fig. 52. The guard-rings are connected 
 directly to the positive pole of the battery, and the galvano» 
 meter, which in this case must be highly insulated from the 
 ground, is placed between the positive pole of the battery and 
 the cable. This, it will be seen, insures that the guard-rings 
 and inner core are at the same potential, and consequently there 
 will be no surface leakage over the insulation at the bared ends 
 
 of the cable. This method has a great advantage over the other 
 in being much simpler and more satisfactory. 
 
 In making the test it will probably be found that the galva- 
 nometer deflection does not remain steady, but gradually 
 diminishes ; this diminution is not due to any alteration in the 
 resistance, but to the effects of electrical absorption in the 
 dielectric, and may continue for some hours. Thus it is 
 possible, by taking readings on the galvanometer with the 
 current left on for different lengths of time, to get a number 
 of results which differ considerably from each other. Strictly 
 speaking, the true resistance should be calculated only from 
 
70 Practical Electricity and Magneiisvi. 
 
 the galvanometer deflection when it has become steady, but 
 since this may not occur for a considerable time it is usual to 
 take the deflection after the current has been running for some 
 definite time, usually one minute, this " time of electrification " 
 of the cable, as it is called, being specified along with the 
 resistance. 
 
 70. The temperature at which the test is made, is important, not 
 so much on account of the variation of the specific resistance of 
 the insulation, but on account of the large variation in the 
 electrical absorption of the dielectric with temperature, which in 
 some cases is as large as two per cent, per 1° C. between 0° C. 
 and 24° C, and is greater at lower than at higher tempera- 
 tures. The rate at which absorption goes on varies very 
 much with the material, being much larger in the case of india- 
 rubber than with guttapercha, while it is very small in the case 
 of mica. 
 
 Since the resistance of the galvanometer enters into the 
 calculation, and since it is likely to be high, the temperature 
 of the coils should be taken, in order to allow for the variation 
 of its resistance with temperature, should the resistance have 
 been measured at some other temperature. The insulation of 
 the galvanometer must be tested before making the experiment, 
 the method of doing this being given in par. 4. 
 
 71. Finally, in testing a long cable, or any resistance the 
 electrostatic capacity of which is large, the galvanometer should 
 be short-circuited during the time the circuit is being complete4 
 at K, otherwise the needle will receive a violent shock, due to 
 the electrostatic charge which the cable receives. 
 
 72. The following test was made of the dielectric resistance 
 between two coils wound on the same bobbin. The battery of 
 two Hellesen cells in series was connected across the ends of a 
 resistance of 10,000 ohms @ 15 '5° C. The galvanometer, of 
 resistance 6091 ohms @ r4° C, was .placed in series with the 
 unknown resistance across the ends of the 10,000 ohms resistance, 
 and the mean of three deflections taken was 4o'i after one 
 minute electrification. The unknown resistance was then 
 removed, and the galvanometer terminals connected across 
 I ohm of the 10,000 ohms, and the mean of three readings was 
 
L oss-of- Charge Method of measuring High Resistance. 7 1 
 
 50-5. The temperature of the galvanometer, boxes of coils, 
 and unknown resistance was 22° C. 
 
 The temperature coefficient of the 10,000-ohm coils was 
 0-026 per cent, per 1° C. ; hence the correct value of this 
 resistance = 10,017 ohms. The temperature coefficient of the 
 galvanometer was 0-38 per cent, per 1° C. ; hence its resistance 
 corrected to 22° C. = 6286 ohms. Then — 
 
 R1O2 
 
 10017 X w< 
 
 = i — ~ X 6286 - 6286 
 
 I X 40' I 
 
 = 78697000 ohms 
 
 = 787 megohms practically 
 
 73. Loss-of- Charge Method of 7neasuring High Resistance. — 
 In cases where the resistance is too high to be measured 
 accurately by means of the direct-deflection method, such as in 
 the case of a very short length of cable dielectric or the dielectric 
 resistance of a condenser, some other more delicate methods 
 have to be employed ; one of these is known as the loss-of- 
 charge method. The principle involved is briefly as follows. 
 If one plate of a condenser is charged with a certain quantity 
 of electricity, its potential rises, and the value of its potential is 
 a measure of the quantity of electricity that has been given. to 
 it. If the insulation was perfect, the charge would remain on 
 the plate, and the potential would keep constant ; if, however, 
 the charge leaks slowly away, the potential will fall, and from the 
 measured fall of potential in a given time the quantity of 
 electricity lost, and therefore the insulation resistance, may be 
 calculated. Thus let R = the dielectric resistance in ohms, 
 K = electrostatic capacity of the condenser in farads, Vj and 
 V2 the values of the potential of the charged plate at the 
 beginning and end of a time = t seconds, q = the instantaneous 
 value of the quantity of electricity in the condenser, and v = 
 the instantaneous value of the potential, then — 
 
 q = Yi-V 
 also dq = Yidv 
 
72 Practical ElectricUy and Magnetism 
 
 But by definition of current we 
 
 have — 
 
 
 
 
 c 
 
 dq 
 - dt- 
 
 V 
 
 R 
 
 
 
 Kdv 
 
 V 
 
 
 
 
 •'■ dt 
 
 = R 
 
 
 
 
 dt 
 andj,^ 
 
 dv 
 
 
 Integrating 
 
 we get — 
 
 t 
 KR- 
 
 ■ R = 
 
 log.v^ 
 t 
 
 X r 
 
 Klog,^^ 
 
 If K is given in microfarads, R will work out in megohms. 
 From this it will also be seen that, since the potential is propor- 
 tional to the quantity of electricity in the condenser, we might 
 measure the quantity of electricity in the condenser at the 
 beginning and end of the time t, instead of the potential. This 
 may be done by means of a ballistic galvanometer, the throws 
 of which may be taken as proportional to the quantity of 
 electricity passing through it, as will be shown later. Calling 
 Si and 82 the throws obtained at the beginning and end of time 
 t, we may write our formula — 
 
 R: 
 
 Kloge| 
 
 74. The measurement of the electrostatic capacity of the 
 resistance, if not too small, may be determined by comparing 
 it with a standard condenser, according to some of the methods 
 given in Chapter V. If the capacity is very small, then it may 
 not be possible to determine it in this way, and the following 
 variation of the method may then be adopted. The resistance 
 of a standard condenser is determined as above; then the 
 resistance to be measured is placed as a shunt across its 
 terminals, and the joint resistance of the two in parallel 
 measured, the electrostatic capacity being taken as that of the 
 
Correction for Leakage. 
 
 73 
 
 condenser alone. From these two measurements the unknown 
 resistance can at once be calculated.' 
 
 75. In making the measurement by observing the fall of 
 potential, an electrometer is employed, the needle being charged 
 heterostatically, the connections being as shown in Fig. 53. R 
 is the resistance to be measured, the terminals of which are 
 connected to the electrometer E, one being also earthed at e, 
 
 Fig. S3. 
 
 whilst the other is connected to the battery B through the high- 
 resistance key K, the other battery terminal being earthed. 
 
 The key K is depressed, and the electrometer reading 
 taken ; K is then opened, and time readings of the electro- 
 meter deflectiori taken. To insure that the leakage does not 
 take place at the electrometer, one pair of quadrants should be 
 charged, and then insulated, the other pair being earthed, and 
 the deflection of the needle 
 observed. If the insula- ' '^ 
 tion is good this will 
 remain constant ; if slight 
 leakage occurs, time read- 
 ings of the deflection 
 should be taken. Two 
 curves are then plotted on Y 
 the same sheet, one giv- 
 ing the fall of potential, 
 due to the combined leakage of R, K, and electrometer, and the 
 other the leakage from K and electrometer only. By adding 
 the difference between this latter curve and the initial value of 
 the potential Vj to the first curve, we get a curve representing 
 the fall due to the resistance R alone (see Fig. 54). The 
 
 "-■-, E + K 
 ~-E + K + R 
 
 Time 
 
 F,G. 54. 
 
74 
 
 Practical Electricity and Magnetism. 
 
 resistance may now be calculated by taking a number of pairs 
 of readings for V , and Vg, the value of /, the time elapsing 
 between them, being obtained from the curve. In each case, 
 also, the total time that has elapsed since insulating R must be 
 specified, in order to give some idea of the effect that electrical 
 absorption may have on the results. 
 
 The particular advantage of the method is that the progress 
 of the leakage can be watched over a considerable interval of 
 time, and a large number of data obtained from which to 
 calculate the resistance. All the disadvantages consequent on 
 the use of an electrometer are, however, attached to the method ; 
 these will be dealt with more fully later on (see Chapter III.). 
 
 Should the guard-ring method of preventing surface leakage 
 be employed, the guard-ring, after being raised to the same 
 
 potential as the core of 
 the cable, must be insu- 
 lated from it and the 
 charging battery, but 
 the paraffining will be 
 more satisfactory in this 
 case. 
 
 76. A slight modifi- 
 cation of this method 
 is sometimes employed 
 when the rate of fall of 
 potential is very small, 
 and is due to the late 
 ^"=- 55- Professor Fleeming 
 
 Jenkin, being known as the " Inferred zero method." ' 
 
 The arrangement of connections are as follows (see Fig. 55). 
 The electrometer E, used heterostatically, is made very sensitive, 
 and the defection produced by i cell of a battery, B, is noted : 
 let it be = 8,. The positive pole of the whole battery B, of 
 n cells, is then connected to terminals 2 and 3, which are at 
 first connected together, the other battery pole being earthed. 
 This brings both pairs of quadrants to the same potential, 
 which we will call Vj, and which is ;/ times that of one cell. If 
 ' See Kemp's "Electrical Testing," p. 362. 
 
Inferred Zero Method of meastiring High Resistance. 7 5 
 
 the electrometer is properly adjusted, no deflection will be 
 obtained. Terminal i is next connected to the two others, and 
 then the connection between 2 and 3 is broken, leaving 2 
 connected to i, and 3 to the battery. Leakage of the charge 
 from the pair of quadrants connected to terminal 2 now takes 
 place through R, and the potential of that pair of quadrants 
 consequently falls ; the other pair, being still connected to the 
 battery, maintain their original potential; the electrometer 
 needle therefore begins gradually to deflect. Let the deflection 
 
 at the end of /seconds be 82, then ~^:^ represents the fraction of 
 
 its original value by which the potential has fallen in that 
 time. We can therefore, knowing the value of K, the electro- 
 static capacity, determine the resistance by the aid of the 
 relation — 
 
 R = 
 
 Klog.^ 
 
 Since the rate of fall of potential is slow, the sensitiveness of 
 the electrometer can be adjusted so that, say, a variation of one 
 per cent, in the potential is represented by a deflection the 
 whole length of the scale, the true zero being a long way past 
 the end of the scale and is said to be inferred. If the cells 
 composing the battery are all in good condition, they should 
 have the same E.M.F. If great accuracy is desired, they must 
 be tested individually on the electrometer, and the E.M.F. of 
 all in series deduced from the separate deflections. 
 
 The same precautions apply to the insulation of the electro- 
 meter and keys as in the last case, and the readings may be 
 taken from curves drawn as before. 
 
 77. In measuring the dielectric resistance of a i -microfarad 
 condenser by means of an electrometer, the following data 
 were obtained. 
 
 The electrometer and key were connected to the battery, 
 and a deflection of 100 scale-divisions was obtained ; the battery 
 was then disconnected, and the following time-readings of the 
 electrometer deflection were taken : — 
 
76 
 
 Practical Electricity and Magnetism. 
 
 Time. 
 
 Electrometer 
 deflection. 
 
 minutes. 
 
 
 o 
 
 lOO'O 
 
 I 
 
 99 '6 
 
 2 
 
 99-4 
 
 3 
 
 99-2 
 
 4 
 S 
 6 
 
 99-0 
 98-8 
 98-6 
 
 The condenser was then placed in circuit, as shown in 
 Fig. 53, and the following readings were got : — 
 
 
 Electrometer 
 
 Time. 
 
 deflection. 
 
 minutes. 
 
 
 O 
 
 lOO'o 
 
 I 
 
 98-4 
 
 2 
 
 97-2 
 
 3 
 
 9>-2 
 
 4 
 
 95-2 
 
 5 
 
 94-2 
 
 6 
 
 93-2 
 
 The curves of fall of potential plotted from these readings 
 
 Q 
 
 U-1 
 
 ^ 
 
 C~--. 
 
 "^^^"^i^isj^r 
 
 *T<5r 
 
 - 
 
 
 X 
 
 ^I--^', 
 ^^'^' 
 
 
 
 
 
 % 
 
 -^ 
 
 *^ 
 
 " • -. 
 
 
 
 
 
 •S'.s 
 
 ^"■^^ 
 
 Z 3 ^ 
 
 Time in Minutes 
 
 Fig. s6. 
 
 are shown in Fig. 56, the dotted curve representing the curve 
 due of the resistance of the condenser alone. 
 
Ballistic Methods of measuring High Resistance, yy 
 
 The resistance of the condenser calculated from these 
 readings by the aid of the relation — 
 
 t 
 
 R = 
 
 Klog. 
 
 V, 
 
 are as follows ; — 
 
 Time of 
 
 Resistance in 
 
 electrification. 
 
 megjhins. 
 
 I 
 
 S5SO 
 
 2 
 
 6o66 
 
 3 
 
 7905 
 
 4 
 
 790s 
 
 5 
 
 7905 
 
 6 
 
 7905 
 
 78. Loss-of- Charge Method, using a Ballistic Galvanometer. — 
 It has been already shown that thq dielectric resistance of a 
 cable or other high resistance may be measured from a know- 
 ledge of the loss of charge in a given time, as well as from the 
 fall of potential, the measurement of quantity of electricity 
 being made on a ballistic galvanometer. 
 
 The arrangement of the apparatus for this measurement is as 
 shown in Fig. 57. G represents a highly insulated sensitive 
 
 r^^y-^\ 
 
 ^. 
 
 B 
 
 \ 
 
 Fig. 57. 
 
 ballistic galvanometer, which is connected in series with R, the 
 unknown resistance, K, a high-resistance key, and B, a battery ; 
 one terminal of the battery and one of the galvanometer being 
 earthed. 
 
 The key K is closed, and the first throw of the ^ballistic 
 needle S^, measured, this is proportional to the quantity of 
 electricity given to R. K is now opened, and the time noted. 
 
78 Practical Electricity and Magnetism. 
 
 Opening K disconnects R from the battery, and consequently 
 the charge on R slowly leaks away. After the lapse of t 
 seconds, K is again closed, and the throw on the galvanometer 
 taken : let it be S, ; this is proportional to the quantity that has 
 leaked away, consequently 8, — §2 will represent the quantity 
 of charge remaining. Determining K as before, we have — 
 
 t 
 R= . 
 
 The object of charging and recharging the resistance, instead of 
 charging and discharging, is in order to have the throws always 
 to the same side of the scale, and to a certain extent to avoid 
 some of the effects produced by electrical absorption. 
 
 All the apparatus, such as keys, galvanometer, and wires, 
 must be highly insulated. It will be found better to place 
 the galvanometer on the earthed side of R rather than between 
 R and B, since then the chances of leakage are diminished. 
 
 79. In all measurements of this kind simplicity and cleanness 
 in connections and apparatus is of the very greatest importance ; 
 elaborate and complicated keys, for instance, are to be avoided, 
 as introducing great opportunities for leakage, unless kept very 
 clean and dry. A key for a make and break, for instance, as 
 in the last experiment, cannot be surpassed for high insulation 
 by a block of ebonite into which two holes are drilled to contain 
 mercury, and connections are brought to these, and the circuit 
 may be made or broken by raising or lowering a bridging con- 
 tact of copper wire, the ends of which dip into the mercury cups, 
 and which is attached to a glass or ebonite rod by silk threads. 
 
 80. The following experiment was made in order to determine 
 the dielectric resistance of a one-third microfarad condenser. 
 Five sets of readings were taken, each set consisting of three 
 readings. In every case the condenser was connected for one 
 minute to the battery terminals before insulating ; it was then 
 insulated for a certain interval of time and recharged, the 
 resistance being calculated from the formula — 
 
 R = ' 
 
Measurement of High Resistance. 
 The battery employed was a single Hellesen cell. 
 
 79 
 
 Throw on 
 
 Time of 
 
 Throw on 
 
 Resistance 
 
 cliarge. 
 
 insulation. 
 
 recharge. 
 
 
 minutes. 
 
 
 uiegohms. 
 
 211 
 
 I 
 
 24 
 
 'S5S 
 
 ZIO 
 
 I 
 
 23 
 
 '5^^ 
 
 209 
 
 I 
 
 23 
 
 1588 
 
 216 
 
 2 
 
 45 
 
 1557 
 
 212 
 
 2 
 
 46 
 
 1506 
 
 210 
 
 2 
 
 43 
 
 1613 
 
 215 
 
 3 
 
 63 
 
 1569 
 
 215 
 
 
 63 
 
 1569 
 
 213 
 
 3 
 
 62 
 
 1569 
 
 215 
 
 4 
 
 78 
 
 1596 
 
 214 
 
 4 
 
 77 
 
 Ibig 
 
 214 
 
 4 
 
 78 
 
 1596 
 
 216 
 
 5 
 
 90 
 
 1677 
 
 214 
 
 S 
 
 89 
 
 1696 
 
 215 
 
 5 
 an value . 
 
 90 
 
 1696 
 
 Me 
 
 
 1 601 
 
 Temperature = 23° C. 
 81. The dielectric resistance of the insulation of a short 
 piece of copper wire covered with indiarubber and two layers 
 of paraffined cotton was then determined according to the 
 modification of the loss-of-charge method mentioned in par. 74. 
 The wire was 78 cm. long, and was immersed in a copper 
 vessel containing water, the ends being carefully insulated 
 and brought out of the water. The following are the means 
 of three separate readings when shunted across the condenser 
 terminals : — 
 
 Throw on 
 
 Time of 
 
 Throw on 
 
 Resistance. 
 
 charge. 
 
 insulation. 
 
 recharge. 
 
 
 minutes. 
 
 
 
 212-3 
 
 I 
 
 31-3 
 
 1 148 
 
 2iro 
 
 2 
 
 S4"o 
 
 1243 
 
 212-6 
 
 3 
 
 82-0 
 
 1 126 
 
 211-3 
 
 4 
 
 98-0 
 
 1 160 
 
 2II-0 
 
 5 
 
 113-6 
 
 1 166 
 
 Mc 
 
 an value . 
 
 
 1 168 
 
8o Practical Electricity and Magnetism. 
 
 Since -^~ 4. -= — 
 A ^ ^ B 
 
 I I i^ 
 lc~''& ~ A 
 AB 
 
 X — 
 
 A-B 
 
 1168 X 1601 
 
 ^ 433 
 
 = 4318 megohms at 23° C. 
 
 Measurement of Liquid Resistance. 
 
 82. With respect to their electrical conductivity, liquids may 
 be divided into three classes : (a) Bad conductors, such as oils, 
 etc.; ifi) conductors such as mercury and fused metals; (<r) 
 electrolytes, or those which conduct and at the same time 
 decompose, as in the case of salt solutions. 
 
 (a) In order to determine the resistance of liquids in the 
 first class, those methods which have already been described 
 for the measurement of dielectric resistance may be employed, 
 the liquid being contained either in a tube into which dip 
 electrodes, or else held as a thin film between two metal plates 
 by its own surface tension. In either case great care must 
 be taken in the measurement of the cross-sectional area of 
 the liquid through which the current flows. This will be 
 discussed later, in the chapter dealing with Specific Resistance 
 Measurement. 
 
 {!}) Good conducting liquids, like mercury, present no par- 
 ticular difficulty in the measurement of resistance, except that 
 of the exact measurement of the length and area of the column 
 tested. 
 
 (c) The last class of liquids — electrolytes — however, present 
 certain difficulties which prevent ordinary methods of measure- 
 ment from being applicable. When a current is sent through 
 an electrolyte the liquid decomposes, and the polarization at 
 the electrodes sets up a back electromotive force, which, if 
 ordinary methods of measurement were employed, would be 
 measured as a resistance ; consequently it becomes necessary 
 
Measurement of Electrolytic Resistance. 8i 
 
 to adopt some method in which polarization is either prevented 
 from taking place or else is allowed for. An example of the 
 latter method has been given in Vol. i. p. 56. It is not, 
 however, very reliable, since the assumption that the polariza- 
 tion remains constant during the various measurements is found 
 not always to be borne out in practice. 
 
 83. Regarding the methods in which the polarization is 
 prevented from taking place, the best known is that due to 
 Kohlrausch.^ In this method the liquid is contained in a tube, 
 into which suitable electrodes are attached. This is connected 
 up to a metre wire bridge, just like an ordinary resistance; 
 instead, however, of using a battery to send a continuous 
 current through the bridge, an alternating current is used, this 
 being usually taken from the secondary of a small induction 
 coil. Currents in opposite directions following one another 
 sufficiently rapidly are found to neutralize each other's polarizing 
 effect. A balancing-point on the stretched wire is obtained, 
 as in the other case, only this cannot be found by using a 
 galvanometer, the current being alternating, not continuous ; 
 a telephone, therefore, is used instead. There should, theo- 
 retically, be silence in the telephone when the balance is 
 obtained. 
 
 The objection to the method is that the ohmic resistance 
 is not the only determining factor in the balance, the self- 
 induction of the various arms of the bridge also having an 
 effect; the resistances might all be the same, yet if the self- 
 inductions of the different arms were not the same, no point 
 of absolute silence in the telephone would be found. The 
 electrostatic capacity of the arms also has an effect, but in 
 the opposite direction to that of self-induction, the two effects 
 tending to counteract one another. In practice, the balancing- 
 point is one where the sound in the telephone is a minimum. 
 
 Again, the induction coil employed to send the alternating 
 current through the bridge must have the condenser attached 
 to it removed, otherwise the currents in reverse directions will be 
 unequal, and consequently polarization will occur. Kohlrausch 
 
 ' ^OSS- Ann., vol. cxxxviii. ; also Kohlrausch, " Physical Measure- 
 ments," p. 317. 
 
 G 
 
82 
 
 Practical Electricity and Magnetism. 
 
 employs a magnet rotating inside a coil of wire to produce the 
 current; and he also finds that large electrodes are necessary 
 to get consistent results. 
 
 84. A modification of Kohlrausch's method, which has been 
 employed successfully at Cambridge/ employs a galvanometer 
 instead of a telephone, but arranged with a commutator so that 
 the galvanometer terminals are reversed at the same instant 
 that the battery current is reversed ; this insures the current in 
 the galvanometer always flotving in the same direction. The 
 difficulty with respect to the self-induction is got over by 
 arranging that the balance is at the same point on the wire at 
 all speeds of reversal of current. 
 
 85. Fall-of-Poteiitial Method. — ^A very satisfactory method of 
 measuring liquid resistance, which is free from many of the 
 objections to which the previous methods are open, is the fall- 
 of-potential method, due originally to Branly. 
 
 Fig. 58 represents the form of apparatus employed by the 
 
 V| 
 
 J" 
 
 Fig. 58. 
 
 author. T ■ represents the tube containing the liquid whose 
 resistance is to be measured. This may consist of a graduated 
 burette tube with the ends cut off. At each end are fitted on 
 glass vessels, V, Vi, also filled with the liquid, and into which 
 dip the electrodes E, E. There are two openings near the 
 bottom of the glass vessels, which are opposite one another. Into 
 one the end of T is fixed by means of a cork, while into the 
 other passes a fine glass tube, g, closed at one end and filled 
 with mercury. These tubes can slide in and out of T, and one 
 is shown enlarged in Fig. 59, contact being made between the 
 
 ' See B. A. Report, 1886, p. 328. 
 
Measurement of Electrolytic Resistance. 
 
 83 
 
 mercury in the tubes and the hquid by means of the platinum 
 wire P fused through the glass. The wires dipping into the 
 mercury are taken to a key, K (see Fig. 60), by means of which 
 
 nercun 
 
 0' 
 
 m 
 
 > p 
 
 s 
 
 Fig. 59- 
 
 they may be connected to the quadrants of the electrometer E. 
 The electrodes are connected to a battery, B, in series with a 
 
 ■III 
 
 I— ^vwwwO ^^ 
 
 (L£ 
 
 W= 
 
 ^ 
 
 U 
 
 Fig. 60. 
 
 known resistance, R (the ends of which are also connected to 
 the key K), and a break-circuit key. 
 
 On completing the battery circuit, a steady current may be 
 
84 Practical Electricity and Magnetism. 
 
 maintained in the tube T. The glass tubes g, g are now 
 adjusted with their ends at a measured distance apart, and on 
 connecting them to the electrometer E by means of the key K, 
 a deflection is obtained which is proportional to the difference 
 of potential between them. The key K is then altered so that 
 the P.D. at the ends of R is measured on the electrometer, 
 the ratio of the deflections in the two cases being proportional 
 to the ratio of the resistances of the liquid between the tubes 
 g g, and of R. The resistance R being known, that of the 
 liquid may be calculated. Since no current is taken from the 
 platinum electrodes of g, g, there will be no polarization. 
 The tube T during the measurement should be placed in a 
 water bath or thermostat and its temperature recorded. 
 
 Battery Resistance. 
 
 86. The measurement of the internal resistance of a battery 
 is a problem closely related to the measurement of liquid re- 
 sistance. It is, however, much more difficult to get consistent 
 results here than in the previous measurements. 
 
 The resistance of a battery has no real meaning unless the 
 current which the battery is supplying is also specified, since it 
 is a well-known fact that the resistance varies with the current 
 supplied, decreasing as the latter increases. As in illustration 
 of this Professor Carhart found ^ that in the case of a Gassnier 
 cell the resistance fell from lo ohms to 2*5 ohms for an 
 increase of current from 0*02 ampferes to 0*2 amperes. 
 
 Then, again, when a current is taken from a battery, the 
 effect of polarization is to lower the E.M.F. of the cell, thus 
 increasing the calculated value of the resistance in methods 
 which depend on the measurement of the relative values of the 
 P.D. at the terminals of the cell when supplying different 
 currents. Consequently, the resistance of a cell not in use is 
 very different from that of the same cell when supplying a 
 current. We must therefore, in measuring the resistance, care- 
 fully specify the conditions under which the battery is working 
 when the measurement is made.^ 
 
 ' See Physical Review, vol. ii. No. 5 ; also Electrician, vol. xxxv. p. 18. 
 ' See Electrician, vol. xxxi. p. 262. 
 
Measurement of Battery Resistance. 
 
 85 
 
 The most satisfactory method of measuring battery resistance 
 is that known as the condenser method. A condenser in series 
 with a ballistic galvanometer is charged from the terminals of 
 the battery when the latter is on open circuit. The first throw 
 of the ballistic needle (Sj) is, therefore, proportional to the 
 E.M.F. (E) of the cell. The battery terminals are now con- 
 nected by a resistance (;■), and the condenser again charged 
 from the battery terminals. Let the second ballistic throw be 
 8^. Then we have — 
 
 (i) 81 oc E 
 (2) 82 oc P.D. 
 
 r-^b- 
 
 where b = battery resistance, therefore from (i) and (2) we 
 get— 
 
 ^^ ;-(8i;-8g) 
 
 87. In practice it is preferable to take the P.D. reading first, 
 and then the open circuit reading immediately on breaking 
 the circuit r, so as to get the value of the E.M.F. as reduced 
 
 Fig. 61. 
 
 by polarization, and therefore the actual E.M.F. in the circuit 
 during the passage of the current. Fig. 61 shows the arrange- 
 ment of apparatus. G represents the ballistic galvanometer in 
 series with the condenser C, the key K^, and the battery B. 
 A separate circuit from the battery goes to the resistance r in 
 series with the key Ki. In making a measurement, Kj is first 
 depressed and then K,, the throw on the ballistic galvanometer 
 
86 
 
 Practical Electricity and Magnetism. 
 
 being recorded. K, is then opened, and the condenser short- 
 circuited for an instant by means of its short-circuit plug. Kj 
 is opened and Kg again depressed, thus getting the throw due 
 to the condenser being charged from B on open circuit. The 
 cell B is now removed, and replac'ed by a standard cell, the 
 condenser being charged from this cell on open circuit only j 
 from the galvanometer throw due to the standard cell the scale 
 of the galvanometer may be calibrated to read in volts, assuming 
 the throws are proportional to the E.M.F.'s. In this way, know- 
 ing the P.D. in volts, and the value of ;■ in ohms, the current 
 in ampbres supplied by the cell during the experiment may be 
 estimated. 
 The results are then tabulated thus — 
 
 Temperature 
 of cell. 
 
 8, 
 
 S., 
 
 Current in 
 amperes. 
 
 (ohms) 
 
 The numbers in the second last column being = 
 
 S2(in volts) 
 
 A curve should now be drawn, showing the connection 
 between the resistance of the cell and the current supplied 
 by it. 
 
 A separate set of experiments should be made to determine 
 the effect on the throw 83, of keeping the key K, depressed for 
 various intervals of time before taking the P.D. reading, and 
 also on the rate of recovery of the E.M.F. after the circuit Kj 
 has been broken. 
 
 88. The following data were obtained from the measure- 
 ment of the resistance of a freshly made-up Daniell cell. 
 
 The E.M.F. of the cell was i-io volts. 
 
Measurement of Galvanometer Resistance. 
 
 87 
 
 Temperature 
 
 K 
 
 8, 
 
 
 b 
 
 Current in 
 
 of cell. 
 
 
 (ohms). 
 
 amperes. 
 
 i6° C. 
 
 123 
 
 23 
 
 I 
 
 4-2 
 
 0-2I 
 
 1 6° 
 
 123 
 
 3i 
 
 2 
 
 4-8 
 
 o-i6 
 
 i6° 
 
 120 
 
 45 
 
 3 
 
 50 
 
 014 
 
 1 6° 
 
 I20 
 
 50 
 
 4 
 
 5-6 
 
 OH 
 
 l6° 
 
 130 
 
 63 
 
 5 
 
 5-8 
 
 O'lO 
 
 i6° 
 
 1 20 
 
 65 
 
 9 
 
 7-6 
 
 o65 
 
 i6° 
 
 130 
 
 70 
 
 10 
 
 8-5 
 
 o'o6 
 
 1 6° 
 
 130 
 
 75 
 
 15 
 
 no 
 
 004 • 
 
 __ 
 
 
 
 
 
 
 _. 
 
 The accompanying curve (Fig. 62) shows the relation between 
 the internal resistance of the cell and the current flowing 
 through it. 
 
 •25 
 
 
 
 < 
 
 
 
 
 JJ.20 
 
 u 
 
 s 
 
 
 
 \ 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 1" 
 
 
 
 
 
 
 
 3(<i 
 U 
 
 
 
 
 
 
 
 Battery Resistance 
 
 Fig. 6z. 
 
 Galvanometer Resistanci:. 
 
 89. To measure this important quantity various methods 
 have been devised, the principal object being to use the 
 deflections of the galvanometer itself, to furnish the data from 
 which to calculate its resistance, so as to avoid the necessity of 
 having to employ a second galvanometer. In general, however, 
 it will be found that these methods do not give consistent 
 results, the only satisfactory method being to measure it as an 
 ordinary wire resistance on a Wheatstone bridge. Before 
 
88 
 
 Practical Electricity and Magnetism. 
 
 making the measurement it is necessary to first remove the 
 suspended needle in the case of a needle galvanometer, or to 
 fix rigidly the suspended coil in the case of a D'Arsonval 
 galvanometer, as in the first case the magnetic effect of the 
 testing current might be sufiScient to reverse the polarity of 
 the needle, and in both cases a very severe strain would be 
 thrown on the suspension, due to the relatively large testing 
 current. The temperature of the coils must be determined at 
 the time of the test. If the galvanometer is supplied with a 
 shunt box, the resistances of the shunt coils should also be 
 determined. To insure the temperature of the shunt coils 
 and galvanometer coils being the same, they should be placed 
 together under a glass shade for some hours before a measure- 
 ment is made. Measurements of this kind should be made on 
 different days, when the temperature is different, in order to 
 determine the temperature coefficient of the galvanometer and 
 shunt coils (see par. 98), but in every case the galvanometer 
 and shunt coils must be kept under the glass shade for at least 
 an hour, at a constant temperature before the measurement is 
 made. 
 
 All the precautions already given in connection with the use 
 of the Wheatstone bridge apply in this case. 
 
 The results may then be tabulated as follows : — 
 
 Temperature. 
 
 Galvanometer 
 resistance. 
 
 Shunt resistance. 
 
 Date. 
 
 Time. 
 
 and from these the following table calculated : 
 
Measurement of Galvanometer Resistance. 89 
 
 Mean temperature coefficient 
 of galvanometer. 
 
 Mean temperature coeffi- 
 cient of shinit coils. 
 
 Ratios temperature coeffi- 
 cient of galvanometer to 
 temperature coefficient of 
 shunt coils. 
 
 
 T3 
 
 T0J 
 
 ■im 
 
 v. 
 
 ife 
 
 TDOO 
 
 
 
 
 
 
 
 
 If the material employed in winding the shunt coils is the 
 same as that on the galvanometer, the ratios will be unity ; if 
 they are of diiferent materials a knowledge of the temperature 
 coefficients is necessary in order to calculate the value of any 
 particular shunt at any stated temperature. 
 
 90. The following data have been calculated from measure- 
 ments of resistance of a 10,000-ohm galvanometer and shunt 
 box on dififerent occasions at different temperatures. 
 
 
 Galvanometer 
 resistance. 
 
 Shunt resistance. 
 
 
 ^ 
 
 i 
 
 TOJ 
 
 10° c. 
 
 12° 
 I^° 
 20° 
 
 ohms. 
 I03S9-6 
 10459-4 
 10589-4 
 10789-2 
 
 1154-4 
 1 163-3 
 II76-6 
 1198-8 
 
 104-93 
 
 J0573 
 106-94 
 108-96 
 
 10-40 
 10-48 
 
 io'6o 
 
 IO-83 
 
 from which data we obtain the following : 
 
 Mean temperature coefficient 
 of galvanometer per cent. 
 per 1° C. 
 
 Mean temperature coefficient 
 of shunt coils per cent, per 
 i°C. 
 
 Ratios temperature coeffi- 
 cient of galvanometer to 
 temperature coefficient of 
 shunt coils. 
 
 
 TU 
 
 TOj 
 
 1 
 
 t'5 
 
 Tn3 
 
 TTiiJlI 
 
 0*40 
 
 0*40 
 
 0-40 
 
 o'40 j I 
 
 X 
 
 I 
 
 The above shows that the copper of which the galvano- 
 
go Practical Electricity and Magnetism. 
 
 meter and the shunt were wound had the same electrical 
 properties. 
 
 Specific Resistance. 
 
 91. The resistance (R) of a substance in the form of a rod 
 or wire is directly proportional to its length (/), inversely as its 
 cross-sectional area (A), and directly as some constant (p), the 
 value of which depends upon the nature and physical con- 
 dition of the material. This constant p is known as the 
 specific resistance of the material, and may be defined as the 
 ohmic resistance, between the opposite faces of a centimetre 
 cube of the substance. The above relations may be expressed 
 algebraically thus — 
 
 and p= - ,- 
 
 In order to determine the constant p of any material, we 
 measure the resistance of a specimen of known length and 
 constant known area, and calculate as above. The measure- 
 ment of resistance may be made by means of a AVheatstone 
 bridge, all the ordinary precautions necessary for accuracy 
 being taken, and the temperature at which the measurement 
 was made also being specified. The specimen of material if 
 insulated wire must first have the insulation removed from it ; 
 this had better be done by hand, as any scraping is apt to 
 injure the uniformity of the wire ; the wire should then be 
 cleaned by rubbing with a dry cloth before being connected to 
 the bridge. The length must be measured between the points 
 where the bridge terminals clamp it. To do this a fine scratch 
 should be made at these points, and the wire removed from the 
 bridge after the resistance has been measured, and stretched 
 over a scale, in order to measure the exact length between the 
 scratches, care being taken in the case of fine wires not to 
 lengthen it by stretching. The ends are then cut off at the 
 scratches. The determination of the area may now be made 
 in two ways — either by measuring the diameter {d), and calcu- 
 lating the area from the relation A = -^, or else it may be got 
 
 4 
 
Measurement of Specific Resistance. 91 
 
 from the length, weight, and specific gravity of the wire. This 
 latter method should be adopted, the first being, if necessary, 
 used to check it, as there is a connection between the specific 
 gravity of a substance and its specific resistance, so that the 
 specific gravity would have to be determined in any case, this 
 being done by weighing in air and water. The wire is coiled 
 up into a loose spiral, and suspended from the beam of a 
 balance by means of a fine glass fibre, the weight of which has 
 been counterbalanced, and the weight of the wire taken. The 
 coil of wire is then plunged into a vessel of boiling distilled 
 water, which, after all air bubbles have come off, is allowed to 
 cool down. When near the temperature of the room the w^ater 
 vessel is supported under the beam of the balance, so as to 
 allow the wire to be weighed when immersed in the water : the 
 temperature of the water must be taken when the weight in 
 water is determined, and this should be as near as possible to 
 the temperature at which the resistance measurement was made, 
 the weight of the partially immersed glass fibre being counter- 
 balanced in a preliminary experiment. 
 
 The specific gravity is then calculated from — 
 
 weight in air 
 specific gravity = weight in air - weight in water 
 
 Should the temperature of the water not be that of its maximum 
 density point, 4° C, the above calculation must be multiplied 
 by the density of water at the temperature at which the 
 weighing was made. Corrections for the different air dis- 
 placements of the weights and wire are very small, and 
 except in cases where very high accuracy is required, may be 
 neglected. 
 
 Then if W = weight of wire, V — its volume, and A = its 
 specific gravity — 
 
 W = VA 
 = /AA 
 
 • • ^ - /A 
 
92 Practical Electricity and Magnetism. 
 
 Combining this equation with the resistance equation, we get 
 for the specific resistance — 
 
 RW 
 
 92. To measure the diameter of a wire directly, the most 
 satisfactory method is by means of the micrometer microscope. 
 A small scale engraved on glass is introduced in front of the 
 eye-piece, so that a slightly magnified image of it is seen, and 
 the relative size of any object looked at through the microscope 
 may be expressed as so many divisions of this magnified scale. 
 To get the absolute size of the object the magnified scale- 
 divisions have to be calibrated, this being done by placing on 
 the stage of the microscope a piece of glass on which a number 
 of lines at a known distance apart, say -jyo or yoVo '^™-> '^^^^ 
 been ruled by a dividing engine, and seeing how many of these 
 divisions correspond to a division on the eye-piece scale, the 
 magnifying power of the microscope being kept the same. 
 
 In a piece of wire 100 cm. long, readings of the diameter 
 should be taken every 2 or 3 cm. along its length, and the 
 mean diameter used in the calculation for the area. 
 
 93. The following measurement of specific resistance of a 
 piece of manganin wire was made : — 
 
 A piece of No. 20 S.W.G. silk-covered manganin wire, about 
 100 cm. long, was cut from a large coil, the insulation was 
 carefully removed, and the wire soldered to thick copper leads. 
 The exact length was then carefully measured, and found to be 
 97 cm. The resistance measured on a Wheatstone bridge, 
 after making all necessary corrections, was found to be 
 0-627 ohms @ 9 '8° C. The specific gravity was determined 
 by weighing in air and water, and was equal to 8' 50, the 
 weight of the wire being S'iso gms. in air. This gave a mean 
 area of 0-062 sq. cm. for the cross-section of the wire, and 
 a specific resistance of 0-00040. 
 
 94. Specific Dielectric Resistance. — In the measurement of 
 specific dielectric resistance, the calculation would be similar to 
 that for a wire, if a rod or slab of material is employed in 
 the measurement, but in the case of a wire covered with 
 insulation, the calculation becomes more difficult. The 
 
Measurement of Specific Dielectric Resistance. 93 
 
 r = 
 
 insulation may be considered as a cylinder surrounding the wire. 
 
 Let the circle A represent the external circumference of the 
 
 ring (see Fig. 63), i.c. the outside of the 
 
 insulated wire, and let r^ be its radius 
 
 (B represents the conductor of radius r^, and 
 
 let c be any layer of insulation of infinitely 
 
 small thickness = dr, and radius r, then, 
 
 calling p the specific resistance, and / the 
 
 length of the cable, since — 
 
 / 
 
 we have for the resistance of the layer ,: — 
 
 pdr 
 X 
 pdr 
 ~ 2Trrl 
 
 but the total resistance of the dielectric is made up of an 
 infinite number of such layers in s'eries, therefore — 
 
 „ ^''^ 
 R = 
 
 or p = 
 
 95. In the case of the measurement of the dielectric resist- 
 ance of the insulation of a length of copper wire, see par. 81. 
 The length of the specimen was 78 cm. and the internal and 
 external radii of the insulating covering were o'o635 cm. and 
 o'i27 cm. respectively. The dielectric resistance was found to 
 be 4318 megohms at 23° C. 
 
 2ir/R 
 Hence p 
 
 log, 
 
 2 X 3"i42 X 78 X 4318 X 10° 
 
 loge: 
 
 = 3 X 10'^ ohms per cub. cm. at 23° C. 
 
94 Practical Electricity and Magnetism. 
 
 96. Specific Resistance of Liquids.— \r\ the determination 
 
 of the specific resistance of liquids, the form of apparatus 
 
 and method employed will be one of those described in 
 
 pars. 82-85. If the liquid is contained in a glass tube, such 
 
 as that shown in Fig. 58, the length between the platinum 
 
 points at the ends of the tubes g, g must be measured, also 
 
 the mean area of the tube between these points must be 
 
 found. If, as suggested, the tube is a part of a burette, then 
 
 the mean area may be found from the graduations, but if not, 
 
 or if it is desirable to measure it directly, the tube T is 
 
 disconnected from the end vessels, one end is corked up, and 
 
 a little mercury poured into the tube, which is then weighed. 
 
 The tube is placed vertically, and the level of the mercury 
 
 noted J more mercury is now poured in, care being taken to 
 
 avoid air bubbles and to keep the temperature constant ; the 
 
 increase in level (/) is measured and the increase in weight iw) 
 
 found. Calling A the specific gravity of mercury at the 
 
 temperature of the experiment, the mean area A of the part of 
 
 W 
 the tube occupied by / cm. of mercury is A = ^. Proceedmg 
 
 in this way until the tube is completely filled, the mean area 
 may be determined for various parts of the tube. 
 
 In all cases of the measurement of specific resistance of 
 liquids, full particulars regarding the preparation of the solution 
 should be given — as, for instance, whether chemically pure or 
 commercial salts were employed, ordinary or distilled water 
 as solvent, and the percentage composition of the solution by 
 weight or volume, also the density of the solution and the 
 temperature at which the measurement was made. 
 
 97. In the following measurement of the specific resistance of 
 a solution of potassium chloride, the strength of which was o'z 
 molecular equivalent per 1000 cub. cm. distilled water, the 
 burette graduations for i cub. cm. were i cm. apart, so that 
 the mean area of the tube was i sq. cm. The electrodes 
 connected to the electrometer were placed 20 cm. apart, and 
 the resistance in series with the liquid was 900 ohms. A 
 Hellesen dry battery consisting of four cells in series was 
 employed. 
 
Variation of Specific Resistance with Temperature. 95 
 
 The circuit was completed, and the electrometer deflection, 
 when placed across the ends of the movable electrodes, was 
 279 scale-divisions; when placed across the 900-ohm resist- 
 ance, it was 270 scale-divisions. On repeating the first deflec- 
 tion it was found to have remained constant. 
 
 Hence^ = ^ 
 goo 270 
 
 X = 930 ohms at 18° C. 
 RA 
 
 930 X I 
 
 20 
 
 = 46'5 ohms per cub. cm. at 18° C. 
 
 The temperature was maintained constant by placing the tube 
 containing the liquid in a thermostat regulated to 18° C. 
 The 900-ohm coil was of manganin, and had no temperature 
 correction. 
 
 Variation of Specific Resistance with Temperature. 
 
 98. In general, when the temperature of a conductor is 
 altered, there is an alteration in its specific resistance, along 
 with alterations of various other physical properties. In order 
 to determine the variation of the specific resistance with tempe- 
 rature, we measure the resistance of a substance at various 
 temperatures, and from these data we can calculate the various 
 values of the specific resistance. In cases where very great 
 accuracy is required, we must take into account in the calcu- 
 lation the fact that both / and A are varying with temperature, 
 since our specific resistance is expressed in terms of a constant 
 volume, I cub. cm. of the substance, whereas the mass only is 
 constant. To do this we would require a knowledge of the 
 coefficient of expansion of the substance with heat, or else we 
 would require to determine the dimensions of the specimen at 
 each different temperature. In the majority of cases, however, 
 on account of the exceeding smallness of the change of volume 
 
g6 Practical Electricity and Magnetism. 
 
 with temperature within the usual ranges of temperature likely 
 to be met with, this correction is unnecessary. "^ 
 
 In making the measurement of resistance, the Wheatstone 
 bridge method had best be employed. The chief difficulties, 
 in addition to those ordinarily met with in bridge work, are (j) 
 the accurate measurement of the temperature of the coil under 
 test, and (2) in allowing for the variation of resistance of the 
 unequally heated leads connecting the coil with the bridge. 
 
 99. In designing the coil of wire to be tested, certain points 
 have to be borne in mind. The mass of wire employed in the 
 coil should be as small as possible, so that it may rapidly arrive 
 at the temperature of the surrounding space ; the wire should 
 not be wound close on a massive bobbin, but should be in an 
 open spiral, or wound into the form of a ring of large diameter, 
 tied loosely together with silk thread, the winding of course 
 to be non-inductive. The advantage of a shallow ring over a 
 long spiral is that in the former every part of the wire is nearly 
 at the same level in the heating vessel, this being important, 
 since the temperature often varies considerably at different 
 depths. 
 
 The wire is cut about 2 cm. longer than is required, and i 
 cm. at each end is soldered to the ends of thick copper leads 
 (No. 12 S.W.G.). The exact length of the wire between the 
 soldered joints is then very carefully measured, and the bared 
 part of the wire at the joint coated over with shellac varnish or 
 other insulating covering which will not be affected by the 
 highest temperature to be used. A second pair of thick copper 
 leads, of exactly the same length, are cut from the same material 
 and soldered together at one end, these being used as com- 
 pensating resistances ; the method of connecting up will be 
 shown presently. 
 
 100. The heater consists of a vessel containing paraffin oil, 
 placed inside and jacketed by a much larger vessel of water, 
 which can be heated by means of a Bunsen burner, to which a 
 sensitive regulator is attached. The large mass of water in- 
 sures slow variation of temperature, and it is therefore possible 
 
 ' See Dewar and Fleming on " Measurement of Resistance at Very Low 
 Temperatures," Electrician, Oct. 7, 1892. 
 
Variation of Specific Resistance with Temperature. 97 
 
 to maintain the temperature steady for long periods of time. 
 The heating vessel must be placed sufficiently far away from 
 the rest of the apparatus to prevent the latter from being heated 
 by radiation. 
 
 The following is the diagram of the connections (see Fig. 64). 
 S represents the standard coil, X the test coil, and C the com- 
 pensating leads ; these are the same length as the leads going to 
 X, and dip into the heater to the same depth, consequently, 
 being on the opposite arm of the bridge, and being under 
 exactly the same physical conditions, they compensate for the 
 leads at X at all temperatures. This is similar to the method 
 employed by Callendar in his platinum thermometer. If it is 
 
 r^ — h r? 
 
 Fig. 64. 
 
 desired to interchange S and X, the compensating leads must 
 also be changed to the opposite end of the bridge. 
 
 In all measurements the temperature must be kept constant 
 for at least a quarter of an hour before a measurement is 
 made, and no two consecutive measurements should be made 
 within this interval of each other. 
 
 loi. In obtaining the variation of resistance by this method, 
 it is assumed that the resistance of the bridge wire is known, 
 and its calibration curve has been obtained. Then, if S is kept 
 at a constant temperature, various balancing-points will be 
 found on the wire corresponding to different temperatures of 
 the coil in the heating vessel ; from the known resistance of the 
 wire, the calibration curve, and the measured displacement of 
 the tapping- contact, the increase in resistance for a given 
 increase of temperature can be calculated. The resistance of 
 the coil and leads is taken at the temperature of the room, and 
 also the resistance of the leads alone at the same temperature ; 
 
 H 
 
98 
 
 Practical Electricity and Magnetism. 
 
 by subtracting this latter value from the former, we obtain the 
 resistance of the coil alone at the temperature of the room, and 
 by adding on the increments of resistance at the different 
 temperatures as determined above, the actual resistance of the 
 coil at the different temperatures can be found, from which the 
 specific resistance is calculated. 
 
 A curve should now be drawn with values of p for ordinates 
 and temperatures for abscissae ; from this curve the law of 
 variation of specific resistance with temperature may be 
 deduced. This law may be expressed in general by the follow- 
 ing relation : — 
 
 PP = Pa (l ± a/° + /3/^) 
 
 where pt = specific resistance at temperature t° C, po = specific 
 resistance at o° C, « and /3 are coefficients which depend on 
 the material of the wire, and may be either positive or negative 
 according to the nature of the material, /3 being always an 
 exceedingly small quantity, and representing the variation of a 
 with temperature. Provided y8 is large enough, it may be 
 deduced from a very carefully plotted curve, thus — 
 
 /Specific 
 
 Temperature. 
 
 Fic. 6s. 
 
 From the temperature scale erect a number of perpendiculars, 
 AB, CD, EF, etc. (see Fig. 65), equidistant from one another. 
 Measure the differences (AB - CD), (CD - EF), etc., and 
 plot these against the mean temperature between A and C, C 
 and E, etc. From this curve, which represents the variation of 
 a with temperature, and which will be practically a straight 
 
Variation of Specific Resistance with Tewperatui'e. gg 
 
 line, determine /3. Putting this value of ^ into the equation 
 pf = Po(i ± at ± pt"^), from two values of p/ we can calculate a. 
 The value of a is positive if the specific resistance increases, 
 and negative if it decreases with temperature. Similarly, (3 is 
 positive if a increases with temperature, and negative if it 
 decreases. 
 
 102. For most substances it is sufficient to calculate a only, 
 from two values of p. 
 
 pt° I + n^° 
 
 pt^ I + 0^2 
 p4 ~ ph 
 
 It has been shown by Mattheissen ' that for all pure metals, 
 except iron and thallium, a has a value about o"oo366. 
 
 103. More recently, Dewar and Fleming,^ in a research on 
 the variation of specific resistance of metals and alloys at low 
 temperatures, in which they employed temperatures down to 
 — 200° C, have shown, that in all pure metals the curves of 
 temperature variation of specific resistance tend to meet at 
 some point near the absolute zero of temperature ; also that 
 three distinct kinds of curves exist — those given by iron, tin, 
 nickle, and copper, in which /3 is positive; those by gold, 
 platinum, palladium, and silver, in which /8 is negative : and that 
 by aluminium, in which /3 = o. 
 
 In the case of alloys they found the curves to be in general 
 straight lines, but differing from those of the pure metals in 
 that they did not run towards the absolute zero on the 
 temperature scale, the direction depending on the metals 
 composing the alloy. If, for instance, the alloy consisted of 
 metals in the same group, such as platinum-iridium or platinum- 
 rhodium, its behaviour more nearly corresponded to that of a 
 pure metal than in the case of an alloy such as platinoid, where 
 the component metals are dissimilar. A very small trace of 
 
 ' FM. Traits., 1862. 
 
 ^ See Dewar and Fleming on " Reiistance of Metals and Alloys at Low 
 Jemperature," FAi/. Mag., vol. xxxiv., Oct., 1892 ; also vol. xxxvi., Sept., 
 1893- 
 
100 Practical Electricity and Magnetism. 
 
 impurity was found to produce a marked efTect on the direction 
 of the curve. 
 
 104. The following experiment illustrates the method of 
 measuring the coefficients of resistance variation of a copper 
 wire. The wire was made up into a loose coil and placed in a 
 thermostat, the temperature of which could be very accurately 
 regulated ; heavy copper leads connected it to a metre bridge, 
 the wire of which had been calibrated and all the corrections 
 determined, compensating leads were also taken to the bridge. 
 The coil was allowed to stand for half an hour in the bath at 
 each temperature before a reading was taken. The tempera- 
 tures were taken by means of thermometers reading to y-j" C., 
 on which j-J-^° could be estimated. The thermometers had all 
 been compared with one another, and with a standard thermo- 
 meter. The standard coils were of manganin. After making 
 all corrections, the following resistances were obtained for the 
 coil at various temperatures ; — 
 
 Temperature. ! 
 
 Resistance. 
 
 
 ohms. 
 
 10° 
 
 103-892 
 
 20° 
 
 107-810 
 
 30° 
 
 "1753 
 
 40° 
 
 115-721 
 
 Tl 
 
 11971S 
 
 60° 
 
 123753 
 
 70° 
 
 127-777 
 
 80° 
 
 131-846 
 
 90° 
 
 1, 
 
 135-940 
 
 From these readings a curve may be plotted, and the values of 
 a and /i deduced, or they may be calculated as follows. 
 
 Assuming the law to be R/ = R„ (1 -f- a/ -f fit'^), we have, 
 taking the first and second readings — 
 
 103-892 = R„ (i + lOa -f 100,3) 
 107-810 = R„(i + 20a -f 400^) 
 from which we get — 
 
 a. = 0-00382 
 and /3 = 0-0000012 
 
Effect of Molecular Change on Specific Resistance. loi 
 From similar equations, we get — 
 
 Temperature. 
 
 Coefficients. 
 
 - 
 
 & 
 
 10-20 
 20—30 
 
 30—40 
 40—50 
 50—60 
 60—70 
 70—80 
 80 — 90 
 
 0-00382 
 0-00384 
 00387 
 0-00389 
 0-00392 
 0-00394 
 0-00397 
 0-00400 
 
 00000012 
 
 0-00000 1 2 
 0-0000013 
 0-0000012 
 
 O-CO0O0I3 
 
 O-00000I2 
 0-00000 I 2 
 0-0000013 
 
 Mean value ... 
 
 0-00390 
 
 0-0000012 
 
 105. Effect of Mokculm- Change. — It will be seen from the 
 table of specific resistances on pp. 272, 273, that the values 
 obtained by different experimenters are by no means identical, 
 even when the chemical composition and method of manu- 
 facture is as nearly as possible the same. These divergences 
 must therefore be put down to some difference in molecular 
 arrangement, due to some slight difiference in handling or in 
 manufacture. 
 
 In the case of copper, it has been shown by Fitzpatrick^ that 
 there is some connection between the specific gravity of the 
 material and its specific resistance, an increase in the former 
 causing a decrease in the latter; this decrease, however, is 
 not proportional to the increase of specific gravity, but changes 
 more rapidly. He also showed that even in wires of the same 
 metal there may be a slight difiference in density due to some 
 difference in drawing. 
 
 106. The permanent efifect of heat on the resistance of wires 
 is very marked, in consequence of the wire becoming annealed, 
 the specific resistance of all metals being diminished by 
 annealing, although in the case Of alloys the efifect is very 
 much less than in that of pure metals. Thus Matthiessen ^ 
 
 B. A. Report, 1890. 
 
 Phil. Trans., 1862. 
 
I02 Practical Electricity and Magnetism. 
 
 found that the resistance of copper wire diminished by about 
 2% on annealing by heating to redness and then cooling slowly ; 
 whereas the platinum-silver alloy (66% Ag., 33% Pt.) did not 
 alter appreciably. 
 
 Partial annealing may, however, occur in a material due to 
 age : this has been found in the case of both hard-drawn silver 
 and copper wires ; but those conductors that do not alter in 
 resistance by baking at a temperature of 100° C. for several 
 days, are found not to alter much with age. 
 
 107. All mechanical operations, such as winding, etc., set up 
 mechanical strains in wires, which harden them and raise their 
 resistance ; this, however, gradually returns to its original value, 
 but the change may be hastened by heating and annealing. 
 This change, for a wire of any given guage, is always greater the 
 smaller the diameter of the spiral into which the wire is coiled. 
 It is therefore important, in the case of standard resistance coils, 
 that the diameter of the bobbin should be large compared with 
 the guage of the wire, in order to prevent excessive strains being 
 set up in the material due to winding. 
 
 108. A frequent cause of alteration of resistance in the case 
 of alloys is the presence of zinc in the alloy. This has been 
 found to slowly crystalhze out, and cause a permanent alteration 
 and rotting of the material in the case of German silver. 
 
 The whole question of molecular change and consequent 
 alteration of specific resistance in metals and alloys is exceed- 
 ingly important, and would form a good line of investigation for 
 more advanced students. 
 
 Fault Testing. 
 
 log. In cables or coils of wire which may be immersed in 
 a conducting liquid, faults are liable to occur the accurate 
 determination of the position of which is of great importance. 
 
 The faults likely to occur in the case of such an insulated 
 cable are the following : — 
 
 {a) Complete fracture of the wire and insulation, the broken 
 end of the wire making a good " earth." 
 
 {b) Local breakdown of insulation, causing partial earth, but 
 no fracture of wire. 
 
The Loop Test for a Fault. 
 
 of 
 
 but 
 
 no 
 
 103 
 
 breakdown of 
 
 (f) Complete fracture 
 insulation. 
 
 The first case is that of a cable which has snapped in two, the 
 bare copper end making contact with the earth, as shown 
 diagrammatically in Fig. 66, where ABC represents the sub- 
 merged cable, fractured at B. To find the distance from A to 
 the fracture we measure the resistance between A and earth on 
 
 Fig. 66. 
 
 a post-office bridge, and thus get the resistance of the part of 
 wire AB (the resistance of the earth being negligible). Con- 
 sequently, knowing from the specification of the wire its re- 
 sistance per mile, we can calculate what length this measured 
 resistance corresponds to. 
 
 no. The second case, which is more important, as occurring 
 more frequently in practice, could be determined in the same 
 
 B, [e] 
 
 B 
 
 Fig. 67 
 
 way, provided the earthing of the wire had been good ; but if 
 there is only a partial earth, the resistance and polarization 
 introduced there would interfere with the results. Of the 
 various methods which may be employed in this case, perhaps 
 that known as the " loop test " is the best ; it however requires 
 a second cable (see Fig. 67). Let AB and CD represent the 
 
I04 Practical Electricity and Magnetism. 
 
 two cables, one of which, AB, has a partial earth at E. The 
 ends B D are joined or " looped " together by a wire of small 
 resistance ; A C are joined by a bridge galvanometer G, and 
 by two resistances, AF and CF, the point F being connected 
 through the key K with the battery B', the other pole of which 
 is earthed at e. 
 
 It will now be seen that this arrangement of conductors con- 
 stitutes a AVheatstone bridge, FA and FC being two arms, 
 whilst AE and CBDE are the others. 
 
 The resistances FA and FC are adjusted until, on depressing 
 the key K, and making the galvanometer circuit, there is no 
 deflection. Then — 
 
 FC _CE 
 FA~ AE 
 
 But CE + AE = R (where R = the resistance of the two 
 cables in series) ; 
 
 .-. CE = R - AE 
 
 FC _ R^E 
 and PA - AE " 
 
 or .Vli - p(-; ^ PA 
 
 So that, knowing the resistance per mile of the cable, we can 
 calculate the distance corresponding to the resistance AE. 
 
 The advantage of this method is that resistance and polari- 
 zation at E do not aifect the measurement. The battery B, 
 however, must have sufficient E.M.F. to overcome any back 
 -E.M.F. due to polarization at E. 
 
 I J I . The third type of fault is the most difficult to localize. 
 Two methods are available, and it is advisable to use one as 
 a check on the other. The first is the measurement of the 
 dielectric resistance of the cable, and the second is the measure- 
 ment of its electrostatic capacity from one end to the break, 
 the other end of the cable being insulated. 
 
 The first method assumes (which may not always be the 
 case) that the dielectric resistance is uniform along the cable. 
 
Artificial Cable. 
 
 105 
 
 In par. 94 we have already shown that in a cable the dielectric 
 resistance is — 
 
 ^^-tl l°g^ 
 
 ri 
 
 where p = specific resistance of the insulating material, / = 
 the length of the copper conductor ?a and i\, the external and 
 internal radii respectively of the insulating covering. The 
 value of R may be found by some of the methods described in 
 the section dealing with high resistance measurement, and p 
 may be found from previous tests, or by testing a known length 
 of the cable. From these measurements / may be calculated 
 approximately. 
 
 112. As a check on this measurement, the following capacity 
 measurement should be made, since it has been shown in 
 par. 262 that the capacity of a cable of length /and S.I.C. = cr, 
 whose external and internal insulation radii are i\ and i\, is — 
 
 o-/ 
 
 K= = 
 
 2l0g/-5 
 
 the capacity being measured between the free end of the cable 
 and earth by some of the methods described in Chapter V., 
 "■ being determined either from previous tests or from an actual 
 test of a measured length cut from the cable, and the length 
 / calculated. These two measurements should be made from 
 each end of the cable, and the mean value of / taken for each 
 end. 
 
 113. In order to practise these tests an artificial cable may 
 be constructed as follows. A number of coils of insulated wire, 
 
 >^AA^«AftAAX XVWAVWK XVXWVWfl* X^/WVWVAX 
 
 A BC DE FG H 
 
 XA/WA/WSSC XVWWWK XWVWWX XVWVWX 
 
 Fig. 68. 
 
 such as No. 40 manganin, are soldered to separate terminals, 
 AB, CD, EF, etc. (see Fig. 68), so that they can be connected 
 
io6 Practical Electricity and Magnetism. 
 
 up in any manner to the terminals T, T, T', T', which represent 
 the ends of the lines ; a wire, E, of low resistance stretched 
 between them represents the " earth." The coils are arranged 
 so that they can be locked up in a box with only the terminals 
 T, T, T', T', E, E projecting out. The demonstrator can thus 
 arrange any fault on the artificial cables — such as a " dead earth," 
 by connecting one of the coil terminals to the wire E ; or a 
 " partial earth," by connecting one terminal of a coil to E through 
 a small battery — and then lock up the box, leaving the student 
 to localize it ; the total length and resistance of the wire on the 
 coils being known. 
 
 Absolute Measurement of Resistance. 
 
 1 14. In making comparisons and measurements of resistance, 
 it is necessary that we should have some standard of resistance 
 in terms of which we are to express the values of the resistances 
 we measure. Several such standards have been proposed from 
 time to time ; the most important, however, was that due to 
 Werner Siemens (i860), who proposed, as a standard of resist- 
 ance, that of a column of pure mercury, 100 cm. long, and 
 I sq. mm. in cross-sectional area, at a temperature of 0° C. 
 This standard was quite empirical, and simply represented a 
 value which was found to be approximately of the order of 
 the resistances which were most commonly measured at that 
 time. Later investigations showed that a resistance could be 
 expressed in terms of the units of length, time, and mass. 
 When this was done for the Siemens unit it was found to be 
 proportional to a velocity of o'gS X 10° cm. per second; it was 
 then decided to call that resistance which had a value pro- 
 portional to a velocity of 10" cm. per second, the practical unit 
 of resistance. 
 
 Starting from this definition, fresh determinations of resistance 
 in absolute measure had now to be made, in order to determine 
 the exact length of a mercury column which would represent the 
 unit of resistance, since the legal definition of the ohm was 
 expressed in terms of a mercury column. Mercury being a sub- 
 stance which, whilst being a good conducter, is a Uquid, and is 
 therefore free from internal strains which might alter its physical 
 
Measurement of Resistance in Absolute Units. 107 
 
 properties, and which are liable to occur in a solid conductor. 
 On account of the difficulty of working with mercury resistances, 
 it is usual to determine in absolute measure the resistance of a 
 wire conductor, which is immediately afterwards compared on a 
 wire bridge with a column of mercury. 
 
 115. We cannot here enter into the full discussion of the 
 various methods of determining resistance in absolute measure, 
 as that would be of little use, since very few physical labora- 
 tories can afford to provide the apparatus necessary for such a 
 purpose ; yet the main principles involved can be illustrated by 
 very simple apparatus, from which the student may be able to 
 grasp the difficulties to be met with in more accurate work. It 
 must, however, be strictly borne in mind that we do not presume 
 to call the measurements made in this way " determinations." 
 
 In order to obtain the resistance of a wire directly in absolute 
 measure, we have the choice of two fundamental principles upon 
 which to base our methods, the first being Joules' law of heating, 
 which establishes a relation between the current, the resistance, 
 and the heat produced in the wire, and the other, some applica- 
 tion of Ohm's law, which defines the resistance of a wire as the 
 ratio between the potential difference at its ends and the 
 current flowing through it. Methods based on both of these 
 principles will be described, but the student is recommended to 
 study the original papers, references to which are given, in 
 order to get the full discussion of these experiments, since we 
 only propose to describe the method sufficiently for measure- 
 ments being made to a first approximation. 
 
 116. Joules' Method. — This method of determining the value 
 of the resistance of a wire in absolute measure, in terms of the 
 amount of heat generated in it when a current of known strength 
 flows through it, cannot rank along with the other methods for 
 accuracy, both on account of the difficulty of making the 
 measurement of the heat evolved, and also since an accurate 
 knowledge of the mechanical equivalent of heat is required, this 
 quantity not being known to the degree of accuracy with which 
 we can determine resistance by the methods based on Ohm's 
 law. The practice of the method, however, forms a splendid 
 test of the ability of a careful student. The apparatus required 
 
io8 Praclical Electricity and Magnetism. 
 
 for this experiment consists of a thin copper calorimeter, which 
 is suspended inside a water-jacketed space ; inside the calori- 
 meter is placed the wire resistance which is to be measured, 
 and which should consist of a coil of bare manganin wire, which 
 has been varnished over to make it insulating. The coil should, 
 if possible, take the form of an open spiral, so that the liquid in 
 the calorimeter is in contact with every part of it, a bobbin 
 being objectionable, on account of the difficulty of ascertaining 
 its exact temperature and calculating its heat capacity. The 
 wire is soldered to two copper leads, the dimensions of which 
 may be so chosen that their rise of temperature shall be 
 approximately equal to that of the liquid, thus avoiding a 
 correction for heat lost by conduction through the leads.^ In 
 series with the manganin resistance is a voltameter or standard 
 galvanometer, a secondary battery, regulating resistance, and 
 break-circuit key. The liquid in the calorimeter may be water 
 or, better still, analine,^ the specific, heat of which is accurately 
 known, and remains more constant with alteration of tempera- 
 ture than that of water. 
 
 If we call R the resistance of the manganin coil in absolute 
 units, C the current in absolute units, t the time in seconds 
 during which the current flows, H the quantity of heat given 
 out in gramme degrees Centigrade, and J the mechanical 
 equivalent of heat, i.e. 4" 2 X 10' ergs, then — 
 
 Heat produced in coil = heat given out 
 C-Ri' = JH 
 and R = ^ 
 
 The accurate measurement of H, the heat given out by the 
 coil, is somewhat difficult, since, in addition to the heat gained 
 by the liquid surrounding the wire, we have to take into account 
 the heat gained by the calorimeter, stirrer, thermometer, and 
 the wire coil itself, besides estimating the heat lost by radiation. 
 The heat gained by the apparatus is calculated from its water 
 
 ' See Ayrton and Haycraft, Proc. Phys. Soc, Dec. 14, 1894. 
 Griffiths, " Specific Ileit of Analine." Phil. Mag., vol. xxxix.,-Tan., 
 1895. 
 
Joules' Method of Measuring Resistance. 109 
 
 equivalent, which may be experimentally determined, as 
 described in vol. i. p. 76; whilst the heat radiated may be 
 found by an experiment similar to that described in vol. i. p. 
 77. The water equivalent of the apparatus may also be 
 calculated, since it is equal to the product of the weight of 
 each part immersed into its specific heat. The correction for 
 radiation may be made negligible by arranging the current 
 so that the temperature at the end of the experiment is as 
 much above the temperature of the air as it was below it at the 
 commencement. 
 
 A known weight of liquid is placed in the calorimeter, and 
 the temperature taken by means of a delicate thermometer 
 graduated to -^^ C, on which the temperature may be estimated 
 to Y5o° C. ; a second thermometer is placed in the jacketed 
 space surrounding the calorimeter. 
 
 The current is now started and maintained constant until the 
 liquid, which has been kept continuously stirred, has risen a 
 few degrees in temperature ; the current is then stopped, and 
 the time during which it has flowed noted accurately on a stop- 
 watch. The thermometer in the calorimeter should, if the 
 stirring has been performed properly, cease rising almost at 
 the same instant that the current was stopped ; this temperature, 
 Tj", is now noted. Then, if Ti° was the initial temperature of 
 the liquid, m its mass in grammes, .f its specific heat, W^ the 
 water-equivalent of the apparatus, and R the number of units 
 of heat lost by radiation during the time t, and which has been 
 determined by a separate experiment, the total quantity of 
 heat given out by the coil is — 
 
 H = ms{T.? - T°) + W,(T,° - 1°) + R 
 
 The current C is now obtained in absolute units from the- 
 current-measuring apparatus employed (see par. 130), and the 
 value of R calculated in absolute units. The student should 
 then proceed to check this result by measuring the resistance 
 of the manganin coil on a wire bridge against a standard 
 resistance, when the resistance in ohms X 10" should give- 
 the same result as that obtained in the above experiment. 
 
1 10 Practical Electricity and Magnetism. 
 
 1 1 "J. In the second class of methods, the principle of which 
 depends on Ohm's law, the resistance is expressed in terms of 
 an electromotive force and a current, the values of which must 
 be known in absolute measure. 
 
 One of the best-known methods of this class is that proposed 
 by Lord Kelvin,' and usually known as the British Association 
 or B. A. method. 
 
 In this method a circular coil of wire is mounted in a frame, 
 so as to be free to rotate about a vertical axis in the earth's 
 magnetic field. At the centre of the coil, and protected, by 
 a closed glass vessel, from air currents, a magnetic needle, with 
 mirror attached, is suspended by a long silk or spider line. 
 
 As the coil rotates in. the earth's field, the flux of lines of 
 force through it is continually altering with the position of the 
 coil with respect to the magnetic meridian ; an E.M.F. will 
 therefore be induced in it, the magnitude of which may be 
 calculated from the known strength of the field, and the rate 
 of change of lines in the coil. If the coil has its circuit 
 completed by joining the ends together, this E.M.F. will induce 
 a current in it, which will be alternately in opposite directions 
 with respect to the coil itself, but will always be in the same 
 direction with respect to the needle at its centre ; this current 
 produces a deflection of the needle, from which its magnitude 
 may be calculated in absolute measure, and hence, by dividing 
 the E.M.F. in absolute measure by the current in absolute 
 measure, we obtain the resistance of the coil also in absolute 
 measure. 
 
 ii8. The following calculation may be regarded as giving 
 the resistance of the coil to a first approximation. 
 
 Let E = E.M.F. set up in the coil in absolute measure; 
 C = current in the coil in absolute measure ; 
 R = resistance of the coil in absolute measure ; 
 II = horizontal intensity of the earth's magnetic field ; 
 A = effective area of the coil, i.e. the sum of the areas 
 
 of each turn ; 
 N — total number of lines passing through the coil ; 
 « = speed of rotation in revolutions per second ; 
 ' See Appendix D., B. A. Report, 1863. 
 
B. A . Method of Measuring Resistance. 1 1 1 
 
 then, when the coil is inclined to the magnetic equator at 
 
 an angle 6 — 
 
 N = HA cos 
 
 de 
 and E = HA sin 6 j 
 
 de 
 
 But -^ = 27r« 
 
 .•. E = ZTT/iHA sin B 
 
 This, by Ohm's law, neglecting the effect of self-induction, 
 
 is = CR. Hence — 
 
 „ 27r«HA sin B 
 C= 5 
 
 Now, the magnetic force/, at right angles to the plane of the 
 coil, acting on a needle of pole strength, ;«, at the centre of 
 the coil, is — 
 
 2irCsm 
 J — ;. 
 
 where j- is the number of spirals on the coil, and ;- is the 
 mean radius of the coil in centimetres ; 
 
 ^Tp'smnYih. sin B 
 '■J^ R? 
 
 and, for simplicity, call this — 
 
 /= A sin 61 
 
 Now, /can be resolved into two components at right angles 
 to one another, one,/, acting at right angles to the magnetic 
 meridian, and the other, f.^, acting along the meridian ; and if 
 B is the angle which/ makes with the meridian, then we have — • 
 / =/sin B 
 = A sin= 6 
 and/ =/cos B 
 
 = A sin ^ cos B 
 
 Integrating the values of/ and/ for a complete revolution 
 of the coil, we get — 
 
 mean value ol f^ = — 
 •' 2 
 
 mean value of/ = O 
 
1 1 2 Practical Electricity and Magnetism. 
 
 Hence the deflection of the magnet at the centre of the coil 
 is due to the component /i, and, calling this deflection 8, we 
 have — 
 
 /i = wH tan S 
 
 therefore :„ = m]A tan S 
 
 . _ 2ifisnK 
 
 and R = ^. ^„„ ^ 
 r tan 6 
 
 or, since A = irr'^'s 
 
 R 
 
 tan S 
 
 which expresses the resistance of the coil in absolute measure. 
 It must be noted that the deflection S is the angular deflection 
 of the needle, and not that of the spot of light. The speed 
 of rotation of the coil may be measured by the stroboscopic 
 disc method, described in par. 120. For the full details of the 
 method, the student is referred to the Report of ths Electrical 
 Standards Committee of the British Association, 1863. 
 
 119. Another method of determining resistance absolutely, 
 which has the additional advantage of being a very good 
 method of comparing low resistances, is that due to Lorenz, and 
 modified by Professor V. Jones.^ 
 
 In this method of determining resistance, the fall of potential 
 ■down the resistance carrying a current is balanced against the 
 E.M.F. of a simple dynamo, so arranged that the E.M.F. may 
 be calculated from its dimensions, and whose magnetic field is 
 produced by the current flowing through the resistance. 
 
 The dynamo consists of a copper disc rotating inside a coil 
 of wire and coaxial with it, as is diagrammatically shown in 
 Fig. 69. The coil is in series with a battery, B, a regulating 
 resistance, ;-, and the resistance R to be determined absolutely. 
 From the ends of R potential wires are taken through the 
 galvanometer G, one to the axis, and the other to the periphery 
 of the copper disc, which is rotated so as to produce an E.M.F. 
 in opposition to the P.D. at the ends of R, the speed of 
 
 ' Phil. Trans., 1891. 
 
Lorenz Method of Measuring Resistance. 1 1 3 
 
 rotation being altered until no deflection is obtained on the 
 galvanometer G. Then, calling C the current in the coil and 
 R its resistance in absolute units, M the coefficient of mutual 
 
 wwyvw — N 
 
 Fig. 69. 
 
 induction between the coil and the disc, and n the speed of 
 rotation of the disc in revolutions per second — 
 
 The E.M.F. produced by the disc is E = M/zC 
 But this, by Ohm's law, is equal to CR ; hence — 
 
 R = M« 
 The value of M may be calculated from the dimensions of the 
 disc and coil,' but the calculation is very complicated; M, 
 however, does not require to be known if it is only desired 
 to compare two resistances. In this case one of the' resistances, 
 say Ri, is placed in the position of R in the diagram, and the 
 disc rotated until there is no deflection on the galvanometer G. 
 Let the speed be n-^ revolutions per second. The coil Rj is now 
 replaced by the other coil, R2, and the disc again rotated till the 
 galvanometer needle shows no deflection. Let the speed now 
 be Wj revolutions per second, then — 
 
 (i) Ri = M«i 
 and (2) R2 = M;z2 
 
 hence ij^ = — 
 
 R2 «2 
 
 ' See Phil. Mag., Jan., 18S9, also July, 1896 ; and B. A. Report, 1894. 
 
1 1 4 Practical Electricity and Magnetism. 
 
 If Ra is a standard resistance, the value of Rj may thus be 
 calculated. 
 
 120. In order to be able to regulate the speed of the disc, 
 and in cases where the highest accuracy is not required, also to 
 measure it, the apparatus known as the stroboscopic disc may 
 be employed. 
 
 This consists of a cardboard disc attached to the same axle 
 that carries the copper disc, and therefore rotating at the same 
 speed. On this are drawn a number of concentric rings, which 
 are divided up into a number of segments, painted alternately 
 white and black, or white and red. The number of segments 
 differing in each ring, say 60 in the first, 34 in the second, 
 20 in the third, 16 in the fourth, etc. In front of this disc is 
 placed a tuning-fork, to the prongs of which are attached thin 
 aluminium plates, overlapping one another and with a long 
 narrow slit cut in each, parallel to the prongs, so that when the 
 fork is at rest the slits coincide, and when it is vibrating they 
 coincide twice in every double vibration. When the disc is 
 rotating, say 10 times a second, the first ring of segments will 
 pass the slits at the rate of 600 per second, the second ring at 
 340 per second, and so on. Now, if the fork vibrates 300 
 times per second, a person looking through the slits at the 
 rings of segments would see them 600 times a second, and 
 consequently the first ring of 60 segments would appear to be 
 at rest whilst the others would appear to be moving in the 
 opposite direction to that of the disc. The speed of rotation 
 of the disc is calculated from the number of teeth on the 
 segment which appears to be at rest, and from the rate of 
 vibration of the tuning-fork. Should none of the segments be 
 absolutely at rest, the speed may be calculated by timing the 
 apparent rate of rotation of the slowest moving one, and from 
 the direction of its motion. For instance, if in the above case 
 the 60-segment ring appeared to move past the slits at the rate 
 of I segment per second, the speed of rotation with the abov^ 
 fork would be 600 -f- i, according as the apparent direction of 
 rotation was with or against that of the disc. 
 
 Some difficulty may be experienced from thermal currents 
 set up at the rubbing contacts ; these, if constant, may be either 
 
Standards rj Resistance. 1 1 5 
 
 allowed for, by observing the galvanometer deflection which 
 they produce, and using that as the true zero, or else they may 
 be annulled by introducing an E.M.F. in opposition to them, 
 which will just balance their effect. In order to obtain good 
 contact at the periphery of the disc, more than one brush may 
 be employed. In Jones's modification of the Lorenz method 
 the brushes consisted of thin metal tubes, through which a 
 stream of mercury was kept flowing, the edge of the copper 
 disc being amalgamated ; this was found to give exceedingly 
 satisfactory results. 
 
 For further details of the method the student is referred to 
 the original papers.^ 
 
 Resistance Standards. 
 
 121. The results of many careful researches have given us 
 data from which the dimensions of the standard ohm may be 
 calculated. This quantity is now legally defined as follows : — ^ 
 
 " The ohm, which has the value 10' in terms of the centi- 
 metre and the second of time, is represented by the resistance 
 offered to an unvarying electric current by a column of mercury 
 at the temperature of melting ice, i4"452i grammes in mass, 
 of a constant ■ cross-sectional area, and of a length of io6'3 
 centimetres." 
 
 Practically, it is found that a mercury column is not a con- 
 venient form in which to embody the unit. The requirements 
 of a practical unit being briefly as follows : (i) it must be easy 
 to construct; (2) it ought to be made of some substance of 
 high specific resistance, in order to avoid having a large bulk ; 
 (3) its temperature variation of resistance should be as small 
 as possible ; (4) it must not be liable to change its value with 
 time, either from oxidation or from molecular change; (5) 
 it must be able to stand handling without injury, and be of 
 such a form that its temperature may be accurately determined. 
 
 122. After many experiments on various substances,' the 
 most satisfactory, both as regards permanency, low temperature 
 
 ' Phil. Trans., 1891 ; Electrician, vol. xxxi. p. 620 ; vol. xxv. pp. 543, 
 562 ; vol. XXXV. p. 351. 
 ^ See London Gazette, Aug. 24, 1894. 
 •■ See B. A. Report, 1862-65. 
 
ii6 Practical Electricity and Magnetism. 
 
 variation, and high specific resistance, was found to be an 
 alloy of 66'6 per cent, silver and 33*4 per cent, platinum, 
 known as platinum-silver, having a specific resistance equal to 
 14 microhms, and a temperature variation of resistance of 
 o"03i per cent, per 1° C. 
 
 123. The accompanying figure (Fig. 70) shows the form 
 taken by the standard. A double silk-covered, well-paraffined 
 platinum-silver wire is wound non-inductively in the space 
 between two brass cylinders, the free ends being soldered to 
 thick copper wires, and the space round the wire filled up with 
 paraffin wax. The case is immersed to the depth of the 
 narrow part in melting ice, and the temperature taken by a 
 
 Fig. 70. FiG. 7,. 
 
 thermometer which passes into the inner tube. The chief 
 
 objections to this form are that it is difficult to insure that 
 
 every part of the coil is at the same temperature, and that the 
 
 thermometer registers this temperature. It also takes a long 
 
 time to settle down to a steady temperature, on account of the 
 
 large mass of paraffin wax surrounding the wires, and it has 
 
 been suspected that the strains set up in the wire, due to 
 
 expansion and contraction of the paraffin wax, may cause a 
 
 permanent alteration in the resistance of the wire.' 
 
 124. A modification of this form of standard, which gets over 
 
 some of these difficulties, is that due to Dr. Fleming ^ (see 
 
 Fig. 71). The coil is wound in the form of a flat spiral, and 
 
 ' Eledncian, vol. xxix. p. 277 ; B.A. Heport, igqo. 
 ^ Electrician, vol. xxii. p. 74. 
 
• 
 
 Standards of Resistance. 1 1 7 
 
 its ends soldered to thick copper leads, which pass up from the 
 coil inside ebonite tubes, insulated from them by air, except 
 at the ends, where they are held by ebonite cups containing 
 paraffin oil. The coil is placed between two annular discs of 
 brass, with flat grooves in their opposite faces, which, when 
 screwed together, form a closed space of rectangular section. 
 The coil is embedded in paraffin wax in the lower groove, but 
 the upper groove is empty, an opening being made into it 
 through which air may be forced when the apparatus is 
 immersed in water, to test for leakage at the joints. The 
 advantages claimed are, better insulation and more uniform 
 temperature throughout. 
 
 125. A description of the standards would, however, be 
 incomplete without mention of the standards employed at the 
 Berlin Reichsanstalt.^ The general form taken by the i-ohm 
 standard being shown in Fig. 72. 
 
 The wire is No. 18 B.W.G. manganin double white silk 
 covered, the ends being soldered to copper washers, which are 
 then screwed and soldered with silver solder to the ends of 
 thick copper leads, the resistance of which together amounts to 
 about 140 microhms. The bobbin on which the wire is wound 
 is a hollow brass cylinder 4 cm. diameter, covered first with 
 a shellaced silk tape dried at 140° C; the wire is also shellac 
 varnished and dried for 10 hours at 140° C. The insulation 
 resistance of the wire is of the order of a million megohms ; 
 the wire when wound on the cylinder being held by a dry 
 cloth and not in the bare hand. The axial length of the coil 
 is 4 cm. The coil hangs by the leads, in a vessel of paraffin 
 oil II cm. broad, 48 cm. long, and filled to a depth of 12 cm., 
 provided with a fan to keep the liquid in circulation. The 
 radiating surface of such a coil is about 100 sq. cms. With 
 such standards a maximum current of i ampfere could be 
 used. 
 
 126. Kelations beticcjii the Various Standards. — Since from 
 time to time, on account of new and more accurate determina- 
 tions of the ohm having been made in terms of a certain length 
 
 " See translation of a paper by Feussner and St. Liiideck, Eliclriciaii, 
 vol. xxxvi. p. 509. 
 
Ii8 Practical Electricity and Magnetism. 
 
 of mercury column, fresh standards have been issued, and 
 different experimenters have used different standards, we here 
 enumerate the more important standards, and give a table 
 
 showing the relation of each to the present legal standard, 
 which is defined in terms of a mercury column io6'3 cm. 
 long. 
 
Relations of the Standards of Resistance. 119 
 
 Standards- 
 
 Siemen's unit 
 
 B.A. unit (1864) 
 Legal ohm (1884) 
 Present legal ohm (1892) 
 
 Centimetres of mercury ^ ^ „f„™^^„fr 
 
 at 0° C. and 1° T?'i° if' 
 
 I sq. mm. section. ' '=SaI ohm. 
 
 1 00 -00 0'94O7 
 
 104-88 0-9866 
 
 io6-oo 0-9972 
 
 106-30 I'OOOO 
 
 In connection with the various standards of resistance that 
 have from time to time been employed, the student must be 
 careful, when studying any published paper or research, to 
 note the particular unit of resistance employed in the measure- 
 ments. If the unit employed was one of the older ones, the 
 measurements must be reduced in terms of the 1892 standard 
 before they can be compared with recent experiments. 
 Thus, if the standard employed was the legal ohm of 1884, 
 any measurement of resistance made in terms of it must 
 
 be reduced in the ratio of t in order to compare it with 
 
 measurements made in terms of the 1892 standard. The same 
 remark applies to the determination of E.M.F., which involves 
 the product of a current into a resistance. Determinations 
 made in terms of the 1884 ohm will, therefore, be about 
 0-3 per cent, too high as compared with those made from the 
 1892 standard. 
 
I20 Practical Electricity and Magnetism. 
 
 127. Refekences to Scientific Papers. 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 I. Galvanometers. 
 
 
 
 Spider's Web Suspensions 
 
 Bennett 
 
 Trans. Roy. Soc, 
 1792. 
 
 Torsional Rigidity of Spider 
 
 Bottomley and 
 
 Fro. Jioy. Soc, vol. 
 
 Lines 
 
 Tanakadate 
 
 46, p. 291. 
 
 SiUc versus Wire Suspensions 
 
 T. Gray 
 
 Phil. Mag., vol. 23, 
 Jan., 1887. 
 
 On the Attacliment of Quartz 
 
 Boys 
 
 /hid., vol. 37, May, 
 
 Fibres 
 
 
 1894. 
 
 On the Shape of Movable Coils 
 
 Mather 
 
 /bid., vol. 29, May, 
 
 in Electrical Instruments 
 
 
 1890. 
 
 On Galvanometers 
 
 Ayrton, Mather, 
 
 /bid., vol. 30, July, 
 
 
 and Sumpner 
 
 1890. 
 
 On a New Reflecting Galvano- 
 
 T. and A. Gray 
 
 Pi'o. /ioy..Soc., vol. 
 
 meter of Great Sensibility 
 
 
 36. P- 4- , 
 
 On Sensitive Galvanometers 
 
 Threlfall 
 
 Phil. /Hag., vol. 29, 
 June, 1890. 
 
 On a Universal Shunt Box for 
 
 Ayrton 
 
 Jour. Elect. En,'., 
 
 Galvanometers 
 
 
 vol. 23, p. 314. 
 
 On a Method of measuring Gal- 
 
 W. Thomson 
 
 Pro. Roy. Soc, vol. 
 
 vanometer Resistance 
 
 
 I9> P- 253 
 
 Calibration of a Galvanometer 
 
 Mather 
 
 Phil. Mag., vol. 21, 
 
 by Constant Current 
 
 
 Jan., 1886. 
 
 Galvaiiometer Calibration 
 
 Thomas 
 
 Jour. Elect. Eng., 
 vol. 13, p. 153. 
 
 >) }} 
 
 Ducretet 
 
 /bid.,vJ\. 13, p. 472- 
 
 
 Edelmann 
 
 Jbid.,yo\. 14, p. 361. 
 
 II. Resistance Measurement. 
 
 
 
 Calibration of a Wire 
 
 Foster 
 
 Ibid., 1872. 
 
 Calibration of Bridge Wires 
 
 Uppenborn 
 
 Ibid., i/o\. 16, p. 598. 
 
 3) IJ 1} 
 
 Braun 
 
 /bid.,vol. 13, p. 281. 
 
 A New Form of Constant 
 
 Nicol 
 
 Phil. /Hag., vol. 15, 
 
 Temperature Bath 
 
 
 May, 1883. 
 
 Best Arrangement of a Wheat- 
 
 T. Gray 
 
 /bid., vol. 12, Oct., 
 
 stone Bridge for the Measure- 
 
 
 1881. 
 
 ment ofa Particular Resistance 
 
 
 
 A New Form of Resistance Bal- 
 
 Fleming 
 
 Ibid., vol. 9, Feb., 
 
 ance for comparing Two Coils 
 
 
 1880. 
 
 Comparison of Standard Coils 
 
 Glazebrook 
 
 /bid., vol. 20, Oct., 
 
 with B.A. and Mercury . 
 
 
 1885. 
 
 Standards 
 
 
 
 Comparison of the Mercury Unit 
 
 Hutchinson and 
 
 irn., vol. 28, July, 
 
 with the B.A. Unit. 
 
 Wilkes 
 
 1889. 
 
 A Practical Point in Connection 
 
 Shaw 
 
 Ibid., vol. 17, May, 
 
 with the Comparison of Re- 
 
 
 1883. 
 
 sistances 
 
 
 
 On the Adjustment of Resist- 
 
 S. P. Thompson 
 
 /bid., vol. 17, Apr., 
 
 ance Coils 
 
 
 188:!. 
 
References to Scientific Papers. 
 
 121 
 
 Tills of Paper. 
 
 On the Construction of Resist- 
 ances 
 
 On the Construction of Non- 
 inductive Resistances 
 
 On an Electro- Dynamic Balance 
 for the Resistance of Short 
 Bars or Wires 
 
 On the use of Bare Wires for 
 Resistance Coils 
 
 On a Design for a Standard 
 Resistance 
 
 On the Reichsanstalt Standards 
 of Resistance 
 
 High Electrical Resistances 
 
 Measurement of High Specific 
 
 Resistances 
 Variation of the Resistance of 
 
 Paraffin Wax and Rosin Oil 
 
 with Temperature 
 Oil as an Insulator 
 
 Electrical Conductivity of Cer- 
 tain Saline Solutions 
 
 Electrical Resistance of Electro- 
 lytes 
 
 Method of Measuring the In- 
 ternal Resistance of a Battery 
 
 On Mance's Method of measur- 
 ing Battery Resistance 
 
 Measurement of the Internal 
 Resistance of Batteries 
 
 Variation of Internal Resistance 
 of Batteries with the Current 
 Specific Resistance. 
 
 On the Specific Resistance of 
 Mercury 
 
 On the Specific Resistance of 
 Mercury 
 
 Specific Resistance of Mercury 
 
 in Absolute Measure 
 On the Electrical Conducting 
 
 Power of Metals 
 On the Electrical Conducting 
 
 Power of Alloys 
 On the Effect of the Presence of 
 
 Metals and Metalloids on the 
 
 Electrical Conducting Power 
 
 of Pure Copper 
 
 Author. 
 
 Morris 
 
 Ayrlon and 
 
 Mather 
 Sir Wm. Thom- 
 
 Burstall 
 
 Fleming 
 
 Feussner and 
 
 St. Lindeck 
 J. Hopkinson 
 
 Threlfall 
 
 Gaze 
 
 Hughes 
 
 Ewing and 
 
 Macgregor 
 Knott 
 
 Mance 
 
 Lodge 
 
 Rimington 
 
 Carhart 
 
 Glazebrook and 
 Fitzpatrick 
 
 Rayleigh and 
 Sedgwick 
 
 Jones 
 
 Matthiessen 
 
 Matthiessen and 
 Holzmann 
 
 Reference. 
 
 Elect., vol. 33, p. 
 
 605. 
 Ibid., vol. 27, p. 254. 
 
 Phil. Mag., vol. 24, 
 p. 149 : 1862. 
 
 Ibid., vol. 42, Sept., 
 
 1896. 
 Ibid., vol. 27, Jan., 
 
 1889. 
 Elect., vol. 36, p. 
 
 509. 
 Phil. Mag., vol. 7, 
 
 Mar. 1879. 
 Ibid., vol. 28, Dec, 
 
 1889. 
 Elect., vol. 36, p. 
 
 473- 
 
 Jour. Elect. Eng., 
 
 vol. 24, ipp. 244, 
 
 267. 
 Trans. H.S.E., vol. 
 
 27, p. 51. 
 Pro.R.S.E.,yo\.i2, 
 
 p. 178. 
 Pro. Roy. Soc, vol. 
 
 19, p. 252. 
 Phil. Mag., vol. 3, 
 
 Supp., 1877. 
 Elect., vol. 31, p. 
 
 263. 
 Ibid., vol. 35, p. 18. 
 
 Trans. Roy. Soc, 
 
 1S88 
 Soc, 
 
 379- 
 Trans. Roy. 
 
 Pro. Roy. 
 vol. 44, p. 
 
 Soc, 
 
 1883 ; Pro. Roy. 
 
 &)f.,vol. 34, p. 27. 
 Pro. Roy. Soc, vol. 
 
 48, p. 434. 
 Trans. Roy. Soc, 
 
 1858.1 
 Ibid., i860. 
 
 Ibid., i860. 
 
122 Practical Electricity and Magnetism. 
 
 Title of Paper, 
 
 Author. 
 
 The Electrical Resistance of 
 
 Platinoid 
 On the Electrical Resistance of 
 
 Iron at High Temperature 
 Absolute Specific Resistance of 
 
 I'ure Electrolytic Copper 
 Specific Resistance of Copper 
 Alloys for Resistance Coils 
 
 Resistance Variation with Tempe- 
 rature. 
 
 Influence of Temperature on the 
 Electric Conducting Power of 
 Alloys 
 
 Influence of Temperature on the 
 Electric Conducting Power of 
 Thallium and Iron 
 
 Influence of Temperature on the 
 Electric Conducting Power of 
 Metals 
 
 Measurement of the Increase of 
 Resistance of Conductors with 
 Rise of Temperature 
 
 Connection between Electrical 
 Resistance and Temperature 
 in the Simple Metals 
 
 Variation with Temperature of 
 the Electrical Resistance of 
 Various Alloys 
 
 On the Temperature Variation 
 of Resistance of Copper 
 
 On the Temperature Variation 
 of Resistance of Mercury 
 
 On the Electrical Conductivity 
 of Solid Mercury and Pure 
 Metals at Low Temperatures 
 
 On the Electrical Resistivity of 
 Mercury at the Temperature 
 of Liquid Air 
 
 On the Electrical Resistance of 
 Metals and Alloys at Tem- 
 peratures near the Absolute 
 Zero 
 
 Effect of Repeated Heating and 
 Cooling on the Electrical Re- 
 sistance of Iron 
 
 Variation of the Electrical Con- 
 ductivity of Glass with Rise 
 of Tempefature 
 
 Bottomley 
 
 J. Hopkinson 
 
 Swan and 
 Rhodin 
 Fitzpa trick 
 St. Lindeck 
 
 Matlhiessen 
 and Vogt 
 
 Matthiessen and 
 Von Bose 
 
 Siemens 
 
 Balfour Stewart 
 
 Knott and 
 Macgregor 
 
 Kenelley and 
 Fessenden 
 Guilleaume 
 
 Cailletet and 
 Bouty 
 
 Devvar and 
 Fleming 
 
 Tomlinson 
 T. Gray 
 
 Reference. 
 
 Pro. Roy. Sac,, vol. 
 
 38, P- 340. 
 /*;«'., vol. 45, p. 457. 
 
 lliii., vol. 56, p. 64. 
 
 Etect.,vol.2i,p.6oS. 
 
 Ibid., vol. 26, p. 493 ; 
 
 vol. 30, p. 119. 
 
 Trans. Roy. Soc, 
 1864. 
 
 /6i<i., 1864. 
 
 Jliid., 1862. 
 
 Pro. Roy. Soc, vol. 
 19. P- 443- 
 
 Pro. R.S.E., vol. 
 4, p, 168. 
 
 Trans. R.SE., vol. 
 29. P- 599- 
 
 Elect., vol. 31, p. 
 
 624. 
 /i/rt'., vol. 29, p.553. 
 
 Phil. Mag., vol. 20, 
 July, 1885. 
 
 Pro. Rctji. Soc, vol. 
 60, p. 76. 
 
 Phil. Mag., vol. 36, 
 Sept., 1893. 
 
 Ibid., vol. 29, Jan., 
 1890. 
 
 Pro. Roy. Soc, vol. 
 
 33. P- 256; vol. 
 
 34, p. 199 : Phil. 
 Mag., vol. 10, 
 Oct., 1880. 
 
References to Scientific Papers. 
 
 123 
 
 Title of Paper. 
 
 Author. 
 
 Electrical Resistance of Glass 
 at Low Temperatures 
 
 Influence of Temperature on the 
 Resistance of Dielectrics 
 
 III. Molecular Change. 
 Increase of Resistance of Wires 
 by Stretching 
 
 Influence of Longitudinal Trac- 
 tion on Electrical Conductivity 
 
 Effect of Mechanical Strain on 
 the Electrical Resistance of 
 Metals 
 
 Effects of Strain on Electrical 
 Conductivity 
 
 On the Alteration Produced on 
 the Electrical Resistance of 
 Wires by Coiling and Un- 
 coiling 
 
 IV. Absolute Measurements. 
 Experiments to determine the 
 Value of the B. A. Unit of Re- 
 sistance in Absolute Measure 
 
 Experiments by the Method of 
 Lorenz for the Determination 
 of the Absolute Value of the 
 B.A. Unit of Resistance 
 
 Experiments on the Value of the 
 B.A. Unit of Resistance 
 
 Results of a Comparison of the 
 
 B.A. Units of Resistance 
 On the Determination of the 
 
 Specific Resistance of Mercury 
 
 in Absolute Measure 
 Determination of the Ohm in 
 
 Absolute Measure 
 Comparison of Methods for the 
 
 Determination of the Ohm in 
 
 Absolute Measure 
 On the Methods employed in 
 
 determining the Ohm 
 Determination of the Ohm by 
 
 Calorimetric Measurements 
 Determination of the Ohm in 
 
 Terms of the Mechanical 
 ' Equivalent of Heat 
 
 Foussereau 
 Appleyard 
 
 Tomlinson 
 
 Gray and 
 Henderson 
 
 Wilkowslii 
 
 Hopps 
 
 Rayleigh 
 
 Rayleigh ahd 
 Sedgwick 
 
 Glazebrook 
 
 Crystal and 
 Saunder 
 J. V. Jones 
 
 Rayleigh and 
 
 Schuster 
 Rayleigh 
 
 Wiedemann 
 
 Weber 
 
 Fletcher 
 
 Reference. 
 
 Phil. Mag., vol. 14, 
 
 Oct., 18S2. 
 Hid., vol. 42, Aug., 
 
 1896. 
 
 Pro. Roy. Soc, vol. 
 
 25, p. 496 ; vol. 
 
 26, p. 401 ; Phi/. 
 Mag., vol. 3, 
 Supp., 1877. 
 
 Pro. Rvy. Soc, vol. 
 
 39. P- 503- 
 /«rtr.,voI.S3, p. 76; 
 vol. 54, p. 2S3. 
 
 Trans. P.S.E.,,vo\. 
 
 30, p. 413. 
 Phil. Mag., vol. 18, 
 
 Nov., 1884. 
 
 Trans. Roy. Soc, 
 1882; Pro. Roy. 
 Soc, vol. 33, p. 
 
 398- 
 Trans. Roy. Soc, 
 1883; /?-(7. Roy. 
 Soc, vol. 34, p. 
 
 438-. 
 Trans. Roy. Soc, 
 
 1883 ; Pro. Roy. 
 
 6ijf.,voI. 34, p. 86. 
 B.A. Report, 1876, 
 
 P- 13- 
 Trans. Roy. Soc, 
 1891. 
 
 P)-o. Roy. Soc, vol. 
 
 32, p. 104. 
 Phil. Mag., vol. 14, 
 
 Nov., 1882. 
 
 Ibid., vol. 14, Oct., 
 
 1882. 
 Jliid., vol. 5> Jan-i 
 
 Feb., Mar., 1878. 
 Ibid., vol, 20, July, 
 
 1885. 
 
124 Practical Electricity and Magnetism. 
 
 Title of Paper. 
 
 On the Lorenz Method of Deter- 
 mining the Ohm 
 
 On the Absolute Determination 
 of the Ohm 
 
 Author. 
 
 Duncan, 
 Hutchinson, 
 and Wilkes 
 
 Glazebrook 
 
 Reference. 
 
 Phil. Mag., vol. 28, 
 Aug., 1889. 
 
 Elect., vol. 25, p. 
 
 128. Referexces to Foreign Scientific Papers. 
 
 Title of Paper. 
 
 Deprez D'Arsonval Galvano- 
 meter 
 
 Best Arrangement for Measure- 
 ments of Resistance by Wheat- 
 stone's Bridge 
 
 Resistance at the Plugs in a. 
 Resistance Box 
 
 Calibration of a Wire 
 
 Calibration of Mercury Resist- 
 ances 
 
 Measurement of Liquid Resist- 
 ance 
 
 Author. 
 
 Reference. 
 
 Deprez 
 Kohlrausch 
 
 Dorn 
 
 Strouhal and 
 
 Barus 
 Foster 
 
 Siemens 
 
 Kohlrausch 
 
 Absolute Measurement of Resist- 
 ance 
 
 Beetz 
 
 Paalzo w 
 
 Wiedemann 
 Branly 
 
 Kirchoff 
 
 Comhtes Hendus, vol. 
 
 94. P- 1347 : 1882. 
 Pogg. Ann,, vol. 142, 
 
 p. 428 : 1872. 
 
 Wied. Ann., vo'. 22, 
 
 p. 558 : 1884. 
 Ibid., vol. 10, p. 326 : 
 
 1880. 
 Ibid., vol. 26, p. 239 : 
 
 1885. 
 Pogg. Ann., ^ol. no, 
 
 p. I : i860. 
 Ibid., vol. 138, p. 280 
 
 1869. 
 IHd., vol. 138, p. 370 
 
 1869. 
 Ibid., vol. 154, p. 3 
 
 1875- 
 yubelband, p. 290 
 
 1874. 
 Wied. Ann., vol- 6, 
 
 pp. 36-49 : 1879. 
 Ibid., vol. II, p. 653 : 
 
 1880. 
 Ibid., vol. 26, p. 168 : 
 
 1885. 
 Pogg. Ann., vol. 117, 
 
 p. I : 1862. 
 Ibid., vol. 137, p. 4S9 : 
 
 1869. 
 Ibil., vol. 99 : 1856. 
 Comp. Rend., vol. 74, 
 
 p 528: 1878. 
 Pogg. Ann., vol. 77 : 
 
 1849. 
 
References to Foreign Scientific Papers. 125 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 Absolute Measurement of R esiit- 
 
 Lorenz 
 
 jPog^. Ann., vol. 149, 
 
 ance 
 
 
 p. 251 : 1870. 
 
 JJ 3J J) 
 
 Mascart 
 
 Ann. de Chim. et lie 
 Phys., vol. 6, p. 5 : 
 1885. 
 
 Ss 1^ It 
 
 Dorn 
 
 Wied. Ann., vol. 17, 
 p. 773: 1882. 
 
 >» 1» J) 
 
 Lorenz 
 
 Ibid; vol. 25, p. I ; 
 1885. 
 
 Temperature Variation of Resist- 
 
 Cailletet and 
 
 Comp. Kend., vol. 100, 
 
 ance 
 
 Bouty 
 
 p. 1188 : 1885. 
 
 Temperature Variation of Liquid 
 
 Grossmann 
 
 Wied. Ann., vol. 18, 
 
 Resistance 
 
 
 p. 119: 1883. 
 
II. 
 
 CURRENT. 
 
 129. The measurement of current should really have been 
 dealt with before the measurement of resistance, since the 
 absolute measurement of the latter quantity involves an abso- 
 lute measurement of current. We have, however, taken resist- 
 ance measurement first, chiefly on account of its very great 
 importance, and also since " measurements " of resistance are 
 in general only comparisons of the resistances of coils with 
 that of some standard coil, and do not involve a measurement 
 of current. 
 
 The absolute unit of current is defined in terms of the 
 magnetic force which it produces, and is that current which, if 
 flowing in a circuit of 1 cm. length, bent into an arc of i cm. 
 radius, would exert unit force (one dyne) on a unit magnetic 
 pole placed at the centre of the arc. 
 
 The practical unit of current is the ampfere, and has a value 
 -^ absolute unit. 
 
 In making absolute measurements of current the magnetic 
 efiect alone is employed, but for ordinary measurements either 
 of the two other effects of the current may be used, viz. the 
 heating effect or the chemical effect, provided they have been 
 once for all standardized in terms of the magnetic effect. 
 
 130. Absolute Determinations of Current. — ^In accordance 
 with the definition of unit current, we must measure the mag- 
 netic force which the current exerts at a certain point ; it is, 
 however, impossible to construct an apparatus for this purpose 
 in terms of the definition, and therefore it is usual to employ 
 circular coils, and calculate the total effect which a current 
 flowing in them will produce at their centre. 
 
Standard Galvanometers. 127 
 
 In general two classes of instruments are employed in 
 making absolute determinations of current strength, these 
 being — 
 
 (i) Standard galvanometers. 
 
 (2) Standard electro-dynamometers. 
 In the first class of instrument the magnetic force set up by 
 the current flowing in a circular coil of known dimensions, acts 
 on a magnetic needle suspended at its centre, under the in- 
 fluence of a magnetic-controlling force of known strength, 
 whilst in the second class the galvanometer needle described 
 above is replaced by a coil of wire of known dimensions 
 through which the current to be measured also passes, its value 
 being deduced from the magnetic force with which the one 
 coil acts on the other. 
 
 131. In constructing an apparatus for the absolute measure- 
 ment of current, there are certain conditions which must be 
 fulfilled in order to insure accuracy, and which it will be as 
 well to enumerate. 
 
 {a) The " constants '' of the apparatus must be of a very 
 permanent character, and should not be affected to any 
 appreciable extent by slight relative displacements of the 
 various parts, such as might occur due to expansion, contraction, 
 or warping. 
 
 {b) There should be no variable quantity introduced into the 
 measurement over which the experimenter has not got complete 
 control. 
 
 {c) The number of measurements which involve readings of 
 the highest possible accuracy, should be as few as possible. 
 
 In the various methods for determining current in absolute 
 measure to be described, we shall point out wherein certain 
 methods have an advantage over others in respect to the above 
 conditions. 
 
 Standard Galvanometers. 
 
 132. Tangent GalvJnometer. — This instrument is so called 
 because the tangents of the angles of deflection of the needle 
 are proportional to the currents producing them, and is one 
 of the simplest and easiest to use of the various standard 
 
128 Practical Electricity and Magnetism. 
 
 current-measuring instruments, although in point of accuracy it 
 may be inferior to some of the others. 
 
 In its simplest form the tangent galvanometer consists of a 
 single coil of wire, at the centre of which a magnetic needle is 
 suspended, so that when under the directive influence of the 
 earth's magnetism only, its magnetic axis is at right angles to 
 the axis of the coil. When a current flows through the coil the 
 needle is deflected out of the meridian, until the moment of the 
 deflecting couple is balanced by the moment of the controlling 
 couple due to the earth. Calling C the value of the current in 
 absolute (C.G.S.) measure, m the strength of the magnetic pole, / 
 the length of the magnet, r the radius of the coil, 8 the angular 
 deflection of the needle, and H the horizontal intensity of the 
 earth's magnetic force, we have — 
 
 zttCot/ „ , „ . 
 ; — cos 8 = deflectmg couple 
 
 and /«/H sin 8 = controlling couple 
 
 zvOnl ,^^ . . 
 
 therefore cos 6 — m/tL sm 8 
 
 r 
 
 r 
 and C = — H tan 8 
 
 27r 
 
 If instead of there being only one turn in the coil there are n 
 turns, and the radius is sufficiently great for them to be assumed 
 to be all at the same distance from the centre of the coil, then — 
 
 C = ^ H tan 8 
 
 2Trn 
 
 The quantity — is, of course, invariable, once the coil is 
 
 wound, and is known as the " constant " of the coil, being 
 usually denoted by the letter G. We may therefore write the 
 above equation — 
 
 Q, = TT tan 8 
 
 Also, for a given place, assuming the earth's magnetism to be 
 constant at the centre of the cc 
 the formula may be written — 
 
 constant at the centre of the coil, — becomes a constant, and 
 
 Ktan8 
 
Standard Tangent Galvanometer. 
 
 129 
 
 The constant G of the coil is usually determined once for all 
 when the coil is wound, although it may be determined electri- 
 cally by comparing it with a coil of known constant at any 
 time, without unwinding it ; the method of doing this will be 
 described later on. 
 
 133. The main objection to the tangent galvanometer as a 
 standard current-measurer is that the calculation involves a 
 knowledge of H, the horizontal intensity of the earth's mag- 
 netic force, this being a somewhat variable and difficult 
 quantity to measure with accuracy. 
 
 In some cases the galvanometer-control is supplied by a 
 permanent magnet, in addition to the earth's control, and the 
 field due to the permanent magnet being much greater than 
 that due to the earth, small variations in the latter do not 
 become so important. Such an arrangement, although better 
 than the previous one, is always open to the objections to the 
 use of permanent magnets, which alter with time and tempera- 
 ture, making it necessary to redetermine the value of H from 
 time to time. 
 
 134. In actual practice, it is found necessary to have con- 
 siderably more than one turn in the galvanometer coil, and 
 since the turns cannot all be at the same distance from the 
 centre, a correction must be applied. If Fig. 73 represents a 
 
 Fig. 73. 
 
 section through the coil, the depth of the windings being 2d'' 
 and their breadth 2/, the mean radius of the coils being ;-, then 
 it can be shown ^ that — 
 
 ' Electrician, vol. xxvii. p. 716 ; a'so Mascart and Joubert's " Electricity 
 and Magnetism," vol. ii. p. 104. 
 
 K 
 
130 Practical Electricity and Magnetism. 
 
 and therefore — 
 
 C = J '-^ ^ H tan S 
 
 The values of d and / should be small compared with r the 
 mean radius of the coils, which latter dimension should not be 
 less than 15 cm. The best proportion between d and. /being — 
 
 d_>Jj, 
 
 I ~ 'sit 
 
 whilst in any case d should not be greater than fV ''■ 
 
 135. The needle of the tangent galvanometer must be small 
 compared with the radius of the coil, in order that when 
 deflected from its zero position it may move in a imiform 
 field. It should not be more than i cm. long, otherwise a 
 correction must be made for its length, this correction, however, 
 disappearing when the deflection is 27". For a needle of 
 length one-tenth the radius of the coil, the divergence from the 
 tangent law is about 0*5 per cent. 
 
 136. Construction. — The bobbin on which the wire is to be 
 wound should be made of some material which, while being 
 an insulator and absolutely non-magnetic, will not be liable to 
 change of shape due to warping, etc. In this latter respect, 
 wood, ebonite, etc., are unsatisfactory; wood well seasoned, 
 however, may be suitable, provided the coil be built up of six 
 or eight pieces inclined to one another, so that at different 
 parts of the bobbin the grain of the wood will run in different 
 directions. Metals, on the other hand, such as brass, 
 aluminium, etc., whilst being easy to turn up accurately, are 
 liable to contain iron, which renders them quite useless. 
 The best material of all would probably be white marble; 
 the expense of manufacture, however, is somewhat prohibitive. 
 Having got a suitable coil properly " turned up," the circum- 
 ference must be accurately measured by means of a steel 
 tape ; this having been recorded, the wire is carefully wound 
 on, the mean diameter inside and outside the insulation having 
 
Correction for Torsion. 1 3 1 
 
 been previously carefully measured by a micrometer gauge, 
 as it is required in order to calculate the mean diameter 
 of the coil when wound, this being the diameter of the bobbin 
 plus the depth of the windings. The number of turns wound on 
 is also recorded. 
 
 137. The needle must be suspended accurately in the centre 
 of the coil (both radially and axially) by means of a spider's- 
 web or quartz-fibre suspension, and may be arranged either to 
 indicate its deflections by means of a light pointer moving over 
 a graduated circular scale, or else more accurately by a mirror 
 lamp and scale. 
 
 Should the latter method be adopted, the needle may be 
 attached to the back of the mirror, and in reading the deflec- 
 tions it must be borne in mind that the angular deflection of 
 the spot of light is twice that of the needle. If a straight scale 
 is used, the deflection in scale-divisions, divided by the distance 
 from the mirror to the scale, also expressed in scale-divisions, 
 gives the tangent of twice the angle of deflection, and the 
 tangent of half that angle may easily be obtained from a table 
 of tangents. For approximate calculation, one-half the tangent 
 of twice the angle may be taken as equal to the tangent of half 
 the angle. 
 
 138. Correction for Torsion. — If the suspension is not too 
 short, and consists of a very fine quartz or spider's thread, it 
 will not be necessary to make any correction for torsion. 
 Should the suspension, however, be stout or short, then a 
 correction will be necessary, and may be made as follows. 
 The needle is allowed to come to rest at zero free from 
 torsion ; it is then, by means of a magnet, caused to rotate once 
 round, and the scale reading when it comes to rest is noted. 
 If there is any torsion, the spot of light will come to rest a little 
 to the opposite side of the zero from the direction of rotation. 
 If this angle is 6, then the coefficient of torsion, r, is — 
 
 e 
 ^ - 360 - e 
 
 so that instead of a deflection, 8, we write — 
 8(1 +t). 
 
133 Practical Electricity and Magnetism. 
 
 139. Adjustment. — In setting up the tangent galvanometer 
 (see Fig. 74), the coil must be set so as to lie in the plane of 
 the meridian, and then be carefully levelled till the needle 
 hangs exactly in the centre of the coil (radially and axially). 
 The lamp and scale is then set up at the proper distance from 
 the mirror, so that the distance from the centre of the mirror to 
 each end of the scale is the same, and the image of the cross 
 wire in the spot of light is at the zero on the scale. 
 
 Tig. 74. 
 
 A current sufficient to give a moderate deflection on the 
 scale is now sent through the instrument, and the deflection 
 noted ; the current is then reversed, and the deflection to the 
 opposite side of the zero noted. If this" is not the same as the 
 previous deflection (the current having been kept constant), it 
 means that the mirror and needle are not parallel to one 
 another, or, in the case of a pointer instrument, that the pointer 
 is not at right angles to the needle. 
 
Sensitiveness of a Tangent Galvanometer. 133 
 
 The mirror, or pointer and needle, must then be adjusted 
 relatively to one another until, on reversing the current, the 
 same deflection is obtained on both sides of zero. 
 
 140. Sensitiveness of a Tangent Galvanometer. — ^The position 
 of maximum sensitiveness in a tangent galvanometer is when 
 the needle is in the plane of the controlling field, as in that 
 position the moment of the deflecting force is a maximum, 
 whilst the controlling moment is a minimum. The minimum 
 error, however, introduced into the calculation by a given error 
 in reading the deflection, occurs when the deflection is between 
 40° and 50°, an error at that portion of the scale of yV 
 producing an error of o"35 per cent, in the result. 
 
 141. Determination of the Controlling Force at the Needk. — 
 The determination of the strength of the controlling force 
 acting on the needle may be made by the method of Gauss, 
 which will be fully described later (see par. 268), and which is 
 applicable both when it is due to the earth's control, or to the 
 earth and a permanent magnet; we will therefore refer the 
 student to the section dealing with the determination of 
 the horizontal intensity of the earth's magnetic force. After 
 having determined H, the time of swing (i D.V.) of the 
 galvanometer needle should be noted and recorded : let it be 
 Ti ; then, should the control be at any time altered, it can always 
 be brought back to the original value by adjusting the permanent 
 magnet, so that the needle has again a time of swing, Tj, or, 
 should it be necessary to reduce or increase the controlling 
 force, the new value of H may be calculated from the new time 
 of swing, for, if H and Hi represent the controlling forces which 
 give times of swing Tj and Tg, then — 
 
 H_TV 
 Hi - Ti= 
 
 142. Helmholtz Tangent Galvanometer. -^-In order to have 
 a more uniform field round the needle, Helmholtz designed a 
 tangent galvanometer having two equal coils, with their axes 
 on the same line and placed parallel to one another, the 
 distance apart of the coils being equal to their mean radius. 
 
 The needle is hung on the line joining the centres of the 
 coils, midway between them. 
 
134 
 
 Practical Electricity and Magnetism. 
 
 This arrangement (see Fig. 75) eliminates the terms of the 
 second order from the calculation, the current in absolute 
 measure beins; obtained from the relation ^ — 
 
 C = 
 
 {r + x^^. 
 
 H tanS 
 
 where 11 = number of turns on each coil ; 
 /• = mean radius of each coil ; 
 2x = mean distance apart of the two coils. 
 
 Flc. 7S. 
 
 In order to see the effect of the two coils on the field round 
 the needle, the student is referred to Maxwell's " Electricity 
 and Magnetism," vol. ii., where plates showing the distribution 
 of the field for one and two coils are given. In this galvano- 
 meter, when the deptli of the windings is small compared with 
 the mean radius of the coils, the correcting factor for the length 
 of the needle practically vanishes. 
 
 143. In a single-coil tangent galvanometer, the following 
 were the dimensions : — 
 
 ' See " Elements of the Mathematical Theory of Electricity and 
 Magnetism" (J. J. Thomson), par. 211. 
 
Standard Sine Galvanometer. 13^ 
 
 Mean radius of coils, r = 9-67 cm. 
 Radial depth of coils, 2d = 0-20 „ 
 Axial length of coil, 2/ = 2*40 „ 
 Number of turns, ti = 2"] 
 Hence the coil-constant G is — 
 2™f d^ ^r-i 
 
 ^ 2 X 3-142 X 27 ( o-io' 1-2^ i 
 
 9'67 r "'"3 X 9'67'~ 2 X 9-674 
 
 = 1755 X o'987 
 
 = i7"32 
 The torsion coefEcient of the silk-thread suspension was 
 determined by setting the spot of light on the scale to zero, then, 
 on rotating the mirror and needle through one complete revolu- 
 tion, the spot of light was found to be ten scale-divisions from 
 the zero. The scale was 1000 mm. from the mirror, and the 
 scale-divisions were half-millimetres. 
 
 10 
 
 Hence — — = o-oo^ = tan o-^" 
 
 2 X 1000 '' -^ 
 
 therefore 5 = 03" 
 and T, the torsion coefEcient, is — 
 
 0-3 
 
 360 -e 360 - 0-3 
 
 = 0-0008 
 
 144. Sine Galvanometer. — The standard sine galvanometer 
 is very similar to the standard tangent galvanometer in con- 
 struction (see Fig. 76), and the remarks which apply to the 
 construction of one apply equally to that of the other, with this 
 difference, however, that in the sine galvanometer the bobbin 
 carrying the coils is capable of rotation about a vertical axis, a 
 separate scale and pointer frequently being attached so that 
 the angular rotation may be measured. 
 
 The adjustment of the sine galvanometer is similar to that 
 of the tangent In using the instrument after the pointer has 
 been set to zero under the influence of the controlling force, 
 the current is sent through it and a deflection of the needle 
 obtained, the coils are then rotated so as to follow up the 
 
1 ^6 Practical Electricity and Magnetism. 
 
 motion of the needle, the current being kept constant; this 
 has the effect- of making the needle deflect still further, but 
 eventually, provided the current is not too strong, the coils will 
 gain upon the needle, and the zero on the scale may be brought 
 
 Fig. 76. 
 
 under the pointer in its new position ; we have then equilibrium 
 between the controlling and deflecting moments, the deflecting 
 force acting at right angles to the needle, and therefore exert- 
 ing its maximum turning moment. 
 
 If we call, as before, C = current in C.G S. units, 
 
 ;■ = radius of the coil, 
 
 n = number of turns on coil, 
 
 $ — angular rotation of coil, 
 
 f/t = pole strength of needle, 
 
 / = length of needle, 
 
 H = strength of controlling field, 
 
 we have — 
 
 , . . 2irnC>nl 
 deflectmg moment = : — 
 
 and controlling moment = Kml sin Q 
 
Statidard Sine Galvanometer. 137 
 
 therefore — 
 
 2irnCml TT , • « 
 
 = timl sin 6 
 
 r 
 
 and C = -^ H sin & 
 2irn 
 
 ^ H . , 
 or C = pSin d 
 
 . 2trlt 
 
 G being — ^, the galvanometer constant ; or, as before (see 
 par. 134), if there are several layers on the galvanometer coil — 
 
 145. For a given controlling field the sine galvanometer does 
 not admit of a very large range of current measurement, since, 
 if the deflection is at all large, on rotating the coils the position 
 of instability of the needle is soon reached, when it turns right 
 round ; so that, if required to measure currents of widely differing 
 values, an adjustable controlling" field must be provided, or else 
 the galvanometer must be shunted. 
 
 The sine galvanometer is, however, more sensitive than the 
 tangent, its maximum sensitiveness being reached just at the 
 point of instability of the needle. 
 
 146. The great advantage of the sine law instrument over the 
 tangent instrument is in the case where the relative values of 
 two or more currents are required to be measured, or where 
 the constant of the instrument is obtained by comparison with - 
 a standard measuring instrument and not calculated from the 
 dimensions of the coils, because all galvanometers used in the 
 above manner follow the sine law independently of the shape of 
 the coil, whilst only ciircular coils will follow the tangent law. 
 We may therefore obtain the relative calibration curve of any 
 galvanometer, by plotting the deflections of the needle when 
 the current is sent through it before the coils are rotated, against 
 the sine of the angular rotation of the coils required to bring the 
 scale zero underneath the pointer. 
 
 Should the galvanometer coil not be provided with a special 
 scale and pointer to register its angular rotation, that may be 
 
138 
 
 Practical Electricity and Magnetism. 
 
 easily measured in the following way. After the current has 
 been sent through the instrument, and the deflection 8 of the 
 needle measured, the coils are rotated round to follow up the 
 motion of the needle, until the zero on the scale stands under- 
 neath the pointer. The current is then broken, and the scale- 
 reading, ^, where the pointer comes to rest noted; then <^ 
 represents the angle through which the coils have been rotated, 
 and sin <^ is proportional to the current ; therefore, by plotting 
 values of 8 against sin ^ we get the relative calibration curve of 
 the instrument. 
 
 147. Gray's Standard Sine Galvanometer. — One of the 
 objections to the ordinary form of standard sine galvanometer is 
 the difficulty of measuring the coil constants accurately; this 
 difficulty has been overcome in a modified form of sine galva- 
 nometer due to Professor T. Gray,^ in which a long solenoid 
 
 Fig. 77. 
 
 is employed instead of a ring-shaped coil. In this way, by 
 using a solenoid whose length is from eight to. ten times its 
 radius, the field produced in the mean plane of the coil is very 
 uniform, and may be calculated with great accuracy. 
 
 The coil of about 10 cm. diameter is mounted in a tube, T 
 
 (see Fig. 77), which is free to rotate about a vertical axis, V, 
 
 attached to the base P, the latter being mounted on three 
 
 levelling screws, L, L, L. A pointer is attached to one end of 
 
 ' Phil. Mag., vol. xxii. Oct., 1886. 
 
Gray's Standard Sine Galvanometer. 139 
 
 the tube, and moves over the scale S, so that the angular rota- 
 tion of the coil may be measured, the end of the coil at S being 
 supported on the small feet/,/ 
 
 The needle which is attached to the back of the small mirror 
 m is suspended at the centre in the mean plane of the coil. 
 At the end of the tube opposite the mirror there is a small slit, 
 s, supplied with cross wires, and above it a plane mirror, M. 
 When the cross wires are illuminated their image is reflected 
 from m to M, and to the telescope t at the other end of the 
 tube. 
 
 The coil is levelled«and turned round until the image of the 
 cross wires at S coincide with those in the telescope ; the needle 
 is then in the meridian, and at right angles to the axis of the 
 coil. On sending a current through the instrument, the needle 
 deflects ; and in order to again make the cross wires coincide 
 the coil must be rotated through an angle, which may be read 
 off on the scale S. Let this angle be ; then, when unit current 
 flows in the coil, the magnetic force at its centre is — 
 
 where 2/ = length of the coil ; 
 
 71 = numbers of turns per centimetre ; 
 
 ;■ = radius of coil. 
 
 Expanding this, we get — 
 
 / r'^ ?'' \ 
 
 /= 4Tr/i( I - ij2 + i/j - etc. J 
 
 Taking as far as the second term in this expression, we get, for 
 the current C in absolute measure — 
 
 Hsin e 
 
 C 
 
 47r« 
 
 (.-4) 
 
 H being the value of the magnetic controlling force at the 
 needle. 
 
 From this formula it will be seen that when the length of 
 the coil is great compared with its diameter, a small error in 
 the determination of ;■ will produce an extremely small effect 
 in the calculation of the current. 
 
140 Practical Electricity and Magnetism. 
 
 Standard Electro-Dynamometer. 
 
 148. This form of current measurer differs from those pre- 
 viously mentioned in that the permanent magnet system is 
 replaced by a small coil connected in series with the large coil, 
 at the centre of which it is suspended by means of a bifilar 
 suspension, which in addition to supplying the controlling force, 
 conveys the current into and out of the small coil. 
 
 The normal position of the small coil, when no current is 
 flowing, is with its axis at right angles to, that of the large coil, 
 and in the magnetic meridian. 
 
 When a current is sent through the instrument, the small coil 
 tends to set itself coaxially with the larger, this tendency being 
 balanced by the controlling couple due to the earth and that 
 due to the suspension. 
 
 If we call C = current in absolute measure ; 
 G = constant of the large coil ; 
 g^ = constant of the small coil ; 
 H = strength of the earth's controlling field ; 
 6 = angular deflection of the small coil ; 
 K = constant of the bifilar suspension ; 
 
 then, when the deflecting and controlling moments are in 
 equilibrium — 
 
 CGg cos 6 = C^H sin 61 + K sin $ 
 from this we have — 
 
 C^H is always small compared with K, so that, on expanding 
 the above, we get — 
 
 tan0 
 
 K 
 
 ^^ K 
 
Standard Electro- Dynamometer. 141 
 
 If we now reverse the current in the coils, we get a shghtly 
 different deflection, 61, and — 
 
 tan (9i = -g- 4- -^, — 
 
 whence, by addition, we get — 
 
 tan Q + tan 6-^ = 
 
 K 
 
 and C = -^ (tan 6 + tan 6,) ^ 
 
 149. We have next to determine K, the constant of the 
 bifilar suspension, and in' order to do this we proceed as fol- 
 lows. The constant depends on the mass (M) of the suspended 
 coil, and varies proportionally with it; hence we may write — 
 
 K = Mt 
 
 where t is a constant depending only on the suspension. To 
 determine t we set the coil vibrating about a vertical axis, and 
 determine its time of double vibration, T. From this we get 
 the well-known relationship — 
 
 T = 27r n/ g. 
 
 I being the moment of inertia of the oscillating system. To 
 the coil is now attached a bar of non-magnetic material of 
 known mass, M', and moment of inertia, I', and the time of one 
 double vibration is again taken : let it be T'. Then — 
 
 r=27r^ 
 
 i + r 
 
 t(M + M') 
 
 from the equations for T and T' we get — 
 
 . , 47r^rM 
 ^ - M(Ti^ - T^) + MiTj^ 
 
 150. In order to insure greater uniformity of field at the 
 
 ' St€ Electrician, vol. xxviii. p. 272. 
 
142 Practical Electricity . and Magnetism. 
 
 suspended coil, the large coil, in some forms of dynamometer, 
 is replaced by two coils arranged after the manner of the coils 
 in the Helmholtz tangent galvanometer, the small suspended 
 
 Fig. 78. 
 
 coil taking the place of the needle in that instrument ; such an 
 arrangement is shown in Fig. 78. 
 The chief objections to the absolute electro-dynamometer 
 
Standard Current Balance. 143 
 
 are the uncertainty introduced by the bifilar suspension, unless 
 its constant is redetermined for each measurement of current, 
 and the difficulty of accurately measuring the mean radii of the 
 two coils and the angular deflection of the suspended coil. 
 
 151. Current Balance. -^Qi all the methods proposed for the 
 absolute measurement of current, the current-balance method is 
 perhaps the most accurate. It has the great advantage over 
 the other instruments previously described, that the final 
 calculation for the current does not involve so many measure- 
 ments requiring first-class accuracy. 
 
 In its simplest form the current balance consists of two coils, 
 one of large and one of small diameter, the coils being placed 
 with their planes horizontal, the smaller one being suspended 
 coaxially above the larger from the beam of a dehcate balance. 
 The coils are arranged in series, the current being calculated 
 from the dimensions and the attraction of one for the other. 
 
 If we call A = mean radius of the large coil ; 
 a = mean radius of the small coil ; 
 X = distance between the mean planes of the coils ; 
 C = current in absolute measure ; 
 fin' = the number of turns on the coils ; 
 m = mass in grammes which counterbalances the 
 
 force between the coils ; 
 g = acceleration of gravity ; 
 then the force acting between the coils can be shown to be ' — 
 
 67rVA^C^a;««' , 
 ■^= (A= + ^)i ^^"^^ 
 = mg 
 
 mg{A^ + x")! 
 Hence C" - g^^a^^^^^w' 
 
 It can also be shown that the best conditions obtain when 
 X = —. Putting this value into the equation, we get — 
 , A^ mg T 
 
 ^ ^ «2 A ^,f X 
 
 a^ ^ mi "• 16-97 
 
 ' Electrician, vol. xxvii. p. 250; also "Mathematical Theory of Elec- 
 tricity and Magnetism " (J. J. Thomson), par. 216, 
 
144 Practical Electricity and Alagnetisrn. 
 
 From this result it will be noted that we do not require to know 
 the exact value of the radius of either coil, but only the ratio 
 of the squares of the radii. This is a very important point, since 
 the exact determination of the mean radius of a coil is a difficult 
 matter, whilst the ratio of the radii of the coils may be deter- 
 mined by an electrical method with great accuracy. All the 
 other quantities in the calculation are capable of measurement 
 with great accuracy. 
 
 In making a measurement, the current may be reversed in 
 both coils, and then first in one and then in the other, the mean 
 of four values of m, two attractions and two repulsions, being 
 taken. 
 
 152. In order to obtain greater sensitiveness and uniformity 
 of field. Lord Rayleigh has constructed a balance having two 
 fixed coils, one placed coaxially above the other, parallel to it, 
 the small coil being suspended midway between the two large 
 coils. For further particulars the student is referred to Lord 
 Rayleigh's paper on the determination of the electro-chemical 
 equivalent of silver,' for which research the balance was con- 
 structed. 
 
 153. Determinatio7i of the Ratio of the Radii of Two Coils by 
 Bosichds Method. — In a coil of wire the cross section of which 
 is rectangular, it is easy to show " that if 2/ represents the axial 
 length of the coil, 2d the radial depth, n the number of turns, 
 and r the mean radius, that the constant G is — 
 
 
 Consequently, if Gj, «i, r-^, d^, and l^ represent these values for 
 one coil, and G2, «2, r^, d^, 4 the values for another coil 
 then — 
 
 G, 
 
 I d^ /"l 
 
 , I "T 3 ,. a — "2 .. 2 ' 
 
 ' Phil. Trans, Part II., 1884. 
 
 = Mascart and Joubert, " Electricity and Magnetism," vol. ii. p. 104. 
 
Tlie Ratio of the Radii of Tiuo Coils. 145 
 and therefore — 
 
 1\ ~ WaGi 
 
 Now, since the ratios — ,■ — , — , — are small, their squares may 
 rx n i\ i\ 
 
 either be neglected or the values calculated from approximate 
 
 data with sufficient accuracy ; we may therefore write — 
 
 In order to get the ratio — ^, the coils are placed with their 
 
 Gi 
 
 planes vertical, one being inside the other and coaxial with it, 
 
 the planes of the coils being parallel to the magnetic meridian. 
 
 A small error in the adjustment of the two coils affects the 
 
 ratio as the square of the displacement.^ 
 
 A very small magnet attached to the back of a mirror is then 
 suspended by a fibre at the common centre of the two coils. 
 The coils are connected in parallel, but in such a way that if 
 the current goes round one in a clockwise direction, it goes in 
 a counter-clockwise direction round the other. Resistances 
 are inserted in the circuit of each coil, the connections being 
 shown diagrammatically in Fig. 79, where, for the sake of 
 clearness, the coils are shown as if they were lying apart from 
 one another. A represents the large and B the small coil, C the 
 battery, K the key, and p^ and /)„ the resistances in series 
 with the coils A and B. 
 
 The resistances in the circuits of the coils are then adjusted 
 until, on closing the circuit at K, no deflection is observed on 
 the small needle suspended at the common centre of the two 
 coils, this being observed with a lamp and scale in the usual 
 way, the whole arrangement being similar to a differential 
 galvanometer. When an exact balance is obtained, if c^ and 
 Ci represent the currents in A and B respectively, and Gj and 
 
 ' P/ii/. Trans, Roy. Soc, 1885. 
 
146 Practical Electricity and Magnetism. 
 
 G2 are the constants of A and B, Rv and Rg being the resist- 
 ances of the two coils, then — 
 
 But Ri<ri = R^fo, where Rj = (p^. + Ra) and R^ = (p„ + Rj) ; 
 therefore ;=r = ^ 
 and'i=^t^ 
 
 Ta 7«2Ri 
 
 It will be noticed that we do not require to know the exact 
 value of the resistance of either coil, but only the ratio of the 
 
 lio. 79. 
 
 two resistances, this being capable of measurement with very 
 great accuracy, as will be seen by referring back to the chapter 
 dealing with resistance measurement. 
 
 154. A neat method of comparing the resistances of the two 
 coils whilst in position is due to Lord Rayleigh. In it the two 
 coils, one inside the other, are connected so as to form the 
 two arms of a Wheatstone bridge, which we will diagrammatically 
 represent as follows (Fig. 80). A and B represent the two 
 coils^ — really one inside the other, but shown separated for the 
 sake of clearness ; the common terminal a is connected to the 
 battery through a reversing key, K ; the other ends of the coils 
 
The Ratio of the Radii of Two Coils. 147 
 
 are connected to the mercury cups b and d, the coil B being 
 connected to d through a known resistance, Rj ; two resistance 
 coils, Ra and R3, in parallel form the third, and a coil, R4, the 
 fourth arm of the bridge ; the bridge galvanometer, G, is placed 
 across bd. A thick copper strap connected to the battery, and 
 shown in dotted lines, can also connect b to d. 
 
 When the ratio of the constants of the coils is to be deter- 
 mined, the copper strap is placed across bd, and Ri is adjusted. 
 
 Fig. 80. 
 
 until, on completing the battery circuit, there is no deflection 
 obtained on the little needle suspended at the common centre 
 ■of the two coils. The strap is then removed, and the battery 
 connected to C ; the coil R3 is then adjusted, till, on completing 
 the battery circuit, the bridge galvanometer G shows no deflec- 
 tion. Then — ■ 
 
 GV 
 
 R4 
 
 -R^' 
 
 < -5> being the resistance of the two coils R2 and R3 in 
 R3 
 
 parallel, which may be calculated.from their separate values. 
 
148 Practical Electricity and Magnetism. 
 
 155. Fig. 81 shows a form of current balance designed by 
 the author for use in the laboratory. It consists of a large coil 
 mounted on a brass stand supplied with levelling screws of such 
 a size that it will go inside the case of a chemical balance. 
 The smaller coil is wound on an ebonite bobbin, and so 
 arranged that it can be suspended from one arm of the balance 
 when the scale-pah is removed. The brass rods supporting it 
 have adjustable screws at their ends, to admit of the coil being 
 
 levelled properly. Inside the ebonite bobbin fits a brass ring, 
 from which a galvanometer mirror and needle may be sus- 
 pended, when it is required to make a measurement of the ratio 
 of the radii by the method of Bosscha. In order to be able 
 to adjust the suspended coil accurately, relatively to the large 
 coil both as regards centering and distance apart of the mean 
 planes, brass templates were made of the exact distance between 
 the outer circumference of the ebonite coil and the inner 
 
Laboratory Form of Current Balance. 149 
 
 circumference of the large coil, and also of the exact distance 
 between the upper flanges of the two coils when their mean 
 planes were the proper distance apart. The following are the 
 data of dimensions and winding of the coils ; — 
 
 2-2 
 
 Fig. 82.— Small Coll. 
 
 Small Coil. — The external circumference of the windings 
 was carefully measured by a steel tape, and was 6^1", the 
 total number of turns was 60, being 10 layers each of 6 turns. 
 The resistance was i'5 ohms. 
 
 Fig. 83. — Large Coil. 
 
 Large Coil. — The external circumference of the windings, as 
 measured by a steel tape, was 15^', the total number of turns 
 was 460, being 20 layers of 23 turns each. The resistance 
 was 4 ohms. 
 
 In setting up the apparatus, the scale-pan is removed from 
 one side of a chemical balance, and the small coil hung in its 
 place. Very thin insulated copper wires are led from the coil 
 to the standard of the balance where they are attached, and then 
 led to the rest of the apparatus. The large coil is now adjusted 
 in position, and both it and the small coil levelled carefully 
 by means of a spirit-level and the templates until they are at 
 the required distance apart.. Weights, are placed ija the other 
 
150 Practical Electricity and Magnetism. 
 
 scale-pan, to counterbalance the coil, and the final adjustment 
 made with sand. 
 
 156. The most convenient way to use the balance is to place 
 known weights in the scale-pan, and then adjust the current in 
 the coils until the attraction between them exactly counter- 
 balances the weight, and brings the pointer of the balance back 
 to the scale zero. It will be found that the balance is extremely 
 sensitive to small variations of current, and it is .better to 
 arrange two stops, one on either side of the pointer, to prevent 
 it swinging too far from the zero position. A liquid or carbon 
 resistance should be placed in series with the balance. 
 
 157. The following comparison was made between a copper 
 voltameter and the above current balance. An accuracy of 
 more than i per cent, was not aimed at, on account of the 
 insensibility of the balance used, which was an old one, and not 
 that for which the coil had been designed to be used. " The two 
 coils were connected in series with each other and with a 
 copper voltameter, carbon resistance, and six secondary cells, 
 and the following data obtained : — 
 
 Weight of cathode before deposition 6o"240 gm. 
 
 ,. after ., 60-903 ,, 
 
 Gain in weight o'663 ,, 
 
 Time of deposit 22 m;n. 12 sec. 
 
 Temperature... ... ... ... ... ... ... 12° C. 
 
 Weight required to counterbalance the attraction of 
 
 the coils 2153 gm. 
 
 Taking e for copper as o'ooo3z79, we get — 
 
 (-. _. °'663 
 
 ■O-0003279 X 1332 
 
 = I 'SI 8 amperes. 
 
 The corrected ratio for the radii of the coils was determined 
 
 by the method of Bosscha, and A = 2-28 
 
 a 
 
 Hence C = r^ZsJ /^SS X 981-3 
 ^ 460 X 60 X 16-97 
 = 0-153 absolute tmits. 
 = 1-53 ampferes. 
 
Voltameters. i 5 1 
 
 This shows a very close agreement between the measurements, 
 and if the balance had been more sensitive, the determination 
 of current would have been much more accurate. 
 
 158. In determining the ratio of the radii of the two coils by 
 the method of Bosscha, we have — 
 
 'I + 3,-2— 2 ,. 2 ; 
 
 The resistances required in series with the coils, in order to 
 obtain a balance when a current was sent through the two in 
 parallel, were 1000 ohms in series Avith the smaller coil of 60 
 turns, and 3392 ohms in series with the coil of 460 turns. The 
 axial lengths and radial depths of the coils were measured, and 
 the following data obtained : — 
 
 2/, = o'375 ' 23'i = o'488' i\ = i'o5o'' 
 
 24 = I".25o' 2i/i = I"92I ' i\ = i'96o' 
 
 From this we get — 
 
 I i (o'244f 1 ( o-i87f 
 ^ + ^ (o-8o6f ~ ^ (o-8o6f 
 
 , 1 (0-960)' i(^25f 
 ^ + ^ (1-960)= ~ 2 (1-960)= 
 
 0-997 
 
 Ti 60 X 3392 I 
 
 Therefore — = —? — — X o'gg? = -— rs 
 
 r^ 460 X 1000 ^" 2'26 
 
 Voltameters. 
 
 159. We have seen in the preceding sections how an electric 
 current may be determined in absolute measure by means of 
 the magnetic effect. For ordinary use in the laboratory, in the 
 standardization of instruments, etc., it is found more convenient 
 to measure current by means of its chemical effect. The value 
 of the current cannot be calculated from the chemical decom- 
 position produced in the same way that it can from the 
 deflection of a standard galvanometer, or from the balancing 
 weights required in a current balance, but the constant depend- 
 ing on the nature of the decomposition, and called the electro- 
 chemical equivalent, is determined experimentally by comparison 
 
152 Practical Electricity and Magnetism. 
 
 with one of the standard current measurers. We are therefore 
 able to employ the chemical decomposition produced by a 
 current as a sort of secondary standard. The object of thus 
 setting up a secondary standard, instead of using one of the 
 standard methods, is that the chemical method of measuring 
 current is much easier to carry out, and requires less complicated 
 apparatus than the other methods, whilst being very accurate. 
 
 Faraday was one of the first to investigate the chemical 
 eflfects produced by an electric current, quantitatively, and the 
 results of his researches may be briefly summed up in his own 
 words. 
 
 " For a constant quantity of electricity, whatever the decom- 
 posing conductor may be, whether water, saline solutions, acids, 
 fused bodies, etc., the amount of electro-chemical action is also 
 a constant quantity, i.e. would always be equivalent to a 
 standard chemical effect founded upon ordinary chemical 
 affinity." ' 
 
 It will therefore be seen that if we can once for all deter- 
 mine the " amount of chemical action " produced by a known 
 quantity of electricity, we can at any time measure a quantity 
 of electricity by finding the amount of chemical action which it 
 produces, and comparing it with that produced by a known 
 quantity of electricity. Also, since the quantity of electricity is 
 equal to the current flowing, multiplied by the time during 
 which it flows, we can, by dividing the quantity of electricity by 
 the time, calculate the current strength. 
 
 The determination of the " amount of chemical action " in a 
 substance produced by a known quantity of electricity is 
 usually called the determination of the electro-chemical equiva- 
 lent of the substance, this being expressed as the number of 
 grammes of substance electrolysed per coulomb of electricity. 
 A coulomb being the quantity of electricity which passes when 
 one ampbre flows for one second. 
 
 1 60. Numbers of researches have been made to determine 
 
 the electro-chemical equivalents of different substances ; some 
 
 of these results are given in the table at the end of the book. 
 
 It is found, however, that comparatively few substances fulfil the 
 
 • Faraday's "Experimental Researches," vol. i. par. 505. 
 
Tfie Silver Voltameter. 153 
 
 conditions necessary for secondary standard current measurers ; 
 of these silver stands out as better than the others, and for this 
 reason it has been adopted as the substance to be used in 
 current measurement. The legal definition of the ampfere — 
 the practical unit of current — being expressed as follows : 
 The ampbre is . that current " which has the value of one-tenth 
 in terms of the centimetre, the gramme, and the second of time, 
 and which is represented by the unvarpng electric current, 
 which, when passed through a solution of silver nitrate in water, 
 in accordance to the specification appended hereto and marked 
 A, deposits silver at the rate of o'ooiiiS gramme per 
 second." ^ 
 
 161. The determination of the electro-chemical equivalent of 
 silver was made by Lord Rayleigh,^ to whose paper the student 
 is referred for details of the measurement. 
 
 Although the silver voltameter affords the most accurate 
 method of measuring a current by chemical means, yet in 
 ordinary laboratory work the copper voltameter is generally 
 employed, on account of greater cheapness and ease of manipu- 
 lation, although there are various sources of error which must 
 be guarded against, and which will be treated of later. 
 
 162. In regard to the silver voltameter, we cannot do better 
 than reproduce the specification for its preparation referred to 
 in the legal definition of the ampfere, since this represents the 
 result of long and patient investigation with varying conditions. 
 
 " In the following specification the term silver voltameter 
 means the arrangement of apparatus by means of which an 
 electric current is passed through a solution of nitrate of silver 
 in water. The silver voltameter measures the total electrical 
 quantity which has passed diuring the time of the experiment, 
 and by noting this time the time average of the current or, if 
 the current has been kept constant, the current itself can be 
 deduced. 
 
 " In employing the silver voltameter to measure currents of 
 about one ampfere, the following arrangements should be 
 adopted. The cathode on which the silver is to be deposited 
 
 ' See London Gazette, Friday, Aug. 24, 1894. 
 = Phil. Trans. Roy. Soc, 1884. 
 
154 Practical Electricity and Magnetism . 
 
 should take the form of a platinum bowl not less than lo cm. 
 in diameter, and from 4 cm. to 5 cm. in depth. The anode 
 should be a plate of pure silver some 30 sq. cm. in area and 
 2 mm. or 3 mm. in thickness. 
 
 " This is supported horizontally in the liquid near the top of 
 the solution by a platinum wire passing through holes in the 
 plate at opposite corners. To prevent the disintegrated silver 
 which is forriied on the anode from falling on to the cathode, 
 the anode should be wrapped round with pure filter paper 
 secured at the back with sealing wax. 
 
 " The liquid should consist of a neutral solution of pure silver 
 nitrate, containing about 15 parts by weight of the nitrate to 
 85 parts of water. 
 
 " The resistance of the voltameter changes somewhat as the 
 current passes. To prevent these changes having too great an 
 effect on the current, some resistance besides that of the volta- 
 meter should be inserted in the circuit. The total metallic 
 resistance of the circuit should not be less than 10 ohms. 
 
 " The platinum bowl is washed with nitric acid and distilled 
 water, dried by heat, and then left to cool in a desiccator. 
 When thoroughly dry it is weighed carefully. 
 
 " It is nearly filled with the solution, and connected to the rest 
 of the circuit by being placed on a clean copper support, to 
 which a binding screw is attached. 
 
 " This copper support must be insulated. The anode is then 
 immersed in the solution so as to be well covered by it, and 
 supported in that position ; the connections to the rest of the 
 circuit are made. 
 
 " Contact is made at the key noting the time of contact. The 
 current is allowed to pass for not less than half an hour, and the 
 time at which the current is broken is observed. Care must be 
 taken that the clock used is keeping correct time during this 
 interval. 
 
 " The solution is now removed from the bowl, and the deposit 
 is washed with distilled water and absolute alcohol, and dried 
 in a hot-air bath at a temperature of 1 60° C. After cooling in 
 a desiccator it is weighed again. The gain in weight gives the 
 silver deposited. 
 
The Copper Voltameter. 155 
 
 " To find the current in ampferes, this weight, expressed in 
 grammes, must be divided by the number of seconds during 
 which the current has passed, and by coon 18. 
 
 " The result will be the time average of the current, if during 
 the interval the current has varied. 
 
 " In determining by this method the constant of an instru- 
 ment, the current should be kept as nearly constant as possible, 
 and the readings of the instrument observed at frequent in- 
 tervals of time. These observations give a curve from which ■ 
 the reading corresponding to the mean current (time average of 
 the current) can be foimd. The current, as calculated by the 
 voltameter, corresponds to this reading." 
 
 163. In connection with the silver voltameter, it is worthy of 
 note that Professor T. Gray,"^ who has had large experience 
 with it, advocates the use of silver plates for both cathode and 
 anode, instead of a silver anode and a platinum bowl cathode. 
 The advantages claimed being that, on account of the lightness 
 of the silver cathode, a more delicate balance may be used on 
 which to weigh it ; also it is much easier to clean the silver 
 plates before the experiment than the platinum bowl. From 
 a large number of experiments he found the best results were 
 obtained from a solution of silver nitrate containing from 5% to 
 10% by weight of silver nitrate, and working at a current density 
 of 200 to 600 sq. cm. per ampfere. If these limits were exceeded, 
 it was found that the silver deposit became unsatisfactory and 
 difficult to wash, on account of its not adhering firmly to the 
 cathode. 
 
 164. Copper Voltameter. — For most purposes not requiring 
 the highest accuracy it is usual to employ the copper volta- 
 meter in preference to the silver one, since the manipulation of 
 the plates do not require the same skill, nor are the conditions 
 for accurate work so limited. The copper voltameter has, 
 however, certain peculiarities which, on account of its frequent 
 use in the laboratory, merit special attention. 
 
 Many experimental researches have been undertaken from 
 time to time with a view to perfecting this instrument, but of 
 these perhaps the most important is the work done by Professor 
 • Pkil. Mag., Nov., 1886. 
 
156 Practical Electricity and Magnetism. 
 
 T. Gray^ in determining the value of tlie electro-chemical 
 equivalent of copper, and we will largely follow his suggestions 
 in connection with this part of the subject 
 
 The voltameter consists of a glass vessel containing a solution 
 of copper sulphate, into ^yhich, suspended from suitable clips, 
 the copper electrodes dip. The plates should be made of good 
 electrolytic copper, the cathodes being as thin as is consistent 
 with strength, in order that the gain in weight due to tjie deposit 
 may represent a larger fraction of the total weight of the plate 
 than it otherwise would if the plates were heavy; also, by 
 having the plates light a more delicate balance may be 
 employed, and the weight determined with greater accuracy. 
 The anodes may be made of considerably thicker copper plate, 
 since they will be gradually dissolved away. 
 
 The plates should be cut with a lug at the top for attach- 
 ment to the clips, all the corners being rounded off, and no sharp 
 edges left, since it is found that the deposit tends to form in a 
 crystalline manner at such places. The clips for holding the 
 plates should be of brass, copper, or platinoid springs, so that a 
 good electrical contact may be obtained, whilst the plates may 
 easily be removed for cleaning and weighing. A convenient 
 form of clip is shown in Fig. 84, consisting of a copper spring 
 pressing on a copper plate fastened to a backing of ebonite. 
 The plates should be about i cm. apart and perfectly parallel 
 to one another, otherwise the deposit will not be uniform over 
 the surface. 
 
 165. Treatment of the Plates. — The plates, before use, must be 
 thoroughly cleaned and polished with sand-paper, the sand 
 being afterwards removed by placing them in running water 
 and rubbing with a clean rag or brush. On no accotmt are the 
 fingers to be placed on that part of the plate which is to receive 
 the deposit. 
 
 If the plates are oxidized they may be cleaned by dipping 
 them into a bath of potassium cyanide, care being taken to have 
 excess of cyanide present. After removal from the cyanide 
 they must be thoroughly washed with water. After cleaning, 
 
 • Phil, ^oj-., vol. xxii., Nov., 1886 ; also Phil. Mag., vol. xxv., Mar., 
 i8?8. 
 
The Copper Voltameter. 
 
 157 
 
 the plates are placed in the voltameter, each cathode being 
 between two. anodes, the number of cathodes required, depend- 
 ing on the current to be measured and the size of the plates ; 
 tables of data will be given later. 
 
 In the case of the standardization of an instrument, the circuit 
 will consist of the voltameter, the instrument to be cahbrated (if 
 necessary so arranged that the current in it may be reversed), a 
 
 Fig. 84. 
 
 variable resistance, break-circuit key, and secondary battery. 
 The resistance is adjusted until the required current is obtained, 
 as indicated roughly by the reading on the instrument ; this is 
 allowed to flow for fifteen minutes, the cathode is removed and 
 at once plunged into a bath of slightly acidulated water, to 
 remove the copper sulphate solution from the surface before 
 
IS8 Practical Electricity and Magnetism. 
 
 the plate can oxidize; it is then thoroughly washed in tap- 
 water and dried in slightly heated clean white blotting-paper ; 
 after allowing to cool in a dessicator, it is weighed, the weight 
 being taken to yg- mg. The plate should be lifted by means 
 of forceps, and, if possible, the fingers should never touch it. 
 The plate is now carefully replaced in the voltameter, and 
 the current started, the time being taken on a stop-watch. 
 During the deposition, which should last one hour, the current 
 must be kept constant, by means of the variable resistance 
 in circuit, so that the reading on the instrument remains the 
 same. When the current is stopped, and the time taken, the 
 cathode is removed and subjected to exactly the same treat- 
 ment as before, and then weighed. If W represents the gain 
 in weight of the cathode in grammes, T the time in seconds 
 during which the deposition has lasted, c the electro-chemical 
 equivalent of copper, and C the current in ampferes, then — 
 
 C =^.p 
 
 1 66. Having described the method of using the voltameter, 
 we will briefly state the causes which effect the value of e. 
 These are found to be — 
 
 {a) The size of the plates, i.e. the current density at the 
 electrodes. 
 
 (p) The density of the solution. 
 
 (c) The chemical action of the solution on the plates. 
 
 (d) The temperature. 
 
 These points have been investigated more or less fully by Gray 
 , and others,'^ and Gray gives a table of apparent electro-chemical 
 equivalents, for various current densities at various tempera- 
 tures, which may be assumed to include any errors that may 
 arise from cause (c). 
 
 The solution for which this table is constructed is made as 
 follows. Pure recrystallized copper sulphate is dissolved in 
 tap-water (distilled water is not necessary) until a density of 
 i'i8 is reached, and to this one per cent, by volume of strong 
 
 ' Gannon, Electrician, vol. xxxii. p. 216 ; Schuster, Electrician, vol. 
 xxxii. p. 216 ; Gore, Nature, vol. xxv. p. 473 ; Vanni, Wied. Ann., vol. 
 44. P- 214- 
 
The Electro-Cliemical Equivalent of Copper. 159 
 
 sulphuric acid is added. The quantity of this used should be 
 about 3 cub. cm. of solution per square centimetre of plate- 
 area immersed, the solution being used for an aggregate time 
 of ten hours and then replaced by fresh. Measurements made 
 in accordance with the above instructions may have a probable 
 error of not more than 0-05 "j. 
 
 167. The following table of values of e for copper are given 
 by Gray, on the assumption that e. for silver is o'ooiiiS : — 
 
 Area of cathode 
 
 
 
 Values of e. 
 
 
 
 m square cen- 
 
 
 timetres per 
 
 
 
 
 
 ampere. 
 
 2=C. 
 
 u° C. 23' c. 
 
 23° C. 
 
 35° C. 
 
 5° 
 
 0-0003288 
 
 0-0003287 
 
 0*0003286 
 
 00003286 
 
 0-0003282 
 
 lOJ 
 
 0*0003288 
 
 0*0003284 
 
 0*0003283 
 
 0*0003281 
 
 00003274 
 
 ISO 
 
 0*0003287 
 
 0*0003281 
 
 0-0003280 
 
 0*0003278 
 
 0*0003267 
 
 200 
 
 0-0003285 
 
 0*0003279 
 
 000327 7 
 
 0*0003274 
 
 0*0003259 
 
 250 
 
 0-0003283 
 
 0-0003278 
 
 0-0003275 
 
 00003268 
 
 0-0003252 
 
 300 
 
 0*0003282 
 
 0*0003278 
 
 0*0003272 
 
 0-0003262 
 
 00003245 
 
 It will be seen from the above table that the apparent 
 electro-chemical equivalent increases with the current density, 
 the increase being, however, small. The effect of change of 
 temperature is also small until a temperature of about 30° C. is 
 reached, when it becomes important. Alterations in the density 
 of the electrolyte produce very little effect between the limits 
 of 1*15 and I "1 8, there being a slight decrease in the value of 
 € as the density increases. 
 
 168. One of the most difficult points to determine is the 
 chemical action of the solution on the cathode. The dis- 
 solving action that appears to go on is attributed by Schuster to 
 the action of the dissolved oxygen in the solution, and on 
 electrolyzing in a vacuum the' gain in weight of the cathode 
 is slightly greater. It was also found to be necessary to have 
 the solution distinctly acid before consistent results could be 
 obtained. Gray has shown that the rate of solution of the 
 cathode is very irregular but never exceeds -^^ mg. per square 
 centimetre per hour, and it is a minimum for solution densi- 
 ties between i*io to i'i5. 
 
i6o 
 
 Practical Electricity and Magnetism. 
 
 Various attempts have been made to obtain a solution which 
 will not act on the cathode, and Vanni gives one made by 
 adding o"oo5 gramme of sulphuric acid per Utre to a perfectly 
 neutral solution of copper sulphate, this giving a value for e of 
 0-0003287 between 50 and 200 sq. cm. per ampfere plate-area. 
 The same result is claimed by Oettel > using alcoholic solutions 
 of copper sulphate. 
 
 169. Iodine Voltameter. — For the measurement of very small 
 currents of electricity, such as those that are employed in 
 ordinary reflecting galvanometers, the usual forms of silver or 
 copper voltameters are hardly applicable, the quantity of 
 electricity passing in an electrolysis, lasting several hours even, 
 being very small, so that the gain in weight of the cathode is 
 very slight, and the effect of the dissolved oxygen in the 
 solution on the deposit may be serious. In order to measure 
 very small currents such as these, an ingenious voltameter has 
 been constructed by Mr. Herroun,^ which 
 depends on the estimation of the amount 
 of iodine liberated from one of its salts 
 during electrolysis. 
 
 The voltameter consists of a tall narrow 
 beaker, at the bottom of which is a platinum 
 anode, A (see Fig. 85), connection to which 
 is made by means of a platinum wire fused 
 into a glass tube containing mercury, B. 
 The cathode consists of a rod of pure zinc, 
 wrapped in filter-paper, in order to pre- 
 vent pieces of zinc which may become 
 detached from falling to the bottom of the 
 vessel ; this rod only dips a few centimetres into the solution, 
 which is a 10-15 P^'' cent, solution of neutral zinc iodide. 
 The zinc iodide solution should be kept in a dark place, in 
 contact with a little pure zinc, in order to prevent its decom- 
 posing. Should zinc iodide not be obtainable, the electrolyte 
 may be made up of a solution containing 15 per cent, zinc 
 chloride, to which 5 per cent, potassium iodide has been added. 
 
 ' Oettel, Electrician, vol. xxxi. p. 59. 
 " Phil. Mao;., vol. xl., July, 1895. 
 
 Fig. 85. 
 
The Iodine Voltameter. i6i 
 
 When a current is sent through the voltameter, iodine 
 separates out at the platinum anode, but, on account of its 
 great density, remains at the bottom of the glass vessel. The 
 current density at the cathode should not exceed i ampfere per 
 200 sq. cm. surface, otherwise insoluble periodate is .liable to 
 be formed. 
 
 After the electrolysis is stopped, the zinc cathode is at once 
 removed, and the solution stirred up ; it will have a reddish- 
 brown colour, due to the liberated iodine. The amount of 
 iodine liberated is o"ooi3i4 gramme per coulomb, and in order 
 to determine the amount liberated by the electrolysis, the 
 brown solution is titrated with a standard solution of pure 
 sodium thiosulphate, a convenient strength being found to be 
 1 2 "845 grammes of pure sodium thiosulphate to 1000 cub. cm. 
 water; i cub. cm. of this solution being equivalent to o"oo657 
 gramme iodine or 5 coulombs of electricity per cubic centimetre 
 of thiosulphate required in the titration. 
 
 The vessel containing the brown solution should be placed 
 on a sheet of white paper, and a burette filled with the sodium 
 thiosulphate solution fixed in a retort stand, so as to be able to 
 run it into the iodine solution. The thiosulphate is run in 
 slowly, and, when the brown solution is nearly decolorized, 
 drop by drop. When the last trace of colour vanishes the 
 burette is closed, and the amount of liquid that has been run 
 out noted. Multiplying the number of cubic centimetres of 
 Uquid used by five, and dividing by the time of the electrolysis 
 in seconds, the current in ampferes is obtained. 
 
 If greater accuracy is desired the burette should be weighed 
 before and after the experiment, and the volume of thiosulphate 
 solution used calculated from its specific gravity and the weight 
 used. 
 
 Also, when the titration has been nearly completed, a little 
 clear starch solution may be added to the iodine solution, and 
 the exact point when the titration has been completed judged 
 by the vanishing of the blue coloration produced by the free 
 iodine on the starch. 
 
 170. When very small currents are to be measured, such as 
 may be employed in the calibration of high resistance sensitive 
 
1 62 Practical Electricity and Magnetism. 
 
 galvanometers, the time of electrolysis must be considerable, 
 and the form of the voltameter must be altered to prevent the 
 diffusion of the iodine through the solution, the most convenient 
 form being a U-tube with an asbestos plug at the bend in the 
 tube. 
 
 If the thiosulphate solution has been standing for any length 
 of time it is liable to decompose, and should be tested with a 
 standard iodine solution. 
 
 171. The foUowiug experiment will illustrate the accuracy of 
 the above method of measuring current. 
 
 A current from three secondary cells was sent through an 
 iodine voltameter in series with a 50-ohm standard coil and an 
 adjustable resistance. The current was regulated by the 
 adjustable resistance, so that the potential difference at the 
 terminals of the so-ohm coil just balanced the E.M.F. of a 
 Clark standard cell of i"434 volts. The current was allowed 
 to flow for 33 min. 20 sec. The zinc was then removed, and 
 the solution stirred up, and titrated with the standard solution 
 of sodium thiosulphate (5 coulombs per cubic centimetre). It 
 was found to require 11 "47 cub. cm. 
 
 Current as determined by Clark cell = _1M _ 0-0286 amp. 
 
 50 
 
 Current as determined by voltameter = \LM. 5 _ o'o286amp. 
 
 Lord Kelvin's Current Balances. 
 
 172. We have already described the principle of the absolute 
 current balance, and from the nature of the calculation for the 
 current passing through it, it will be seen that its indications are 
 independent of the magnetic force of the earth, the controlling 
 moment being due to a weight in the scale-pan of the balance. 
 
 This principle has been employed by Lord Kelvin in a series 
 of instruments for the measurement of currents, the instruments 
 being called current balances. These instruments, however, 
 differ from the absolute current balance in this respect : that 
 whilst in the former the current is calculated from the dimen- 
 sions of the apparatus and the balancing weight employed j in 
 
Lord Keh'in's Current Balance. 163 
 
 the latter, the constant of the instrument is determined by the 
 aid of a copper voltameter. On account of the invariability of 
 this constant, the balances may be employed as secondary 
 standard current measuring instruments. 
 
 In each of the balance instruments, except the kilo-ampfere 
 balance, each movable ring is actuated by two fixed rings — 
 all three approximately horizontal (see Fig. 86). There are 
 two such groups of three rings — two movable rings attached to 
 the two ends of a horizontal balance arm pulled, one of them 
 up and the other down, by a pair of fixed rings in its neighbour- 
 hood. The current is in opposite directions through the two 
 movable rings to practically annul disturbance due to horizontal 
 components of terrestrial or local magnetic forces. In all the 
 instruments the balance arm is supported by two trunnions, 
 each hung by an elastic ligament of fine wire, through which 
 the current passes into and out of the circuit of the movable 
 rings. In all the balance instruments in which the movable 
 ring is between the two fixed rings, the mid-range position of 
 each movable ring is in the horizontal plane nearly midway 
 between the two fixed rings which act on it. The current goes 
 in opposite directions through the two fixed rings, so that the 
 movable ring is attracted by one and repelled b.y the other. 
 The position of the movable ring equidistant frorii the two 
 fixed rings, is a position of minimum force, and the sighted 
 position, for the sake of stability, is above it at one end of the 
 beam and below it at the other, in each case being nearer to 
 the repelUng than to the attracting ring by such an amount as 
 to give about o'2 per cent, more than the minimum force. 
 
 The balancing is performed by means of a weight which 
 slides on an approximately horizontal graduated arm attached 
 to the balance ; and there is a trough fixed on the right-hand 
 end of the balance, into which a proper counterpoise weight is 
 placed, according to the particular one of the sliding weights in 
 use at any time. For fine adjustment of the zero, a small metal 
 flag is provided, as in an ordinary chemical balance. This flag 
 is actuated by a fork, having a handle below the case outside. 
 To set the zero, the left-hand weight is placed with its pointer 
 at the zero of the scale, and the flag is turned to one side or 
 
164 Practical Electricity and Magnetism. 
 
Lord Kelvins Current Balance. 
 
 165 
 
 the other until it is found that, with no current in the rings, the 
 balance rests in its sighted position. 
 
 To measure a current, the weight is slipped along the scale 
 until the balance rests in its sighted position. The strength of 
 the current is then read off approximately on the fixed scale 
 (called the inspectional scale) with the aid of the finely divided 
 scale for more minute accuracy, according to the explanations 
 given below. Each number on the inspectional scale is twice 
 the square root of the corresponding number on the fine scale 
 of equal divisions. 
 
 The slipping of the weight into its proper position is 
 performed by means of a self-releasing pendant, hanging from 
 a hook carried by a sliding platform, which is pulled in two 
 directions by two silk threads passing through holes to the 
 outside of the glass case. Four pairs of weights, sliding and 
 counterpoise, of which the sledge and its counterpoise constitute 
 the first pair, are suppUed with each instrument. These 
 weights are adjusted in the ratios of i : 4 : 16 : 64, so that 
 each pair gives a round number of ampferes, half-ampferes, 
 quarter-ampferes, or of decimal sub-divisions or multiples of 
 these magnitudes of current, on the inspectional scale. The 
 useful range of each instrument is from i to 100 of the smallest 
 current for which its sensibility suffices, these ranges in the 
 centi-ampfere, deci-ampfere, and deka-ampere balances being from 
 I to 100 centi-ampferes, deci-ampbres, andampbres, respectively. 
 The balances are designed to carry 75 per cent, of their maxi- 
 mum current continuously, and their maximum current long 
 enough for all standard purposes. The following table gives for 
 each type of instrument the value per division of the inspectional 
 scale corresponding to each of the four pairs of weights : — 
 
 
 I. 
 
 II. 
 
 III. 
 
 Weights. 
 
 Centi-amperes 
 
 Deci-ampferes 
 
 Amperes per 
 
 
 per division. 
 
 per divii>ion. 
 
 division. 
 
 1st pair 
 
 25 
 
 0-25 
 
 0-25 
 
 2nd ,, 
 
 0'50 
 
 0-50 
 
 0-50 
 
 3rd „ 
 
 I-O 
 
 I-O 
 
 I-O 
 
 4th „ 
 
 20 
 
 2-0 
 
 2-0 
 
1 66 Practical Electricity and Magnetism. 
 
 The fixed inspectional scale shows, approximately enough for 
 most purposes, the strength of the current ; the notches in the 
 top of the aluminium scale show the precise position of the 
 weight corresponding to each of the numbered divisions on 
 the fixed scale, which practically annuls error of parallax due to 
 the position of the eye. When the pointer is not exactly below 
 one of the notches corresponding to integral divisions of the 
 inspectional scale, the proportion of the space on each side, to 
 the space between two divisions, may be estimated inspec- 
 tionally with accuracy enough for almost all practical purposes. 
 Thus we may readily read off 34"2 or 347 by estimation, with 
 little chance of being wrong by one in the decimal place. But 
 when the utmost accuracy is required, the reading on the fine 
 scale of equal divisions must be taken, and the strength of the 
 current calculated by the aid of a table of doubled square roots. 
 Thus, for example, if the reading is 292, we find 34" 18, or say 
 34'2, as the true scale reading for the strength of current ; or, 
 again, if the balancing position of the pointer be 301 on the 
 fine scale, we find 3470 as the true reading of the inspectional 
 scale. The centi-ampfere balance, with a thermometer to test 
 the temperature of the rings, and with platinoid resistances up 
 to 1600 ohms, serves to measure potentials from 10 to 400 
 volts, the following being the constants when so used : — 
 
 Weights. 
 
 Resistance in 
 
 circuit. * 
 
 Volts per division 
 of fixed scale. 
 
 First pair 
 J) 
 J) 
 
 400 
 
 800 
 
 1200 
 
 1600 
 
 i-o 
 
 2-0 
 30 
 4-0 
 
 If the second pair of weights is used, the constants will be 
 double those given above. 
 
 173. Adjustment of tlie Ealances. — The instrument should be 
 levelled, in accordance with the indications of the attached 
 spirit-level, by means of the levelling screws on which the sole 
 plate of the instrument stands. 
 
 ' Including the resistance of the instrument, which is about 50 ohms. 
 
Adjustment of Current Balance. 167 
 
 In the centi- and deci-ampfere balances, the beam can be 
 Ufted off its supporting ligaments by turning a handle attached 
 to a shaft passing under the sole-plate of the instrument. 
 
 This shaft carries an eccentric, on the edge of which rests the 
 lower end of a vertical rod, which is fixed at its upper end to a 
 tripod lifter. When the instrument is to be removed by hand 
 from place to place, the lifter should be raised ; but when it is 
 fixed up for regular use, it is advisable to keep the beam always 
 hanging on the ligaments. 
 
 A set of four sliding weights, of which the carriage constitutes 
 one, is supplied with each instrument. The carriage is fitted 
 with an index to point to the movable scale, and is intended to 
 remain always on the rail. One or other of the weights is to 
 be placed on the carriage in such a way that the small hole and 
 slot in the weights pass over the conical pins. The weights 
 are moved by means of a slider, which slides on a rail fixed to 
 the sole-plate of the instrument, and carries a pendant with a 
 vertical arm intended to pass up through the rectangular recess 
 in front of the weight and carriage. The slider and weight are 
 shown in position in the figure. The slider is moved by silk 
 cords, which pass out at the ends of the glass case. When the 
 cords are not being pulled for shifting the weight, their ends 
 should be left free so that the pendant may hang clear of the 
 weight. 
 
 When a weight is to be placed on or removed from the 
 carriage, the slider should be drawn forward at the top until it is 
 clear of the weight, and then pushed to one side until the weight 
 is adjusted, when it may be replaced in position in a similar 
 manner. 
 
 Cylindrical counterpoise weights, with a cross-bar passed 
 through them, are supplied for the purpose of balancing the 
 sliding weights when they are placed at the zero of the scale. 
 The sliding weight should be placed so that the index of the 
 carriage points to the zero of the scale, and the proper counter- 
 poise weight should be placed in the trough, fixed to the right- 
 hand end of the beam, with its cross-bar passing through the 
 hole in the bottom of the trough. The flag, which is attached 
 to the cross trunnion of the beam, should then be turned by 
 
1 68 Practical Electricity and Magnetism. 
 
 means of the handle projecting from under the sole-plate until 
 the index on the end of the movable scale points to the middle 
 one of the fine black lines on the fixed scale opposite to it. 
 Care must be taken when making this adjustment that the fork 
 which moves the flag is not left in contact with it, as this would 
 impede the free swing of the beam. The fork should be turned 
 back a little after each adjustment of the flag, and, when the 
 flag is being adjusted, it is better to watch the flag itself, and 
 make successive small adjustments until the beam stands at 
 zero, than to make successive trials by pushing the handle 
 round while watching the position of the index. 
 
 If the ligament has stretched since the instrument was 
 standardized, the index at one end of the movable scale will be 
 found to be below the middle line on its vertical scale, when the 
 index at the other end is correctly pointing to its zero position. 
 The error so introduced would be a small one, but it may easily 
 be put right by slightly loosening the movable beam to the base- 
 plate, and raising it by slipping one or two thicknesses of paper 
 below it until the indexes simultaneously point to their zero 
 position. 
 
 In using the centi-ampbre balance as a voltmeter when great 
 accuracy is required, care must be taken that the effect of 
 change of temperature in changing the resistance of the coils of 
 the instrument, and of the external resistance coils is allowed 
 for ; and in this use of the instrument it is advisable to employ 
 currents such as can be measured by the lightest weight on the 
 beam. When the instrument is to be used as a voltmeter, 
 four resistances are provided, three of which are 400 ohms, and 
 the fourth is less than 400 ohms, by the resistance of the coils 
 of the instrument at a certain specified temperature. The 
 smallest resistance is intended to be included by itself in the 
 circuit when the lowest potentials are being measured, and in 
 series with one or more of the others when the potential is so 
 high as to give a stronger current than can be measured with 
 the lightest weight on the beam. The correction for tempera- 
 ture is, for the copper coils of the balance, about 0-38% per 
 1° C, and for the platinoid resistances, about 0-024% pei' 1° C. 
 
Siemens' Electro-Dynamometer. 169 
 
 Siemens' Electro-Dvnamometer. 
 
 174. A most useful piece of apparatus in the laboratory for 
 the measurement of currents is the Siemens' electro-dynamo- 
 meter. It also partakes of the nature of a secondary standard 
 current measurer, inasmuch as it is independent of variations 
 of the earth's magnetism or of the magnetism of permanent 
 magnets, the controlling force being supplied by the torsion of 
 a spiral spring, the constant of which is determined once for 
 all, and, provided the instrument is carefully used, will remain 
 unaltered for years. The instrument is shown in Fig. 87, and 
 consists essentially of two coils — one fixed to the framework of 
 the instrument, and the other suspended by a silk thread so 
 that it hangs with its axis at right angles to that of the other 
 coil. To the upper end of the suspended coil is also fixed one 
 end of the spiral spring, the other end of which is attached to a 
 torsion head, which has a pointer moving over a circular scale 
 divided into degrees. A pointer is also attached to the 
 movable coil, which is free to move between two stops on 
 either side of the scale zero. The free ends of the suspended 
 coil dip into mercury cups, the connections being made so that 
 the fixed and movable coils are in series. In most forms of 
 this instrument there are two fixed coils — one consisting of a 
 few turns of thick wire, and the other of many turns of finer 
 wire, thus enabling two degrees of sensibility to be obtained. 
 The central terminal (3) is common to both coils, terminals 
 I and 3 are those of the suspended coil in series with the 
 thin fixed coil, while terminals 2 and 3 are those of the 
 suspended coil in series with the thick fixed coil. A spring not 
 shown in the figure, actuated by a screw at the back of the 
 instrument, can be released when the instrument is not in use, 
 and supports the weight of the suspended coil. 
 
 In setting the instrument up, it must be placed so that when 
 the suspended coil swings free its axis is in the , magnetic 
 meridian, so that the earth's magnetism will not produce any 
 deflecting effect on it. The instrument is levelled until the 
 coil swings free, and its pointer is at the zero on the scale, the 
 torsion head pointer also being at zero. The winding of 
 
I/O Practical Electricity and Magnetism. 
 
 Fig. 87. 
 
The Constant of a Siemens' Dynamometer. 17 1 
 
 the instrument is such that when a current traverses the two 
 coils, the suspended coil rotates in a counter-clockwise direction, 
 the distance through which it can move being Umited by the 
 stops. The torsion head is now rotated in a clockwise direction, 
 until the pointer attached to the coil is brought back to zero. 
 Since the same current flows through both coils, the resultant 
 magnetic effect between them must be proportional to the 
 product of the currents in each— that is, to the square of the 
 current. Also, since in the torsion control the controlling force 
 is proportional to the angle of torsion <^, we have — 
 
 and C = K v'^ 
 
 where K is a constant depending on the torsion coritrol and 
 winding of the instrument, and must be determined experiment- 
 ally for the dynamometer. 
 
 175. Determitiation of tlu Constant of the Dynamometei: — To 
 determine the constant K, we must send a known current 
 through the instrument, and find the angle of torsion <f>, required 
 to bring the suspended coil back to zero. The current may be 
 measured by means of a current balance, standard galvano- 
 meter, or voltameter. The dynamometer, after having been 
 levelled as described, is connected in series with a variable 
 resistance, secondary battery, voltameter, and break-circuit key. 
 The current is first adjusted until the deflection of the torsion 
 head required to produce equilibrium is pretty large, the 
 instrument being much more sensitive at high than at low 
 readings, since the squares of the readings increase less rapidly 
 at high than at low values. The voltameter plates are then 
 removed, and after being prepared as described previously, 
 (par. 165), are weighed, and replaced in the voltameter; the 
 electrolysis should be allowed to go on for about one hour, the 
 current being kept constant by means of the variable resistance. 
 
 The current C is calculated from the gain in weight of the 
 cathode, and then the constant calculated, since — 
 
 <f> being the angle of torsion as read off on the dial. Two or 
 
172 Practical Electricity and Magnetism. 
 
 three such determinations may be made at various parts of the 
 scale, and the mean of the values thus obtained taken as the 
 constant of the instrument. 
 
 176. Calibration of a Direct Current Reading Instrument. — 
 The Siemens' electro-dynamometer affords a very simple 
 means of absolutely calibrating a direct-reading ammeter, the 
 ammeter being connected in series with the dynamometer, and 
 with a variable resistance, battery, and break-circuit or reversing 
 key. Simultaneous readings are taken on both instruments with 
 various currents, the currents being calculated from the reading 
 on the Siemens' electro-dynamometer, by multiplying the square 
 root of the reading of the torsion angle by the dynamometer 
 constant. Thus a calibration curve may be plotted for the 
 direct-reading instrument, or a table of corrections made out. 
 
 In using the electro-dynamometer in connection with other 
 apparatus, care must be taken that the magnetic effect of the 
 current in the other apparatus does not affect it. 
 
 177. In order to determine the constant of a Siemens' 
 electro-dynamometer, it was connected in series with a copper 
 voltameter, carbon resistance, break-circuit key, "and six 
 secondary cells. The copper cathode was carefully weighed 
 before the deposit. 
 
 Weight of cathode before deposit S8'654gm. 
 
 ,, ,, after ,, 6o'244 ,, 
 
 Gain in weight i'S90 n 
 
 The time of deposit was one hour, and the temperature of the 
 bath was 11° C. The dynamometer reading was kept constant 
 
 at 59"3- 
 
 Taking £ for copper as equal to o'ooo3279 — 
 
 current = "^ ; — 
 
 o"ooo3279 X 3600 
 
 = I "347 amperes 
 
 Hence, since C = KV<A 
 
 1-347 = KVsg's 
 and K = o'i75 
 
References to Scientific Papers. 
 
 173 
 
 178. References to Scientific Papers. 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 Improved Construction of the 
 Movable Coil Galvanometer for 
 the Absolute Measurement of 
 Current 
 
 On a New Standard Sine Galvano- 
 meter 
 
 On an Electrodynamic Balance 
 
 Absolute Determination of a Current 
 by a Balance 
 
 On a Method of determining ex- 
 perimentally the Constant of an 
 Electro-Dynamometer 
 
 Calculation of the Constants of the 
 Absolute Electro-Dynamometer 
 
 Determination of the Electro- 
 Chemical Equivalent of Silver 
 
 Copper Electrolysis in Vacuo 
 
 Action of CUSO4 and H2SO4 on 
 
 Metallic Copper 
 On the Electrolysis of Copper and 
 
 Silver 
 On the Electrolysis of Copper 
 
 On the Accuracy of the Copper 
 
 Voltameter 
 On the Apparent Variation of the 
 
 Electro-Chemical Equivalent of 
 
 Copper 
 Electro- Chemical Equivalent of Zinc 
 
 Use of an Iodine Voltameter for 
 the Measurement of Small 
 Currents 
 
 Methods of Measuring Currents of 
 Great Strength 
 
 Electro-Chemical Equivalent of 
 
 Copper 
 Heating Effects of Electric Currents 
 
 Obach 
 
 T. Gray 
 Helmholtz 
 Blyth 
 Chattock 
 
 Gray 
 Rayleigh 
 
 Gannon 
 Schuster 
 T. Gray 
 
 »j 
 Blount 
 Vanni 
 
 Murray 
 Herroun 
 
 Trowbridge 
 
 Shaw 
 Preece 
 
 Phil. Mag., vol. 16, 
 Aug., 1883. 
 
 Ibid., vol. 22, Oct., 
 
 1886. 
 Pro. Roy. Soc, vol. 
 
 32. p. 39- 
 Pro. R.S.E., vol. 
 
 13, p. 650. 
 Phil. Mag., vol. 16, 
 
 Feb., 1883. 
 
 Ibid., vol. 33, Jan., 
 
 1892. 
 Trans. Roy. Soc, 
 
 1885 ; Pro. Roy. 
 
 Soc, vol. 36, p. 
 
 448 ; Ibid., vol. 
 
 37. P- 142. 
 Pro. Roy. ioc, vol. 
 
 55, p. 66. 
 Ibid., vol. 55, p. 84. 
 
 Phil. Mag., vol. 22, 
 
 Nov., 1886. 
 Ibid., vol. 25, Mar., 
 
 1888. 
 Elect., vol. 51, p. 
 
 59- 
 Ibid.,\o\. 28, p. 12. 
 
 Ibid., vol. 31, p. 
 
 125. 
 Phil. Mag., vol. 40, 
 
 July, 1895. 
 
 Ibid., vol. 7, Mar., 
 1879; vol. 19, 
 May. 1885. 
 
 B.A. Report, 1886, 
 
 p. 3l8- 
 Pro. Roy. Soc, vol. 
 36, p. 464; vol. 
 
 43, p. 280; vol. 
 
 44, P- 109 ; vol. 
 48, p. 68. 
 
174 Practical Electricity and Magnetism. 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 On the Permanent Temperature of 
 Conductors through which 
 Currents are flowing 
 
 Sine Galvanometer 
 
 Electro-Magnetic Balance 
 
 Bottomley 
 
 Pouillet 
 Becquerel 
 
 Pro. Roy. Soc, vol. 
 36, p. 464. 
 
 Comptes Rendus, 
 
 vol. 4, 1837. 
 Ibid., vol. S, 1837. 
 
III. 
 
 ELECTRO-MOTIVE FORCE. 
 
 179. The absolute determination of an electro-motive force in 
 electro-magnetic measure is made from the relations which sub- 
 sist between electro-motive force,' current, and resistance, the 
 produce of current and resistance being equal to electro-motive 
 force. This being so, if current and resistance are expressed 
 in absolute units, the electro-motive force will be in absolute 
 units ; if current is in ampbres and resistance in ohms, electro- 
 motive force wiU be in volts, the volt being the practical 
 unit of electro-motive force, and being that necessary to send 
 a current of one ampfere through a resistance of one ohm. 
 
 Standards of E.M.F. 
 
 180. A standard of E.M.F. might be arranged by sending 
 a known current through a known resistance, and using the 
 E.M.F. at the ends of the resistance as the standard. This is 
 done in some cases, but it is found much more convenient to 
 employ some form of voltaic cell whose E.M.F. is a known 
 multiple of the unit. 
 
 if such secondary standards are to be employed, they must 
 fulfil certain conditions. 
 
 (a) They must be easily made and reproduced. 
 
 {b) They must remain absolutely constant under constant 
 physical conditions. 
 
 \c) Their alteration with the alteration of conditions must be 
 accurately known. 
 
 {d) They must always return to their original value when the 
 original conditions are reproduced. 
 
176 Practical Electricity and Magnetism. 
 
 Standard Cells. 
 
 181. Very few, of numbers of primary batteries that have 
 been brought out from time to time, fulfil the above condi- 
 tions. One or two cells, however, stand out from the others 
 as being suitable for use in this direction. Of these we pro- 
 pose to describe the three best known — 
 
 (i) The Clark cell. 
 
 (2) The Daniell cell. 
 
 (3) The Weston cell. 
 
 Of these the first is by far the most important, both on account 
 of its constancy, and on account of the exact knowledge we 
 possess of its behaviour under varying conditions, since its 
 action has been the subject of numerous researches. It is 
 also important because the legal definition of the volt is ex- 
 pressed in terms of the E.M.F. of one of its modifications, 
 the definition being as follows : — 
 
 182. " The volt which has the value 10* in terms of the 
 centimetre, the gramme, and the second of time, being the 
 electrical pressure that, if steadily applied to a conductor wjiose 
 resistance is one ohm, will produce a current of one ampfere, 
 and which is represented by 0-6974 ( ^°°° ) of the electrical 
 pressure at a temperature of 15° C. between the poles of the 
 voltaic cell known as Clark's cell, set up in accordance with 
 the specification appended and marked B." ^ 
 
 183. The following is the specification for the construction 
 of the above-mentioned standard : — 
 
 " The cell consists of zinc or an amalgam of zinc with mer- 
 cury and of mercury in a neutral saturated solutioh of zinc 
 sulphate and mercurous sulphate in water, prepared with 
 mercurous sulphate in excess. 
 
 " Pteparation of the Materials. 
 " I. The Me7-cury. — To secure purity it should be first 
 treated with acid in the usual manner, and subsequently dis- 
 tilled in vacuo. 
 
 "2. The Zinc. — Take a portion of a rod of pure redistilled 
 ' London Gazette, Av^. 24, 1894. 
 
Standard Clark Cells. 177 
 
 zinc, solder to one end a piece of copper wire, clean the whole 
 with glass-paper or a steel burnisher, carefully removing any 
 loose pieces of the zinc. Just before making up the cell dip 
 the zinc into diluted sulphuric acid, wash with distilled water, 
 and dry with a clean cloth or filter-paper. 
 
 " 3. TJu Merairous Sulphate. — Take mercurous sulphate 
 purchased as pure, mix with it a small quantity of pure mercury, 
 and wash the whole thoroughly with cold distilled water by 
 agitation in a bottle : drain off the water, and repeat the pro- 
 cess at least twice. After the last washing, drain oflF as much 
 of the water as possible. 
 
 " 4. Th: Zinc Sulphate Solution. — Prepare a neutral satiurated 
 solution of pure (' pure recrystallized ') zinc sulphate by mixing 
 in a flask distilled water with nearly twice its weight of crystals 
 of pure zinc sulphate, and adding zinc oxide in the proportion 
 of about two per cent, by weight of the zinc sulphate crystals 
 to neutralize any free acid. 
 
 " The crystals should be dissolved with the aid of gentle 
 heat, but the temperature to which the solution is raised should 
 not t^ceed 30° C. Mercurous sulphate, treated as described in 
 3, should be added in the proportion of about twelve per cent. 
 by weight of the zinc sulphate crystals, to neutralize any free 
 zinc oxide remaining, and the solution filtered, while still 
 warm, into a stock-bottle. Crystals should form as it cools. 
 
 " 5. Mercurous Sulphate and Zinc Sulphate Paste. — Mix the 
 washed mercurous sulphate with the zinc sulphate solution, 
 adding sufficient crystals of zinc sulphate from the stock-bottle 
 to insure saturation, and a small quantity of pure mercury. 
 Shake these up well together to form a paste of the consistence 
 of cream. Heat the paste, but not above a temperature of 
 30° C. Keep the paste for about an hour at this temperature, 
 agitating it from time to time, then allow it to cool ; continue 
 to shake it occasionally while it is cooling. Crystals of zinc 
 sulphate should then be distinctly visible, and should be dis- 
 tributed throughout the mass ; if this is not the case add more 
 crystalls from the stock-bottle, and repeat the whole process. 
 This method insures the formation of a saturated solution of 
 zinc and mercurous sulphates in water. 
 
 N 
 
178 Practical Electricity and Magnetism. 
 
 " To set tip the Cell. 
 
 184. "The cell may be conveniently set up in a small test- 
 tube of about 2 cm. diameter and 4 cm. or 5 cm. deep. Place 
 the mercury in the bottom of this tube, filling it to a depth of, 
 say, o"S cm. Cut a cork about o'5 cm. thick to fit the tube; at 
 one side of the cork bore a hole through which the zinc rod can 
 pass tightly ; at the other side bore another hole for the glass 
 tube which covers the platinum wire ; at the edge of the cork 
 cut a nick through which the air can pass when the cork is 
 pushed into the tube. Wash the cork thoroughly with warm 
 water, and leave it to soak in water some hours before use. 
 Pass the zinc rod about i cm. through the cork. 
 
 " Contact is made with the mercury by means of a platinum 
 wire about I^o. 22 guage. This is protected from contact with 
 the other materials of the cell by being sealed into a glass tube. 
 The ends of the wire project from the ends of the tube; one 
 end forms the terminal, the other end and a portion of the 
 glass tube dip into the mercury. 
 
 " Clean the glass tube and platinum wire carefully," then 
 heat the exposed end of the platinum red hot, and insert it in 
 the mercury in the test-tube, taking care that the whole of the 
 exposed platinum is covered. Shake up the paste and intro- 
 duce it without contact with the upper part of the walls of the 
 test-tube, filling the tube above the mercury to a depth of 
 rather more than i cm. 
 
 " Then insert the cork and the zinc rod, passing the glass rod 
 through the hole prepared for it. Push the cork gently down 
 until its lower surface is nearly in contact with the liquid. The 
 air will thus be nearly all repelled, and the cell should be left 
 in this condition for at least twenty-four hours before sealing, 
 which should be done as follows : — 
 
 " Melt some marine glue until it is fluid enough to pour by 
 its own weight, and pour it into the test-tube above the cork, 
 using sufficient to cover completely the zinc and soldering. 
 The glass tube containing the platinum-wire should project 
 some way above the top of the marine glue. 
 
 " The cell may be sealed in a more permanent manner by 
 
Standard Clark Cells. 179 
 
 coating the marine glue when it is set with a solution of sodium 
 silicate, and leaving it to harden. The cell thus set up may be 
 mounted in any desirable manner. It is convenient to arrange 
 the mounting so that the cell may be immersed in a water bath 
 up to the level of, say, the upper surface of the cork. Its 
 temperature can then be determined more accurately than is 
 possible when the cell is in air. In using the cell sudden varia- 
 tions of temperature should as far as possible be avoided. 
 
 " The form of the vessel coiitaining the cell may be varied. 
 In the H form the zinc is replaced by an amalgam of 10 parts of 
 zinc to 90 of mercury. The other materials should be pre- 
 pared as already described. Contact is made with the 
 amalgam in one leg of the cell, and with the mercury in the 
 other, by means of platinum ^\-ires sealed through the glass." 
 
 185. The E.M.F. of the Clark cell was determined by Lord 
 Rayleigh,^ by balancing its E.M.F. against the fall of potential 
 of a known current down a known resistance, the current being 
 measured on the current balance, and was found to be i"4345 
 volt at 15° C. This value, however, varies with temperature 
 and from a long series of experiments Lord Rayleigh deduced 
 the following expression, from which the E.M.F. may be 
 calculated at any temperature f C. : — 
 
 E = 1-4345 {i - o-ooo77(/^ - 15)} 
 
 186. One objection to the form of cell just described is that 
 after use for some time it is found that the mercury is apt to 
 come into contact \vith the zinc, if the cell has been carried 
 about a good deal, and consequently shaken ; also that local 
 action is apt to take place between the mercury salt and the 
 zinc, the zinc becoming amalgamated and altering the E.M.F. 
 slightly. Also, on account of its comparatively high tempera- 
 ture coefficient, special precautions have to be taken to insure 
 the exact temperature of the cell being known, this temperature 
 being kept constant for some time before the cell is used, since 
 the alteration of E.M.F. is found to lag behind the alteration of 
 •temperature.^ 
 
 • Phil. Trans. Roy. Soc., Part II., 1885. 
 
 ° Ayrton and Cooper, Prec. Roy. Soc, vol. lix. p. 368. 
 
i8o Practical Electricity and Magnetism. 
 
 Again, on referring to the method of construction, it will be 
 seen that the zinc sulphate solution is saturated at 30° C. ; hence 
 at ordinary temperatures some of the salt must crystallize out, 
 and a slight variation of temperature will mean a redissolving 
 or a crystallizing out of a small portion of the salt. This is found 
 to give rise to variations in density throughout the liquid, which 
 might even become supersaturated, thus of course affecting the 
 constancy of the cell. To remedy these defects Professor 
 Carhart has devised a form of Clark cell which differs some- 
 what from the above form.' 
 
 187. In the Carhart Clark cell the electrodes are zinc and 
 mercury, the construction of the cell differing slightly from the 
 B.A. form, and is shown in Fig. 88. The solution and salts are 
 
 ^- - - Marine Glue 
 Cofk 
 
 Qlass Sleeve 
 Zinc Sulphate 
 Zinc 
 
 --Asbestos Plu^ 
 ---Metcurous Sulphate 
 
 ---Mercuiy 
 
 --Platinum Wire 
 
 Fig 
 
 prepared in the same way as previously described, except that the 
 zinc sulphate solution is saturated at 0° C. instead of at 30° C. 
 For this cell Professor Carhart claims that it gets over the objec- 
 tions just mentioned, and also that its temperature coefficient is 
 about half that of the B.A. form. The electro-motive force and 
 its alteration with temperature being expressed as follows : — 
 
 E= r438{i - 0-000387 (iP - 15)} 
 
 When using this cell on a circuit of over 30,000 ohms resist^' 
 ' Phil. Mag., vol. xxviii., Nov., 1889. 
 
Standard Daniell Cell. i8i 
 
 ance no polarization was detected, and when the resistance was 
 diminished to 10,000 ohms it only amoimted to ^oooo P^"^ '" 
 five minutes. 
 
 Slandard Daniell Cell. 
 
 188. The Daniell cell, although much simpler to construct 
 than the Clark cell, has the disadvantage that it is not portable. 
 If, however, care is exercised in making it up, it will be found 
 a very convenient form of standard for laboratory work, and 
 has also the great advantage that its temperature coefficient is 
 practically negligible. 
 
 The best form in which to embody this standard, and the 
 best conditions for working, have been carefully investigated by 
 Dr. J. A. Fleming.^ 
 
 The cell consists of a glass U-tube fixed to a wooden base. 
 Into one limb of the tube dips the copper, and into the other 
 the zinc plate ; the copper plate being surrounded by a solution 
 of copper sulphate, and the zinc by zinc sulphate solution, the 
 liquids being carefully poured into the tube so that the interface 
 between them is sharp. The interface may be arranged so as 
 to be at the level of an escape-tube provided with a glass 
 tap, so that when it becomes bliured as the liquids diffuse into 
 one another, a httle of the solution may be run off and the inter- 
 face again made sharp. Two additional glass reservoirs, one 
 connected to each limb of the U-tube, may be provided, in 
 which a supply of solution, CUSO4 in one and ZnSOj in the other, 
 may be kept. The copper plate should consist of a thin copper 
 wire 4" long, on which copper has been plated until it is about 
 \' thick, the rod being always freshly plated before use. 
 The zinc rod, of about the same dimensions, should be of 
 twice-distilled zinc, which has been cast into rods and then 
 amalgamated with pure mercury. 
 
 189. Solutions. — ^The solutions may be made of various 
 strengths, but the E.M.F. of the cell varies slightly with the 
 density of the solution. The two sets of solutions most 
 frequently employed are equi-dense solutions of specific gravity 
 
 • Fleming " On the use of Daniell's Cell as a Standard of Electro-Motive 
 Force," Fhil. Mag., vol. xx., Aug., 1885. 
 
1 82 Practical Electricity and Magnetism. 
 
 1-2 at 15° C, and ZnSOi solution specific gravity i'4 and 
 CUSO4 solution specific gravity i"i at 15° C. 
 
 Of these the equi-dense solutions of specific gravity 1-2 are 
 made by dissolving 28'25 parts by weight of pure crystallized 
 CuSOi in 71-75 parts of distilled water, and 32 parts by weight 
 of ZnSOi in 68 parts of distilled water; whilst the ZnSOj 
 solution of density i'4 takes 55*5 parts by weight of ZnSO^ 
 crystals to 44-5 parts of distilled water, and the CUSO4 of density 
 i"i, i6*5 parts by weight CuSOj to 83'S parts by weight of 
 distilled water. 
 
 190. Electro-Motive Force} — Careful measurements of E.M.F. 
 were made, using the above solutions, the standard of reference 
 being the Clark cell, which was taken as i'43S volt at 15° C, 
 the results being as follows : — 
 
 Using equi-dense solutions of i'2 sp. gr. — 
 
 E.M.F. = I-I02 volt 
 Using ZnSOj = i'4 sp. gr. and CuSOj = I'l sp.gr. — 
 
 E.M.F. = 1-072 volt 
 
 These representing the values as obtained from a freshly 
 made up cell. After standing about an hour these values in- 
 crease by about 0-003 volt. 
 
 191. Causes of Variation of E.M.F. in tJie Daniell Cell. — The 
 purity of the electrodes greatly affects the E.M.F. of the cell ; 
 if, for instance, the copper rod is cleaned with sand-paper instead 
 of being electroplated, the E.M.F. increases by about o-6%. 
 Oxidation of the copper raises the E.M.F. by uncertain 
 amounts. 
 
 The difference between pure zinc and pure zinc amalgamated 
 is very small ; oxidation of the zinc lowers the E.M.F., whilst 
 the smallest deposit of copper on it, which will occur if any of 
 the CuSOj solution diffuses into the ZnSOi, lowers the E.M.F. 
 by from 2% to 3%. As regards the solutions, an increase in 
 density of the ZnS04 lowers the E.M.F., whilst an increase in 
 density of the CUSO4 raises it. 
 
 The effect of temperature on the cell is small, the E.M.F. 
 
 ' See also par. 220. 
 
The Weston Cadniiwu Standard Cell. 
 
 183 
 
 falling by about 0-00015 ^o^' P^r degree Centigrade rise of 
 temperature. 
 
 When not in use the rods should be removed from the 
 solutions and kept in separate vessels. 
 
 Weston Cadmium Cell. 
 
 192. This cell, invented by Mr. Weston,^ represents an 
 endeavour to obtain a cell with a small or negligible tempera- 
 ture coefficient, which at the same time will fulfil the require- 
 ments of a standard cell. 
 
 The electrodes of the cell are pure mercury and an amalgam 
 of mercury and cadmium, the electrolyte being a saturated 
 solution of cadmium and mercurous sulphates. One form of 
 cell is shown in Fig. 89. 
 
 Perforalal 
 
 Cork- 
 
 Muslin 
 
 Cadtnium 
 Amalfam 
 
 PtWin 
 
 ^mm 
 
 -■miin 
 
 niiiii - 
 
 - -Cork 
 
 -Muslin 
 -^e/ruiy 
 
 Pt Wire 
 
 Fig. 89. 
 
 The cadmium amalgam is made by adding cadmium to 
 mercury heated in a crucible, in the proportion of about 7 of 
 mercury to 1 of cadmium by weight. The amalgam on cool- 
 ing should solidify. 
 
 The E.M.F. of this cell is given as I'oig volt, and the tem- 
 perature coefficient, which is practically negligible, o'oi% per 
 i°C. 
 
 The salts of cadmium are equally soluble in hot and cold 
 water, so that variations of temperature do not tend to alter the 
 density of the solution. 
 
 An elaborate research on the behaviour of this form of 
 standard, extending over two years, has been made by Messrs. 
 ' Patent No. 22482, 189 1. 
 
184 Practical Electricity and Magnetism. 
 
 Jaeger and Wachsmuth.^ The E.M.F. was determined in terms 
 of that of a Clark cell, and was found to be i-oig volt at 20° C, 
 the composition of the cadmium amalgam being iCd to 6Hg. ; 
 the mercurous sulphate (HgaSOj) was rubbed into a stiff 
 paste with crystals and a concentrated solution of cadmium 
 sulphate together with mercury. 
 
 The results of tests made on six cells gave, as the variations 
 of E.M.r., with temperature between the limits of 5° C and 
 25° C- 
 
 E^ = Ejo - 3-8 X ]o-=(/ - 20) - 0-065 X i°~°(^ - 20)2 
 (E20 = I '01 9 volt) 
 
 Tests were also made on the effect of impurities in the materials 
 employed. The cadmium should be prepared electrolytically 
 from sulphate of cadmium solution, ordinary commercial 
 cadmium containing traces of zinc. An increase of one per 
 cent, of cadmium in the composition of the amalgam gave an 
 increase of o-ooi volt in the E.M.F. 
 
 The saturated solution of cadmium sulphate, 3Cd + SHjO, 
 is made by shaking the powdered salt with water and allowing 
 to stand for a considerable time. 
 
 The cadmium sulphate must be tested for free acid, which if 
 present is neutralized by digesting with cadmium hydrate 
 (obtained by precipitating from CdS04 by sodium hydrate and 
 repeated washing), filtering, and evaporating until the crystals 
 separate. 
 
 The mercurous sulphate is similar to that used in the Clark 
 cell. 
 
 The cells were found very constant over a period of two 
 years, only one out of thirty-three varying more thano'oooi volt. 
 They were also found to remain CDnstant after having been 
 carried about for some time. 
 
 Care of Standard Cells. 
 
 193. In using standard cells the greatest care must always be 
 exercised to prevent the possibility of a short circuit, and it is 
 
 ' Wied. Ann., vol. lix., No. 11, p. 575. 
 
Comparison of E.M.F's. 
 
 i8s 
 
 advisable to connect in series with a cell whenever it is used, a 
 resistance of the order of 10,000 ohms. 
 
 In general it is found convenient to get a case containing two 
 cells (see Fig. 90), one being used for testing, etc., and marked 
 in some suitable manner, the other being only employed to 
 check the first from time to time, and its termi- 
 nals should be protected so as to prevent 
 the possibility of its being accidentally used 
 for testing purposes. 
 
 In the case of those cells having an appreci- 
 able temperature coefficient, it is necessary to 
 keep the cell at some definite temperature 
 for a little while before using. This may be 
 done by packing it in a box with cotton wool, 
 when the temperature will alter very slowly, 
 or, still better, by placing it in a thermo- 
 stat. The temperature of the cell is usually recorded by a 
 thermometer which dips into it. These thermometers are of 
 two kinds— one with a straight stem, the other with the stem 
 bent so as to lie along the top of the cell. The former, 
 although the more fragile, is by far the better form, as the 
 mercury column is liable to break at the bend in the tube, and 
 the temperature cannot be accurately taken. 
 
 Fig. 90.' 
 
 Comparison of E.M.F.'s. 
 
 194. vVny method of comparing E.M.F.'s of cells to be 
 accurate must be a null method, i.e. one which does not involve 
 the taking of a current from the cell, thus avoiding polarization 
 which would otherwise occur and vitiate the results. 
 
 One of the most accurate methods of comparing two electro- 
 motive forces is probably one or other modification of the 
 potentiometer principle. Since a special section is devoted to 
 the application of the potentiometer principle to the com- 
 parison of resistances, currents, and electro-motive forces, 
 description of this method will be held over. 
 
 Another null method available is one employing a condenser. 
 
 195. Condenser Methods of Comparing E.M.F.'s. — If Q 
 
1 86 Practical -Electricity and Magnetism. 
 
 represents the quantity of electricity required to charge a con- 
 denser of capacity, K, to a potential diflference, V, then— 
 
 Q= KV 
 
 and V = £ 
 
 We may therefore, using the same condenser, compare two 
 E.M.F.'s by comparing the quantities of electricity with which 
 they charge the condenser, the plates of which are initially at 
 zero potential. To measure the quantity of electricity, a 
 ballistic galvanometer is employed, the amptitude of the first 
 swing of the needle being to a first approximation proportional 
 to the quantity of electricity which has passed through the 
 galvanometer. (The student is here recommended to refer to 
 the section dealing with the ballistic galvanometer for full 
 details of its theory and action.) 
 
 The arrangement of apparatus for the comparison of the 
 E.M.F.'s of two cells by this method is shown in the accom- 
 panying diagram (see Fig. 91). The ballistic galvanometer 
 
 Fig. 91. 
 
 BG and the condenser C are connected in series with the 
 tongue of the two-way key K. One of the cells, B„ is con- 
 nected to the condenser and the upper contact of K, the other 
 cell, B.2, from the condenser to the lower contact of K. The 
 condenser C is provided with a short-circuiting plug, by means 
 of which the plates may be connected together. The con- 
 denser must be short-circuited (K being midway between the 
 
Temperature Coefficient of a Battery. 187 
 
 upper and lower contact), and the plug removed before every 
 reading. 
 
 When K js raised, the condenser is charged by the battery Bj, 
 and the throw 81 is observed on the ballistic galvanometer. 
 After the condenser has been discharged by means of the short- 
 circuit plug, and the galvanometer needle has been brought to 
 rest, K is depressed and a throw, 83, is obtained, the condenser 
 being charged from Bo. If Vj and V, are the E.M.F.'s of Bj 
 and B., respectively, then — ■ 
 
 83 "" KV2 ~ Va 
 
 If one of the cells, Bj, is a standard cell, the E.M.F. of B2 may 
 be calculated from the above relation and the known value 
 ofV,. 
 
 The capacity of the condenser employed may be about \ 
 microfarad. 
 
 Temperature Coefficient of a Battery. 
 
 196. The total effect of change of temperature on the 
 electro-motive force of a cell is made up of several separate 
 temperature effects, and it may either be determined as a 
 whole, or synthesized out of its component parts. It forms an 
 interesting experiment to determine it both ways and compare 
 the results. 
 
 A good example of the synthetic process has been worked 
 out by Professor Carhart, in connection with the Daniell and 
 Clark cells,^ which, since it is of considerable interest, we pro- 
 pose to describe. 
 
 In an ordinary cell having two electrodes and two electro- 
 lytes, there are three quantities which may be separately 
 affected b}' change of temperature, and which will affect the 
 total E.M.F. of the cell ; these are (i) the E.M.F. between the 
 positive electrode and the electrol3^e round it, (2) that between 
 the negative electrode and its electrolyte, and (3) the E.M.F. 
 between the two electrolytes, these forming, as it were, three 
 
 • Carhart, Electrician, vol. xxvii. p. 167 ; also Bouty, Jour. de. Phys., 
 i88o, p. 229. 
 
i88 
 
 Practical Electricity and Magtietism. 
 
 thermal couples. The effect of change of temperature may be 
 determined separately in each case, the algebraic sum repre- 
 senting the total effect. « 
 
 197. We shall first describe the method of measuring the 
 effect of change of temperature between the electrodes and the 
 electrolytes. For this a special form of cell must be employed, 
 consisting of two tubes communicating with each other near 
 their upper ends by means of a long tube of narrow bore (see 
 Fig. 92). Both tubes and the connecting tube are filled with 
 
 B 
 
 \-J 
 
 <^ 
 
 Fig. 92. 
 
 one of the electrolytes; the electrodes A and B are made of 
 the same metal that is placed in that electrolyte in the cell. 
 The two tubes containing the electrodes are put each into a 
 thermostat, T, so that the temperature may be regulated and 
 kept constant at any desired value. 
 
 First the temperatures of both are kept the same, and then 
 there should be no difference of potential between A and B if 
 tested on a potentiometer or other instrument for measuring 
 E.M.F.'s ; then, keeping one tube at constant temperature, the 
 other is raised to different temperatures, and the E.M.F. 
 between A and B noted for the various differences of tempera- 
 ture. If the connecting tube is of sufficiently fine bore, and 
 the apparatus is tilted so that the heated tube is at a slightly 
 higher level than the cold one, there will be no fear of liquid 
 convection currents being set up inside the tubes. 
 
 A curve should be plotted from the results, showing the con- 
 nection between the E.M.F. and the difference of temperature 
 between A and B ; from this curve the mean E.M.F. per 1° C. 
 
Temperature Coefficient of a Battery. 
 
 189 
 
 difference of temperature may be calculated. Record must 
 also be kept of which electrode is at the higher potential. 
 
 The above is then repeated, using electrodes of the other 
 metal employed in the cell, and for electrolyte the liquid in 
 which it is immersed. 
 
 198. Lastly, there is the determination of the thermo-electric 
 eflfect at the junction of the two electrolytes. This is more difficult 
 to carry out experimentally than the other. Fig. 93 showing a 
 
 A : 
 
 Fig. 93. 
 
 form of apparatus which may be employed. Into the tubes A 
 and B, through tightly fitting corks, pass the tubes C and D, 
 the other ends of which terminate in the vessels E and F. 
 The tubes A and B are filled with one electrolyte, V^ilst EC 
 and DF are filled with the other. The junctions between the 
 electrolytes are near the ends of the tubes C and D, where they 
 dip into A and B. The tubes A and B are placed as before 
 into thermostats. Contact is made by means of platinum wires 
 dipping into E and F, the temperatures at E and F being kept 
 constant during the experiment. The tube A is then heated 
 up, B being kept at constant temperature, and the mean E.M.F. 
 per 1° C. rise of temperature calculated from a curve plotted 
 as in the last case. The total thermal effect in the cell will be 
 the algebraic sum of these separate effects. 
 
 199. In the case of the Daniell cell, Carhart found that the 
 E.M.F. generated at the zinc-zinc-sulphate junction (the zinc 
 sulphate being saturated at 0° C.) was 0*00079 volt per 1° C, 
 the cold zinc rod being at the higher potential, whilst that of 
 copper in copper sulphate (density I'li) was 0*0007 3 volt 
 per 1° C, the cold plate again being positive to the heated 
 
I go Practical Electricity and Magnetism. 
 
 one; the E.M.F. generated at the junction of the two liquids 
 was found to be negligible, being less than o'oooo3 volt per i° C. 
 The total temperature coefficient should therefore be — 
 
 o'ooo79 — 0*00073 = o'oooo6 volt per 1° C. 
 
 A Daniell cell was then made up with the same solutions, and 
 its E.M.F. measured at different temperatures. This may be 
 done by placing it in a thermostat, and steadying the tempera- 
 ture for some time before each reading. This was found to 
 give a variation of 0-000073 '^olt per 1° C., and dividing this 
 by i"og, the E.M.F. of the cell, we get the true temperature 
 coefficient — 
 
 O'oooo73 
 
 = o"oooo6^ 
 
 i"o9 -^ 
 
 which agrees closely with the value as determined by the first 
 method. The student is recommended to make the same in- 
 vestigation for the other standard cells. 
 
 Testing a Primary Battery. 
 
 200. In order to make a complete test of a primary battery 
 so as to report upon its merits, the following data must be 
 obtained : — 
 
 (i) The E.M.F. of the cell on open circuit ; 
 
 (2) The temperature coefficient of the cell ; 
 
 (3) The internal resistance ; 
 
 (4) The behaviour of the cell when sending a current ; 
 
 (5) The life and cost of working the cell. 
 
 In order to make the above measurements, any of the fofe-- 
 going methods may be employed ; but probably the simplest 
 will be the potentiometer method, for the details of which see 
 par. 219. 
 
 The methods of measuring E.M.F., temperature coefficient, 
 and internal resistance, have already been dealt with, and need 
 not be repeated. In (4), however, we have the most important 
 test of all. 
 
 201. This test should be made in two parts, first taking a 
 relatively small, current from the cell, and another test taking 
 a large current from it. In this, as in all the other tests,- the- 
 
Complete Test of a Primary Battery. 191 
 
 results obtained from experiments on one cell cannot be taken 
 as conclusive ; a number of cells must be tested individually, 
 and from all the results thus obtained the general behaviour of 
 the battery may be deduced. 
 
 In the first test, using small currents, after the E.M.F. and 
 temperature coefficient have been determined, the cell is con- 
 nected Up to a resistance, which may have any value between 
 20 and 100 ohms — about 50 ohms will be found convenient. 
 Time readings of the P.D. of the cell are then taken at intervals 
 of five minutes at first, and afterwards at longer intervals ; 
 also the circuit of the cell is broken, from time to time, just long 
 enough to allow of a measurement of its E.M.F. being made. 
 This test may be continued for twelve hours, the circuit is then 
 broken, and time-readings of the recovery of the E.M.F. are 
 taken. 
 
 Curves showing the variations of E.M.F., P.D., and internal 
 resistance with time of discharge may then be plotted from the 
 data obtained, also a recovery of E.M.F. curve. 
 
 From these curves may be deduced the mean percentage fall 
 of P.D. per minute and the mean rate of recovery. The 
 approximate value of the current flowing should be stated on 
 the curves. Several determinations of the above are made, 
 using fresh cells each time. Another batch of cells are then 
 tested for the high rate of discharge. In this case, the external 
 resistance may have any value between i and 5 ohms, and 
 readings similar to those made above are taken. The discharge 
 may be kept on for 15 min. at a time, and recovery-readings 
 taken after each discharge until the cell is completely exhausted j 
 also, some cells may be discharged right on to exhaustion. 
 The time-intervals between the readings must be smaller in 
 this experiment than in the slow-discharge experiment, for 
 obvious reasons. 
 
 From the curves plotted from the above, the percentage fall 
 of P.D. and recovery of E.M.F. per minute may be calculated, 
 and also the number of ampere hours supplied by the cell may 
 be estimated from a curve showing the variation of the current 
 with the time. 
 
 202. In the case of cells required to give out intermittent 
 
192 Practical Electricity and Magnetism. 
 
 currents in practice, such for instance as cells required to ring 
 bells, etc., a test should be made in which as nearly as possible 
 the actual working conditions are reproduced. The circuit 
 may easily be arranged so as to be made and broken at fixed 
 intervals, by driving the key by clockwork, such for instance as 
 a series of radial spokes attached to the main spindle of the 
 minute hand of an American clock, which may, by dipping into 
 mercury cups, be arranged to make and break the circuit at 
 any desired interval, readings of E.M.F. and P.D. being 
 taken from time to time, and the life of the cell calculated. 
 
 203. The question of cost of working the cell is not very 
 important, as, except for running bells, etc., primary cells are 
 not much used commercially. But an estimate of the cost may 
 be got by (in the case of cells requiring fresh electrolyte, etc.) 
 supplying the cell with fresh chemicals until the plates are worn 
 out ; then, adding the price of the chemicals to the price of the 
 cell, and dividing by the number of ampfere hours supplied, we 
 can get the price per ampere hour. This, of course, will only 
 give a rough estimate, and is only intended to be done in the 
 case of isolated cells ; when large numbers of cells are employed, 
 other things must be taken into consideration. But even with 
 the best conditions obtaining, Ayrton has shown that the cost 
 per Board of Trade unit, with the best-known cells, varies from 
 \s. to ij. bd., whilst from engines and dynamos the cost price 
 is about id. 
 
 Electrometers. 
 
 204. Since electrostatically charged conductors attract or 
 repel one another according as they are oppositely or similarly 
 charged, and since, other things being the same, the force 
 between the charged bodies depends^ on their diiference of 
 potential, the value of this difference of potential may be 
 calculated, provided the value of the force is known. 
 
 It is on this principle that the absolute electrometers of Lord 
 Kelvin have been constructed. 
 
 It must, however, be distinctly borne in mind that any such 
 measurement of potential difference will be expressed in electro- 
 static units, whilst the practical unit of potential difference — the 
 
The Absohite Electrometer. 193 
 
 volt— is expressed in electro-magnetic units ; so that if we desire 
 to convert the potential difference as given by the absolute 
 electrometer into volts, we must know what relation the one 
 system of units bears to the other. This ratio is not known 
 with great accuracy, but is very approximately — 
 
 Potential in the electrostatic system o f imits i 
 
 Potential in the electro-magnetic system of units ~ 3 X 10'" 
 
 The volt, it must be remembered, is 10^ electro-magnetic units. 
 205. Absolute Electrometer. — ^The value of the attractive force 
 between two charged conducting plates can be shown ' to be 
 expressed by the following equation — 
 
 where / is the force in dynes, A the area of one of the 
 attracting plates in square centimetres, d the distance between 
 the plates in centimetres, and V the potential difference 
 between them in electrostatic units. This relation assumes 
 that the distribution of the electric intensity is uniform over the 
 plates. This latter condition, however, does not obtain in the 
 case of a flat plate of finite area, the distribution at the edges 
 not being the same as that over the surface. To overcome this 
 difficulty. Lord Kelvin adopts an ingenious device known as a 
 " guard-ring." This consists of a ring of metal, G, which sur- 
 rounds one of the plates, B, the plates being circular (see Fig. 94), 
 
 C 
 
 Fig. 94. 
 
 and leaving a very small gap between them all round the 
 circumference of B. If G and B are raised to the same 
 potential, the inductive action of G on B prevents the distribu- 
 tion of the electric intensity at the edge of B from differing 
 
 ' See J. J. Thomson's "Elements of ihe Mathematical Theory of 
 Electricity and Magnetism," p. 97. 
 
194 Practical Electricity and Magnetism. 
 
 from that over its surface. The other plate, C, is much larger 
 than B, so that although the distribution at the edges may not 
 be uniform, that over the part that attracts B may be assumed 
 quite uniform. 
 
 One effect of the air gap between G and B is to increase the 
 effective area of B by an amount equal to half the area of the 
 
 Fii;. 95- 
 
 annular gap between G and B, the effective area of B being 
 the quantity A referred to in the above equation. 
 
 In one form of Lord Kelvin's absolute electrometer (see 
 Fig- 95)> the attracted plate is held in position by means of a 
 delicate spring, so that it rests a littie above the level of the 
 guard-ring, and can be charged by means of a dry pile or 
 Leyden jar. The attracting plate is movable vertically by 
 
The Quadrant Electrometer. 195 
 
 means of a screw of accurately known pitch, the head of the 
 screw being suppUed with a graduated disc, so that fractions of 
 a revolution may be read off accurately. 
 
 In order to use the instrument, the plate C is earthed 
 (Fig. 94), and G and B are charged. The plate C, still earthed, 
 is then gradually raised by means of the screw until the plate 
 B is pulled into the plane of the guard-ring by the force of the 
 attraction between them, the exact position of B being indicated 
 by sights attached to the instrument. The position occupied by 
 C is noted, it is then insulated from earth and connected to the 
 body whose potential (V) is required, and again adjusted until, 
 by means of the sights, the plate B is seen to occupy a position 
 in the plane of the guard-ring. The position of C is again 
 noted. 
 
 Let d = distance between the first and second position of C ; 
 Yb = the potential of B, and A its efifective area ; 
 then, firstly — 
 
 and secondly — 
 from this we sret — 
 
 A(Vb - o)^ 
 
 A(Vb - V)^ 
 
 = d\/ 
 
 Sirf 
 A 
 
 In order to determine 7^ which is a constant of the apparatus, 
 and depends on the spring supporting B, the plates and guard- 
 ring are discharged, and known weights are placed on B until it 
 sinks to the plane of the guard-ring ; if w represents this weight 
 in grammes, then — 
 
 y = w X 981 dynes 
 
 From the formula it will be seen that, since the force varies as 
 the square of the potential difference, it will be very small for 
 small potential differences. The instrument can therefore only 
 be used to measure large potential differences. 
 
 206. Quadrant Electrometer. — For the measurement of small 
 potential differences, the quadrant electrometer is generally 
 
iqS 
 
 Practical Electricity and Magnetism. 
 
 employed. This, in its simplest form, consists of four metal 
 quadrants supported on insulating stands, adjacent quadrants 
 being insulated from one another, but the alternate quadrants 
 are connected together. Over the top of these is suspended a 
 flat paddle-shaped needle of aluminium, which, in its normal 
 condition, occupies a position symmetrically over the gaps 
 between the quadrants (see Fig. 96). When the needle is 
 
 charged, the quadrants being all 
 connected to earth, there should 
 be no deflection. If the needle 
 does deflect, it shows that it is 
 not symmetrical with regard to 
 the quadrants, and the position 
 of one of the latter, which is ad- 
 justable, is altered until equi- 
 librium is obtained. If now alter- 
 nate quadrants are brought to 
 different potentials, a deflection 
 of the needle will result, it being 
 attracted by one pair of quadrants 
 and repelled by the other ; the controlling force acting on the 
 needle being the torsion of the suspending fibre, the earth's 
 magnetic force, or gravity, according to circumstances. 
 
 207. In the better forms of electrometer the quadrants com- 
 pletely enclose the needle. Fig. 97 shows one of the most 
 modern forms of electrometer, designed by Messrs. Ayrton, 
 Perry & Sumpner.^ This instrument has been designed espe- 
 cially with a view to sensitiveness and accessibility of the 
 working parts. 
 
 The quadrants, which are smaller than in most electrometers, 
 are mounted on long glass rods which are attached to the base- 
 plate of the instrument, one of the quadrants being free to 
 move out or in by means of a screw attached to the sUde which 
 carries the glass stem and quadrant. The needle is suspended 
 by a silk fibre attached to the movable piece N; from the 
 lower side of the needle projects a piece of platinum wire which 
 dips into a lead vessel, L, containing strong H2SO4, the lead 
 ' Pro. Roy. Soc, June, 1891. 
 
 Fig. 96. 
 
The Quadrant Electrometer. 
 
 197 
 
 vessel being supportecLby a strong wire, X, from a screw, Y, on 
 the vertical rod R, which ends in a glass stem, and is therefore 
 insulated from the base. The rods attached to the terminals 
 T, T pass through the holes in the base of the instrument and 
 are fixed to the ends of two glass stems, contact being made 
 from these points to the quadrants. 
 
 The charging is performed through the charging rod cc, which 
 is attached to a glass stem, but hinged so that its upper end, 
 
 Fig. 97. 
 
 on which there is a lead ball, may either be in contact with 
 RR or not, as desired. To charge the needle, the rod cc is 
 turned so as to bring the ball into contact with RR ; to the 
 other end of the rod there is applied a charged electrophorus, 
 or other source of electrification. After charging, the rod is 
 disconnected (by a tap from the charging body) from RR, thus 
 minimizing the chance of leakage. The mirror attached to the 
 needle is surrounded by a guard-ring, made in two halves, 
 carried by the arms A, A. 
 
 The whole of the parts are enclosed in a glass shade, which 
 also serves as a Leyden jar to keep the needle charged. 
 
198 Practical Electricity and Magnetism. 
 
 The controlling force in this instrumeiafls magnetic, a small 
 magnet being attached to the back of the" mirror. 
 
 208. One of the chief difficulties in connection with an electro- 
 meter is the difficulty of getting good insulation for the needle 
 and quadrants. To insure this, the glass-supporting stems 
 must be perfectly clean and well dried, the air inside the case 
 of the instrument being kept dry by the strong H2SO4. 
 
 Before using an electrometer it must be tested for insulation. 
 To do this the needle should be charged, the quadrants being 
 earthed, then one pair of quadrants is charged and the deflec- 
 tion noted ; if this remains steady (the quadrant being insulated 
 after charging) the leakage from this pair of quadrants is 
 negligible. The other pair of quadrants are then tested in the 
 same way, the first pair being earthed. 
 
 Should the deflection fall off, it shows that either the needle 
 or quadrants are leaking. If the quadrants are then connected 
 to the poles of a cell, the needle being charged, and the deflec 
 tion falls, it points to leakage from the needle, since the quad- 
 rants are at a constant potential. Where this leakage has been 
 stopped a repetition of the first test will show whether the 
 quadrants are leaking. 
 
 2og. One other point must be borne in mind when using the 
 quadrant electrometer, namely, that if part of the circuit is 
 brass and part copper, as the quadrants and the leads, or in 
 fact any pair of dissimilar metals, a deflection may be obtained 
 due to the contact difference of potential between the dissimilar 
 metals ; this can be determined experimentally. 
 
 210. With regard to the law of the instrument, it can be shown' 
 that, if Vi and Vj are the potentials of the quadrants, V„ the 
 potential of the needle, and 6 the angular deflection of the needle, 
 then — 
 
 K being a constant depending on the construction of the 
 instrument. From this it will be seen that, if V„ is very large 
 
 ' " Elements of Mathematical Theory of Electricity and Magnetism " 
 (J. J. Thomson), p. 98. 
 
Ttie Quadrant Eteciyometer. 199 
 
 compared with Vj aqd Vo, then the angular deflection B is 
 approximately proportional to the difference of potentials of 
 the quadrants. This method of using the electrometer with 
 the needle highly charged is called the heterostatic method. 
 On the other hand, if the needle is connected to one pair of 
 quadrants, V„ becomes equal to Vj or Vo and — 
 
 e = K(V, - Vsf 
 
 the deflection being proportional to the square of the potential 
 difference. The instrument in this form is said to be used 
 idiostatically, and is not so sensitive, but is adapted for measur- 
 ing greaiter P.D.'s ; it is also capable of measuring alternating 
 E.M.F.'s, since the needle changing its potential with the quad- 
 rants causes the deflection to be always on the same side of zero. 
 
 211. In some forms of electrometer a special arrangement, 
 known as a replenisher, is employed to charge the needle, the 
 potential of the latter being roughly indicated by means of a 
 small auxiliary guage, working on the principle of the absolute 
 electrometer ; if leakage occurs the guage falls, and may be 
 brought back to its original position by recharging the needle 
 from the replenisher. Many experimenters^, however, object to 
 using it, and prefer to charge the needle some time before the 
 instrument is required for use, so as to give the potential time 
 to settle down to a steady value. The potential of the needle 
 can always be tested for constancy by taking the deflection pro- 
 duced by a standard cell. 
 
 The most usual way of using the electrometer for the 
 comparison of E.M.F.'s is to charge the needle to a high 
 potential by means either of an electrophorus or of a dry pile 
 kept at a constant temperature. 
 
 The E.M.F. to be measured is then applied, and the 
 deflection obtained recorded; the connections are then 
 reversed and the deflection to the other side taken. The 
 mean deflection is calculated. The same process is repeated, 
 using a standard cell of known E.M.F., and from the ratio 
 of the two mean deflections the ratio of the E.M.F.'s may be 
 obtained. 
 
 212. In order to measure the E.M.F. of a Daniell cell an 
 
200 Practical Electricity and Magnetism. 
 
 electrometer was employed the needle of which had been 
 charged to a high potential by means of an electrophorus some 
 hours previously, and had retained its charge, as was proved 
 by the instrument giving always the same deflection when con- 
 nected up to a standard Carhart Clark cell. 
 
 The temperature of the Clark cell was i8° C, and the 
 following readings were obtained : — 
 
 Deflection to right with standard cell 
 left 
 
 Mean deflection ... 
 Deflection to right with Daniell cell 
 
 Mean deflection ... 
 
 123 scale-divisions. 
 I2S » 
 
 124 
 
 9i'5 scale-divisions. 
 93'6 .. 
 
 92-5 
 
 E.M.F. of standard cell = i-438{i - 0-000387(18° - 15°)} 
 = i'436 volt 
 
 1-43^ 124 
 Hence - - = — .- 
 X 92-5 
 
 X = I '07 1 volt 
 
 The Potentiometer. 
 
 213. We have several times referred to this instrument as 
 one suitable for making comparisons of resistances, currents, 
 and electro-motive forces. But instead of -describing it in each 
 of the chapters dealing with these measurements, we have 
 considered it better to devote a special section to it and its 
 applications. 
 
 In all cases the comparisons, whether of resistance, current, 
 or E.M.F. , are reduced to a comparison of E.M.F.'s, resist- 
 ances being compared in terms of the fall of potential in them 
 when carrying the same current, and currents by comparing the 
 potential differences which they produce at the ends of a 
 standard resistance. 
 
 Of the various experimenters who have worked at this 
 instrument, it has perhaps gained more at the hands of Mr. 
 
TJie Potentiometer. 201 
 
 R. E. Crompton than any one else, and the arrangement 
 which we describe here is a modification of the form of 
 potentiometer constructed by him.^ 
 
 214. The potentiometer consists of a wire which is divided 
 up into fifteen- parts, such that all the segments have the same 
 electrical resistance. The last segment at one end consists of 
 bare wire stretched over a graduated scale, and is provided 
 with a sliding tapping-contact similar to that on a wire bridge. 
 The other fourteen segments may be wound into coils, 
 potential wires being brought away from the ends of the 
 segments (see Fig. 98). 
 
 RQPONMLKHGFEDCB 
 
 Fig. 98. 
 
 In constructing the potentiometer, it is not necessary to 
 make the fifteen lengths separately and then join them in 
 series ; it will, in fact, be found much more convenient to 
 cut off one long piece of wire sufficient for the whole instrument 
 and then solder on the potential wires A, B, C, D, etc., so as 
 to divide up the whole wire into fifteen parts of equal resistance. 
 The first fourteen segments may then be coiled up and the 
 fifteenth stretched over its scale. In this way we avoid 
 soldered joints in the wire itself. The wire should be made 
 of some substance possessing high specific resistance and low 
 temperature coefficient, such, for instance, as manganin. The 
 resistance of each segment being somewhere near 2 ohms. 
 
 After the coils have been adjusted and the wire stretched, 
 the latter must be carefully calibrated by some of the methods 
 described already (see par. 32), and then each of the fourteen 
 coils accurately compared with the stretched wire, by sending 
 a current through them and comparing the fall of potential 
 down each coil with that down the stretched wire. In this 
 way, taking the length of the stretched wire as a standard, the 
 effective lengths of the coils may be obtained in terms of it, 
 
 ' See Electrician, vol. xxxi. p. 32, and vol. xxxvi. p. 158. 
 
202 Practical Electricity and Magnetism. 
 
 and a table of corrections constructed if necessary. A con- 
 venient method of arranging the coils is shown in Fig. 99. 
 This can be used in conjunction with a metre bridge, the wire 
 of which, if of equal resistance to the coils^ may be used as the 
 stretched wire. 
 
 The fourteen coils are wound on bobbins and enclosed in 
 a box, the potential wires being soldered to the contact studs 
 I, 2, 3, 4, etc. The sliding arm A makes contact between the 
 
 
 
 
 
 B 
 
 R 
 
 
 ■ ** 
 
 19 
 
 4 
 
 A a' 
 a* 
 
 1 »- 
 
 
 
 
 
 
 
 ** 
 
 
 ^' 
 
 
 
 
 
 .---• 
 
 S' 
 
 •-■-. 
 
 
 
 
 
 
 
 E 
 
 >t- 
 
 
 -X 
 
 E, 
 
 
 
 c 
 
 r 
 
 
 
 Si 
 
 
 
 
 
 
 
 
 T 
 
 
 
 
 
 . ^ 
 
 
 
 
 1 
 
 . 
 
 
 
 Fig. 99. 
 
 Studs and the two-way switch Si, to which one terminal of E 
 and El is connected, the switch 83 making contact between the 
 other terminals of E and Ej and the galvanometer G. 
 
 In using the potentiometer, a steady current must be main- 
 tained in the fourteen coils and stretched wire, from a secondary 
 battery, B, in series with a variable resistance, R. Primary 
 batteries cannot be used satisfactorily for this purpose, on 
 account of their E.M.F. falling oif rapidly when supplying 
 a current. 
 
 215. Comparison of Resistances. — In order to compare two 
 resistances by means of the potentiometer, the resistances are 
 connected in series with one another and a current from a 
 secondary battery sent through them. Should the resistances 
 be small it is necessary to have a variable resistance in series 
 with them, to prevent the current from the battery being too 
 
Comparison of Resistances by Potentiometer. 203 
 
 great. Potential wires are then taken from the ends of the 
 resistances to terminals E and Y^ respectively of the potentio- 
 meter. The switches Si and S2 are then adjusted so as to 
 connect the terminals E to the potentiometer wire, and by 
 adjusting the rotating arm A and the tapping-contact connected 
 to G, the fall of potential down the coil may be balanced 
 against the fall of potential down a portion of the potentiometer 
 wire, as shown by the galvanometer G giving no deflection. 
 The positions of A and of the tapping-point are noted; the 
 switches Sj and 83 are then altered so as to connect the 
 terminals Ej to the potentiometer wire, and another pair of 
 balancing-points are found for the other coil. The ratio of the 
 eflFective lengths of potentiometer wire between the balancing- 
 points in the two cases is the ratio of the two resistances, and 
 if one of the resistances is a standard coil, the resistance of the 
 other coil can be calculated. 
 
 In making such a comparison of resistances, care must be 
 taken to insure that the direction of the current in the coils to 
 be compared is such that the potential difference at their ends 
 is in opposition to the potential difference on the wire, and 
 also that the fall of potential down the larger of the two • 
 resistances is not greater than the total difference of potential 
 at the ends of the potentiometer wire. 
 
 The accuracy of a comparison such as the above depends on 
 the sensitiveness of the galvanometer and on the accuracy of 
 calibration of the potentiometer wire and coils, and may be 
 made very high. The standard resistances employed with the 
 potentiometer should be of the form described in par. 21, 
 which is specially designed for this kind of measurement. The 
 galvanometer used may be of any sensitive type, but one or 
 other modification of the D'Arsonval galvanometer will be 
 found most convenient to work with, on account of its high 
 sensibility and freedom from magnetic control. 
 
 All precautions respecting the measurement of the tempe- 
 rature of the coils which have been previously dealt with, apply 
 equally in this case. 
 
 216. The following comparison of resistances was made on 
 a potentiometer. 
 
204 Practical Electricity and Magnetism. 
 
 The wire and coils of the potentiometer were calibrated by 
 the fall of potential method, the galvanometer employed 
 having been previously tested and found to have a straight-line 
 law within the limits of the deflections employed in the' 
 experiment. The following readings were obtained in the 
 calibration : — 
 
 Coil. 
 
 Deflection. 
 
 1 — 2 
 
 IIO-O 
 
 2—3 
 
 IIOO 
 
 3-4 
 4—5 
 
 UO'O 
 IIO-O 
 
 5-6 
 
 IIO-O 
 
 6-7 
 
 IIOO 
 
 7-8 
 
 iio-o 
 
 8-9 
 
 IIOO 
 
 9 — 10 
 
 HOC 
 
 10 — It 
 
 IIO'O 
 
 It — 12 
 
 iio-o 
 
 12 — 13 
 
 13—14 
 14—15 
 
 Stretched wire 
 
 109-5 
 
 109-0 
 
 109-5 
 
 III-O 
 
 A calibration of the stretched wire, which, like the coils, was 
 of manganin, proved it to be perfectly uniform. 
 
 The first balance, when the coil connected to the terminals 
 E was tested, was obtained with the arm A at stud 4, and the 
 tapping-contact at 97 "50, and therefore the fall of potential 
 was proportional to — 
 
 (3 X "°)+^rrt-X 111=43822. 
 
 The second balance of the fall of potential down a standard 
 i-ohm coil (manganin) against the fall of potential in the wire 
 gave arm A at stud 8, and the tapping-point at 83-55, O"^ ^ ^^1 
 of potential proportional to — 
 
 83-55 
 7 X no -f •'^^ X III =862-74 
 100 ' ^ 
 
Measurement of Current by Potentiometer. 205 
 
 And the ratio of resistances is — 
 
 X 438"22 
 I ~ 862-74 
 ox X = o'5o8 ohm. 
 The temperature of the coil x was 12° C. 
 
 217. Measuretnefit of Cwrent. — In the measurement of 
 current by the potentiometer, the current is sent through a 
 standard resistance, which, if the current is large, must be 
 specially constructed to carry heavy currents without seriously 
 heating. The P.D. at the ends of the coil is balanced against 
 the fall of potential in the potentiometer wire, as in the last 
 case. The E.M.F. of a standard cell is then balanced against 
 the fall of potential in the wire ; from these two readings, and 
 the known E.M.F. of the standard cell, the P.D. in volts at the 
 terminals of the known resistance, may be calculated, and, divid- 
 ing this by the resistance in ohms, we get the current in ampferes. 
 
 The limit of accuracy in the measurement of current by this 
 method depends on the accuracy to which the E.M.F. of the 
 standard cell is known, z.^. to about i part in 1400, the E.M.F. 
 of the cell being of course corrected for temperature in the 
 above measurement. 
 
 218. In measuring a current on the potentiometer, the cali- 
 bration table of which is given in par. 216, the current was 
 sent through a standard i-oHm manganin coil, and the potential 
 difference at its ends was balanced against the fall of potential 
 in the wire and coils, and gave the following reading : Rotating 
 arm A at stud 10, sliding contact at 98'3o on the scale. A 
 Clark cell was then balanced on the potentiometer, and gave 
 sliding arm A at stud 14, tapping-contact at 38"oo. The 
 temperature of the cell was 15° C. 
 
 The E.M.F. ofthe Clark cell was = 1-438(1 --ooo387(/- 15°) 
 
 = 1-438 volt 
 
 The first fall of potential was proportional to — 
 
 98-30 
 
 g X no -1- X III = logo-ii 
 
 ^ 100 
 
 The second fall of potential was proportional to — 
 
 1640 -f- -^ X III = i6qi-i8 
 ^^ 100 ^ 
 
2o6 Practical Electricity and Magnetism. 
 
 Hence, calling E the potential difference at the ends of the 
 
 i-ohm coil, we have — 
 
 E 1099-11 
 
 = —i. — r^ = 0-940 volt 
 
 1-438 i6oi'i8 '^ 
 
 E 0-940 . 
 
 Therefore the current = ^ = — - — = 0-940 ampere 
 
 K. I 
 
 219. Comparison of E.M. F.'s. — The comparison of E.M.F.'s 
 is made in exactly the same manner as that just described 
 above, only two cells are compared instead of a cell and the 
 P.D. at the terminals of a resistance. The instrument, how- 
 ever, may be adjusted so as to read directly in volts, and thus 
 saves time and calculation. To do this the E.M.F. of the 
 standard cell corrected for temperature is calculated ; suppose 
 it is 1-4346 volt. The sliding contact A is then turned to 
 the extreme left-hand stud 15, and the tapping-point on the 
 wire is adjusted to read 34-6 (the wire being divided into a 
 hundred parts) on the scale. The length of wire between the 
 tapping-points is then 14-346 segments. The variable resistance 
 R is now adjusted until the fall of potential downthe 14-346 
 segments is equal to the E.M.F. of the standard cell. If any 
 other cell is balanced against the fall of potential down the 
 wire now, the number of segments between the balancing- 
 points will be ten times the E.M.F. in volts. Thus supposing, 
 in order to get a balance with another cell, the arm A had to 
 be turned so as to make contact with stud 12, and the tapping- 
 point gave a balance when placed at 57-0, the E.M.F. required 
 
 would be — 5Z_ = 1-257 volt. It is assumed here that, after 
 10 
 
 the instrument has been standardized by the standard cell, the 
 current in the wire keeps perfectly constant. In order to test 
 this, readings with the standard cell should be taken from 
 time to time. It is also assumed that the resistance of each 
 of the fourteen coils is equal to that of the stretched wire, and 
 that the stretched wire is perfectly uniform. 
 
 The accuracy which the potentiometer is capable of, is very 
 great when comparisons only are to be made ; but if a measure- 
 ment of E.M.F. or current is desired, it is limited to the 
 
Comparison of E.M.F's by Potentiometer. 207 
 
 accuracy of our knowledge of the E.M.F. of the standard 
 cell. 
 
 220. As an exercise on the use of the potentiometer the 
 following investigation was carried out on a Daniell cell in 
 which dilute svilphuric acid was used as the electrolyte surround- 
 ing the zinc instead of zinc sulphate solution. The cell was 
 prepared in the following manner. The copper-plate was 
 freshly plated with copper before using. The zinc rod em- 
 ployed was of double distilled zinc, and free from impurity ; 
 it was well amalgamated with pure mercury before using. The 
 copper sulphate solution was made by dissolving crystals of 
 CUSO4 in tap-water till the solution was saturated at 15°, C. 
 First of all, experiments were made by adding acid to the 
 water in the porous pot surrounding the zinc only ; the results 
 were, however, not satisfactory, on account of the alteration 
 of E.M.F. produced by the slow passage of the acid into the 
 copper sulphate solution. After many experiments, the best 
 and most constant results were obtained by adding acid to 
 both the liquids. 
 
 The table below gives the E.M.F.'s of the cell when made 
 up with different quantities of acid. In making these measure- 
 ments, after the liquid had been made up, the cell was short- 
 circuited for two minutes, and then allowed to rest, its E.M.F. 
 being measured on the potentiometer against a standard Clark 
 cell. After two minutes it was found to reach a steady value, 
 the numbers recorded in the table representing the value of 
 the E.M.F. after fifteen minutes' rest. 
 
 Is 
 
 
 Electrolyte surrounding copper 
 plate. 
 
 Electrolyte surrounding zinc 
 
 rod. 
 
 SI 
 
 0-° 
 
 fi-° 
 
 
 
 
 wa 
 
 
 
 
 
 Volts. 
 
 150-2 
 
 114-8 
 
 CuSOj dissolved in tap-water 
 
 tap-water 
 
 
 1-095 
 
 150-2 
 
 119-7 
 
 loo c.c. CuS04+o-5 C.C. HjSOj 
 
 too C.C. H,0+o-5 c.c. H^SO, 
 
 1-142 
 
 150-2 
 
 119-5 
 
 „ + I .. .. 
 
 „ + I ,. 
 
 
 1-140 
 
 150-2 
 
 118-3 
 
 ,. + 2 „ „ 
 
 „ + 2 „ 
 
 
 1-128 
 
 150-2 
 
 116-8 
 
 ,, + 5 ., .. 
 
 „ „ + 5 .. 
 
 
 1-114 
 
 150-2 
 
 iiS-9 
 
 „ -l-io „ „ 
 
 ., +10 „ 
 
 
 1-106 
 
 150-2 
 
 115-2 
 
 „ +15 .. ,. 
 
 „ +15 .. 
 
 
 1-099 
 
 150-2 
 
 "47 
 
 „ +20 „ „ 
 
 „ +20 „ 
 
 
 1-094 
 
208 Practical Electricity and Magnetism. 
 
 The E.M.F. of the Clark cell corrected for temperature was 
 I "434 volt. The following curve (Fig. loo) shows the varia- 
 tions in voltage with amount of acid added. 
 
 I-ISO 
 
 ly 
 
 
 
 
 
 
 
 I-IW 
 
 \ 
 
 
 
 
 
 
 
 J I'ISO 
 
 .E iiio 
 
 *-• t-IIO 
 
 <0 
 
 \ 
 
 y 
 
 
 
 
 
 
 
 \ 
 
 s 
 
 
 
 
 
 
 
 ^ 
 
 
 "^ 
 
 
 
 
 
 
 
 
 
 
 -- 
 
 12 tS IB SJ 
 
 CC of acid in 100 CC. of fclectrolyte 
 
 Fig. ioo. 
 
 The first reading, with no acid in the cell, is very variable 
 and liable to sudden changes if the cell is shaken. 
 
References to Scientific Papers. 
 
 209 
 
 221. References to SciENTrpic Papers. 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 Standard Cells. 
 
 ^ 
 
 
 A Standard Voltaic Battery 
 
 Clark 
 
 Trans. Roy. Soc, 
 1874; /V«. Roy. 
 Soc, vol. 20, p. 
 444; Itid., vol. 
 21, p. 421. 
 
 On the Electro-Motive Force of 
 
 Rayleigh and 
 
 Trans. Roy. Soc, 
 
 Standard Cells 
 
 SedgAvick 
 
 1885. 
 
 On the Clark Cell as a Standard 
 
 Rayleigh 
 
 Ibid., 1886. 
 
 of Electro-Motive Force 
 
 
 
 *» »» >) 
 
 Glarebrook and 
 Skinner 
 
 Ibid., 1892. 
 
 Variation of the E.M.F. of 
 
 Ayrton and 
 
 Pro. Roy. Soc, vol. 
 
 Clark Cells with Temperature 
 
 Cooper 
 
 59. P- 368- 
 
 On the Clark Cell as a Source 
 
 Threlfall and 
 
 Phil. Mag., \o\.2'&. 
 
 of Small Currents 
 
 Pollock 
 
 Nov., 1S89. 
 
 The Clark Cell when pro- 
 
 Skinner 
 
 Ibid., vol. 38, Sept., 
 
 ducing a Current 
 
 
 1894; vol. 39, 
 April, 1895. 
 
 S3 )» H 
 
 Threlfall 
 
 Ibid., vol. 39, Mar., 
 1895. 
 
 An Improved Standard Clark 
 
 Carhart 
 
 Ibid., vol. 28, Nov., 
 
 Cell 
 
 
 1889. 
 
 Instructions for preparing Clark 
 
 Kahle 
 
 Elect., vol. 31, p. 
 
 Cells 
 
 
 625. 
 
 On the Variation of Clark Cells 
 
 Swinburne 
 
 Ibid., vol. 27, p. 
 
 Standard Clark Cells 
 
 Glazebrook 
 
 500. 
 Ibid., vol. 27, p. 98. 
 
 Clark Cells* 
 
 Kahle 
 
 Ibid., vol. 29, p. 
 
 516. 
 Phil. Mag. vol. $, 
 
 On a Form of Daniell Cell con- 
 
 Lodge 
 
 venient for a Standard of 
 
 
 Jan., 1878. 
 
 Electro-Motive Force 
 
 
 
 On the use of Daniell's Cell 
 
 Fleming 
 
 Ibid., vol. 20, Aug., 
 
 as a Standard of E.M.F. 
 
 
 1885. 
 
 Weston Standard Cell 
 
 
 Elect., vol. 30, p. 74 1 . 
 
 E.M.F. and Temperature Co- 
 
 Dearlove 
 
 Ibid., vol. 31, p. 
 
 efficient of Cadmium Cells 
 
 
 645. 
 
 The Cadmium Standard Cell 
 
 Jaeger and 
 
 Jour. Elec. Eng., 
 
 
 Wachsmuth 
 
 vol. 25, p. 726. 
 
 E.M.F. of Daniell Cell 
 
 Carhart 
 
 Am. Jour. of 
 Science, vol. 28, 
 1884. 
 
 Electrometers. 
 
 
 
 Quadrant Electrometers 
 
 Ayrton, Perry, 
 
 Traits. Roy. Soc, 
 
 
 Sumpner 
 
 1892. 
 
 ■*— 
 
 
 P 
 
2IO Practical Electricity and Magnetism. 
 
 Title of Paper. 
 
 A Guard-Ring Electrometer 
 On the Quadrant Electrometer 
 
 An Absolute Spherical Electro- 
 meter 
 
 A Method of determining the 
 Value of Rapid Variations of 
 Potential Difference by the 
 Capillary Electrometer 
 
 The Time Relations of the 
 Excursions of the Capillary 
 Electrometer 
 
 Capillary Electrometer 
 
 Capillary Electrometer in 
 
 Theory and Practice 
 Contact Difference of Potential 
 
 On Dry Charging Piles 
 
 Battery Tests, etc. 
 
 On a Constant Daniel! Battery 
 
 Effects of Temperature on the 
 E.M.F. and Resistance of 
 Batteries 
 
 Determination of Chemical 
 Affinity in Terms pf E.M.F. 
 
 On the E.M.F. of Alloys 
 
 On the Application of Standard 
 Current Balances to the De- 
 termination of the E.M.F. of 
 Voltaic Cells 
 
 Divergence of E.M.F.'s from 
 Thermo-Chemical Data 
 
 Temperature Coefficient of a 
 Battery 
 
 Report on the Hellesen Dry 
 Battery 
 
 Report on the Lessing Dry 
 Battery 
 
 The Crompton Potentiometer 
 
 Use and Capabilities of the 
 Crompton Potentiometer 
 
 Author. 
 
 Reference. 
 
 Fitzgerald 
 
 Phil. Mag., vol. 10, 
 
 
 July,' 1880. 
 
 Hopkinson 
 
 Ibid., vol. 19, Apr., 
 
 
 1885. 
 
 Lippmann 
 
 .e/frf., vol. 22,Tuly, 
 
 
 1886. 
 
 Burch 
 
 ^0. Roy. Soc, vol. 
 
 
 48, p. 89. 
 
 
 Trans. Roy. Soc, 
 
 
 1892. 
 
 Berget 
 
 Elect., vol. 27, p. 
 
 
 252. 
 
 Burch 
 
 Ibid., vol. 37, p. 
 
 
 380, etc. 
 
 Clifton 
 
 Pro. Roy. Soc, vol. 
 
 
 26, p. 299. 
 
 Elster and Geitel 
 
 Phil. Mag., vol. 16, 
 
 
 Aug., 1883. 
 
 Sir Wm. Thom- 
 
 Pro. Roy. Soc, vol. 
 
 son 
 
 19. p. 253- 
 
 Preece 
 
 /*zV/.,vol.3S,p.48; 
 
 
 vol. 35, p. 250 ; 
 
 
 vol. 36, p. 464. 
 
 Alder Wright 
 
 Phil. Mag., vol. 9, 
 
 
 Apr., 1880, etc. 
 
 Trowbridge and 
 
 Ibid., vol. 16, Nov. 
 
 Stevens 
 
 1883. 
 
 Sir Wm. Thom- 
 
 Ibid., vol. 24, Dec, 
 
 son 
 
 1887; vol. 25, 
 
 
 Feb., 1888. 
 
 Herroun 
 
 Ibid., vol. 27, Mar., 
 
 
 1S89. 
 
 Carhart 
 
 Elect., vol. 27, p. 
 
 
 167. 
 
 Krehbiel 
 
 Ibid., vol. 26, p. 
 
 Walmsley 
 
 419. 
 Ibid., vol. 36, p. 
 
 
 589. 
 
 
 Ibid., vol. 31, p. 32. 
 
 Fisher 
 
 Ibid., vol. 36, p. 
 
 
 158, etc.; vol.37. 
 
 
 p. 5. etc- 
 
IV. 
 
 QUANTITY. 
 
 2 2 2. The practical unit of quantity of electricity is the 
 coulomb, and is the quantity of electricity conveyed along a 
 conductor by a current of one ampfere flowing for one second. 
 
 The practical measurement of quantity of electricity may be 
 made by means of a voltameter, which is really a quantity, and 
 not a current measuring- apparatus, since the weight of the ions 
 liberated by electrolysis is proportional to the quantity of 
 electricity which has passed through the instrument. 
 
 For the measurement of very small quantities of electricity 
 the voltameter is not a convenient instrument to use, a ballistic 
 galvanometer being employed instead. 
 
 223. In general construction, the main difference betw.een a 
 ballistic galvanometer and an ordinary reflecting galvanometer 
 is in the shape of the needle, which in the former instrument is 
 made very massive, and shaped so as to ofier as small a surface 
 to air friction as possible. When a discharge of electricity 
 takes place through the coils of the galvanometer, the needle 
 receives a sudden impulse, which causes it to deflect, the 
 amplitude of the first swing of the needle being proportional 
 to the quantity of electricity which has passed through the coils, 
 provided that the whole of the discharge took place before the 
 needle started on its swing. It is with the object of insuring 
 this latter condition that the galvanometer needle is made so 
 massive. 
 
 Theory of the Ballistic Galvanometer. 
 
 224. The following is the proof of the relation between the 
 constants of the ballistic galvanometer and the quantity of 
 electricity producing a given swing of the needle. 
 
212 Practical Electricity and Magnetism. 
 
 Let M = magnetic moment of the galvanometer needle ; 
 
 I = moment of inertia of galvanometer needle ; 
 G = galvanometer coil constant \ 
 
 (J) = angular velocity of the needle at any instant ; 
 Q = total quantity of electricity in absolute units ; 
 
 a = amplitude of the first swing of the needle ; 
 H = horizontal intensity of the earth's magnetic field ; 
 T = periodic time of swing of the needle. 
 
 The quantity of electricity Q = ct where c is the current 
 strength, and t the time during which it flows. 
 
 Moment of the force on the needle due to ^ = MG^r 
 The total impulse given to the needle = MGct 
 
 = lo) 
 
 MGcT 
 :. <- - — ^ (i) 
 
 If the pole strength of the needle is m, and its length 2/, 
 the work done in deflecting the needle through an angle a 
 is — 
 
 2/»/H(i — cos a) = MH(i — cos a) 
 
 But the work done on the needle = kinetic energy in needle 
 
 = h 1-= 
 .-. i W = MH(i - cos a) 
 
 / 2MH(i - cos a) 
 and (0 = V ^ (2) 
 
 Now, combining equations (i) and (2), we get — 
 
 MGQ _ / 2MH(i - cos a) 
 
 I ~ V I 
 
 M 
 
 (3) 
 
 Now, the periodic time T = 2w'\/ jrjjT. and equation (3) 
 may be written — - 
 
 _ HT . a 
 Q = 2l^ ' '''^ r 
 
Theory of Damping. 213 
 
 Multiplying this result by 10 gives the number of coulombs, 
 since there are 10 coulombs in i C.G.S. unit of quantity. 
 
 225. Correction for Dampittg. — In the above calculation we 
 have assumed that the only retarding force acting on the 
 needle is that due to the controlling-field H. There are, how- 
 ever, other causes which affect the amplitude or the swing of 
 the needle. Of these the most important are (i) the induced 
 currents set up in the galvanometer coil, due to the motion of 
 the needle inside it, in accordance with Lenz's law; (2) the 
 resistance to the motion of the needle due to the friction of the 
 air; (3) the temporary alteration in M due to the field pro- 
 duced by the current in the galvanometer coil ; and (4) the 
 torsion of the suspending fibre. 
 
 The effect of (3) may be made small by sending discharges 
 through the galvanometer, which will only produce small swings ; 
 and (4) may be allowed for in the same manner as described in 
 par. 138. 
 
 The effects of (t) and (2), which tend to diminish the ampli- 
 tude of the swings, are usually termed the " damping " effects, 
 and may be determined experimentally. 
 
 If the needle of a ballistic galvanometer is set swinging, 
 the amplitude of the swings gradually diminishes, on account 
 of this damping action, till finally the needle is brought to 
 rest. 
 
 If the damping is not too great, then it is found that the 
 amplitudes of successive swings diminish in a geometrical 
 series, calling aj, a^, a^, . . . a„ successive swings, then — 
 
 ?}_ f^V_ r^y_ f-O'rz f'^O"^ = 
 Oj Vos' \a^/ ^fg/ ^"■n'' 
 
 or, calling »„, = amplitude of the ;«th swing 
 and a„ = amplitude of the «th swing 
 
 then 
 
 (::)"- 
 
 hence log^ c = ^-^7^ (loge ",« - log. «„) 
 
214 Practical Electricity and Magnetism. 
 
 logj c is the log of the constant ratio, and is known as the 
 logarithmic decrement (\) ; 
 
 or X = ^^-;^ (log, a„, - loge a„) 
 
 226. In order, to correct any swing for damping we can now 
 proceed as follows. Let a represent the, observed swing of the 
 ballistic galvanometer needle. If there had been no damping, 
 the swing would- have been greater, say a', so that the effect of 
 the damping has been, during a half-period of a complete swing 
 .of the needle, to reduce the swing from a value a' to a value a. ; 
 therefore, since — 
 
 \ = J (loge «' - log, a) 
 and - = loge a' - loge a 
 
 or loge a = - + loge a 
 
 from which, by the theory of logarithms— 
 
 a = £ 
 
 X . X 
 
 but £2 when expanded gives i + " + etc., and since X is 
 
 small it is not necessary to go further up the series than the 
 second term ; also — 
 
 hence a' = ali-|- — j \ 
 
 So that, in order to correct the observed swing of a ballistic 
 galvanometer = needle for damping, we must multiply by 
 
Measurement of Logarithmic Decrement. 215 
 
 ( I H — ) . We may therefore write the full formula for 
 the ballistic galvanometer as — 
 
 ^ HT . a/ X\ 
 
 This simple correction for damping may be applied in all 
 cases when A. does not exceed 0*5, without introducing any 
 sensible error. 
 
 Measurement of A.. 
 
 227. From what has just been said, it will be seen that we 
 may determine \ from the observation of a series of swings of 
 the needle. It must, however, be borne in mind that A. is not a 
 constant for all conditions under which the galvanometer may 
 be used ; since it depends partly on the induced currents set up 
 in the coil in accordance with Lenz's law, and these depend on 
 the resistance of the circuit in which they are induced, the 
 damping must obviously vary with the resistance of the 
 galvanometer circuit. The measurement of X should therefore 
 be made for a number of cases, the galvanometer circuit resist- 
 ance being specified in each case, starting with the galvano- 
 meter on open circuit and ending with it short circuited. A 
 curve may then be plotted showing the relation between X and 
 the resistance of the galvanometer circuit. From such a curve 
 the value of A. for any particular experiment may be deduced. 
 
 To measure X the galvanometer needle should be set swinging, 
 either by means of a magnet, which is afterwards removed from 
 its vicinity, or else by passing a current momentarily through 
 the galvanometer coils. The successive swings to right and 
 left of the scale zero are then noted by two observers. Let these 
 be tti, Oj, 03, Oj, . . . a,,. 
 
 These should be tabulated in two parallel columns, and for 
 convenience a zigzag line connecting them, thus — 
 
2i6 Practical Electricity and Magnetism. 
 
 (l-t) 
 
 If the zero lies between % and 0.2, then the arithmetical sum 
 of aj + "2 gives the amplitude of the first swing, or, in general, 
 calling readings to the left hand of zero minus, and to the 
 right hand plus — 
 The algebraic diff. of a^ — a^ is the amplitude of the ist swing 
 
 "2 — <hi 
 
 «.■! — '^i 
 
 2nd 
 
 3rd 
 
 («-i)th 
 
 The logarithinic decrement may be calculated from the above, 
 taking various combinations of swings, and using the relation — 
 
 , (loge a„, - loge a„) 
 
 228. In order to determine the logarithmic decrement of a 
 ballistic galvanometer, the needle was adjusted so that the 
 spot of light was at the zero on the scale. ■ The needle was 
 then started swinging by bringing a magnet near the galvano- 
 
Calibration of a Ballistic Galvanometer. 217 
 
 meter and then removing it. The following readings were 
 taken with the galvanometer on open circuit : — 
 
 + 135 
 
 -127 
 
 +77 
 
 Amplitude of ist swing = 135 + 127 = 262 
 2nd „ = 127 + 112 = 239 
 3rd ,, = 112 + 102 = 2r4 
 4th „ = 102 + 93 = 195 
 Sth „ = 93 4- 83 = 176 
 6th „ = 83 + 77 = 160 
 
 Taking swings i and 3, X = ._ ^ ('oge 262 — log^ 214) = 0-103 
 
 , 5.^^ 
 
 (loge 214 - log. 1 7 6) = 0-098 
 
 4 „ 6,X: 
 
 6-4 
 
 I 
 
 (loge 195 - loge i6o)=o-io3 
 
 2 „ 6, A. = g-^ri (log' ^39 - loge ioo)=o-ioi . 
 Mean value of logarithmic decrement A = o-ioi 
 
 Calibration of a Ballistic Galvanometer. 
 
 229. The calibration of a ballistic galvanometer, in order to 
 obtain the relation between the quantity of electricity dis- 
 charged through the coils and the throw produced on the 
 needle, may be carried out in three ways — 
 
2[S Practical Electricity and Magnetism. 
 
 HT . a/ , A 
 (i) From the relation Q = — r< 2 sin -I i +— ) 
 ^ ' ^ 27rG 2 \ 2 ' 
 
 (2) By means of an earth inductor or standard current 
 inductor. 
 
 (3) By means of a standard cell and condenser. 
 
 230. First Method. — In calibrating the ballistic galvanometer 
 
 TTrp 
 
 by means of the relation O = — 7^ 2 sin - we have to first find 
 ' ^ 2-!rG 2 
 
 XT 
 
 the value of — . This may be got by sending a steady current 
 
 of known value, C (in G.G.S. units), through the galvanometer, 
 and observing the steady deflection, 8, produced'by it ; the con- 
 ditions of control, etc., remaining the same as when the galva- 
 nometer is used baUistically. Assuming the deflection 8 of the 
 galvanometer to follow a tangent law for steady currents, we 
 have — 
 
 
 TT 
 
 C = 7=; tan 8 
 
 and therefore — 
 
 
 
 H C 
 
 
 G ~ tan 8 
 
 So that— 
 
 
 ^ " 2"^ ^ to7S X 2 sin ^ (i + -) (absolute units) 
 or Q (coulombs) = ^^^g sm " I, i + J 
 and the constant K of the galvanometer is — 
 
 TT tan 8 
 
 The measurement of C, the steady current producing the de- 
 flection 8, which in general will be very small, may be effected 
 either by means of a zinc iodide voltameter in series with it, or 
 by connecting the galvanometer in series with a large resist- 
 ance and battery, and measuring the fall of potential between 
 the galvanometer terminals on a potentiometer, against the 
 
Calibration of a Ballistic Galvanometer. 219 
 
 E.M.K. of a standard cell, then if the galvanometer resistance 
 be known, the current passing through it may be calculated. 
 
 In order to determine T, the periodic time of swing of the 
 galvanometer needle, the latter should be set swinging through 
 a small angle, and the time required for the spot of light on the 
 scale to make fifty transits in the same direction across the zero, 
 should be taken on a stop-watch. The time between two suc- 
 cessive transits in the same direction may then be calculated, 
 and is the periodic time of swing T. 
 
 In reading the steady deflection S of the galvanometer needle, 
 should the instrument be a mirror-galvanometer, it must be borne 
 in mind that the angular deviation of the spot of light will be 
 twice that of the needle. The same remark applies to the 
 swing u. In connection with this point the student should 
 refer back to par. 137, where a similar case is dealt with. 
 
 231. In order to determine the constant of a 6000-ohm 
 reflecting ballistic galvanometer, it was set up with its mirror a 
 distance of 1000 mm. from the scale on which the divisions 
 were measured and found to be half-millimeters. A battery of 
 constant E.M.F. was connected to the ends of a resistance of 
 10,000 ohms, and the galvanometer terminals were connected 
 across 20 ohms of the 10,000-ohm resistance. The deflection 
 of the spot of light was 303 scale-divisions. The potential 
 difference at the ends of a sooo-ohm coil, which made up part 
 of the 10,000-ohm resistance, was measured on a potentiometer 
 against a standard Clark cell, and found to be exactly i volt. 
 
 The galvanometer needle was then set swinging, and by 
 means of a stop-watch it was found to make fifty swings in 260 
 seconds. Then — 
 
 loTC 
 
 K = 
 
 TT 
 
 tan S 
 
 T = — = i;'2o seconds 
 50 ^ 
 
 If the fall of potential was 1 volt per 5000 ohms, the fall of 
 
 potential between the galvanometer terminals will be — 
 
 20 
 — = o'oo4 volt 
 5000 ^ 
 
220 Practical Electricity and Magnetism. 
 and the current C = -? = o'oooooo66 ampfere 
 
 0000 
 
 = o'ooooooo66 absolute unit 
 
 The distance between the scale and the mirror of the galvano- 
 meter = looo mm. = 2000 scale-divisions, hence -^^ = 
 o"is I = tangent of angular deflection of spot of light ; 
 
 Hence the angular deviation of the spot of light = 8-6° 
 and angular deviation of galvanometer needle 8 = 4*3° 
 
 and tanS = 0752 
 
 therefore K = — : 5, 
 
 TT tan 6 
 
 10 X S'20 X o'ooooooo66 
 
 ~ 3-142 X 0-0752 
 
 = 0-0000145 
 
 Inductor Methods of Calibration. 
 
 232. (a) Earth Indtutor. — When a coil of wire is rotated in 
 a magnetic field so as to cut lines of force, an electro-motive 
 force is induced in it, the value of which in volts is — 
 
 ^ - io» dt 
 
 dN 
 
 ^-t: denoting the rate of cutting the lines of force. If the Cir- 
 cuit of the coil is completed through a baUistic galvanometer a 
 current will be produced, and if R is the total resistance of the 
 circuit in ohms (galvanometer and coil), the current, according 
 to the above equation, must be — 
 
 R "" ^ ~ lo'Rrf/ 
 From this we have — 
 
 10* R 
 integrating this expression, we get — 
 
 ^ io«R 
 
Calibration of a Ballistic Galvanometer. 221 
 
 where Q is the quantity of electricity discharged through the 
 galvanometer in coulombs. 
 
 If the earth inductor consists of a coil of n turns, each of 
 area A sq. cm., and placed with its plane parallel to the 
 direction of the horizontal component of the earth's mag- 
 netic field H, then, when the coil is rotated through 180°, the 
 total number of lines of force cut will be — • 
 
 and therefore — 
 
 We can now proceed to calibrate the galvanometer for quan- 
 tity of electricity. Thus, if on turning the earth inductor 
 through 180°, we obtain on the galvanometer needle a swing a„ 
 we have — 
 
 ^ 2«HA HT .a,/- X\ 
 
 •Q = — fn" = — 7^ 2 sm — I I + r J 
 ^ lo'R 27rG 2 V 2 J 
 
 and therefore the constant of the galvanometer (K) will be — 
 
 _ HT _ 2«HA 
 
 ~ ttG ~ 
 
 lo'R sin 
 
 ?(-+^) 
 
 So that in order to calculate the quantity of electricity in 
 coulombs which produces any swing, a, of the galvanometer 
 needle, we may write — 
 
 Q=Ksin^(i+^) 
 
 It must again be noticed here that u, refers to the angular 
 swing of the needle, and not that of the spot of light. 
 
 In making a calibration of this kind, a separate experiment 
 must be performed in order to determine the value of H, the 
 horizontal intensity of the earth's magnetic field, at the earth 
 inductor coil (this may be done by some of the methods 
 described in par. 268) ; also the relative value of the controlling 
 field H, acting on the ballistic galvanometer needle, must be 
 specified by determining its periodic time of swing. 
 
222 Practical Electricity and Magnetism. 
 
 233. Earth inductors, of a more or less elaborate kind, 
 mounted so as to admit of rotation about two axes, and pro- 
 vided with graduated dials and stops such as that shown in 
 Fig. loi, may be used. A much simpler form, however, with 
 
 Fig. ioi. 
 
 which very satisfactory results may be obtained, consists of a 
 large circular coil of many turns wound on a wooden ring, the 
 ends of the coil being brought to two terminals on the inner 
 side of the ring, and the whole entirely unmounted. 
 
 The total area is calculated by very carefully measuring the 
 circumference of each layer of wire, and adding . together the 
 areas of each turn so calculated. To use this coil, it is placed 
 on a flat table with its plane parallel to the. table, and turned 
 over .through 180° by hand, thus cutting the vertical component 
 of the earth's field, V. The value of V may be found from a 
 measurement of H and the angle of dip S, since — 
 
 V = H tan 8 
 A convenient value for the total area of such a coil is about 
 200,000 sq. cm. 
 
Calibration by means of the Earth Inductor. 223 
 
 In using the earth inductor, care must be taken that the 
 whole rotation of the coil has taken place before the ballistic 
 needle commences to deflect. 
 
 234. The following data were obtained in an experiment to 
 determine the constant of a ballistic galvanometer : — 
 
 Distance of mirror from scale = 2000 scale-divisions 
 Swing of spot of light = 36 „ „ 
 
 Effective area of earth inductor nk. = 192,300 sq. cm. 
 Resistance of galvanometer = 6000 ohms 
 
 The earth inductor was laid flat on a table and turned 
 through 180°, so as to cut the vertical component of the earth's 
 magnetic field. The strength of the horizontal component H 
 had been carefully, measured previously, and was found to be 
 = o"384 C.G.S. unit. The angle of dip was also measured, 
 and found to be 70*3°. 
 
 Hence the vertical component V = H tan 70*3 
 
 = o"i84 X 2"793 
 = o"5i4 C.G.S. unit 
 
 The logarithmic decrement of the galvanometer was deter- 
 mined, and was A = 0*30. 
 
 The angular deviation of the spot of light was the angle 
 
 whose tan = -? — = o"oi8 = i'05°. 
 2000 
 
 Hence a = 
 
 I "OS 
 
 2 
 
 = 0-525° 
 
 
 
 and sin - = 
 2 
 
 sin 
 
 ■262° = o'oo45 
 
 
 )nstant K = 
 
 
 2«VA 
 
 
 
 lo^R 
 
 sin^(. 
 
 ^\) 
 
 
 
 
 2 X 192300 X 
 
 ■514 
 
 10' X 6000 X o'oo45 (i + o'ls) 
 = o'oooo633. 
 
 235. (b) Current Inductor. — Instead of using a coil moving in 
 the earth's field, we may produce a field of known strength 
 
224 Practical Electricity and Magnetism. 
 
 inside a coil by means of a long solenoid through which a 
 measured current is flowing. The form of apparatus employed 
 usually consists of a solenoid of length (2/), which is at least 
 ten times its radius (;-), mounted horizontally on a stand (see 
 Fig. 102). Round the centre portion of the solenoid a small 
 
 Fig. 102. 
 
 test-coil is wound, consisting of many turns of fine wire, the 
 number of turns, n^, being known. 
 
 The test-coil is connected to the ballistic galvanometer. If 
 now a current of C ampferes is sent through the solenoid, the 
 intensity of the field produced inside it at its centre is ' — 
 
 H = 
 
 
 
 11 being the number of turns per centimetre on the solenoid. 
 If A is the mean area of the solenoid, the total number of 
 lines of force (N) produced is — 
 
 ^^ 10 V^ 2/V 
 
 and this magnetic flux cuts the test-coil «i times. Hence the 
 effective number of lines, Ni, cut by the test-coil during the 
 starting or stopping of the current in the solenoid is — 
 
 N,= 
 
 47r««iCA 
 
 (■-1?) 
 
 If R is the total resistance in ohms of the ballistic galvano- 
 meter and test-coil, the number of coulombs of, electricity (Q) 
 discharged through the circuit will be — 
 
 „_Ni 47r««jCA/' jf^N 
 
 ^~io'R~ lo'R V^~2/V 
 
 ' Phil. Mag., vol. xxii., Oct. 1886, p. 369. 
 
Calibration by Means of the Current Inductor. 225 
 
 Hence, if when a current of Cj ampferes is started in the solenoid 
 we get a throw, Uj, on the ballistic galvanometer, we have — 
 
 4a- wwiCiA ( ^i^\ HT . ai/ X 
 
 Q= i^^'a^-^70 = ^;;g'''"^^ ^+- 
 
 2 
 
 and the constant of the galvanometer K will then be- 
 
 K = 
 
 HT 47r;?»iCiA ( ^ 1^ 
 
 TT' 
 
 ■G 
 
 io«Rsin;'(i+^) 
 
 
 so that the quantity of electricity in coulombs required to 
 produce any swing, a, of the ballistic needle will be — 
 
 Q = K sin ^ f I + -^ 
 
 2 
 
 236. The method of calibration by means of the current 
 inductor has certain advantages over the earth inductor method, 
 since it is independent of the earth's magnetic field, if the axis 
 of the coil is placed at right angles to the magnetic meridian. 
 It is also very useful for calibrating ballistic galvanometers of 
 the D'Arsonval type where the ordinary ballistic formula does 
 not necessarily apply ; in this case it is usual to observe the 
 ballistic throws produced by starting or stopping currents of 
 different values in the coil, and plotting an absolute calibration 
 curve with the scale-readings for abscissae against the corre- 
 sponding quantities of electricity in coulombs which produced 
 them. 
 
 The time of swing of the galvanometer coil, the distance of 
 the mirror from the scale, and the length of a scale-division in 
 centimetres should be carefully specified. 
 
 The current sent through the large coil of the solenoid may 
 be measured on a tangent galvanometer, or on some form of 
 ammeter which has been calibrated absolutely by comparison 
 with a standard current measuring instrument. 
 
 In using the apparatus, it will be found more satisfactory to 
 take the ballistic swing on " breaking " a known current in the 
 solenoid, rather than on " make." 
 
 237. The following are the dimensions of the coil shown in 
 Fig. 102 : — 
 
 Q 
 
226 Practical Electricity and Magnetism. 
 
 Large Col,. — 640 turns of No. 20 wire in two layers — 
 Internal diameter = 5*00 cm. 
 External „ = 5-53 „ 
 Total length = 41-85 „ 
 Resistance = 2'82 ohms. 
 
 Small Coil.— 1^0 turns No. 36 wire one layer — 
 Resistance = 2*35 ohms. 
 
 238. In the calibration of a ballistic galvanometer by the 
 current inductor, the dimensions of which have just been given, 
 the galvanometer needle was 2000 scale-divisions from the 
 scale zero, and on breaking a current of 5 ampbres in the large 
 coil, a ballistic swing of 50 scale-divisions was obtained on the 
 galvanometer. The logarithmic decrement of the galvanometer 
 was determined, and was X = 0-30. 
 
 The number of turns per cm. on large coil n = — - — = i5"3o 
 
 The number of turns on test-coil w, = 130 
 
 Mean area of coil A = 21-74 sq. cm. 
 
 Value of r I -I- ii j = 0-992 
 
 Resistance of galvanometer = 6000 ohms 
 
 Tangent of angular deflection of spot of light = -^— 
 
 2000 
 
 = 0-025 
 
 Angular deflection of spot of light = 1-41° 
 
 Angular deflection of needle a = 0-70° 
 
 Value sin - = o-oo6i 
 
 4-;r««iCiA 
 ro»Rsin5(,+^) 
 
 Hence.K = -^"'"T .x i^+V^. 
 
 ^/V 
 
 _ 4 X 3'i42 X 15-30 X 130 X 5 X 21-7 X 0- 992 
 
 10' X 6000 X o-oo6i X 1-15 
 — 0-0000640 
 
Calibration by Standard Cell and Condenser. 227 
 
 Calibration of Ballistic Galvanometer by Means of a Standard 
 Cell and Condenser. 
 239. If a condenser having a capacity of K farads is charged 
 to a potential of V volts, the quantity of electricity stored in it 
 
 in coulombs is — 
 
 Q = KV 
 
 Since the capacity of standard condensers is usually measured 
 in microfarads, it will have to be divided by 10" to bring it 
 to farads before using in the above formula. 
 
 In order to calibrate the balUstic galvanometer by this method, 
 it is connected up with a battery, condenser, and suitable key, 
 so that the condenser may be charged or discharged through 
 the galvanometer. The following diagram (Fig. 103) shows 
 
 BG 
 
 ^P- 
 
 V2 
 
 A i 
 
 B 
 
 Fig. 103. 
 
 one method of arranging the circuit. The condenser C is 
 placed in series with the ballistic galvanometer BG and the 
 discharge key K ; the other terminal of the galvanometer being 
 connected to the upper contact (2) of K, to the lower contact 
 (i) of K through the standard cell B, and to earth. 
 
 When Kis depressed on contact (i), the condenser is charged, 
 and the amplitude of the first swing of the needle noted. By 
 raising K to the upper contact (2) the condenser is discharged. 
 This should be repeated several times, and the mean of the 
 swings taken ; also, if necessary, with several cells in series. 
 The E.M.F. of the cell is corrected for temperature, and the 
 value of Q calculated as shown above. K may be measured by 
 a method to be described in the next chapter, or if a standard 
 condenser is used it will be known. The cell B and the key 
 K must be carefully insulated from earth. If the galvanometer 
 
228 Practical Electricity and Magnetism. 
 
 to be calibrated is one in which the sine of half the angle of 
 swing of the needle is proportional to the quantity of electricity 
 discharged through it, then — 
 
 HT . aJ A\ 
 Q = KiVi = -y, 2 sin -( I + - ) 
 
 ^ ' ' 27rG 2\ 2/ 
 
 where the capacity potential and swing of the needle have the 
 values K], Vi, and a.^ respectively, and the constant of the 
 galvanometer K will therefore be — 
 
 HT KiVi 
 
 2 V 2/ 
 
 TTVjr 
 
 Sin 
 
 2 \ 2> 
 
 and the quantity of electricity corresponding to any throw of 
 the needle will be— - 
 
 a/ X\ 
 Q = Ksin-(i+-j 
 
 Thexondenser method of calibration may also be employed 
 in obtaining the absolute calibration curve of a D'Arsonval 
 galvonometer for quantity of electricity. The condenser in this 
 case being charged to different potential differences, either by 
 employing several standard cells in series or by standardizing, 
 by means of a Clark cell, the fall of potential down a long 
 uniform wire, and charging the condenser across various known 
 lengths of the wire. A curve may then be plotted with scale- 
 readings for abscissae and corresponding quantities of electricity 
 in coulombs for ordinates, the distance of the mirror from 
 the scale-division and relative value of the controlling force 
 being specified. 
 
 240. In a calibration of a ballistic galvanometer by means 
 of a standard condenser and known E.M.F., a battery of two 
 Leclanche cells in series was connected to a resistance of 10,000 
 ohms; whilst a condenser of 10 microfarads capacity was 
 charged through the ballistic galvanometer across 300 ohms of 
 the io,ooo-ohm resistance. A ballistic swing of 230 scale- 
 divisions was obtained, the scale being 2000 scale-divisions 
 distant from the mirror. The difference of potential across 
 a coil of 1000 ohms resistance, which formed part of the 
 
Use of Ballistic Galvanometer. 229 
 
 10,000-ohm resistance, was then carefully measured on a poten- 
 tiometer against a standard Clark cell, and was found to be 
 o'28o volt. The diiference of potential employed to charge the 
 condenser was therefore — 
 
 ^00 X 0"28 
 
 ^ = 0-084 volt 
 
 1000 
 
 The angular swing of the spot of light = 6'55° 
 The angular swing of the galvanometer needle a = 3' 2 7 
 
 Hence sin - = o'o28 
 
 2 
 
 The logarithmic decrement was found to be X = 0-40 
 
 K. V 
 Hence K = ,' — r^ 
 
 ''" ?(^ + 2) 
 
 10 X o'o84 
 
 ~ 10" X 0'028 X I'2 
 
 = 0-0000250 
 
 Remarks on t/ie Use of the Ballistic Galvanometer. 
 
 241. In using a ballistic galvanometer, where great accuracy 
 is not required, and the swings of the needle are small, the 
 sines of the angles of swing are nearly proportional to the 
 angles themselves, and therefore, in a comparison of two 
 quantities of electricity, the ratio of the amplitudes may be 
 taken instead of the ratio of the sines of the half-angles. 
 
 The student will probably experience some difficulty at first 
 in reading the exact value of the swing of the needle, this 
 becoming more difficult the quicker the periodic time of swing 
 of the needle. A few trial swings should be taken first, and 
 thus the part of :the scale reached by the spot of light will be 
 located ; the observer may then at once turn his attention to 
 this spot after having completed the circuit. A sliding pointer 
 moving over the scale may be used with advantage to locate the 
 exact position of the swing. It is advisable, however, for 
 measurements of this kind, that there should be at least two 
 observers. 
 
230 Practical Electricity and Magnetism. 
 
 242. As stated before, the needle of the galvanometer must 
 be perfectly stationary before the discharge is sent through the 
 coils. To enable this to be attained quickly, a small auxiliary 
 coil, connected to a battery through a reversing key, and known 
 as a damping coil, may be set up near the galvanometer, so that 
 when a current is flowing in it, its magnetic field will act on the 
 galvanometer needle. By properly timing the duration and 
 direction of the current in this coil, the needle may be rapidly 
 brought to rest. It has been stated that almost any type of 
 galvanometer may be used ballistically ; for accurate work, 
 however, a special design is necessary, since the damping m 
 
 Flu 104. 
 
 most galvanometers is pretty large, and when this is so, the 
 correction is difficult, and cannot be made by taking the 
 logarithmic decrement, but the galvanometer must be cali- 
 brated directly in coulombs per degree of swing, throughout 
 the scale, by some of the inductor methods which are inde- 
 pendent of the damping. A special form of ballistic galvano- 
 meter, designed by Messrs. Nalder Bros., is shown in Fig. 104; 
 in it the magnets, which are bell-shaped to diminish air-friction, 
 are arranged astatically, two being at the centre of the coil and 
 two outside, one above and one below the coil. 
 
Ballistic Galvanometer. 231 
 
 The coils, which are hinged to give easy access to the needle, 
 are mounted in ebonite cases, and the whole highly insulated ; 
 there is as little solid metal as possible in the vicinity of the 
 needles, as eddy currents might be set up, which would increase 
 the damping action. The mirror is also reduced in size as 
 much as possible, on account of air-friction. 
 
V. 
 CAPACITY. 
 
 243. The capacity of a conductor is defined as the quantity of 
 electricity required to raise its potential from zero to unity. 
 The electro-magnetic practical unit of capacity is the farad, 
 which is that capacity requiring one coulomb of electricity to ' 
 raise the potential from zero to i volt. 
 
 In practice this unit is much too large, and the millionth 
 part of it, called the microfarad, is used instead. 
 
 I farad = lo"" C.G.S. electro-magnetic units of capacity. 
 I farad = 10° microfarads. 
 I microfarad = io~'* C.G.S. electro-magnetic units of capacity. 
 
 Measurement of the Capacity of a Condenser. 
 
 244. In order to measure the capacity of a condenser we 
 
 take advantage of the relation K = ^, where K is the capacity 
 
 in farads, Q the quantity of electricity in coulombs, and V the 
 potential in volts. The quantity of electricity, Q, in the 
 condenser may be found by discharging it through a ballistic 
 galvanometer, and calculating from the balUstic galvanometer 
 formula ; whilst the potential difference, V, of the plates may 
 be found in terms of a current and resistance. 
 
 The following diagrams (Figs. 105 and 106) show the con- 
 nections of the apparatus. 
 
 The condenser C, whose capacity is required to be measured, 
 is connected in series with a ballistic galvanometer, BG, and 
 high-resistance key, K, across the ends of a large resistance R^ 
 
Measurement of the Capacity of a Condenser. 233 
 
 which is at least 10,000 ohms (see Fig. 105). A battery, B, of 
 low internal resistance and constant E.M.F. is also connected 
 to the ends of R. 
 
 When the circuit of the condenser is completed by pressing 
 the key K, the condenser is charged to a potential, V, which 
 
 B6 
 
 Fig. 105. 
 
 represents the fall of potential between the ends of R, and the 
 amplitude of the first swing of the galvanometer needle (a) is 
 noted. This should be repeated several times, the condenser 
 being discharged, after each throw, by means of its short- 
 circuiting plug, after the circuit at K has beeri broken. Using 
 the same symbols as in par. 224, we have — 
 
 Q = 
 
 Q being in coulombs. 
 
 loHT 
 
 2\ 2/ 
 
 Fig. io5. 
 
 In order to get the value of V, the charging potential, the 
 connections are now altered as below (see Fig. 106). 
 
 The condenser is removed and a resistance box, Rj, is 
 
234 Practical Electricity and Magnetism. 
 
 placed in series with the galvanometer, and the terminals are 
 brought to two points on R, so as to include a known fraction, 
 ;■, of the resistance R, in order to get a steady deflection, S, on 
 the ballistic galvanometer. The value of 8 should not be very 
 different from that of a. This may be obtained by altering the 
 resistance Rj or the resistance r, or, if necessary, both of them. 
 Then, if R„ = galvanometer resistance, c = potential difference 
 at the ends of the resistance r, and assuming that for steady 
 currents the galvanometer follows a tangent law, we have — 
 
 (=13-, _=io„ tan h 
 J<» + -Ki ij 
 
 
 ., V R 
 
 But -=- 
 e r 
 
 tfR 
 
 therefore V = ^ 
 
 io(R„ + RQHR 
 
 tan 8 
 
 and K = 
 
 Q 27rG 
 
 =(-+;) 
 
 loHT . a/ A'^ 
 
 2 sin ■ 
 
 V- io(R„ + R,)HR ^^^g 
 i^-G 
 
 T?-sin 
 
 j(.+^) 
 
 ~ ■7rR(Rj, + Ri) tan 8 
 
 245. In the second part of the experiment the resistance r 
 must be small compared with the resistance of the galvanometer 
 and Ri, otherwise the fall of potential down R will not be the 
 same as in the first part of the experiment. Also the controlling 
 force acting on the ballistic galvanometer is supposed to be the 
 same in the two experiments. Both a and 8 represent actual 
 angular deflections of the galvanometer needle, and must be 
 calculated from the scale-readings of the spot of light as shown 
 in par. 137. The logarithmic decrement should be determined 
 when the apparatus is connected up as in Fig. 105. The 
 
Methods of Comparing Capacities. 235 
 
 result will give the capacity of the condenser in farads. 
 Multiplying by 10^ we get it in microfarads. 
 
 246. The following are the details of a measurement of 
 capacity of a condenser by the above method. Three constant 
 cells were connected to the ends of a resistance of 10,000 ohms, 
 and the condenser was charged from two points, the resistance 
 between which was 4700 ohms, and a ballistic throw of 
 151 scale-divisions was obtained on a scale 100 cm. distant 
 from the mirror ; the scale-divisions were in millimetres. The 
 condenser was then removed and the galvanometer terminals 
 connected across two points on the 10,000 ohms, the resistance 
 between which was 20 ohms, and a steady deflection of 
 303 scale-divisions was obtained. The galvanometer resistance 
 was measured and found to be 6000 ohms. The periodic 
 time of swing and logarithmic decrement of the galvano- 
 meter needle were taken and were T = 5'2o seconds and 
 A. = 0*300. 
 
 The angle of swing of the needle a is half the angle whose 
 
 tangent is -^ — = i' X 8'6° = 4"3° ; and the angle of steady 
 deflection is half the angle whose tangent is — ^ = \ X i6'8" 
 
 8-4° 
 
 Tr sm I I + - I 
 
 Hence K = — — ;^ — 
 
 irRR, tan 8 
 
 5'2o X 20 X o"o377(i -j- o'i5) 
 
 ~ 3"i4i6 X 4700 X 6000 X o'i477 
 
 = o'344 X 10 * farads 
 
 = o'344 microfarads 
 
 The temperature was 11 "8° C. 
 
 Methods of comparing Capacities. 
 
 247. If it is desired to compare two capacities with one 
 another, one of them being a standard, there are several 
 methods which may be adopted. Since the capacity is the 
 
236 Practical Electricity and Magnetism. 
 
 ratio of the quantity of electricity in a condenser to the 
 potential difference at its terminals, we may either charge both 
 condensers to the same potential and compare the quantities 
 of electricity in them, or we may give them a charge of known 
 quantity and measure the potential differences. 
 
 248. Comparison of Condensers by Means of a Ballistic 
 Galvanometer. — In this method both are charged to the same 
 potential and the quantities compared by means of a ballistic 
 galvanometer. 
 
 The two condensers, A and B (see Fig. 107), have each one 
 terminal connected to earth and to the upper and lower 
 contacts of a Kemp discharge key K, a battery C being 
 inserted between the lower contact of K and the common 
 terminal. The other terminals of the condensers are connected 
 
 Fig. 107. 
 
 to the terminals (2) and (3) of a high-resistance two-way 
 plug key respectively J the terminal (i) of the two-way key 
 being connected through the ballistic galvanometer BG, to the 
 movable tongue of the key K. The terminals (i) and (2) of 
 the two-way key are connected together, and the key K 
 depressed, thus charging condenser A from the cell C ; the 
 ballistic swing o.^ is noted. The key K is then raised to the 
 upper contact, discharging the condenser. The terminals (i) 
 and (2) are now disconnected, and (i) and (3) are connected to- 
 gether and the process repeated, charging condenser B this time, 
 and obtaining a ballistic swing, Oj. If Y^^ and K^ are the 
 capacities of A and B respectively, and V is the potential of 
 
Covtpartson by means of the Electrometer. 237 
 
 the charging battery, which is assumed to have remained 
 constant during the experiment, then — 
 
 Q,oc a, oc Y^iS 
 
 Q.J oc a, oc KjV 
 
 therefore ^= — 
 Kb a^ 
 
 If Kb is a standard condenser, then Y^^ = — ^— 
 
 249. Comparing a condenser with a standard -3- microfarad 
 condenser, by the above method a ballistic swing of 183 scale- 
 divisions was obtained with the standard, and 273 scale-divisions 
 with the condenser whose capacity was required. 
 
 XT °'333 183 
 
 Hence - = — 
 
 X 273 
 
 X = o"498 microfarads 
 
 The charging battery employed was a Hellesen dry cell. 
 
 250. Comparison of Condensers by the Electrometer Method. — 
 In this method, one of the condensers is charged and its 
 potential measured, the charge is then divided between the 
 two condensers, and from the alteration in potential the ratio 
 of their capacities may be calculated. 
 
 The following diagram (Fig. 108) shows the arrangement 
 of the apparatus. 
 
 The condensers A and B are connected in parallel with each 
 other and with the battery C and electrometer E,, highly in- 
 sulating keys, Ki and K,, being placed in the battery circuit 
 and between the two condensers respectively. It will also be 
 advisable to keep the battery well insulated and to earth one 
 pair of plates, as at e. 
 
238 Practical Electricity and Magnetism. 
 
 In making the measurement, the keys Ki and Kg are opened 
 and the electrometer needle adjusted to zero. The key Ki is 
 then closed, thus charging condenser A with a certain quantity 
 of electricity, Q, and raising its potential to Vi, which is 
 measured by the deflection of the electrometer needle, Sj. 
 The battery circuit is now broken at the key Kj, and key Kj 
 is closed; this allows the charge Q to mix between the two 
 condensers, and the potential will fall to Vj, as shown by the 
 deflection 83 of the electrometer needle. Now, calling K^ and 
 Ke the capacities of A and B respectively, we have — • 
 
 (i) Q = K^Vi oc KA 
 
 (2) Q = (K^ + K3)V, « (K^ + K,)S3 
 
 whence ^^ = 5 5- 
 
 •is-B 61 — 62 
 
 In comparing two condensers in this way, a preliminary test 
 must be made to see that the apparatus is insulated properly. 
 This may be assumed to be the case if the deflections 81 and 82 
 remain constant for at least two minutes. The electrometer 
 deflections are assumed proportional to the potential, and 
 therefore the instrument must be used heterostatically, with the 
 needle potential largely in excess of that of the quadrants. 
 The battery C should consist of a sufficient number of cells, to 
 give a large deflection on the electrometer for 81. 
 
 251. In a comparison of two condensers, one of which was 
 a standard condenser of capacity o'333 microfarads, when the 
 standard alone was charged the electrometer deflection was 
 240 scale-divisions; on dividing the charge between the two 
 condensers, the deflection fell to 121 scale-divisions. 
 
 Hence ~ — .EL333 
 
 Kb Kb 240 — 121 
 
 and Kb = 0*336 microfarads 
 
 The battery employed consisted of six Lessing dry cells 
 in series, and the electrometer needle was charged to a high 
 potential by means of a Zamboni dry pile. 
 
 252. Comparison of Capacities by the Method of Mixtures. — 
 One of the most satisfactory methods of comparing the 
 
Comparison by the Method of Mixtures. 239 
 
 capacities of two condensers, and one which may be applied 
 even when there is a considerable difference in their values, 
 is the method of mixtures. The principle of the method 
 being the measurement of the potentials required to give 
 equal charges to the two condensers. The condensers are 
 charged to different potential differences, the charges being 
 afterwards mixed, by placing the condensers in series, and 
 the resultant charge tested for; when this latter is zero the 
 condensers must have contained equal quantities of electricity, 
 and therefore the capacities will be in the inverse ratio of the 
 charging potentials. 
 
 One method of arranging the apparatus is shown in Fig. 109. 
 
 Fig. 109. 
 
 The battery C, consisting of several cells of constant E.M.F., 
 is connected in series with the two resistance boxes r-^ and r^. 
 The condensers A and B, by an arrangement of keys — ^a 
 reversing key, RK, supplied with cams by means of which 
 the keys can be insulated midway between the upper and 
 lower contacts, and a Kemp discharge key, K — may be charged 
 across the ends of r-^ and r^ respectively, connected in series 
 so as to allow the charges to mix, and finally connected so 
 that the resultant charge (if any) may discharge through the 
 ballistic galvanometer BG to earth, the junction of r-^ and /o 
 being connected to one plate of each of the condensers and 
 to earth. Calling (i), (2), and (3) the contacts of the two 
 keys, the following operations are gone through : — 
 
 (i) The contacts (i), (2), and (3) are all insulated ; i.e. they 
 are held midway between the upper and lower contacts. 
 
240 Practical Electricity and Magnetism. 
 
 (2) Key (i) is allowed to rise to the upper contact, thus 
 charging condenser A across the ends of r^. 
 
 (3) Key (3) is depressed so as to touch the lower contact 
 and charge condenser B across the ends of r^. 
 
 (4) Key (3) is insulated and keys (i) and (2) are depressed, 
 to make contact with the lower contact, which is common to 
 both; by this means the positively charged plate of A is 
 connected to the negatively charged plate of B, and vice vers&, 
 and the two charges mix. 
 
 (5) Key (3) is now raised to the upper contact, and the 
 resultant charge (if any) in the condensers escapes to earth 
 through the ballistic galvanometer. 
 
 Should there be any swing on the ballistic needle after the 
 last operation, it signifies that one of the charges must have 
 been greater than the other. From the direction of the swing 
 the direction of the discharge, and consequently the condenser 
 having the larger charge, can be found, and the potential dif- 
 ference to which it is charged must be diminished ; this may be 
 effected either by diminishing the resistance, between the ends 
 of which it is connected, or by increasing the resistance across 
 which the other condenser is charged. The above operations 
 are repeated, and 7\ and r^ altered until there is no resultant 
 charge left, and the ballistic galvanometer shows no deflection 
 after operation (5). Then, since the charging potential differ- 
 ences are proportional to the resistances r, and r^, we have — ■ 
 
 ^_'_:^ 
 
 K„ - r. 
 
 The value of r-^ + r^ must always be large, never being less 
 than 10,000 ohms. The keys may be arranged close together, 
 so that all the above connections may be made in a second or 
 two, thus, as far as possible, eliminating the effects of dielectric 
 absorption. The keys must be highly insulated, as also must 
 all the connecting wires and the condensers. 
 
 The ballistic galvanometer must be arranged so as to be 
 sufficiently sensitive to detect small discharges. The accuracy 
 of the comparison may be tested by finding the range through 
 which ri or r.^ may be altered without the galvanometer showing 
 
specific Inductive Capacity. ^'^^ 
 
 a deflection due to the resultant charge. This should be done, 
 and the percentage accuracy of the comparison stated. 
 
 253. In a comparison between the standard one-third micro- 
 farad condenser and a large condenser, the method of mixtures 
 was employed. For the resistances i\ and 7-^, two large resist- 
 ance boxes wound with platinoid wire were used. The ballistic 
 galvanometer employed was very sensitive, but in order to make 
 the test still more sensitive, instead of having the galvanometer 
 needle at rest before sending the resultant charge through it, 
 the following procedure was adopted. The needle was set 
 swinging through a small arc, and the amplitudes of two 
 successive swings observed ; the resultant charge was then sent 
 through the galvanometer when the needle was in the middle of 
 its third swing, and the eifect on the amplitude of that swing 
 noted. If the discharge was in such a direction as to deflect 
 the needle in the same direction as that in which it was moving, 
 the amplitude was greater that it otherwise would have been as 
 might be calculated from the two observed swings, allowing for 
 damping ; if the discharge was in the opposite direction, the 
 amplitude would be smaller than the theoretical ; and if the dis- 
 charge was zero, i.e. if the charges in the two condensers were 
 exactly equal, there would be no effect on the swing at all. 
 
 The condenser A was the standard 0*333 microfarad, and in 
 order to get zero effect on the ballistic galvanometer needle, 
 the resistances were i\= 15,015 ohms, and ^3 = 500 ohms. A 
 change of about 5 ohms could be detected. The coils were 
 correct at 12 "5° C, and the temperature during the test was 
 15-5° C. Taking o"o2i per cent, per 1° C. as the temperature 
 variation of resistance of platinoid, the resistances corrected ^r 
 temperature are i\ = 15024-4, and r.^ = 500-315. 
 
 Hence °^ = ^^°^^ 
 B 15024-4 
 
 B = 9-99 microfarads at 15-5° C." 
 
 Specific Inductive Capacity. 
 
 254. The ratio of the capacity of a condenser with some 
 medium other than air as dielectric to the capacity of the same 
 
 R 
 
242 
 
 Practical Electricity and Magnetism. 
 
 condenser with air as the dielectric, is termed the specific 
 inductive capacity (S.I.C.) of that medium. 
 
 In order to determine the value of the S.I.C. of various 
 substances, we must therefore have some form of condenser 
 between the plates of which various dielectrics may be intro- 
 duced. The method here described is that due to Dr. John 
 Hopkinson.' 
 
 The apparatus required are a guard-ring condenser, an adjust- 
 able condenser, and a quadrant electrometer. 
 
 The guard-ring condenser consists of a brass disc about 15 
 cm. in diameter (see Fig. no), surrounded by a guard-ring, 
 
 'T. 
 
 yHF' 
 
 s.^A. 
 
 ff 
 
 [a 
 
 ■^^ 
 
 5 
 
 z I 
 
 I " I I 
 
 Ji-r 
 
 Fig. 
 
 A, also of brass, about 7-5 cm. wide, and separated from it by a 
 space of o"i cm., the ring h being supported from the b^se of 
 the instrument by means of ebonite rods. 
 
 The other plate of the condenser, e, is parallel to h, and 
 
 ' Phil. Trans., 1878 
 
Measurement of Specific Inductive Capacity. 243 
 
 mounted on a rod which has a screw of -^" pitch accurately 
 cut on it, and passes through a nut, / the circumference of 
 which is graduated into 100 parts. The plate e is free to 
 move up and down in a vertical plane, but is prevented from 
 rotating. 
 
 The disc is supported from two rods of ebonite, the ends 
 of which rest on the guard-ring, and is thus insulated from it. 
 A metal shield attached to gq, covers and protects the upper 
 surface of the plate hh. When in use, a vessel containing 
 pumice and strong sulphuric acid is placed on the upper surface 
 of h, inside the metal shield, in order to keep the insulation 
 perfect. The rods supporting gg must also be kept well 
 insulated. 
 
 255. The adjustable condenser used is of the form due to 
 Lord Kelvin,! and consists essentially of two hollow brass 
 cylinders about 5 cm. diameter, one 26*5 cm. long, the other 3S'3 
 cm. long, supported on ebonite insulators with their axes in the 
 same line and their ends separated by a small air gap. Inside 
 these, and insulated from them, is a third brass cylinder, placed 
 coaxial with the other two, 36'6 cm. long and 2'3 cm. diameter, 
 arranged so that it can sUde backwards and forwards, and so 
 project by different amounts inside the interior of the larger 
 tubes. On the outside of all is a metal cover, which can be 
 earthed to protect the tubes within from outside induction. In 
 using this instrument, the outside cover d, the cylinder i, and 
 the inner cylinder c are all earthed, whilst the cylinder a is 
 charged. By sliding i- in or out of a, the capacity of the 
 condenser may be increased or decreased. If necessary, the 
 capacity of this sliding condenser may be calculated from its 
 dimensions ; in the following measurements, however, the know- 
 ledge of its capacity is not required. 
 
 The electrometer employed may be any form of sensitive 
 electrometer. 
 
 256. In making a measurement of the S.I.C. of a substance, 
 the following diagrams show the connections. GR represents 
 the guard-ring condenser, SC the sliding condenser, E the 
 electrometer, and -h and — the poles of the charging battery. 
 
 ' Phil. Trans., 1871, p. 573. 
 
244 Practical Electricity and Magnetism. 
 
 Fig. Ill shows the connections in the first part of the experi- 
 ment. The plates of the sliding condenser are connected to + 
 and earth respectively, the disc and guard-ring of GR to 
 -, whilst the movable plate of GR is earthed, as are also the 
 
 SC 
 
 GR 
 
 r 
 
 -© e- 
 
 W^ 
 
 H 
 
 E 
 
 Fig. 111. 
 
 quadrants of the electrometer E. In this way the condensers 
 will both receive a charge of opposite sign. The connections 
 are now altered to those of Fig. 112. The battery is entirely 
 disconnected from the guard-ring and disc, and from the 
 
 SC 
 
 GR 
 
 e 
 
 — \ 
 
 e 
 
 charged plate of SC, and the guard-ring is earthed. This 
 allows the charges in the condensers to mix. 
 
 On connecting now, as in Fig. 113, one quadrant of the 
 electrometer, being connected to the condensers, will be charged 
 to the same potential. If the two condensers are of equal 
 
Measurement of Specific Inductive Capacity. 245 
 
 capacity, this charge will be zero, and there will be no electro- 
 meter deflection, because the -f and — charges would mix and 
 neutralize. 
 
 If the electrometer does deflect, it proves that one of the 
 condensers has a greater capacity than the other, the sliding 
 condenser SC is then adjusted until no deflection is obtained. 
 
 257. It is important that the connection in Fig. 113 should 
 follow that of Fig. 112 within a fraction of a second, and it may 
 be found advisable to earth both poles of the battery during the 
 second and third part of the experiment. The connections in 
 
 these experiments may be made by means of separate keys. 
 It is, however, desirable to devise a special form of key, so that 
 the various changes may be made rapidly. One such key, 
 devised by Dr. Hopkinson, and described by him,' is shown 
 below (Fig 114). It consists of an ebonite plate, qq, which is 
 screwed to the shielding cover of the condenser ; a steel spring, 
 r, connected to earth ; a similar spring, s, connected to one pole 
 of the battery ; tv are segments of brass connected to the brass 
 cover ; wti are similar segments insulated from the cover and 
 guard-ring, and connected to the sliding condenser and electro- 
 meter respectively ; / is an ebonite handle and brass pin which 
 turns in an insulated brass socket connected with the disc k ; 
 the ebonite piece x moves the springs ;■ and s from t and v to 
 u and w respectively ; the spring yy may connect tv with k, or w 
 
 ' Phil. Trans., 1878, p. 19. 
 
246 Practical Electricity and Magnetisrn. 
 
 Sliding \ 
 Condenser 
 
 with k, and immediately after, both with the electrometer ; the 
 
 other battery pole is connected to hh. 
 
 258. Having got the slid- 
 ing condenser and guard-ring 
 condenser adjusted until they 
 have equal capacities, a plate 
 of dielectric, carefully cut 
 with its opposite faces paral- 
 lel, is then placed on the 
 movable plate of the guard- 
 ring condenser, its diameter 
 being greater than that of the 
 disc opposite to it. On re- 
 peating the process of com- 
 paring the condensers, it will 
 be found that they are no 
 longer equal, since the 
 capacity of GR has been 
 increased by the plate of 
 dielectric. The movable plate 
 of GR is now lowered by 
 means of the screw until a 
 balance of the capacities is 
 again obtained, and the dis- 
 tance s, through which it has 
 been lowered, is noted. 
 Then, if A is the effective 
 area of the guard-ring con- 
 denser, (/the distance between 
 the plates in the first experi- 
 ment, and t the thickness of 
 the slab of dielectric of S.I.C. 
 Harth n <=jElecCromeCer = K, the capacity in the first 
 r" y case is — 
 
 Fig. 114. ^ 1 
 
 ' See J. J. Thomson, "Mathematical Theory of Eleclricity and Magne- 
 tism," pp. 91 and 129. 
 
Measurement of Specific Inductive Capacity. 247 
 and in the second case — 
 
 and since these capacities are equal, we have- 
 
 and K = ^ 
 
 t - s 
 
 The thickness t of the dielectric slab may also be measured 
 by means of the guard-ring condenser as follows. The 
 dielectric slab is removed, and the movable plate is screwed 
 up until it comes into contact with the disc S, the exact position 
 of contact being judged by inserting slips of tissue paper 
 between them and adjusting the screw until they just become 
 loose. The plate is then screwed down, the dielectric slab 
 placed on it, and again screwed up until the upper surface of 
 the slab makes contact with the disc, the tissue paper being 
 again used to test the exact position; from the difference in 
 the readings of the screw in the two positions the thickness of 
 the slab may be calculated. 
 
 The battery employed in the above experiment consists of 
 from 50 to 100 cells, and the electrometer should be sufficiently 
 sensitive to give a large deflection with one cell. 
 
 259. The specific gravity of the slab of dielectric should be 
 determined, and also, if possible, its index of refraction for 
 light of known wave-length, since, according to the electro- 
 magnetic theory of light — 
 
 /i^ = K 
 
 where /* = index of refraction for waves of infinite wave-length. 
 A sufficiently close approximation may, however, be obtained 
 by using the sodium D line in the spectrum in the measure- 
 ment of ju.. Since K increases slightly with temperature, 
 a measurement of the temperature of the slab should be 
 recorded. 
 
 260. There are many other methods of measuring the S.I.C. 
 
248 Practical Electricity and Magnetism. 
 
 of dielectrics, references being given at the end of the chapter, 
 one of considerable importance being the five-plate method of 
 Gordon. By this method Gordon obtained results which 
 differed widely from those of Hopkinson. This difference has 
 been discussed by Hopkinson,' who showed that it was due to 
 the distance between the plates not being small compared with 
 their diameter. 
 
 261. The following data have been taken from Hopkinson's 
 results : — 
 
 Material. 
 
 Density. 
 
 t 
 
 (revs, of screw ) 
 
 (tevs, of screw.) 
 
 Battery. 
 
 K 
 
 Light flint glass 
 
 3-2 
 3-2 
 
 15-01 
 15-01 
 
 12-83 
 12-78 
 
 48 cells 
 72 „ 
 
 689 
 6-76 
 
 262. Measurement of the Specific LidudivC Capacity of the 
 Dielectric of a Cable. — A wire of circular cross-section covered 
 uniformly with insulating material may be looked upon as a 
 cylindrical condenser, and if the dimensions of the cable are 
 known, and its total capacity measured, the specific inductive 
 capacity of the dielectric may be calculated. The measurement 
 of the capacity of the cable is made by some of the methods 
 described in par. 247, the cable being immersed in a tank of 
 water as in the measurement of dielectric resistance, when the 
 metal tank forms one of the plates of the condenser and the 
 wire core the other. 
 
 In order to calculate the specific inductive capacity, we pro- 
 proceed as follows. Let a represent the radius of the 
 wire core, and b the radius of the outside of the dielectric 
 surrounding the wire ; then, if we consider any ring of 
 dielectric of radius ;-, and of infinitely small thickness dr, 
 whose length in the direction of the length of the cable is / 
 (centimetres), its capacity is practically that of two parallel 
 plates of length / and breadth 27rr, which are separated by 
 a distance dr, and is — 
 
 ' = ^r='^r (electrostatic units) 
 ' Phil. Trans., i88i, p. 366. 
 
S.I.C. of the Dielectric of a Cable. 249 
 
 But since the measurement of the capacity of the cable will 
 be made in microfarads, this had better be expressed in micro- 
 farads, and will be — 
 
 2dr '^ 9 X 10^ 
 
 c = Tv7. X „ ^, ^ 20 microfarads 
 
 the ratio of the electrostatic imit of capacity to the electro- 
 magnetic unit being proportional to 7', where z' = 3 X 10" cm. 
 per second; the microfarad being io~"^ C.G.S. electro-magnetic 
 units of capacity. 
 
 If the specific inductive capacity of the dielectric of the 
 cable be denoted by K, the above expression becomes — 
 
 Kr/ 10" . , , 
 
 c ■= —r X — r sj microfarads 
 2dr 9 X 10-' 
 
 Now, this gives the capacity of a single ring of dielectric, but 
 the insulation of the cable may be looked upon as consisting of 
 a number of such rings in series, and if C denotes the total 
 capacity in microfarads of the whole cable, by the laws of 
 condensers in series, we get — 
 
 9 X io"° [^ dr 
 
 C~K/^ 10'^ ^ I r 
 a 
 
 2 9X IO20 , l> 
 from which — 
 
 K/^~^^5^ ^'°^'<? 
 
 2C lOge- 
 
 &6 
 
 a 
 
 K= J X 9 X 10' 
 
 In making a measurement of the specific inductive capacity 
 in this way, the temperature of the water in the tank contain- 
 ing the cable should be taken. 
 
 263. In order to measure the specific inductive capacity of 
 G.P. insulating covering, experiments were made with three 
 wires of different diameter, all covered with the same material. 
 
 The length and the ratio of the external to the internal radius 
 was first measured, and was — 
 
250 Practical Electricity and Magnetism. 
 
 No. I. Length = o'25 mile, - = r^o 
 
 No. 2. Length = 0-25 mile, - = 2*62 
 
 b 
 No. 3. Length = 0-25 mile, - = 3'oo 
 
 The capacities were next measured by comparison with a 
 standard one-third microfarad condenser, the method of mixtures 
 being employed. The following capacities were found : — 
 
 No. I = o"o986 microfarads at 23'8° C. 
 No. 2 = 0-0938 „ „ 
 
 No. 3 = o'o'Soo „ „ 
 
 From these data we get, expressing the length in centi- 
 metres — 
 
 No. I. 
 
 No. 
 
 2 X 0-0986 X loge 2-50 X 9 X 10' 
 
 K = = 4'°'? 
 
 40233 
 
 K 
 
 2 X 0-0938 X loge 2-62 X 9 X 10^ 
 
 = 4'03 
 
 No. 3. K 
 
 IMIMMMM»JM.>M^^^Jjr7jm^ p 
 
 ' = 4-02 
 
 40233 
 
 2 X q-qSoo X logs 3-00 X 9 X 10' 
 
 40233 
 
 264. Measuremetit oftJie Specific Inductive Capacity of Liqidds. 
 
 The method of Hopkinson for the measurement of the S.LC. 
 
 of solids may also be applied to 
 the measurement in the case of 
 liquids. 
 
 The liquid is contained in a 
 double metal cylinder, C (see 
 Fig. 115), into which, but insu- 
 lated from it, hangs the metal 
 cylinder P, the position of P 
 C relatively to C being fixed by 
 means of ebonite stops. 
 
 ill 
 i i ^ 
 
 i 
 
 -Pi 
 
 Fig. 115. 
 
 The cylinder P forms one plate, and C the other, of the 
 condenser. This condenser is balanced against the sliding 
 condenser, first without and afterwards with the dielectric liquid 
 
S.I.C. of Liquids. 251 
 
 between the plates, the sliding condenser being in this case 
 adjusted until a balance is obtained, the capacities being calcu- 
 lated from the positions of the sliding tube in the cylindrical 
 condenser, which, for two concentric cylinders of length / and 
 external and internal radii b and a respectively, air being the 
 medium between them, is — 
 
 1 l> 
 2 logs - 
 
 (7 
 
 265. Siloitis Method. — A much simpler method is a modifi- 
 cation of one due to Silow.^ 
 
 A cylindrical quadrant electrometer is made by pasting four 
 tinfoil strips, each 10 cm. X 10 cm., symmetrically round the 
 inner sides of a glass jar 10 cm. deep and 15 cm. diameter. In 
 the interior, suspended by a wire, zo, from a torsion head, hangs 
 a platinimi needle, consisting of two curved 
 plates of platinum P, P (see Fig. 116) con- 
 nected together by a platinum wire, the whole 
 needle being well within the vessel. The 
 opposite tinfoil quadrants of the glass jar are 
 connected together as in an electrometer ; one 
 pair are connected to one pole of a battery the other pole of 
 which is earthed, whilst the other pair and the needle are both 
 earthed. 
 
 The needle is first brought to zero with the quadrants both 
 earthed, one pair are then connected to the battery, and the 
 needle deflects ; by turning the torsion head at the top of .the 
 wire w, the needle may be brought back to zero, and the angle 
 tti through which the wire has been turned noted. The vessel 
 is then filled up with the liquid and the above repeated, a 
 torsion ou will now have to be applied to restore the needle to 
 its original position, the E.M.F. of the battery being assumed 
 constant. 
 
 The ratio - is the ratio of the capacities, and therefore equal 
 
 to the S.I.C. of the liquid. 
 
 ' Pogg. Ann., 155, p. 389. 
 
252 Practical Electricity and Magnetism. 
 
 266. References to Scientific Papers. 
 
 Tide of Paper. 
 
 Author. 
 
 Reference, 
 
 An Instrument for reproducing 
 
 Deprez 
 
 Phil. Mag., vol. 21, 
 
 an Invariable Quantity of Elec- 
 tricity 
 Measurements of S.I.C. of Di- 
 
 
 June, 18S6. 
 
 Gibson and 
 
 Trans. Roy. Sac, 
 
 electrics 
 
 Barclay 
 
 1871. 
 
 S.I.C. of Glass 
 
 Hopkinson 
 
 /did., 1878, 1881 ; 
 Pro. Roy. Soc, 
 vol. 26, p. 298. 
 
 Dielectric Properties of Different 
 
 .J 
 
 Trans. Roy. Soc, 
 
 Glasses 
 
 
 1878; Pro. Roy. 
 Soc, vol. 25, p. 
 496. 
 
 Measurement of Electrostatic 
 
 ,, 
 
 Fro. Roy. Soc, vol. 
 
 Capacity of Glass 
 
 
 31, p. 148. 
 
 Dielectric Capacity of Liquids 
 
 ,, 
 
 /*«■(/., vol. 3 1, p. 347. 
 
 Note on S.I.C. 
 
 ., 
 
 7«af., vol.41, p. 453. 
 
 On S.I.C. 
 
 9 J 
 
 Jbid., vol. 43, p. 88. 
 
 Measurement of S.I.C. 
 
 Gordon 
 
 Ibid., vol. 27, p. 270 ; 
 vol. 28, p. 155. 
 
 S.I.C. of Dielectrics where acted 
 
 J. J. Thomson 
 
 Ibid., vol. 46, p. 
 
 on by Very Rapidly Alternating 
 
 
 292. 
 
 Currents 
 
 
 
 Effect of Temperature on the 
 
 Cassie 
 
 Ibid., vol. 46, p. 
 
 S.I.C. of a Dielectric 
 
 
 357- 
 
 A New Form of Air Leyden with 
 
 Kelvin 
 
 Ibid., vol. 52, p. 6. 
 
 Application to Measurements 
 
 
 
 of Small Electrostatic Capacity 
 
 
 
 Dielectric Hysteresis 
 
 Porter 
 
 Ibid., vol. 57, p. 
 469. 
 
 On a Condenser of Variable 
 
 Boys 
 
 Phil. Mas:., vol. 7, 
 
 Capacity 
 
 
 Feb., 1879. 
 
 The Refractive Index and S.I.C. 
 
 Hopkinson 
 
 Ibid., vol. 13, Apr., 
 
 of a Transparent Media 
 
 
 1882. 
 
 On a Method of Measuring the 
 
 Glazebrook 
 
 Ibid., vol. 18, Aug., 
 
 Electrostatic Capacity of a 
 
 
 1884. 
 
 Condenser 
 
 
 
 Researches on the Dielectric Con- 
 
 Klemencic 
 
 Ibid., vol. 19, May, 
 
 stants of some Gases 
 
 
 1885. 
 
 Specific Inductive Capacity of 
 
 Rosa 
 
 /bid., vol. 31, Mar., 
 
 Electrolytes 
 
 
 1891 ; vol. 34, 
 Oct., 1892. 
 
 On a Method of determining the 
 
 Trouton and 
 
 Ibid., vol. 33, June, 
 
 S.I.C. of a Dielectric 
 
 Lilly 
 
 1892. 
 
 Determination of the S.I.C. of 
 
 Slschegtiaeff 
 
 Ibid,, vol. 34, Oct., 
 
 Conducting Liquids 
 
 
 1892. 
 
 On Dielectrics 
 
 Appleyard 
 
 Ibid., vol. 38, Oct., 
 1894. 
 
References to Scientific Papers. 
 
 253 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 Measurement of the S.I.C. of ; Fessenden 
 
 Water and Alcohol 1 
 
 On a Method of Comparing the Northrup 
 
 Values of S.I.C. under Slow 
 
 and Rapidly Changing Fields 
 ModiBcation of the B.A. Method Womack 
 
 of determining the Capacities 
 
 of Condensers 
 Effect of Temperature on the Appleyard 
 
 S.I.C. of Dielectrics 
 Experiments with Condensers Leblanc 
 
 Absorption in Mica Condensers 
 
 S.I.C. of Mica 
 Capacity of Coils 
 S.I.C. Ice 
 
 Platymeter 
 
 Determination of the Capacity of 
 
 a Condenser 
 Comparison of Capacities by 
 
 Method of Mixtures 
 
 Bouty 
 
 Cauro 
 
 Ayrton and 
 Perry 
 
 Sir Wm. Thom- 
 son 
 
 Jenkin 
 
 Sir Wm. Thom- 
 son 
 
 Phil. Mag., vol. 38, 
 
 Dec, 1894. 
 Ibiti., vol. 39, Feb., 
 
 1895. 
 
 Ibid., vol. 39, Feb., 
 1895- 
 
 Ibid., vol. 42, Aug., 
 
 1896. 
 Elect., vol. 27, p. 
 
 306. 
 Jour. Elec. Eng., 
 
 vol. 19, p. 755. 
 Ibid., vol. 20, p. 499. 
 Ibid., vol. 24, p. 378. 
 Phil. Mag., vol. 5, 
 
 Jan., 1878. 
 B.A. Report, 1855. 
 
 IbiJ., 1867. 
 
 Jour. Elect. Eng., 
 vol. I, p. 394. 
 
 267. References to Foreign Scientific Papers. 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 Measurement of S.I.C. 
 
 Boltzmann 
 
 Wien. Ber., vols. 
 66, 67 : 1872. 
 
 S.I.C. of Mica 
 
 Klemencic 
 
 BeibliUler, vol. 12 : 
 1888. 
 
 S.I.C. of Liquids 
 
 Silow 
 
 Pogg. Anil., vol. 
 156: 1875. 
 
 J, >» 
 
 Quincke 
 
 Wied. Ann., vol. 19, 
 .|3 . vol. 33. 
 
 )» >i 
 
 N^reano 
 
 Comptes Pendtts, 
 vol. 104: 1887. 
 
 S.I.C. of Gases 
 
 Boltzmann 
 
 Pogg. Ann.,yo\. 1^: 
 
 
 Ayrton and 
 
 Trans. Asiatic Soc. 
 
 
 Perry 
 
 Japan, 1877. 
 
VI. 
 MAGNETISM. 
 
 Determination of H. 
 
 268. In very many of the experiments performed in a 
 physical laboratory, especially those involving the deflec- 
 tion of a magnetic needle, which is either wholly or partly 
 controlled by the magnetic field of the earth, it becomes neces- 
 sary to determine the value of this controlling field. Also, since 
 the direction of this field varies from place to place, and since 
 ordinarily it is only the component of the total force which acts 
 parallel to the surface of the earth that we have to do with, we do 
 not require to know the total force, but only H, the horizontal 
 component of it. The relation connecting the value of H 
 with the total force is — 
 
 H = T cos 8 
 
 where T is the total force in C.G.S. units, and 8 is the angle 
 between H and T, usually known as the angle of dip. 
 
 269. Gauss's Method of determining H. — Several methods 
 have been devised for the measurement of H in absolute units, 
 but that most frequently employed is due to Gauss. This 
 method has been thoroughly investigated, and improved by 
 Professor T. Gray, to whose resea!rches the student is referred.' 
 
 A small magnetic needle is suspended so as to swing freely 
 in the earth's field, and a permanent magnet, the length between 
 the poles of which is 2/, is placed with its centre at a distance, d, 
 from the centre of the needle, so that the line joining the 
 centres is at right angles to the magnetic meridian, and also 
 ' Phil. Mag., vol. vi., Nov., 1878 ; Vol. xx., Dec, 1885. 
 
Determination of H— Gauss's Method. 255 
 
 the magnetic axis of the magnet is at right angles to the 
 magnetic meridian (see Fig. 117); or so that the line joining 
 the centres lies in the magnetic meridian, and the magnetic 
 axis of the magnet at right angles to the meridian (see Fig. 
 118). 
 
 In each case the eflfect of the magnet is to deflect the needle 
 out of the plane of the magnetic meridian. Then, in the first 
 
 H 
 
 ■«---2 
 
 /_.. 
 
 d, 
 
 <r-- 
 S 
 
 d — 
 
 Fig. 117. 
 
 _L. 
 
 <-—zl > 
 
 Fig. 1 18. 
 
 case, usually called the A position of Gauss, we have,^ calling M 
 the magnetic moment of the deflecting magnet, and the 
 angular deflection of the needle — 
 
 M {d^ - py- 
 
 H ~ 2d 
 
 tan 6 
 
 and in the second case, the B position of Gauss, if d^ is the 
 distance apart of the centres, and 0i the angular deflection of 
 the needle — 
 
 '^ = {d^- + py tan e. 
 
 If we now remove the needle, and suspend the deflecting 
 magnet in its place, so that it is free to oscillate about a vertical 
 axis, then, calling I the moment of inertia of the magnet, and 
 T its periodic time of swing, we have — 
 
 kMH = 
 
 477^1 
 
 K being a constant depending on the inductive action of the 
 
 earth's' field on the magnetic moment of the magnet. Where 
 
 ' See Appendix. 
 
256 Practical Electricity and Magnetism. 
 
 great accuracy is not required, k may be taken as equal to 
 unity. By combining this last equation with either of the 
 previous ones, we can get an expression for either M or H. 
 
 270. Coirediojis. — If a very accurate determination of H is 
 to be made, there are certain corrections that must be taken 
 into account in the above calculation. Of these the most 
 important are — 
 
 (i) The correction for the distribution of magnetism in the 
 bar. 
 
 (2) The correction for the inductive action of the earth's 
 field. 
 
 271. (i) The correction for the distribution of magnetism in 
 the bar magnet resolves itself into the measurement of 2/, the 
 distance between the poles of the magnet, as distinguished from 
 /o, the actual length of the magnet. The difference between 
 these two quantities is very small, especially in the case of 
 short magnets, and for ordinary measurements they may be 
 taken as identical. Kohlrausch has shown that the distance of 
 the poles from the ends of a bar magnet is about one-twelfth 
 the length of the bar. The exact length, 2/, may be deter- 
 mined from observations of the deflections produced by the 
 magnet on the needle in the A and B positions, as follows : — 
 
 In the A position ^s^ = -'^ 7—— tan 
 
 ±1 2a. 
 
 M 3 
 
 In the B position g = (d^ -f l'^^ tan ^i 
 
 Hence ^'^' " ^^)' ^an 6 
 ^^"^^ 2d{di + Pf = tST^ 
 
 and expanding, we get, neglecting small terms— 
 
 ^ ed' - 2 v/ 
 2ed + 3 Vi 
 
 the value / thus calculated is used in the calculation for H. 
 
 272. (2) The correction for the inductive effect of the earth's 
 field on the magnetic moment of the deflecting magnet is due 
 to the fact that during the first part of the experiment the 
 magnet is placed with its axis at right angles to the field of the 
 
Correction for Induction. 257 
 
 eaTth, -whilst during the determination of its periodic time of 
 swing it lies in the magnetic meridian, and is therefore liable 
 to a slight increase in magnetization by induction from the 
 earth's field. In order to find the relation between the mag- 
 netic moment of the magnet in the two positions, the following 
 method may be adopted. The magnet is placed inside a long 
 solenoid, with its centre at the centre of the solenoid, and at a 
 distance, d, from the centre of the needle, so that it is in the 
 A Gauss position with respect to it. In series with the 
 solenoid is a compensating coil, which is adjusted with respect 
 to the needle so as to neutralize the magnetic effect due 
 to the solenoid alone ; this adjustment is made before the 
 magnet is placed in position, and with a much stronger current 
 than is to be used afterwards. The magnet is now placed in- 
 side the solenoid, and the deflection, <^, of the needle noted ; 
 a current of C amperes is sent through the solenoid sufficient 
 to produce a magnetic field at the centre of the coil approxi- 
 mately equal to the field H to be measured, and the deflection 
 ^ of the needle noted. 
 Now, since — 
 
 M oc tan <^ 
 and kM « tan ^1 
 tan <^i 
 ~ tan <^ 
 
 The solenoid should be considerably longer than the needle, 
 and may consist of a coil of wire one layer deep, wound on a 
 long glass tube of a diameter slightly larger than the magnet. 
 Then, if n represents the total number of turns on the coil, 
 and / its length in centimetres, the strength of the field at the 
 centre is — ■ 
 
 4?r«C 
 10/ 
 
 Puttmg in the values for //, /, and the strength of field H 
 required, we get — 
 
 47r« 
 The cuirent may be measured on a tangent galvanometer 
 
 s 
 
25S Practical Electricity and Magnetism. 
 
 whose constant has been determined by copper deposition, and 
 which is sufficiently far off from the other apparatus not to 
 exercise any magnetic effect upon it, or else measured by its 
 fall of potential down a known resistance. 
 
 273. The Magnetometer.— Ihn magnetometer needle should 
 consist of two steel magnets about i cm. long and o-o8 cm. 
 diameter, cemented to the back of a light galvanometer mirror 
 0-8 cm. diameter, so that they are parallel to one another and 
 have like poles pointing in the same direction. This mirror is 
 suspended by a silk or spider line inside a glass tube, so that 
 the magnets lie in the horizontal plane. 
 
 The glass tube may be mounted on a stand provided with 
 levelling screws, great care being taken to remove all iron or 
 other magnetic substance from the neighbourhood, and for 
 this reason the apparatus must be entirely constructed of non- 
 magnetic material; any brass work should be tested, as it 
 sometimes contains iron as an impurity. 
 
 An ordinary galvanometer lamp and scale are placed at a 
 carefully measured distance from the mirror, the scale being 
 parallel to the plane of the mirror, and adjusted so that when 
 the needle is in tlie magnetic meridian the spot of light is at 
 the zero. To test the scale for accuracy of adjustment, the 
 deflection produced by the magnet when its centre is at a given 
 distance from the needle should be the same as when it is 
 turned end for end. 
 
 Two lines at right angles to one another should now be 
 marked out to indicate the positions to be occupied by. the 
 centre of the magnet in the A and B positions. 
 
 If the galvanometer scale is not graduated in centimetres its 
 length must be measured, and the distance from the mirror to it 
 expressed in scale-divisions, instead of in centimetres ; also, in 
 reading off the angle 6 from the deflection of the spot of light, 
 the remarks in par. 137 apply here. 
 
 If necessary, the coefficient of torsion t of the suspending 
 fibre of the mirror must be measured by the method described 
 previously (par. 138), and the angle 6 increased in the ratio 
 e{i + r). 
 
 274. Deflecting Magnets. — It is advisable that readings should 
 
Deflecting Magnets for Gauss's Method. 259 
 
 be taken with more than one deflecting magnet, in order to 
 eliminate errors in the determination of 2/ and of I. 
 
 These magnets should have a length at least forty times 
 their diameter, and may be made as follows. From a long, 
 uniform steel rod of about o"2 cm. diameter, which has been 
 soflened, lengths from 8 to 10 cm. are cut off, and the ends 
 carefully filed flat, these lengths are then glass hardened by 
 heating to a bright red heat and at once plunging into cold 
 water; after drying, they are magnetized by placing them 
 inside a solenoid through which a strong current is flowing 
 and at the centre of which there is an intense magnetic field, 
 or else by placing them between the poles of a powerful electro- 
 magnet. After having been magnetized in this way, it is 
 advisable to artificially age the magnets, as they are in a very 
 unstable condition, and liable to lose a large amount of their 
 magnetism on getting a slight knock ; they are therefore 
 plunged once or twice into a vessel of boiling w-ater, being 
 allowed to cool between each immersion. In this way they 
 lose some of their magnetism, but are in a much more 
 permanent condition than before. 
 
 275. Method. — ^The needle having been carefully adjusted, 
 one of the magnets is taken and placed ■ndth its centre at a 
 distance, d, from the centre of the needle in the A position of 
 Gauss, and the deflection noted ; it is then turned end for end, 
 its centre being still kept at the same point, and the deflection 
 to the opposite side of the zero on the scale noted. This is 
 now repeated with the magnet at the opposite side of tlie 
 needle, and at such a distance from it that the deflection pro- 
 duced is the same. The exact distance of the centre of the 
 magnet from the centre of the needle being half the distance 
 between the positions of the centre of the magnet on either 
 side of the needle. 
 
 The above should be repeated at two or three different 
 distances for each of the magnets. Care must be taken to 
 keep the magnets apart and to note the particular magnet 
 which gave each set of readings. 
 
 Similar sets of readings are taken for the B position of Gauss. 
 The readings may be tabulated thus — 
 
26o Practical Electricity and Magnetism. 
 
 Magnet. 
 
 Gauss 
 position. 
 
 d 
 
 Deflections. 
 
 Mean 
 deflection 
 
 Value of (^ 
 calculated 
 
 Right. 
 
 Left. 
 
 Right. 
 
 Left. 
 
 from mean 
 deflection. 
 
 
 
 
 
 
 
 
 
 
 From readings in the A and B positions the effective lengths 
 of the various magnets may be calculated. 
 
 276. After having got all the deflection experiments performed, 
 the determinations of the periodic times of swing of the various 
 magnets must be made. To do this the magnetometer is 
 removed and the magnets one after another are suspended 
 in its place, inside a glass shade, or box with 
 glass sides, the .suspension consisting of a 
 long silk fibre with two loops at one end to 
 hold the magnet in (see Fig. 119). It is 
 not advisable to use a brass clip at the end 
 of the suspension to hold the magnet, as it 
 introduces an additional amount of mertia 
 which would have to be determined by a 
 separate experiment and allowed for. 
 On the bottom of the box, underneath the suspended magnet, 
 there should be placed a strip of mirror glass with a line 
 scratched on it, so that when the magnet lies in the magnetic 
 meridian it is exactly over this line. The magnet is set 
 oscillating about the fibre as a vertical axis by bringing a 
 piece of soft iron near and then removing it, care being taken 
 not to set up a pendulum swing ; and when the amplitude of 
 oscillation has fallen to about 3° on either side of the zero 
 line, a stop-watch with centre seconds hand reading to o'z sec. 
 is started just as the magnet crosses the line. After ten such, 
 transits in the same direction the watch is stopped and the 
 time of one complete double vibration calculated (T). The 
 accuracy with which this measurement can be made increases 
 
 Fig. 119. 
 
Calculatioti of H — Gauss's Metlwd. 261 
 
 with the number of swings observed, so that, if possible, more 
 than ten complete swings should be timed. 
 
 The periodic time of swing of each of the magnets is taken 
 in this way. The amplitude of the swing should always be kept 
 small, otherwise a correction will have to be made for the arc 
 of oscillation, this, however, being less than o'02 % for an arc 
 of 3° on either side of zero. 
 
 277. Having obtained the time of swing of the magnets, we 
 next proceed to determine their moments of inertia (I). This, 
 for a cylindrical rod of length /„ (actual length of magnet), 
 radius r, and weight w, is — 
 
 \I2 4/ 
 
 4- 
 
 The length and diameter of each bar should be measured in 
 centimetres by a pair of bar-callipers, the diameter being taken 
 at various points along the bar, and the mean taken in the 
 calculation of the radius. The weight in grammes of each 
 bar is taken on a delicate balance. 
 
 278. The magnetometer needle should now be replaced in 
 position and the induction constant k determined for each 
 magnet, in the manner described above. 
 
 The values — 
 
 M (d- - F) 
 
 H ~ id 
 
 tan 6 
 
 M 3 
 
 and g = (-^^ + l-f tan 0^ 
 
 are calculated for each set of observations on each magnet 
 and the mean for each of the Gauss positions obtained ; also 
 the value of 
 
 kI 
 
 being known for each magnet, the value of H as determined 
 
 by each of the magnets may now be calculated for the two 
 
 Gauss positions by combining the two equations. Thus we 
 
 have — 
 
 %i^\d , ^ ^ 
 
 H- = — zr-. ;;:2 ;; for the A posiUon 
 
252 
 
 Practical Electricity and Magnetism. 
 
 and H' = 
 
 «TV + /')itani9a 
 
 for the B position 
 
 .M 
 
 279. The values of — , or the magnetic moment per gramme 
 
 Aveight of the different magnets, should also be calculated, in 
 order to give some idea of the quality and magnetic condition 
 of the magnets employed. The following table, taken from 
 Gray's paper, is of interest in this respect : — 
 
 M 
 
 h 
 
 
 
 IV 
 
 2/ 
 
 Diameter. 
 
 
 centimetres. 
 
 centimetres. 
 
 centimetre. 
 
 44'9 
 
 8-03 
 
 6-91 
 
 0-2S 
 
 58-S 
 
 8-os 
 
 7"io 
 
 0'25 
 
 352 
 
 4-00 
 
 3-12 
 
 o'2i; 
 
 S5-8 
 
 14-93 
 
 13-22 
 
 0-25 
 
 71-0 
 
 lo-oi 
 
 9-14 
 
 0'20 
 
 280. The effect of change of temperature on the magnetic 
 moments of the magnets is very slight, being only about 
 °'°°5 % per 1° C, and since the alteration in temperature of 
 the room during the determination is never likely to be more 
 than a few degrees, it may be neglected. 
 
 In order to show the magnitude of the error due to the 
 inductive action of a field on a magnet, the following numbers 
 are taken from a table by Gray for glass-hard steel magnets, 
 showing the percentage alteration of magnetic moment due 
 to a field about five times the strength of the earth's field : — 
 
 I 
 
 'o 
 
 Per cent, varia- 
 tion of moment 
 in unit field. 
 
 M 
 
 
 diameter 
 
 IV 
 
 centimetres. 
 
 
 
 
 3 
 
 10 
 
 0-85 
 
 27 
 
 4 
 
 16 
 
 070 
 
 32 
 
 6 
 
 20 
 
 0-69 
 
 35 
 
 \ 
 
 31 
 
 0-54 
 
 39 
 
 32 
 
 0-S4 
 
 54 
 
 10 
 
 34 
 
 0-51 
 
 40 
 
 10 
 
 44 
 
 0-48 
 
 43 
 
 10 
 
 50 
 
 0-51 
 
 67 
 
 10 
 
 '°S 
 
 0-44 
 
 66 
 
Measurement of H. 
 
 263 
 
 281. The following determination of H was made by the 
 foregoing method. The deflecting magnets, of which there 
 were three, were glass hard, and had the following dimensions : — 
 
 (A) Length, 8'o5 cm.; diameter, 0-25 cm. 
 
 (B) Length, iS'03 cm. ; diameter, o'25 cm. 
 
 (C) Length, 10 cm. j diameter, o'2 cm. 
 
 The mirror of the magnetometer needle was 1087 cm. distant 
 from the scale, which was graduated in millimetres. The deflec- 
 tion magnets were adjusted until the readings on each side of 
 zero were equal, and their distance from the mirror of the 
 magnetometer was taken as half the distance between the 
 positions on either side of the needle where they produced 
 the same deflection. From the readings in the A and B 
 positions of Gauss, the effective lengths of the magnets were 
 * calculated. 
 
 Number of 
 experiment. 
 
 Magnet. 
 
 Gauss 
 position. 
 
 d 
 
 Scale 
 reading. 
 
 (of needle). 
 
 tanO 
 
 2/ 
 
 I 
 
 A' 
 
 A 
 
 cm. 
 32-06 
 
 IS47 
 
 4-05° 
 
 0-0707 
 
 7-11 
 
 2 
 
 A 
 
 B 
 
 2875 
 
 97-6 
 
 2-31° 
 
 0-0406 
 
 7-11 
 
 3 
 
 B 
 
 A 
 
 51-90 
 
 77-6 
 
 2-04° 
 
 0-0356 
 
 12-82 
 
 4 
 
 B 
 
 B 
 
 38-85 
 
 72-9 
 
 2-26° 
 
 0-0395 
 
 12-82 
 
 S 
 
 C 
 
 A 
 
 35 -oo 
 
 99-1 
 
 275° 
 
 o"048i 
 
 9-14 
 
 6 
 
 C 
 
 B 
 
 30-00 
 
 83-0 
 
 2-lS° 
 
 0-0381 
 
 9-14 
 
 In order to determine the efiect of the inductive action of the 
 earth's field on the magnets, the magnetizing coil, with its 
 compensator, were fitted up in the A Gauss position, and the 
 deflection noted, when a field of approximately 0-15 C.G.S. 
 units was produced in the interior of the coil. This gave the 
 following readings : — 
 
 Magnet. 
 
 Deflection. 
 
 Deflection with 
 current on. 
 
 A 
 B 
 C 
 
 IS47 
 
 77-6 
 99-1 
 
 iS4'9 
 
 777 
 99-2 
 
264 Practical Electricity and Magnetism. 
 
 The effect of the earth's field on the magnet as calculated 
 from these readings was taken as 0-08 per cent. 
 
 The following table gives the results of the experiments to 
 determine the time of swing of the magnets, and the final calcu- 
 lated value of H. 
 
 Magnet. 
 
 I. 
 
 T. 
 
 E 
 
 M. 
 
 A 
 B 
 C 
 
 16-58 
 
 106-50 
 
 19-32 
 
 seconds. 
 5-177 
 8-074 
 
 5-549 
 
 C.G.S. units. 
 0-1522 
 0-1525 
 0-1526 
 
 160-4 
 4228 
 162-2 
 
 The mean value of H may be taken as 0-1524. 
 
 Vibration Method of determining H. 
 
 282. The following method of determining the value of* 
 the horizontal component of the earth's magnetic force, due to 
 the author, is very much simpler to carry out than the method 
 of Gauss. 
 
 The method consists in comparing, by the method of vibra- 
 tion of a magnetic needle, the strengths of two magnetic fields, 
 one due to the earth's magnetism alone, and the other made up 
 partly of the earth's field and partly of an artificial field of 
 known strength produced in the interior of a long solenoid. 
 Knowing the strength of field due to the solenoid the value of 
 the earth's field may be calculated. 
 
 283. The apparatus for carrying out this experiment consists of 
 a solenoid containing a known number of turns and of a length at , 
 least ten times its radius, similar to that described in par. 237. 
 Inside tliis solenoid there fits a cardboard tube, rather longer 
 than half the length of the solenoid, near the end of which a 
 mark is made, so that when the mark is flush with one end of 
 the solenoid the other end of the cardboard tube is exactly at 
 the centre of the solenoid. 
 
 At the end of the tube which fits into the solenoid there is 
 suspended, by means of a fine silk fibre or spider-line, a small 
 magnetic needle so that it hangs exactly in the axis of the coil. 
 A concave galvanometer mirror is attached to the needle so 
 
Determination of H — Vibration Method. 265 
 
 that the plane of the mirror is at right angles to the axis of the 
 needle. \Vhen the tube has been adjusted in position, glass 
 plates are cemented over the open ends of the coil, to prevent 
 air-currents from affecting the needle. The solenoid is mounted 
 on a horizontal stand, so that its axis lies in the magnetic 
 meridian, and a galvanometer lamp and scale are arranged so 
 as to indicate the motions of the mirror attached to the needle. 
 The solenoid is connected through a reversing key to a con- 
 stant battery, resistance-box, and tangent, or other galvanometer 
 whose constant has been obtained by a copper deposition 
 experiment. The needle is set oscillating about the fibre as 
 a vertical axis before the current is sent through the coil, and 
 the periodic time of swing taken by means of a stop-watch, the 
 oscillations of the needle being taken from the movement of 
 the spot of light across the galvanometer scale. In no case 
 should the angle of oscillation exceed three degrees on either 
 side of zero, and should be kept as small as possible. Let 
 T, represent the periodic time of swing in this experiment. 
 
 A current of known strength is now sent through the solenoid, 
 sufficient to produce a field at the centre of strength rather less 
 than that due to the earth, and in the same direction as the 
 earth's field. The periodic time of swing Tj is again observed. 
 
 The current, still of the same value, is reversed in the solenoid 
 so as to produce a field in opposition to the earth's field, and 
 the time of swing Tj of the needle is taken. 
 
 284. Now, calling H the horizontal component of the earth's 
 magnetic field, Hi the strength of field produced by the current 
 in the solenoid, M the magnetic moment of the needle, and k a 
 constant depending on the moment of inertia of the needle, and 
 equal to 4T-I, we have — 
 
 (i) MH = 
 
 T' 
 
 (2) M(H + H.) = ;j^ 
 
 (3) M(H - HO = ~ 
 
266 Practical Electricity and Magnetism. 
 
 From (i) and (2) or (i) and (3) we can calculate the value 
 of H, since — 
 
 (i) MH T/ 
 
 (2) - M(H + H,) - Ti^ 
 
 and H = Hj ,^^2 fftj 
 ii — I2 
 
 (i) MH _-Yl 
 
 °' (3) - M(H - H,) - 1? 
 
 rp2 
 
 and H = H, 
 
 ; ^^2 _ ^^2 
 
 The value of Hi, the field strength at the centre of the solenoid 
 due to a current of c ampbres, is — 
 
 H.-^(.-*7:) 
 
 n being the number of turns per centimetre on the solenoid, r 
 its radius in centimetres, and 2/ its length. 
 
 285. Corrections. — In the above calculation it is assumed that 
 the magnetic moment of the needle remains constant in the 
 different fields, this, however, is hardly true, on account of 
 the inductive action on the needle of the field in which it is 
 placed. This error, however, may be eliminated by taking the 
 readings with the directon of the current in the solenoid re- 
 versed as above, provided we assume that the increase in 
 magnetic moment {m) of the needle, when placed in a field of 
 the same direction as the earth's, is the same as the decrease 
 when placed in a reverse field of equal strength. This has 
 been proved to be the case for strongly magnetized steel bars 
 in fields of strength up to 0-2 C.G.S. unit.^ Re-writing the 
 above equations, we get — 
 
 (i) MH = ^2 
 
 (2) (M -f »0(H + Hi) = ^ 
 
 * See Sack on the " Specific Induction Constants of Magnets," Phil. 
 Mag., vol. xxii., Oct., 1886. 
 
Determination of H — Vibration Method. 267 
 (3) (M - ;«)(H - H,) = ^ 
 
 -■■3 
 
 From (2) and (3) we get — 
 
 Mi I 
 
 ''T=1V(H + H,) + T,^(H-H,) 
 
 and from (3) — 
 
 2 T/(H + HQ + T/(H - H,) 
 
 from which H = 
 
 T,'H, (T3' - T,')+ VTi'tli'iT/ - T3')' - ST/Tj'Hi^CI-i'T^HTin'^' - 2 I'/Ta' J 
 2(T,n'/ + T,2T3= - 2T2'T3=) 
 
 286. In a measurement of H by the vibration method, the 
 coil employed consisted of a solenoid 417 cm. long, containing 
 640 turns of wire, the mean radius of the coil being 2 '63 crn. 
 The oscillations of the magnet suspended at the centre of the 
 solenoid were observed through a telescope with spider-lines in 
 the eye-piece, instead of by means of a mirror, lamp, and scale. 
 The solenoid was connected in series with a standard resistance- 
 coil of 300 ohms, a battery, and reversing-key. In order to 
 measure the current in the coil, the potential difference at the 
 ends of the 300-ohm coil was measured against a standard cell 
 by the potentiometer, and during the experiment it was i'434 
 volt. The following readings were obtained : — 
 
 Field in 
 
 Total number of 
 
 Time in seconds 
 
 Time of one double 
 
 solenoid. 
 
 swings observed. 
 
 for total swings. 
 
 vibration. 
 
 H 
 
 40 
 
 114-6 
 
 ^■^^5 „„ 
 
 H 
 
 40 
 
 ii6-o 
 
 2-900 2-880 mean 
 
 H 
 
 40 
 
 115-0 
 
 2-875 
 
 H + H, 
 
 100 
 
 232-6 
 
 2-326 
 
 H + H, 
 
 100 
 
 2320 
 
 2-320 2-325 mean 
 
 H + H, 
 
 100 
 
 233-0 
 
 2-330 
 
 H-H, 
 
 30 
 
 129-6 
 
 4-320 
 
 H-H, 
 
 30 
 
 129-6 
 
 4-320 4-310 mean 
 
 H-H, 
 
 30 
 
 128-6 
 
 4-290 
 
268 Practical Electricity and Magnetism. 
 
 The current C = -rrr ampbres 
 
 300 
 
 Hence H = Hi 
 
 T,^ 
 
 T/ - T2 
 
 Hy t; 
 
 — -til 'p 2 'p 2 
 
 2 VTi^ - T. 
 
 2 I T 2 
 
 -T,V 
 
 Therefore- 
 4X3; 
 417 X 300 
 
 4 X3'i4x 640x1^34/ j^:^"" \( .(^'325)" 
 
 " - - ■■ ' X 3 X 10 V^-2X(4i7W M2-88)^-(2-325)' 
 
 + 7 
 
 .(4-3i°)° 
 
 = 0-162 C.G.S. unit 
 
 ■(4-3ior-(2-88of. 
 and, calculating from the larger formula, we get — 
 H = 0:161 C.G.S. unit 
 
 Determination of the Angle of Dip. 
 287. The measurement of the angle which the direction of 
 
 the total force of the earth's magnetic field makes with the 
 
 horizontal is usually made 
 with an apparatus known 
 as a dip circle, in which a 
 magnetic needle is supported 
 so as to move freely in a 
 vertical plane. One form of 
 this apparatus is shown in 
 Fig. 120. 
 
 The needle, which is made 
 of steel, is supported on a 
 perfectly cylindrical axis be- 
 tween two agate knife-edges 
 in front of a graduated circle ; 
 the angle of inclination being 
 read off on the graduated 
 circle by means of the lenses 
 /, /, the vernier attached to 
 
 the scale being adjusted until the needle points cover their 
 
 Fig. 120. 
 
 reflections in the concave mirrors 
 the vernier. 
 
 M, M, which move with 
 
Determination of the Angle of Dip. 269 
 
 288. Adjustments. — The first adjustment is to get the plane 
 of the graduated circle accurately at right angles to the plane 
 of the table. This is effected by means of a plumb-line, which 
 is hung in front of the circle, and the vernier adjusted until the 
 line covers its reflection in the two mirrors. The vernier 
 reading indicates the position when the line joining the centres 
 of the mirrors is exactly vertical. The spirit-level attached to 
 the base will assist in the levelling process. The next adjust- 
 ment is to get the plane of motion of the needle in the plane 
 of the magnetic meridian. Here advantage is taken of the fact 
 that when the plane of the needle's motion is at right angles 
 to that of the magnetic meridian, the needle sets exactly 
 vertical. The needle is therefore magnetized and placed in 
 position, and the dip circle turned about its vertical axis until 
 the readings at the ends of the needle correspond with the 
 vertical axis as determined above, and the whole circle is then 
 rotated through 90°, thus placing the plane of motion of the 
 needle in the plane of the magnetic meridian. 
 
 Another cause of error is the eccentricity of the axis of 
 rotation of the needle with respect to the centre of the circle, 
 since, as the inclination of the needle alters, it tends to roll 
 along the agate knife-edges. To prevent this, the axis can be 
 caught in V-shaped notches, which raise it off the knife-edges 
 and then replace it exactly at the centre of the circle, the 
 screw E raising and lowering the notches. 
 
 In taking the readings, both ends of the needle are observed ; 
 the circle is then rotated through 180° about its vertical axis, 
 and the readings repeated. The needle is reversed on its 
 bearings, and above readings taken again. The needle is then 
 removed and its polarity reversed by placing it in a long 
 solenoid through which a current is flowing, and a second set 
 of readings similar to the above taken again. The mean of 
 several such sets of readings gives the angle of dip. 
 
 289. Errors. — The principal errors to be guarded against in 
 the above determination are — 
 
 (i) Defective centreing of the needle with respect to the dial. 
 
 (2) Want of coincidence between the magnetic and geometrical 
 axes of the needle. 
 
2/0 Practical Electricity and Magnetism. 
 
 (3) Eccentricity of the axis of suspension of the needle with 
 respect to its centre of gravity. 
 
 (4) Want of level of the agate knife-edges. 
 
 Of these errors the first is eliminated by taking readings at 
 both ends of the needle, the second by reversing the position 
 of the needle in its pivots, the third by reversing the polarity 
 of the needle, and the fourth by rotating the dial 180° about 
 a vertical axis. Thus, the mean of all the readings gives the 
 mean value of the angle of dip. 
 
 290. Determination of the Angle of Dip by Induced Currents. 
 — The angle of inclination may also be determined by experi- 
 ments with the earth inductor, a form such as that shown in 
 Fig. loi being used, which is capable of being rotated about 
 two axes at right angles to one another. Either of the following 
 methods may be employed. 
 
 (i) The earth inductor is set up on a horizontal table and 
 carefully levelled, so that the axis of rotation of the coil is 
 perpendicular to the plane of the table, this adjustment being 
 made by means of a plumb-line and spirit-level. The coil is 
 connected to the terminals of a ballistic galvanometer placed 
 at some distance off, and on suddenly rotating the coil through 
 180° a swing, ySj, will be observed on the galvanometer. The 
 coil, in rotating through 180°, has twice cut the horizontal 
 component of the , earth's magnetic field. Calling this H, and 
 A the total effective area (the sum of the number of turns on 
 the coil multiplied by the area of each) of the coil, we get, 
 for the quantity of electricity, Qj, discharged through the 
 galvanometer — 
 
 ^ _ 2HA . A/ , ^'^ 
 
 R being the resistance of the coil and galvanometer. 
 
 The axis of rotation of the earth inductor is now turned 
 through 90°, by. the help of the graduated circle attached to it, 
 and the coil again suddenly rotated through 180°, this time 
 cutting the vertidal component (V) of the earth's field twice, 
 and discharging a quantity of electricity, Qj, through the 
 galvanometer, which produces a swing, p.^. Then — 
 
Determination of the Dip by Earth Inductor. 271 
 
 Q.,= ?VA^^i„^/ ^,M 
 R 2 V 2/ 
 
 
 hence y = 
 
 sin 
 sin 
 
 2 
 2 
 
 
 
 
 H 
 
 But y = 
 
 cos 
 sin 
 
 8 
 8- 
 
 I 
 
 
 
 tan 
 
 8 
 
 dip; 
 
 
 
 
 
 
 there: 
 
 fore = 
 
 tan 8 
 
 sin 
 
 2 
 
 
 
 
 sin 
 
 a" 
 
 
 when the swing of the needle is small, from which, by the help 
 of a table of tangents, the value of 8 may be found. 
 
 In making this experiment, care must be taken that the whole 
 rotation of the coil has taken place before the ballistic needle 
 commences to move. 
 
 291. (3) The second method of using the earth inductor to 
 determine the angle of inclination is to arrange it so that its 
 axis of rotation lies in the direction of the lines of force of the 
 earth, i.e. so that the angle of inclination of the axis of the coil 
 to the horizontal is the angle of dip. When this is the case, 
 then, when connected to a ballistic galvanometer, and rotated 
 through 180°, no swing of the galvanometer needle will be 
 observed, since the coil in rotating does not cut any lines of 
 force of the earth. This method may be made very sensitive 
 by starting the ballistic galvanometer needle swinging slightly, 
 and then timing the rotations of the coil to be synchronous 
 with the swings of the needle, when any increase in the 
 amplitude of the swing will denote a small quantity of electricity 
 produced in the coil. 
 
 292. In a measurement of the angle of dip by means of the 
 earth- inductor method, the following ballistic throws were 
 observed : — 
 
2/2 Practical Electricity and Magnetism. 
 
 Component cut by 
 
 Ballistic 
 
 Mean 
 
 tana = ''l 
 
 earth inductor. 
 
 swing. 
 
 deflection. 
 
 lii 
 
 Vertical 
 
 206-5 
 
 . 
 
 
 Vertical 
 
 207-0 
 
 > 206-0 
 
 
 Vertical 
 
 206-5 
 
 
 Vertical 
 
 205-0 
 
 J 
 
 
 Horizontal 
 
 79 "0 
 
 
 2-6042 
 
 Horizontal 
 
 8o-o 
 
 79-1 
 
 
 Horizontal 
 
 777 
 
 
 Horizontal 
 
 800 
 
 
 
 tan 8 = 2-6042, corresponds to 8 = 69° 
 
 This value of 8 was verified by placing the axis of rotation of 
 the earth inductor inclined to the horizontal at an angle of 69° 
 in the direction of the magnetic meridian, and on rotating the 
 coil no deflection of the galvanometer needle was observed. 
 
 Measurement of the Magnetic Qualities of Iron 
 AND Steel. 
 
 293. When a current of electricity flows through a solenoid, 
 a magnetic field is produced inside it, the strength of which, 
 expressed in terms of lines of force per square centimetre, is 
 usually denoted by the letter H. If now, keeping the current 
 still the same, a bar of unmagnetized soft iron is introduced 
 into the interior of the solenoid, the number of lines of force is 
 very greatly increased, and the iron bar becomes strongly 
 magnetized. The number of lines of force per square centi- 
 metre in the iron bar is denoted by the letter B, and is generally 
 termed the magnetic induction in the iron, or the flux density. 
 If A represents the cross-sectional area of the bar, then 
 BA = N the total flux of lines through it. The ratio of B tO' 
 H in the above case, or the flux density with iron inside the 
 solenoid to the flux density with air, the current being the 
 same, is termed the magnetic permeability of the iron, and is- 
 denoted by the letter /i. So we have — 
 
 B 
 
 '^ = H 
 
 or /tH = B 
 
The Measurement of Permeability. 273 
 
 The permeability of the iron is sometimes termed its specific 
 conductivity for lines of force, and its value is a measure of 
 the magnetic quality of the iron. Another term sometimes 
 employed in connection with the magnetization of iron is the 
 " intensity of magnetization " (I), which is defined as the mag- . 
 netic moment per unit volume, and the ratio between I and 
 H is called the magnetic susceptibility (K) of the bar, the 
 relations connecting these quantities being — 
 
 -I 
 
 M = magnetic moment, V = volume ; 
 
 --\ 
 
 B = H + 47rl 
 
 = H(i + 47rK) 
 And since B = /xH 
 
 /i = I + 47rK 
 
 and K = ■ 
 
 47r 
 
 Measurement of Permeability. 
 
 294. In order to measure the permeability of a substance, 
 we have to measure both B and H. The methods generally 
 employed for this may be classed under three heads — 
 (i) Magnetometric methods. 
 
 (2) Inductive or ballistic methods. 
 
 (3) Traction methods. 
 
 In (i) the intensity of magnetization of the bar is deduced 
 from the effect which it exerts on a magnetic needle placed in 
 its vicinity. The magnetic induction is measured in (2) by 
 the quantity of electricity which it induces in a circuit surround- 
 ing the specimen, when its value is suddenly altered ; whilst in 
 (3) the magnetic induction can be calculated from the force 
 required to overcome the magnectic attraction between two 
 bars. 
 
 T 
 
274 Practical Electricity and Magnetism. 
 
 N 
 
 X 
 
 (i) The Magnetometric Method of Measuring Permeability. 
 295. In this method the specimen to be experimented upon 
 is placed so that it can act on a magnetic needle under the 
 influence of the earth's or other controlling force of known 
 value. The position occupied by the bar with respect to the 
 needle differs with different experimenters ; thus, for instance, 
 one or other of the Gauss positions may be chosen. We will, 
 however, follow Ewing, and take what 
 he terms the one-pole method, in which 
 one pole is so much nearer the needle 
 than the other that it practically pro- 
 duces the entire effect ; a small correc- 
 tion, however, is made for the effect 
 of the other pole. The relative positions 
 of the specimen and the needle N are 
 shown in Fig. 121. AB represents 
 the distance between the poles of the 
 magnet, which is slightly less than the 
 length of the bar : the method of finding 
 these points experimentally will be de- 
 scribed later. Let x represent the 
 distance NA, y the distance NB, and a the cross-sectional area 
 of the rod in square centimetres — 
 
 la 
 Then the force exerted by A at the needle = - ^ 
 
 :)' 
 
 B 
 
 Fig. 121. 
 
 and 
 
 B 
 
 But the horizontal component of the 
 force due to B at N 
 
 Hence the total horizontal force 
 acting at N is 
 
 
 la 
 f 
 la X 
 
 y y 
 
 This force exerts a turning moment on the needle N, which 
 is balanced by the moment of the controlling force. Hence, 
 calling 6 the angular deflection of the needle, and F the value 
 of the controlling force, we have — 
 
Magnetometric Method. 275 
 
 
 /= F tan (9 
 
 - - ) = F tan e 
 
 , , a:^ F tan e 
 and I = 
 
 y?\ 
 
 
 In order to calculate B from this we use the relation — 
 
 B = H + 47rl 
 
 and the value of the magnetizing force H, at the centre of a 
 long solenoid such as that used in this experiment, is— 
 
 10 
 where S is the number of turns per centimetre length of the 
 solenoid, and C the current in ampferes. 
 
 296. Apparatus. — In carrying out a measurement of perme- 
 ability by means of the magnetometric method, the length of 
 the specimen must be great compared with its diameter, in 
 order to reduce the demagnetizing effect of the poles at the 
 ends of the bar to a minimum. Thus, for instance, if a short 
 bar of soft iron is strongly magnetized", and the magnetizing 
 force removed, the specimen will be found to have lost all its 
 magnetism, owing to the large demagnetizing action of the ends ; 
 if, however, the bar has a length about four hundred times its 
 diameter, this demagnetizing action is very much diminished. 
 In order to experiment on specimens whose length is four 
 hundred times their diameter, we have practically to work with 
 wires, and this may be looked on as one of the disadvantages 
 of this method of measuring permeability, since there is reason 
 to believe that wires may not behave in the same way as large 
 masses of metal. 
 
 The wire to be experimented upon is placed inside a long 
 solenoid, so that the coils project a little way beyond the ends 
 of the specimen, in order that the magnetizing force acting on 
 the wire may be sensibly uniform all along its length. The 
 solenoid may be wound on a thin tube of some non-magnetic 
 substance, which fits closely over the specimen. The arrange- 
 ment of the apparatus is shown in Fig. 122. S represents 
 
276 Practical Electricity and Magnetism. 
 
 the magnetizing solenoid with the iron wire inside, clamped in 
 a vertical position near the magnetometer needle N ; in series 
 with S is the compensating coil C, which neutralizes the 
 magnetic effect of the solenoid on the needle N, a reversing 
 key K, variable liquid resistance R, secondary battery B, and 
 current measuring galvanometer G. A second solenoid S', in 
 
 N 
 
 y<^^ 
 
 series with a smgle secondary cell i, and large resistance ;-, is 
 wound over the solenoid S in order to neutralize the inductive 
 action of the vertical component of the earth's magnetic force 
 on the specimen. 
 
 The magnetometer needle N may be the same as that em- 
 ployed in the measurement of H (see par. 273), the deflections 
 being read by means of a lamp and scale, it being borne in 
 mind that the angle ^ in the formula above is the angular 
 deflection of the needle, and not that of the spot of light. In 
 order to determine the value of the controlling force F acting 
 on the needle, which may be due to the earth alone, or to 
 the earth and a permanent magnet, the method of Gauss 
 
Magnetometric Method. iTf 
 
 employed In the determination of H may be used ; or, if H 
 has been determined for any particular part of the laboratory, 
 the periodic time of swing of the magnetometer needle should 
 be taken at that place, and then again, when under the con- 
 trolling force, to be used during the experiment. If H 
 represents the strength of the earth's field, and Tj the periodic 
 time of swing in it, while Ta represents the periodic time of 
 swing under the controlling force F, then — 
 
 H ~ IV 
 and E" = H^ 
 
 The reversing key K employed must be of a form which 
 will allow very rapid reversals to be made, and may con- 
 veniently take the form of a commutator mounted so as to 
 rotate on an axle, contact being made by springs or brushes. 
 
 The variable resistance R must be so arranged that its value 
 can be altered without breaking circuit, and for this reason a 
 liquid resistance, such as that described in par. 22, is very 
 suitable; or else a carbon resistance may be used (see Fig. 14). 
 
 The current measuring instrument G may be a sensitive 
 ammeter which has been carefully calibrated throughout its 
 scale, or a sensitive low resistance galvanometer whose 
 absolute calibration curve has been obtained. In any case 
 the instrument must be so placed that its indications are not 
 affected by the other pieces of apparatus ; also all the wires 
 leading to the various pieces of apparatus should be twisted 
 together, to neutralize any magnetic effect they might have. 
 
 297. Adjustments. — The first adjustment is to place the iron 
 wire so that its upper pole is in a line with the needle of the 
 magnetometer ; it is therefore placed into the solenoid, and a 
 current sent through the coils; the solenoid, with the wire 
 inside, is then raised or lowered until the maximum deflection 
 is obtained on the magnetometer ; it is then clamped firmly in 
 this position. The iron rod is now removed, a strong current 
 sent through the solenoid, and the compensating coil C 
 adjusted relatively to the needle until it neutralizes the 
 
2/8 Practical Electricity and Magnetism. 
 
 magnetic effect of the latter on the needle, this being the 
 case when the magnetometer needle returns to zero; this 
 adjustment will be perfect for all currents, but should be made 
 for the strongest current likely to be employed. 
 
 298. As it is desirable to start with the bar in a perfectly 
 unmagnetized condition, any residual magnetism must be 
 removed. To do this the wire is adjusted inside the solenoid 
 till it produces its maximum effect at the needle, and starting 
 with the strongest current in the solenoid, the reversing key is 
 worked rapidly, so as to magnetize the bar, first in one direction 
 and then in the opposite direction, at the same time the resist- 
 ance R is gradually increased until the current is diminished to 
 zero. This treatment should completely wipe out any residual 
 effects of previous magnetizations. It may be found, however, 
 that the bar is always left feebly magnetized in one direction, 
 this being due to the inductive effect of the earth's magnetic 
 field acting on the bar. To neutralize this, a current is sent 
 through the solenoid S' so as to produce a magnetic field in 
 opposition to that of the earth's, the current being regulated 
 until, after the demagnetizing process described above has been 
 carried out, the bar is left perfectly unmagnetized, as evidenced 
 by the magnetometer needle setting at zero. 
 
 299. Method of Measurement. — A very weak current is now 
 sent through the solenoid S, and readings taken on the mag- 
 netometer and galvanometer G, the current is gradually 
 increased by slowly diminishing the resistance R, and a set of 
 simultaneous readings on the two instruments taken until the 
 current has reached its maximum value, it is then gradually 
 diminished to zero, reversed, and carried to a maximum in the 
 opposite direction, and finally reduced again to zero. The 
 readings may be tabulated thus — 
 
 Galvano- 
 meter 
 reading. 
 
 Current in 
 amp&res. 
 
 Magneto- 
 meter 
 deflection. 
 
Correction for End Effects. 279 
 
 Great care must be taken during the above operations not to 
 shake the specimen, since, especially in the case of soft iron, a 
 shght jar is sufficient to seriously affect its magnetic condition. 
 
 From the above table a curve should be plotted, having 
 values of B for ordinates, and H for abscissae ; also a curve 
 with /u. for ordinates and B for abscissae. 
 
 300. Correction for the End Effects. — In the above it has 
 been assumed that the value of H, as calculated from the dimen- 
 sions of the solenoid and the current flowing in it, represents 
 the actual magnetizing force acting on the specimen ; this 
 assumption, however, is not strictly true, on account of the 
 demagnetizing action of the poles at the ends of the bar, which 
 tend to set up lines of force in the reverse direction to those in 
 the solenoid, and therefore diminish the magnetizing force inside 
 the specimen. If we call H' the effective magnetizing force 
 acting on the specimen, then it has been shown by Ewing^ that 
 in the case of ellipsoids — 
 
 H' = H - NI 
 
 The following table giving values of N for various sizes of 
 specimen : — 
 
 „ .. lensth 
 '^^t'" diameter 
 
 N 
 
 5° 
 
 o'oi8i7 
 
 ICXJ 
 
 0-00540 
 
 200 
 
 0.00157 
 
 300 
 
 0-00075 ■ 
 
 400 
 
 0-00045 
 
 Soo 
 
 0-00030 
 
 The corrections may be made for each value of H employed, 
 and a separate column of values of H' added, or the correction 
 may be made graphically on the curve connebting B and H. 
 
 301. The following example of the method is taken from 
 Ewing's " Experimental Researches in Magnetism." ^ The 
 specimen consisted of annealed iron wire 0-07 7 cm. diameter and 
 
 ' See Electrician, vol. xxiv. pp. 313, 341. 
 ^ Phil. Trans,, 1S85, p, 539. 
 
28o Practical Electricity and Magnetism. 
 
 30-5 cm. long. The curve plotted from these readings is shown 
 in Fig. 123. 
 
 
 '] 
 
 ^ 
 
 -^, 
 
 
 J 
 
 i 
 
 
 
 n 
 
 00 
 
 
 
 no 
 
 B 
 
 00 
 
 
 
 ro 
 
 00 
 
 
 
 ■0 
 
 oc 
 
 
 
 •0 
 
 OL 
 
 
 
 80 
 
 
 
 -zo 
 
 -10 
 
 
 
 ,0 H 
 
 
 -ZO 
 
 a. 
 
 
 
 -n 
 
 V. 
 
 
 
 60 
 
 H 
 
 
 
 ■80 
 
 H 
 
 
 
 -100 
 
 r 
 
 
 
 -120 
 
 \ 
 
 ! I 
 
 
 
 )^ 
 
 r 
 
 
 
 
 
 -jei 
 
 100 
 
 G. 123. 
 
 
Magtietisatton of Iron. 
 
 281 
 
 Galvano- 
 
 
 Magnelc- 
 
 
 Galvano- 
 
 
 Magneto- 
 
 
 meter 
 
 H 
 
 mcter 
 
 B 
 
 meter 
 
 H 
 
 meter 
 
 B 
 
 * reading. 
 
 
 reading. 
 
 
 reading. 
 
 
 reading. 
 
 
 
 
 o-oo 
 
 
 
 
 
 141 
 
 -499 
 
 -278 
 
 -11490 
 
 9 
 
 0-32 
 
 I 
 
 41 
 
 162 
 
 -573 
 
 -306 
 
 — 12650 
 
 24 
 
 0-85 
 
 4 
 
 165 
 
 211 
 
 -7-47 
 
 -338 
 
 — 14400 
 
 39 
 
 1-38 
 
 10 
 
 413 
 
 261 
 
 -923 
 
 -352 
 
 — 14700 
 
 59 
 
 2-l8 
 
 28 
 
 1460 
 
 312 
 
 -11-05 
 
 -373 
 
 -14920 
 
 79 
 
 2-8o 
 
 89 
 
 3680 
 
 411 
 
 — iS-90 
 
 -376 
 
 -15550 
 
 99 
 
 3-50 
 
 175 
 
 7230 
 
 311 
 
 — II-OI 
 
 -375-5 
 
 -15530 
 
 119 
 
 4-21 
 
 239 
 
 9880 
 
 211 
 
 -7-47 
 
 -375 
 
 -15500 
 
 139 
 
 492 
 
 279 
 
 1 1540 
 
 III 
 
 -3-93 
 
 -372 
 
 -15380 
 
 159 
 
 5-63 
 
 304 
 
 12570 
 
 61 
 
 — 2'l6 
 
 -368 
 
 -15210 
 
 189 
 
 6-69 
 
 327 
 
 13520 
 
 23 
 
 -o-8i 
 
 -361 
 
 -14930 
 
 239 
 
 8-46 
 
 348 
 
 J4390 
 
 ii'5 
 
 -0-41 
 
 -358 
 
 — 14800 
 
 289 
 
 10-23 
 
 359 
 
 14840 
 
 
 
 0-00 
 
 -352 
 
 -14550 
 
 342 
 
 12 n 
 
 365 
 
 15090 
 
 9 
 
 032 
 
 -348 
 
 - 14390 
 
 441 
 
 15-61 
 
 373 
 
 15420 
 
 19 
 
 0-67 
 
 -339 
 
 — 14010 
 
 574 
 
 20-32 
 
 378 
 
 15630 
 
 29 
 
 1-02 
 
 -327 
 
 -13520 
 
 629 
 
 22-27 
 
 3S0 
 
 15710 
 
 39 
 
 1-38 
 
 -306 
 
 - 12650 
 
 <t64 
 
 16-42 
 
 379 
 
 15670 
 
 49 
 
 -173 
 
 -274 
 
 -1 1330 
 
 239 
 
 846 
 
 375 
 
 15560 
 
 59 
 
 — 2-C9 
 
 -195 
 
 -8060 
 
 139 
 
 4-92 
 
 372 
 
 15380 
 
 69 
 
 2-44 
 
 -103 
 
 — 4260 
 
 89 
 
 3-15 
 
 369 
 
 15270 
 
 79 
 
 2-80 
 
 4 
 
 165 
 
 39 
 
 t-38 
 
 363 
 
 15010 
 
 86 
 
 3-04 
 
 58 
 
 2400 
 
 
 
 — 0-00 
 
 350 
 
 14470 
 
 99 
 
 3-50 
 
 130 
 
 5370 
 
 "•s 
 
 —0-41 
 
 .•^42 
 
 14140 
 
 114 
 
 403 
 
 199 
 
 8230 
 
 23 
 
 -o-8i 
 
 329 
 
 13360 
 
 140 
 
 4-96 
 
 279 
 
 11530 
 
 31 
 
 — i-io 
 
 3'8 
 
 13150 
 
 190 
 
 6-72 
 
 332 
 
 13720 
 
 41 
 
 -1-45 
 
 295 
 
 12200 
 
 240 
 
 8-50 
 
 351 
 
 14510 
 
 51 
 
 -r8o 
 
 253 
 
 10460 
 
 291 
 
 10-30 
 
 362 
 
 14970 
 
 62 
 
 — 2-20 
 
 166 
 
 6860 
 
 339 
 
 12-00 
 
 368 
 
 15210 
 
 71 
 
 -2-51 
 
 70 
 
 2890 
 
 340 
 
 12-04 
 
 374 
 
 15460 
 
 81 
 
 -2-87 
 
 — 12 
 
 -496 
 
 579 
 
 12-50 
 
 381 
 
 15750 
 
 91 
 
 — 3"22 
 
 -83 
 
 -3430 
 
 630 
 
 22-30 
 
 383 
 
 15830 
 
 lOI 
 
 -3-50 
 
 -142 
 
 -5870 
 
 
 
 0-00 
 
 353 
 
 14590 
 
 121 
 
 -4-28 
 
 — 226 
 
 -9340 
 
 
 
 
 
 Values of ^. 
 
 B 
 
 ^ 
 
 B 
 
 U 
 
 41 
 
 128 
 
 12570 
 
 2230 
 
 165 
 
 194 
 
 13520 
 
 2020 
 
 413 
 
 299 
 
 14390 
 
 1700 
 
 1460 
 
 670 
 
 14840 
 
 1450 
 
 3680 
 
 1310 
 
 15090 
 
 1250 
 
 7230 
 
 2070 
 
 15420 
 
 990 
 
 9880 
 
 2350 
 
 15630 
 
 770 
 
 11540 
 
 2350 
 
 15710 
 
 705 
 
282 Practical Electricity and Magnetism. 
 
 (2) Inductive or Ballistic Methods of Meastiring Permeability. 
 
 302. In the inductive methods of measuring /«., advantage is 
 taken of the fact that if the magnetic flux through any closed 
 conducting circuit is suddenly altered, a current is induced in 
 the circuit during the change, the total quantity of electricity 
 induced being proportional to the change in the magnetic flux. 
 By placing a ballistic galvanometer in series with the conducting 
 circuit, we can measure the quantity of electricity induced, and 
 so get a measure of the change in the number of the lines of 
 force. Also, since we can calculate the magnetizing force H, 
 required to produce this induction change, in terms of the number 
 of spirals in the magnetizing solenoid and the current flowing 
 in them, we have all the necessary data from which to obtain /x. 
 
 The specimen to be experimented upon by this method may 
 take the form of either a straight bar or a closed ring. If the 
 former is chosen, a correction must be applied for the demagne- 
 tizing action of the ends similar to that employed in the 
 magnetometric method just described; the advantage of the 
 ring form is that, not having any poles, there is no such 
 demagnetizing effect. The ring has usually a circular cross- 
 section, but this is not necessary ; in fact, in the case of rings of 
 small diameter, it is better that the section should rather 
 approach that of a rectangle, since, on account of the difference 
 between the external and internal circumferences of the ring, 
 the magnetizing force will be stronger on the inner side where 
 the spirals of the magnetizing coil lie closest together. For 
 this reason, the diameter of the ring should always be large 
 compared with its thickness, and may in the case of a ring of 
 circular cross-section be from forty to fifty times its cross- 
 sectional diameter. 
 
 The ring to be experimented upon should have a perfectly 
 uniform cross-section all round, and after its sectional area and 
 mean circumference have been carefully measured, must be 
 uniformly overwound with the magnetizing coil. A small 
 secondary or induction coil of fine wire is wound over a part of 
 the magnetizing coil. The numbers of turns on the two coils 
 must be known. 
 
Ballistic Method. 283 
 
 The apparatus is then arranged as shown in Fig. 124. The 
 primary or magnetizing coil P is connected through a reversing 
 key, K, to a secondary battery, B, current-measuring galvano- 
 meter, G, and variable liquid or carbon resistance, R. The 
 secondary or induction coil S is connected in series with the 
 ballistic galvanometer BG and the earth inductor E. Care 
 must be taken in setting up the apparatus that the various 
 
 instruments have no direct effect on one another. The ballistic 
 galvanometer should be supplied with a damping coil, in order 
 to be able to quickly bring the needle to rest. 
 
 303. The method of experiment is what is termed the step- 
 by-step method, the current in the magnetizing spiral being 
 increased suddenly by small amounts, by diminishing the resist- 
 ance R, and the ballistic throws observed corresponding to the 
 increments in the lines of force in the ring, the total number of 
 lines of force in the ring at any instant being proportional to 
 the algebraic sum of the ballistic throws from the commence- 
 ment up to that point. When the current in the magnetizing 
 coils has been increased to its maximum value, it is then 
 reduced by steps to zero, reversed, carried to a negative maxi- 
 mum and again reduced to zero, thus carrying the specimen 
 through a complete cycle of magnetization. 
 
 The ballistic thro~ws may be calibrated directly, in terms of 
 values of B, by means of the earth inductor. If the earth 
 inductor has a total effective area. A, and is arranged so as to 
 cut the horizontal component of the earth's lines of force, then, 
 when it is rotated through 180°, let it produce a throw, B, on 
 the ballistic galvanometer, then — 
 
 61 « 2HA 
 
 and — ^— will represent the number of lines cut per scale-division 
 
284 Practical Electricity and Magnetism. 
 
 throw of the ballistic needle. Hence, if B^ be any throw due 
 to an increment of current in the magnetizing coils, and a be 
 the total effective area of the induction coil S — 
 
 2HA6'i 
 a6 
 will represent the induction B. 
 
 It is necessary here to point out a small source of error in 
 the above calculation. The value of B required is the induc- 
 tion in lines per square centimetre in the iron. What has been 
 calculated above is the induction in lines per square centimetre 
 in the induction coil J. Now, the area of the induction coil, if 
 wound outside the primary coil, will be slightly larger than the 
 area of the iron core. Generally speaking, this correction is so 
 small as to be negligible, but if a correction is to be applied, 
 then from the total flux of lines through the test-coil must be 
 subtracted the number of lines flowing through the air-space 
 between the test-coil and the iron core ; or, calling this area a, 
 and the magnetizing force due to the current in the primary H, 
 Ha lines of force must be subtracted from the total flux of lines, 
 and the remainder, when divided by a, will give the true value 
 of B. The magnetizing force H is calculated exactly as in the 
 last case, or — 
 
 10 
 
 where n is the number of turns per centimetre on the ring, and 
 C is the current in ampferes. The same method of measuring 
 the current C may be employed as before. 
 
 304. One of the disadvantages of the ring method of measur- 
 ing /x is, that if an error isl made in reading the ballistic throw 
 at any point, it is carried through all the subsequent readings, 
 and errors of this kind are therefore cumulative ; also, the 
 method will give no indication of slow creeping up of the in- 
 duction, as has been observed by Ewing using the magneto- 
 metric method, so that the value of B may really be somewhat 
 larger than will be calculated from the sum of the ballistic throws. 
 
 305. The following example of the ballistic method of 
 measuring /t is due to Ewing : — ^ 
 
 • Phil. Trans., 1885, p. 530. 
 
Ballistic Method. 
 
 285 
 
 The ring was welded out of a piece of soft annealed iron wire. 
 Diameter of wire forming the ring = o'248 cm. 
 
 Area of section of iron = o"o483 sq. cm. 
 Mean radius of ring = S"o cm. 
 Mean circumference of ring = 3i"4 cm. 
 Number of turns in magnetizing coil = 474 
 Number of turns in secondary coil =167 
 
 Area of earth inductor = 1 2 1 6 sq. cm. 
 Number of turns in earth inductor =10 
 Earth's force cut by the earth inductor = 0-34 C.G.S. units 
 Ballistic throw on turning over earth inductor = 42*9 scale div. 
 
 From the above it follows that the change in B per scale- 
 division ballistic throw = 23-89. 
 
 H 
 
 Ballistic 
 
 Sum of 
 
 B 
 
 
 
 tlirow. 
 
 throws. 
 
 /' 
 
 0-13 
 
 I-I 
 
 I-I 
 
 26 
 
 
 0*26 
 
 fl 
 
 2-2 
 
 S3 
 
 — 
 
 0-30 
 
 o"5 
 
 2-7 
 
 65 
 
 — 
 
 0-40 
 
 0-8 
 
 3-S 
 
 84 
 
 — 
 
 0-S3 
 
 i-o 
 
 4'S 
 
 107 
 
 — 
 
 071 
 
 2- 1 
 
 6-6 
 
 158 
 
 — 
 
 0-93 
 
 2-9 
 
 9-S 
 
 227 
 
 — 
 
 1-31 
 
 3'9 
 
 13-4 
 
 320 
 
 245 
 
 1-69 
 
 9-2 
 
 22-6 
 
 540 
 
 320 
 
 1-89 
 
 6-9 
 
 29-S 
 
 705 
 
 370 
 
 278 
 
 77-5 
 
 107-0 
 
 2560 
 
 920 
 
 3-36 
 
 787 
 
 185-7 
 
 4440 
 
 1320 
 
 4-01 
 
 82-0 
 
 267-7 
 
 6400 
 
 1600 
 
 4'9S 
 
 91-5 
 
 359'2 
 
 8580 
 
 1740 
 
 S"86 
 
 S7-0 
 
 416-2 
 
 9940 
 
 1700 
 
 7-20 
 
 57-0 
 
 473-2 
 
 11300 
 
 1570 
 
 8'io 
 
 23-5 
 
 496-7 
 
 11870 
 
 1460 
 
 9-14 
 
 24-0 
 
 520-7 
 
 12440 
 
 1360 
 
 7-83 
 
 -4"4 
 
 516-3 
 
 12330 
 
 — 
 
 6-21 
 
 -67 
 
 509-6 
 
 12170 
 
 — 
 
 475 
 
 -7-1 
 
 502-5 
 
 12000 
 
 — 
 
 270 
 
 — i4'o 
 
 488-5 
 
 11670 
 
 — 
 
 O'OO 
 
 -33-2 
 
 455-3 
 
 10880 
 
 ~ 
 
 Fig. 125 shows the curve plotted from these readings. 
 306. An interesting modification of the ring method just 
 described, due to Messrs. Evershed and Vignoles,' gets rid of 
 • Electrician, vol. xxvii. pp. 49, 77. 
 
286 Practical Electricity and Magnetism. 
 
 the cumulative errors which may occur, and at the same time 
 subjects the specimen to treatment more closely approximating 
 
 
 ., 
 
 
 
 -P^ 
 
 -^ 
 
 
 
 / 
 
 
 
 
 / 
 
 
 
 B 
 
 / 
 
 / 
 
 
 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 ^ 
 
 / 
 
 H 
 
 
 
 O Z 4- 6 8 10 
 
 Fig. 125, 
 
 to what would occur in actual practice ; also, the method admits 
 of any reading being repeated at any time, and as often as 
 desired, under exactly similar conditions, which is not possible 
 with the other methods. 
 
 In Evershed and Vignoles' method the ring of iron is over- 
 wound in opposite directions with two magnetizing spirals, one 
 having twice the number of turns that are on the other; the 
 connections are shown in Fig. 126, where, for the sake of clear- 
 ness, the coils are shown as wound on separate parts of the ring, 
 but in reality both coils completely surround the- ring, as does 
 also the induction coil T. The magnetizing coils Sj and S2 are 
 each connected to reversing keys Kj and Kj, variable resist- 
 ances Ri and R,, ammeters Aj and A2, and batteries Bj and 
 B2, the test-coil T being connected by a pair of long leads 
 to the ballistic galvanometer BG. 
 
 The method of working is as follows. A current is always 
 kept flowing in coil Sj of sufficient strength to produce the 
 
Ballistic Method. 
 
 287 
 
 maximum amount of magnetization desired in one direction ; 
 if now, the current in Sj being kept on, a current of the same 
 strength is sent through the coil S^, since the number of spirals 
 on Sa is twice that on Sj, and the direction of winding is 
 opposite to that on Sj, the iron will be magnetized to the same 
 extent but in the opposite direction, so that by keeping a given 
 current flowing in Si'the iron may be carried through a com- 
 
 B, 
 -I'M' 
 
 t I'I'I'I' <^ 
 
 III 
 
 www^-(7)- 
 Rz A, 
 
 BG 
 
 Fig. 126. 
 
 plete cycle of magnetization by simply starting and stopping an 
 equal current in S^. This is one of the principal features of 
 the method, the specimen being [carried through a complete 
 cycle of magnetization between each pair of readings. 
 
 In order to obtain data for the BH curve, currents of various 
 strengths are started in Sa and the corresponding ballistic throw 
 noted, the cyclic process being carried through between each 
 pair of readings. The falling curve of magnetization is ob- 
 tained in the same way by reversing the currents in both coils. 
 
 It will also be seen that any reading on the curve can at 
 once be reproduced. The rings employed in the measurement 
 were about 13 cm. diameter and i sq. cm. cross-sectional area. 
 The ballistic galvanometer was calibrated by means of a current 
 inductor. 
 
 307. The following examples of the method are taken from 
 their paper : — 
 
288 
 
 Practical Electricity and Magnetism. 
 
 Wrought Iron Annealed. 
 
 Rising curve. 
 
 Falling 
 
 curve. 
 
 H 
 
 B 
 
 H 
 
 B 
 
 -IS7S 
 
 — 12500 
 
 22-0 
 
 12900 
 
 -14-5 
 
 -12350 
 
 21-5 
 
 12900 
 
 -II-4 
 
 -I180O 
 
 18-4 
 
 12700 
 
 -9'3 
 
 — 1 1400 
 
 16-95 
 
 12600 
 
 ~7'9 
 
 — 1 1000 
 
 14-9 
 
 12300 
 
 — S'8 
 
 — 10400 
 
 13-1 
 
 I2I0O 
 
 -4'4 
 
 — 9800 
 
 12-8 
 
 12000 
 
 0-35 
 
 -7200 
 
 II-4 
 
 1 1800 
 
 I'D 
 
 -6600 
 
 87s 
 
 1 1 200 
 
 I '4 
 
 -5900 
 
 7-35 
 
 10800 
 
 1-9 
 
 — 5000 
 
 5-2 
 
 lOOOO 
 
 2-S 
 
 -3400 
 
 37 
 
 9^00 
 
 2-6 
 
 — 2900 
 
 07 
 
 7900 
 
 3-3 
 
 -250 
 
 0-2 
 
 7600 
 
 4-0 
 
 1800 
 
 -07 
 
 7000 
 
 47 
 
 3400 
 
 -37 
 
 -250 
 
 f3 
 
 5800 
 
 -4-0 
 
 — 1400 
 
 8-2 
 
 7900 
 
 -s-i 
 
 -4800 
 
 10-3 
 
 9100 
 
 -7-2 
 
 — 7200 
 
 I4'0 
 
 logoo 
 
 — 103 
 
 -9400 
 
 19-3 
 
 12400 
 
 -"5 
 
 -9800 
 
 For curve, see Fig. 127, 
 
 
 
 
 . -3 
 
 *• 
 
 
 ISOOO 
 
 B 
 
 A 
 
 y 
 
 
 
 8000 
 
 1 
 
 
 
 
 
 7 
 
 \ 
 
 
 
 -!0 
 
 / 
 
 "JH 
 
 JO 
 
 20 
 
 
 A 
 
 r 
 
 
 
 J' 
 
 y 
 
 -8000 
 
 
 
 ^ 
 
 
 -IZOOO 
 
 
 
 1^'1G.,I27. 
 
The Magnetization of Iron. 
 
 289 
 
 308. Magnet Bar Iron. 
 Average length, 35 cm. ; sectional area, 0'877 sq. cm. 
 
 Rising 
 
 curve. 
 
 Falling 
 
 curve. 
 
 H 
 
 B 
 
 H 
 
 B 
 
 -39-S 
 
 -15550 
 
 48-3 
 
 15800 
 
 — 29-6 
 
 — 15100 
 
 3|1 
 
 15400 
 
 -I97S 
 
 -14500 
 
 28-6 
 
 15050 
 
 -9-9 
 
 -13400 
 
 19-75 
 
 14400 
 
 -0-4 
 
 -775° 
 
 9-9 
 
 13200 
 
 i-o 
 
 -3700 
 
 -0-6 
 
 — 6900 
 
 3-0 
 
 4400 
 
 — 2-0 
 
 — 2000 
 
 99 
 
 11800 
 
 -4-9 
 
 — 8900 
 
 207 
 
 14000 
 
 -9-9 
 
 — 12150 
 
 29-6 
 
 15000 
 
 -29-6 
 
 - 14950 
 
 39S 
 
 15600 
 
 -39-S 
 
 -15550 
 
 48-3 
 
 15800 
 
 
 
 For curve, see Fig. 128. 
 
 
 16000 
 
 ^ 
 
 1^ 
 
 
 
 12000 
 
 B 
 
 '1 
 
 
 
 
 soaa 
 
 
 
 
 
 ^000 
 
 
 
 
 fo 
 
 zo 
 
 -woo 
 
 20 
 
 40 
 
 
 
 -8000 
 
 
 
 
 J 
 
 -JZOOO 
 
 
 
 
 / 
 
 -16000 
 
 
 
 Fig. 128. 
 
290 Practical Electricity and Magnetism. 
 
 . 309. Cast Iron. 
 
 " ' '\ ♦ ■ 
 
 . " *■ Average length, 32"6 cm. ; sectional area, 0'8S cm. 
 
 Rising 
 
 curve. 
 
 Falling 
 
 curve. 
 
 H 
 
 B 
 
 H 
 
 B 
 
 -32-3 
 
 -5400 
 
 437 
 
 5700 
 
 -24-5 
 
 -5100 
 
 33-2 
 
 5300 
 
 -157 
 
 —4600 
 
 24-9 
 
 4750 
 
 -6-6 
 
 -3970 
 
 i6-6 
 
 4300 
 
 -17 
 
 -3360 
 
 8-3 
 
 3660 
 
 3'3 
 
 — 2200 
 
 O'O 
 
 2550 
 
 7-5 
 
 500 
 
 -8-3 
 
 — 1000 
 
 10-4 
 
 2000 
 
 -i6-6 
 
 -3400 
 
 157 
 
 3400 
 
 -24-9 
 
 -4500 
 
 207 
 
 4100 
 
 -33 '2 
 
 -S160 
 
 28-8 
 
 5000 
 
 
 — 
 
 37'3 
 
 5440 
 
 — 
 
 — 
 
 43-5 
 
 5700 
 
 
 
 For curve, see-Fig. 129. 
 
 
 6000 
 
 ^ 
 
 y 
 
 ^ 
 
 
 4000 
 
 B 
 
 /7 
 
 
 
 
 zoooj 
 
 / 
 
 / 
 
 
 
 to 
 
 7 
 
 "Iw 
 
 20 
 
 40 
 
 
 y 
 
 -400O 
 
 
 
 J- 
 
 y 
 
 -600O 
 
 
 
 Fig. 129. 
 
 310. Hopkinsorls Method. — One of the drawbacks to the ring 
 method of measuring the magnetic properties of iron, is that a 
 ring of each different sample must be carefully overwound with 
 a magnetizing coil, this being somewhat laborious when many 
 
Measjtremeni of Perrkeability — Hopkinson's Method. 29 1 
 
 samples are to be examined. This drawback is to some extent 
 overcome in the ballistic method due to Dr. J. Hopkinson,' 
 and usually known as the divided-bar method. 
 
 The apparatus consists of a block of soft annealed wrought 
 iron, 18'' long by 6^" wide and 2" thick, out of the middle of 
 which a rectangular groove is cut, into which are placed the 
 magnetizing coils, The sample to be tested is in the form of a ' 
 rod, i"265 cm. diameter, carefully turned to slide through the 
 holes at the ends of the block, and cut in two at the centre, the 
 ends being faced up true where they meet in the space between 
 the two magnetizing coils (see Fig. 130). To one end of 
 
 Maanetizinf Coil 
 Test Coil 
 
 Mafnetizina Coil 
 
 Fig. 130. 
 
 one of the rods a collar, C, is attached, in order to prevent it 
 from slipping through the hole ; to the other end, which passes 
 through the hole at the opposite end of the wrought iron 
 yoke, a ring to which weights can be fixed is attached. The 
 test-coil encircles the bars at the point where they butt up 
 against one another, and is held by means of stretched india- 
 rubber bands. In order to make a measurement, a magne- 
 tizing current of known strength is sent through the coils ; 
 it is then broken and the rods pulled apart by the weight W ; as 
 ' Phil. Trans., 1885. 
 
292 Practical Electricity and Magnetism. 
 
 the rod slips out of the test-coil, the indiarubber bands pull the 
 latter right out of the rectangular space, thus causing it to cut 
 all the lines of force in the iron bar. The reading on the 
 ballistic galvanometer is then noted, and the process repeated 
 for various magnetizing currents. In this way the entire cycle 
 tnay be gone through, and various specimens examined rapidly. 
 The object of having a heavy wrought-iron yoke or rectangle, 
 is to provide a return path for the lines of force, so that the 
 magnetic circuit is closed, and there are no free poles and 
 consequent demagnetizing effect. The magnetic circuit is not 
 so good, however, as in the ring form, on account of the 
 various joints in it. 
 
 311. The two following tables are taken from Hopkinson's 
 results. 
 
 Annealed Wrought Iron. 
 
 H 
 
 B 
 
 V- 
 
 1-66 
 
 5000 
 
 3000 
 
 4-0 
 
 9000 
 
 2250 
 
 5-0 
 
 lOOOO 
 
 2000 
 
 ^■5 
 
 1 1000 
 
 1692 
 
 8-5 
 
 12000 
 
 I4I2 
 
 I2-0 
 
 13000 
 
 1083 
 
 17-0 
 
 14000 
 
 823 
 
 28-s 
 
 15000 
 
 526 
 
 50-0 
 
 16000 
 
 320 
 
 105-0 
 
 17000 
 
 161 
 
 200"0 
 
 18000 
 
 90 
 
 350-0 
 
 19000 
 
 54 
 
 666-0 
 
 20000 
 
 30 
 
 Grey Cast Iron. 
 
 H 
 
 B 
 
 J* 
 
 5 
 
 4000 
 
 800 
 
 10 
 
 5000 
 
 500 
 
 21-5 
 
 6000 
 
 279 
 
 42- 
 
 7000 
 
 133 
 
 80 
 
 8000 
 
 100 
 
 127 
 
 9000 
 
 71 
 
 188 
 
 lOOOO 
 
 53 
 
 292 
 
 1 1000 
 
 37 
 
Traction Method of measuring Permeability. 293 
 
 (3) Traction Method of measuring Permeability. 
 
 312. The last method of measuring the magnetic quality of 
 a sample of iron to be described is the traction method, in 
 which the induction is calculated from the force of attraction 
 between two magnets. If A represents the area of contact of 
 the magnets in square centimetres, and B the magnetic induction 
 in lines per square centimetre, it can be shown^ that the force 
 in dynes required to separate them is — 
 
 W (dynes) = —— 
 
 OTT 
 
 B^A 
 or W (grammes) 
 
 981 X 8 X TT 
 
 Hence 
 
 / W X 981 X Stt 
 
 313. Measurements of induction by this method have been 
 carried out by Bidwell ^ both on divided bars and rings. The 
 ring was made of very soft charcoal iron rod, o'i>\ cm. thick, 
 the joint being carefully welded. It was then turned in a lathe 
 to a uniform circular section, and when finished its external 
 diameter was 8 cm. and its diameter of cross-section 0-482 cm. 
 The ring was sawn into two equal portions, and the cut faces 
 ground so as to be perfectly flat. Two brass tubes, o'5 cm. long, 
 were fitted over the ends so that they projected i mm. beyond 
 the faces, in order to serve as guides in placing the two parts of 
 the ring together. Each half ring was then overwound with 
 magnetizing coils, the total number of turns for the whole ring 
 being 1929. The mean radius of the ring was 3'76 cm., and 
 the mean circumference 23'6 cm. The upper half of the ring 
 was rigidly fixed, while to the lower part a scale-pan was 
 attached, the apparatus being connected as shown in Fig. 131. 
 The battery B is connected in series with the magnetizing coils 
 on the ring, the galvanometer or ammeter G, variable resistance 
 R, and key K. The weights required in the scale-pan in order 
 
 ' " Mathematical Theory of Electricity and Magnetism " (J. J. 
 Thomson), p. 73. 
 ' Pro. Roy. Soc, vol. xl. 
 
294 Practical Electricity and Magnetism. 
 
 to separate the two portions of the ring when various mag- 
 netizing currents are flowing in the coils, are noted, and the 
 corresponding vakie of B calculated as above. 
 
 'W\NW\AN^ 
 
 Fig. 131. 
 
 314. The following data were obtained for the above 
 specimen : — 
 
 Grammes 
 
 
 
 
 per sq. cm. 
 
 H 
 
 B 
 
 M 
 
 area. 
 
 - 
 
 
 
 3210 
 
 3-9 
 
 7390 
 
 1899-1 
 
 5400 
 
 10-3 
 
 11550 
 
 II2I-4 
 
 9680 
 
 40-0 
 
 15460 
 
 386-4 
 
 I2170 
 
 iiS'o 
 
 17330 
 
 150-7 
 
 1 38 10 
 
 208 
 
 18470 
 
 88-8 
 
 15130 
 
 427-0 
 
 19330 
 
 45-3 
 
 15905 
 
 585-0 
 
 19820 
 
 33-9 
 
 Magnetic Hysteresis. 
 
 315. In plotting out the rising and falling curves of 
 magnetic induction in iron and steel, it will have been observed 
 that the falling curve does not coincide with the rising one, 
 and when a sample of iron is carried through a complete 
 cycle of magnetization, the curves connecting the values of B and 
 H enclose an S-shaped space. This phenomena of the lagging 
 of the induction behind the magnetizing force has received 
 from Professor Ewing the name of hysteresis. One important 
 consequence of magnetic hysteresis is that it involves a dissipa- 
 tion of energy, or, in other words, a certain amount of energy is 
 
Measurement of Magnetic Hysteresis. 295 
 
 absorbed by the sample under test during a cyclic magnetiza- 
 tion and goes to raise its temperature by a very small amount. 
 The quantity of energy so absorbed can be calculated from the 
 cyclic curves connecting the magnetizing force H with the 
 intensity of magnetisation I and also from the BH cyclic curves. 
 
 Suppose that in any experiment I is increased by a very 
 small amount, dl, the work done per unit volume of the material 
 is that required to bring unit volume of the material with a 
 magnetization dl fronti an infinite distance to that point. This 
 work equals the product dl into the mean value of the force H, 
 that is, dw = 'E.dl. Now, if in Fig. 132 SP 
 represents the value of d\. on the magneti- 
 zation curve OP, then the area OPQ 
 
 represents the value H</I or the work 
 
 done in increasing the magnetization by 
 
 an amount d\. If the falling curve of 
 
 magnetization coincided with the curve 
 
 OP, then all this work would be recovered, _, 
 
 and there would be no hysteresis loss. 
 
 But the falling curve follows the path PR, hence the amount of 
 
 work recovered is only the area RPQ. Therefore the difference 
 
 (area OPQ — area RPQ) equals area OPR, and is the measure 
 
 of the energy absorbed by the specimen. 
 
 If instead of a small part of the curve OP we take the com- 
 plete cycle, then the area enclosed by it represents the work 
 done during the magnetization, and by plotting the curves of H 
 against I, both being expressed in C.G.S. units, the area gives us 
 the work done in ergs per cubic centimetre on the material. 
 
 In order to calculate the work done during the cycle from 
 the BH curve, we employ the relation — 
 
 B = H 4- 41-1 
 Hence rfB = ^H + 4WI 
 and H^B = H^H + 47rH^I 
 
 ^ ^ HrfB - H^H 
 
 therefore Hal = 
 
 4ir 
 
 But for a cyclic process fUdB. vanishes ; 
 
296 Practical Electricity and Magnetism. 
 
 therefore Hrfl = — H^B 
 J 4- J 
 
 and the work done per cubic centimetre on the material is — 
 
 times the area of the BH curve. In general, in plotting the 
 
 curve the scale of B or I will be many times that of H, thus in 
 
 Fig. 128 B is 200 times the scale of H, so that the value of one 
 
 . 2 X 200 400 100 , .„ ^, 
 
 small square is = — = — ergs, and it there are n 
 
 4^ 4^ '^ 
 
 small squares enclosed by the curve — n = ergs per cubic 
 
 TT 
 
 centimetre absorbed by the iron. 
 Thus in Fig. 128 the value 
 
 «'/ 
 
 H(/Ior— I H<^Bis 16,300. 
 
 47r 
 
 ergs per cubic centimetre. 
 
 316. The following table, taken from Ewing's researches, 
 give the approximate hysteresis loss for various materials per 
 complete cycle of magnetization : — 
 
 Material. 
 
 Loss in ergs per c.c. 
 
 Iron, soft annealed 
 
 ... lOOOO 
 
 Iron, annealed wrought 
 
 16000 
 
 Iron, cast 
 
 
 
 30000 to 40000 
 
 Mild steel 
 
 
 40000 to 60000 
 
 Pianoforte steel wire, annealed 
 
 
 94000 
 
 Pianoforte steel wire, commercial 
 
 
 1 16000 
 
 Pianoforte steel wire, glass-hard 
 
 
 1 17000 
 
 Chrome steel (i "/„ Cr.), annealed 
 
 
 65000 
 
 Chrome steel (i 7,, Cr.), oil-hardened ... 
 
 
 167000 
 
 Tungsten steel (3-4 "/„ Tungsten, 0-5 "/„ C, 
 
 o-6«/oMn'.j 
 
 216800 
 
 Nickel, annealed 
 
 
 1 1000 
 
 Nickel, hardened 
 
 
 25000 
 
 Effects of Temperature on Magnets. 
 
 317. In permanent steel magnets the alteration in tempera- 
 ture produces a very slight diminution in the magnetic moment 
 of the magnet. This alteration may be determined experi- 
 mentally, and expressed as a temperature coefficient. Thus 
 
Temperature Coefficient of a Magnet. 297 
 
 calling M, the magnetic moment at a temperature t° C, Mo the 
 magnetic moment at 0°, and /8 the temperature coefRcient of 
 the magnet, we may write — 
 
 M, = M„ (i - ^f) 
 In order to determine the value of /3 for a particular 
 magnet, it should be placed in a vessel of oil or water, the 
 temperature of which may be altered by heating. A magneto- 
 meter needle is then arranged near it, so as to be in one of the 
 
 M 
 Gauss positions, say the A tangent position, then ■^= d^ tan a, 
 
 where M is the magnetic moment, H the controlling force at 
 the needle, d the distance of the magnet from the needle, and 
 a the angle of deflection of the needle. The temperature 
 of the vessel must be steadied for some time before the 
 reading is taken, in order to allow the magnet to acquire the 
 temperature of the liquid, which may be taken by a thermo- 
 meter. The deflections for various temperatures are taken, 
 and the coefficient calculated from them. If aj and a^ repre- 
 sent two deflections for temperatures t° C. and t.° C. then — ■ 
 
 M^j oc tan a-y and M^^ °^ tan a^ 
 
 therefore -. -'- = ik-} 
 
 tan 02 I — pti 
 
 . „ tan oo — tan a, 
 
 and y8 = -5 ; 75 
 
 ' tan o^t-y — tan 0.^1^ 
 
 In good steel magnets /? varies from 0-0003 to o-ooi. 
 
 318. The effect of temperature on the permeability of soft 
 iron and steel is very interesting, and depends to a certain 
 extent on the magnetizing force acting on the material. In 
 order to investigate this, a ring of the material should be made 
 and wound with magnetizing and secondary coils, which should 
 be carefully insulated with asbestos paper, since the tempera- 
 tures to which the specimen is to be carried would destroy 
 ordinary insulation. The ring is then placed in an iron box, 
 which can be heated by a gas furnace, the temperature 
 of the ring being estimated from the resistance of the secondary 
 coil, which is wound completely round the ring and should 
 
298 
 
 Practical Electricity and Magnetism. 
 
 be next the specimen, the magnetizing coil being wound 
 over it. The temperature coefficient of the wire forming the 
 secondary or test-coil should be previously determined by a 
 separate experiment. The BH curve is then taken by the 
 ballistic method, but before and after each reading on the 
 ballistic galvanometer the resistance of the secondary coil is 
 taken on a Wheatstone bridge, and the temperature calcu- 
 lated from the mean resistance, and the known temperature 
 coefficient. Curves should be taken for both strong and weak 
 magnetizing fields at various temperatures, since the variation 
 of the permeability greatly depends on this. 
 
 319. The following tables, taken from curves given by 
 Hopkinson, will give an idea of the variation of /«, in the case 
 of soft iron : — 
 
 Magnetizing Force, 0-3 C.G.S. Unit. 
 
 Temperature. 
 
 (* 
 
 20° C. 
 
 500 
 
 480° 
 
 700 
 
 580° 
 
 900 
 
 730° 
 
 2500 
 
 750° 
 
 3800 
 
 770° 
 
 7700 
 
 775° 
 
 1 1000 
 
 785° 
 
 I 
 
 320. Magnetizing Force, 45 C.G.S. Units. 
 
 Temperature. 
 
 20° c. 
 
 300 
 
 370° 
 
 300 
 
 570' 
 
 260 
 
 670- 
 
 250 
 
 720 
 
 240 
 
 7SOo 
 
 200 
 
 78S 
 
 I 
 
 321. In strong magnetic fields the value of fx. steadily de- 
 creases as the temperature increases, and then drops suddenly 
 
Effect of Temperature on Permeability. 299 
 
 to unity at 785° C. ; whilst in the weak fields it first increases 
 slowly, then very suddenly, until it reaches the value 11,000, 
 when it drops as suddenly again to unity at 785° C. At this 
 temperature the iron becomes non-magnetic. In steel bars in 
 weak and strong fields, the behaviour is similar, the tempera- 
 ture at which the steel loses its magnetic properties being about 
 700° C. 
 
 322. When the bar is allowed to cool down it is found to 
 regain its magnetic properties at about the above temperatures. 
 A most interesting physical change also occurs in the iron at 
 this temperature, which was first observed by Professor Barrett,i 
 and investigated by Dr. Hopkinson.^ If a piece of iron is 
 heated to a temperature above 785" C, and then allowed 
 slowly to cool, it is observed to get perceptibly hotter and 
 brighter red at a temperature about 785° C, this being obviously 
 due to a disengagement of heat at this temperature. This 
 phenomenon has received the name of recalescence. 
 
 In order to investigate it, a rod of steel is over-wound with 
 an asbestos insulated coil of copper wire, and then jacketed 
 with a number of layers of asbestos paper. The copper coil, 
 the temperature variation of resistance of which should be 
 accurately known, is connected up to a Wheatstone bridge and 
 its resistance measured. The steel bar is then heated in a gas 
 furnace to bright redness and allowed to cool slowly, time- 
 measurements of the resistance of the wire being taken; on 
 plotting a curve of temperature of the bar, as calculated from 
 the resistance of the copper coil, against time as abscissae, a 
 curve of the shape shown in Fig. 133 is obtained, the tempera- 
 ture faUing to 680° C. and then rising to 712° C, when it again 
 falls. From this curve, also, the amount of heat disengaged 
 may be estimated by finding the time corresponding to the 
 distance between the two straight parts of the curve ; during 
 this time the steel bar has been giving out heat without fall of 
 temperature, and, comparing this with the time-rate of fall of 
 temperature of the curve after the bend, we can therefore esti- 
 mate the relative quantities of heat given out in the two cases, 
 
 ' Phil. Mag; vol. xv., April, 18S3. 
 ' Roy. Soc. Pro., March, 1 889. 
 
300 Practical Electricity and Magnetism. 
 
 and if the latter quantity can be measured, knowing the mass 
 of the metal and its specific heat at that temperature, the 
 quantity disengaged during recalescence may be calculated. 
 Thus Hopkinson showed that in a case where the rate of 
 
 Temp 
 
 Time 
 
 Fig. 133. 
 
 fall of temperature was 0*2 1° C. per second after the bend, the 
 quantity of heat disengaged was 173 times the heat liberated 
 during a fall of 1° C. 
 
 Self and Mutual Induction. 
 
 323. When a current of electricity is started in a circuit, it 
 does not at once reach its steady value, but gradually increases 
 from zero to its maximum value, the time required for the 
 growth of the current depending on the nature of the circuit. 
 When a current is established in a circuit it sets up a magnetic 
 field, but this magnetic field, according to Lenz's law, reacts 
 inductively on the circuit during the time it is being set up, and 
 induces a back E.M.F., which tends to stop the current. This 
 is known as the back E.M.F. of self-induction, and its magni- 
 tude depends on the strength of the magnetic field set up by 
 the given current, it therefore depends on the magnetic 
 permeability of the field round the circuit, and will be greater, 
 the larger the value of /a. 
 
 In the case of a coil of wire consisting of n turns carrying a 
 current C and producing a total flux of N lines of force, the 
 total cutting of lines by the coil itself when the current is 
 started in it, assuming all the lines to cut the coil, will be N«, 
 
Measurement of L — Rayleighs Method. 301 
 
 and the ratio of this to the current C is defined as the coefficient 
 of self-induction (L) of the coil, or the " inductance." 
 
 The coefficient of self-induction of a circuit may also be 
 defined as the ratio of the back E.M.F. to the time rate of change 
 of the current in the coil, or as the electro-kinetic energy 
 possessed by the current. All these definitions lead to the 
 same result when the permeability of the circuit is constant ; 
 but when ^i. varies with different values of B, as in the case of 
 iron, they lead to different results. The following methods 
 to be described of measuring the coefficient of self-induction 
 are based on the first definition. 
 
 324. The practical unit of self-induction is called the 
 " secohm," or sometimes the " henry," and corresponds to 
 lo^C.G.S. units, i.e. to the cutting of 10^ magnetic lines when 
 unit current (10 ampbres) is sent through the circuit. 
 
 If two coils are placed close together, and a current is 
 started in one, the magnetic field set up will induce an 
 E.M.F. in the other, the total number of lines cut by the second 
 coil, when unit current is started in the first, is defined as the 
 coefficient of mutual induction (M) between them. This 
 quantity, like the other, also depends on the nature of the cir- 
 cuits and on the magnetic permeability of the medium between 
 them. 
 
 325. Lo7-d RayldgKs Method of measuring L. — In the follow- 
 ing method, due to Lord Rayleigh,^ the coil whose coefficient of 
 self-induction it is required to measure, is connected up to three 
 other coils which are non-inductive, so as to form a Wheatstone 
 bridge (see Fig. 134). A, B, and S represent the three non- 
 inductive resistances, R the resistance whose self-induction is 
 required, BG a ballistic galvanometer, Bj a battery, and K and 
 Ki break-circuit keys. 
 
 The resistances are adjusted so as to give a balance for 
 steady currents, i.e. so that on closing the battery key and then 
 closing the galvanometer key no deflection is obtained on BG. 
 Great care must be taken to get this balance exact ; the final 
 adjustment may be made by slipping a bare wire under the 
 terminal at the junction of R and B until balance is obtained. 
 ' Trans. Roy. Soc, 1882. 
 
302 
 
 Practical Electricity and Magnetism. 
 
 If now the galvanometer circuit is kept closed and the battery 
 circuit is opened, a ballistic swing, 6, will be obtained on BG, 
 due to the E.M.F. of self-induction in the coil R. 
 
 Fig. 134. 
 
 The quantity of electricity so discharged may be calculated 
 as follows. An E.M.F. of value e in the coil will produce a 
 current equal to, say, m, but — 
 
 dt 
 
 where c = current in the Qoil. The current g in the galvano- 
 meter will therefore be — 
 
 ■^ dt 
 
 and the total quantity of electricity, Q, discharged through the 
 galvanometer is — _ 
 
 Q.=Jgdt 
 
 d{L,c)dt 
 IT 
 
 = - kLC 
 
 where C = value of c when it has reached a steady value. But 
 from the theory of the ballistic galvanometer (see par, 224)— 
 
Measurement of L — Rayleigh's Method. 303 
 
 HT . e 
 
 —^ 2 sin - 
 27rG 2 
 
 (. + 9 
 
 „ _ HT . «/ , M 
 
 Hence kLC =^-/^ 2 sin -( i + - ) 
 
 2 V 
 
 HT . e 
 and L = — >,-;=, 2 sin - 
 
 27rLrKC 2 
 
 (.+^) 
 
 The steady current-balance of the bridge is now upset by 
 inserting in series with the coil R a very small resistance of 
 known value, ;-. This will produce a steady deflection, S, on 
 the galvanometer, due to a small current, g', flowing through it. 
 Also, on account of the smallness of r, we may assume that the 
 current flowing in the coil R is still C. 
 
 Hence g' = kiC 
 
 XT 
 
 But g' = y^ tan 8 
 
 where H and G have the same meanings as in the ballistic 
 galvanometer formula — 
 
 therefore wC = 7^- tan 8 
 
 , ^ H tan 8 
 and kC = — 7^ 
 
 putting this value for kC in the equation for L, we get — 
 
 .61/ A.\ 
 
 L = 
 
 T;-2 sin 2 
 
 (■^j) 
 
 tan S 
 
 If the periodic time of swing of the needle (T) is taken in 
 seconds, and r in absolute units of resistance, the value of L 
 will be obtained in C.G.S. units. If r is taken in ohms, L 
 will be expressed in henrys. The angles Q and 8 are the 
 angular deflections of the needle, not the spot of light, and 
 the logarithmic decrement is determined as described in 
 par. 227. 
 
 The resistance ;■ may either be a small coil of known resist- 
 ance, or a piece of straight wire of known length, area, and 
 specific resistance. 
 
304 Practical Electricity and Magnetism. 
 
 326. In order to measure the coefficient of self-induction of 
 a coil of wire wound on a bobbin with non-magnetic core, it 
 was connected up as a Wheatstone bridge to three other coils of 
 nearly the same resistance, but wound non-inductively, and a 
 steady current-balance was obtained by slipping the bare end 
 of one of the coils through the terminal until an exact balance 
 was obtained. The galvanometer circuit being closed, the battery 
 circuit was suddenly opened and a ballistic swing was obtained ; 
 the mean of six such swings was 52*4 scale-divisions. A 
 standard o'loo-ohm resistance was now inserted in series 
 with the coil whose self-induction was required, and a steady 
 deflection of 48 scale-divisions was obtained. The periodic 
 time of swing and logarithmic decrement of the galvanometer 
 rieedle were then determined by the methods previously de- 
 scribed, and were T = 20-15 seconds and A = q-ioi respec- 
 tively. The mirror was i metre distant from the scale, which 
 was graduated in millimetres. 
 
 C2'4 
 
 Hence tan 26 = — = o"oi;24. 
 1000 ^ ^ 
 
 and .-. (9 = 3° 
 
 and sm - = o'oi^i 
 2 •* 
 
 also tan 28 = = o'o4.8 
 
 .-. S = 1-27° 
 and tan 8 = 0*0253 
 
 Therefore L = '°''^ ^ °'^°° ^ ^°° x ' ^ °'°'^' ^' + °'°^°^ 
 2 X 3-142 0-0253 
 
 = 3-49X 108 C.G.S. units 
 = 0-349 henrys 
 
 327. Maxwell's Method of measuring L. — In Maxwell's 
 
 method ^ the coefficient of self-induction of a coil is determined 
 
 in terms of the capacity of a condenser. The coil R is, as in 
 
 Lord Rayleigh's method, connected up to three non-inductive 
 
 ' See Maxwell's " Electricity and Magnetism," vol. ii. 
 
Measurement of L — Maxwell's Method. 305 
 
 coils, so as to form a Wheatstone bridge (see Fig. 135). The 
 arms A, B, and S represent the non-inductive resistances, across 
 one of which is 'placed''the condenser of capacity K. A 
 
 Fig. 135. 
 
 balance is first obtained for steady currents, when the following 
 relation holds : — 
 
 RS = AB 
 
 A balance must now be obtained for both steady and tran- 
 sient currents, i.e. no deflection or swing will be obtained on 
 the galvanometer BG, whether the battery key is pressed and 
 then the galvanometer key, or vice versA. There is only one 
 possible arrangement of resistances which will fulfil this con- 
 dition, and therefore the method is an exceedingly tedious one, 
 since the experimenter must go on trying various combinations 
 until he discovers the right one. 
 
 When a balance is obtained for both steady and transient 
 currents, the rate of rise of the potential at each of the galvano- 
 meter terminals must be the same, since there is no current 
 flowing through it. The rate of rise of potential of the ter- 
 minal connected to the condenser is proportional to the time- 
 constant of the condenser, t.e. it is proportional to KS, where K 
 is the capacity of the condenser; also the rate of rise of 
 
 X 
 
30^ Practical Electricity and Magnetism. 
 
 potential at the terminal connected to the coil R is propor- 
 tional to its time-constant, which is — "^ 
 
 Hence - 
 
 R 
 
 KS 
 
 and L = KRS 
 If K, C, and S are all in C.G.S. units, L will be expressed in 
 C.G.S. units. 
 
 This method of Maxwell's, although it leads to a very simple 
 result, is so difificult to carry out in practice, on account of the 
 adjustments required to get the balance, that it is almost use- 
 less in the laboratory, so that the following modifications are 
 recommended. 
 
 328. Rimingtoris Modification of Maxwell's Method of 
 measuring L.^ — The arrangement of the apparatus in this 
 modification is the same as in Maxwell's method, with the 
 
 Fig. 136. 
 
 exception that the condenser, instead pf being connected across 
 the whole of the resistance S, is only connected across a part 
 of it, r (see Fig. 136). The bridge is adjusted so as to balance 
 for steady currents. It is then tested for balance with transient 
 
 ' See Fleming's " Alternate Current Transformer,'' vol. 
 2 P/ul. Mag., vol. xxiv., July, 1887. 
 
 1. p. loi. 
 
Measurement of L — Riinington's Method. ■ 307 
 
 currents by closing the galvanometer circuit before the battery 
 circuit ; should a throw be obtained on the galvanometer, the 
 resistance r must be altered, without altering the total resistance 
 S of the arm, until a balance is obtained for transient, as well 
 as for steady currents. 
 
 If we call the steady value of the current flowing in A and R 
 X, andjc that in B and S, also G the galvanometer resistance, 
 then the quantity of electricity discharged through the 
 galvanometer due to the self-induction of the coil R is — 
 
 ^g' •"■ G;+.A + S 
 
 since AB = SR 
 
 S(B + R) + G(S + B) 
 
 Also, the quantity which passes through the galvanometer due 
 to the discharging of the condenser K is — 
 
 Q = Kv; '- X S + ^ 
 
 A+^+G+B+R 
 
 S(B + R) + G(S + B) 
 But these quantities flow through the galvanometer in opposite 
 directions, and, since there is no ballistic swing, they must be 
 equal to one another. 
 
 La:S Kji>;-'B 
 
 Hence 
 
 S(B + R) + G(S + B) S(B + R) + G(S + B) 
 and LxS = Y^yr'^ 
 Kjr'^B 
 
 or L ; 
 
 ^ y ^ 
 
 But- = ^ 
 X B 
 
 icS 
 
 . . . K;-'R 
 therefore L = ' — — - 
 
 In order to simplify the adjustment, part of the resistance S 
 
3o8 Practical Electricity and Magnetism. 
 
 should include a straight calibrated wire on which a sliding 
 contact from the condenser may make contact. 
 
 In carrying out the method, the values of R and G are 
 usually fixed, and it can be shown mathematically ' that for the 
 most sensitive arrangement the following relation should 
 obtain, viz. — 
 
 RS(G + R)(r + S) 
 (G + S)(^+S) 
 
 329. Sumpner's Method of measuring L. — The connections 
 in Dr. Sumpner's modification of Maxwell's method are shown 
 in Fig. 137. S and B represent non-inductive resistances of 
 
 B^ = 
 
 Fig. 137. 
 
 about 10,000 ohms each ; A is a non-inductive resistance the 
 value of which can be altered ; BG is a ballistic galvanometer, 
 also of about 10,000 ohms resistance ; and K is a condenser of 
 one-third microfarad capacity. 
 
 The resistances are first adjusted until the bridge is balanced 
 for steady currents, the final adjustments being made, as in the 
 other methods, by slipping a piece of bare wire, in series with R, 
 under the terminal until the galvanometer gives no deflection 
 when its circuit is completed afttr the battery circuit. 
 ' Gray's "Absolute Measurements,'.' vol. ii. p. 493. 
 
Measurement of L — Sumpner's Method. 309 
 
 Keeping the galvanometer circuit closed, the battery circuit 
 is now opened and a ballistic throw may be obtained of value 
 ^1. This swing represents a quantity of electricity which is 
 proportional to the difference between the time-constants of the 
 coil and condenser, and is — 
 
 ^ - KS oc e. 
 
 One of the condenser terminals is now disconnected and the 
 above operation repeated, the throw B«, this time being pro- 
 portional to the time-constant of the coil alone, since the 
 condenser is out of circuit ; 
 
 L 
 
 therefore Oi<^ — 
 
 is. 
 
 Hence, from these we get- 
 
 e, L 
 
 R 
 
 and L = KRS 
 
 6,-e, 
 
 This method is very much simpler to carry out than Maxwell's 
 method, and is found to give the most satisfactory results, 
 when — 
 
 B = S, B -t- R = 2G, KRS := 2L 
 
 (G = galvanometer resistance) 
 
 330. It must be borne in mind that in all these measurements 
 of L the value obtained has no definite meaning, if the perme- 
 ability of the medium is not constant, unless the permeability 
 corresponding to the particular conditions under which the test 
 was made is stated. This is well illustrated in the following 
 measurements by Sumpner, of the coefficient of self-induction 
 of an electro-magnet with an iron core for various magnetizing 
 currents. The resistance of A in the following was 5 ohms, 
 and K = ^ microfarad, SB and G being 10,000 ohms each. 
 
3IO Practical Electricity and Magnetism. 
 
 Current in R 
 
 H 
 
 H 
 
 L 
 
 (amperes). 
 
 (henrys) 
 
 0*220 
 
 688 
 
 508-0 
 
 0-0637 
 
 0'200 
 
 627- 
 
 462-0 
 
 0-0634 
 
 0-I47 
 
 444 
 
 321 
 
 0-0606 
 
 « Olio 
 
 367 
 
 261-0 
 
 0-OS77 
 
 0-073 
 
 230 
 
 160-0 
 
 0-0547 
 
 0-055 
 
 164 
 
 III'O 
 
 0-0517 
 
 0'020 
 
 49-5 
 
 29-75 
 
 0-0419 
 
 o-oi6 
 
 38 
 
 22-7 
 
 0-0414 
 
 0-0IC5 
 
 24 
 
 13-5 
 
 0-0381 
 
 0-0092 
 
 20-5 
 
 II-Q 
 
 0-0360 
 
 The value of L when the iron core was removed was 0-0028 henry. 
 
 331. Joubert's Method of measuring Self-Lidnction . — The 
 following modification of Joubert's method of measuring the 
 coefficient of self-induction, due to Professor S. P. Thompson,^ 
 depends on the measurement of the apparent increase of resist- 
 ance of a coil possessing self-induction when traversed by an 
 alternating current. It can easily be shown '' that, when a coil 
 of resistance R ohms (as measured by a steady current on a 
 bridge) is traversed by an alternating current which is a simple 
 sine function of the time, its apparent resistance R' is expressed 
 by the following relation — 
 
 R' = VR^ + i,T^rn} 
 
 where n is the frequency of alternation of the current, and L 
 
 the coefficient of self-induction of the coil; the expression 
 
 VR"^ + A'^ii'iJ' being known as the "impedance '' of the coil. 
 
 In order to produce an alternating current of known frequency. 
 
 Professor Thompson employs a tuning-fork of known pitch as 
 
 the interrupter of the current in the primary of an induction 
 
 coil, the current in the secondary then alternating according 
 
 to a simple sine function. 
 
 The following diagram (Fig. 138) shows the method of 
 
 arranging the apparatus. The coil R is that of which the 
 
 coefficient of self-induction is required, PQ and S are non- 
 
 inductively wound bridge coils. The key Kj is arranged so 
 
 ' Jour. Elect. Eng., vol. xvi. p. 385. 
 
 = "Alternate Current Transformer," Fleming, vol. i. p. 105. 
 
Measurement of L—Jcnberfs Method. 311 
 
 that either the battery B, can be connected to the bridge or 
 the secondary Si of the induction coil. Key K2 connects 
 either the bridge galvanometer G or the telephone T across 
 the other diagonal. The battery B^ is connected in series 
 with the primary coil P and the tuning-fork interrupter F, 
 
 .1 
 
 Fig. 138. 
 
 which may be arranged so as to be vibrated electro-magneti- 
 cally by inserting an electro-magnet in series with P between 
 the prongs of the fork. 
 
 In making a measurement, Kj is connected to the battery B,, 
 and K2 to G, this allows a steady current to flow through the 
 bridge, and by adjusting the coils a balance is obtained in the 
 ordinary way, and the resistance to steady currents calculated. 
 Key Ki is now connected to Sj, and Kg to T, and the current in 
 the induction coil started. A series of alternating currents 
 now flow through the resistances, the frequency of which will 
 be the same as that of the fork F. Balance must now be 
 obtained by adjusting the non-inductive coils until there is a 
 
312 Practical Electricity and Magnetism. 
 
 minimum of sound in the telephone, and the apparent resist- 
 ance, Ri, calculated. Then — 
 
 j^_ 'J^' - R^ 
 
 lirll 
 
 The value of n may be determined experimentally as described 
 in vol. i. p. 103. 
 
 332. The following example will illustrate the method. A 
 solenoid was placed in the bridge, and its resistance to steady 
 currents was found to be 4*06 ohms. The tuning-fork in the 
 primary circuit of the induction coil had a frequency of 100 
 vibrations per second; and the apparent resistance to alter- 
 nating currents, as found by the arrangement of resistances 
 required to give minimum sound in the telephone, was i3'8 
 ohms. 
 
 Hence L = ^ 
 
 2irn 
 
 _ (i3-8f - (4-o6f 
 2 X 3'i42 X 100 
 = 0*0209 henrys 
 
 Comparison of Coefficients of Self-Induction. 
 
 333. If a coil of variable known self-induction is constructed, 
 comparisons of the coefficients of self-induction of coils may be 
 made on the Wheatstone bridge, in much the same way as 
 comparisons of resistances, and the coefficients expressed in 
 terms of that of the standard. 
 
 Thus if coils A and S, whose coefficients of self-induction are 
 to be compared, are connected up to the non-inductive coils Rj 
 and R2 so as to form a Wheatstone bridge (see Fig. 139), and 
 the values of Rj and R2 adjusted so as to give a balance for 
 steady currents, then, by adjusting the variable standard of self- 
 induction S, we can also obtain a balance for transient currents, 
 without altering the steady current balance. When the double 
 balance is obtained, then — 
 
The Secohmmeter. 
 
 313 
 
 where L^ and Lg are the coefficients of self-induction of the 
 coils A and S respectively ; and since Lg is known — 
 
 In order to simplify the test for transient currents, and make it 
 
 Fig. 139. 
 
 more sensitive, Professors Ayrton and Perry have devised an 
 instrument known as a secohmmeter (see Fig. 140), in which 
 
 Fig. 140. 
 
 there are two commutators mounted on the same axis, one of 
 which is placed in the galvanometer, and the other in the 
 
314 
 
 Practical Electricity and Magnetism. 
 
 battery circuit, so that by rotating a handle the galvanometer 
 and battery connections are periodically reversed at rates vary- 
 ing from 200-6000 reversals per minute, the galvanometer 
 terminals being reversed between each of the battery reversals, 
 thus sending the transient current (if any) always in the same 
 direction through the galvanometer, and so producing a steady 
 deflection. 
 
 The resistances A and S are first balanced for steady currents 
 with the secohmmeter at rest. The commutator is then 
 rapidly rotated, and if the coefificients of self-induction of the 
 coils are not also balanced, a steady deflection will be obtained 
 in the galvanometer. The variable standard of self-induction 
 S is now adjusted without altering the resistance of the arm S, 
 until the galvanometer deflection has been reduced to zero, 
 when the self-inductions balance, and — 
 
 L, 
 
 R2 
 
 Fig. 141 represents the connections as applied to an ordinary 
 
 Fig. 141. 
 
 P.O. bridge. BC and GC represent the battery and galvano- 
 meter commutators respectively, ;-, and r., the ratio arms, L, 
 and L2 the coil whose coefficient is to be measured, and the 
 
Standards of Self- Induction. , 3i5 
 
 standard of self-induction. Exact balance for steady currents 
 is in this case obtained by means of a slide wire adjustment. 
 
 Standards of Self-Induction. 
 334. The variable standard of self-induction designed by 
 Ayrton and Perry is shown in Fig. 142, and consists of two 
 
 Fig. 142. 
 
 coils in series, one of which revolves inside the other about a 
 vertical axis. The coils are wound on frames which are parts 
 of spheres, so as to allow the coils to be very close together when 
 they lie in the same plane. When the coils are in the same 
 plane, and the current circulates in the same direction in both, 
 the coefficient of self-induction has its maximum value. By 
 rotating the inner coil through 180°, the coefficient passes 
 through all values from the maximum to zero. 
 
 The coils are made of copper wire, so as to have a low 
 resistance, and on the top of the instrument there is a dial over 
 which moves a pointer connected to the movable coil, which 
 indicates either the angular separation of the coils, or, if desired, 
 the dial may be graduated directly in henrys, the usual range 
 being from o"oo4-o'o4 henry. 
 
3i6 Practical Electricity and Magnetism. 
 
 For use in conjunction with the variable standard, there are 
 supplied boxes of fixed standards (see Fig. 143) which are 
 arranged in much the same way as resistances, and contain 
 inductances of 10, 20, 30, and 40 millihenrys. TRese coils 
 may be placed in series with the variable standard when the 
 
 Fig. 143. 
 
 latter is not sufficiently large. In adding additional coils, care 
 must be taken to rebalance the bridge for steady currents. 
 
 Great care must be exercised, when comparing the coefficients 
 of self-induction of two coils, to place them sufficiently far apart 
 so that they do fiot affect one another, and so produce a mutual 
 induction effect in addition to that of self-induction. 
 
 Comparison of Coefficients of Self-Induction by 
 Sumpner's Method. 
 
 335. A comparison of the coefficients of self-induction of two 
 coils may also be made by a method similar to that employed 
 by Dr. Sumpner, in comparing the self-induction with the 
 capacity of a condenser. 
 
 The two coils, A and B, whose coefficients of self-induction 
 are to be compared, are connected up to the non-inductive 
 coils R and S so as to form a Wheatstone bridge (see Fig. 144). 
 The non-inductive coils R and S are then adjusted so as to 
 give a balance for steady currents. If, keeping the galvano- 
 meter circuit closed, the battery circuit is now opened, a swing, 
 ^1, will be obtained on the ballistic galvanometer EG which is 
 proportional to the difference between the time-constants of 
 the two coils, and — 
 
 L. _ L„ 
 A B 
 
Comparisons of Coefficients of Self-Indtiction 317 
 
 where A and B are the resistances of the coils of self-induction 
 L^ and Lb. 
 
 One of the coils, say B, is now removed, and replaced by a 
 
 ' Fig. 144. 
 
 non-inductive resistance of equal value,, so that the balance for 
 steady currents remains undisturbed, and the above operation 
 is repeated, this time the ballistic swing B^ is proportional to 
 the time-constant of coil A, and — 
 
 oc e. 
 
 and 
 
 A 
 
 
 
 
 
 L/ Lb 
 
 A B 
 
 ^1 
 
 
 
 
 A 
 
 
 
 
 
 BLi - ALb 
 BL^ 
 
 e. 
 e. 
 
 
 
 
 or- 
 
 Lb 
 
 _A 
 ~ B 
 
 ^; 
 
 A 
 
 Or 
 
 
 R 
 
 ~ S 
 
 ^^ 
 
 e 
 
 \ 
 
31^ 
 
 Practical Electricity and Magnetism. 
 
 336. Two coils exactly similar were wound, one on a bar of 
 iron and the other on a rod of wood, the resistances of both 
 were exactly the same, 5 ohms. These coils were then 
 connected up to two coils, R and S, each of 10,000 ohms, and 
 a ballistic swing of 665 scale-divisions was obtained. When the 
 coil wound on the wooden rod was replaced by an inductionless 
 resistance of 5 ohms, a ballistic swing of 700 scale-divisions 
 was obtained. Hence — 
 
 0^ 
 
 La R 
 — = — X ■ 
 
 Lb S ^2 — 6-i, 
 
 La 
 L„ 
 
 1 0000 
 
 X 
 
 700 
 
 loooo 700 — 665 
 
 20 
 
 I 
 
 337. The following table of coefficients of self-induction of 
 common pieces of apparatus has been compiled from data 
 given in a paper on inductance by A. E. Kennelly.^ 
 
 
 Approximate 
 
 Instrument. 
 
 value of L 
 
 
 (henrys). 
 
 Cardew voltmeter 
 
 0-ooooor 
 
 Doubly wound resistance coil 
 
 o-oooooi 
 
 Standard lo-ohm telegraphic relay 
 
 0-2 to O'S 
 
 Mirror-speaking galvanometer, 2250 ohms 
 
 3-6 
 
 An 80-ohm telephone call-bell ... 
 
 I '4 
 
 Bell telephone receiver, 75 ohms 
 
 o'07 to o'l 
 
 Dynamo field magnets 
 
 I to 900 
 
 Dynamo armature 
 
 0-02 to 50 
 
 Primary of alternating current transformer 
 
 0'4 to 40 
 
 Secondary ,, „ „ 
 
 o"ooi to O'l 
 
 Primary small medical coil 
 
 o-ooS 
 
 Secondary,, ,, ,, 
 
 O'lOO 
 
 Primary large induction coil 
 
 0-013 
 
 Secondary,, ,, „ 
 
 20O0-O 
 
 Astatic mirror galvanometer, 5000 ohms 
 
 2-0 
 
 Electric bell, 2-5 ohms 
 
 0-012 
 
 ' Electrician, vol. xxvi. p. 267. 
 
Measurement of Coefficient of Mutual Induction. 319 
 
 Determination of the Coefficient of Mutual Induction 
 OF Two Coils. 
 
 338. The following method of determining the coefficient of 
 mutual induction of two coils is due to Professor Carey Foster.' 
 
 The two coils A and B are connected up, one to a ballistic 
 galvanometer BG (see Fig. 145), and the other to a non- 
 
 ^ — ''"''^ — JzmCSe- ^ 
 
 ^I'l— ^ —T 
 
 B' K 
 
 Fig. 14s. 
 
 inductive resistance R, battery B', and break-circuit key K. 
 On opening or closing the key K, a quantity of electricity, Qi, 
 is induced in the coil B, which will produce a swing of 
 amplitude, 61, on the ballistic galvanometer needle. If y 
 represents the strength of the current in the coil A, M the 
 coefficient of mutual induction, and g the resistance of the coil 
 B and the ballistic galvanometer, then — 
 
 If now a condenser of capacity K is connected in series with 
 the ballistic galvanometer across the ends of resistance R, as 
 
 K BG 
 
 Fig. 146. 
 
 in Fig. 146, then, on closing or opening the key Kj, the 
 ' Phil. Mag., vol. xxiii., February, 1887. 
 
320 
 
 Practical Electricity and Magnetism. 
 
 condenser will be charged or discharged, a quantity of electri- 
 city, Qj, passing through the ballistic galvanometer and pro- 
 ducing a swing of amplitude 0^, and — 
 
 Q, = yRK 
 
 yM 
 
 hence 
 
 Qi 
 0.' 
 
 g 
 yRK 
 
 M 
 ~RK^ 
 
 
 But 
 
 
 . ^1 
 
 sin — 
 
 2 
 
 Sin - 
 
 2 
 
 
 (ar<af*-\Vj: 
 
 » A/r 
 
 RK^ sin 
 
 2 
 
 sm — 
 
 2 
 
 The experimental conditions may be so chosen that B-, = 6.^ 
 when the formula simpUfies to — 
 
 M = RK^ 
 
 In order to do this, and make the method a zero one, the 
 connections are altered to those in Fig. 147. 
 
 HF 
 
 5 
 
 BG 
 
 riT ^ (JMy) (nm 
 
 R. 
 
 K, 
 
 A 
 
 Fig. 147. 
 
 B 
 
 An additional resistance, Rj, is inserted in circuit with the 
 coil B, this being adjusted till, on opening and closing the 
 
Measurement of M — Carey Fosters Method. 321 
 
 key Ki, no swing is observed on the galvanometer needle 
 when the induced quantity of electricity in B just balances the 
 quantity passing in or out of the condenser K. Then — 
 
 M = RK(5- + R,) 
 
 ^ being the resistance of the coil B and galvanometer. 
 
 Should it not be found possible to get a balance for any 
 value of Ri, then the resistance R must be altered and the 
 test repeated. In order to save time when making the test, 
 Ri should first be made infinite, and the direction of the 
 galvanometer swing noted on opening or closing the key Kj, 
 this swing being entirely due to the condenser K ; Rj is then 
 short circuited, and, if the throw is still to the same side, it 
 shows that the quantity of electricity given to the condenser 
 is greater than the maximum induced quantity in B, so that 
 the charging' potential of K must be diminished by diminishing 
 the resistance of R between the condenser terminals. This 
 should be done without altering the total resistance in the 
 circuit of coil A, otherwise the current in the inducing circuit 
 will be altered, and this will alter the quantity of electricity 
 induced in B. The sensitiveness of this test depends on the 
 sensitiveness of the ballistic galvanometer. If the values of 
 R, K, and {g + Ri) are expressed in absolute units, M will be 
 calculated in absolute units. 
 
 339. The following example is taken from Professor Foster's 
 paper. 
 
 A measurement was made of the coefficient of mutual 
 induction between the primary and secondary coils of an 
 Apps 6-inch spark induction coil of the following dimensions : — 
 
 I,ength of secondary coil =21 cm. 
 Outside diameter of secondary coil = ii'3 cm. 
 
 Resistance of primary coil = o'278 ohm (i6'5° C). 
 Resistance of secondary coil = 7394 ohms (i6'5° C). 
 
 The condenser used had a capacity of 4'92 6 microfarads. 
 The following results were obtained : — 
 
322 
 
 Practical Electricity and Magnetism. 
 
 R 
 (ohms). 
 
 (ohms). 
 
 (absolute .units). 
 
 27 
 28 
 29 
 30 
 31 
 32 . 
 
 7394+1550 
 7394 + 125,0 
 7394+ 940 
 7394+ 650 
 7394+ 390 
 7394+ 150 
 
 2-41S X 10" 
 
 2-420 
 2-417 
 
 - 2"4I3 
 2-413 
 2-414 
 
 [ean - = 
 
 2-415 X 10^ 
 
 4-926 X 10-'= 
 1-1896 X 10' 
 
 X 2-415 X 10^ 
 
 Comparison of a Capacity and Coefficient of Mutual 
 Induction. 
 
 340. A comparison may be effected between the capacity of 
 a condenser and the coefiScient of mutual induction between 
 two coils in a similar manner to the comparison of a capacity 
 
 Fig. J48. 
 
 and coefficient of self-induction. One of the coils, A, is 
 connected with three non-inductive coils, P, Q, and R, so as 
 to form a Wheatstone bridge (see Fig. 148). The other coil, 
 
Comparison of M with a Capacity. 323 
 
 B, is connected between the junction of A and P and the 
 battery C. A condenser of known capacity, K, is placed 
 across the end of coil Q. 
 
 The bridge is first balanced for steady currents in the usual 
 ■way, and then, keeping the galvanometer circuit closed, the 
 battery circuit is opened, when a ballistic swing, 0-^, will be 
 obtained \ and — 
 
 (9i oc QJl^ M- OAK 
 Q 
 
 The condenser is now disconnected from the coil Q, and the 
 galvanometer swing again taken ; let this be B^, then — 
 
 • e, ^^M-QAK 
 Hence -^ =^ v 
 
 ^2 Q + R 
 
 Q 
 
 M 
 
 ^ M Q^A 6., 
 
 and — = „ „ X - 
 
 K Q + R B.i-e^ 
 If the capacity of K is known, the value of M may be found. 
 
324 Practical Electricity and Magnetism. 
 
 341. References to Scientific Papers. 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 Experimental Determination of 
 
 T. Gray 
 
 Phil. Mag., vol. 6, 
 
 Magnetic Moments in Absolute 
 
 
 Nov., 1878. 
 
 Measure 
 
 
 
 Complete Theory of Determina- 
 
 Wiedemann 
 
 Ibid., vol. q, June, 
 
 tion of H 
 
 
 1880. 
 
 On the Measurement of H 
 
 T. Gray 
 
 Ibid., vol. 20, Dec, 
 1885. 
 
 On a Galvanometric Method of 
 
 Lehfeldt 
 
 Ibid., vol. 33, Jan., 
 
 determining H 
 
 
 1892. 
 
 A New Form of Magnetometer 
 
 Stroud 
 
 Elect., vol. 25, p. 
 658; Pro. Roy. 
 Soc, vol. 48, p. 
 260. 
 
 U >' >> 
 
 Smith 
 
 Phil Mag., vol. 14, 
 Sept., 1882. 
 
 On a. New Method for the De- 
 
 Kruger 
 
 Ibid.,vo\. 22, Sept., 
 
 termination of the Vertical 
 
 
 1886. 
 
 Compound of the Earth's Mag- 
 
 
 
 netic field 
 
 
 
 Additions to the Kew Magneto- 
 
 Thorpe and 
 
 Ibid., vol. 26, Aug., 
 
 meter 
 
 Riicker 
 
 1888. 
 
 Magnetic Survey of the British 
 
 )» 
 
 Pro. Roy. Soc, vol. 
 
 Isles 
 
 
 4S. P- 546. 
 
 Specific Induction Constants of 
 
 Sack 
 
 Phil. Masr.. vol. 22, 
 
 Magnets in Magnetic Fields of 
 
 
 Oct., 1886. 
 
 Different Strengths 
 
 
 
 Temperature Corrections and In- 
 
 Whipple 
 
 Pro. Roy. Soc, vol. 
 
 duction Coefficients of Magnets 
 
 
 26, p. 218. 
 
 Effects of Percussion and Anneal- 
 
 Brown 
 
 Phil. Mag., vol. 23, 
 
 ing in changing the Magnetic 
 
 
 Mar., 1887, May, 
 
 Moments of Steel Magnets 
 
 
 1887. 
 
 Permanent Magnetism 
 
 Bosanquet 
 
 Ibid., vol. IS, Mar., 
 1883. 
 
 Distance of the Poles of a Magnet 
 
 Hallock and 
 
 Ibid., vol. 18, Oct., 
 
 
 Kohlrausch 
 
 1884. 
 
 Determination of Magnetic Mo- 
 
 Helmholtz 
 
 Ibid., vol. 17, Jan., 
 
 ments by the Balance 
 
 
 1884. 
 
 Experimental Researches in 
 
 Ewing 
 
 Iratu. Ray. Soc, 
 
 Magnetism 
 
 
 1886. 
 
 Magnetic Qualities of Nickel 
 
 ii_ 
 
 Ibid., 1888. 
 
 Magnetic Qualities of Iron 
 
 Evring and 
 Klaassen 
 
 Ibid., 1894. 
 
 On the Magnetization of Iron and 
 
 Ewing and Low 
 
 /bid., 1889. 
 
 other Magnetic Metals in very 
 
 
 
 Strong Fields 
 
 
 
 Magnetization of Iron 
 
 Hopkinson 
 
 /bid., 1885. 
 
References to Scientific Papers. 
 
 32s 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 Magnetic and other Physical 
 
 Hopkinson 
 
 Trans. Roy. Sec. , 
 
 Properties of Iron at High 
 
 
 1889. 
 
 Temperatures 
 
 
 
 Measurement of the Magnetic 
 
 T. Gray 
 
 Ibid., 1893. 
 
 Properties of Iron 
 
 
 
 Effects of Stress on Magnetization 
 
 Sir Wm. Thorn- 
 
 /i;rf.,i876 and 1879. 
 
 Magnetic Induction 
 
 son 
 Tomlinson 
 
 Ibid., 1891. 
 
 Experimental Determination of 
 
 Shida 
 
 Fro. Roy. Soc, vol. 
 
 Magnetic Susceptibility and 
 
 
 34. P- 28s ; vol. 
 
 Maximum Magnetization in 
 
 
 3S> P- 404- 
 
 Absolute Measure 
 
 
 
 Time Lag in the Magnetization of 
 
 Ewing 
 
 Ibid., vol. 46, p. 
 
 Iron 
 
 
 269. 
 
 Magnetic Properties of Alloys of 
 
 Hopkinson 
 
 Ibid., vol. 47, p. 23 ; 
 
 Iron and Nickel 
 
 
 vol. 48, p. I. 
 
 Measurement of Magnetic Per- 
 
 Rowland 
 
 Pliil. Mag., vol. 46, 
 
 meability 
 
 
 1873, p. 140; vol. 
 48, 1874, p. 321. 
 
 Effects of Retentiveness in the 
 
 Ewing 
 
 Ibid., vol. 16, Aug., 
 
 Magnetization of Iron and 
 
 
 i883;Nov.,i883. 
 
 Steel 
 
 
 
 On Electro-Magnets 
 
 Bosanquet 
 
 Ibid., vol. 19, Feb., 
 1885 ; vol. 20, 
 Oct., 1885 ; vol. 
 17, supp., 1884. 
 
 On Magneto-Motive Force 
 
 " 
 
 Ibid., vol. 15, Mar., 
 1883. 
 
 Magnetic Investigations 
 
 Wiedemann 
 
 Ibid., vol. 21., May, 
 1886, vol. 22, 
 July, 1886. 
 
 Behaviour of Iron and Steel 
 
 Rayleigh 
 
 Ibid., vol. 23, Mar,, 
 
 under the Operation of Feeble 
 
 
 1887. 
 
 Magnetic Forces 
 
 
 
 Magnetization of Iron in Strong 
 
 Bidwell 
 
 Ibid., vul. 29, May, 
 
 Fields 
 
 
 1890. 
 
 On Magnetism 
 
 Hopkinson 
 
 Jour. Elect. Eng., 
 vol. 19, p. 10 ; 
 Elect., vol. 24, p. 
 245- 
 
 Testing Iron 
 
 Swinburne 
 
 Elect., vol. 25, p. 
 648. 
 
 Researches in Magneto-Electric 
 
 Vignoles 
 
 Ibid., vol. 27, p. 
 
 Induction 
 
 
 54 
 
 Measurement of the Permeability 
 
 Barton and 
 
 Ibid., vol. 29, p. 
 
 of Magnetite 
 
 Williams 
 
 432. 
 
 Magnetic Research at Low Tem- 
 
 Fleming 
 
 Ibid., vol. 37, p. 
 
 perature 
 
 
 301- 
 
 Permeability Bridge 
 
 Ewing 
 
 /*/</., vol. 37, p. 41. 
 
 Testing Iron and Steel 
 
 ,, 
 
 yfov/., V01.37.P 115. 
 
326 Practical Electricity and Magnetism. 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 I 
 Magnetic Permeability and Hyste- 
 
 Dewar and 
 
 Fro. Roy. Soc, vol. 
 
 resis of Iron at Low Tempera- 
 
 Fleming 
 
 60, p. 81. 
 
 ture 
 
 
 
 Measurement of Hysteresis Loss 
 
 Ewing 
 
 Elect., vol. 34, p. 
 
 in Iron 
 
 
 7^^- . 
 
 Measurement of Hysteresis Loss 
 
 Kennelly 
 
 Ibid., vol. 28, p. 
 
 in Nickel 
 
 
 666. 
 
 Magnetometric Method of 
 
 Moore 
 
 Jbid., vol. 30, p. 
 
 measuring Iron Losses 
 
 
 3*^- , ^ 
 
 Hysteresis Loss in rapidly alter- 
 
 Baily 
 
 Ibid., vol. 36, p. 
 
 nating Fields 
 
 
 116. 
 
 Hysteresis Measurement 
 
 Searle 
 
 Ibid., vol. 36, p. 
 8oo. 
 
 Hysteresis 
 
 Du Bois and 
 
 Jbid., vol. 37, p. 
 
 
 Jones 
 
 595- 
 
 On the Alleged Luminosity of the 
 
 Barratt 
 
 Phil. Mag., vol. 1$, 
 
 Magnetic Field 
 
 
 Apr., 1883. 
 
 Recalescence in Iron 
 
 Tomlinson 
 
 Ibid., vol. 25, Feb., 
 1888. 
 
 Temperature at which Nickel 
 
 J) 
 
 Ibid., vol. 25, May, 
 
 begins to lose its Magnetic 
 
 
 1888. 
 
 Properties 
 
 
 
 Recalescence in Steel 
 
 Newall 
 
 Ibid., -vol. 25, June, 
 1888. 
 
 Magnelization in Strong Fields at 
 
 Du Bois 
 
 Ibid., vol. 29, Apr., 
 
 Different Temperatures 
 
 
 1890. 
 
 Method of measuring the Re- 
 
 Smith 
 
 Ibid., vol. 31, May, 
 
 calescence in Iron and Sieel 
 
 
 1891. 
 
 Recalescence in Iron 
 
 Hopkinson 
 
 Pro. Roy. Soc, vol. 
 45. P- 4SS- 
 
 Magnetism and Recalescence 
 
 )j 
 
 7ij(/., vol. 48, p. 442. 
 
 Recalescence 
 
 j» 
 
 £/i?rf.,vol.23,p.63. 
 
 On the Changes produced by 
 
 Bidwell 
 
 Trans. Roy. Soc, 
 
 Magnetization in the Dimen- 
 
 
 1888. 
 
 sions of Rings and Rods of Iron 
 
 
 
 Hysteresis attending the Change 
 
 Nagaoka 
 
 PMl. Mag., vol. 37, 
 
 of Length of Iron and Nickel 
 
 
 Jan., 1894. 
 
 by Magnetization 
 
 
 
 Villari Critical Points in Iron and 
 
 Tomlinson 
 
 Ibid., vol. 29, May, 
 
 Nickel 
 
 
 1890, vol. 30, 
 Aug., 1890. 
 
 342-— 
 
 
 
 Determination of H by Bifilar 
 
 Kohlrausch 
 
 IVied. Ann., vol. 
 
 Method 
 
 
 17, p. 76s : 1882. 
 
 Determination of the Dip by an 
 
 Weber 
 
 Pogg. Ann,, vol. 
 
 Earth Inductor 
 
 
 43, p. 293 : 1886. 
 
 Determination of Dip by an Earth 
 
 Mascart 
 
 Comples Rendus, yo\ . 
 
 Inductor 
 
 
 97, p. 1 191: 1883. 
 
References to Scientific Papers. 
 
 32? 
 
 Title of Paper. 
 
 Author. 
 
 Reference. 
 
 Poles of a Magnet 
 
 Jamin 
 
 Ibid., vol. 80, p. 
 1553: 1875- 
 
 )j j> 
 
 Bouty 
 
 Jour, de Phys., vol. 
 4, p. 367 : 1875. 
 
 " 
 
 Kohlrausch 
 
 Wied.Ann.,so\.22, 
 p. 411 : 1884. 
 
 Dissipation of Energy in a Mag- 
 
 Warburg 
 
 Ibid., vol. 13, p. 
 
 netic Cycle 
 
 
 141: 1881. 
 
 Modes of Measuring Self and 
 
 Ayrton 
 
 J.E.E., vol. 16, p. 
 
 Mutual Induction 
 
 
 292; Elect., vol. 
 19, p. 17- 
 
 Measurement of Self-induction 
 
 Sumpner 
 
 J.E.E., vol. t6, p. 
 344; Elect., vol. 
 19, p. 127. 
 
 Modification of Maxwell's Method 
 
 Rimington 
 
 Phil. Mag., vol. 24, 
 
 of measuring L 
 
 
 July, 1887. 
 
 Some Methods of Comparing 
 
 Niven 
 
 Ibid., vol. 24, Sept., 
 
 Coefficients of Self and Mutual 
 
 
 1887. 
 
 Induction 
 
 
 
 Variation of the Coefficients of 
 
 Sumpner 
 
 Ibid., vol. 25, June, 
 
 Induction 
 
 
 1888. 
 
 Calculation of the Coefficient of 
 
 Perry 
 
 yfoV/.,voI.30, Sept., 
 
 Self-induction of a Coil 
 
 
 1890. 
 
 Measurements of L 
 
 Steinmetz 
 
 Elect., vol. 26, p. 79. 
 
 Measurement of Inductance 
 
 Kennelly 
 
 Ibid., vol. 26, p. 267. 
 
 Inductance of Common Instru- 
 
 S) 
 
 Ibid., vol. 26, p. 290. 
 
 ments 
 
 
 
 Measurements of the Cofficients 
 
 Russell 
 
 Ibid., vol. 33, p. 5. 
 
 of Induction 
 
 
 
 Determination of the Coefficient 
 
 Rayleigh 
 
 Trans. Roy. Soc, 
 
 of Self-induction 
 
 
 1882; Pro. Roy. 
 Soc, vol. 32, p. 
 116. 
 
 A Method of Measuring the Co- 
 
 Foster 
 
 Phil. Maz.. vol. 23, 
 
 efficient of Mutual Induction 
 
 
 Feb., 1887. 
 
 On the Determination of the Co- 
 
 Bosanquet 
 
 Ibid., vol. 23, Mav, 
 
 efficient of Mutual Induction 
 
 
 1887. 
 
 On Carey Foster's Method of 
 
 Swinburne 
 
 Ibid., vol. 24, July, 
 
 measuring M 
 
 
 1887. 
 
 Calculation of M for a Circle and 
 
 Jones 
 
 Ibid., vol. 27, Jan., 
 
 Coaxial Helix 
 
 
 1889. 
 
 On Coefficients of Induction 
 
 Anderson 
 
 Ibid., vol. 31, April, 
 1891. 
 
VII. 
 
 ELECTRO-MAGNETIC WAVES. 
 
 343. Within recent years the experimental part of this 
 section of the subject has been developed to such an extra- 
 ordinary extent, and its importance in connection with the 
 relation of light to electricity is so great, that we do not 
 consider it necessary to offer any apology for adding a chapter 
 descriptive of experiments on electro-magnetic waves. It will 
 also be found that in general the apparatus required is of such 
 a simple character that the experiments can easily be repeated, 
 provided a little care is exercised in making them. 
 
 344. Before describing the apparatus required for producing 
 and detecting electro-magnetic waves, it will be as well to give 
 a short account of the phenomena itself, and by describing 
 some of the phenomena peculiar to wave-motions in general 
 indicate the nature of the experiments required to establish the 
 wave-motion nature of electro-magnetic effects. 
 
 345. Suppose that an insulated metal sphere is set up at 
 some distance from an uncharged electrophorus, and the sphere 
 suddenly charged; the gold leaves of the electrophorus will 
 diverge, since they are charged by induction, but in separating 
 the leaves and causing them to diverge, work must be done 
 against the force of gravity, which will be given out again when 
 the leaves collapse. Now, the energy necessary to make the 
 leaves diverge must have come from the sphere, and since in 
 any case of the propagation of energy some medium is neces- 
 sary to transmit it, we have to answer the questions, What is 
 this medium in the case above mentioned ? and how does the 
 medium transmit the energy ? 
 
Electro-Magnetic Waves. 329 
 
 346. Again, suppose we set up a coil of wire, and connect it 
 to a battery through a reversing key. At some distance from 
 the coil let us place a pivoted magnetic needle, then when we 
 send a current through the coil it becomes an electro-magnet, 
 and will deflect the needle ; by properly timing the direction of 
 the current in the coil we can cause the needle to spin round 
 on its pivot. Here, again, energy is transmitted across space, 
 namely, from the coil to the magnet, and for the same reason 
 as before we must assume the existence of some medium. In 
 the first experiment mentioned above we were transmitting what 
 was almost entirely an electrostatic effect, and in the second 
 a magnetic effect. Strictly speaking, however, electrostatic 
 and magnetic effects were propagated in both cases — a magnetic 
 effect in charging the sphere, since that involved a flow of 
 electricity for a very short interval of time ; and an electrostatic 
 effect in starting a current in the coil, since its potential was 
 raised slightly above that of the earth. The same two effects 
 are propagated in the charging and discharging of a Leyden jar 
 or other condenser, but instead of considering each separately, 
 the two are combined in the term electro-magnetic effect. 
 
 347. As regards the medium which transmits this electro- 
 magnetic effect, we can prove that it is not atmospheric air, 
 since the effect can be propagated through a vacuum ; a deter- 
 mination of the velocity of propagation of the effect, how^ever, 
 gives us a valuable clue to the nature of the transmitting 
 medium. The velocity of propagation of an electro-magnetic 
 effect has been shown to be the same as the velocity of propa- 
 gation of light, and therefore we might expect that the medium 
 by which light is transmitted — the luminiferous ether — is also 
 that which transmits electro-magnetic effects. The experimental 
 proof of this supposition was the great work of the late 
 Professor Hertz, to whose treatise on "Electric Waves," 
 translated by Professor Jones, and also to the " Work of 
 Hertz," by Professor Lodge, the student is recommended to 
 refer. 
 
 348. Given a supply of energy and the means of transmitting 
 it from one place to another we have next to consider how the 
 transmission is effected. Here we find great assistance by 
 
330 Practical Electricity and Magnetism. 
 
 considering the method of the propagation of sound energy. 
 Sound is propagated in its media — solids, hquids, or gases— by 
 means of waves, and the experimental study of sound is on this 
 account of great importance, apart from its musical interest, 
 since we can study it as a typical case of energy propagation 
 by waves. If, then, waves are a means of transmitting energy 
 in a medium, we must examine our electro-magnetic phenomena 
 for effects which we should expect from a wave-motion. 
 
 349. Firstly, with regard to the propagation of waves in 
 homogeneous media, we find that the transmission is rectilinear, 
 and that the velocity of propagation depends on the medium, 
 and can be calculated, provided we know some of its physical 
 properties. Newton was the first to annunciate the law respect- 
 ing the velocity of propagation, which is — 
 
 ^=\/ 2 
 
 V = velocity of propagation ; e = elasticity of the medium ; 
 d = density of the medium. 
 
 This law has been experimentally proved to hold in the 
 propagation of sound-waves in ordinary matter ; if, however, 
 we wish to apply it to the case of electro-magnetic waves, we 
 must remember that e and d will be the elasticity and density 
 of the ether respectively. Now, elasticity is the reciprocal of 
 pliability, and the pliability of a dielectric, we call its specific 
 
 inductive capacity, K, hence — will represent the elasticity of 
 
 the ether ; also, the density of the ether is what we call perme- 
 ability, //,, and rewriting Newton's law with the above constants 
 we get — 
 
 I 
 
 350. Secondly, when a wave which is being propagated 
 through a homogenous medium arrives at the interface between 
 that medium and another of different density, the following 
 phenomena may occur : — 
 
 1. Reflection. 
 
 2. Transmission. 
 
 3. Absorption. 
 
Calculation of Velocity of Propagation. 331 
 
 The laws which govern the reflection of waves from plane or 
 curved surfaces are well known, and do not require any further 
 mention. 
 
 351. As regards the transmission of waves in the new 
 medium, if the latter is of greater density than the first medium, 
 the velocity of propagation is decreased ; if of smaller density 
 it is increased. The consequence of this alteration of velocity 
 is, that should the wave impinge normally on the surface of the 
 new medium its velocity alone will be affected; should it, 
 however, meet it at some other angle, then both velocity and 
 direction will be altered. This phenomena is usually, in the 
 case of light or sound waves, termed refraction ; the amount 
 of bending for a given ray entering a medium at a given angle 
 depending on the index of refraction of the medium, ;•, which 
 we can define as follows — 
 
 velocity of wave in space 
 
 velocity of wave in medium 
 
 Now, we have just seen how to calculate the velocity of propa- 
 gation of electro-magnetic waves, and the above definition of 
 the refractive index gives us a means of proving that electro- 
 magnetic waves and light waves are propagated by the same 
 t medium, and are very similar. Let K be the S.I.C. of a 
 substance, fx. its permeability, and r its index of refraction for 
 light, then, since K and /x for air are both unity, we have — 
 
 velocity of wave in space •v j. , — 
 ' = V Kii 
 
 velocity of wave in medium ' 
 
 or r"^ = K^ 
 Taking as an example carbon bi-sulphide (CSg), we find that 
 ;-^ = 2"678, and K/x = 2"67, /a being = i, which shows a 
 remarkable agreement. There are, however, many cases in 
 which experiment has failed to show that the above law holds ; 
 this, however, may be due to a difference in degree rather than 
 in kind of the two wave-motions compared. 
 
 When the new medium which the wave meets is of smaller 
 density than the other, and the angle of incidence is greater 
 
332 Practical Electricity and Magnetism. 
 
 than the critical angle, the wave is totally reflected with the 
 loss of half a wave-length. 
 
 352. The third effect that may occur in the new medium is 
 absorption, and we wish more particularly to call attention to 
 one particular kind, namely, selective absorption, in which one 
 particular frequency is absorbed more readily than any other ; 
 just as when a tuning-fork of definite pitch, placed in the 
 vicinity of a second fork of the same pitch which is vibrating, 
 absorbs these vibrations and itself commences to vibrate, 
 exhibiting the phenomena of resonance ; if the forks were of 
 different pitch this would not occur. 
 
 353. One other effect peculiar to wave-motions, and one 
 which is well illustrated in the case of sound-waves, is that of 
 interference, which deals with the resultant effect of two or 
 more waves at a point. If two waves of the same frequency, 
 amplitude, and wave-length meet at a point so that one is half 
 a wave-length in advance of the other, they completely annul 
 one another and produce interference. Thus if waves are 
 transmitted so as to fall normally on a reflecting surface, the 
 outgoing waves meet the reflected waves, and at certain points 
 half a wave-length apart produce interference. In the case of 
 sound-waves this means silence, and by measuring the distance 
 between these points the wave-length can be calculated. 
 
 We have now enumerated some of the more important effects 
 produced by a typical wave-motion, and if we can reproduce 
 these effects in our electro-magnetic phenomena, we will have 
 strong evidence in favour of the view that it also is propagated 
 by a wave-motion. 
 
 Apparatus for producing Electro-Magnetic Waves. 
 
 354. In making experiments with waves of any kind, it is 
 necessary that we should suit our apparatus to the length of 
 the waves with which we are working, or, in cases where we can 
 employ waves of different lengths, that we should choose those 
 lengths which can be most conveniently experimented upon ; 
 thus in sound-waves we should not think of trying interference 
 experiments with waves twenty or thirty feet long but with 
 
Production of Electro- Magnetic Waves. 333 
 
 those of a few inches in length, so also with our electro-magnetic 
 waves, they must be of a convenient length. We could, of 
 course, produce electro-magnetic waves by means of our coil 
 battery and reversing key, but the length would be so great — 
 many miles in this case — that they would be perfectly useless ; 
 the more rapid the alternations of current the shorter the waves 
 would be, but no commutator could be made to alter the 
 direction of the current so rapidly as to produce waves of a few 
 inches in length. 
 
 In order to get a discharge which is alternating with sufficient 
 rapidity to produce waves of useful length, we have to fall back 
 on the discharge of a. Leyden jar. A charged Leyden jar 
 corresponds very closely to a stretched spring, which, if released, 
 oscillates backwards and forwards about its position of rest for 
 some time before it finally stops. If, however, the motion of 
 the spring was damped by placing it in some viscous liquid 
 such as glue, there would be no oscillations, it would simply 
 return slowly to its position of rest. A Leyden jar discharging 
 behaves in the same way if the oscillations are not damped by 
 self-induction, and the frequency of the oscillation produced 
 can be calculated as follows ^ — 
 
 where n = frequency of oscillation in vibrations per sec. ; 
 
 K = capacity of the jar in farads ; 
 
 L = coefficient of self-induction in henrys ; 
 
 R = resistance of discharge circuit in ohms ; 
 the conditions for the production of an oscillating discharge 
 being that — 
 
 R<v'f 
 
 In cases where R is small, the expression for frequency may 
 be written — 
 
 I 
 «= — 1=^ 
 airVLK 
 
 ' " Electricity and Magnetism " (Joubert, Foster, and Atkinson), ars. 
 
 329. 33°- 
 
334 Practical Electricity and Magnetism. 
 
 355. If ordinary Leyden jars are employed for producing 
 the oscillations, it is found that the wave-length is still far too 
 great, being from 150-300 feet with ordinary sized jars. A 
 better arrangement for producing oscillations is that first 
 employed by Hertz, and known as a Hertz oscillator. This 
 consists simply of two brass spheres attached to the ends of 
 two rods which are attached to the secondary of an induction 
 coil, the spark passing between the two brass knobs. In order 
 to give capacity to this, metal spheres or plates are sometimes 
 attached to the other ends of the rods. 
 
 356. The following formula is given by Lodge ^ for the 
 calculation of L in the formula for the frequency of the 
 oscillation : — 
 
 L= 2/log^ 
 a 
 
 where / = length of the entire rod portion of the oscillator in 
 cases where the oscillator consists of metal spheres mounted on 
 the ends of brass rods, to the other ends of which are attached 
 small plates to give capacity, and d = diameter of the rod. In 
 measuring /, it is best to include the knobs and spark-gap, and 
 about a quarter the diameter of each end plate. 
 
 The capacity K is practically half the static capacity of the 
 plate or sphere at either end of the oscillator, since the two 
 capacities are really in series. Thus for two oppositely charged 
 spheres of radius r and distance j- from centre to centre — 
 
 K: 
 
 i '•(■-;) 
 
 357. For isolated bodies, the following values of K are given 
 by Lodge : — 
 
 Condenser. 
 
 Value of K 
 
 Globe 
 
 Thin circular disc 
 Thin square disc 
 Thin oblong disc • 
 
 radius 
 - radius 
 
 0-36 side of square 
 fslightly greater than square 
 \ of same area 
 
 ' Lodge, Phil. Mag., vol. xxviii., July, 1889. 
 
Oscillators. 335 
 
 358. The following example will illustrate the calculation, 
 and applies to an oscillator consisting of two rods each 6 cm. 
 long and i cm. diameter, to which knobs 2 cm. diameter and 
 plates 8 cm. diameter have been attached (see Fig. 149.) The 
 spark-gap is o'S cm. 
 
 00 
 
 Fig. 149. 
 
 Static capacity K = i"4 cm. 
 Self-induction L = 190 cm. 
 
 Log ^ = 4-5 
 
 Rate of vibration n = 300 million per sec. 
 Wave-length X = 100 cm. 
 
 359. A still more convenient form of oscillator, originally due 
 to Professor Lodge, consists of two small brass spheres con- 
 nected to the secondary of an mduction coil, and arranged so 
 as to spark across the diameter of a larger brass sphere placed 
 between them (see Fig. 150). This form has also been adopted 
 by Bose.i The small spheres in his case 
 
 -o 
 
 o 
 
 -o 
 
 Fig. ISO. 
 
 were o"3 cm. diameter, and the large one " 
 078 cm. diameter. The distance from 
 centre to centre of the small spheres was 
 I '2 cm. The waves produced by this - 
 oscillator were i'84 cm. long. It is ad- 
 visable to hinge the supports of the small 
 spheres so that their distance apart may be varied. 
 
 360. Another form of oscillator due to Lodge consists of 
 two metal spheres sparking to the interior of a copper cylinder ; 
 with the dimensions in Fig. 151 waves of 3" long are produced. 
 
 361. A form of oscillator used by Marconi, and described by 
 Preece,^ consists of two solid brass spheres fixed in an oil-tight 
 
 ' See Pro. Roy. Soc, vol. Ix. p. 167. 
 Electrician, vol. xxxix. p. 217. 
 
336 Practical Electricity and Magnetism. 
 
 insulating case filled with vaseline oil, two smaller spheres 
 placed close to the outside of the others are connected to the 
 induction coil, the spark passing across the three gaps. With the 
 large spheres, 4" diameter, waves r2 metres long were obtained. 
 362. The great difficulty with all these oscillators is that 
 when the sparking surfaces get roughened, as they do with use, 
 
 /N 
 
 •^ 
 
 Fig. 151. 
 
 the discharge ceases to be oscillatory ; the brass knobs should 
 therefore be made easily removable, to admit of their being 
 burnished up. It is also necessary in all quantitative experi- 
 ments to completely enclose the oscillating apparatus — battery, 
 coil, and wires — inside a copper-lined box, leaving only the 
 spark-gap outside, since oscillations may escape from other 
 parts of the apparatus. For this reason the form of apparatus 
 designed by Bose ' is recommended. 
 
 363. The oscillator is fixed to the outside of a box contain- 
 ing the coil and battery ; a copper tube open at both ends is 
 slipped over the oscillator so as to direct the path of the waves. 
 
 ' See Phil. Mag., vol. xliii., Jan., 1897; also see Fro. Roy. Soc, 
 vol. lix. p. 160, 
 
Apparatus for detecting Electro- Magnetic Waves. 337 
 
 The box containing the battery and coil is encased inside two 
 metal cases — an inner one of iron, which forms a magnetic shield 
 round it, and an outer one of copper, which prevents stray 
 radiation from escaping. Also, instead of having the usual 
 vibrating make and break on the coil, an ordinary break- 
 circuit key is employed, the condenser being connected to it as 
 in the vibrating break ; the object of this is to enable single 
 flashes of radiation to be obtained instead of a continuous 
 stream, thus saving the sparking surfaces. The key can be 
 manipulated from the outside of the box by a string. 
 
 Apparatus for detecting Electro-Magnetic Waves. 
 
 364. If the electro-magnetic waves are produced by a Leyden 
 jar discharging through a wire circuit, the presence of the 
 waves may be detected by placing a second similar jar, pro- 
 vided with a discharging circuit of the same dimensions, in 
 the vicinity of the first. On the first jar being discharged, the 
 oscillations produced set up electro-magnetic waves which, on 
 arriving at the second jar, induce discharges in it, which can 
 be detected by the presence of a spark at the discharging 
 knobs. This effect, however, is only possible when the jars 
 and their discharging circuits are identical with one another, 
 the second jar having to be " tuned into resonance " with the 
 first before it will act as a detector. 
 
 365. A much more sensitive form of detector, and one which 
 is easier to work with, is that known as a coherer. 
 
 The action of the coherer was first demonstrated by Branly,' 
 who showed that a tube containing metallic filings, and which 
 had a very high resistance, became a tolerably good con- 
 ductor when placed near a sparking coil. This phenomena 
 has been utilized by Lodge, Bose, Marconi, and others, as 
 a means of detecting electro-magnetic waves. 
 
 A simply made form of coherer consists of a glass tube about 
 eight inches long and half-inch diameter, filled with small 
 clippings of copper. Passing into the clippings through corks 
 at either end of th^ tube are copper wires, which connect the 
 
 ' Branly, Comptes Rendus, vol. cxi. p. 785 ; vol. cxii. p. 90. 
 
 Z 
 
338 Practical Electricity and Magnetism. 
 
 coherer in series with a galvanometer and battery. The tube 
 in its normal state is a very bad conductor, due to the number 
 of bad contacts at the copper clippings, and no deflection will 
 be obtained on the galvanometer. If a sparking coil is now 
 set a-going in the vicinity of the tube, the galvanometer will 
 be observed to deflect, this being due to the sudden acquisition 
 of conducting power by the tube of clippings. On slightly 
 tapping the tube, it returns at once to its badly conducting 
 condition. This action, according to Lodge, is due to the 
 breaking down of the insulating layer of oxide between the 
 clippings by the electrical surgings setup in them by the electro- 
 magnetic waves. 
 
 366. A much more satisfactory form of coherer is that 
 devised by Professor Bose,^ in which the copper clippings are 
 
 I , replaced by steel springs. Tho construction 
 
 of the instrument is as follows. A rectangular 
 
 block of ebonite, about 3 cm. long, of square 
 
 cross-section, with I'o cm. side, has a small 
 
 rectangular groove cut in one side, 2 cm. 
 
 ^ long by o'4 cm. broad. Into this groove 
 
 ^Plates ^'■^ placed eight or nine short steel wire 
 
 ' springs, o'4 cm. in length, of No. 34 wire. A 
 
 glass plate is fixed over them to prevent 
 
 them from dropping out of the groove. Fig. 
 
 T] 152 shows the spirals in position. 
 
 Fig. isz. Two brass plates make contact with the 
 
 upper and lower spirals respectively, the upper 
 brass plate being attached to a screw, by means of which 
 it can be made to compress the spirals. The upper and 
 lower brass plates are connected in series with a battery 
 and sensitive galvanometer. If the spirals are loose, no 
 deflection will be obtained on the galvanometer. The screw 
 is now turned so as to , compress the spirals until a small 
 deflection is obtained; the coherer is now in a sensitive 
 condition, and if exposed to electro-magnetic radiation the 
 galvanometer will deflect violently, indicating a great diminu- 
 tion of resistance at the spirals. In order to restore them 
 ' Pkil. Mag., vol. xliii., January, 1897. 
 
Apparatus for detecting Electro-Magnetic Waves. 339 
 
 to their normal state, the coherer may be tapped, or the 
 compressing screw may be slightly unscrewed. It will be 
 found, as has been pointed out by Bose, that there is a certain 
 value for the E.M.F. acting in the circuit, for which the coherer 
 is most sensitive when detecting waves of given length. This 
 must be found experimentally, and the battery, instead of being 
 connected directly to the coherer, should be joined up to the 
 ends of a high resistance, from which the coherer circuit is 
 taken as a shunt; thus, by altering the resistance included 
 between the coherer wires, various E.M.F.'s may be obtained, 
 as has already been explained in par. 20. 
 
 367. In experimenting with the above, or, in fact, with any 
 form of coherer, the greatest difficulty will be experienced in 
 protecting it from stray radiation. Thus, covering the face 
 of the spirals with a copper plate, which is opaque to electro- 
 magnetic waves, will not be found sufficient to prevent the 
 coherer from working, since the waves impinge on the connect- 
 ing wires, or on the battery or galvanometer, and transmit their 
 effects through the circuit to the coherer. We must therefore 
 completely shield all the instruments in the coherer circuit. 
 
 The following Fig. (153) illustrates the method adopted by 
 the author. 
 
 Fig, 153. 
 
 The battery B, consisting of a small iron-zinc cell (the iron 
 in a solution of ferric chloride and the zinc in sulphuric acid. 
 
340 Practical Electricity and Magnetism. 
 
 this giving an E.M.F. of about 0*5 volt, which is best for 
 all-round work), and the galvanometer coil G, are enclosed 
 inside a square brass box, the needle attached to the mirror 
 M being suspended outside. The coherer C is enclosed 
 inside a circular brass box attached to one side of the square 
 box, an opening, protected by the brass tube T, being made 
 in front of the sensitive spirals. On the open end of T various 
 diaphragms, D, can be attached, so as to limit the amount of 
 radiation received by C. The wires from the coherer pass 
 through two ebonite bushed holes into the interior of the brass 
 box, where they are attached to the b3.ttery and galvanometer 
 respectively. An opening, O, provided with a sliding brass 
 lid in the circular box containing the coherer, admits of entrance 
 to the adjusting screw S. The whole apparatus can be attached 
 to a spectrometer circle, taking the place of the observing 
 telescope. Before using, the apparatus should be tested for 
 electro-magnetic wave tightness, by placing a brass cap with 
 no hole in it over the end of the tube, T, and adjusting the 
 coherer to its maximum sensitiveness. Then, on producing 
 electro-magnetic waves in its vicinity, no effect _ should be 
 obtained on the coherer. This receiver is used iii conjunction 
 with the oscillator described in par. 363. 
 
 368, In performing the various optical experiments with the 
 electro-magnetic waves, it is necessary that we should employ 
 a parallel beam, and not a divergent one. In order to obtain 
 this, a lens of some refracting material must be placed in front 
 of the oscillator, so that the spark-gap is at its principal focus. 
 The lens should be in the form of a cylinder or semi-cylinder, 
 and should be enclosed in a square metal tube, which fits on 
 to the tube surrounding the oscillator. Such a lens may be 
 made of sulphur, the index of refraction of which is 173. 
 Employing the usual lens formula — ■" 
 
 where F = focal length ; 
 
 R = radius of curvature of side nearest source ) 
 S = radius of curvature of side farthest from source ; 
 ' Vol. i , Exp'. 72. 
 
Transmission, Reflection, & Absorption Experiments. 341 
 
 a surface concave to the incident rays being reckoned positive, 
 we can calculate the necessary curvature for our sulphur 
 lens, and, making a metal mould, proceed to cast it to the 
 required size. 
 
 Transmission, Reflection, and Absorption. 
 
 369. (i) Rectilinear Propagation. — In order to demonstrate 
 the rectilinear propagation of the rays, the oscillator and coherer 
 should be set up opposite each other, and at some distance 
 apart, as A B in Fig. 154. The coherer B is then adjusted so 
 
 A B 
 
 O d 
 
 Fig. 154. 
 
 as to respond to the waves from A, which are supposed to 
 consist of a parallel ray. If a thick plate of some good 
 conductor, such as copper, is now interposed between A and B 
 no effect will be obtained at B, because a conductor is opaque 
 to the electro-magnetic waves. If, however, a non-conductor, 
 as ebonite sheet, or a wooden board, etc., be interposed the 
 effect is obtained as if there was nothing between A and B. 
 Non-conductors are transparent to the waves. Various sub- 
 stances should be tried, and the relative thicknesses of the 
 plates required to just stop the passage of the waves determined. 
 If now a number of copper plates are taken, and holes of the 
 same diameter as the radiator tube drilled in them, and the 
 plates placed parallel to each other between A and B, then, if 
 the holes coincide, the effect will still be obtained at B ; but if 
 they are displaced, so that the holes are not opposite one 
 another, then no effect will be obtained at the coherer, thus 
 demonstrating the rectilinear propagation of the waves. 
 
 370. (2) Reflection and Absorption. — In order to prove experi- 
 mentally the laws of reflection of the waves, a spectrometer 
 should be employed ; but they may be demonstrated without 
 in the following manner. On a large flat board a graduated 
 circle of about 12" diameter is drawn (see Fig. 155). The 
 
342 Practical Electricity and Magnetism. 
 
 oscillator and coherer A and B are placed opposite each other, 
 and B adjusted to respond to the waves from A. The lens 
 must be placed in front of A, in order to get a parallel beam, 
 and the smallest diaphragm which can be used should be put in 
 front of A, so as to get a small ray. B is then rotated through 
 90° to B', and on working the oscillator, no eifect should be 
 obtained. A small copper plate, C, is now placed at the centre 
 of the circle, and rotated about a vertical axis until an effect is 
 
 B 
 
 obtained at B'. On measuring the angles it will be found that 
 the angles of incidence and reflection are equal. The effect 
 should be tried with B in various other positions, also using 
 different substances for the reflector C. It will be found that 
 while almost every substance reflects more or less of the 
 incident waves, the effect is much more marked in the case of 
 the good conductors than of the insulators. Tests should be 
 made for substances that are electrically black to the waves ; 
 i.e. substances which do not either reflect or transmit to any 
 marked extent, but absorb the waves; Blotting-paper soaked 
 in water will be found to do this to some extent. 
 
 Refraction. 
 
 371. Of all the experiments with the electro-magnetic waves 
 the measurement of the index of refraction is one of the most 
 important. The methods employed are similar to those when 
 
Refraction of Electro- Magnetic Waves. 343 
 
 using light waves. We may, for instance, make a prism of the 
 refracting substance, and determine the angle of minimum 
 deviation of the electro-magnetic lay and the angle of the prism, 
 when — 
 
 . A + D 
 
 . A 
 
 sm — 
 
 2 
 
 where A = angle of prism, and D = angle of minimum 
 deviation. Such measurements have been made by Hertz, 
 Lodge, and others, but, as Bose ^ has shown, it is much more 
 satisfactor)' to measure the index of refraction by determining 
 the critical angle a of the medium, from which we get — 
 
 sin u 
 
 In the case of solids, the refracting medium should take the 
 form of two semi-cylinders with an air-space between them. 
 This shape is chosen because, in order that all the ray should 
 be totally reflected at the same instant, it should impinge 
 normally on the air film, this being obtained by using semi- 
 c}-lindrical lenses as the medium, and placing the spark-gap at 
 the principal focal line. 
 
 In the case of substances whose index of refraction is not 
 approximately known, a rough measurement is first made, with 
 the spark-gap somewhere near the semi-cylinders; from the 
 approximate value of r thus obtained, the focal length of the 
 lens may be calculated, and, placing the spark-gap at that point, 
 a new and more accurate value of r may be obtained. In order 
 only to utilize the central rays, a copper diaphragm, with a slit 
 cut in it, is placed between the semi-cylinders, thus stopping all 
 rays but those that get through the slit. 
 
 The arrangement of the apparatus is shown in Fig. 156. 
 The oscillator A and coherer B are placed diametrically 
 opposite one another, and tested for proper adjustment. The 
 semi-cylinders, with the diaphragm D between them, are then 
 placed in position, so that the diaphragm is at right angles to 
 ' Pro. Roy. Soc., vol. lix.|p. i6o. 
 
344 Practical Electricity and Magnetism. 
 
 the line joining AB. The tubes surrounding the oscillator and 
 coherer should end quite close up to the semi-cylinders. On 
 starting the oscillator the coherer will at once respond, due to 
 the rays which pass through the hole in the copper diaphragm. 
 The diaphragm D and semi-cylinders are now rotated in a clock- 
 wise direction, and when they reach some position, D"D", it 
 
 Fig. 156. 
 
 will be found that the coherer no longer responds, the ray being 
 just totally reflected in that position. This position is noted 
 on the graduated circle. The semi-cylinders are then rotated 
 in a counter-clockwise direction, till, on reaching the position 
 D'D', it is found that total reflection just occurs again, this 
 position being noted, the angle D'OD" is twice the critical 
 angle, and from this r may be calculated. In practice it is 
 better to start with the ray totally reflected, .and find the 
 position where the coherer just begins to act ; this gives, of 
 course, the same result. The dimensions of the semi-cylinders 
 depend considerably on the size of the rest of the apparatus, 
 but they may be about 25 cm. diameter, and 10 cm. high. 
 Solids like wood, ebonite, etc., should be turned up in a lathe, 
 and then cut in two, whilst substances like sulphur should be 
 cast in a mould. 
 
 Liquids are placed in a cylindrical glass vessel divided by 
 two glass partitions, which enclose an air-space between them. 
 
 372. Measurement of Wave-Length. — In order to measure 
 the length of the waves emitted by a Hertz oscillator, there are 
 
Measurement of Wave-Length. 345 
 
 several methods available. We might, for instance, employ the 
 relationship v = n\ where v = velocity of propagation, n = 
 frequency, and X = wave-length ; this would involve a measure- 
 ment of velocity and frequency, both of which measurements 
 present great difficulties, rendering this method unsuitable. 
 
 373. Another method that has been adopted, and successfully 
 carried out, by Hertz, was to produce a stationary wave by 
 reflecting a wave back on itself from a metal sheet ; then, by 
 finding the positions in front of the reflector where interference 
 occurs, and measuring the distance between them, a value may 
 be obtained for the wave-length. Some doubts have, however, 
 been thrown on this method of measuring wave-length, as some 
 experiments tend to show that the wave-length measured 
 depends to some extent on the free period of vibration of the 
 resonator employed. 
 
 374. A much more satisfactory method, and one that has 
 given good results, is that employed by Bose,' in which a curved 
 diffraction grating is used. The theory of the curved diffraction 
 grating has been fully worked out,^ since it is employed for the 
 measurement of the length of light waves, _ x 
 
 so that only the dimensions of the / 
 
 srratinat have to be altered to meet the 1/ >^ c' 
 
 changed conditions of using it with much /_^ — 'Z^^^^^'^\ 
 
 longer waves. M ru^*^^ J ^ 
 
 Let G in Fig. 157 represent the curved r^ "^--^^^::aiy 
 grating, and c the centre of curvature. vv J/ ^ 
 
 The circle/, drawn with Mf as diameter, V^ "^ 
 
 is the focal curve. Any source of radi- \ ■' 
 
 ation placed on this curve will give a Fig. 157. 
 
 diffracted spectrum situated somewhere on the same curve, 
 defined by the equation — 
 
 n\ = (a -\- V) (sin / + sin S) 
 where n\ = //th spectrum ; 
 (a -\- h) = sum of the breadths of a strip and space in grating ; 
 
 i = angle of incidence ; 
 
 B = angle of diffraction. 
 
 ' Pro. Roy. Soc, vol. Ix. p. 167. 
 ° See Preston's " Light," par. 136. 
 
346 Practical Electricity and Magnetism. 
 
 The sign of 6 is + if the diffracted image / lies on the same 
 side of M.C as s the source, and — if it Ues on the opposite side. 
 
 375. The grating employed consists of a sheet of ebonite on 
 which a number of strips of tinfoil have been pasted, these 
 being about 2 '5 to 3 cm. broad, and the space between two 
 adjacent tinfoil strips is equal in breadth to the strips. The 
 ebonite sheet before pasting on the strips must be bent into a 
 curve of about 100-150 cm. radius. 
 
 This grating is then placed vertically on a flat surface, as G 
 (Fig. 157), and the circle f drawn to give the positions for j 
 and /. The coherer should be placed at c, thus making 6 in 
 the calculation equal to zero. The radiator must now be 
 moved about on the curve/, until the diffracted image falls on 
 the coherer, when it will respond. The angle of incidence, /, 
 must now be measured, and the wave-length calculated. 
 
 376. The following numbers obtained by Bose for a 3 cm. 
 grating will serve to illustrate the measurement : — 
 
 i 
 
 e 
 
 \ 
 
 21-5 
 
 
 
 1-8321 
 
 
 29-5 
 
 -7 
 
 1-852 
 
 mean 
 
 33*o 
 
 — 10 
 
 I -8541 
 
 i-«4S 
 
 34"o 
 
 — II 
 
 I -841 J 
 
 
 The radiator employed in the above measurement was the 
 same as that described in par. 359. 
 
 Polarization of Electro-Magnetic Waves. 
 
 377. All the phenomena previously described are common to 
 sound, light, and radiant heat waves ; the phenomena of polari- 
 zation, however, is common only to light and radiant heat 
 waves, and indicates that the nature of the wave-motion in 
 these two cases differs essentially from that in sound propaga- 
 tion. Sound is propagated in solids, liquids, or gases, by 
 means of longitudinal waves whose direction of vibration is the 
 same as the direction of propagation. The phenomena of 
 polarization indicates that light and radiant heat waves are 
 
Polarization of Electro-Magnetic Waves. 347 
 
 propagated by transverse vibrations, or those whose direction 
 of motion is at right angles to the direction of propagation, and 
 moreover that the direction of vibration is not confined solely 
 to one plane. 
 
 378. If a ray of light falls on a plate of tourmaline cut 
 parallel to the axis of the crystal, it is transmitted with a slight 
 diminution of intensity. The light that is transmitted, however, 
 differs essentially from the incident light on the tourmaline, 
 since it will only pass through a second plate of tourmaline if 
 the latter is parallel to the first ; if placed at right angles, the 
 ray is totally stopped by the second plate. The ray of light 
 after passing through the first tourmaline plate has therefore 
 acquired a two-sided property that it did not possess before, 
 and is said to be plane polarized. The first tourmaline is 
 called the " polarizer," and the second one which is used to 
 discover the polarization is called the analyzer. 
 
 379. In order to polarize the electro-magnetic waves, they 
 are sent through a screen of parallel copper wires, which should 
 be constructed as described by Bose,^ by winding copper wire 
 of 2 mils, diameter round a thin piece of mica, which is then 
 immersed in melted paraffin, the object being to fix the wires. 
 Four circular pieces are cut out of this, and two are fixed 
 parallel to each other at the ends of a tube which will fit into 
 the tube surrounding the spark-gap, this constituting the 
 polarizer. There are about twenty-five wires to the centimetre. 
 This polarizer only transmits vibrations at right angles to the 
 wires, and by means of it, it can be shown that the waves from 
 the sparking knobs are partly plane polarized parallel to the 
 spark-gap. 
 
 The analyzer is made in the same way as the polarizer, and 
 can be attached to the tube surrounding the coherer. 
 
 When the polarizer is placed with the wires at right angles to 
 the direction of the spark, the ray proceeding from the oscillator 
 will be completely plane-polarized, as can be tested by turning 
 the analyzer so that its wires are (i) parallel, (2) perpendicular 
 to the direction of the wires in the polarizer, when in the 
 second case, no effect will be obtained at the coherer. 
 ' Bose, Electrician, vol. xxxvi. p. 291. 
 
348 Practical Electricity and Magnetism. 
 
 We have now seen that all the ordinary optical experiments 
 can be performed with electro-magnetic waves as produced 
 above, and that light and these electro-magnetic waves are 
 identical, except in so far as regards frequency and wave-length, 
 the latter vibrating more slowly and being considerably longer 
 than the visible ether waves ; they will therefore occupy a place 
 far to the left of the red in the spectrum. 
 
References to Original Papers. 
 
 349 
 
 380. References to Original Papers. 
 
 Title of Paper. 
 
 Author. 
 
 Discharge of Leyden Jars 
 
 Indices of Refraction for Electric 
 
 Rays 
 Determination of Wave-Length of 
 
 Electric Radiation 
 Selective Conductivity 
 Kerr's Experiments on Relation 
 
 between Light and Electricity 
 Influence of Light on Electrical 
 
 Resistance of Metals 
 Influence of Light on the Electric 
 
 Tension of Metals 
 Electrical Absorption of Crystals 
 
 Transmission of Radiation of Low 
 Refrangibility through Ebonite 
 Refractive Index of Ebonite 
 
 Reflection of Electric Rays 
 
 Indices of Refraction of Metals 
 
 Calorimetric Investigations on 
 Electric Discharge 
 
 Investigation of Electric Vibra- 
 tions with Thermo-Elements 
 
 Refraction and Dispersion in 
 Metals 
 
 Measurement of Electro- Magnetic 
 Phenomena 
 
 Experiments on Photo-Electricity 
 
 Method of Determining Electro- 
 magnetic Radiation 
 
 Electro-Magnetic Radiation on 
 Films 
 
 Sudden Acquisition of Conducting 
 Power by Metallic Particles 
 
 On a Complete Apparatus for the 
 Study of Electric Waves 
 
 Concentration of Electric Radia- 
 tion by Lenses 
 
 Variations of Electrical Conduc- 
 tivity under Electric Influence 
 
 Sensitive Cells 
 
 Photo-Electric Researches 
 
 Lodge 
 Bose 
 
 Gordon 
 Bornstein 
 
 Rowland and 
 
 Nichols 
 Abney 
 
 Ayrton and 
 
 Perry 
 Goldstein 
 
 Kund 
 
 Staub 
 
 Klemencic 
 
 Du Bois 
 
 Boys, Briscoe, 
 and Watson 
 Minchin 
 
 Klemencic 
 
 Minchin 
 
 Lodge 
 
 Bose 
 
 Lodge 
 
 Branly 
 
 Minchin 
 Right 
 
 Reference, 
 
 Fro. Roy. Soc, vol. 
 
 SO, p. 2. 
 Jitd., vol. 59, p. 
 
 160, 
 Hid., vol. 60, p. 
 
 167. 
 Jliid., vol. 60, p. 433. 
 jPAil. Mag., vol. 2, 
 
 Sept., 1876. 
 Ibid., Supp., vol. 3, 
 
 1877. 
 Ibid., vol. 4, Nov., 
 
 1877. 
 Ibid., vol. II, June, 
 
 1881. 
 Ibid., vol. n, June, 
 
 18S1. 
 Ibid., vol. 12, Sept., 
 
 18S1. 
 Ibid., vol. 14, Dec, 
 
 1882. 
 Ibid., vol. 26, July, 
 
 1888. 
 Ibid., vol. 30, Sept., 
 
 l8go; 
 Ibid., vol. 30, Sept., 
 
 1890. 
 Ibid., vol. 30, Nov., 
 
 i8go. 
 Ibid., vol. 31, Jan., 
 
 1891. 
 Ibid., vol. 31, Mai., 
 
 1891. 
 Ibid., vol. 33, April, 
 
 1892. 
 Ibid., vol. 37, Jan., 
 
 1894. 
 Ibid., vol. 37, Jan., 
 
 1894. 
 Ibid., vol. 43, Jan., 
 
 1897. 
 Elect., vol. 23, p. 
 
 32- 
 Ibid., vol. 27, pp. 
 
 221, 448. 
 Ibid., vol. 28, p. 85. 
 Ibid.,vo\. 33, p. 297, 
 
3SO Practical Electricity and Magnetism. 
 
 Title of Paper. 
 
 Author. 
 
 Reference, 
 
 Electric Waves 
 
 Trowbridge 
 
 Elect., vol. 
 712, 812. 
 
 35. pp. 
 
 Electric Oscillations 
 
 Klemeucic 
 
 Ibid., vol. 35 
 
 ,p.8l2. 
 
 Resistance of Filings of Metals 
 
 Lhuillier 
 
 Ibid., vol. 35, p. 570. 
 
 Contact Resistance of Metals 
 
 Branly 
 
 Ibid., vol. 35, p. 4. 
 
 Heat developed by Electric 
 
 Cardani 
 
 Ibid., vol. 36, p. 292. 
 
 Oscillations in Wires 
 
 
 
 
 Tinfoil Grating for Electric Oscil- 
 lations 
 Apparatus for the Study of 
 
 Mezuno 
 
 IbU., vol. 36, p. 187. 
 
 Bose 
 
 Ibid; vol. 37, p. 788. 
 
 Electric Waves 
 
 
 
 
 Multiple Resonance in connection 
 
 Sarasin and De 
 
 Ibid., vol. 
 
 24. P 
 
 Vfith Hertz Experiments 
 
 la Rive 
 
 317- 
 
 
 An Electric Radiation Meter 
 
 Gregory 
 
 Ibid., vol. 24, p. 16. 
 
 ExDeriments to determine Wave 
 
 Trouton 
 
 Ibid., vol. 
 
 25. P- 
 
 Velocity in certain Dielectrics 
 
 
 557- 
 
 
 On the Speed of Propagation of 
 
 Arons and 
 
 lUd., vol. 
 
 26, p. 
 
 Electric Waves in Insulating 
 
 Rubens 
 
 157- 
 
 
 Dielectrics 
 
 
 
 
 Experimental Determination of 
 
 Blondlot 
 
 Ibid., vol. 
 
 28, p. 
 
 the Rate of Propagation of 
 
 
 85- 
 
 
 Electro-Magnetic Waves 
 
 
 
 
 Refractive Index of Electric Waves 
 
 EUinger 
 
 Ibid., vol. 
 
 3O) P' 
 
 in Alcohol 
 
 
 387- 
 
 
 The Equality of the Velocities of 
 
 Dufour 
 
 Ibid., vol. 
 
 33. P- 
 
 Propagation of very Short 
 
 
 485. 
 
 
 Electric Waves in Free Space 
 
 
 
 
 and along Conductors 
 
 
 
 
 On the Refraction and Dispersion 
 
 Garbasso 
 
 Ibid., vol. 
 
 34> P- 
 
 of Rays of Electric Force 
 
 
 39- 
 
 
 A Carbon Detector for Hertz 
 
 Jervis-Smith 
 
 Ibid.,, vol. 
 
 40, p. 
 
 Waves 
 
 
 84. 
 
 
 The History of the Coherer 
 
 Lodge 
 
 Ibid., vol. 
 
 40, p. 
 
 Principle 
 
 
 87. 
 
 
 The Practical Applications of the 
 
 A. C. Brown 
 
 Ibid. , vol. 
 
 40, p. 
 
 Coherer 
 
 
 91. 
 
 
APPENDIX 
 
 THE TANGENT POSITIONS OF GAUSS. 
 
 {\) A Tangent Position. — The relative positions of magnet 
 and needle in this position are shown in Fig. 158. 
 
 N 
 
 s 
 
 ..zl... 
 
 < d ^1 
 
 Fig. 158. 
 
 Calling in and iti the pole-strengths of the deflecting magnet 
 and the needle respectively, then — 
 
 iHjn 
 The repulsion of n for N = rr^pi 
 
 the attraction of 5 for N = — 
 
 therefore the total force of ') mm' 
 
 repulsion acting on N j {d — I)- (d + /)- " 
 
 ^mm'dl 
 
 ^ (d"" - Pf 
 
 And in the same way it can be shown that — 
 
 ™, , ,- ,- • „ /imnidl 
 
 The total force of attraction acting on S = — / jz _ j-is i. 
 
 Now, if 2X represents the length of the needle, and Q the 
 angular deflection of the needle from the magnetic meridian, 
 then the turning couple acting on the needle is^ 
 imftidl 
 
352 Practical Electricity and Magnetism. ' 
 
 But the controlling couple due to the earth's magnetism, H, 
 which balances the deflecting couple,- is — 
 
 2H«'\ sin B 
 
 Ztrnddl 
 
 and therefore - 
 
 I \ cos B = 2'H.m' A. sin 6 
 
 ■ {d^ - ly 
 
 And since 2////= M, the magnetic moment of the deflecting 
 magnet, then — 
 
 M id' - Pf ^ , 
 
 ^ = -i -^ tan e 
 
 H 2d 
 
 which, when / is very small compared with d, simplifies to — 
 
 M d^ 
 
 iT- = — tan a 
 
 H 2 
 
 (2) B Tangent Fosition. — The relative positions of deflector 
 and needle are shown in Fig. 159. 
 
 V fS 
 ^ Jii \ 
 
 V St 
 
 \IiH 
 
 a 
 
 n' 
 
 \r 
 
 Fig. 159. 
 
 The force at the needle due to pole « = — , where m = pole- 
 strength of n ; also the force due to s will be equal to — -^ 
 
 Therefore the resultant force F of /i and_/^ will be parallel to 
 the deflecting magnet, and — 
 
Appendix. 353 
 
 F 2/ 
 
 -: = 2 COS a =- 
 /i r 
 
 and F 
 
 r 
 
 
 2ml 
 
 
 ~ r- 
 
 
 M 
 - ,.3 
 
 Butr 
 
 = (</^ + 0i 
 
 ■ TT 
 
 M 
 
 If the needle has been deflected through an angle, 6, from 
 the meridian, and 2A. is the length of the needle, m' being its 
 pole-strength, then — 
 
 2M^' 
 
 the deflectinar couple = — — , A. cos 6 
 
 This is balanced by the controlling couple due to the earth's 
 field, which is — 
 
 2Hw'X sin & 
 
 2M?w' 
 Hence 73r"r~m» cos 6 = 2'H.m'X. sin 6 
 
 M 
 H 
 
 and^ = (^^ + /^)itan6l 
 
 Here, ^gain, when / is small compared with d, the expression 
 simplifies, and may be written — 
 
 M 
 
 ^ = d^X3ne 
 
 ix 
 
 2 A 
 
LOGARITHMS — ANTILOGARITHMS — TABLES OF 
 SQUARES— RECIPROCALS— TANGENTS— SINES 
 —CONVERSION TABLES— CONSTANTS. 
 
3s6 Practical Electricity and Magnetism. 
 
 Logarithms. 
 
 183456789 
 
 12 3 4 6 6 7 
 
 8 9 
 
 0000 0043 
 
 0414 
 0792 
 
 "39 
 1461 
 1761 
 
 2041 
 
 2304 
 
 2SS3 
 2788 
 3010 
 
 3222 
 
 3424 
 3617 
 3802 
 3979 
 
 4150 
 4314 
 4472 
 4624 
 
 4771 
 
 4914 
 5051 
 5185 
 S315 
 5441 
 
 SS63 
 5682 
 5798 
 59" 
 6021 
 
 6128 
 6232 
 6335 
 643s 
 6532 
 
 6628 
 6721 
 68i2 
 6902 
 6990 
 
 7076 
 7160 
 7243 
 7324 
 
 0453 
 0828 
 
 (173 
 1492 
 1790 
 
 2068 
 2330 
 
 2*77 
 2810 
 3032 
 
 3243 
 3444 
 3636 
 3820 
 
 3997 
 
 4166 
 4330 
 4487 
 4639 
 4786 
 
 4928 
 S0S5 
 5198 
 5328 
 5453 
 
 5575 
 5694 
 5809 
 5922 
 6031 
 
 6138 
 6243 
 
 6345 
 6444 
 
 0086 0128 
 
 0492I0531 
 08640899 
 1206 1239 
 
 1523 1553 
 18181847 
 
 2095 2122 
 2355 2380 
 26012625 
 28332856 
 30543075 
 
 3263 3284 
 3464 3483 
 365SI3674 
 3838'3856 
 4014 4031 
 
 4i83'4200 
 43464362 
 4502,4518 
 46544669 
 48004814 
 
 49424955 
 
 017c 
 
 3212 0253 
 
 0569 0607 
 
 0934 0969 XOO4 
 
 1271 
 1584 
 187s 
 
 2148 
 240; 
 
 1303 1335 
 1614 1644 
 1903 I93I 
 
 220I 
 
 2455 
 2695 
 
 2175 
 
 2430 
 2648 267; 
 2878 290012923 
 3096 3II8 3139 
 
 3304 3324 3345 
 3502 3522 
 3692 3711 
 
 3874 3892 3909 
 404S 4065 4082 
 
 4216 4232 4249 4265 
 
 4378 
 4533 
 4683 . 
 4829 4843 
 
 5092 
 5224 
 
 5353 
 5478 
 
 5599 
 5717 
 5832 
 5944 
 6053 
 
 6160 
 6263 
 
 6365 
 6464 
 6561 
 
 6637 6646 6656 
 
 6730 6739 
 6821 
 
 6920 
 
 7007 
 
 6542655: 
 
 5079 
 521 1 
 S340 
 6465 
 
 5587 
 5705 
 ,821 
 
 5933 
 6042 
 
 6149 
 6253 
 6355 
 6454 
 
 6911 
 6998 
 
 7084 
 7168 
 7251 
 
 7093 
 7177 
 7259 
 
 73327340 
 
 4969 
 5105 
 
 5237 
 5366 
 
 5490 
 5611 
 
 5843 
 5955 
 
 68306839 
 6938 
 7016 
 
 7101 
 
 7185 
 7267 
 
 734SJ356I7364 
 
 727, 
 
 0645 
 
 3541 
 3729 
 
 0294 
 
 C682 
 1038 
 1367 
 1673 
 I9S9 
 
 2227 
 2480 
 2718 
 
 2945 
 3160 
 
 3365 
 3560 
 
 3747 
 39"7 
 4099 
 
 0334 
 
 0719 
 1072 
 1399 
 
 0374 
 
 "755 
 1 106 
 
 1430 
 
 170311732 
 1987 2014 
 
 2253 2279 
 25042529 
 274^2765 
 2967 2989 
 3181 3201 
 
 3385 
 3579 
 
 3404 
 3598, 
 
 37663784I 
 39453962 
 4116 4133 
 
 4393 4409 
 
 4548 4564 
 
 4698 4713 
 
 4857 
 
 4983 4997 
 5"9 5132 
 5250 5263 
 5378 5391 
 5502 5514 
 
 5623 
 
 5635 
 
 5729 5740 5752 
 
 5855 
 5966 
 
 5866 
 5977 
 
 6064 6075 6085 
 6170 6180 6191 
 
 44^25 
 4579 
 4728 
 4871 
 
 son 
 
 5145 
 5276 
 
 5403 
 
 5527 
 
 5647 
 5763 
 5877 
 5988 
 
 4281 
 4440 
 4^4 
 4742 
 
 5024 
 5159 
 5289 
 5416 
 SS39 
 
 5658 
 S77S 
 5888 
 
 5999 
 
 4298 
 4456 
 4609 
 4757 
 4900 
 
 5038 
 5172 
 5302 
 5428 
 5551 
 
 5670 
 578,6 
 
 5899 
 6010 
 
 6201 
 
 6212 
 
 6274 6284 
 
 6375 5385 
 6474 5484 
 
 6294 
 639s 
 &493 
 
 657J 5580 6590 6599 
 
 6096 6107 6117 
 
 6222 
 6325 
 6425 
 6522 
 6618 
 
 6304 6314 
 
 6405 
 6503 
 
 6665 6675 
 
 5684 6693 
 
 674916758 6767 5776 6785 
 
 6848 6857 5866 6875 
 
 6937 &946 
 7024 7033 
 
 7110 7118 
 7193 7202 
 
 S9S5 
 7042 
 
 6964 
 7050 
 
 6415 
 6513 
 6609 
 
 6702 
 6794 
 6884 
 6972 
 7059 
 
 7126:713517143 
 
 72iOj72i8l7226 
 
 7284 7292J7300I7308 
 
 73721738017388 
 
 I I 
 
 6712 
 6803 
 6893 
 6981 
 7067 
 
 7152 
 7235 
 7316 
 7396 
 
 48 
 
 48 
 
 3 7 
 3« 
 
 3 6 
 
 3 5 
 
 3 5 
 
 2 5 
 
 2 4 
 
 2 4 
 
 2 4 
 
 2 4 
 
 2 4 
 
 2 4 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 I 3 
 
 I 3 
 
 I 3 
 
 I 3 
 
 I 3 
 
 I 3 
 
 I 2 
 I 2 
 
 12 17 
 
 " 1= 
 
 10 14 
 
 10 13 
 
 9 12 
 
 8 II 
 
 8 II 
 
 7 10 
 7 9 
 7 9 
 6 8 
 
 S 71 
 
 S 6 
 
 S 6 
 
 4 6 
 
 4 6 
 
 3 3 
 
 25 29 
 
 23 26 
 21 24 
 19 23 
 18 21 
 17 20 
 
 16 18 
 
 IS 17 
 
 14 16 
 13 16 
 13 15 
 
 12 14 
 12 14 
 II 13 
 
 II J2 
 10 12 
 
 10 II 
 
 9 
 
 9 11 
 
 9 
 9 10 
 
 8 10 
 8 9 
 8 9 
 8 9 
 7 9 
 
 33 37 
 
 3034 
 28 31 
 26 29 
 24 27 
 22 25 
 
 21 24 
 20 22 
 19 21 
 18 20 
 17 19 
 
 16 18 
 15 17 
 15 17 
 14 16 
 
 14 IS 
 
 '3 IS 
 13 14 
 12 14 
 12 13 
 II 13 
 
 II 12 
 II 12 
 10 12 
 10 II 
 10 II 
 
 10 II 
 9 10 
 9 10 
 9 10 
 9 10 
 
 6 7 
 
 S 6 
 
 5 6 
 
 5 6 
 
 S 6 
 
 5 6 
 
 5 6 
 
 5 6 
 
 S 6 
 
 7 8 
 
 7 7 
 
 6 7 
 
 6 7 
 
Logarithms. 
 
 3S7 
 
 Logarithms. 
 
 12 3 4 
 
 6 7 8 g 12 3456 
 
 ss 74047412 
 
 56 
 S7 
 58 
 59 
 60 
 
 61 
 62 
 
 63 
 64 
 
 65 
 
 66 
 67 
 68 
 69 
 70 
 
 71 
 72 
 
 73 
 74 
 
 75 
 
 76 
 
 77 
 78 
 
 7482 
 7559 
 
 76347642 
 
 7490 
 7566 
 
 7709 
 7782 
 
 7853 
 7924 
 
 7993 
 8062 
 8129 
 
 819s 
 8261 
 
 8325 
 8388 
 
 8451 
 
 8513 
 8573 
 8633 
 8692 
 8751 
 
 8808 
 886s 
 8921 
 
 7716 
 
 77897796 
 7860 7868 
 
 7419 
 
 7497 
 7574 
 
 7427 
 
 7505 
 7582 
 
 7435 7443 745' 
 
 7459 
 
 7513 7520 7528 
 
 7589 7597 
 
 . _ 7536 
 7604 7612 
 
 7649 7657 7664 7672 7679 
 
 7723 
 
 7931 
 8000 
 S069 
 8136 
 
 8202 
 8267 
 8331 
 8395 
 8457 
 
 8519 
 
 8579 
 863a 
 8698 
 
 7938 
 8007 
 
 807s 
 8142 
 
 8209 
 8274 
 8338 
 8401 
 8463 
 
 8525 
 8585 
 8645 
 8704 
 
 8756 8762 
 
 79 8976 
 
 80 9031 
 
 81 
 82 
 
 83 
 84 
 
 85 
 
 86 
 
 87 
 88 
 89 
 90 
 
 91 
 92 
 93 
 94 
 95 
 
 95 
 97 
 98 
 99 
 
 9085 
 9138 
 9191 
 
 9243 
 9294 
 
 9345 
 9395 
 9445 
 9494 
 9542 
 
 959° 
 9638 
 968s 
 9731 
 9777 
 
 9823 
 9868 
 9912 
 
 8814 
 8871 
 8927 
 8982 
 9036 
 
 SO90 
 
 9143 
 9196 
 9248 
 9299 
 
 9350 
 9400 
 
 9450 
 9499 
 9547 
 
 9595 
 9643 
 9689 
 
 9736 
 9782 
 
 9827 
 9872 
 9917 
 
 8820 
 8876 
 8932 
 8987 
 9042 
 
 9096 
 9149 
 92CI 
 9253 
 9304 
 
 9956 9961 
 
 7731 
 7803 
 
 7875 
 7945 
 8014 
 8082 
 8149 
 
 8215 
 8280 
 8344 
 8407 
 8470 
 
 8531 
 8591 
 8651 
 8710 
 8768 
 
 8825 
 8382 
 8938 
 8993 
 9047 
 
 9IOI 
 
 9154 
 9206 
 
 9258 
 
 9309 
 
 7738 7745 7752 
 7810 7818 7825 
 
 7882 7889 7896 
 
 7952 7959 7966 
 8028 8035 
 
 8021 
 
 8089 8096 3 
 8156 3162 
 
 102 
 8169 
 
 822218228 3235 
 8287 8293 3299 
 
 8351 
 
 8357 8363 
 
 9355 
 9405 
 9455 
 9504 
 9552 
 
 9647 
 9694 
 9741 
 9786 
 
 9832 
 9877 
 9921 
 
 8414 342c S426 
 8476 3482 
 
 8537 8543 3549 
 8597 8603 8609 
 86S7 8663 8669 
 8716 B722 8727 
 
 8774 
 883J 
 
 8779 8785 
 8837 8842 
 
 8887 8893 8899 
 
 8943 8949 8954 
 8998 9004 9009 
 9053 90s8|go63 
 
 9106 g: 
 
 9159 9i65|9i7< 
 9212 9217 9222 
 9263 9269 9274 
 9315 9320 932s 
 
 9360 
 9410 
 
 9365 9370 937S 
 
 9415 
 
 9652 
 9699 
 9745 
 9791 
 
 9836 
 9881 
 9926 
 
 <;84i 
 
 „ .. 9930 9934 9939 
 9965 9969 9974 9978 9983 
 
 7686 
 7760 
 7832 
 
 7903 
 7973 
 8041 
 8109 
 8176 
 
 8241 
 8306 
 8370 
 8432 
 8494 
 
 8555 
 8615 
 8675 
 8733 
 8791 
 
 7466 
 
 7543 
 7619 
 7694 
 7767 
 7839 
 
 7910 
 7980 
 8048 
 8it6 
 8182 
 
 8248 
 8312 
 8376 
 
 8439 
 8500 
 
 8561 
 8621 
 8681 
 8739 
 8797 
 
 7474 
 
 7551 
 7627 
 7701 
 
 7774 
 7846 
 
 7917 
 7987 
 8055 
 8122 
 8189 
 
 8254 
 8319 
 8382 
 
 8445 
 8506 
 
 8567 
 8627 
 8686 
 
 8745 
 8S02 
 
 9117 
 ■0 
 
 942c 
 
 9425 
 9460 9465I9469I9474 
 
 9509 
 9557 
 
 9513 9518 9523 
 9562 9566 9S7I 
 
 9600 9605 9609 9614 9619 9624 9628 
 
 9657 9661 9666 
 9703 9708 9713 
 
 9750 9754 9759 
 9795 980Q 9805 
 
 8904 
 8960 
 
 9015 
 9069 
 
 9122 
 
 917s 
 9227 
 
 9279 
 9330 
 
 9380 
 9430 
 9479 
 9528 
 9576 
 
 8SS4 8859 
 
 9845 98SO 
 
 9886 989a 9894 9899 
 
 8910 
 8965 
 9020 
 9074 
 
 912S 
 9180 
 9232 
 9284 
 9335 
 
 9385 
 9435 
 9484 
 
 9533 
 9581 
 
 9671 
 9717 
 9763 
 9809 
 
 9854 
 
 9943 
 99b7 
 
 8915 
 8971 
 502s 
 9079 
 
 9133 
 918b 
 9238 
 9289 
 9340 
 
 9390 
 9 HO 
 9489 
 9538 
 9586 
 
 9533 
 9680 
 9727 
 9773 
 
 9675 
 9722 
 9768 
 98149816 
 
 98599863 
 99039908 
 9948 9952 
 9991 9996 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 5 
 5 
 4 
 4 
 4 
 
 4 
 4 
 4 
 4 
 4 
 
 4 
 4 
 4 
 4 
 4 
 
 4 4 
 
 4 
 
 4 
 
 4 
 
 3 
 
 3 
 3 
 3 
 3 
 3 
 
 3 
 3 
 3 
 3 
 3 
 
 3 
 3 
 3 
 3 
 3 
 
 3 
 3 
 3 
 3 
 3 
 
 3 3 
 
 3 3 
 
 3 3 
 
 3 3 
 
3 58 Practical Electricity and Magnetism. 
 
 Antilogarithms. 
 
 4 6 6 
 
 8 9 1234 6 67 
 
 ■03 
 •04 
 
 •OS 
 
 ■06 
 •07 
 ■08 
 •09 
 •10 
 
 ■13 
 ■14 
 •IS 
 
 •16 
 
 ■17 
 ■18 
 
 ■19 
 
 ■20 
 
 "23 
 ■=4 
 •2S 
 
 •a6 
 •27 
 -28 
 •29 
 ■30 
 
 •31 
 ■32 
 •33 
 ■34 
 •3S 
 
 •36 
 ■37 
 •38 
 ■39 
 ■40 
 
 •4> 
 •42 
 •43 
 ■44 
 ■45 
 
 1000 
 
 1023 
 1047 
 1072 
 1096 
 1122 
 
 1148 
 
 "75 
 1202 
 1230 
 '2S9 
 
 1288 
 1318 
 
 1349 
 1380 
 
 1413 
 
 144s 
 1479 
 1514 
 IS49 
 <S8S 
 
 1622 
 1660 
 1698 
 1738 
 1778 
 
 1820 
 1862 
 190S 
 1950 
 1995 
 
 2042 
 2089 
 2138 
 2188 
 2239 
 
 1002 
 
 1026 
 1050 
 1074 
 1099 
 1125 
 
 "SI 
 1 178 
 1205 
 
 1233 
 1262 
 
 1291 
 1321 
 1352 
 1384 
 1416 
 
 1449 
 1483 
 1517 
 
 1552 
 IS89 
 
 1626 
 1663 
 1702 
 1742 
 1782 
 
 1005 
 
 1028 
 1052 
 1076 
 1 102 
 1127 
 
 "S3 
 1 180 
 1208 
 1236 
 1265 
 
 1294 
 1324 
 I3S5 
 1387 
 1419 
 
 I4S2 
 i486 
 1521 
 
 1556 
 1592 
 
 1629 
 1667 
 1706 
 1746 
 1786 
 
 1824 1828 
 
 1866 
 1910 
 
 I9S4 
 2000 
 
 2046 
 2094 
 
 2143 
 2193 
 2244 
 
 2291 
 2344 
 2399 
 24SS 
 2512 
 
 2570 
 2630 
 2692 
 
 27S4 
 2818 
 
 -46 2884 
 
 ■47 2951 
 
 •48 3020 
 
 •49 3090 
 
 2296 
 23SO 
 2404 
 2460 
 2518 
 
 2576 
 2636 
 2698 
 2761 
 2825 
 
 2891 
 2958 
 3027 
 3097 
 
 1871 
 1914 
 
 I9S9 
 2004 
 
 2051 
 2099 
 2148 
 219B 
 2249 
 
 2301 
 
 2355 
 2410 
 2466 
 2523 
 
 2582 
 2642 
 2704 
 2767 
 2831 
 
 2897 
 2965 
 
 3034 
 310S 
 
 1007 
 
 1030 
 1054 
 1079 
 1 104 
 1130 
 
 1156 
 1183 
 1211 
 1239 
 1268 
 
 1297 
 1327 
 I3S8 
 1390 
 1422 
 
 I4SS 
 1489 
 
 1S24 
 1S60 
 
 1596 
 
 1633 
 1671 
 1710 
 
 1750 
 1791 
 
 1832 
 
 1875 
 1919 
 1963 
 2009 
 
 2056 
 2104 
 
 2153 
 2203 
 2254 
 
 2307 
 2360 
 
 2415 
 2472 
 2529 
 
 icx>9 1012 
 
 '033 
 1057 
 1081 
 
 103s 
 1059 
 1084 
 
 1014 
 
 1038 
 1062 
 1086 
 
 1107 1109 
 
 1 132 1135 1 138 
 
 IIS9 "61 
 
 1 164 
 
 1186 [189 IZ91 
 
 1213 
 
 1242 
 
 1271 
 
 [2l6 
 
 1245 
 1274 
 
 1219 
 
 1247 
 1276 
 
 1300 1303 
 1330 1334 
 
 1361 
 
 1393 
 1426 
 
 I4S9 
 1493 
 1528 
 
 1563 
 1600 
 
 1637 
 167s 
 1714 
 1754 
 
 1365 
 
 1306 
 1337 
 1368 
 
 1393 1396 1400 
 1429 1432 
 
 1466 
 1500 
 IS35 
 1570 
 1607 
 
 1644 
 
 1462 
 1496 
 1531 
 •567 
 i6o^ 
 
 1496 1500 
 
 1567 1570 
 
 1641 
 1679 1683 
 
 1718 
 '7S8 
 
 1795 '799 
 
 1722 
 1762 
 1803 
 
 1837 
 1879 
 1923 
 1968 
 2014 
 
 1841 1845 
 
 " 1888 
 
 1932 
 
 1977 
 
 2023 
 
 1016 
 
 1040 
 1064 
 1089 
 1114 
 1140 
 
 1167 
 
 "04 
 1222 
 1250 
 1279 
 
 1309 
 1340 
 1371 
 1403 
 
 143s 
 
 1469 
 1503 
 1538 
 IS74 
 161 1 
 
 1019 
 
 1042 
 1067 
 1091 
 1117 
 1143 
 1 169 
 
 "97 
 1225 
 
 125^ 
 1282 
 
 1312 
 1343 
 1374 
 1406 
 
 1439 
 
 1472 
 IS07 
 1542 
 1578 
 1614 
 
 1648 1652 
 1687 1690 
 
 1928 
 1972 
 
 20l8 
 
 2158 
 
 2208 
 
 2065 
 2II3 
 
 2163 
 
 2061 
 
 2109 211312: 
 
 2070 
 
 :Il8 
 
 2i68 
 
 Z213 2218 
 
 2259 Z265 Z270 
 
 2317 2323 
 
 1366 2371 2377 
 
 2427 2432 
 
 231a 
 
 2; ' 
 
 2421 
 
 2477 
 
 2483 
 
 2535 2541 
 
 2588 
 2649 
 2710 
 
 2773 
 2838 
 
 2904 
 2972 
 
 3041 
 3112 
 
 2594 
 
 2600 
 
 2655 2661 
 
 2716 27: 
 
 2780 2786 
 2844 
 
 2489 
 2547 
 
 2606 
 2667 
 2729 
 
 2793 
 2858 
 
 2911 
 
 2979 
 3048 
 
 3119 
 
 23 
 
 86 
 
 2851 
 
 2917 2924 
 
 2985 
 3055 
 
 2992 
 3062 
 
 3126 3133 
 
 1726 
 1766 
 1807 
 
 1849 
 1892 
 1936 
 1982 
 2028 
 
 2075 
 2123 
 
 2173 
 2223 
 
 2275 
 
 2328 
 2382 
 2438 
 2495 
 2SS3 
 
 2612 
 2673 
 
 273s 
 2799 
 2864 
 
 2931 
 
 2999 
 3069 
 
 314I- 
 
 1730 
 1770 
 1811 
 
 1854 
 1897 
 1941 
 1986 
 2032 
 
 2080 
 
 2128 
 2178 
 2228 
 2280 
 
 2333 
 2388 
 
 2443 
 2500 
 
 2559 
 
 2618 
 2679 
 2742 
 2805 
 2871 
 
 2938 
 3006 
 3076 
 3148 
 
 lOZI 
 
 1045 
 1069 
 1094 
 1119 
 1 146 
 
 1 172 
 
 "99 
 1227 
 1256 
 1285 
 
 131S 
 1346 
 1377 
 1409 
 
 1442 
 
 1476 
 1510 
 545 
 1581 
 1618 
 
 1656 
 1694 
 1734 
 1774 
 1816 
 
 1901 
 
 1945 
 1991 
 2037 
 
 2084 
 
 2133 
 2183 
 2234 
 2286 
 
 2339 
 2393 
 2449 
 2506 
 
 2564 
 
 2624 
 2685 
 2748 
 2812 
 2877 
 
 2944 
 3013 
 3083 
 31SS 
 
 o o 
 o o 
 o o 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 O I 
 
 I 
 
 1 I 
 I I 
 
 X I 
 
 I I 
 
 I I 
 I I 
 I I 
 I I 
 I I 
 
 I I 
 I I 
 
 I 2 
 
 I 2 
 
 I 2 
 
 I 2 
 
 I 2 
 
 I 2 
 
 I 2 
 
 I 2 
 
 1 2 
 
 2 2 
 2 2 
 
 2 2 
 
 2 2 
 
 2 2 
 
 2 2 
 
 2 2 
 
 2 2 
 
 2 2 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 I 2 
 
 I 
 I 
 
 1 2 
 
 2 2 
 2 2 
 
 2 2 
 2 2 
 2 2 
 2 
 2 
 
 2 
 3 
 
 3 
 
 2 3 
 3 
 3 
 3 
 3 
 
 3 
 
 3 3 
 3 3 
 3 3 
 3 3 
 
A ntilogarithmi. 
 
 3S9 
 
 Antilogarithms. 
 
 2 3 4 
 
 6 7 8 9 1234 
 
 6 7 
 
 8 9 
 
 3162 
 
 3=36 
 33" 
 
 •S3 13388 
 ■54 3467 
 •55 3548 
 
 •56 
 •57 
 •58 
 •59 
 •60 
 
 •61 
 •62 
 
 •63 
 •64 
 
 ■65 
 
 •66 
 •67 
 ■68 
 •69 
 
 70 
 
 ■71 
 ■72 
 
 •73 
 74 
 •75 
 
 •76 
 
 "77 
 •78 
 
 •79 
 •80 
 
 •81 
 -82 
 
 •83 
 •84 
 •8s 
 
 •86 
 •87 
 •88 
 •89 
 ■90 
 
 •91 
 •92 
 
 •93 
 •94 
 "95 
 
 •96 
 
 ■97 
 •98 
 •99 
 
 3631 
 3715 
 3802 
 3890 
 3981 
 
 4074 
 4169 
 4266 
 
 4365 
 4467 
 
 4571 
 4677 
 4786 
 4898 
 501Z 
 
 5129 
 5248 
 5370 
 5495 
 5623 
 
 5754 
 5888 
 6026 
 6166 
 
 3170 
 
 3243 
 3319 
 3396 
 3475 
 3556 
 
 3639 
 3724 
 3811 
 
 3899 
 3990 
 
 4083 4093 
 4178 4188 
 4276 4285 
 4375 
 4477 
 
 3177 
 
 3251 
 3327 
 3404 
 3483 
 356: 
 
 3648 
 3733 
 3819 
 3908 
 3999 
 
 6310 6324 
 
 6457 
 6607 
 6761 
 
 5508 
 5636 
 
 5768 
 5902 
 
 7079 
 7244 
 7413 
 7586 
 7762 
 
 7943 
 
 8128 
 8318 
 8511 
 8710 
 8913 
 
 9120 
 9333 
 9550 
 9772 
 
 4385 
 4487 
 
 4581 4592 
 4688^4699 
 479714808 
 49094920 
 5023J503S 
 
 514015152 
 5260:5272 
 
 53835395 
 5521 
 5649 
 
 3184 
 
 3258 
 3334 
 3412 
 3491 
 3573 
 
 3656 
 
 3741 
 3828 
 
 3917 
 4009 
 
 4102 
 4198 
 4295 
 4395 
 4498 
 
 3192 3199 3206 
 3266 3273 3281 
 
 3214 
 
 3221 
 
 32893296 
 
 3342 3350 
 
 3357 
 
 3420 3428 3436 
 
 3499 3Soi 
 3581 ■ 
 
 3664 
 3750 
 
 8 3516 
 3589 3597 
 3673 3681 
 
 3365 
 3443 
 3524 
 3606 
 
 3373 
 3451 
 3532 
 3614 
 
 3758 
 
 3767 
 
 3837 3846 3855 
 3926 3936 394S 
 
 36903698 
 37763784 
 
 4018 
 
 ^.. , 3954 
 
 ^027 4036 4046 
 
 4111 4121 
 
 4130 
 
 4207 4217 4227 
 4305 4315 4325 
 4406 4416 4426 
 4508 4519 4529 
 
 4603 4613 4624 4634 
 4732 4742 
 4842 4853 
 4943 4955 49^6 
 5058 5070 5082 
 
 5781 
 5916 
 
 6039,6053 
 6i8o|r 
 
 6339 
 
 4710 
 4819 
 4932 
 5047 
 
 5164 
 5284 
 5408 
 5534 
 5662 
 
 5794 
 5929 
 6067 
 
 61946209 
 6353 
 
 6471 
 6622 
 6776 
 
 6486 
 6637 
 6792 
 
 6918 6934 6950 
 
 7096 
 
 7261 
 7430 
 7603 
 
 77807798 
 
 8147 
 8337 
 8531 
 8730 
 8933 
 
 9141 
 9354 
 9572 
 9795 
 
 7112 
 
 7278 
 7447 
 7621 
 
 796279807998 
 
 8551 
 8750 
 
 9162 
 9376 
 9594 
 9817 
 
 4721 
 4831 
 
 5176 5188 
 5297 
 
 5420 
 
 5297 5309 5321 
 
 j82i 5834 
 5943 5957 597° 
 
 5808 
 
 5943 
 6081 
 
 6501 
 
 6653 
 680S 
 6966 
 7129 
 
 729s 
 7464 
 7638 
 
 8166 
 8356837s 
 
 8570 
 8770 
 
 89548974 
 
 9183 
 9397 
 9616 
 9840 
 
 5433 5445 
 j546 5559 5572 
 5675 5689 57 
 
 5095 6109 
 6223 6237 6252 
 0368 5383 5397 
 
 6516 5531 6546I6561 
 
 66686683 
 
 6823 6839 58; 
 6982 5998 7015 
 
 7816 7834 
 
 8017 
 
 4140 
 4236 
 4335 
 4436 
 4539 
 
 4645 
 
 4753 
 4864 
 
 4977 
 5093 
 
 3873 
 3963 
 4055 
 
 4150 
 4246 
 
 4345 
 4446 
 
 4550 
 
 3228 
 
 3304 
 3381 
 3459 
 3540 
 3622 
 
 3707 
 3793 
 3882 
 
 3972 
 4064 
 
 4159 
 4256 
 4355 
 4457 
 4560 
 
 46564667 
 
 47''4 
 4875 
 4989 
 5105 
 
 5200 5212 5224 
 5333 5346 
 
 5699 
 
 55 
 
 7145 7161 
 
 7178 
 
 73" 7328 7345 
 7482 7499 7516 
 
 7656 7674 769: 
 
 81858204 3222 3241 
 
 839s 8414 8433 
 8590 8610 3630 
 8790 8810 3831 
 8995 9016 9036 
 
 9419 J441 
 9638 9661 
 9863 9886 9908 
 
 5458 
 5585 
 5715 
 
 5848 
 5984 
 6124 
 6266 
 6412 
 
 5470 
 5598 
 5728 5741 
 
 7852 7870 
 B03S 8054 
 
 6714 
 6871 
 7031 
 7194 
 
 7362 
 
 '7534 
 7709 
 
 4775 
 4887 
 5000 
 5117 
 
 5*36 
 5358 
 5483 
 5610 
 
 5861 
 5998 
 6138 
 6281 
 6427 
 
 6577 
 6730 
 6887 
 7047 
 72" 
 
 7379 
 7551 
 7727 
 
 7889i79°7 
 80728091 
 
 86508670 
 8851I8872 
 905719078 
 
 9204 9226 9247 9268 
 
 9462 
 19683 
 
 8260 
 8453 
 
 8279 
 8472 
 
 9290 
 9484I9506 
 9705I9727 
 99319954 
 
 587s 
 6012 
 6152 
 6295 
 6442 
 6592 
 
 674s 
 6902 
 7063 
 7228 
 
 7396 
 7568 
 
 7745 
 7925 
 8110 
 
 8299 
 8492 
 8690 
 8892 
 9099 
 
 93" 
 9528 
 9750 
 9977 
 
 2 
 3 
 3 
 3 
 
 3 
 3 
 3 
 3 
 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 4 
 
 2 4 
 
 2 4 
 
 2 4 
 2 4 
 
 2 4 
 2 4 
 2 4 
 
 2 4 
 
 2 4 
 
 2 4 
 
 2 S 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 3 3 
 3 3 
 3 4 
 3 4 
 34 
 
 3 4 
 34 
 3 4 
 3 4 
 3 4 
 
 3 4 
 
 3 4 
 
 3 4 
 
 3 5 
 
 4 5 
 
 4 S 
 
 4 5 
 46 
 46 
 46 
 
 56 
 56 
 
 5 6 
 5 7 
 
 5 7 
 5 7 
 S 7 
 
 5 7 
 67 
 
 6 8 
 6 8 
 6 8 
 6 8 
 6 8 
 
 4567 
 
 8 9 
 8 10 
 
 8 16 
 
 9 
 
 9 10 
 
 9 " 
 9 " 
 9 II 
 10 II 
 10 12 
 
 10 12 
 1012 
 
 11 12 
 II 12 
 II 13 
 
 6 
 6 
 6 
 6 
 7 
 
 7 
 7 
 7 
 7 
 7 
 
 89 
 89 
 8 9 
 8 9 
 
 8 9 
 
 9 10 
 9^ lO 
 9 10 
 9 10 
 9 II 
 
 lO II 
 
 10 II 
 
 10 IX 
 10 12 
 
 10 12 
 
 11 12 
 II 12 
 II 13 
 
 11 13 
 
 la 13 
 
 12 14 
 
 12 14 
 
 13 14 
 
 13 IS 
 13 IS 
 
 13 IS 
 
 14 16 
 14 16 
 
 14 16 
 
 15 17 
 
 11 13 15 17 
 
 12 14 15 17 
 12 -I4I16 18 
 
 12 14I16 l3 
 
 12 is|i7 19 
 
 13 IS 17 19 
 
 13 15 17 20 
 
 13 16 18 20 
 
 14 16 18 20 
 
36p 
 
 :al Electricity and Magnetism. 
 Squares. 
 
 IT 
 I "2 
 
 I "3 
 I '4 
 I'S 
 
 1-6 
 
 17 
 1-8 
 1'9 
 
 2'0 
 
 2T 
 2 '2 
 2'3 
 
 2-4 
 
 2'S 
 
 2-5 
 
 27 
 
 2-8 
 2-9 
 3-0 
 
 3"i 
 31 
 3'2 
 3*3 
 3'4 
 
 3-6 
 37 
 3-8 
 3"9 
 4-0 
 
 I '210 
 
 1-440 
 I "690 
 I '960 
 2-250 
 
 2-560 
 2-890 
 3-240 
 3-610 
 4-000 
 
 4-410 
 4-840 
 5-290 
 5-760 
 6-250 
 
 6-760 
 7-290 
 7-840 
 8-410 
 9"ooo 
 
 9-610 
 
 10-24 
 10-89 
 11-56 
 
 3"S I2'2S 
 
 12-96 
 13-69 
 14-44 
 15-21 
 16-00 
 
 ^ 16-81 
 
 •2 17-64 
 
 ■3 18-49 
 
 ■4 I9'36 
 
 •5 20 
 
 S'3 
 S-4 
 
 ■25 
 
 21-16 
 22-09 
 23-04 
 24-01 
 
 4-6 
 
 47 
 4-8 
 
 4"9 
 
 5-0 25-00 
 
 26-01 
 27-04 
 28-09 
 29-16 
 
 1-232 
 1-464 
 1-716 
 1-988 
 2-280 
 
 2-592 
 2-924 
 3-276 
 3-648 
 4-040 
 
 4'4S2 
 4-884 
 
 S'336 
 5-808 
 §•300 
 
 6-812 
 
 7"344 
 7-896 
 8-468 
 9"o6o 
 
 9-672 
 
 10-30 
 10-96 
 11-63 
 12-32 
 
 r3'03 
 13-76 
 
 I4'S2 
 IS '29 
 16 -oS 
 
 16-89 
 17-72 
 18-58 
 
 I9-4S 
 20-34 
 
 21-25 
 22-18 
 
 2314 
 24-11 
 25-10 
 
 26-11 
 27-14 
 28-20 
 29-27 
 
 2 3 
 
 1-040 
 
 I '254 
 1-488 
 1-742 
 2 '016 
 2-310 
 
 2-624 
 2-958 
 3312 
 3-686 
 4-080 
 
 4"494 
 4-928 
 5-382 
 5-'i56 
 6 '350 
 
 6-864 
 T\ 
 
 7 '952 
 3-526 
 9-120 
 
 9734 
 
 10-37 
 ii-oa 
 11-70 
 12-39 
 
 13-10 
 13-84 
 14-59 
 15-37 
 16-16 
 
 16-97 
 17-81 
 18-66 
 
 19-54 
 20-43 
 
 21-34 
 22-28 
 23-23 
 24-21 
 25-20 
 
 26.21 
 27-25 
 28-30 
 29-38 
 
 I -061 
 
 1-277 
 
 1-513 
 1-769 
 2-045 
 2-341 
 
 2-657 
 2-993 
 3-349 
 3-725 
 4 -121 
 
 4-537 
 4-973 
 5-429 
 5-905 
 6-401 
 
 6-917 
 
 7-453 
 8-009 
 
 8-585 
 9-181 
 
 9-797 
 
 1-30011 
 
 1-538 
 
 1-796 
 
 '323 
 1-563 
 1-823 
 2-074 2-103 
 ■372 2-403 
 
 2-69o|2-723 2-756 
 
 V '- 3-098 
 
 ■386 3-423 3-460 
 
 .'.. -3-842 
 162 4*203 4-244 
 
 10-43 
 11-09 
 11-76 
 
 1-082 
 
 I -103 
 
 4-580 4 
 5-018 
 5-476 5 
 5-954 6 
 6-452 5 
 
 -623 4-666 
 
 5-063)5-108 
 
 ■570 
 
 052 
 
 ■554 
 
 '970I7 -02317 -076 
 
 " _' 7-618 
 
 8 06618 -12318 -180 
 
 -762 
 
 •364 
 
 8-644 B-703J8 
 9-242 9,-303 9 
 
 13-18 
 13-91 
 14-67 
 15-44 
 16-24 
 
 17-06 
 17-89 
 
 18-75 
 19-62 
 
 13-25 
 13-99 
 14-75 
 15-52 
 16-32 
 
 19-71 
 
 20-5223-61 
 
 26-32 
 
 27-35 
 28-41 
 29-48 
 
 26-42 
 
 1*124 
 
 1-346 
 1-588 
 1*850 
 2-132 
 2-434 
 
 -5235 
 -003 5 
 
 -503 5 
 
 9-860 3-923 9;986 
 
 10-63 
 
 16 11-22 1I-! 
 
 11-90 11*97 
 12*46 i2-5;^i2-6o|i2-67 
 
 10-50I10-56 
 11-: ' 
 11-83 
 
 13-32 
 14 
 14 -i 
 
 15-' 
 16 
 
 -06 14 
 82 14 
 ■60 15 
 4c 16 
 
 I7-I4|I7'22 17 
 17-98 i8-o6|i8-: 
 
 18-84 18-92 19-01 
 
 21-44I21-53 21-62 
 22-37122-47 22-56 2! 
 
 23-33123-43 23 _ _ 
 24-30,24-40 24-50 24-60 
 25-30I25-40 25-50 25-60 
 
 26-52 
 
 27 _' 
 28-62 
 
 27-46 
 28-52 
 29-59 29-70I29 
 
 13-40 
 
 -14 
 ■90 
 68 
 48 
 
 19-80 19-89 
 20 -7c 20-79 
 
 21*72 
 2*66 
 62 
 
 I -145 
 
 1-369 
 i'6i3 
 1-877 
 2 '161 
 2-465 
 
 2-789 
 3-133 
 3-497 
 3-881 
 4-285 
 
 4*709 
 5-153 
 5-617 
 6*101 
 6*605 
 
 7*129 
 7-673 
 8-237 
 8-821 
 
 9-425 
 
 10*05 
 10-69 
 11*36 
 12*04 
 12*74 
 
 I3'47 
 14*21 
 14*98 
 15-76 
 16*56 
 
 17-39 
 18-23 
 19-10 
 19-98 
 20 -88 
 
 8 9 1 S 3 4 
 
 ri66 
 
 1-392 
 1-638 
 1-904 
 2-190 
 2-496 
 
 2-822 
 3*168 
 
 3*534 
 3*920 
 4*326 
 
 4-752 
 5-198 
 5-664 
 6-150 
 6-656 
 
 7-182 
 7-728 
 8-294 
 8-880 
 9-486 
 
 IO*II 
 
 10-76 
 11*42 
 12-11 
 12-82 
 
 I3"S4 
 14-29 
 
 15-05 
 16*65 
 
 17-47 
 18*32 
 19*18 
 20*07 
 20*98 
 
 ■81 21 
 
 ;*75 22 
 *72 23 
 
 -7024 
 
 21 
 
 22 
 
 23 
 
 24. . 
 
 25-70 25-81 
 
 -56 27 
 
 26*63 
 •67 
 
 28*73 
 81 
 
 26*73 
 
 27-77 
 28*84 
 29*92 
 
 26*83 
 27-88 
 28-94 
 
 I -188 
 
 I -416 
 1-664 
 1*932 
 
 2*2^ 
 2*528 
 
 2*856 
 3*204 
 3-572 
 3*960 
 4*368 
 
 4*796 
 5-244 
 5-712 
 6-200 
 6-708 
 
 7-236 
 7-784 
 8-352 
 8-940 
 9*548 
 
 io*i8 
 10*82 
 H"49 
 
 I2*l8 
 
 12*89 
 
 i3"62 
 14-36 
 
 15-13 
 15-92 
 i6*73 
 
 17-56 
 18*40 
 19*27 
 
 20'l6 
 
 21*07 
 
 22'00 
 22-94 
 23-91 
 24-90 
 25*91 
 
 26:94 
 27*98 
 29*05 
 
 30"033°-M 
 
 5 " 
 
 5 " 
 
 6 II 
 6 12 
 6 
 
 6 13 
 
 I 
 I 
 I 
 I 
 I 
 
 6 8 
 
 7 9 
 
 7 10 
 
 8 II 
 
 9 12 
 9 12 
 
 10 13 
 10 14 
 
 " 15 
 12 16 
 
 12 16 
 
 13 17 
 
 13 18 
 
 14 19 
 
 15 20 
 
 15 20 
 
 16 21 
 
 16 22 
 
 17 23 
 
 18 24 
 
 18 24 
 
 19 2J 
 2 
 2 
 
 2 3 
 
 2 3 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 13 IS 
 
 14 16 
 
 15 17 
 
 13 16 18 
 
 14 17 20 
 
 15 19 22 
 
 20 23 
 
 21 24 
 
 22 26 
 
 23 27 
 
 25 29 
 
 26 30 
 
 27 31 
 
 28 33 
 
 29 34 
 
 31 36J41 46 
 
 32 37 42 48 
 
 6 7 8 9 
 
 17 19 
 
 18 21 
 20 22 
 
 22 24 
 
 23 26 
 
 25 28 
 
 26 30 
 28 31 
 
 30 33 
 
 31 35 
 
 33 37 
 
 34 39 
 36 40 
 
 38 42 
 
 39 44 
 
 33 38 
 
 34 40 
 
 35 41 
 
 37 43 
 
 38 44 
 4 5 
 4 
 4 
 4 
 4 
 
 44 49 
 
 46 51 
 
 47 S3 
 
 49 55 
 
 50 57 
 ■S 6 
 
 6 7 
 
 6 7 
 
 6 7 
 
 7 8 
 
 8 9 
 
 8 9 
 .9 16 
 
 9 10 
 
Squares. 
 
 Squares. 
 
 36i 
 
 S-6 
 57 
 S-8 
 S'9 
 60 
 
 6-1 
 6-2 
 6-3 
 6-4 
 6-S 
 
 6-6 
 67 
 6-8 
 6-9 
 7-0 
 
 7'i 
 7-a 
 
 7"3 
 7'4 
 7S 
 
 30"2S 
 
 31 "36 
 3Z"49 
 33"64™ 
 34 "81 34'' 
 36-0036 
 
 30"36 
 
 37'2i 37-33 
 38-44138-56 
 39-69J39-82 
 40-96,41-09 
 
 42-2542-38 
 
 43 "56 43 '69 
 44 '8945 
 46-24-46-38 
 47-6i!47-75 
 49-00:49-14 
 
 SO-4i|so-SS 
 51-84151-98 
 
 53"44 
 S4'9i 
 56-40 
 
 41-2241-34 
 42-5142-64 
 
 S3 '29 
 5476 
 56-25 
 
 7-6 57-76 
 
 77 59 '29 
 7-8 60-84 
 7-9 62-41 
 8'o 64-00 
 
 57'9i 
 59"44 
 61 -oo 
 62-57 
 64-16 
 
 8-1 165-6165-77 
 8-2 67-24.67-40 
 68-8969-06 
 
 8-3 
 8-4 
 
 8-5 
 
 8-6 
 8-7 
 8-8 
 8-9 
 9-0 
 
 9-1 
 9-2 
 93 
 94 
 
 9'S 
 
 9-6 
 
 97 
 9-8 
 
 9'9 
 
 70-56 
 72-25 
 
 73'96 
 75"69 
 77'44 
 79-21 
 81-00 
 
 82-81 
 84-64 
 86-49 
 88-36 
 90-25 
 
 92-16 
 94-09 
 96-04 
 98-01 
 
 7073 
 72-42 
 
 74'I3 
 75-86 
 77-62 
 
 79-39 
 8i-i8 
 
 82-99 
 84-82 
 86-68 
 
 88-55 
 90-44 
 
 92-35 
 94-28 
 96-24 
 98-21 
 
 30-47 
 
 31-58 
 3272 
 33-87 
 35-05 
 36-24 
 
 37-45 
 38-69 
 
 39-94 
 
 30-58 
 
 31-70 
 32-83 
 33-99 
 35-16 
 3636 
 
 37-58 
 38-81 
 40-07 
 
 43-82I43-96 44-09 
 4S"i6i45-29 45-43 
 
 30-69 30-80 30-91 
 
 31-8131 
 32-95 33 -o6|33-: 
 34'" 34 
 35-28 35 
 36-48 36 
 
 37 
 
 3894 
 
 40-1 
 
 41 
 42-77 
 
 7o|37 82 37-9538-07 
 
 39-19I39-31 
 
 '20I40-32 40-45140-58 
 
 47J4i-6c|4i -73141 -86 
 
 -03J43-16 
 
 ■2244-: 
 
 46-51 
 47-89 
 
 49-23 
 
 46-6514679 
 48-0248-16 
 49-42 49 
 
 56 45 -; 
 
 50-69 50-84 
 52-I3I52-27 
 53-5853-73 
 
 55-06 
 56-55 
 
 55-20 
 56-70 
 
 58-0658-22 
 59-60 59-75 
 
 61-15161-31 
 62-73;62-88 
 64-32164-48 
 
 65-93.66-10 
 
 67-57167-73 
 69-2269-39 
 70-90I71-06 
 72-5972-76 
 
 74-30 
 76-04 
 77-79 
 79-57 
 81-36 
 
 83-17 
 85-01 
 86-86 
 88-74 
 90-63 
 
 92-54 
 94-48 
 
 96-43 
 98-41 
 
 74-- 
 76-21 
 
 77-97 
 79-74 
 81-54 
 
 83-36 
 85-19 
 87-05 
 88-92 
 90-82 
 
 92-74 
 94-67 
 96-63 
 98-60 
 
 -36I44-49 
 
 . -7045-83 
 
 47-0647-20 
 
 . _ 48-4448-58 
 
 ■56J49-7c[49 -84 49-98 
 
 44 -i 
 45-. 
 46*92 
 48-30 
 
 8-42 52 
 
 50-98 
 
 52 
 
 53-88 
 
 55 
 
 56-85 57-00 
 
 !-35 55 
 
 51-12 
 
 56 
 54-02 
 50 
 
 '•3976 
 
 83-54 83 
 85-38 85 
 87-24 87 
 89-11 89 
 91-01 
 
 '92 32 
 
 ■2234 
 
 ;-4o 35 
 
 ■6036 
 
 '9° 43 ■' 
 
 58-37 58 -52158 -68 
 59-91 6o-o6|6o-22 
 61 -47 Si 
 63-04 53 
 64-64 
 
 ;-62 Si 
 
 i-42 56 
 
 66-26 66 
 67-9058-06168-: 
 69-56 S9-72 
 71-2371-40171 
 72-93 73 
 
 74-65 
 
 76. 
 
 78-15 
 
 79-92 
 
 81 -72 3l -90 3: 
 
 74-82 
 
 •56 
 78-32 
 80 "10 
 ii-( 
 
 92-93l93'i2|93 
 94-8795-06 
 96-83 97-02 97 
 98-80 99 -oc 
 
 31-02 
 
 32-15 
 33-29 
 34-46 
 3564 
 36-84 
 
 44-6244-76 
 45-9746-10 
 47-3347-47 
 
 51-27 
 5271 
 54-17 
 55-65 
 57-15 
 
 78 
 
 _ .36 
 
 54-80 54-96 
 
 (-20 S3-; 
 
 l-ic 73 
 
 ■59 
 
 23 
 
 59-89 
 
 ■57 
 ■27 
 
 75-00 
 76-74 
 78-50 
 80-28 
 2 -08 
 
 72 83 -( 
 
 91 
 75 
 •61 
 
 ■49 
 91-20191-39 
 
 ■56 B5-; 
 ■42 87-6 
 -30 89 -< 
 
 -32 
 95-26 
 
 -22 
 99-20 
 
 51-41 
 52-85 
 54-32 
 55-80 
 57-30 
 
 58-83 
 60-37 
 61-94 
 63-52 
 65-12 
 
 66-75 
 68-39 
 70-06 
 71-74 
 73-44 
 
 75-17 
 76-91 
 78-68 
 80-46 
 82-26 
 
 84-09 
 
 85-93 
 87-8' 
 89-68 
 91-58 
 
 93-51 
 95-45 
 97-42 
 99-40 
 
 48-72 
 50-13 
 
 51-55 
 53-00 
 
 54-46 54-61 
 55-95 56-10 
 57-46 57-61 
 
 31-14 
 
 32-26 
 33-41 
 34-57 
 3576 
 36-97 
 
 38-19 
 39"44 
 40-70 
 41-99 
 43-30 
 
 31-25 
 
 34-69 1 2 
 
 1 2 
 
 -38 I 
 52 I 
 
 -88 I 
 •09 I 
 
 38-32 
 39"56 
 40-83 
 42-12 
 43*43 
 
 48-86 
 50-27 
 
 5170 
 53-14 
 
 58-98 
 
 60-53 
 62-09 
 63-68 
 65-29 
 
 66-91 
 68-56 
 70-22 
 71-91 
 7362 
 
 7S"34 
 77-09 
 78-85 
 80-64 
 
 82-45 
 
 84-27 
 86-12 
 87-98 
 
 89-87 
 91-78 
 
 93-70 
 95-65 
 97-61 
 99-60 
 
 59-14 
 6o-68 
 62-25 
 63-84 
 65-45 
 
 67-08 
 68-72 
 
 70-39 
 72-08 
 
 7379 
 
 7S'52 a 
 77-26 2 
 79-03 2 
 80-82 2 
 82-63 2 
 
 84-46 
 86-30 
 88-17 
 90*06 
 91-97 
 
 93-90 
 95-84 
 97-81 
 99-80 
 
 3 4 
 
 I 3 
 
 2 3 
 
 3 
 4 
 4 
 
 4 
 
 3 4| 
 
 3 
 3 
 4 
 4 
 4 
 
 S 
 S 
 5 
 5 
 5 
 
 S 
 S 
 S 
 5 
 S 
 
 S 
 S 
 5 
 5 
 S 
 
 S 
 6 
 6 
 6 
 6 
 
 6 
 6 
 6 
 6 
 
 6 7 
 
 8 9 
 
 7 8 9 10 
 
 8 
 8 
 8 10 
 8 lo 
 
 8 10 
 
 9 10 
 9 10 
 9 
 9 
 9 II 
 
 9 " 
 
 9 
 
 9 " 
 10 II 
 10 II 
 
 10 11 
 10 12 
 10 12 
 10 12 
 
 10 12 
 
 ro 12 
 
 11 12 
 II 12 
 II 13 
 11 13 
 
 II 13 
 II 13 
 II 13 
 II 13 
 II 13 
 
 9 ia 
 9 10 
 
 9 " 
 10 11 
 10 II 
 
 10 II 
 10 II 
 10 II 
 10 12 
 
 10 12 
 
 11 12 
 II 12 
 II 12 
 
 13 
 II 13 
 
 11 13 
 
 12 13 
 12 13 
 12 13 
 12 14 
 
 12 14 
 
 12 14 
 
 13 14 
 13 14 
 13 H 
 
 13^5 
 3 IS 
 
 13 IS 
 
 14 IS 
 14 IS 
 
 14 16 
 14 .16 
 14 .i6 
 14 16 
 
 14 16 
 
 15 16 
 IS 17 
 IS 17 
 15 17 
 15 17 
 
 12 14 IS 17 
 
 4 
 
 12 14 
 12 14 
 
 16 18 
 16 18 
 l6 18 
 
362 Practical Electricity and Magnetism. 
 
 Reciprocals of Numbers from iooo to 9999. 
 
 1 2 3 4 6 6 7 8 9123466789 
 
 0'00I0300 
 
 0*0009091 
 0-0008333 
 0*0007692 
 0*0007143 
 0*1 
 
 9901 
 
 9009 
 8264 
 
 7634 
 7092 
 
 *ooo6667 6623 
 
 9804 
 
 8929 
 8197 
 
 7576 
 7042 
 
 9709 
 
 8850 
 8130 
 7519 
 6993 
 
 9<5i5 9524 
 
 8772 
 8065 
 
 9434 
 B621 
 
 8696 
 
 8000 7937 
 7463 7407 7353 
 
 0*00:6250 
 0*0005882 
 0*0005556 
 0*0005263 
 0*0005000 
 
 0*0004762 
 0*0004545 
 0*0004348 
 0*0004167 
 0*0004000 
 
 0*0003846 
 0*0003704 
 0*0003571 
 0*0003448 
 0*0003333 
 
 0*0003226 
 00003125 
 0*0003030 
 0*0302941 
 00002857 
 
 0*0002778 
 0*0002703 
 0*0002632 
 0*0002564 
 0*0002500 
 
 0*0002439 
 0*0002381 
 0*0002326 
 0*0002273 
 00002222 
 
 0*0002174 
 0*0002128 
 0*0002083 
 0*0002041 
 0*0002000 
 
 0*0001961 
 00001923 
 0*0001887 
 0*0001852 
 
 6211 
 
 5848 
 5525 
 5236 
 497S 
 
 4739 
 4525 
 4329 
 4149 
 
 3984 
 
 3831 
 3690 
 3SS9 
 3436 
 3322 
 
 3215 
 3"S 
 3021 
 
 2933 
 2849 
 
 2770 
 2695 
 2625 
 2558 
 2494 
 
 2433 
 237s 
 2320 
 2268 
 2217 
 
 2169 
 2123 
 207Q 
 2037 
 1996 
 
 I9S7 
 1919 
 1883 
 1848 
 
 657965366494 
 6173 6135 609B 
 
 6897 5849 
 
 9346 
 
 8547 
 7874 
 7299 
 
 9259 
 
 8475 
 7813 
 7246 
 
 6452 
 5o6i 
 
 5814 
 5495 
 5208 
 
 4950 
 
 5780 
 
 5464 
 5181 
 
 6410 
 6024 
 
 6803 6757 
 6369 6329 6289 
 
 5747 
 5435 
 5155 
 
 5714 568: 
 
 5435 5405 
 5128 
 
 5376 
 5102 
 
 5988 
 5650 
 5348 
 5076 
 
 5952 
 5618 
 5319 
 
 4926 4902 4878 4854 4831 
 
 4695 4673 
 
 4651 
 4444 
 
 ♦255 
 40B2 
 
 , . . _ 4630 
 4464 4444 4425 
 4274 4255 4237 
 
 3922 3906 
 
 4098 
 3937 
 
 3788 
 3650 3636 
 
 4717 ._ 
 4505I4484 
 4310:4292 
 
 41324115 
 
 39683953 
 
 38173802 
 36763663 
 35463534 
 34253413 
 33113300 
 
 32053195 
 31063096 
 30123003 
 2924J2915 
 2841 2833 2825I2817 
 
 4065 
 
 3521 
 3401 
 3289 
 
 3774 3759 
 
 3623 
 
 3509 3497 
 
 3390 
 3279 
 
 2762275s 
 2688 2681 
 
 2618 
 
 2551 
 2488 
 
 2611 
 
 2545 
 2481 
 
 3378 
 3268 
 
 3185I3175 3165 
 3086 3077 3067 
 1994 2985 2976 
 2907 2899 2890 
 2809 
 
 2732 
 2660 
 2591 
 2525 
 
 2747 274c 
 2667 
 2597 
 2532 
 
 2674 
 
 2604 2597 
 2538 
 247s 
 
 2427 2421 
 2370J2364 
 
 2315 2309 
 2257 
 2208 
 
 2262 
 2212 
 
 2165 
 2119 
 
 207s 
 2033 
 1992 
 
 1953 
 1916 
 1880 
 1845 
 
 2469 2463 
 
 2410 2404 
 
 2353 2347 
 2304 2299 2294 
 
 2415 
 2358 
 
 2252 
 2203 
 
 224; 
 
 2160 
 2114 
 
 2070 
 2028 
 
 552151 
 
 2242 
 2193 
 
 Z146 
 
 2110 
 2066 
 
 2105 2101 
 2062 2058 
 2024 2020 2016 
 1984 1980 1976 
 
 1949 
 1912 
 1876 
 1842 
 
 1946 
 1908 
 
 1873 l86g 
 1838 ' 
 
 1942 II 
 
 1905 
 l86g 
 
 1835 
 
 938 
 1901 
 1866 
 1832 
 
 4608 
 
 4405 
 4219 
 4049 
 3891 
 
 3745 
 3610 
 
 3484 
 3367 
 3257 
 
 3155 
 3058 
 2967 
 2882 
 2S01 
 
 2725 
 2653 
 2584 
 2519 
 2457 
 
 2398 
 2342 
 228B 
 2237 
 2188 
 
 2141 
 2096 
 
 2053 
 2012 
 1972 
 
 1934 
 1898 
 1862 
 1828 
 
 S051 5025 
 4785 
 
 4587 4566 
 4386 - 
 4202 
 4032 
 3876 
 
 9174 
 
 8403 
 7752 
 7194 
 671 1 
 
 9 18 
 
 8 15 
 6 13 
 5 " 
 5 
 4 8 
 
 5917 
 5587 
 5291 
 
 3731 
 3597 
 3472 
 3356 
 3247 
 
 3145 
 3049 
 2959 
 2874 
 
 2793 
 
 2717 
 2646 
 2577 
 2513 
 2451 
 
 4367 
 4184 
 4016 
 3861 
 
 3717 
 3584 
 3460 
 3344 
 3236 
 
 3135 
 3040 
 2950 
 2865 
 2786 
 
 2710 
 2639 
 
 2571 
 2506 
 
 2445 
 
 2392 2387 
 
 2336 
 2283 
 2232 
 2183 
 
 2137 
 2092 
 2049 
 2008 
 1969 
 
 1931 
 1894 
 
 1859 
 1825 
 
 2331 
 2278 
 2227 
 2179 
 
 2132 
 2088 
 
 204s 
 2004 
 
 1965 
 
 1927 
 1890 
 
 1855 
 1821 
 
 27 36 
 
 23 30 
 19 26 
 16 22 
 14 19 
 13 17 
 
 " 15 
 10 13 
 9 12 
 8 II 
 7 
 
 10 
 
 5564 
 
 45 53 
 3845 
 33 38 
 29 33 
 25 29 
 
 22 26 
 
 20 23 
 17 20 
 16 18 
 14 17 
 
 13 15 
 12 14 
 II 13 
 10 12 
 9 
 
 8 10 
 8 9 
 
 I 
 I 
 
 I 2 
 
 73 82 
 
 61 68 
 51 58 
 44 49 
 3843 
 3338 
 
 *9 33 
 26 29 
 23 26 
 
 2t 24 
 1$ 21 
 
 I? 20 
 1$ 18 
 14 16 
 
 13 15 
 12 14 
 
 II 13 
 II 12 
 
 Id II 
 9I 10 
 9 io 
 
 N.B. — Three zeros follow the decimal point in the reciprocal of any four figure whole number 
 
 except the number looo. 
 
 Note. — Numbers in difTerence columns to be subtracted, not added. 
 
Reciprocals of Numben from looo to 9999. 3^3 
 Reciprocals of Numbers from iooo to 9999. 
 
 2 3 4 6 9 7 8 91S3466788 
 
 o'oooi8i8 
 
 o'cxxii786 
 o'030i7S4 
 0*0001724 
 o'oooi6qs 
 o"oODi667 
 
 o '0001639 
 0*0001613 
 0*0001587 
 0*0001563 
 0*0001538 
 
 0*0001515 
 0*0001493 
 0*0001471 
 0*0001449 
 0*0001429 
 
 0*0001408 
 0*0001389 
 0*0001370 
 0*0001351 
 0*0001333 
 
 0*0001316 
 0*000x299 
 0*0001282 
 0*0001266 
 0*0001250 
 
 0*0001235 
 0*0001220 
 0*0001205 
 ■0001190 
 0*0001176 
 
 0*0001163 
 0*0001149 
 0*0001136 
 0*0001124 
 0*0001111 
 
 0*0001099 
 0*0001087 
 0*0001075 
 0*0001064 
 0*0001053 
 
 0*0001042 
 0*0001031 
 0*0001020 
 0*0001010 
 
 181S 
 
 1783 
 1751 
 1721 
 1692 
 1664 
 
 1637 
 1610 
 
 1585 
 1560 
 
 1536 
 
 iS'3 
 1490 
 1468 
 
 1447 
 1427 
 
 1406 
 
 '387 
 1368 
 
 "35° 
 1332 
 
 1314 
 1297 
 \Q.io 
 [264 
 1248 
 
 233 
 
 12l8 
 
 1203 
 1189 
 "75 
 
 n6i 
 1 148 
 "35 
 
 1122 
 1110 
 
 1098 
 
 io86 
 1074 
 1063 
 1052 
 
 1041 
 1030 
 1019 
 1009 
 
 l8l2 
 
 1779 
 1748 
 
 1718 
 1689 
 1661 
 
 1634 
 1608 
 1582 
 
 1558 
 1534 
 
 15" 
 
 1488 
 1466 
 
 144s 
 1425 
 
 1404 
 1385 
 
 1366 
 
 1348 
 1330 
 
 I3I2 
 
 1279 
 1263 
 1247 
 
 1232 
 
 1217 
 1202 
 
 II88 
 
 "74 
 
 1160 
 "47 
 "34 
 
 1121 
 
 iiog 
 
 1096 
 1085 
 
 1073 
 1062 
 1050 
 
 1040 
 1029 
 1018 
 1008 
 
 1S08 
 
 1776 
 1745 
 171S 
 [686 
 1658 
 
 1631 
 1605 
 1580 
 1555 
 1531 
 
 1508 
 i486 
 1464 
 
 1443 
 1422 
 
 1403 
 1383 
 1364 
 1346 
 1328 
 
 13" 
 
 1294 
 1277 
 1261 
 1245 
 
 123D 
 1215 
 1200 
 1186 
 1 172 
 
 "59 
 "45 
 "33 
 1120 
 1107 
 
 1095 
 1083 
 1072 
 1060 
 1049 
 
 1038 
 1028 
 1017 
 1007 
 
 1805 
 
 1773 
 1742 
 1712 
 
 1802 
 1770 1 
 
 1799 
 767 
 
 1739 1736 
 
 1684 1681 
 
 1709 1706 
 
 1656 
 
 1653 
 
 1678 
 1650 
 
 1629 1626 1623 
 1603 1600 1597 
 1577 1575 1572 
 
 1553 
 
 1550 
 
 529 1527 
 506 1504 
 
 1484 
 146: 
 1441 
 142c 
 
 1401 
 1381 
 1362 
 
 1344 1342 
 1326 1325 
 
 1307 
 
 I: 
 
 176 1274 
 1258 
 
 1309 
 1292 
 12 
 
 1259 
 1244 1242 
 
 1229 1227 
 1214 1212 
 1199 1198 
 
 1548 
 1524 
 
 1502 
 1479 
 1458 
 1437 
 1416 
 
 1397 
 1377 
 1359 
 1340 
 1323 
 
 _ 1305 
 290 1289 
 
 1481 
 460 
 
 1439 
 1418 
 
 1399 
 1379 
 1361 
 
 1185 
 1171 
 
 1183 
 1170 
 
 571 
 
 156 
 
 "43 
 113c 1129 
 
 11 
 
 "44 
 
 1131 
 
 iiig "17I1 
 
 1106 1105 
 
 1272 
 1256 
 1241 
 
 1225 
 1211 
 1196 
 1182 
 1168 
 
 "55 
 1142 
 
 116 
 1104 
 
 1094 1093 
 
 1082 
 1071 
 
 1059 
 1048 
 
 1037 
 1027 
 1016 
 1006 
 
 1081 
 1070 
 1058 
 1047 
 
 1036 
 1026 
 1015 
 1005 
 
 1092 
 io8o 
 1068 
 1057 
 
 1795 
 
 1764 
 
 1733 
 1704 
 
 1675 
 1647 
 
 1621 
 1595 
 1570 
 1546 
 1522 
 
 1499 
 1477 
 1456 
 1435 
 1414 
 
 1395 
 1376 
 
 1357 
 1339 
 1321 
 
 1304 
 1287 
 1271 
 
 1255 
 1239 
 
 1224 
 1209 
 
 "95 
 1181 
 1167 
 
 "53 
 1140 
 1127 
 
 "15 
 "03 
 
 1091 
 1079 
 1067 
 1056 
 
 1046 1045 
 
 1792 
 
 1761 
 1730 
 1701 
 1672 
 1645 
 
 1618 
 1592 
 1567 
 1543 
 1520 
 
 1497 
 1475 
 1453 
 1433 
 1412 
 
 1393 
 1374 
 I3S5 
 1337 
 1319 
 
 1302 
 1285 
 1269 
 
 1253 
 1238 
 
 1222 
 1208 
 "93 
 "79 
 1166 
 
 "52 
 "39 
 1126 
 1114 
 1101 
 
 1089 
 1078 
 1066 
 1055 
 
 1789 
 
 1757 
 1727 
 1698 
 1669 
 642 
 
 1616 
 1590 
 1565 
 1541 
 1517 
 
 1495 
 1473 
 1451 
 1431 
 1410 
 
 1391 
 1372 
 
 1353 
 1335 
 1318 
 
 1300 
 12S4 
 1267 
 1252 
 1236 
 
 1221 
 1206 
 1192 
 1178 
 1164 
 
 "51 
 1138 
 1125 
 1112 
 1 100 
 
 1088 
 1076 
 1C65 
 1054 
 
 o o 
 o 
 
 1044 1043 
 
 1035 
 1025 
 
 1014 
 1004 
 
 1034 
 
 1024 
 1013 
 1003 
 
 1033 
 
 1022 
 1012 
 1002 
 
 1032 
 
 1021 
 1011 
 lOOI 
 
 N.B. — Three zeros follow the decimal paint in the reciprocal of any four figure whore number 
 
 except the number looo. 
 
 Note.— Numbers in difference columns to be subtracted, not added* 
 
364 Practiced Electricity and Magnetism. 
 
 Natural Tangents. 
 
 
 •0° 
 
 •1° 
 
 .go 
 
 .30 
 
 .40 
 
 •5° 
 
 .go 
 
 .70 
 
 .go 
 
 .90 
 
 0° 
 
 •0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 0157 
 
 I 
 
 •OI7S 
 
 0192 
 
 0209 
 
 0227 
 
 0244 
 
 0262 
 
 079 
 
 0297: 
 
 0314 
 
 0332 
 
 2 
 
 '0349 
 
 0367 
 
 0384 
 
 0402 
 
 0419 
 
 0437 
 
 0454 
 
 0472 
 
 0489 
 
 0507 
 
 3 
 
 •0524 
 
 0542 
 
 0559 
 
 0577 
 
 0594 
 
 0612 
 
 0629 
 
 0647 
 
 0664 
 
 0682 
 
 4 
 
 •o6q9 
 
 0717 
 
 0734 
 
 0752 
 
 0769 
 
 0787 
 
 080s 
 
 0822 
 
 0840 
 
 0857 
 
 S 
 
 •0875 
 
 0892 
 
 0910 
 
 0928 
 
 0945 
 
 0963 
 
 0981 
 
 0998 
 
 1016 
 
 '033 
 
 6 
 
 •105 1 
 
 1069 
 
 1086 
 
 1 104 
 
 1 122 
 
 "39 
 
 "57 
 
 "75 
 
 1192 
 
 1210 
 
 7 
 
 •1228 
 
 1246 
 
 1263 
 
 1281 
 
 1299 
 
 1317 
 
 1334 
 
 1352 
 
 1370 
 
 1388 
 
 8 
 
 •f40S 
 
 1423 
 
 1441 
 
 1459 
 
 1477 
 
 1495 
 
 1512 
 
 1530 
 
 1548 
 
 1566 
 
 9 
 
 •1584 
 
 1602 
 
 1620 
 
 1638 
 
 1655 
 
 1673 
 
 1691 
 
 1709 
 
 1727 
 
 1745 
 
 10 
 
 ■1763 
 
 1781 
 
 1799 
 
 1817 
 
 1835 
 
 1853 
 
 1871 
 
 1890 
 
 1908 
 
 1926 
 
 II 
 
 •1944 
 
 1962 
 
 1980 
 
 1998 
 
 2016 
 
 2035 
 
 2053 
 
 2071 
 
 2089 
 
 2107 
 
 12 
 
 •2126 
 
 2144 
 
 2162 
 
 2180 
 
 2199 
 
 2217 
 
 223s 
 
 2254 
 
 2272 
 
 2290 
 
 13 
 
 •2309 
 
 2327 
 
 234s 
 
 2364 
 
 2382 
 
 2401 
 
 2419 
 
 243s 
 
 2456 
 
 2475 
 
 J4 
 
 ■2493 
 
 2512 
 
 2530 
 
 2549 
 
 2568 
 
 2586 
 
 2605 
 
 2623 
 
 2642 
 
 2661 
 
 'S 
 
 ■2679 
 
 269S 
 
 2717 
 
 2736 
 
 2754 
 
 2773 
 
 2792 
 
 281 1 
 
 2830 
 
 2849 
 
 16 
 
 •2867 
 
 2886 
 
 2905 
 
 2924 
 
 2943 
 
 2962 
 
 2981 
 
 3000 
 
 3019 
 
 3038 
 
 17 
 
 ■3057 
 
 3076 
 
 3096 
 
 3115 
 
 3134 
 
 3153 
 
 3172 
 
 3191 
 
 3211 
 
 3230 
 
 18 
 
 •3249 
 
 3269 
 
 3288 
 
 3307 
 
 3327 
 
 3346 
 
 3365 
 
 3385 
 
 3404 
 
 3424 
 
 19 
 
 •3443 
 
 3463 
 
 3482 
 
 3502 
 
 3522 
 
 3541 
 
 3561 
 
 3581 
 
 3600 
 
 3620 
 
 20 
 
 •3640 
 
 3659 
 
 3679 
 
 3699 
 
 3719 
 
 3739 
 
 3759 
 
 3779 
 
 3799 
 
 3819 
 
 21 
 
 •3839 
 
 3859 
 
 3879 
 
 3899 
 
 3919 
 
 3939 
 
 3959 
 
 3979 
 
 4000 
 
 4020 
 
 22 
 
 •4040 
 
 4061 
 
 4081 
 
 4101 
 
 4122 
 
 4142 
 
 4163 
 
 4183 
 
 4i04 
 
 4224 
 
 23 
 
 •4245 
 
 4265 
 
 4286 
 
 4307 
 
 4327 
 
 4348 
 
 4369 
 
 4390 
 
 441 1 
 
 4431 
 
 24 
 
 •4452 
 
 4473 
 
 4494 
 
 451S 
 
 4536 
 
 4557 
 
 4578 
 
 4599 
 
 4621 
 
 4642 
 
 25 
 
 •4663 
 
 4684 
 
 4706 
 
 4727 
 
 4748 
 
 4770 
 
 4791 
 
 4813 
 
 4834 
 
 4856 
 
 26 
 
 ■4877 
 
 4899 
 
 4921 
 
 4942 
 
 4964 
 
 4986 
 
 5008 
 
 5029 
 
 5051 
 
 5073 
 
 27 
 
 •5095 
 
 SI17 
 
 S»39 
 
 S161 
 
 5184 
 
 5206 
 
 5228 
 
 5250 
 
 5272 
 
 5295 
 
 28 
 
 ■S3 '7 
 
 S340 
 
 5362 
 
 5384 
 
 5407 
 
 543° 
 
 5452 
 
 5475 
 
 5498 
 
 5520 
 
 29 
 
 ■5543 
 
 5566 
 
 5589 
 
 5612 
 
 5635 
 
 5658 
 
 5681 
 
 5704 
 
 5727 
 
 575° 
 
 30 
 
 •5774 
 
 5797 
 
 5820 
 
 5844 
 
 S867 
 
 5890 
 
 5914 
 
 5938 
 
 5961 
 
 59?5 
 
 31 
 
 •6009 
 
 6032 
 
 6056 
 
 6080 
 
 6104 
 
 6128 
 
 6152 
 
 6176 
 
 6200 
 
 6224 
 
 32 
 
 •6249 
 
 6273 
 
 6297 
 
 6322 
 
 6346 
 
 6371 
 
 639s 
 
 6420 
 
 6445 
 
 6469 
 
 33 
 
 •6494 
 
 6519 
 
 6544 
 
 6569 
 
 6594 
 
 6619 
 
 6644 
 
 6669 
 
 6694 
 
 6720 
 
 34 
 
 ■674s 
 
 6771 
 
 6796 
 
 6822 
 
 6847 
 
 6873 
 
 6899 
 
 6924 
 
 6950 
 
 6976 
 
 35 
 
 •7002 
 
 7028 
 
 7054 
 
 7080 
 
 7107 
 
 7133 
 
 7159 
 
 7186 
 
 7212 
 
 7239 
 
 36 
 
 •7265 
 
 7292 
 
 7319 
 
 7346 
 
 7373 
 
 7400 
 
 7427 
 
 7454 
 
 7481 
 
 7508 
 
 37 
 
 •7536 
 
 7563 
 
 7S90 
 
 7618 
 
 7646 
 
 7673 
 
 7701 
 
 7729 
 
 7757 
 
 7785 
 
 38 
 
 •7813 
 
 7841 
 
 7869 
 
 7898 
 
 7926 
 
 7954 
 
 7983 
 
 8012 
 
 8040 
 
 8069 
 
 39 
 
 ■8098 
 
 8127 
 
 8156 
 
 8185 
 
 8214 
 
 8243 
 
 8273 
 
 8302 
 
 8332 
 
 8361 
 
 40 
 
 •8391 
 
 8421 
 
 8451 
 
 8481 
 
 8511 
 
 8541 
 
 8571 
 
 8fcoi 
 
 8632 
 
 8662 
 
 41 
 
 •8693 
 
 8724 
 
 8754 
 
 8785 
 
 8816 
 
 8847 
 
 8878 
 
 8910 
 
 8941 
 
 8972 
 
 42 
 
 ■9004. 
 
 9036 
 
 9067 
 
 9099 
 
 9131 
 
 9163 
 
 9195 
 
 9228 
 
 9260 
 
 9293 
 
 43 
 
 •9325 
 
 93 S8 
 
 939> 
 
 9424 
 
 9457 
 
 9490 
 
 9523 
 
 9556 
 
 959° 
 
 9623 
 
 44 
 
 •9657 
 
 9691 
 
 9725 
 
 9759 
 
 9793 
 
 9827 
 
 9861 
 
 9896 
 
 9930 
 
 996s 
 

 
 
 Natural Tangents. 
 
 1 
 
 
 36S 
 
 Natural Tangents. 
 
 
 •0° 
 
 •1° 
 
 .go 
 
 .30 
 
 .40 
 
 •6° 
 
 .go 
 
 .70. 
 
 ■8° 
 
 , -S" 
 
 45° 
 
 I-CXXX3 
 
 003s 
 
 0070 
 
 0105 
 
 0I4I 
 
 0176 
 
 0212 
 
 0247 
 
 0283 
 
 0319 
 
 46 
 
 I -0355 
 
 0392 
 
 0428 
 
 0464 
 
 0501 
 
 0538 
 
 0575 
 
 0612 
 
 0649 
 
 0686 
 
 47 
 
 1-0724 
 
 0761 
 
 0799 
 
 0837 
 
 0875 
 
 0913 
 
 0951 
 
 0990 
 
 1028 
 
 1067 
 
 48 
 
 i'iio6 
 
 1 145 
 
 1 184 
 
 1224 
 
 1263 
 
 1303 
 
 1343 
 
 1383 
 
 1423 
 
 1463 
 
 49 
 
 1-1504 
 
 1544 
 
 1585 
 
 1626 
 
 ^^l 
 
 1708 
 
 1750 
 
 1792 
 
 1833 
 
 1875 
 
 5° 
 
 1-1918 
 
 i960 
 
 2002 
 
 2045 
 
 2088 
 
 2131 
 
 2174 
 
 2218 
 
 2261 
 
 2305 
 
 51 
 
 1-2349 
 
 2393 
 
 2437 
 
 2482 
 
 2527 
 
 2572 
 
 2617 
 
 2662 
 
 2708 
 
 2753 
 
 52 
 
 1-2799 
 
 2846 
 
 2892 
 
 2938 
 
 2985 
 
 3032 
 
 3079 
 
 3127 
 
 3175 
 
 3222 
 
 S3 
 
 1-3270 
 
 33 '9 
 
 3367 
 
 3416 
 
 3465 
 
 3SH 
 
 3564 
 
 3613 
 
 3663 
 
 3713 
 
 54 
 
 1-3764 
 
 3814 
 
 386S 
 
 3916 
 
 3968 
 
 4019 
 
 4071 
 
 4124 
 
 4176 
 
 4229 
 
 55 
 
 I -4281 
 
 4335 
 
 4388 
 
 4442 
 
 4496 
 
 4550 
 
 4605 
 
 4659 
 
 4715 
 
 4770 
 
 56 
 
 I -4826 
 
 4882 
 
 4938 
 
 4994 
 
 5051 
 
 5 108 
 
 5166 
 
 5224 
 
 s?f^ 
 
 5340 
 
 57 
 
 I '5399 
 
 5458 
 
 5517 
 
 5577 
 
 5637 
 
 5697 
 
 5757 
 
 .5818 
 
 5880 
 
 5941 
 
 58 
 
 1-6003 
 
 6066 
 
 6128 
 
 6191 
 
 6255 
 
 6319 
 
 6383 
 
 6447 
 
 6512 
 
 6577 
 
 59 
 
 1-6643 
 
 6709 
 
 6775 
 
 6842 
 
 6909 
 
 6977 
 
 7045 
 
 7113 
 
 7182 
 
 7251 
 
 60 
 
 17321 
 
 7391 
 
 7461 
 
 7532 
 
 7603 
 
 7675 
 
 7747 
 
 7820 
 
 7893 
 
 7966 
 
 61 
 
 I -8040 
 
 ^I'J 
 
 8190 
 
 8265 
 
 8341 
 
 8418 
 
 8495 
 
 8572 
 
 8650 
 
 8728 
 
 62 
 
 1-8807 
 
 8887 
 
 8907 
 
 9047 
 
 9128 
 
 9210 
 
 9292 
 
 9375 
 
 9458 
 
 9542 
 
 63 
 
 1-9626 
 
 97" 
 
 9797 
 
 9883 
 
 9970 
 
 0057 
 
 0145 
 
 0233 
 
 0323 
 
 0413 
 
 64 
 
 20503 
 
 0594 
 
 0686 
 
 0778 
 
 0872 
 
 0965 
 
 1060 
 
 "55 
 
 1251 
 
 1348 
 
 65 
 
 2; 1445 
 
 1543 
 
 1642 
 
 1742 
 
 1842 
 
 1943 
 
 2045 
 
 2148 
 
 2251 
 
 2355 
 
 66 
 
 2-2460 
 
 2566 
 
 2673 
 
 2781 
 
 2889 
 
 2998 
 
 3109 
 
 3220 
 
 3332 
 
 3445 
 
 67 
 
 2'3559 
 
 3673 
 
 3789 
 
 3906 
 
 4023 
 
 4142 
 
 4262 
 
 4383 
 
 4504 
 
 4627 
 
 68 
 
 2-4751 
 
 4876 
 
 5002 
 
 5129 
 
 5257 
 
 5386 
 
 5517 
 
 5649 
 
 5782 
 
 5916 
 
 69 
 
 2-6051 
 
 6187 
 
 6325 
 
 6464 
 
 6605 
 
 6746 
 
 
 7034 
 
 7179 
 
 7326 
 
 70 
 
 2-7475 
 
 7625 
 
 7776 
 
 7929 
 
 8083 
 
 8239 
 
 8397 
 
 8556 
 
 8716 
 
 8878 
 
 71 
 
 2-9042 
 
 9208 
 
 9375 
 
 9544 
 
 9714 
 
 9887 
 
 0061 
 
 0237 
 
 0415 
 
 059s 
 
 •72 
 
 3-0777 
 
 0961 
 
 1146 
 
 1334 
 
 1524 
 
 1716 
 
 1910 
 
 2106 
 
 2305 
 
 2506 
 
 73 
 
 32709 
 
 2914 
 
 3122 
 
 3332 
 
 3544 
 
 3759 
 
 3977 
 
 4197 
 
 4420 
 
 4646 
 
 74 
 
 3-4874 
 
 5>oS 
 
 5339 
 
 5576 
 
 5816 
 
 6059 
 
 6305 
 
 6554 
 
 6806 
 
 7062 
 
 75 
 
 3-7321 
 
 7583 
 
 7848 
 
 8118 
 
 8391 
 
 8667 
 
 8947 
 
 9232 
 
 9520 
 
 9812 
 
 76 
 
 4-0108 
 
 0408 
 
 0713 
 
 1022 
 
 1335 
 
 1653 
 
 1976 
 
 2303 
 
 2635 
 
 2972 
 
 77 
 
 4-3315 
 
 3662 
 
 4015 
 
 4374 
 
 4737 
 
 5107 
 
 5483 
 
 5864 
 
 6252 
 
 6646 
 
 78 
 
 4-7046 
 
 7453 
 
 7867 
 
 8288 
 
 8716 
 
 .9152 
 
 9594 
 
 0045 
 
 0504 
 
 0970 
 
 79 
 
 5-1446 
 
 1929 
 
 2422 
 
 2924 
 
 3435 
 
 3955 
 
 44:86 
 
 5026 
 
 5578 
 
 6140 
 
 80 
 
 5-6713 
 
 7297 
 
 7894 
 
 8502 
 
 9124 
 
 9758 
 
 0405 
 
 io66 
 
 1742 
 
 2432 
 
 81 
 
 6-3138 
 
 3859 
 
 4596 
 
 5350 
 
 6122 
 
 6912 
 
 7920 
 
 8548 
 
 9395 
 
 0264 
 
 82 
 
 7-1154 
 
 2066 
 
 3002 
 
 3962 
 
 4947 
 
 5958 
 
 6996 
 
 8062 
 
 9158 
 
 0285 
 
 83 
 
 8-1443 
 
 2636 
 
 3863 
 
 5126 
 
 6427 
 
 7769 
 
 9"52 
 
 0579 
 
 2052 
 
 3572 
 
 84 
 
 9-5144 
 
 9-677 
 
 9-845 
 
 1O-02 
 
 10-20 
 
 16-39 
 
 10-58 
 
 10-78 
 
 10-99 
 
 n-20 
 
 85 
 
 "•43 
 
 11-66 
 
 11-91 
 
 12-16 
 
 12-43 
 
 12-71 
 
 13-00 
 
 13-30 
 
 13-62 
 
 13-95 
 
 86 
 
 14-30 
 
 14-67 
 
 15-06 
 
 15-46 
 
 15-89 
 
 16-35 
 
 '^1l 
 
 17-34 
 
 17-89 
 
 18-46 
 
 87 
 
 19-08 
 
 19-74 
 
 20-45 
 
 21-20 
 
 22-02 
 
 22-90 
 
 23-86 
 
 24-90 
 
 26-03 
 
 27-27 
 
 88 
 
 28-64 
 
 30-14 
 
 31-82 
 
 33-69 
 
 35-80 
 
 38-19 
 
 40-92 
 
 44-07 
 
 '^27* 
 
 52-08 
 
 89 
 
 57-29 
 
 •63--66 
 
 71-52 
 
 8i^S 
 
 95-49 
 
 114-6 J 143-2 
 
 191-0 
 
 286-5 
 
 "S73-0 
 
366 Practical Electricity and Magnetism. 
 
 Natural Sines. 
 
 
 •0° 
 
 •1° 
 
 •2° 
 
 •3° 
 
 .40 
 
 •5° 
 
 •6° 
 
 .70 
 
 •8° 
 
 .90 
 
 o° 
 
 0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 00S7 
 
 0105 
 
 0122 
 
 0140 
 
 0157 
 
 I 
 
 0175 
 
 0192 
 
 0209 
 
 0227 
 
 0244 
 
 0262 
 
 0279 
 
 6297 
 
 0314 
 
 0332 
 
 2 
 
 0349 
 
 0366 
 
 0384 
 
 Q401 
 
 0419 
 
 0436 
 
 0454 
 
 9471 
 
 0488 
 
 0506 
 
 3 
 
 0523 
 
 0541 
 
 oss» 
 
 0576 
 
 0593 
 
 0610 
 
 0628 
 
 0645 
 
 0663 
 
 0680 
 
 4 
 
 0698 
 
 071S 
 
 0732 
 
 0750 
 
 0767 
 
 0785 
 
 0802 
 
 0819 
 
 0837 
 
 0854 
 
 5 
 
 0872 
 
 0889 
 
 0906 
 
 0924 
 
 0941 
 
 0958 
 
 0976 
 
 0993 
 
 lOII 
 
 1028 
 
 6 
 
 1045 
 
 1063 
 
 1080 
 
 1097 
 
 "Ji 
 
 1132 
 
 1149 
 
 1167 
 
 1 184 
 
 1201 
 
 7 
 
 1219 
 
 1236 
 
 1253 
 
 1271 
 
 1288 
 
 1305 
 
 •323 
 
 1340 
 
 1357 
 
 "374 
 
 8 
 
 1392 
 
 1409 
 
 1426 
 
 1444 
 
 146 1 
 
 1478 
 
 1495 
 
 1513 
 
 1530 
 
 1547 
 
 9 
 
 1564 
 
 1582 
 
 •599 
 
 1616 
 
 1633 
 
 1650 
 
 1668 
 
 1685 
 
 1702 
 
 1719 
 
 lO 
 
 1736 
 
 I7S4 
 
 1771 
 
 1788 
 
 iSos 
 
 1822 
 
 1840 
 
 1857 
 
 1874 
 
 1891 
 
 II 
 
 1908 
 
 192s 
 
 1942 
 
 1959 
 
 1977 
 
 1994 
 
 201 1 
 
 2028 
 
 204s 
 
 2062 
 
 12 
 
 2079 
 
 2096 
 
 2113 
 
 2130 
 
 2147 
 
 2164 
 
 2181 
 
 2198 
 
 2215 
 
 2232 
 
 13 
 
 2250 
 
 2267 
 
 2284 
 
 2300 
 
 2317 
 
 2334 
 
 2351 
 
 2368 
 
 2385 
 
 2402 
 
 H 
 
 2419 
 
 2436 
 
 2453 
 
 2470 
 
 2487 
 
 2504 
 
 2521 
 
 2538 
 
 2554 
 
 2571 
 
 IS 
 
 2588 
 
 2605 
 
 2622 
 
 2639 
 
 2656 
 
 2672 
 
 2689 
 
 2706 
 
 2723 
 
 2740 
 
 i6 
 
 2756 
 
 2773 
 
 2790 
 
 2807 
 
 28«3 
 
 2840 
 
 2857 
 
 2874 
 
 2890 
 
 2907 
 
 'Z 
 
 2924 
 
 2940 
 
 2957 
 
 2974 
 
 2990 
 
 3007 
 
 3024 
 
 3040 
 
 3057 
 
 3074 
 
 i8 
 
 3090 
 
 3107 
 
 3123 
 
 3140 
 
 3156 
 
 3173 
 
 3190 
 
 3206 
 
 3223 
 
 3239 
 
 19 
 
 3256 
 
 3272 
 
 3289 
 
 3305 
 
 3322 
 
 3338 
 
 3355 
 
 3371 
 
 3387 
 
 3404 
 
 20 
 
 3420 
 
 3437 
 
 3453 
 
 3469 
 
 3486 
 
 3502 
 
 3518 
 
 3535 
 
 3551 
 
 3567 
 
 21 
 
 3584 
 
 3600 
 
 3616 
 
 3633 
 
 3649 
 
 366S 
 
 3681 
 
 3697 
 
 3714 
 
 3730 
 
 22 
 
 3746 
 
 3762 
 
 3778 
 
 3795 
 
 3811 
 
 3827 
 
 3843 
 
 3859 
 
 3875 
 
 3891 
 
 23 
 
 3907 
 
 3923 
 
 3939 
 
 3955 
 
 3971 
 
 39S7 
 
 4003 
 
 4019 
 
 4035 
 
 4051 
 
 24 
 
 4067 
 
 4083 
 
 4099 
 
 41 15 
 
 4131 
 
 4147 
 
 4163 
 
 4179 
 
 4195 
 
 4210 
 
 25 
 
 4226 
 
 4242 
 
 4258 
 
 4274 
 
 4289 
 
 43°5 
 
 4321 
 
 4337 
 
 4352 
 
 4368 
 
 26 
 
 4384 
 
 4399 
 
 4415 
 
 4431 
 
 4446 
 
 4462 
 
 4478 
 
 4493 
 
 4509 
 
 4524 
 
 27 
 
 4540 
 
 4555 
 
 4571 
 
 4586 
 
 4602 
 
 4617 
 
 4633 
 
 4648 
 
 4664 
 
 4679 
 
 28 
 
 4695 
 
 4710 
 
 4726 
 
 4741 
 
 4756 
 
 4772 
 
 4787 
 
 48OZ 
 
 4818 
 
 4833 
 
 29 
 
 4848 
 
 4863 
 
 4879 
 
 4894 
 
 4909 
 
 4924 
 
 4939 
 
 4955 
 
 4970 
 
 4985 
 
 3° 
 
 5000 
 
 .5015 
 
 5030 
 
 5045 
 
 5060 
 
 S075 
 
 5090 
 
 5105 
 
 5120 
 
 513s 
 
 3' 
 
 S150 
 
 5165 
 
 5180 
 
 5 195 
 
 5210 
 
 5225 
 
 5^12 
 
 5255 
 
 5270 
 
 5284 
 
 32 
 
 5299 
 
 5314 
 
 5329 
 
 S344 
 
 5358 
 
 5373 
 
 5388 
 
 5402 
 
 5417 
 
 5432 
 
 33 
 
 5446 
 
 5461 
 
 5476 
 
 5490 
 
 5505 
 
 5519 
 
 5534 
 
 5548 
 
 5563 
 
 5577 
 
 34 
 
 S592 
 
 5606 
 
 5621 
 
 563s 
 
 5650 
 
 5664 
 
 5678 
 
 5693 
 
 5707 
 
 5721 
 
 35 
 
 5736 
 
 5750 
 
 5764 
 
 5779 
 
 5793 
 
 5807 
 
 5821 
 
 583s 
 
 S850 
 
 5864 
 
 36 
 
 5878 
 
 5892 
 
 5906 
 
 5920 
 
 5934 
 
 5948 
 
 5962 
 
 5976 
 
 5990 
 
 6cx>4 
 
 37 
 
 6018 
 
 6032 
 
 6046 
 
 6060 
 
 6074 
 
 6088 
 
 6101 
 
 611S 
 
 6129 
 
 6J43 
 
 38 
 
 6157 
 
 6170 
 
 6184 
 
 6198 
 
 6211 
 
 6225 
 
 6239 
 
 6252 
 
 6266 
 
 6280 
 
 39 
 
 6293 
 
 6307 
 
 6320 
 
 6334 
 
 6347 
 
 6361 
 
 6374 
 
 6388 
 
 6401 
 
 6414 
 
 40 
 
 6428 
 
 6441 
 
 6455 
 
 6468 
 
 6481 
 
 6494 
 
 6508 
 
 6521 
 
 6534 
 
 6547 
 
 4» 
 
 6561 
 
 6574 
 
 6587 
 
 6600 
 
 6613 
 
 6626 
 
 6639 
 
 6652 
 
 6665 
 
 6678 
 
 42 
 
 6691 
 
 6704 
 
 6717 
 
 6730 
 
 6743 
 
 6756 
 
 6769 
 
 6782 
 
 6794 
 
 6807 
 
 43 
 
 6820 
 
 6833 
 
 6845 
 
 685S 
 
 6871 
 
 6884 
 
 6896 
 
 6909 
 
 6921 
 
 6934 
 
 44 
 
 6947 
 
 6959 
 
 6972 
 
 6984 
 
 6997 7009 
 
 7022 
 
 .7034 
 
 7046 
 
 7059 
 

 
 
 
 Natural Sines. 
 
 
 
 
 367 
 
 Natural Sines. 
 
 
 1 
 •0° 
 
 •1° 
 
 ■a° 
 
 .30 
 
 •40 
 
 •SP 
 
 •6° 
 
 ■7° 
 
 .go 
 
 .90 
 
 45 
 
 7071 
 
 7083 
 
 7096 
 
 7108 
 
 7120 
 
 7133 
 
 7145 
 
 7157 
 
 7169 
 
 7181 
 
 46 
 
 7193 
 
 7206 
 
 7218 
 
 7230 
 
 7242 
 
 7254 
 
 7266 
 
 7278 
 
 7290 
 
 7302 
 
 47 
 
 7314 
 
 7325 
 
 7337 
 
 7349 
 
 7361 
 
 7373 
 
 7385 
 
 7396 
 
 7408 
 
 7420 
 
 48 
 
 7431 
 
 7443 
 
 7455 
 
 7466 
 
 7478 
 
 7490 
 
 7501 
 
 7513 
 
 7524 
 
 7536 
 
 49 
 
 7547 
 
 755« 
 
 7570 
 
 7581 
 
 7593 
 
 7604 
 
 761S 
 
 7627 
 
 7638 
 
 7649 
 
 50 
 
 7660 
 
 7672 
 
 7683 
 
 7694 
 
 7705 
 
 7716 
 
 7727 
 
 7738 
 
 7749 
 
 7760 
 
 51 
 
 7771 
 
 7782 
 
 7793 
 
 7804 
 
 7815 
 
 7826 
 
 7837 
 
 7848 
 
 7859 
 
 7869 
 
 52 
 
 7880 
 
 7891 
 
 79Q2 
 
 7912 
 
 7923 
 
 7934 
 
 7944 
 
 7955 
 
 7965 
 
 7976 
 
 S3 
 
 7986 
 
 7997 
 
 8007 
 
 8018 
 
 8028 
 
 8039 
 
 8049 
 
 8059 
 
 8070 
 
 8080 
 
 54 
 
 S090 
 
 8100 
 
 8III 
 
 8121 
 
 8131 
 
 8141 
 
 8151 
 
 8161 
 
 8171 
 
 8I8I 
 
 Si 
 
 8192 
 
 8202 
 
 821 1 
 
 8221 
 
 8231 
 
 8241 
 
 8251 
 
 8261 
 
 8271 
 
 8281 
 
 56 
 
 8290 
 
 8300 
 
 8310 
 
 8320 
 
 8329 
 
 8339 
 
 8348 
 
 8358 
 
 8368 
 
 8377 
 
 57 
 
 8387 
 
 8396 
 
 8406 
 
 8415 
 
 8425 
 
 8434 
 
 8443 
 
 8453 
 
 8462 
 
 8471 
 
 58 
 
 8480 
 
 !4i° 
 
 8499 
 
 8508 
 
 8517 
 
 8526 
 
 8536 
 
 8545 
 
 !554 
 
 8563 
 
 59 
 
 8572 
 
 !§!' 
 
 8590 
 
 8599 
 
 8607 
 
 8616 
 
 8625 
 
 8634 
 
 8643 
 
 8652 
 
 60 
 
 866Q 
 
 8669 
 
 8678 
 
 8686 
 
 8695 
 
 8704 
 
 8712 
 
 8721 
 
 8729 
 
 8738 
 
 61 
 
 8746 
 
 im 
 
 8763 
 
 8771 
 
 8780 
 
 8788 
 
 8796 
 
 8805 
 
 8813 
 
 8821 
 
 62 
 
 8829 
 
 f38 
 
 8846 
 
 8854 
 
 8862 
 
 8870 
 
 8878 
 
 8886 
 
 8894 
 
 8902 
 
 63 
 
 8910 
 
 8918 
 
 8926 
 
 8934 
 
 8942 
 
 8949 
 
 8957 
 
 896S 
 
 8973 
 
 8980 
 
 64 
 
 8988 
 
 8996 
 
 9003 
 
 9011 
 
 9018 
 
 9026 
 
 9033 
 
 9041 
 
 9048' 
 
 9056 
 
 65 
 
 9063 
 
 9070 
 
 9078 
 
 9085 
 
 9092 
 
 9100 
 
 9107 
 
 9114 
 
 9121 
 
 9128 
 
 66 
 
 9135 
 
 9«43 
 
 9150 
 
 9157 
 
 9164 
 
 8171 
 
 9178 
 
 9184 
 
 9191 
 
 ■9198 
 
 67 
 
 9205 
 
 9212 
 
 9219 
 
 922s 
 
 9232 
 
 9239 
 
 9245 
 
 9252 
 
 9259 
 
 9265 
 
 68 
 
 9272 
 
 9278 
 
 928s 
 
 9291 
 
 9298 
 
 9304 
 
 93" 
 
 9317 
 
 9323 
 
 9330 
 
 69 
 
 9336 
 
 9342 
 
 9348 
 
 9354 
 
 9361 
 
 9367 
 
 9373 
 
 9379 
 
 9385 
 
 9391 
 
 70 
 
 9397 
 
 9403 
 
 9409 
 
 9415 
 
 9421 
 
 9426 
 
 9432 
 
 9438 
 
 9444 
 
 9449 
 
 71 
 
 9455 
 
 9461 
 
 9466 
 
 9472 
 
 9478 
 
 9483 
 
 9489 
 
 9494 
 
 9500 
 
 9505 
 
 72 
 
 9511 
 
 ^^lo 
 
 9521 
 
 9527 
 
 9532 
 
 9537 
 
 9542 
 
 9548 
 
 9553 
 
 9558 
 
 73 
 
 9563 
 
 9568 
 
 9573 
 
 9578 
 
 9583 
 
 9588 
 
 9593 
 
 9598 
 
 9603 
 
 9608 
 
 74 
 
 9613 
 
 9617 
 
 9622 
 
 9627 
 
 9632 
 
 9636 
 
 9641 
 
 9646 
 
 9650 
 
 9655 
 
 75 
 
 9659 
 
 9664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9686 
 
 9690 
 
 9694 
 
 9699 
 
 76 
 
 9703 
 
 9707 
 
 971 1 
 
 9715 
 
 9720 
 
 9724 
 
 9728 
 
 9732 
 
 9736 
 
 974° 
 
 77 
 
 9744 
 
 9748 
 
 9751 
 
 9755 
 
 9759 
 
 9763 
 
 9767 
 
 9770 
 
 9774 
 
 9778 
 
 78 
 
 9781 
 
 9785 
 
 9789 
 
 9792 
 
 9796 
 
 9799 
 
 9803 
 
 9806 
 
 9810 
 
 9S13 
 
 79 
 
 9816 
 
 9820 
 
 9823 
 
 9826 
 
 9829 
 
 9833 
 
 9836 
 
 9839 
 
 9842 
 
 9845 
 
 80 
 
 9848 
 
 9851 
 
 9854 
 
 9857 
 
 9860 
 
 9863 
 
 9866 
 
 9869 
 
 9871 
 
 9874 
 
 81 
 
 9877 
 
 9880 
 
 9882 
 
 9885 
 
 9888 
 
 9890 
 
 9893 
 
 9895 
 
 9898 
 
 9900 
 
 82 
 
 9903 
 
 9905 
 
 9907 
 
 9910 
 
 9912 
 
 9914 
 
 9917 
 
 9919 
 
 9921 
 
 9923 
 
 83 
 
 9925 
 
 9928 
 
 9930 
 
 9932 
 
 9934 
 
 9936 
 
 9938 
 
 9940 
 
 9942 
 
 9943 
 
 84 
 
 9945 
 
 9947 
 
 9949 
 
 9951 
 
 9952 
 
 9954 
 
 9956 
 
 995? 
 
 9959 
 
 9960 
 
 85 
 
 9962 
 
 9963 
 
 9965 
 
 9966 
 
 9968 
 
 9969 
 
 9971 
 
 9972 
 
 9973 
 
 9974 
 
 86 
 
 9976 
 
 9977 
 
 9978 
 
 9979 
 
 9980 
 
 9981 
 
 9982 
 
 9983 
 
 9984 
 
 9985 
 
 87 
 
 9986 
 
 9987 
 
 9988 
 
 9989 
 
 9990 
 
 9990 
 
 9991 
 
 9992 
 
 9593 
 
 9993 
 
 88 
 
 9994 
 
 9995 
 
 9995 
 
 9996 
 
 9996 
 
 9997 
 
 9997 
 
 9,997 
 
 9998 
 
 9998 
 
 89 
 
 9998 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 I'OOO 
 
 I 000 
 
 i"ooo 
 
 I'OOO 
 
 I'OOO 
 
 
 
 
 
 
 
 nearly. 
 
 nearly. 
 
 nearly. 1 nearly. ' nearly. 
 
CONVERSION TABLES, CONSTANTS, ETC. 
 
 To convert inches to centimetres 
 
 X 
 
 2-S4 
 
 
 centimetres to inches 
 
 X 
 
 0-3937 
 
 
 grains to grammes 
 
 X 
 
 0'o648 
 
 
 grammes to grains 
 
 X 
 
 15*432 
 
 
 oz. (avoir.) to grammes 
 
 X 
 
 28-35 
 
 
 grammes to oz. (avoir.) 
 
 X 
 
 003S3 
 
 
 lbs. to graimmes 
 
 X 
 
 453-59 
 
 
 grammes to lbs. 
 
 X 
 
 0-0022 
 
 
 gallons to c.c. 
 
 X 
 
 4541 
 
 The weight of i grain = 63-57 dynes 
 
 „ I oz. = 2-78 X 10' ,, 
 
 I lb. = 4-45 X io» „ 
 
 ,, I gramme = 981 „ 
 
 I foot-pound = I -356 X 10' ergs 
 
 I kilogrammetre = 9*81 x lo' „ 
 
 I joule = 10' ,» 
 
 I Hp. = 7-46 X 10° ergs per sec. 
 
 I watt (volt ampere) = 10' „ 
 
 To convert common into Naperian logs x 2-3026 
 ,, Naperian into common „ X 0-4343 
 
 "^" (Manchester) = 981-34 
 
 Latitude (Manchester) = 53° 29' 
 
 Length of seconds pendulum (Manchester) = 99-430 cm. 
 
 IT = 3-I4I6 
 
 Area of a circle = 0-7854 (diam.)' 
 
Tables of Physical Constants. 
 
 369 
 
 Table of Specific Gravities. 
 
 Substance. 
 
 Specific 
 gravity. 
 
 Substance. 
 
 Specific 
 gravity. 
 
 Platinum 
 
 2I-S 
 
 Sulphur 
 
 2-0 
 
 Mercury at 0° C. ... 
 
 IS'.W6 
 
 Ivory 
 
 1-9 
 
 Lead 
 
 11-3 
 
 Sand 
 
 1-42 
 
 Silver 
 
 lo-s 
 
 Hooper's I.R. 
 
 I -18 
 
 Nickel 
 
 8-9 
 
 Ebony 
 
 I-I-I-2 
 
 Copper 
 
 8S-8-9 
 
 Ebonite 
 
 1-iS 
 
 German silver 
 
 8-5 
 
 Boxwood 
 
 0-9I-I-03 
 
 Brass 
 
 8- 1-8-6 
 
 Oak 
 
 0-7-1-0 
 
 Steel (cast) 
 
 7-8 
 
 Guttapercha 
 
 . 0-97-0-98 
 
 Iron (wrought) 
 
 7-8 
 
 Wax 
 
 0-96 
 
 » (wire) 
 
 77 
 
 Indiarubber (pure) 
 
 0-93 
 
 .. (cast) 
 
 71-7-6 
 
 Cork 
 
 0-24 
 
 Tin 
 
 7-3 
 
 HjSO, ato°C. 
 
 1-85 
 
 Zinc 
 
 71 
 
 HNO3 
 
 1S6 
 
 Glass (flint) 
 
 3 0-3-S 
 
 HCl 
 
 1-27 
 
 „ (crown) 
 
 2-5-2-7 
 
 cs. 
 
 1-293 
 
 Marble 
 
 2 -5-2 -8 
 
 Glycerine ,, 
 
 127 
 
 Aluminium ... 
 
 2-6 
 
 Linseed oil ,, 
 
 0-94 
 
 Porcelain 
 
 2-4 
 
 Oil of turpentine „ 
 
 0-87 
 
 Chalk 
 
 1-8-2-8 
 
 Alcohol „ 
 
 o-8o6 
 
 Carbon (graphite) ... , 
 
 2-3 
 
 Mineral oil ,, 
 
 0-76-0-83 
 
 Gas carbon 
 
 1-9 
 
 Ether. „ 
 
 0-736 
 
 Brick 
 
 2-1 
 
 
 
 Density of Water at different Temperatures. 
 
 4° C. = I -QOOGO 
 
 6° C. = o'99997 
 10° C. = 0-99974 
 12° C. = 0-999SS 
 14° tJ. = 0-99930 
 
 15° C. = 0-9991S 
 17° C. = 0-99884 
 20° C. = 0-99827 
 25° C. = 0-99713 
 30° C. = 0-99577 
 
 Manchester, 1900 a.d. 
 
 Dip = 69° 
 
 Declination = 18° West of North 
 H = 0-1712 
 
 2 B 
 
370 Practical Electricity and Magnetism. 
 
 p 
 1-1 
 o 
 
 tn 
 
 ■ P 
 
 O 
 
 < 
 
 > 
 
 < 
 
 O 
 
 Weight of subs. 
 
 in lOo parts by 
 1 weight of solu- 
 
 tioD. 
 
 OinowioinotoomomoiociootnQinQ 
 
 *OS"H 
 
 * O .0 H H w « «. m m CO ^•*- lo in\p \p t^ t^op oo oo 
 Ommhmhh'mm'h'i-i'm'mhm'mhmhh m 
 
 "ONH 
 
 O* S, lOOO M lOOO « lOOO M in ^. O W ■<t^ fN.OtH CO 
 
 o*.o .0 ,0 M M H H « « mmtn ■«■.■*.■*,■*■,■*,* ."^ ,*" 
 Ohhmhhmhmh'm'mhmm'mhmhhm 
 
 OH 
 
 lllfllffl.llllll 
 
 •OSnO 
 
 ^^?f 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
 
 H M H 
 
 'OS«Z 
 
 l^sllHl Mill 
 
 b M *M M M M '« 
 
 'a^o 
 
 lllSifflHiiiiiiiiiiii 
 
 "OD'^K 
 
 Ol « »o 
 
 ss.s'.si II 1 1 1 1 1 1 1 1 1 1 1 1 I.I 1 
 
 O M « 
 
 ID«N 
 
 •!-!*!•:•!•!' 1 1 1 1 1 1 1 1 1 1 1 1 II . 
 
 ID'HN 
 
 .ifllf 1 1 1 II 1 1 II 11 1 II II 
 
 H M H M M 
 
 "hn 
 
 OOO oo M ■* g" ig) 
 
 .m.o;pJ.5.ch.o.^oo 1 1 1 1 1 1 11 1 11 M 
 
 oooooooo 
 
 
 .s;.?.? .2 1 II 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
 
 M M « 
 
 mfp 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
 
 H M M H 
 
 llll^l 1 1 1 II 1 1 1 1 1 1 1 1 1 1 
 
 HOa 
 
 iH?&iHHi,iii 
 
 OHMlHMHHHHMn 
 
 Weight of subs, 
 in loo parts by 
 weight of sohi- 
 tion. 
 
 
Wire Gauge Tables. 
 
 371 
 
 Wire Table No. i. 
 
 Gauge. 
 
 Diameter. Copper. | Manganin. 
 
 B.W.G. 
 
 Inches. 
 
 mm. 
 
 Yds per 
 
 ibr 
 
 Cbare), 
 
 Ohms pel- 
 lb. 
 (bare). 
 
 Ohms 
 per yd. 
 
 Yds. per 
 (bare). 
 
 Ohms per 
 lb. 
 
 (bare). 
 
 Ohms 
 pet ydi 
 
 10 .' 
 
 12 
 
 1^:::::: 
 
 18 
 
 20 
 
 22 
 
 24 
 
 26 
 
 28 
 
 30 
 
 3 
 
 34 
 
 36... .. 
 
 38 
 
 40 
 
 0*134 
 0*109 
 0-083 
 0065 
 0*049 
 0*035 
 0*028 
 0*022 
 o*oi8 
 0*014 
 
 0*012 
 
 o-oii 
 
 0*0095 
 0*0079 
 0*0068 
 0-0058 
 
 3*403 
 2*770 
 2*108 
 1*651 
 
 0-889 
 0-711 
 0*560 
 o*457 
 0-3S5 
 0*304 
 0*279 
 0*241 
 0*200 
 0*172 
 0-147 
 
 6*1 
 
 l6*o 
 26*0 
 46*0 
 . 89*9 
 140*4 
 227*5 
 339"9 
 5618 
 764-7 
 908*0 
 1226*0 
 1760-0 
 2375-0 
 3190-0 
 
 0*01057 
 0*02416 
 0*07186 
 0-X9104 
 
 0*59157 
 2*2725 . 
 
 5*5484 ' 
 14-557 
 
 88-772 
 ,164*46 
 334*24 
 421*07 
 880*62 
 1604*1 
 3031*1 
 
 0*0017 
 0*0026 
 0*0045 
 0*0073 
 0*0129 
 0*0253 
 0*0397 
 0*0643 
 0*0961 
 0*1580 
 0*2150 
 0-2574 
 0*3451 
 0*4991 
 0*6736 
 0*9260 
 
 45-85 
 89*89 
 140*44 
 227|53 
 
 561*83 
 761*71 
 910-06 
 1336*14 
 1764*42 
 3381*32 
 3»73-4l 
 
 I '9545 
 5*1964 
 16*0907 
 61-8131 
 150-918 
 395-97 
 S8563 
 2414-61 
 4473-36 
 6372*67 
 11455*1 
 23954;3 
 
 82445-9 
 
 0*0971 
 
 0*1994 
 
 0*3508 
 
 0*4109 
 
 1*075 
 
 1*740 
 
 2*6o6 
 
 4-296 
 
 5-349 
 
 7*001 
 
 9-389 
 
 13-58 
 
 18*32 
 
 25-19 
 
 Wire Table No. 2. 
 
 Gauge-, Diameter, 
 
 German silver. 
 
 Platinoid. 
 
 1 
 
 1 ^ ■ 
 L.S.G. 1 Inches. 
 
 mm. 
 
 Yds. per 
 
 lb. 
 (bare). 
 
 ■ Ohms per 
 
 lb. 
 
 (bare). 
 
 Ohms 
 per yd. 
 
 Yds per 
 
 lb. 
 (bare). 
 
 Ohms per 
 
 lb. 
 
 (bare). 
 
 Ohms 
 per yd. 
 
 10 ...... 
 
 13 
 
 1^ :::::: 
 
 18. 
 
 20 
 
 22 
 
 24 
 
 26 
 
 28 
 
 30 
 
 32 
 
 34 
 
 36 
 
 38 
 
 40 
 
 0*128 
 
 0*104 
 
 0*080 
 
 0-064 
 
 0*048. 
 
 0*036 
 
 0*028 
 
 0*022 
 
 0*018 
 
 0*0148 
 
 0*0124 
 
 0*0108 
 
 0*0092 
 
 0-0076 
 
 00060 
 
 0-0048 
 
 3-351 
 3*643 
 3*032 
 1*626 
 1*219 
 0-914 
 0-711 
 
 0-S59 
 o*4S7 
 0*376 
 0-315 
 0*274 
 0*233 
 0*193 
 0*152 
 0-123 
 
 6-7 
 lO'S 
 
 36*9 
 . 47-9 
 
 15-1 
 140-S 
 237-7 
 340|2 
 
 1303*0 
 1908-0 
 3058*0 
 4785*0 
 
 0-167 
 0*384 
 
 X^ 
 
 8*46 
 36*75 
 
 73*11 
 191*8 
 , 438*1 
 936*6 
 1900*0 - 
 3303*0 
 6272*0 
 13469*0 
 34673^0 
 84652-0 
 
 0*0249 
 
 0-0377 
 0*0637 
 
 0-099S 
 0*1769 
 0*3146 
 0*5201 
 
 , °-2si6. 
 
 1*8517 
 2*6521 
 3-4962 
 4*8181 
 7*0603 
 
 11*347 
 17*699 
 
 6-7 
 
 10*2 
 
 ip 
 
 26*9 
 
 47-9. 
 
 72-5 
 
 85*o 
 
 I40"5 
 
 . 327*7 
 
 340*3 
 
 503*2 
 
 946-1 
 
 1302-0 
 
 1908*0 
 
 3058-0 
 
 4785*0 
 
 0*3031 
 
 • °'^ 
 - 4*8520 
 
 : %T 
 132*43 
 
 ^47*47 
 
 1696*49 
 3443*80 
 5982*72 
 11362*0 
 24398-0 
 ■ 62805-0 1 
 153333-0 
 
 0*0450 
 0-0682 
 
 0-I154 
 0-1803 
 0-3206 
 0*5699 
 0-9422 
 1*5261 
 2*2798 
 3*3722 
 4-8039 
 6*3329 
 8*7271 
 12-7896 
 20*5183 
 32*0601 
 
 A single cotton covering adds o*oo6 to O*oo8 inch to the diameter 
 A double „ „ „ 0*004 .> » >. 
 
 A single silk „ „ 0*002 to 0*003 » >. » 
 
 A double „ » „ 0*004 to o*oo6 ji >> d 
 
372 
 
 Practical Electricity and Magnetism. 
 
 
 V 
 
 ^ 
 
 
 J3 
 
 c 
 CO 
 
 e 
 
 till 
 PQ 5 
 
 w .jC iW 5>. 
 
 I I I I I I I I 
 
 Mil 1 1 
 
 1 1 
 
 o 
 
 + 
 
 a 
 
 
 u 
 
 
 r, 
 
 O. 
 
 < 
 
 R 
 
 tt 
 
 f5 
 
 
 
 
 
 
 
 Vi 
 
 
 
 o o o o o 
 
 I I M I I I 
 
 I 
 
 ^ ao 
 
 ooco «nN ^O ^t^ts.-«hir> 
 O •-• ►-00 m^^r*. r^co CO 
 
 8 8 8 8.8 8 8 8 8 8 8 
 
 bbbbbbbbbbb 
 
 Sci 
 
 ■*0 I M , 
 
 »H M ^4 I 
 
 ^ ^ r^ r^. i-< mvo 
 t>. i>- 1-* r^ r* jN. r* 
 
 »0 mioON 
 
 g^ I I 
 
 ill 
 
 p\p I 
 
 00 c» 
 
 ^ t-* o^ o^ o\ ^ •- 
 
 op 00 00 00 00 p\ C^ 
 
 oo do oo oo 00 do cb 
 
 00 00 00 
 
 «n»nm . , 
 pvps o^in I I 
 
 oododo ^ 
 
 \^\\\ li"l I I 
 
 i 
 
 o 
 
 CIS 
 
 i 
 
 •a i.o 
 
 >»CO ^ 
 I g CJ « 
 
 la 
 
 CO ^ 
 
 u o 
 
 28 •? 
 
 ■"^rt^ « o a o o 
 
 ggi3g.a85 
 5 5 o ^ g a « 3 
 
 
 
 i^'J 
 
 o o o 
 « woo 
 ►" o>« 
 
 I 
 
 o o o 
 
 X C C c^ ^ 
 
 S-p-a 2 ' -V^ %° 
 
 I a 
 
 a 
 
 i g e; 
 
 CO 
 
 3 jj « 
 
 n 
 
 c I 
 
 ^ - 
 
 41 
 
 pq 
 
 
 - > 
 
 CO 
 
Tables of Physical Constants. 
 
 373 
 
 S.iS 
 
 — ■£ 
 3 <J 
 OS 
 
 3 
 O 
 
 n 
 
 bo 
 
 bS 
 
 a 
 
 u s ^ 3i %> 
 
 .w rt rt w '^ 
 
 
 HUPqSQuScn 
 
 ■a 
 
 ES 
 C4 
 
 •St., 
 
 ■« 
 
 § 2 
 
 « o 
 
 Qn 
 
 QWco 
 
 
 I I I I I I I 
 
 in '^ ^»*i w I 
 
 88888§8 
 
 bbbbbbobbb 
 
 boooobooo 
 I I I 
 
 O N 
 MOO I 
 
 W CO 
 
 N Q O ti , O Q 
 
 00 5 Q ro vo Q 
 
 ON O O t^ I O O 
 
 0\ CT^^N •-» ON 
 
 « N to* w -i- 
 
 I I I I I I IS. «£• 
 
 ■* ■* 
 
 i!:i I I I I I I I I I I I I I 
 
 I I M I I I 
 
 c 
 
 <: 
 
 I 
 
 O 
 
 o 
 O 
 
 a 
 
 » 
 
 Ph 
 
 o_ o o o o_ o p_ 
 
 w .*_ u u w ti w v_ 
 
 oo O »^0 "^O »no ri_o 
 
 — « COtO*1-»0«OVOi5o~ 
 
 ."I 
 
 I o o o o o' 
 
 -~2or5 ooaoaooo ^S 
 o^r-H O o O m Q »o o i^S^H 
 Q° MM roto**"^m° 
 
 00 
 
 "5 
 
 3 
 U 
 
 .a - - > 
 
 6 
 s 
 a s 
 
 -S 
 
 o 
 
 • n. ■ 
 p. 
 o 
 U 
 
 c a 
 
 % 
 
 •<?■ 
 
 ^ 
 
 -§ 
 
 ^ 
 
 (tl 
 
 "5 
 
 c4 
 
 •<! 
 
 
 
 u 
 
 •5i d 
 
 
 1 
 
 t^ 
 
 
 
 S 
 
 •^ 
 
 £ 
 
 O 
 
 ■■. 
 
 ;i 
 
 .. 
 
 pq 
 
 o 
 
 
 un 
 
 
 
 ■^ 
 
 §8 
 
 c^ 
 
 d 
 
 
 
 
 
 •H 
 
 >. 
 
 o. 
 
 00 > 
 
 n 
 
 3 
 
 > 
 
 00 N 
 
 ■"•■a .00 
 
 .<i o 51 ^ 
 
 ;;^°^|. 
 
 •«3 rt •« ^ 1^ 
 S *. « S .-• 
 
 ■§ ^ ^-i^^- 
 ■«^r, u a o 
 
 ,S .. o - . 
 O mo >: >: 
 
 „. SO -^ .§ 
 
 M i I "^ I 
 
 «3 -J g! ^ "T 
 09 O "S - l-l 
 
 " *■ -.s ^- 
 
 Pn T3 m 
 
 
 c3 a< 
 
 s 3 S S >< 
 
 •s P."S ^>*) 
 
 ^ ^^"^. i, 
 
 >• "ii '^ I— 1 
 
374 Practical Electricity and Magnetism. 
 
 Table of Specific Resistances of Liquids. 
 
 
 
 
 Specific 
 
 
 Liquid. 
 
 Density. 
 
 Tempera- 
 ture. 
 
 resistance 
 (ohms 
 per c.c.) 
 
 Authority. 
 
 HaSOi 
 
 0-9985 
 
 22° C. 
 
 70-41 
 
 Kohlrausch * 
 
 ,1 
 
 i-pooo 
 
 J> 
 
 41 -OS 
 
 
 >» 
 
 I -0^04 
 
 )> 
 
 3-25 
 
 
 i> 
 
 1-0989 
 
 >f 
 
 1-787 
 
 
 ,, 
 
 1-1430 
 
 Jf 
 
 1-414 
 
 
 ,, 
 
 I -2045 
 
 J» 
 
 1-239 
 
 
 >» 
 
 1-3163 
 
 j» 
 
 1-347 
 
 
 
 I -3994 
 
 )> 
 
 1-672 
 
 a ' 
 
 »» 
 
 1-4482 
 
 19 
 
 1-962 
 
 
 1) 
 
 1-5026 
 
 «l 
 
 2-412 
 
 
 CuSO, 
 
 1-0167 
 
 10° c. 
 
 164-4 
 
 Ewing and Macgregorf 
 
 1) .-. 
 
 I -0216 
 
 » 
 
 134-8 
 
 
 11 
 
 1-0318 
 
 >» 
 
 98-7 
 
 
 11 
 
 1-0622 
 
 >J 
 
 59-0 
 
 
 )1 
 
 I -1174 
 
 »l 
 
 38-0 
 
 
 „ (saturated) 
 
 1-2054 
 
 )> 
 
 29-6 
 
 
 ZnSO^ 
 
 I -4220 
 
 9> 
 
 33-7 
 
 
 HCl 
 
 1-109 
 
 If 
 
 1-31 
 
 
 HNO3 
 
 1-185 
 
 )» 
 
 1-28 
 
 
 Table of Specific Dielectric Resistances. 
 
 Material. 
 
 Specific 
 resistance. 
 
 Density. 
 
 Temp. 
 
 Authority. ■ 
 
 Glass (Bohemiar 
 
 ) 4-25 X lo" 
 
 2-427 
 
 60° C. 
 
 Gray 
 
 
 7-15 X lo" 
 
 2-587 
 
 60° 
 
 
 „ (test-tube) 
 
 I-44X id" . 
 
 2-435 
 
 60° 
 
 
 >> it 
 
 . 3-50 X 10" 
 
 2-44 
 
 60° 
 
 
 „ (flint) . 
 
 . 3 89 X 10™ 
 
 2-7S3 
 
 60° 
 
 
 » 3> 
 
 1-02 X 10" 
 
 3-172 
 
 60° 
 
 
 Mica 
 
 . 8-40 X 10" 
 
 
 20° 
 
 Ayrton and Perry 
 
 Guttapercha 
 
 . 4-50 X 10" 
 
 0'97 to 0-98 
 
 24" 
 
 Latimer Clark 
 
 Shellac ... 
 
 9-00 X 10" 
 
 — 
 
 28° 
 
 Ayrton and Perry 
 
 Ebonite ... 
 
 . 2-80 X 10'' 
 
 i-iS 
 
 46° 
 
 1) 1) 
 
 Paraffin ... 
 
 . 3-40 X io»' 
 
 
 46° 
 
 «> >f 
 
 Indiarubber 
 
 i-ogxio" 
 
 — 
 
 24" 
 
 Jenkin 
 
 Hooper's core . 
 
 . 1-50 X 10" 
 
 — 
 
 24" 
 
 »i 
 
 Parchment 
 
 . 3-00 X 10" 
 
 — 
 
 20° 
 
 Uppenborn 
 
 Ordinary paper 
 
 4-85 X 10" 
 
 ~^ 
 
 20° 
 
 i» 
 
 * Kohlrausch and Nippoldt, "Leilfaden der Praktschfin Physick, 
 p. 298. 
 t Ewing and Macgregor, Trans. Jiffy. Sue, Edinburgh, vol, xxvii,, 1873. 
 
Tables of Physical Constants. 
 
 375 
 
 Table of Dielectric Resistance and Capacity of G.P. Covered 
 Wire after One Minute Electrification. 
 
 External diameter 
 
 Resistance per mile 
 in megohms. 
 
 Capacity per mile in 
 microfaTads. 
 
 Internal diameter. 
 
 25 
 
 366-1 
 
 0-394 
 
 2-6 
 
 384-8 
 
 0375 
 
 2-7 
 
 396-9 
 
 • 0-363 
 
 2-8 
 
 411-4 
 
 0-351 
 
 2-9 
 
 425-4 
 
 .0-339 
 
 3-0 
 
 439-0 
 
 0-329 
 
 3-S 
 
 500-6 
 
 0-288 
 
 4-0 
 
 553-9 
 
 0-260 
 
 Table of Electro-Motive Forces of Cells. 
 
 Cell. 
 
 E.M.F. 
 (volts). 
 
 Cell. 
 
 E.M.F. 
 (volte). 
 
 Grove 
 
 Bunsen 
 Daniell 
 
 Bichromate ... 
 Leclanche 
 Gassner 
 
 1-9 to I -95 
 1-9 to I -95 
 1-07 to 1-14 
 
 2 
 15 
 13 
 
 Hellesen ... 
 E.C.C. 
 Obach 
 Lessing 
 Secondary ... 
 Claik 
 
 I -45 
 1-45 
 
 1-5 
 
 1-85 to 2-1 
 1434 
 
 Table of Contact Differences of Potential. 
 
 Metals. 
 
 Difference of potential 
 Volts at 18° C. 
 
 Copper to iron 
 
 Copper to platinum ... 
 
 Copper to zinc 
 
 Copper to brass 
 Platinum to brass 
 
 0-146 
 -0-238 
 0-750 
 0-087 
 0-287 
 
376 Practical Electricity and Magnetism. 
 
 Table of Potential Difference required to Spark between 
 TWO Parailel Plates. 
 
 ' Distance 
 
 E.M.F. 
 
 Distance. 
 
 E.M.F. 
 
 (in cms.). 
 
 Cvolts). 
 
 (in cms,). 
 
 (in volts). 
 
 0-0205 
 
 1003 
 
 0-1800 
 
 7000 
 
 0-0430 
 
 2000 
 
 0-2146 
 
 8000 
 
 0-0660 
 
 3000 
 
 0.2495 
 
 9000 
 
 0-0914 
 
 4000 
 
 0-2863 
 
 ICOOO 
 
 0-1176 
 
 5000 
 
 0-3245 
 
 IICOO 
 
 0-1473 
 
 6000 
 
 0-3378 
 
 II330 
 
 Table of Electro-Chemical Equivalents. 
 
 Element, 
 
 Grammes per 
 coulomb. 
 
 Element. 
 
 Grammes per 
 coulomb. 
 
 Silver 
 
 Copper (cupric) 
 
 „ (cuprous) 
 Iron (ferric) 
 „ (ferrous) ... 
 
 Nickel 
 
 Zinc 
 
 O-OOII18 
 
 0-0003279 
 
 0-000655 
 
 0-0001934 
 
 0-0002900 
 
 0-0003042 
 
 0-0003367 
 
 Lead 
 
 Tin (staunic) ... 
 „ (staunous) 
 Hydrogen 
 Oxygen 
 
 Water 
 
 Iodine 
 
 0-00107 1 
 
 0-000304 
 
 0-000609 
 
 0-00001035 
 
 0-00008286 
 
 0-00009321 
 
 O-OOI3134 
 
 Element. 
 
 Cub. cm. per coulomb 
 at 0° C. and 760 mm. 
 
 Hydrogen 
 
 Oxygen 
 
 Water 
 
 0-1156 
 0-0578 
 
 0-1734 
 
Tables of Physical Constants. 
 Table of Specific Inductive Capacities. 
 
 377 
 
 Material. 
 
 Glass (flint) 
 
 (crown) 
 
 (plate) 
 
 Mica 
 
 Ebonite 
 
 Guttapercha (best) 
 Indiarubber (black) 
 
 » Csi'cy vulcanized) 
 Paraffin wax (M.P. 68° C) 
 
 Shellac "...^ 1- 
 
 Petroleum spirit 
 
 » oil 
 
 Turpentme 
 
 Castor oil 
 
 Sperm oil 
 
 Olive oil 
 
 Eenzene 
 
 Water (distilled) 
 
 Alcohol (ethyl) 
 
 Density. 
 
 S.I.C. 
 
 4'5 
 
 9-896 
 
 3-66 
 
 7 '376 
 
 3-20 
 
 6'72 
 
 2-87 
 
 6-61 
 
 2-48 
 
 6-96 
 
 — 
 
 8-45 
 
 — 
 
 6-64 
 
 I'lS 
 
 /4-2 
 (2-284 
 
 0*91 to 0*98 
 
 2-462 
 
 0-93 
 
 2-220 
 
 — 
 
 *'497, 
 
 0*9109 
 
 1-9936 
 
 0-9080 
 
 1-977 
 
 — 
 
 2-740 
 
 — 
 
 1-92 
 
 o'88 
 
 2*10 
 
 087 
 
 2-23 
 
 — 
 
 4-78 
 
 — 
 
 3-02 
 
 o'gi 
 
 ,-i6 
 
 0-85 
 
 2-198 
 
 i-oo 
 
 76-0 
 
 o'8o 
 
 26-5 
 
 Authority. 
 
 Hopkinson, Phil. Trans.t 1881 
 
 Klemencic» Beibl&tter., vol, xii., 18 
 
 Faraday 
 
 Gordon, Phil. Trans., 1879 
 
 Gibson and Barclay, Phil. Trans., 1871 
 Gordon, Phil. Trans., 1879 
 Hopkinson, Phil. Trans., 1881 
 
 Silow, Po£rff- Ann., 1875 
 Quincke, ^ied. Ann., 1888 
 
 Table of Coefficients of Cubical Kxpansion. 
 
 Material. 
 
 Glass 
 Alcohol 
 Water at 5° C. 
 
 „ at ico° C. 
 Mercury 
 
 Coefficient. 
 
 0-0009258 
 
 O-OOII 
 
 0-000022 
 
 o'oo7SS 
 0-00018 
 
 Table of Specific Heats. 
 
 Substance. 
 
 Specific heat. 
 
 Substance. 
 
 Specific heat. 
 
 Brass 
 
 0-094 
 
 Glass 
 
 O-iq 
 
 Copper 
 
 0-094 
 
 Alcohol 
 
 0-58 
 
 Zinc 
 
 0-094 
 
 Mercury 
 
 0034 
 
 Iron 
 
 0-113 
 
 Turpentine 
 
 o'43 
 
 Silver 
 
 0057 
 
 Paraffin oil 
 
 o'434 
 
 Platinum 0° to 100° 
 
 0-0335 
 
 Aniline 
 
 0-49 
 
 „ 0° to 300° 
 
 0-0343 
 
 Water 0° to 40° ... 
 
 1-0013 
 
 „ 0° to 500° 
 
 0-0352 
 
 „ o°to8o° ... 
 
 I -0035 
 
 „ 0° to 1000° 
 
 0-0373 
 
 
 
378 Practical Electricity and Magnetism. 
 
 Table of Melting-Points. 
 
 Substance. 
 
 Melt:ng-point. 
 
 Subs tan -e. 
 
 Melting-point. 
 
 Platinum 
 
 1775° C. 
 
 Zinc 
 
 412° C. 
 
 Iron 
 
 1600° 
 
 Lead 
 
 326° 
 
 Nickel 
 
 1450° 
 
 Cadmium 
 
 315° 
 
 Steel 
 
 1370° 
 
 Bismuth 
 
 260° 
 
 Copper 
 
 ioS4° 
 
 Tin 
 
 230° 
 
 Glass 
 
 1100° 
 
 Selenium 
 
 217° 
 
 Silver 
 
 954° 
 
 Sulphur 
 
 115° 
 
 Aluminium 
 
 600° 
 
 Paraffin 
 
 54° 
 
 Antimony 
 
 432° 
 
 
 
 Table of Heat Constants. 
 
 Latent heat of water (gramme degree Centigrade) = 80 
 „ „ „ (pound ,, Fahrenheit) = 143 
 
 ,, ,, steam (gramme „ Centigrade) = 536 
 ,, „ ,, (pound ,, Fahrenheit) = 966 
 
 Mechanical equivalent of heat = 772 ft. -lbs. per ° F. 
 
 >. » ,, = 1390 .. .. ° c. 
 
 » >. =42400 gramme „ ° C. 
 
 Table of Indices op Refraction for Sodiu.m Light. 
 
 Material. 
 
 Density. 
 
 Index. 
 
 Glass (crown) 
 
 2-S 
 
 JSi 
 
 ., (flint) 
 
 2-8 to 4*2 
 
 1-5410 I -71 
 
 Water 
 
 I'O 
 
 I '33 
 
 Carbon bisulphide ... 
 
 1-27 
 
 1-6303 
 
 Benzol 
 
 0-877 
 
 1-4972 
 
 Alcohol 
 
 0-800 
 
 1-3633 
 
 Ether 
 
 0-713 
 
 I '3594 
 
 Chloroform 
 
 1-501 
 
 1-4492 
 
INDEX 
 
 Subject. Par. Page 
 
 Absolute electrometer 205 . 193 
 
 measurement of current 130 . 126 
 
 ■ of electro-motive force 179 . 175 
 
 ofresistance 114 . 106 
 
 (B.A. metliod) . .117,118 . no 
 
 (Joules' method) . . . 116 . 107 
 
 (Lorenz* method) .119,120 . 112 
 
 Adjustment and care of apparatus ... 3 . 2 
 
 of standard tangent galvanometer . '. . . 139 . 132 
 
 Amperi, l^al definition of the 160 . 152 
 
 Angular deflection of galvanometer needle 137 131 
 
 Apparatus for detecting electro-magnetic waves .... 364 . 337 
 
 for producing electro-magnetic waves .... 354 . 332 
 
 Artificial cable , 113 • '°S 
 
 B.A. method of determining resistance absolutely . . .117,118 . no 
 
 B.A. standard ohm . . 123 . 1 16 
 
 Ballistic galvanometer 223 . 211 
 
 calibration 227 . 215 
 
 , by condenser .... 239 . 227 
 
 , by current inductor . . 235 . 223 
 
 , by earth inductor . . . 232 . 220 
 
 , by steady current . . . 230 . 218 
 
 .theory of 224 . 211 
 
 , use of 241 . 229 
 
 Batteries .... 19 . 14 
 
 Battery, complete test of 200-203 • '9° 
 
 resistance, measurement of 86-88 . 84 
 
 , temperature coefficient of 196-199 . 187 
 
 Bifilar suspension, constant of 149 . 141 
 
 Bose's coherer 366, 367 . 338 
 
 oscillator 359-335 
 
 Bosscha's method of determining the ratio of the ladii of 
 
 two coils 153 . 144 
 
380 Index. 
 
 Subject. Par. 
 
 Branly's coherer 365 
 
 Bridge wire calibration . 32 
 
 ■ , fall of potential method . . . 34-37 
 
 , Mattheissen and Hockin's method . 33 
 
 Cable, artificial ■ • -113 
 
 Cadmium standard cell 192 
 
 Calibration of a bridge wire 32 
 
 • — , fall of potential method . 34-37 
 
 , Mattheissen and Hockin's 
 
 method 32 
 
 of a direct-current reading instrument . . . . 176 
 
 of a sensitive galvanometer 23 
 
 Capacities, comparison of 247 
 
 , by ballistic galvanometer . . 248, 249 
 
 , by electrometer 250, 251 
 
 , by method of mixtures . . . 252, 253 
 
 Capacity, absolute measurement of 244-246 
 
 , definition of . ; . . 243 
 
 , original papers on . 266, 267 
 
 , specific inductive 254 
 
 Carbon resistance . 22 
 
 Carey Foster bridge 53 
 
 Carhart Clark standard cell 187 
 
 Clark standard cell * 183-186 
 
 Coefficient of self-induction of a Hertz oscillator .... 356 
 
 Coherer, Bose's 366-367 
 
 , Branly's 365 
 
 Comparison of a capacity and an inductance 340 
 
 of electro-motive forces by condenser . . 194, 195 
 
 by potentiometer . . 219 
 
 of resistances by potentiometer 215,216 
 
 Condenser adjustable 255 
 
 Connecting wires 13 
 
 Constant of bifilar suspension 149 
 
 of Siemens' electro-dynamometer 175 
 
 of standard tangent galvanometer 134 
 
 Contact, E.M.F. of 209 
 
 Contacts 13-15 
 
 Control, magnetic, for galvanometers 6 
 
 Copper voltameter 164-168 
 
 treatment of plates .... ... 165 
 
 Correction for end contacts in metre bridge 38-41 
 
 Cost of working a primary battery 203 
 
 Coulomb, definition of 222 
 
Index. 381 
 
 Subject. Par. 
 
 Current, absolute measurement of ... . .... 130 
 
 balance 151-158 
 
 , laboratory form . 155 
 
 , Lord Kelvin's . . . 172, 173 
 
 inductor, dimensions of . . 237 
 
 measurement .... . . 129 
 
 by potentiometer . .217,218 
 
 ' , original papers on ... . 178 
 
 Damping of ballistic galvanometer needle . . 225 
 
 Daniell standard cell . 188-191 
 
 , measurement of E.M.F 212 
 
 , variations of E.M.F. . . . . 220 
 
 Demagnetizing effect of poles of a magnet 300 
 
 Determination of H ... 268 
 
 by Gauss's method 269-281 
 
 by vibration method . . . . .282-286 
 
 of magnetic inclination by induced currents 290-292 
 
 by needle . . . 287-289 
 
 Dial bridge 30 
 
 Dielectric resistance of cable .68 
 
 , specific ... . ... 94-95 
 
 Directing magnet stand for galvanometer . 6 
 
 Earth inductors . 233 
 
 Effective liagth of a bar magnet .... . . . 271 
 
 Electro-chemical equivalent of copper . . .166,167 
 
 Electro-dynamometer, absolute . . 148 
 
 , Siemens' 174-177 
 
 , determination of constant . 175 
 
 Electro-magnetic waves 343 
 
 , apparatus for producing .... 354 
 
 , calculation of frequency .... 354 
 
 , measurement of wave-length . 372 
 
 , original papers on 380 
 
 , polarization of 377 
 
 , reflection of ... . ... 370 
 
 , refraction of 371 
 
 , velocity of propagation .... 349 
 
 Electro-motive force, absolute determination of .... 179 
 
 , contact 209 
 
 , conditions for a standard of . . . 180 
 
 , method of obtaining a small .... 20 
 
 , original papers on 221 
 
 forces, comparison by potentiometer . . . 219 
 
382 
 
 Tndex. 
 
 Subject. Pak- Page 
 
 Electrometer, absolute . . 205 . 193 
 
 — ; insulation testing . ... 208 . 198 
 
 needle, charging of ... . ... 211 . 199 
 
 , quadrant ... 206 . 195 
 
 Emmissivity constant of an insulated wire coil 47 ■ 49 
 
 Farad, definition of 243 . 232 
 
 Fault-testing ... 109-113 102 
 
 Fleming standard ohm 124 , 116 
 
 — tlieory of 
 -, use of . 
 
 Galvanometer adjustment . 3 
 
 , ballistic, calibration of . . 229 
 
 , by condenser . . 239, 240 
 
 , by current inductor 235-238 
 
 , by earth inductor 232-234 
 
 -, by steady currents 230, 231 
 
 . 224 
 
 24J 
 
 23,24 
 
 II 
 
 6 
 
 4 
 
 7-10 
 
 89,90 
 
 144 
 
 147 
 132 
 
 135 
 "34 
 136 
 138 
 137 
 12 
 2 
 
 375 
 69 
 
 - calibration 
 
 - connecting wires . 
 
 - control 
 
 - insulation ... 
 
 - needle suspension . . . 
 
 - resistance, measurement of 
 
 -, standard sine 
 
 , Gray's form . 
 
 tangent . . ... 
 
 — , best dimensions for 
 
 , constant of . . . 
 
 , construction . . 
 
 , correction for torsion 
 
 , needle deflection . 
 
 vibration, method of preventing . 
 
 General instructions for experimental work . . 
 , Grating for measurement of wave-length . 
 Guard-ring for preventing leakage . . 
 
 2 
 217 
 227 
 223 
 220 
 
 2l8 
 
 211 
 229 
 19 
 9 
 4 
 3 
 6 
 
 87 
 13s 
 138 
 127 
 130 
 129 
 130 
 131 
 131 
 9 
 2 
 
 346 
 68 
 
 H, determination of, Gauss's method 268-281 . 254 
 
 , vibration method . . ... 282-286 . 264 
 
 Helmholtz standard tangent galvanometer 142 133 
 
 Hertz oscillator ... 355 . 334 
 
 , calculation of inductance of 356 . 334 
 
 .capacity of 357,358 • 334 
 
Index. 383 
 
 Subject. Par. Pace 
 
 High-resistance keys 18 . 13 
 
 measurement by direct deflection . 66-72 65 
 
 by inferred zero ... 76 . 74 
 
 by loss of charge 7S-8i . 73 
 
 of small capacity . . 74 72 
 
 Index of refraclion for electro-magnetic waves . . 
 Inductive effect of a magnetic field on a bar magnet 
 Inferred zero method of measuring high resistance 
 Insulation test for electrometers 
 
 galvanometers 
 
 Interference of waves ... . . 
 
 Iodine voltameter . . . . 
 
 Iron and steel, measurement of magnetic qualities 
 
 Joules' method of determining resistance . . . . 
 
 Kelvin current balances . . 
 
 Keys 
 
 Kohlrausch method of measuring liquid resistance 
 
 L^al definition of the ampere . 
 
 of the ohm . 
 
 of the volt . . 
 
 Lens for electro-magnetic waves 
 Liquid resistance, measurement of . 
 
 371 
 . 272 
 
 ■ 76 
 . 208 
 
 4 
 
 • 353 
 169-171 
 
 • 293 
 
 • "7 
 
 Cambridge method 
 -, by fall of potential 
 -, Kohlrausch method 
 -, specific .... 
 
 resistances 
 
 Lodge's oscillator . .... 
 
 Logarithmic decrement 
 
 Loop test for cables . . 
 
 Lorenz method of determining resistance 
 
 Low resistance measurement by Carey Foster bridge . 
 • by fall of potential . . 
 
 ' standards , 
 
 172, 173 
 16-18 
 ■ 83 
 
 160 
 
 121 
 
 182 
 
 368 
 
 82 
 
 . 84 
 
 . . 85 
 
 . 83 
 
 96. 97 
 
 , . 22 
 
 3S9, 360 
 
 . 227, 228 
 
 . . no 
 
 . 119, 120 
 
 . . 56 
 
 57-60 
 
 . 61-63 
 
 342 
 256 
 
 74 
 198 
 
 3 
 332 
 160 
 272 
 
 no 
 
 162 
 It 
 81 
 
 152 
 "5 
 176 
 340 
 80 
 
 82 
 
 82 
 
 81 
 
 94 
 
 17 
 
 335 
 
 215 
 
 103 
 
 112 
 
 56 
 
 57 
 61 
 
 Magnetic hysteresis .... 315, 316 
 
 inclination, measurement by induced currents . 290-292 
 
 by needle 289 
 
 qualities of iron and steel 293 
 
 Magnetism, original papers on 341, 342 
 
 Magnetometer .... 273 
 
 294 
 270 
 269 
 273 
 324 
 258 
 
384 Index. 
 
 Subject. Par. Page 
 
 Magnets, deflecting 274 . 258 
 
 , moments of inertia . . 277 . 261 
 
 Make-and-break key . . 16 . 11 
 
 Making and adjusting a resistance coil ... 64, 65 . 64 
 
 Manganin resistances, ageing of 21 16 
 
 standards of low resistance . . 63 . 63 
 
 Marconi's oscillator ... 361 . 335 
 
 Measurement of battery resistance ... .... 86-88 84 
 
 of capacity 244, 245 . 232 
 
 of current 129 . 126 
 
 by potentiometer . ... 217, 218 . 205 
 
 of galvanometer resistance 89 . 87 
 
 of inductance, Joubert's method . . . .331,332 . 310 
 
 , Maxwell's method .... 327 . 304 
 
 , Rayleigh's method . . . 325, 326 . 301 
 
 , Rimington's method .... 328 . 306 
 
 , Sumpner's method . . . 329 . 308 
 
 of liquid resistance 82-85 ■ 80 
 
 of permeability 294 . 273 
 
 , Evershed and Vignole's 
 
 method 306-309 . 285 
 
 '■ , Hopkinson's method . .310,311 290 
 
 , magnetometer method .295-301 . 274 
 
 , ring ballistic method . .302-305 . 282 
 
 , traction method . . . 312-314 . 293 
 
 ' of resistance 25, 26 . 23 
 
 by P.O. bridge 28 . 26 
 
 Wheatstone bridge . ... 27 24 
 
 of specific resistance 91-97 • 90' 
 
 of temperature of bridge coils 46 . 48 
 
 of very high resistance by direct deflection 66-72 65 
 
 by loss of charge . 73-81 71 
 
 of very low resistance 57-60 . 57 
 
 Mercury contacts 14 • 10 
 
 Metallic contacts IS • 1 1 
 
 Metre bridge 31 . 28 
 
 , end contacts ... ... . 38-41 . 40 
 
 , extra resistance for ... 44 . 47 
 
 , practical test of sensitiveness 45 • 48 
 
 , j-AaOT^ of method of using . . ... 51, 52 . 51 
 
 , tapping error 42 . 44 
 
 , thermo-electric effect 43 ■ 46 
 
 — ^ , wire calibration 32 . 31 
 
 Microfarad, definition of the 243 . 232 
 
 Molecular change and resistance 105-108 . loi 
 
Index. 385 
 
 Subject. Par. Page 
 
 Mutual induction coefficient 324 . 301 
 
 , determination of . . • 33**! 339 • 3'9 
 
 Ohm, legal definition of the . . 
 Original papers on capacity . . . 
 
 on current . ... 
 
 on electro-magnetic waves 
 
 on electro-motive force . 
 
 ■ on magnetism . 
 
 -; on resistance . . . . 
 
 Oscillator, Bose's . 
 
 , Hertz's . . 
 
 , Lodge's 
 
 , Marconi's . . . 
 
 . 121 . 115 
 
 266, 267 . 252 
 
 • 178 . 173 
 
 • 380 • 349 
 . 221 . 209 
 
 341, 342 • 324 
 
 127, 128 . 119 
 
 359 335 
 
 • 356 . 334 
 359> 360 . 335 
 
 ■ 361 • 335 
 
 Permeability, measurement of, by magnetometer . . . 295-301 . 274 
 
 ■ , by ring ballistic method . 302-305 . 282 
 
 , by Evershed and Vignole's 
 
 method . ... 306-309 286 
 
 , by Hopkinson's method . 310, 3H . 290 
 
 . by traction method . . 312-314 . 293 
 
 Permeability and temperature . 317-322 . 296 
 
 Polarization of electro-magnetic waves .... . 377 . 346 
 
 Polarizing apparatus for electro-magnetic waves ... 379 . 347 
 
 P.O. bridge 28 . 26 
 
 Potentiometer ... . .... . 213, 214 . 200 
 
 •, comparison of resistance by ... . . 215 . 202 
 
 of currents by .... 217 . 205 
 
 ofE.M.F.'sby . . .219 206 
 
 Quadrant electrometer . . .... 206, 207 . 195 
 
 , law of . 210 . 198 
 
 Quantity of electricity, measurement of ... . . 222 211 
 
 Quartz fibre suspension .... 9 . 7 
 
 Radii of two coils, method of measuring the ratio of . . . 153 . 144 
 
 Rayleigh's current balance 152 . 144 
 
 method of measuring inductance 325, 326 . 301 
 
 Recalescence 322 . 299 
 
 Recording results . . .... i . i 
 
 Reichsanstalt standard ohm 125 . 117 
 
 Relations of the various standards of resistance . . . 126 . 117 
 
 Relation of S.I. C. to index of refraction . . . ... 259 . 247 
 
 2 C 
 
386 
 
 Resistance, battery . . 
 
 , galvanometer . 
 
 , liquid . . . 
 
 measurement . 
 
 Index. 
 
 Subject. 
 
 interval between tests 
 , P.O. bridge . . . 
 , resumJ 
 
 substitution method . 
 , Wheatstone bridge method . 
 
 -, specific . . 
 -, standardization 
 -, standards . . 
 -. very high 
 -, very low ., . 
 
 Resistances 
 
 -, carbon . . 
 
 ' , coil making and adjusting 
 
 -, liquid .... 
 
 -, sizes of wires for . 
 
 Reversing keys 
 
 Pae. 
 . 86 
 
 . 89 
 82-85 
 25, 26 
 
 5° 
 28 
 
 SI 
 27 
 
 27 
 91-97 
 
 54,55 
 
 121-126 
 
 66-81 
 
 • 56 
 
 21, 22 
 
 22 
 
 64,65 
 
 22 
 
 21 
 
 17 
 
 Page 
 84 
 
 87 
 80 
 
 23 
 51 
 26 
 
 SI 
 
 24 
 24 
 90 
 
 54 
 "5 
 65 
 56 
 16 
 
 17 
 64 
 17 
 16 
 
 Secohm, definition of . . 
 Self and mutual induction 
 Self-induction coefficient . 
 
 measurement by Rayleigh's 
 
 327 
 323 
 323 
 
 method 325, 326 
 
 by Maxwell's method 327 
 
 by Rimington's 
 
 328 
 329 
 
 method 
 
 by Sumpner's method 
 
 by Joubert's 
 
 method 331, 332 
 
 comparison of coefficients 333 
 
 by Sumpner's 
 
 • • 335 
 
 • ■ • 334 
 
 • • 337 
 
 • • 330 
 ■ ■ 44 
 
 • ■ 174. 175 
 162 
 
 . . 160 
 
 .... 144 
 
 Gray's form . . 147 
 
 method ... 
 
 standards 
 
 of various instruments . . 
 
 , variation with permeability 
 
 Sensitiveness of Wheatstone bridge 
 Siemens' electro-dynamometer . . 
 
 Silver, specification of 
 
 — — voltameter . . . . 
 
 Sine galvanometer 
 
 304 
 300 
 300 
 
 301 
 304 
 
 306 
 308 
 
 310 
 312 
 
 316 
 315 
 318 
 309 
 47 
 169 
 
 153 
 152 
 135 
 138 
 
Index. 387 
 
 Subject. Par. Page 
 
 Specific inductive capacity 254 . 241 
 
 , measurement of, by Hopkinson's 
 
 method . . 256-261 243 
 
 ■ of cable dielectric .... 262, 263 . 248 
 
 and index of refraction . • • . 3Si 33 1 
 
 of liquid . 264, 265 . 250 
 
 Specific resistance . . ... ... 91-97 90 
 
 of dielectrics 94, 95 • 92 
 
 of liquids ... 96, 97 94 
 
 and molecular change 105-108 loi 
 
 variation with temperature .... 98-104 . 95 
 
 Standard cell, Cahart-CIark . 187 . 180 
 
 , care of ... 193 . 184 
 
 , Clark's 183, 184 . 176 
 
 , Daniell 188 . 181 
 
 , Weston 192 . 183 
 
 current balance 151-158 . 143 
 
 electro-dynamometer . . . . .... 148 . 140 
 
 Standards of resistance . . 121-126 . 115 
 
 of low resistance . 61-63 • 61 
 
 Standardization of resistances . 54, 51;. 54 
 
 Stationary electro-magnetic waves .... ... 373 345 
 
 Suspensions, metallic . . ... ...10. 9 
 
 , quartz fibre .... . 9-7 
 
 , silk . . ... . 7 6 
 
 , spider's web . . . 8 . 7 
 
 Tangent galvanometer adjustment . . . ... 139 . 132 
 
 constant . . . . 134 . 129 
 
 construction ... ... 136 130 
 
 , Helmholtz form . . . 142 . 133 
 
 sensitiveness . . 140 . 133 
 
 standard . 132 . 127 
 
 Tapping error in metre bridge . . . 42 . 44 
 
 Temperature coefficient of a battery . . 196-199 . 187 
 
 and dielectric resistance .... ... 70 . 70 
 
 measurement of bridge coils .... 46 . 48 
 
 variation of specific resistance 98-104 . 95 
 
 Test of a primary battery . ... . 200-203-. '9° 
 
 Thermostat . . . 48 . 50 
 
 Thermometer standardization . . 49 . 5 1 
 
 Thermo-electric effect in metre bridge . 43 ■ 46 
 
 Torsion in galvanometer needle suspension ... . . 138 131 
 
 Velocity of propagation of electro-magnetic waves . . . 349 . 330 
 
388 Index. 
 
 Subject, Pak. Pace 
 
 Vibration of galvanometer, method of preventing . ... 12 . 9 
 
 Volt, legal definition of ... 182 . 176 
 
 Voltameters IS9-I7I • 151 
 
 , copper 164-168 . 155 
 
 , iodine. ... . ... 169-171 . 160 
 
 , silver .... ... 160-162 . 152 
 
 Waves, electro-magnetic ... 343 . 328 
 
 Wave-length of electro-magnetic waves 372 . 344 
 
 •, measurement of . 374 . 345 
 
 Weston standard cadmium cell 192 . 183 
 
 Wheatstone bridge 27 . 24 
 
 , sensitiveness of 44 . 47 
 
 Wire calibration, fall of potential method 34-37 ■ 32 
 
 , Mattheissen and Hockin's method ... 33 . 31 
 
 I'tllNtED UV WILLIAM CLOWES AND SONS, LIMIfED, LONDON AND DECCLES. 
 

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