THE FIRST TRANSFORMER. 
 
 The Iron Ring, with insulated Primary and Secondary Coils wound on 
 it, with which Faraday made the discovery of Magneto-Electric 
 Induction. Photographed from the original, preserved in the Museum 
 of the Royal Institution. 
 
 \Fronti*piett.
 
 THE ALTERNATE CURRENT 
 TRANSFORMER 
 
 IN THEORY AND PRACTICE. 
 
 J. A. FLEMING, 
 
 M.A., D.SC. (LOND.), F.R.S., 
 
 FROKKSSOK OK ELECTRICAL F.NG I N KICKING IN UNIVERSITY 
 
 COLLEGE, LONDON ; 
 I.ATK FF.I.LOXV AND SCHOLAR OF ST. JOHN'S COI.I.F.GE, CAMBRIDGE 
 
 FELLOW OF UNIVERSITY COLLEGE, LONDON ; 
 MEMltER OK THE INSTITUTION OF EI.ECTKICAI. ENGINEERS; 
 
 MEMliEK OK THE PHYSICAL SOCIETY OF LONDON; 
 MK.MISEU OK THE ROYAL INSTITUTION OF GREAT liRITAIN; 
 
 F.TC., El'C. 
 
 VOLUME I. 
 
 THE INDUCTION OF ELECTRIC CURRENTS. 
 
 NEW EDITION. 
 
 LONDON: 
 
 1 THE ELECTRICIAN " PRINTING AND PUBLISHING COMPANY, 
 LIMITED, 
 
 SALISBURY COURT, FLEET STREET. 
 
 [All Rights Reserved.]
 
 Printed ami Published by 
 
 "THK KLKCTUIOIAN " PRINTING AND PUBLISHING CO.. 
 
 1, 2 and 3, Salisbury Court, Fleet Street, 
 London, K.C.
 
 PREFACE. 
 
 IN the seven years which have elapsed since the first 
 edition of this Treatise was published, the study of 
 the properties and applications of alternating electric 
 currents has made enormous progress. At the outset 
 the aim of the author was to collect, and present in a 
 form suitable for students, a general statement of the 
 facts and principles of electromagnetic induction, and the 
 manner in which these are applied in the design and 
 construction of the Induction Coil and Transformer. 
 At that time most of the practical information on the 
 subject was embedded in technical journals and original 
 papers. Confident that alternating electric currents 
 would play a very important part in the evolution of 
 the electrical industry, the author believed that service 
 would be rendered to engineering students by an 
 attempt, even if an imperfect one, to place a brief 
 systematic treatise on the subject of the Alternating 
 Current Transformer within reach. The result, so far, 
 has justified the belief. At the present time, however, 

 
 iv. Preface. 
 
 much of the subject matter in the first edition has 
 become antiquated, and it became necessary to revise 
 the book thoroughly, to eliminate those parts which had 
 been seen to be imperfect or unnecessary, and to bring 
 the remainder of the information as far as possible into 
 line with recent views and experience. 
 
 The author has, accordingly, rewritten the greater 
 part of the chapters, and availed himself of various 
 criticisms, with the desire of removing mistakes and 
 remedying defects of treatment. In the hope that this 
 will be found to render the book still useful to the 
 increasing numbers of those who are practically engaged 
 in alternating-current work, he has sought, as far as 
 possible, to avoid academic methods and keep in touch 
 with the necessities of the student who has to deal with 
 the subject not as a basis for mathematical gymnastics 
 but with the object of acquiring practically useful 
 knowledge. 
 
 It is, perhaps, in some ways, a positive disadvantage 
 that alternating-currents lend themselves so easily to- 
 mathematical treatment and, by a few assumptions akin 
 to that of the perfectly frictionless machine, offer 
 an attractive field for mathematical ingenuity. Real 
 difficulties are thus often passed over, and an intimate 
 knowledge only gained of a perfectly hypothetical 
 transformer. 
 
 The ever-increasing progress of electrical knowledge, 
 and the constant necessity for recasting electrical 
 theories, renders it a most difficult matter to secure in 
 an electrical treatise of any length, uniformity of treat- 
 ment ; whilst views must always differ as to the mode 
 in which any special subject should be approached.
 
 Preface. v. 
 
 The author ventures to hope, however, that in its 
 revised form, the information here collected may be of 
 use to those who are in any way concerned with 
 alternating-current practice or investigations, and that 
 it may be effective as an introduction to treatises of a 
 more advanced character, which deal with the properties 
 of periodic currents and their utilization in various 
 technical applications. 
 
 J. A. F. 
 
 University College, London, 
 April, 1896. 
 
 A 2
 
 CONTENTS. 
 
 CHAPTER I. 
 
 PAGS 
 
 HISTORICAL INTRODUCTION 1 
 
 1. Faraday's Discoveries ; His Electrical Researches ; The Discovery 
 of Induced Currents ; The Decisive Experiment with the Iron 
 Ring ; Lines of Magnetic Force. 2. Faraday's Theories ; 
 Magneto-Electric Induction ; The Electrotonic State ; Number 
 of Lines of Magnetic Force ; Conduction and Induction ; Views 
 of Maxwell, Helmholtz and Kelvin ; Action at a Distance ; The 
 Electromagnetic Medium. 3. Joseph Henry's Investigations ; 
 Henry's Electromagnet ; His Independent Discovery of Electro- 
 magnetic Induction ; Production of Induced Currents from 
 Henry's Large Electromagnet. 
 
 CHAPTER II. 
 ELECTROMAGNETIC INDUCTION 13 
 
 1. Magnetic Force and Magnetic Fields ; The Magnetic State ; 
 Magnetic Fields ; Magnetic Axis ; Direction of Magnetic Force ; 
 Magnetic Force near a Straight Conductor Conveying a Current ; 
 Ampere's Fundamental Law ; Magnetic Force at the Centre of 
 a Circular Conductor Conveying a Current ; Magnetic Force Due 
 to a Circular Current at a Point on the Axis ; Magnetic Force due 
 to a Straight Helix at Points on Axis ; Magnetic Force of 
 a Circular Solenoid. 2. Magnetomotive Force and Magnetic 
 Induction ; Flux of Force ; Magnetic Permeability ; Lines of 
 Induction ; Definition of Unit of Magnetic Induction ; Faraday's 
 Law of Induction ; Electromotive Force due to the Change 
 of Induction. 3. The Magnetic Circuit ; Magnetic Resist- 
 ance ; Definition of Magnetic Resistance or Reluctance ; Case 
 of a Circular Ring ; Lines of Induction of Closed and Open 
 Magnetic Circuits; Fundamental Relation between Magnetic
 
 viii. CONTENTS. 
 
 CHAPTER II. ( Continued.) 
 
 Force and Magnetic Induction ; A Magnetic Shell ; Intensity of 
 Magnetisation ; Magnetic Moment. 4. Lines and Tubes of 
 Magnetic Induction ; Equipotential Surface ; Number of Tubes 
 of Induction ; Surface Integral of Induction ; Linkage of Lines 
 of Induction and Circuits ; Rate of Change of Induction Linked 
 with Circuit. - 5. Curves of Magnetisation ; Curves of Mag- 
 netisation of Soft Iron Ring ; Permeability Curves ; Determina- 
 tion of Permeability. 6. Magnetic Hysteresis ; Hysteresis 
 Curves ; H-B Diagram ; Values of Hysteresis for Various Kinds 
 of Iron and Steel ; Variation of Hysteresis with Speed of Cycle ; 
 Effect of Temperature on Hysteresis ; Experiments of Kunz. 
 7. The Electromotive Force of Induction ; Graphical Repre- 
 sentations ; Electromotive Force of Self-induction. 
 
 CHAPTER III. 
 
 PAGE 
 
 THE THEORY OF SIMPLE PERIODIC CURRENTS 79 
 
 1. Variable and Steady Flow ; Periodic Motion. 2. Current 
 and Electromotive Force Curves ; Simple and Multiple Valued 
 Functions ; Simple Periodic Curves. 3. Simple and Com- 
 pound Periodic Curves. 4. Fourier's Theorem ; Mechanical 
 Harmonographs. 5. Mathematical Sketch of Fourier's 
 Theorem ; Practical Application of Fourier's Theorem to the 
 Harmonic Analysis of a Periodic Curve. 6. Simple Periodic 
 Currents and Electromotive Forces ; Analysis and Synthesis of 
 Compound Periodic Curves. 7. Description of a Simple 
 Periodic Curve. 8. The Mean Value of the Ordinate of a 
 Sine Curve. 9. The Square Root of the Mean of the Squares 
 of the Ordinates of a Simple Periodic Curve ; Mean-Square 
 Value. 10. Derived Curves. - H. Inductance and Inductive 
 Circuits ; Inductance, Resistance, and Capacity of Circuits ; 
 Analogy of Inductance and Inertia ; Inductive and Non- 
 Inductive Circuits ; Faraday's and Henry's Experiments on 
 Self-induction ; Edlund's and Maxwell's Arrangement for 
 Exhibiting the Inductance of Circuits. 12. Electromagnetic 
 Momentum ; Electrotonic State and Electromagnetic Momen- 
 tum ; Coefficient of Self-Induction. 13. Electromagnetic 
 Energy ; Coefficient of Mutual Induction;. Energy of Two 
 Circuits. 14. The Unit of Inductance ; The Henry ; Value 
 of the Self-Induction in Henrys for Various Instruments. 15. 
 Current Growth in Inductive Circuit* ; Analogy of Current and 
 Velocity Change : Fundamental Equations for Current Growth 
 in Inductive Circuits. 16. Equation for the Establishment of
 
 CONTENTS. ix. 
 
 CHAPTER III. (Continued.) 
 
 a Steady Current in an Inductive Circuit ; Time Constant of an 
 Inductive Circuit. 17. Logarithmic Curves ; Law of Growth 
 of Current in Inductive Circuit. 18. Instantaneous Value of 
 Simple Periodic Current ; Solution of Current Equation ; Impe- 
 dance of Inductive Circuit; Relations of Impressed Electromotive 
 Force, Current and Impedance. 19. Geometrical Illustrations; 
 Clock Diagrams. 20. Graphical Representation of Periodic 
 Currents ; Laws of Vector Addition. 21. Impressed and 
 Effective Electromotive Forces ; Clock Diagram of Electromotive 
 Forces in Inductive Circuit ; Triangle Representing Resistance, 
 Impedance, and Reactance of Circuit. 22. The Mean Value 
 of the Power of a Periodic Current ; Geometrical Theorem. 
 23. Power Curves for Inductive and Non-inductive Circuits. 
 24. The Experimental Measurement of Periodic Currents and 
 Electromotive Forces ; Mean-Square Value. 25. Method of 
 Measuring the True Mean Power Given to an Inductive Circuit ; 
 Theory of the Wattmeter ; Divided Circuits ; Important Trigo- 
 nometrical Lemma. 26. Impedance of Branched Circuits. 
 27. Wattmeter Measurement of Periodic Power. 28. Correct- 
 ing Factor of a Wattmeter. 29. Mutual Induction of Two 
 Circuits of Constant Inductance ; Effect of Closed Secondary 
 on the Resistance and Inductance of a Primary Circuit. 30. 
 The Flow of Simple Periodic Currents into a Condenser ; Time- 
 Constant of a Condenser ; Charging a Condenser Through a 
 Resistance ; Admittance ; Condenser Equation. 31. A Shunted 
 Condenser in Series with an Inductive Resistance ; Annulment 
 of Inductance by Capacity. 32. Representation of Periodic 
 Currents by Polar Diagrams ; Determination of Mean-Square 
 Value of a Periodic Quantity from the Polar Diagram. 33. 
 Initial Conditions on Starting Current Flow in Inductive 
 Circuits ; Increased Initial Amplitude of Current. 34. Initial 
 Conditions in Circuits having Capacity, Inductance and Resis- 
 tance. 35. Complex Periodic Functions ; Apparent and True 
 Power Given to Inductive Circuits ; Power Factor. 
 
 CHAPTEE IV. 
 
 PACK 
 MUTUAL AND SELF-INDUCTION 207 
 
 1. Researches of Prof. Joseph Henry on Self and Mutual Induc- 
 tion ; Experiments with Coils and Bobbins ; Discovery of Self- 
 induction. 2. Mutual Induction. 3. Induction at a Dis- 
 tance ; Induction between Telephone Circuits ; Induction Over 
 Great Distances ; Communication with Moving Railway Trains. 
 4. Induced Currents of Higher Orders ; Secondary. Tertiary
 
 x. CONTENTS. 
 
 CHAPTER TV. (Continued.) 
 
 and Quarteuary. Currents. 5. Inductive Effects Produced by 
 Transient Electric Currents, such as Leyden Jar Discharges ; 
 Magnetic Screening ; Direction of Induced Currents ; Various 
 Qualities of an Induced Current. 6. Elementary Theory of 
 Mutual Induction of Two Circuits ; Theory of Induction Coil 
 with Non-Magnetic Core. 7. Comparison of Theory and 
 Experiment ; Duration of Induced Currents ; Researches of 
 Blaserna, Bernstein, and Helmholtz. 8. Magnetic Screening 
 of Good Conducting Masses ; Faraday's and Henry's Experi- 
 ments ; Wilioughby Smith's Investigations on Magnetic Screen- 
 ing ; Dove's Experiments on the same ; Henry's Views on 
 Magnetic Screening. 9. The Reaction of the Secondary 
 Currents on the Primary Circuit in the Case of an Induction 
 Coil ; Elihu Thomson's Experiments. 10. Hughes's Induction 
 .Balance and Sonometer. 11. The Transmission of Rapidly 
 Intermittent or Alternating Currents through Conductors ; 
 Induction Bridges ; Hughes's Experiments with the Induction 
 Bridge ; Lord Rayleigh's Researches ; Experimental Confirmation 
 of Theory ; Flow of Current Through a Conductor ; Surface 
 Flow of Alternating Currents ; Increased Resistance of Con- 
 ductor for Alternating Currents of High Frequency ; Lord 
 Rayleigh's Form of Induction Bridge ; Limiting Size of Con- 
 ductors for Conveyance of Alternating Currents ; Stefan's 
 Analogies. 12. Electromagnetic Repulsion ; Elihu Thomson's 
 Experiments on Electromagnetic Repulsion ; Copper Rings 
 Projected from the Pole of an Alternating Current Magnet ; 
 Copper Disk Galvanometer; Electromagnetic Rotations Produced 
 by a " Shaded Pole " ; Electromagnetic Gyroscope ; Rotations 
 Produced by Magnetic Leakage Field 13- Symmetry of 
 Current and Induction. 
 
 CHAPTER V. 
 
 PAGE 
 
 DYNAMICAL THEORY OF INDUCTION 833 
 
 1. Electromagnetic Theory ; Faraday's Conception of an Electro- 
 magnetic Medium ; Maxwell's Suggestion ; The Lumeuiferous 
 Ether ; Maxwell's Theory of Electric Displacement ; Electric 
 Elasticity of the Medium ; The Displacement Current ; The Con- 
 duction Current : Electromotive Intensity. 2- Displacement 
 Currents and Displacement Waves. 3. Maxwell's Theory of 
 Molecular Vortices ; Mechanical Analogy. 4- Comparison of 
 Theory and Experiment ; Maxwell's Law Connecting Dielectric 
 Constant and Refractivity of a Dielectric ; Tables of Comparison. 
 5. Velocity of Propagation of an Electromagnetic Disturbance ;
 
 CONTENTS. xl. 
 
 CHAPTER V. (Continued.) 
 
 Two Systems of Measurement of Electric Quantities; Electromag- 
 netic and Electrostatic ; Ratio of the Units Defined as a Velocity ; 
 Velocity of Light ; Determinations of the Electromagnetic "V" ; 
 Table of Results ; Explanation of Meaning of Vector Potential ; 
 Operation called " Curling " ; Vector Potential of a Current ; 
 Relation of Vector Potential and Induction ; Fundamental 
 Equation for Propagation of Vector Potential. 6. Electrical 
 Oscillations ; Charge and Discharge of Leyden Jar ; Oscillations 
 in Discharging Circuit ; Experiments on Discharge of Jar ; 
 Oscillatory and Dead-beat Discharge : Influence of Self-induction 
 on Rate of Discharge ; Conditions of Quickest Discharge. 7. 
 The Function of the Condenser in an Induction Coil ; Time of 
 Free Electrical Oscillation in a Circuit ; Conditions under which 
 Inductance can Neutralize Capacity. 8. Impulsive Discharges 
 and Relation of Inductance Thereto ; Lodge's Experiments on 
 the Alternative Path of Discharge ; Lightning Protectors ; 
 Side Flashing ; Conditions of Impulsive Discharge ; Electrical 
 Surgings. 9. Theory of Experiments on the Alternative Path ; 
 Electromagnetic Radiation from Oscillating Discharges ; Syntonic 
 Circuits. 10. Impulsive Impedance. 11. Hertz's Experi- 
 ments, Researches on the Propagation of Electromagnetic Induc- 
 tion ; Preliminary Experiments ; Electric Resonator Circuit ; 
 Experiments with Induction Coil and Oscillating Discharge ; 
 Induction Phenomena in Open Circuits ; Resonance Phenomena ; 
 Oscillations in Iron Wires ; Propagation of Electromagnetic and 
 Electrostatic Effects ; Electromagnetic Waves ; Influence of 
 Dielectrics ; Interference Phenomena ; Reflection ; Refraction 
 and Polarization of Electromagnetic Waves. 13. Propagation 
 of Electromagnetic Energy through Space ; Poynting's Views 
 and Law ; The Importance of the Effects in the Medium. 14. 
 Propagation of Currents along Conductors ; Hertz's Experiments ; 
 Energy Penetrates into the Conductor from the Dielectric. 15. 
 Experimental Determination of Electromagnetic Wave Velocity ; 
 Photographing Electromagnetic Waves. 
 
 CHAPTER VI. 
 
 PAGE 
 
 THE INDUCTION COIL AND TRANSFORMER 514 
 
 1. General Description of the Action of the Transformer or Induction 
 Coil ; Air Core and Iron Core Transformers ; Classification of 
 Transformers ; Problems of the Transformer. 2. The Delinea- 
 tion of Periodic -Curves of Current and Electromotive Force ; 
 Curve Tracing ; Joubert's Method of Delineating Alternating 
 Current Curves ; Fleming's Curve Tracer ; Transformer Diagrams ;
 
 lii. CONTENTS. 
 
 CHAPTEE VI. (Continued.) 
 
 Bloudel's Oscillograph ; Various Curve Tracers. 3. Discussion 
 of Transformer Diagrams ; Curves of Electromotive Force, 
 Current and Induction in Case of Various Transformers taken 
 off Various Alternators ; Open Circuit Current of Transformers ; 
 Various Diagrams ; Symmetry of Transformer Curves ; Ryan's 
 Diagrams ; Effect of Loading-up Transformer on Position of the 
 Curves of Current ; Harmonic Analysis of Transformer Curves ; 
 Only Odd Harmonics Present. 4. Derivation of Curves of 
 Power and Hysteresis ; Hysteresis Curves of Various Trans- 
 formers. 5. The Efficiency of Transformers ; Efficiency 
 'Curves of Closed and Open Circuit Transformers ; Table of 
 Efficiencies. 6. Current Diagrams of a Transformer ; Open 
 and Closed Circuit Transformers ; Tables of Complete Tests. 
 7. Power Factor of Transformers ; Table of Power Factors ; 
 Influence of Form of E.M.F. Curve on the Power Factor ; Tables 
 of Tests ; Relation of Power Factor to Output 8. Magnetic 
 Leakage and Secondary Drop ; Various Causes of Secondary 
 Drop ; Transformation Ratio of a Transformer ; Determination 
 of Magnetic Leakage. 9. Effect of the Form of the Curve of 
 Primary Electromotive Force on the Transformer Efficiency and 
 Currents ; Investigations of the Author and of Dr. Roessler. 
 10. The Form Factor and Amplitude Factor of a Periodic 
 Curve. 11. General Analytical Theory of the Transformer 
 and Induction Coil. 
 
 APPENDIX. 
 
 NOTE A. (page 19) 597 
 
 The Magnetic Force at any point in the Plane of a Circular 
 Current 
 
 NOTE B. (page 35) 599 
 
 The Total Induction in a Circular S.jleuoM. 
 
 -NOTE C. (page 52) 600 
 
 The Calibration of the Ballistic Galvanometer.
 
 ERRATA, 
 
 
 63 In the table at the bottom of page 63, for the power 
 wasted in watts per cubic centimetre per 100 cycles 
 per second, in all the laumerical values except the 
 first for very soft annealed iron, the decimal point is 
 displaced. The numbers should be divided by ten. 
 
 117. Line 6 from top, for lies down at b read dies down at b. 
 
 122. Line 7 from top, for 10 9 linkages of current and magnetic 
 force read 10 s linkages of current and magnetic force ; 
 
 ^ 
 
 B S \ 
 also, in equation (25), for _ ~ read -- 
 
 and in line 7 from bottom, 
 
 162. The equation numbered (12) should be numbered (39). 
 184. Equation (47), instead ofv = V(l+e~ ^ 
 read t = V(l + ~5o). 
 
 _ R 
 
 195. Last line equation (64),/br i = A.e L + , &c., 
 read i = Ae~^- t +, <fec. 
 
 203. Line 11 from bottom, for of the mean is - read of the 
 mean is J. 
 
 219. In the first line of paragraph 4, for 1888 read 1838. 
 
 375. Line 7 from bottom, for then Sir William Thomson 
 read then Prof. William Thomson. 
 
 377. In equation (116), for q = A -'" + B e m * read q = \ e "^ 
 
 38:?. At the end of line 7 insert a full stop, and in line 8 for 
 when t = Q read When t = 0. 
 
 58H. Equation (156), for N, S ^ mid N, S ^1 ; 
 (It dt 
 
 also, in equation (157), for + N, S 1^1 rid - N 2 S ^?, 
 
 in line 7 from bottom, /or N, 8^ read N S^l 
 '/ d f ' 
 
 and in line 4 from bottom, for - N, S-^rawJN S^? 
 
 rf'
 
 CHAPTER I. 
 
 HISTOEICAL INTRODUCTION. 
 
 1. Faraday's Discoveries. The autumn days of the year 
 1831 are ever memorable in the annals of electrical discovery. 
 At that time Faraday, then in the prime of his intellectual 
 powers, began the continuous series of " Electrical Researches," 
 which enriched physical science with discoveries of far-reaching 
 importance, and laid the firm foundations on which much of 
 the modern applications of electricity rests. Looking on the 
 whole of electrical phenomena with an eye eager to see physical 
 analogies, and confident that where these exist they may prove 
 suggestive for further research, he had already asked himself 
 if it were possible there was any effect in the case of electric 
 currents analogous to that known as electrostatic induction. 
 An insulated conductor possessing an electric charge when 
 introduced into a closed chamber having conducting walls calls 
 forth upon them an equal charge of an opposite sign. This 
 induced electrification is invariably present, 110 matter how far 
 off the walls of the enclosing chamber may be, and all sur- 
 rounding conductors share in the duty of carrying a portion of 
 the induced charge. At a later date, when Faraday viewed 
 this phenomenon of electrostatic induction by the aid of the 
 education he had received in dealing with magnetic lines of 
 force, he was able to picture to himself lines of electrostatic 
 force proceeding in all directions from the surface of a charged 
 body. Wherever they terminated, whether on neighbouring con- 
 ductors or on the walls of an enclosing chamber, they developed 
 on these " corresponding points " a charge equal and opposite 
 to that of the surface at the point from which they took their 
 rise. Just eleven years previously H. C. Oersted had made
 
 2 HISTORICAL INTRODUCTION. 
 
 Copenhagen famous as the birthplace of the discovery that an 
 electric current passing through a metallic wire magnetises it 
 circularly and creates round it a magnetic field, the direction 
 of the lines of magnetic force being closed curves surrounding 
 the axis of the wire. Faraday placed these two phenomena 
 side by side before his mental vision, and he asked himself 
 whether it was possible that the magnetic field of force gene- 
 rated round a current-carrying conductor could develop in an 
 adjacent circuit an induced current just as the charged body 
 calls forth an induced electrostatic charge on the neighbouring 
 conductors. 
 
 Some notions on this subject of electric current induction 
 had before 1831 occupied his mind at intervals, but these 
 early experiments did not lead to any satisfactory results. 
 
 In 1825, in the month of November, Faraday stretched 
 alongside of a wire connected with a galvanometer another 
 through which an electric current was flowing, but both then 
 and on December 2, 1825, and on April 22, 1828, he had to 
 record of his experiment that it gave "no result." A very 
 little step in experimental research often separates failure from 
 success. A reversal of operations, a change of some dimension, 
 an alteration of some proportion, is often all that is needed to 
 step from the region of failure into the field of discovery and 
 achievement. In this case it may have been the apparently 
 trivial one of starting the electric current in one wire before 
 completing the circuit of the galvanometer. 
 
 The one thing, it seemed, that this preliminary work did dis- 
 prove was the notion that a continuous steady current in one 
 conductor could generate a continuous current in another 
 adjacent conductor relatively at rest to the first. It is possible 
 that some conception of the above nature had been dominant 
 in the mind of Faraday before these trials had convinced him 
 that the effect, if existing at all, was not detectable with his 
 apparatus. Three years later he returned to the attack, and 
 we cannot describe the experimental results of the autumn 
 months of 1831 better than they have been given in Faraday's 
 own words in the laboratory note-books of the Royal Insti- 
 tution.* On the 29th day of August, 1831, he thus records 
 the epoch-making discovery by which he will be for ever 
 
 * See Dr. Bence Jones's " Life of Faraday,' Vol. II., p. 2.
 
 HISTORICAL INTRODUCTION. 3 
 
 known. He wrote : " I have had an iron ring made (soft 
 iron), iron round and fin. thick, and ring Gin. in external 
 diameter. Wound many coils of copper round one-half of it, 
 the coils being separated by twine and calico ; there were three 
 lengths of wire, each about 24ft. long, and they could be con- 
 nected as one length or used as separate lengths. By trials 
 with a trough, each was insulated from the other. Will call 
 this side of the ring A. On the other side, but separated by 
 .an interval, was wound wire in two pieces, together amounting 
 to about 60ft. in length, the direction being as with former 
 coils. This side call B. Charged a battery of ten pairs of 
 plates 4in. square. Made the coil B side one coil, and con- 
 nected its extremities by a copper wire passing to a distance 
 and just over a magnetic needle (3ft. from wire ring), then 
 connected the ends of one of the pieces on A side with battery ; 
 immediately a sensible effect upon needle. It oscillated, and 
 settled at last in original position. On breaking connection of 
 A side with battery, again a disturbance of the needle." (See 
 Frontispiece.) 
 
 On September 24th he resumed his attack. He prepared an 
 iron cylinder and wound on it a helix of insulated wire. The 
 ends of the helix were connected with a galvanometer. The 
 iron was then placed between the poles of bar magnets. Every 
 time the magnet poles were brought in contact with the ends 
 of the iron cylinder the galvanometer needle indicated a 
 current, the effect being, as in former cases, not permanent, 
 but a mere momentary impulse or deflection. 
 
 But the full meaning of this hardly appeared clear, and on 
 October 1st he once more laid siege to the fortress. Preparing 
 a battery of 100 pairs of plates, each 4in. square, and charged 
 with a mixture of nitric and sulphuric acids, he arranged to 
 send the current from this through a wire of copper 203ft. long 
 wound round a block of wood. Bound the same block, and 
 wound parallel to the first, was a second wire, of equal length 
 to the first, but insulated from it. This second wire he joined 
 up to the terminals of his galvanometer, and then when the 
 battery connection was made or broken with the first wire he 
 noticed a small but sudden jerk of the needle, one way when 
 the current was made, the other way when it was broken. 
 The clue to the real phenomenon was now in his hand, and 
 
 B2
 
 4 HISTORICAL INTRODUCTION. 
 
 guided by it he stepped over a series of confirmatory experi- 
 ments, and entered as a triumphant conqueror into the 
 stronghold wherein the whole truth lay hid. 
 
 Writing on November 29th to his friend, Mr. E. Phillips, he 
 says : " Now, the pith of all this I must give you very briefly. 
 When the electric current is passed through one of two 
 parallel wires it causes at first a current in the same direction 
 through the other, but this induced current does not last a 
 moment, notwithstanding the inducing current (from the voltaic 
 battery) is continued. All seems unchanged except that the 
 principal current continues its course. But when the current 
 is stopped, then a return current occurs in the wire under 
 induction of about the same intensity and momentary duration, 
 but in the opposite direction to that first formed. Electricity 
 in currents, therefore, exerts an inductive action like ordinary 
 electricity, but subject to peculiar laws. The effects are a 
 current hi the same direction when the induction is estab- 
 lished, a reverse current when the induction ceases, and a 
 peculiar state in the interim." 
 
 The path for valuable discovery now lay open. Fully 
 familiar with the work of Ampere and Arago, Faraday knew 
 that a closed circuit conveying an electric current affects all 
 surrounding space with magnetic force, and that, in particular, 
 a small closed circular current can, as far as magnetic action 
 is concerned, be exactly replaced by a very thin disc of steel, 
 whose edge coincides with the line of the closed current, 
 and which is magnetised everywhere in a direction perpen- 
 dicular to its surface. Such a normally magnetised disc is 
 called a magnetic shell. It follows that a helix of wire, which 
 may be regarded as a number of closely approximate circular 
 currents nearly in the same plane, should be magnetically 
 equivalent to a number of magnetic shells piled one above the 
 other, with similar polar faces turned the same way. But such 
 an arrangement of shells would form a cylindrical magnet, and 
 therefore a helix of wire or solenoid in which a current is flow- 
 ing is for all external space the magnetic equivalent of a cylinder 
 of steel of the same dimensions magnetised uniformly in a 
 longitudinal direction. It remained, therefore, to test this 
 hypothesis. The fifth day of his experiments was October 
 17th, and on that day he thus notes in the laboratory book the
 
 HISTORICAL INTRODUCTION. 5 
 
 results: "A cylindrical bar magnet fin. in diameter and 8|in. 
 in length had one end just inserted into the end of a helix of 
 wire 220ft. long. It was then quickly thrust in the whole 
 length, and the galvanometer needle moved ; then pulled out 
 again, and again the needle moved, but in the opposite direc- 
 tion. This effect was repeated every time the magnet was 
 put in or out, and therefore a wave of electricity was so pro- 
 duced from mere approximation of a magnet." 
 
 Exactly twenty years afterwards, in the 28th and 29th series 
 of his " Researches," Faraday illuminated, by the exactness and 
 clearness of his experimental method, the whole behaviour of 
 magnets towards closed conducting circuits. It is probable 
 that even at this time he had learned to think of a magnet as 
 carrying with it, as part of itself, a whole system of lines of 
 magnetic force, which emanate from it and surround it. The 
 system of lines of force moves with the magnet wherever it 
 goes. Regarding the production of a current in the helix by a 
 magnet thrust into it, Faraday pictured to himself the advanc- 
 ing magnet as pushing its lines of magnetic force across the 
 coils of wire of the helix, and "cutting " or intersecting them in 
 its progress towards its final position in the coil. The conclusion 
 to which he was led by this reflection seemed to be that the 
 very essence of the effect was the movement across one another 
 of a line of force and a portion of a conducting circuit. If this 
 was so, then the result could be obtained by a more simple and 
 obvious method. The ninth day, October 28th, saw these ideas 
 put to further crucial test. Taking the great permanent horse- 
 shoe magnet of the Royal Society, he placed a copper disc so 
 that it was free to revolve on an axis placed in the line of the 
 poles. Soft iron pole pieces were then adjusted to create a 
 powerful magnetic field, the lines of force of which passed 
 through the disc at right angles to its surface. The wires of 
 the galvanometer were made to press against the disc, one 
 near the axis, and the other near the edge. When the disc 
 remained stationary, no current whatever was manifested, but 
 on causing the disc to revolve on its axis a permanent and 
 steady current traversed the galvanometer. This experiment 
 was conclusive. The operation taking place during the revolu- 
 tion of the disc could be viewed as consisting simply in the 
 continual movement of any radial section of the disc across a
 
 6 HISTORICAL INTRODUCTION. 
 
 stream of lines of magnetic force flowing at right angles to its 
 surface. The continuous current resulted from the fact that 
 the motion of that radial section of the disc was always the 
 same relatively to the stream of force. On November 4th. 
 Faraday reduced the conception to its utmost simplicity. 
 Taking in his hand the mere closed galvanometer wires, he 
 passed a portion of the loop between the poles of his large 
 permanent magnet in such a way that the direction of that 
 part of the loop between the poles was at right angles to the 
 direction of the magnetic force, and the direction of the move- 
 ment was at right angles to the direction of the force and that 
 portion of the conductor. The galvanometer deflected, and 
 showed the presence of a momentary current at the instant 
 when the intersection took place. 
 
 2. Faraday's Theories. In ten days of splendid and con- 
 clusive experiment in the autumn of 1831, Faraday had there- 
 fore not only discovered the law of induction of currents, but 
 the facts of magneto-electricity as well ; and more, for he had 
 not merely accumulated a mass of experimental results, but 
 had reduced the whole valuable store of knowledge to one 
 fundamental principle of exquisite simplicity, namely, that the 
 passage of a line of magnetic force across a line of a conducting 
 circuit generates in that portion of the circuit an electromotive 
 force, or a force setting electricity, or tending to set electricity, 
 hi motion. 
 
 The subsequent work of all experimentalists and mathema- 
 ticians has been to work out the applications of this principle 
 in countless forms ; but no one has since added any essential 
 discovery of fact which is not implicitly contained in the series 
 of discoveries by which, in this short space, Faraday stepped 
 from happy conjectures into possession of facts, which have 
 proved more fertile in far-reaching practical consequences than 
 any of those which even his genius bestowed upon the world. 
 Faraday's theoretical views, however, on the phenomena under- 
 went, in process of time, some modification. He apparently 
 distinguished at first between the induction of currents by 
 a current, which he called i-olta-eleciric induction, and the pro- 
 duction of currents by a conductor moving in a magnetic field, 
 which he called magneto-electric induction. That which seemed
 
 HISTORICAL INTRODUCTION. 7 
 
 to impress him most forcibly was, however, the fact that it 
 was only the beginning and ending of the inducing current 
 which had any effect upon the other circuit. He considered 
 that, since the mere cessation of the inducing current was 
 accompanied by a wave of induced current, that could only 
 be because the induced current circuit was, meantime, in a 
 peculiar condition, to which he gave the name of the electro- 
 tonic state, the annulment of which gave rise to a current in 
 the circuit. The same state he considered to be found in 
 a wire or circuit at rest in a magnetic field. The circuit was 
 in the electrotonic state whilst in the field, but withdrawing 
 the circuit or removing the magnetic field annulled the electro- 
 tonic state and gave rise to a current. To use his own words 
 at a later date (Ser. XXVIIL, 3172, Exp. Eesearches "), 
 "Mere motion would not generate a relation which had not 
 a foundation in the existence of some previous state ; " and 
 (Ser. XXIX., 3269, ibid.} " Again and again the idea of an 
 electrotonic state has been forced upon my mind." The mere 
 motion of an external body, such as a copper wire, in a mag- 
 netic field cannot, he considers, be the sole cause of the 
 current, unless there is a previous peculiar state as regards 
 the wire which, when motion is superadded, produces the 
 current. When, however, subsequent thought and diverse 
 experiment had clarified his ideas and adjusted facts in proper 
 relation, he came to see that that which he had denominated 
 the electrotonic state is really the amount of electromagnetie 
 momentum which the circuit possesses in virtue of its being in 
 a magnetic field. In modern language, it is the equivalent of 
 that which is now called the number of lines of magnetic force 
 passing through the circuit. Every line of magnetic force is 
 a closed loop or continuous line, and if we set out at any point 
 on a line of magnetic force and travel forwards along that line 
 we shall come back to that same point again. If this line of 
 force is originated by a permanent magnet or an electromagnet, 
 then part of our journey will be performed through the iron or 
 steel and part through the air or othe'r diamagnetic surround- 
 ing it. If, then, a closed conducting circuit is so situated that 
 the line of force considered passes through it or is linked with 
 it, the line of force and the closed circuit form, as it were, two 
 links of a chain, and cannot be separated except by pulling
 
 8 HISTOEICAL INTRODUCTION. 
 
 one through the other (Fig. 1). When they are so pulled 
 through one another the line of force " cuts " and is cut by 
 the circuit. The number of lines of force, therefore, which at 
 any instant are linked with a given circuit represent poten- 
 tially the greatest amount of " cutting " possible. The exist- 
 ence of lines of magnetic force linked with the circuit is an 
 essential antecedent to the appearance of a current of induction 
 in that circuit when removed from the magnetic field. At 
 a later stage of his investigations Faraday was able to modify 
 his earlier notions of the electrotonic state, and learnt to look 
 on the induced current appearing under these circumstances as 
 due not to a state of things in the circuit, but to a condition 
 of things outside the circuit, or, more precisely, to the relation 
 in which the circuit stands to the magnetic field of force 
 around it. 
 
 FIG. 1. 
 
 In the 28th and 29th series of his "Experimental Ee- 
 searches," Faraday exhausted all possible means of experi- 
 ment in proving that this conception of the linking or un- 
 linking of loops of force and loops of conducting circuits was 
 an unerring guide to the solution of all problems of electro- 
 magnetic induction. The circuit being given, he was able to 
 show by a course of rigid demonstration that the process of 
 linking with it a loop of magnetic force was always accompanied 
 by the passage of a wave of current round the circuit in one 
 direction, and the unlinking was invariably associated with the 
 flow of an opposite pulsation of electricity. Moreover, and 
 most important of all, he built up a quantitative conception 
 around the term a line of magnetic force," so that it came to 
 him to mean not merely a geometrical line or a direction, but 
 a definite physical magnitude, which represented the product of 
 a certain area of space, and a certain mean intensity of mag-
 
 HISTORICAL INTRODUCTION. 9 
 
 netic force over that area.* Armed with this idea, he proceeded 
 to show that the quantity of electricity represented by each 
 current of induction is the numerical equivalent of the " number 
 of lines of force " which are linked or unlinked with the circuit 
 by any operation. He found that this hypothesis never failed 
 to enable him to render a satisfactory and a logical explanation 
 of all his results, and with this clue in hand he could find his 
 way about amidst the entanglements of experimental inquiry, 
 and return always from each fresh excursion after fact with 
 new confirmation of its consistency, and with fresh power to 
 predict the results of other experiments. 
 
 So strong became at last his conviction that these lines of force 
 could hardly have such powers if they were mere geometrical 
 conceptions, like lines of latitude and longitude, that he gives 
 expression to it by speaking of them as physical lines of force. 
 He intends to imply that he thinks " a line of force " must be 
 taken to be a definite action going on in a certain region of 
 space, and that, whatever may be its real nature, we must accord 
 to it a definite physical character in some sort or sense, as much 
 as we do an electric current of unit strength flowing along a 
 prescribed circuit. Faraday was not a professed mathematician, 
 and it was perhaps fortunate that his inability to employ the 
 mechanical aid of symbolic reasoning forced him to make clear 
 to himself each step by experimental demonstration. He was 
 thereby compelled to keep to the main track of discovery, and 
 prevented from deviating into the more abstract lines of thought. 
 The special abilities of Kelvin and Helmholtz, and subse- 
 quently those of Clerk Maxwell, were, however, directed to the 
 complete elucidation of these conceptions of Faraday, and the 
 great treatise of Maxwell, as he himself has stated, was under- 
 taken mainly with the hope of making these ideas the basis of a 
 mathematical method. The one cardinal principle which may 
 be said to be at the base of the mode of viewing electrical and 
 magnetic phenomena introduced by these investigators is the 
 denial of action at finite distances, and accounting for the 
 phenomena by the assumption of the existence of a medium 
 
 * Faraday's notion of " a line of force " was at first merely a geometrical 
 conception, representing a certain line of action, but his ultimate applica- 
 tions of the term showed that he had come to think of it as a surface 
 integral.
 
 10 HISTORICAL INTRODUCTION. 
 
 which is the active agent in the transmission of energy from 
 one place to another, and which is itself capable of storing up 
 energy in a potential and kinetic form. 
 
 The mathematical methods and hypotheses of the French 
 school of physicists, represented chiefly by Ampere, Arago, 
 Poisson, and Coulomb, consisted in the assumption that material 
 particles in special states, called electric and magnetic, could 
 act on one another at finite distances without any intervening 
 mechanism according to certain laws of force varying with the 
 distance. Faraday may be said to have raised the standard of 
 revolt against this notion, and indeed he was able to quote in 
 his support the great authority of Newton in rejecting the idea 
 that matter could act on matter across intervening distance 
 without aid from any mechanism. He never considers bodies 
 as existing with nothing between them but their distance, and 
 acting on one another according to some function of that dis- 
 tance. He conceives all space as a field of force, the lines of 
 force being in general curved, and those due to any body ex- 
 tending from it on all sides, their direction being modified by 
 the presence of other bodies. A magnet, an electrified conductor, 
 or a wire conveying an electric current, are thus the focus and 
 source of a system of radiations of force lines or loops which 
 are to be thought of as part and parcel of it. This force- 
 system is capable of deformation or change by the presence of 
 other bodies, but it moves with the magnet, electrified body, or 
 current-carrying wire. These force radiations penetrate sur- 
 rounding bodies, and the apparent actions between bodies at a 
 distance are in reality actions due to immediate action of the 
 field of force of one body upon the other at the place where 
 it is. Then rises for solution the important problem : What 
 are these lines of force ? Faraday answered the question by 
 saying that they consist in some sort of operation or action 
 going on in a medium along certain lines or axes, and Maxwell 
 added to this the suggestion that the electromagnetic medium 
 must be identical with the medium postulated to account for 
 the phenomena of light. 
 
 The question which yet remains unanswered is : What is 
 the nature of the action or operation along certain lines in this 
 medium which causes a line of force to exist ? The future of 
 electric and magnetic investigation will, perhaps, conduct us
 
 HISTORICAL INTRODUCTION. 11 
 
 step by step to the solution of this supremely important 
 problem. 
 
 3. Henry's Investigations. At the same time that Fara- 
 day was pursuing in England a career of triumphant discovery 
 in the field of electromagnetic science a young philosopher of 
 hardly less intellectual power, but more limited opportunities 
 for research, was following hard on the same path of investi- 
 gation in America. The name of Joseph Henry is one which 
 we must link with that of our own great countryman as a co- 
 worker, nay, even an anticipator in some things, in the region 
 of fundamental discovery in electromagnetism. 
 
 To Henry clearly belongs the credit of having improved 
 Sturgeon's electromagnet by substituting for the single layer 
 of copper wire wound on the iron horse-shoe a spool or bobbin 
 of insulated copper wire. By this means he made what he 
 then called intensity magnets, or electromagnets, suitable for 
 excitation by an intensity battery or battery of many cells. 
 Henry in this manner constructed in 1829 or 1830 a very large 
 electromagnet, capable of supporting a weight of GOOlb. or 
 700lb. Before having any knowledge of Faraday's experi- 
 ments, and guided apparently by the notion that as electric 
 currents can produce magnetism, so magnetism should be able 
 to generate electric currents, Henry experimented as follows : 
 A piece of wire about 30ft. long and covered with an elastic 
 varnish was closely coiled round the middle of the soft iron 
 armature of this large electromagnet. The wire was wound 
 upon itself so as only to occupy about lin. in length of 
 the armature which was 7in. in all its length. The ends 
 of this wire were connected by long copper wires with a 
 distant galvanometer. The armature with its coil was laid 
 upon the poles of the electromagnet, and the galvanic plates 
 connected with the helix of the electromagnet immersed in the 
 trough of acid. At the moment of immersion the needle of 
 the galvanometer was seen to be deflected about 30deg., but it 
 immediately returned to its normal position. On withdrawing 
 the battery plates from the acid it was noticed that the gal- 
 vanometer needle made a sudden deflection in the opposite 
 direction of about 20deg. A similar effect was produced by 
 pulling off or putting on the armature whilst the magnet
 
 12 HISTORICAL INTRODUCTION. 
 
 remained excited. Henry, in Ms account of this experiment, 
 savs ;_ From the foregoing facts it appears that a current of 
 electricity is produced for an instant in a helix of copper wire 
 surrounding a piece of soft iron whenever magnetism is induced 
 in the iron, and a current in the opposite direction when the 
 magnetism ceases ; also, that an instantaneous current in one 
 or other direction accompanies every change in the magnetic 
 intensity of the iron." 
 
 This very lucid statement of experiments, made probably in 
 August, 1831, shows that Henry was at least an independent 
 discoverer of the induction of electric currents. In April, 1832, 
 an account reached him of Faraday's discovery in the previous 
 year, and Henry then repeated his former experiments, and 
 was able by means of larger helices of wire wound on the 
 armature of his electromagnet to greatly increase the magni- 
 tude of the induced current. Henry, therefore, not only dis- 
 covered independently the facts of electromagnetic induction, 
 but correctly interpreted them as well. He early laid a firm 
 grasp upon the essential principles involved, and he came 
 almost within reach of anticipating that discovery which is, 
 and will remain, the crowning glory of his illustrious rival. 
 Between 1831 and 1840, or later, Henry continued to add 
 fresh knowledge to the original facts, and in a later chapter 
 a description will be given of his important investigations on 
 the self and mutual induction of conducting circuits.
 
 CHAPTER II. 
 
 ELECTKO-MAGNETIC INDUCTION. 
 
 1. Magnetic Force and Magnetic Fields. Certain sub- 
 stances, such as iron, nickel, cobalt, steel, and some of their 
 compounds, particularly a native oxide of iron, possess peculiar 
 physical properties, and either exist in, or can be put into, a 
 condition in which they are said to be magnetised. When in 
 this condition they exhibit physical qualities which are called 
 magnetic properties, the most obvious of which is the power 
 of producing attraction and repulsion upon other magnetic 
 substances. Some bodies, notably hardened steel, can acquire 
 marked permanent magnetic qualities. The neighbourhood 
 round these bodies when in this state, and within which they 
 exercise these actions, is called a magnetic field. If a small 
 magnetised steel needle is suspended freely at its centre of 
 gravity and held in a magnetic field it is found that it takes up- 
 a certain direction under the influences of forces acting upon 
 it. If disturbed from this position it returns to it again. 
 
 It is found that there is a line in the needle round which it 
 can be revolved without changing the set of that line when 
 the needle is left free to obey the forces acting upon it. The 
 direction of this line in the needle is called its magnetic axis. 
 Oersted discovered that a magnetic field exists in the neigh- 
 bourhood of a conductor conveying an electric current, and 
 that it imposes a certain directive influence upon a magnetic 
 needle held near to it. If a small steel magnetised needle is 
 placed in any region containing either conductors carrying 
 electric currents or substances in a permanent magnetic state- 
 it is found that at every point of the field the magnetic axis of 
 this small exploring needle takes up a definite position if it is
 
 14 ELECTRO-MAGNETIC INDUCTION. 
 
 freely suspended so as to be removed from the influence of 
 gravity. The direction so assumed by its magnetic axis is 
 called the direction of tlie magnetic force at that point. The 
 magnetic force has at every point in the magnetic field of 
 these active agents a certain direction. On examining the 
 behaviour towards one another of two magnetised steel needles 
 we find that their magnetic properties are exhibited chiefly at 
 the two extremities, and these are called the magnetic poles. 
 The two poles of a magnetic needle are not identical in quality. 
 If a uniformly magnetised steel needle is broken in the middle, 
 the ends where it is broken immediately become new magnetic 
 poles, whereas before rupture that portion of the needle exhibited 
 no apparently active magnetic properties. If the two poles 
 which make their appearance at the broken ends are tested it 
 will be found that they attract one another. If these poles 
 are placed one centimetre apart and the force with which they 
 attract one another measured in absolute units, the square root 
 of the number which expresses this attraction is called the 
 numerical value of the strength of these poles. Hence, a unit 
 magnetic pole is a pole which at a unit of distance attracts 
 another unit pole of opposite kind with a unit of force. The 
 earth as a whole is a magnetic body, and if a small magnetic 
 needle is freely suspended at its centre of gravity, its magnetic 
 axis assumes a certain position at each point on the earth's 
 surface which is called the direction of the terrestrial magnetic 
 force at that point. The pole of the needle which points in 
 our latitude in any direction north of the true east and west 
 line is called the north pole or north-seeking pole of the 
 needle. If we take a very long thin magnetised needle, 
 called for shortness a magnetic filament, we can employ one 
 pole of it, say the north pole, for exploration in a field, whilst 
 the other pole is so far removed as not to be affected. If such 
 a pole, called for shortness a free north pole, is placed in any 
 magnetic field it is acted upon by the magnetic force and 
 urged to move in the direction of this force. If this free 
 north pole is a pole of unit strength, then the force dynami- 
 cally measured in absolute units which acts upon it is called 
 the numerical measure of the magnetic force at that point. 
 
 The direction in which a free north pole tends to move 
 is called the positive direction of the magnetic force at that
 
 ELEGTEO-MAGNETIG INDUCTION. 15 
 
 point. The magnetic force at any point in the magnetic 
 field of magnetic bodies, whether magnetised substances or 
 conductors conveying electric currents, is thus a quantity 
 which has direction as well as magnitude, and we have 
 defined above how both of these can be measured. By 
 means of a free north magnetic pole of unit strength we 
 may thus explore and define a magnetic field at every point. 
 
 A magnetic field in which the magnetic force is the same 
 in magnitude and direction at every point is called a uniform 
 magnetic field. The magnitude of the magnetic force at 
 any point is a measure of the strength of the magnetic field 
 at that point. 
 
 There are several simple and yet important cases in which 
 it is possible to calculate the strength of the magnetic field 
 or the magnetic force at certain assigned points in the 
 neighbourhood of conductors conveying electric currents. 
 The pre-determination of the field strength at points near 
 to magnets and conductors conveying electric currents is, 
 generally speaking, except in these simple cases, a very 
 difficult matter. 
 
 The Magnetic Force near to a very long Straight Wire 
 conveying an Electric Current. 
 
 If a current flows in a thin circular wire we may call a 
 very short length of this conductor, denoted by d s, an 
 element of the circuit or of the current. Ampere showed 
 by a classical series of experiments that the magnetic force 
 due to an element of a current at any point near it was 
 numerically equal to the product of the strength of the 
 current, the length of the element, and the sine of the 
 angle between the direction of the element of the circuit 
 and the line joining the centre of that element with the point, 
 and inversely as the square of this distance. Thus, if ds 
 (Fig. 2) represents the element P of a circuit in which is 
 flowing a current of strength I in absolute electromagnetic 
 measure, and if a is the angle which any line P makes with 
 the direction of the element, and r is the length of the line 
 OP, then the magnetic force at the point due to that 
 element of the current is numerically equal to 
 I d s sin a
 
 16 ELECTRO-MAGNETIC INDUCTION. 
 
 and this force is in a direction at right angles to the plane- 
 containing the element of the circuit and the line joining it to 
 the given point. Starting with this fundamental law, we can 
 deduce expressions for the strength of the magnetic field due 
 to currents flowing in conductors of certain forms at certain 
 assigned points. Consider, for instance, a very long, practically 
 infinite straight wire in which a current is flowing, the return. 
 
 wire being at a very great distance. Take any point P in the 
 neighbourhood of this conductor (Fig. 3). It is required to 
 find the magnetic force at the point P. Draw P M perpen- 
 dicular to the wire from P. Let N N' be any element of the 
 conductor. Then the magnetic force at P due to the element 
 ds = N N' of the conductor is in a direction at right angles to 
 the plane of the paper, and if the length N P is called r and. 
 
 FIG. 3. 
 
 the angle P N M is called a, the magnetic force at P due to tha 
 element N N' is equal numerically to 
 
 d s sin a 
 
 (1) 
 
 where I is the current flowing in the element. Let P M be 
 denoted by p. In order to find the magnetic force due to the 
 whole wire at P we have to integrate the above expression 
 throughout the whole length of the wire. To do this we
 
 ELECTRO-MAGNETIC INDUCTION. 17 
 
 transform it as follows : Let the angle M P N = 0, then N P N' 
 is the increment of this angle, call it d6. From the geometry 
 of the figure it is easily seen that sin a = cos 0, and that when 
 
 ds is very small J^ - I ,or ^ = d . 
 rdv p r 2 p 
 
 Hence substituting these values for sin a and ds/r? in equation 
 (1), we have as the expression for the value of the magnetic 
 force at P, duo to the element N N' of the current, the formula 
 
 rf F - 1 cos ^ cld 
 
 P 
 
 The magnetic force due to the whole infinitely long straight 
 current is obtained by integrating this expression between the 
 
 limits 6 = and 6 = 1L and then doubling this value. Hence 
 the magnetic force of the whole wire at P is equal to 
 
 f^cos8dO=. . (2) 
 
 P J P 
 
 In other words, the magnetic force at any point due to the 
 current I flowing in an infinitely long straight conductor is 
 in magnitude inversely proportional to the perpendicular dis- 
 tance of the point from the wire ; and, as regards direction, it 
 is everywhere perpendicular to the plane containing the wire, 
 and the perpendicular let fall on it from the given point. This 
 conclusion was experimentally verified by Biot and Savart by 
 vibrating a small magnetic needle at different distances from 
 a long straight current, and counting the square of the numbei 
 of oscillations in a given time made by the said small needle 
 in these different positions. If the current in the wire is 
 measured in amperes, then, since ten amperes equal one 
 unit current in absolute electromagnetic measure, and if A 
 is the current so measured in amperes, the magnetic force at 
 any point p centimetres from the wire is equal to 
 
 A^=l A 
 
 10 p 5 p ' 
 
 Thus the magnetic force due to a current of one ampere 
 flowing in a long straight wire at a point one centimetre from 
 the wire is equal to one-fifth of a unit of magnetic force. This 
 is nearly equal to the value of the earth's horizontal magnetic
 
 18 ELECTRO-MAGNETIC INDUCTION. 
 
 force in England. It will be seen that the magnetic force due 
 to powerful currents in long straight cables may be sensible at 
 points very far removed from the cable. This magnetic force 
 is at every point perpendicular to the conductor, and hence the 
 direction of the force of such a straight conductor must be 
 everywhere a tangent to a circle drawn round the wire with 
 its plane perpendicular to the axis of the wire and its centre 
 in that axis. Hence a freely suspended magnetic needle tends 
 to stand perpendicular to a straight conductor when this last 
 is traversed by a current. If a magnetic pole of strength m is 
 placed at any point in the field of such a straight conductor 
 the magnetic force tends to drive the pole in a circle round the 
 
 wire with a force equal to m dynamical units or dynes, 
 
 P 
 
 where p is the distance of the pole from the axis of the wire 
 and I is the absolute value of the current flowing in it. It 
 follows that the lines of magnetic force of such a linear current 
 are circles described round the wire with planes perpendicular 
 to it and centres in the axis of the wire. Oersted was aware 
 of this fact, and he expressly says,* " The electric conflict " 
 (that is, magnetic field) " performs circles round the wire." 
 We may next proceed to determine 
 
 The Magnetic Force at the Centre of a Circular Current. 
 If a thin wire is bent into a circle, and a current of slrength 
 I is sent round it, the magnetic force, estimated at the centre 
 of the circle, due to each element of the length of the current, 
 is in a direction at right angles to the plane of the circle. Let 
 ds be an element of length and let r be the radius of the 
 circular wire, then the magnetic force due to ds at the centre 
 
 is equal to 5- . But, since the force due to each element is 
 r* 
 
 the same, the magnetic force due to the whole length of the 
 circular wire is equal to 
 
 <) 
 
 If the current is measured in amperes and denoted by A, 
 then the magnetic force at the centre of the circular current is 
 
 * Annals of Philosophy, Oct., 1820 Vol. XVI., p. 274,
 
 ELECTRO-MAGNETIC INDUCTION. 19 
 
 This magnetic force at the centre of a circular current is in 
 a direction perpendicular to the plane of the circle. If the 
 circular current consists of a current of A amperes flowing 
 in a very thin wire wound n times round a circular groove of 
 mean radius r, then the magnetic force at the centre is 
 equal to 
 
 ^ (5) 
 
 5 T 
 
 The expression for the magnetic force at a point in the plane 
 of the circle not in the centre is less simple (see Appendix, 
 Note A). 
 
 The Magnetic Force due to a Circular Current of n turns at a 
 
 point on its axis out of its own plane. 
 
 The third case of importance is to find the value of the 
 magnetic force due to a circular current at a point on a line 
 
 FIG. 4. 
 
 drawn through its centre and perpendicular to its plane. Let 
 the circular current be X Y Z (Fig. 4), and let P be any 
 point on a line P drawn through the centre and perpen- 
 dicular to the plane of X Y Z. The magnetic force at P, 
 due to an element ds of the circuit at X, acts along a line 
 perpendicular to X P, and is in the plane of X P. If r 
 stands for X, and x for P, the magnetic force due to the 
 element ds at P resolved in the direction P is equal to 
 
 ( 
 
 where I is the strength of the current in the element. The 
 above is equal to 
 
 Irds
 
 20 ELECTRO-MAGNETIC INDUCTION. 
 
 and hence the magnetic force due to one whole turn of the 
 
 conductor is equal to 
 
 I 2 TT r 2 
 
 If the circuit makes n turns, the magnetic force at P, due 
 to the current I flowing n times round the circular conductor 
 X Y Z estimated in the direction P, is equal to 
 
 27TI ^_, (6) 
 
 This, then, is the expression for the magnetic force, due to 
 a circular current of n turns at a point on its axis but outside 
 of its own plane. The calculation of the magnetic force due 
 to the circular currrent at points other than those on the axis 
 P, is a much more difficult matter. In the above formula I 
 is measured in the electromagnetic units. If the current is 
 measured in amperes and denoted by A, then (6) becomes 
 
 ^A__^ (7) 
 
 The above expression may be put into another useful form. 
 
 r 2 
 ^ mce ?~2 2\I 
 
 is the differential with respect to x of 
 x 
 
 which last, as can be seen from Fig. 4, is equal to cos X P 0, 
 we may write (7) in the form 
 
 *nAA(cos0) (8) 
 
 where stands for the angle X P 0. 
 
 The Magnetic Force due tn a long closely-coiled Helical Current 
 
 at point* on the axis near the centre. 
 
 Another useful case in which it is possible to calculate the 
 magnetic force due to a current is in the case of points in the in- 
 terior of a very long closely-coiled helical current called a sole- 
 noid. Such a case is practically realised by coiling insulated 
 wire round a tube. Let the length of the helix be I, and let 
 there be N turns of wire per unit of length. Then if a slice of
 
 ELECTRO-MAGNETIC INDUCTION. 21 
 
 this helix is considered of thickness d x, the number of turns of 
 wire in this slice is N d x. Let a current I flow through the 
 wire. Take a point on the axis of the helix somewhere near 
 the centre (see Fig. 5), and take any element of length of the 
 helix at a distance x from this point. Then by (8) the mag- 
 netic force due to this element of length of the helix at the 
 point P is 
 
 dx 
 where 6 is the angle P X. 
 
 Let the slice of the helix be taken at successive distances 
 from the point P, bsginning with x = and ending with the 
 end of the helix. The sum of all the magnetic forces due to 
 each element of the helix to the left of the point P is then 
 equal to the integral of (9) taken between the limits 6 = 90deg. 
 or cos = and 0=0,, where O l is the angle 0' P X', or the 
 angle subtended by half the mean diameter of the end of the 
 
 FIG. 5. 
 
 helix at the point P. Similarly, to obtain the whole force at P 
 due to the elements of the helix lying to the right of P we have 
 to integrate (9) from 6 = to 6 = 2 , where 9 is the angle sub- 
 tended by half the aperture of the other end of the helix at P. 
 Adding these forces together we have as value of the whole 
 magnetic force of the whole helix at P the expression 
 
 F = 27rNI(cos fli + cos 2 ). 
 
 If the helix is so long that the half diameter of the aperture of 
 the ends of the helix, as seen from the point P, is practically 
 zero, then : and 2 are both practically zero, and therefore 
 cos 0J + COS 2 = 2, nearly, or 
 
 F = 47rNI (10) 
 
 The magnetic field at P is then equal to 4?r times the absolute 
 current-turns per unit of length. If the current is measured 
 
 o O 
 
 in amperes, the force is equal to -?rN A, or to -TT times the 
 
 5 5
 
 22 ELECTRO-MAGNETIC INDUCTION. 
 
 ampere-turns per unit of length of the coil. Since - TT is nearly 
 
 1-25, the approximate practical rule for the magnetic force 
 in the neighbourhood of the centre of a long helix of this kind 
 is that the magnetic force is numerically equal to 1 times 
 the ampere-turns per unit of length of the helix. The above 
 formula is only strictly true for points on the axis of the helix 
 and for helices very long compared with their diameters. It 
 is very nearly true for all points in the interior of a fairly long 
 helix. Thus, for instance, if the helix is twelve diameters 
 long, the magnetic force in the interior throughout one quarter 
 of its length on either side of the central point does not differ 
 by much more than one per cent, from the value it has at the 
 central point. Hence this fact presents us with an easy and 
 practical method of procuring a magnetic field of known 
 strength. On a long pasteboard or metal tube provided 
 with cheeks wind covered copper wire carefully and evenly 
 in any number of layers. Count the turns and layers of 
 wire, and measure the length between the cheeks ; this 
 gives us the turns per unit of length. Then pass a known 
 current through the wire, and calculate by formula (10) the 
 field at the centre. The coil should be at least twelve 
 diameters long. We may approximately apply (10) to calcu- 
 late the magnetic force in the interior of such long bobbins 
 as are used in winding the field-magnets of dynamos. The 
 magnetic force in the interior of a long bobbin is strongest in 
 the centre of the bobbin, and falls off towards either end, 
 and it is slightly stronger at points nearer the wire than 
 on the central axis even at the centre. The complete 
 calculation of the field at any point in the interior or 
 exterior of a not very long helix is a rather difficult matter, 
 but the above formula (10) will be sufficient for most practical 
 purposes. 
 
 A final, practical, and useful case is that of the predetermina- 
 tion of the magnetic force in tJie interior of a circular closed 
 solenoid or endless helical current. Let a wooden ring of 
 circular cross-section be wound over closely with insulated 
 wire so that the turns of the wire are contiguous and one 
 or more layers are put on. This is called a circular solenoid, 
 and we can calculate the magnetic force for points in the
 
 ELECTRO-MAGNETIC INDUCTION. 23 
 
 interior when the circular solenoid is traversed by a current. 
 Let R be the mean radius of the solenoid, and a that of the 
 mean circular section. If the wire is wound in one layer 
 on a wooden ring, then a will be the mean between the 
 half diameter of the section of the ring and the half diameter 
 measured over all after the wire is wound on it. 
 
 The magnetic force is not the same at all points over the 
 circular cross-section of the solenoid. To find out what-it is 
 at any point we may proceed as follows : A solenoid of any 
 size, meaning by that a spiral current with turns closely 
 adjacent, is electrically equivalent to a bundle of elementary 
 solenoids or spiral currents of exceedingly small cross-section. 
 Consider such a very small-sectioned solenoid, which may be 
 called a spiral filament. It may be obtained in practice by 
 winding insulated wire of small size on a very fine knitting 
 needle as a core, and then withdrawing the needle. The section 
 of this solenoid being very small, the magnetic force in its 
 interior is everywhere nearly the same over the cross-section, 
 and if the spiral is long the force in the centre in the interior 
 is equal to 47rral, where I is the absolute current flowing 
 in the wire, and n is the number of turns per unit of length. 
 Let this long elementary solenoid be bent round into a circle 
 so as to form a closed or endless solenoid, let x be the mean 
 radius of the circle which it forms and let n v be the number 
 of turns of wire of the spiral in an arc of the solenoid equal 
 to one unit angle in circular measure. Then, the number 
 of turns per unit of length of the spiral being n, we have 
 n x = n v and we may write the expression for the magnetic 
 force in the interior of the solenoid as 
 
 (11) 
 
 A little consideration will then show that the magnetic effect 
 of any circular solenoid must be the same as that of a bundle 
 of elementary solenoids, so wound and arranged as that the 
 number of turns of wire of each spiral per length of arc 
 subtending one unit angle in circular measure is the same. 
 The magnetic force in the interior of the circular solenoid 
 at any point in the cross-section is then equal to the product 
 of im l I, and the reciprocal of the perpendicular distance
 
 24 ELECTRO-MAGNETIC INDUCTION. 
 
 of this point from the axis of the circular solenoid. The 
 force over the cross-section is not uniform, but has a 
 particular value for every point, but the same value at all 
 points at an equal radial distance from the axis of the 
 circular solenoid or ring coil. 
 
 2. Magnetomotive Force and Magnetic Induction. We 
 have in the foregoing section denned magnetic force, and 
 shown how it can be determined in a few simple cases from 
 a fundamental principle. If any line is drawn in a magnetic 
 field of force, and we sub-divide this line into very small 
 elements of length, and estimate the magnitude of the mag- 
 netic force at the centre of each element resolved in the 
 direction of this element, and then sum up all the products 
 obtained by multiplying the length of each element by the 
 strength of the magnetic force along its direction, we obtain 
 the line integral of magnetic force along that line. This is also 
 called the magnetomotive force along that line. 
 
 In mathematical language, if H is the magnetic force at any 
 point on the line, and the angle this force makes with the 
 line and d s, an element of length of that line at that point, then 
 
 I H cos d s is the line integral of magnetic force along that 
 
 line. From the definition of magnetic force, it is clear that 
 this line integral is the ivork done in carrying a free unit 
 magnetic pole along that line. This magnetomotive force 
 along a line is likewise called the difference of magnetic potential 
 between the two ends of the line. In those cases in which the 
 magnetic force has a uniform value and is in the direction of 
 the path chosen, it becomes a simple matter to calculate the 
 magnetomotive force along that Ihie, for it is the simple 
 product of the numerical values of the magnitude of the 
 magnetic force and the length of the line. 
 
 Let us consider two simple cases. First, when the line integral 
 is taken along a closed line or loop in a magnetic field drawn 
 hi air or other non-magnetic medium, but not linked with 
 or encircling a circuit conveying a current. In this case the 
 value of the line integral is zero, because no work is done in 
 carrying a free pole around a closed path in an air field. 
 Second, when the line integral is taken along a path which is
 
 ELECTRO-MAGNETIC INDUCTION. 25 
 
 a closed loop, and which surrounds or is linked with a circuit 
 conveying an electric current. Consider the simplest case. 
 Let a straight wire convey an electric current C, the return 
 being at a great distance. Describe a circular line round the 
 wire at a distance r from the axis of the wire. The length of 
 this line is 2ir r ; the magnetic force at a distance r from a 
 
 2C 
 straight wire is units ; and the line integral along this 
 
 op 
 
 line is TT r x - = 4?r C. Hence the line integral of the 
 
 magnetic force taken once round the circuit is 4?r times the 
 total current through the line of force. This can be shown 
 to be generally true, and is the general relation between 
 magnetic force and current.* 
 
 If a looped line is taken through a helical current which 
 links itself round the line n times, then, if A amperes traverse 
 the conductor, the total quantity of current flowing through 
 the loop is n A (equal to the ampere-turns), or in absolute 
 
 C. G. S. measurement is ; hence the line integral of the 
 magnetic force taken along any closed line threading n times 
 through the circuit of a current A is ^-nA., or 1^ times the 
 
 ampere-turns of the current which are linked with the closed 
 
 4?r 
 
 line. It is useful to remember that the value of ^ is very 
 
 nearly 1-25. 
 
 We have here introduced the student to the notion of a 
 line integral. Another similar mathematical idea which has 
 to be grasped is that of a surface integral. If, in any field of 
 magnetic force, we describe a surface of any form bounded by 
 a closed line, the magnetic force at all points of this surface 
 will have a certain value, call it H. Let the surface be 
 supposed to be divided in a number of very small elements 
 of surface each equal to d S. At the centre of each element 
 estimate the value of the magnetic force perpendicularly or 
 normally to that element, take the product of the value of 
 this normal value and that of the element of area. If 6 is 
 the angle between the direction of the force and that of the 
 
 * See Electrician, Vol. X., p. 7 : Mr. Oliver Heaviside on " The Relation 
 between Magnetic Force and Electric Current."
 
 26 ELECTRO-MAGNETIC INDUCTION. 
 
 normal to the surface, then the product of the normal force, 
 or H cos 6 and d S is to be taken for all elements of the 
 surface. The sum of all such products is called the surface 
 integral of tlie force, and is expressed in mathematical language 
 by the integral /H cos Od S. 
 
 This is also called the flux of (he force through the area, for 
 if we suppose that, instead of dealing with magnetic force, we 
 were considering the velocity of a moving fluid, the surface 
 integral of the velocity over any area would represent the 
 whole quantity of liquid which flows in one second through 
 the line bounding the area or the flux of the fluid. 
 
 Returning to the measurement of magnetomotive force, the 
 reader will notice that, as a consequence of the above general 
 theorem, in those cases in which we are dealing with the 
 magnetic force due to a spiral current or solenoid making a 
 number of turns round, or linkages with, the line of the 
 magnetomotive force, the measurements of this magneto- 
 motive force is practically made in ampere-turns, or by the 
 products of the number of turns of the wire and the ampere 
 current conveyed by it. 
 
 Owing to the fact that the circumference of a circle is 2?r 
 
 times its radius in length we get a numeric -^ introduced 
 
 which makes the magnetomotive force, measured along a 
 line linked with a line of current making n turns round it, 
 numerically equal to 1^ times the ampere-turns, but it is not 
 difficult to remember or to use this simple factor. 
 
 When magnetomotive force acts on any body, whether 
 magnetic like iron or non-magnetic h'ke wood or air, it 
 produces in it an effect called magnetic induction. The student 
 must think of magnetic induction as something which is 
 produced by magnetomotive force, just as electric current 
 is produced by electromotive force. Magnetic induction is a 
 quantity which is called a flux, and the magnetic induction 
 in magnetic bodies results from magnetomotive force or 
 difference of magnetic potential, just as the flow of water 
 results from difference of pressure or head of water, and the 
 flow of electricity in conductors from difference of electric 
 potential, and the flow of heat in thermal conductors from 
 difference of temperature.
 
 ELECTRO-MAGNETIC INDUCTION. 27 
 
 The quality in virtue of which magnetomotive force can 
 produce magnetic induction in a magnetic body is called its 
 magnetic inductivity, or magnetic permeability. A body having 
 large permeability is one in which a given magnetic force 
 produces a relatively large magnetic induction. Similarly, 
 we might say that the quality of bodies in virtue of which a 
 hydrostatic force or pressure produces a flow of fluid through 
 them is called their porosity, and a very porous body is one 
 in which a given hydrostatic pressure produces a relatively 
 great flow of liquid. Quite similarly we define electric and 
 thermal conductivity as those qualities of bodies in virtue of which 
 electromotive force and difference of temperature produces 
 in them electric current or flow of heat. Hence magnetic 
 permeability, electric conductivity, thermal conductivity, 
 porosity, and, we may add, specific inductive capacity, in 
 electrostatics are all analogous qualities of bodies numerically 
 capable of being measured which are of importance in that 
 they determine the amount of the flux produced by a unit of 
 force of the corresponding kind. 
 
 The magnetic induction, like magnetic force, has a definite 
 direction as well as magnitude at every point where it exists- 
 Both magnetic induction and force belong to that class of 
 quantities which in mathematics are called vector quantities, 
 and possess both magnitude and direction. In order to 
 define the direction and amount of magnetic induction we 
 fall back upon the fundamental discovery of Faraday. The 
 ground fact of all his investigations is that if a conducting 
 circuit is placed in a field of magnetic induction any change 
 in the magnitude or strength of the magnetic induction 
 will create an electromotive force in that circuit urging, 
 or tending to urge, an electric current round it, provided 
 that the plane of this circuit has any but one particular 
 direction. We may, therefore, use a small conducting circuit 
 to explore a field of magnetic induction, just as we employed a 
 free magnetic pole to explore a field of magnetic force. Let a 
 small circular conducting circuit, formed, say, of one turn of 
 very thin wire, and enclosing one unit of area, be placed 
 in a field of magnetic induction. Let this small loop or 
 unit circuit be held in various positions, and let changes 
 be made in the induction by varying the magnetomotive
 
 28 ELECTRO-MAGNETIC INDUCTION. 
 
 force. It will be found that there is a particular position 
 or positions of the circuit in which no change in the strength 
 of the induction produces any electromotive force in this 
 small circuit. The direction of the induction at the centre 
 of the circuit is then parallel to the plane of the circuit. 
 The axis round which the circuit can be revolved without 
 affecting this inactive condition of the circuit is the direction 
 of the induction at that place. Hence we can map out at 
 all points of the field of induction the direction of the magnetic 
 induction. If the circuit is turned into any other position 
 such that a change in the induction does produce an electro- 
 motive force acting in the circuit, we may find by trial 
 another position of the circuit at any point in the field in which 
 the total suppression of the induction, or its instantaneous 
 reversal in direction, produces the maximum electromotive 
 force in the circuit. The direction of the induction is then 
 normal to the plane of that exploring circuit. At every 
 point in the field of induction the induction has a certain 
 magnitude as well as direction. If we suppose any plane 
 surface placed normally to the direction of the induction in a 
 field of uniform induction, the product of the strength of the 
 induction and the area of the surface is called the total 
 induction through that area. If we take any surface placed 
 in any position, and suppose it divided up into small elements 
 of surface d S, and at the centre of each element of the surface 
 estimate the magnitude of the induction in a direction normal 
 to the surface at the centre of the element, and sum up all the 
 products obtained by multiplying the normal value of the 
 induction and the area of the element, the sum so obtained is 
 called the surface integral of induction. If we draw in a field 
 of induction a line such that the direction of its tangent 
 at any point is the direction of the induction at that point, 
 such a line is called a line of induction. Suppose a surface of 
 any kind, for simplicity a plane surface, placed in a field 
 of induction, and let it be divided up into small areas such 
 that over each of them the surface integral of induction is 
 equal to unity, and if through the centre of each of these 
 small areas we draw a line of induction, then we may make 
 the following statements: The bounding line of the surface 
 is said to be perforated by induction, or to have a flux of
 
 ELECTRO-MAGNETIC INDUCTION. 29 
 
 induction taking place through it, or to have lines of induction 
 passing through it. Since the total surface integral of induc- 
 tion through this surface is, by definition, equal numerically to 
 the number of lines of induction passing through the boundary 
 of the surface, we may also speak of the number of lines of 
 induction passing through the surface. Faraday used the 
 phrase number of lines of magnetic force instead of induction. 
 Hence the student should notice that the following expressions 
 all denote the same thing : 
 
 1. The surface integral of induction over a surface. 
 
 2. The total induction through the surface (Maxwell). 
 
 3. The flux of induction through the surface. 
 
 4. The number of lines of induction passing through the 
 surface. 
 
 5. The number of lines of force (Faraday) passing through 
 the surface. 
 
 If a circuit be placed normally in a field of uniform 
 induction, then the numerical product of the area of the 
 circuit and the strength of the induction gives us the total 
 induction through that surface, or total number of lines of 
 induction (force) passing through that circuit. 
 
 In the twenty-eighth series of his "Experimental Eesearches 
 on Electricity," Faraday examined afresh with elaborate care 
 the notion of lines of magnetic force which had guided him 
 at all stages of his electromagnetic discoveries. He there 
 gathers together his ideas on this subject, and by a series 
 of researches, inimitable for physical insight and exquisite 
 experimental skill, he has shown how the definition of a line 
 of force or induction can be raised from a merely qualitative 
 or directive one into a quantitative conception by which not 
 only the direction but the magnitude of the induction can 
 be denoted. Having placed clearly before his mind the idea 
 of the surface integral of induction or the total induction 
 through any surface or circuit as the important one to hold 
 in view, he proceeds in the latter part of his investigations 
 ( 3,152 and 3,199 " Exp. Res.") to show experimentally 
 that when any closed circuit, such as a loop of wire, is placed 
 in a field of magnetic induction, and if in any way the total 
 induction through that circuit is changed, there is a flow of 
 electricity round the circuit, and the total quantity so set
 
 30 ELECTRO-MAGNETIC INDUCTION. 
 
 flowing is proportional to the conducting power of the circuit, 
 and to the change in the total magnetic induction passing 
 through it. Employing for this purpose a ballistic galvano- 
 meter, or a galvanometer with a needle having a long periodic 
 time of vibration inserted in the circuit, he placed circuits 
 of various forms in fields of induction, and exhausted every 
 possible method of experimental proof that in every case 
 any change which altered the total induction through the 
 circuit was accompanied by the production of an electro- 
 motive force in the circuit, and that the product of the total 
 quantity of electricity so set in motion, and the number repre- 
 senting the resistance of the whole circuit, was in every case 
 proportional to the change in the total induction through 
 the circuit. This provides us with the means of defining 
 the strength or magnitude of induction and stating what 
 is meant by a unit of total induction. 
 
 A unit of magnetic induction is a flux or amount of induction 
 such that, when passing through or linked once with a circuit of 
 unit resistance, it gives rise, if suppressed, to a flow of a unit 
 quantity of electricity round tliat circuit. 
 
 In other words, the above is the definition of what is 
 meant by OTM line of induction (or force), linked once with 
 a single turn of a circuit of unit resistance. We can then 
 define the meaning of the term density of magnetic induction 
 or induction density. It is the total induction through a unit 
 of area taken normally to the lines of induction at that place. 
 Instead of the term density of induction, it is usual to speak 
 simply of the induction at any point in a field of induction, 
 or the number of lines of induction (force) passing normally 
 through a unit of area. 
 
 We shall employ the letter B to stand for the induction at 
 any point in a field of magnetic induction. Hence in the 
 notation of the differential calculus d B will stand for any 
 small change in the induction. Let a circuit formed of one 
 single turn of wire whose resistance is B be placed in a 
 uniform field of induction of strength B, and let S be the 
 area of that circuit ; the total induction through the circuit 
 will be B S. Let the induction be changed by an amount 
 d B, then the total induction is changed by an amount 
 d (B S). According to Faraday's experiments the change
 
 ELECTRO-MAGNETIC INDUCTION. 31 
 
 will set a small quantity of electricity, which may be denoted 
 by d q, flowing round the circuit, and we have 
 
 If the whole of this change of induction takes place in a 
 very short interval of time, which may be denoted by dt, 
 then during that time the total flow of electricity is equiva- 
 lent to an average current of strength ?', and we have 
 
 idt = dq. 
 Hence also d ( B 8) = E id t. 
 
 But E i is the instantaneous value of the induced electro- 
 motive force in the circuit ; let this be denoted by e, and 
 we have 
 
 or S = e ....... (12) 
 
 Therefore the magnitude of the induced electromotive force 
 in the circuit is at any instant expresssed by the time-rate 
 of change of the total induction passing through the circuit. 
 
 If we bear in mind clearly the meaning which Faraday 
 attached to the phrase " a line of force " or " the number of 
 lines of force " as expressing the total induction through any 
 area or conducting circuit, we can express in his language 
 the above fact in a statement which may be called Faraday's 
 Law of Induction ; it is as follows : 
 
 If there be in any field of magnetic induction a circuit 
 which is traversed by induction, then any change either 
 of the size or position of the circuit or of the direction or 
 strength of the induction which changes the number of lines 
 of force (induction) passing through the circuit creates an 
 induced electromotive force in this circuit which is numeri- 
 cally equal at any instant to the rate of change in the number 
 of lines of force (induction) so passing through it. The 
 above defines the magnitude of the electromotive force ; we 
 have next to define its direction in the circuit. To do this 
 we must make certain conventions. If a watch-face is held 
 in front of the observer, then the positive direction through that 
 watch-face is away from the observer, and the positive direction 
 round that disc is in the direction in which the hands rotate. 
 The connection between positive direction round and positive
 
 32 ELECTRO-MAGNETIC INDUCTION. 
 
 direction through can be also fixed by thinking of the 
 direction of the thrust and twist of a corkscrew or other 
 right-handed screw. The negative direction of rotation is 
 therefore the counter-clockwise direction. 
 
 In regard to the direction of the induced electromotive 
 force, the following is the law : The insertion of lines of 
 induction into a circuit in the positive direction, or the in- 
 crease of positively-directed lines of induction, gives rise to 
 negatively-directed electromotive force. Let the reader bear 
 in mind the following rule : Imagine a magnetic north pole 
 and a magnetic south pole placed opposite to each other, as in 
 the case of a horse-shoe magnet or dynamo field-magnet. 
 The positive direction of the magnetic force in the interspace 
 is by convention from the north pole to the south pole. This 
 field of force creates magnetic induction in the air-space, 
 and the positive direction of the lines of induction is in the 
 same direction from the north pole to the south pole in the 
 air-space. Place a watch (non-magnetic) in this space with 
 its watch-face facing the north pole ; the watch-face is 
 traversed by a certain flux of induction. Let the rim of this 
 face be thought of as a conducting circuit. Then demagne- 
 tise the magnets or withdraw them so as to diminish the 
 induction perforating through the watch- face ; it will generate 
 an induced electromotive force in the rim which will tend to 
 set an induced current flowing round the rim in the direc- 
 tion in which the hands of the watch usually rotate, or in 
 the positive direction. Hence as time increases induction 
 diminishes, and we get positively- directed electromotive force. 
 If, then, N represents the number of lines of induction passing 
 
 positively at any instant through the watch-face and - 
 
 d t 
 
 the rate of decrease of induction at any instant, since this 
 creates positively-directed electromotive force, we must write 
 
 -=, 
 
 dt 
 
 The differential coefficient must, therefore, have the negative 
 sign. 
 
 The above rules accordingly settle the magnitude and the 
 direction of the induction and of the rate of change of induc- 
 tion, and hence of the induced electromotive force.
 
 ELECTRO-MAGNETIC INDUCTION. 33 
 
 3. The Magnetic Circuit. Magnetic Resistance. We 
 have in the foregoing section denned magnetic force, magneto- 
 motive force, and magnetic induction, and explained how they 
 are measured. We have in the next place to examine the 
 conditions under which magnetic induction is produced by 
 magnetomotive force. The same magnetomotive force will 
 not always produce the same magnetic induction. What it 
 will produce depends upon the nature ot the magnetic circuit. 
 The path of a line of induction is always a closed loop that 
 is, it is an endless line. In its path it passes either through a 
 magnet or is linked with an electric circuit, and the path of a 
 line of induction is called a magnetic circuit. This circuit 
 may pass wholly through air, or it may pass partly or wholly 
 through iron masses, or of masses partly of iron, air, brass, 
 wood, &c. There are three principal kinds of magnetic 
 circuits. First, those in which the path of a line of induction 
 lies wholly in iron or magnetic metals. Second, those in 
 which it travels wholly in air or in non-magnetic metals. 
 Third, those in which it passes partly through iron masses 
 and partly through air or non-magnetic materials. This 
 distinction is founded upon the fact that there is a very 
 great difference between circuits in which the lines of in- 
 duction lie wholly in iron, wholly in air or non-magnetic 
 materials, or partly in magnetic and partly in non-magnetic 
 materials. 
 
 A given magnetomotive force produces an enormously 
 greater induction when the path of the lines of induction is 
 wholly in iron masses than it does when they are wholly in 
 air. This is analogous to the fact that a given electromotive 
 force produces a much greater current when the electric 
 circuit of given dimensions is composed wholly, say, of copper, 
 than it does when it is composed wholly, say, of carbon. The 
 number expressing the ratio between the numerical value of 
 the electromotive force and that of the total current produced 
 by it in any electric circuit is a measure of the value of the 
 electrical resistance of that circuit, and briefly we may say that 
 for unvarying or continuous currents the electrical resistance 
 of a circuit 
 
 _ The electromotive force acting in the circuit 
 The total current in circ uit 
 
 r
 
 34 ELECTRO-MAGNETIC INDUCTION. 
 
 Similarly the term magnetic resistance is applied to that 
 quality of the magnetic circuit in virtue of which a given 
 magnetomotive force produces a certain definite total mag- 
 netic induction, and the numerical measure of this magnetic 
 resistance of a circuit, is obtained by taking the quotient of the 
 numbers representing the magnetomotive force and the total 
 induction, or the magnetic resistance of a circuit 
 
 The magnetomotive force acting in the circuit 
 The total induction in circuit 
 
 There is, however, a great difference between electric and 
 magnetic circuits. In the case of electric circuits, though the 
 electric resistance is affected by temperature changes and 
 other physical alterations, yet, when all necessary corrections 
 are made, it is found practically that the electric resistance of 
 a circuit does not depend upon the current flowing through 
 it, whereas in the case of magnetic circuits formed wholly 
 or partly of magnetic metals the magnetic resistance of the 
 circuit does depend essentially upon the value of the induction. 
 
 It has not yet been found necessary to coin words analogous 
 to volt, ampere, and ohm, as names for the practical units 
 of magnetomotive force, magnetic induction, and magnetic 
 resistance. 
 
 All magnetic circuits which are wholly composed of non- 
 magnetic substances, such as air, wood, brass, &c., have the 
 same magnetic resistance for the same dimensions, and the 
 specific magnetic resistance of these substances, or the magnetic 
 resistance per unit length and unit cross-section, is taken as 
 unity. As an instance of the great difference in magnetic 
 resistance between an air circuit and an iron circuit let us 
 consider the following simple case. 
 
 Let two rings be made, one of wood and the other of the 
 best soft iron. Let these rings have a circular cross-section of 
 radius a, and let the mean radius of the circular axis of the ring 
 be denoted by R, the value of E being large compared with that 
 of a. Let each of these rings be wound uniformly and closely 
 over with insulated wire, and let there be N turns on each 
 ring. We have then two circular solenoids, and if a current is 
 passed through each coil of which the strength is A amperes, 
 we have equal magnetomotive forces acting round the axis of
 
 ELECTRO-MAGNETIC INDUCTION. 36 
 
 -each circular solenoid. This magnetomotive force along the 
 .circular axis of the ring is equal to the product of the length 
 2?rR of the path of the line of induction and the strength of 
 the field along this line, and it is therefore approximately equal 
 
 47T NA 
 
 The sectional area of the magnetic circuit is IT a 2 , since all the 
 lines of induction are confined to the inside of the circular 
 solenoid and from endless circular lines. The mean length of 
 the magnetic current is 2n-R. The total magnetie resistance 
 is numerically obtained by multiplying the specific magnetic 
 resistance (which in the case of the wooden ring is equal to 
 unity) by the length and dividing by the section of the path. 
 
 Hence, the magnetic resistance is equal to 1 x ^ = 2 , and 
 
 TT a- a- 
 
 this is the total magnetic resistance of the circuit. The total 
 induction is numerically equal to the quotient of the magneto- 
 motive force, viz., ' N A, by the magnetic resistance, viz., 
 
 , and is therefore equal to =^_ , and the induction or 
 
 a* ' 10 . R 
 
 1 N A 
 
 induction density is equal to - , since the section is jr 2 . 
 
 5 R 
 
 The above is very nearly true if a is very small compared 
 with R, but if this is not the case, we have to take account of 
 the fact that in this last instance the magnetic force has 
 different values at different points over the cross-section of the 
 solenoid (see Appendix, Note B). 
 
 As an actual example we may take the case of a wooden 
 ring in which a = 1-27015 centimetres, R= 12-8291 centi- 
 metres,' and N = 623. If a current of 1 ampere is sent 
 round this wire, the magnetomotive force on the axis is 
 
 ^ X N A = 1 x 623 = 779, and the induction density along the 
 central axis is \ = _ 62B =9-7 C.G.S. units nearly, or 
 
 about 10 lines of induction per square centimetre. 
 
 If, however, a soft iron ring of the same dimensions is 
 employed and wound over with the same number of turns and 
 the same current employed, it would have been found that the 
 
 D2
 
 38 ELECTRO-MAGNETIC INDUCTION. 
 
 this additionally complicates the case. These magnetic poles 
 produce a reverse magnetising force in the interior of the iron 
 which opposes the magnetic force due to the magnetising 
 solenoid. Such an imperfect iron circuit is often called an 
 open magnetic circuit, whilst the complete iron ring core would 
 be called a closed magnetic circuit. The wood, or air, or other 
 body of unit specific magnetic resistance is said to form a gap 
 in the iron magnetic circuit. 
 
 ' 
 
 FIG. 8. 
 
 Diagram showing the paths of the Lines of Induction in and near an 
 Iron Ring with wider air-gap in it, when magnetised by a solenoid wound 
 on it. The dotted lines show the paths of the lines of induction in the 
 iron core, and the exterior field is supposed to be delineated by iroa 
 filings. 
 
 Speaking generally, the subject of closed magnetic circuits 
 is more easy to treat and deal with than that of open magnetic 
 circuits. The particular reason why magnetic problems are 
 more difficult to manage than the corresponding electrical 
 problems is because the magnetic resistance even of a closed 
 magnetic circuit is not a constant quantity. It is not only 
 affected by temperature, but is also determined, within very 
 wide limits, by the value of the magnetic induction itself, by
 
 ELECTRO-MAGNETIC INDUCTION. 39 
 
 the direction of the induction, in that it is not the same for 
 increasing as for diminishing induction when the magneto- 
 motive force, and therefore the induction, is periodic ; and, in 
 fact, it is dependent upon the whole past magnetic history of 
 the iron. It is impossible, therefore, to draw up a table of 
 exact specific magnetic resistances of magnetic metals as we 
 can draw up a table of specific electrical resistances, and 
 reduce the calculation of the induction produced by a given 
 steady magnetomotive force to the same simplicity that 
 results from the application of Ohm's law to electric circuits. 
 
 The reciprocal of magnetic resistance is magnetic conductance. 
 Faraday used the phrase " conducting power for lines of 
 magnetic force" to express the reciprocal notion to that of 
 magnetic resistance. The reciprocal quality to specific magnetic 
 resistance is also called magnetic permeability . This is generally 
 denoted by the symbol fi. Hence the magnetic permeability 
 is the magnetic conductance of a magnetic circuit of unit 
 length and unit cross- section. If p stands for specific magnetic 
 resistance and p. for magnetic permeability, we can say that 
 
 Consider a closed magnetic circuit of uniform specific magnetic 
 conductance p- and of length / and section s, and let a uniform 
 magnetic force H act in this circuit, producing a uniform 
 induction B. Then by definition the magnetic resistance 
 R of the circuit is given by 
 
 -D _ H I _ Magnetomotive force 
 B s Total induction 
 
 where H I is the magnetomotive force, and B s the total 
 induction ; but 
 
 therefore we have B^H ....... (14) 
 
 This equation (14) is not only a fundamental equation in 
 magnetism, just as Ohm's law is in electro-kinetics, but it 
 furnishes a definition of the method of measuring magnetic 
 permeability.
 
 38 ELECTRO-MAGNETIC INDUCTION. 
 
 this additionally complicates the case. These magnetic poles 
 produce a reverse magnetising force in the interior of the iron 
 which opposes the magnetic force due to the magnetising 
 solenoid. Such an imperfect iron circuit is often called an 
 open magnetic circuit, whilst the complete iron ring core would 
 be called a dosed magnetic circuit. The wood, or air, or other 
 body of unit specific magnetic resistance is said to form a gap 
 in the iron magnetic circuit. 
 
 .^/ /V> :;: 
 
 FIG. 8. 
 
 Diagram showing the paths of the Lines of Induction in and near an 
 Iron Ring with wider air-gap in it, when magnetised by a solenoid wound 
 on it. The dotted lines show the paths of the lines of induction in the 
 iron core, and the exterior field is supposed to be delineated by iron 
 
 Speaking generally, the subject of closed magnetic circuits 
 is more easy to treat and deal with than that of open magnetic 
 circuits. The particular reason why magnetic problems are 
 more difficult to manage than the corresponding electrical 
 problems is because the magnetic resistance even of a closed 
 magnetic circuit is not a constant quantity. It is not only 
 affected by temperature, but is also determined, within very 
 wide limits, by the value of the magnetic induction itself, by
 
 ELECTEO-MAGNETIG INDUCTION. 39 
 
 the direction of the induction, in that it is not the same for 
 increasing as for diminishing induction when the magneto- 
 motive force, and therefore the induction, is periodic ; and, in 
 fact, it is dependent upon the whole past magnetic history of 
 the iron. It is impossible, therefore, to draw up a table of 
 exact specific magnetic resistances of magnetic metals as we 
 can draw up a table of specific electrical resistances, and 
 reduce the calculation of the induction produced by a given 
 steady magnetomotive force to the same simplicity that 
 results from the application of Ohm's law to electric circuits. 
 
 The reciprocal of magnetic resistance is magnetic conductance. 
 Faraday used the phrase " conducting power for lines of 
 magnetic force " to express the reciprocal notion to that of 
 magnetic resistance. The reciprocal quality to specific magnetic 
 resistance is also called magnetic permeability . This is generally 
 denoted by the symbol /*. Hence the magnetic permeability 
 is the magnetic conductance of a magnetic circuit of unit 
 length and unit cross- section. If p stands for specific magnetic 
 resistance and /* for magnetic permeability, we can say that 
 
 (13) 
 
 Consider a closed magnetic circuit of uniform specific magnetic 
 conductance /* and of length I and section s, and let a uniform 
 magnetic force H act in this circuit, producing a uniform 
 induction B. Then by definition the magnetic resistance 
 R of the circuit is given by 
 
 T} _ H I _ Magnetomotive force 
 B s Total induction 
 
 where H / is the magnetomotive force, and B s the total 
 induction ; but 
 
 therefore we have B = /^H ....... (14) 
 
 This equation (14) is not only a fundamental equation in 
 magnetism, just as Ohm's law is in electro-kinetics, but it 
 furnishes a definition of the method of measuring magnetic 
 permeability,
 
 40 ELECTRO-MAGNETIC INDUCTION. 
 
 Experiment shows, as stated, that the ratio B to H, 
 
 p 
 
 expressed by the equation - = p., is not of a determinate 
 H 
 
 character, and that the value of /*, so far from being constant, 
 is dependent on the whole previous magnetic history of the 
 iron, on the value of B, and on the nature of the magnetic 
 changes the iron is undergoing, viz., whether H is increasing 
 or diminishing. In fact, p for iron varies from a value not 
 far from unity for strongly magnetised iron up to a value of 
 2,000 or 2,500 at a maximum for closed magnetic circuits of 
 best soft iron, and down to a value of about 100 for very 
 feeble magnetising forces. 
 
 In a cycle of magnetic operations, during which a bar of 
 iron is exposed to increasing magnetising forces and then 
 afterwards to decreasing ones, beginning and ending with the 
 same force, the value of B is always greater on the descend- 
 ing than on the ascending course, and hence the value of p. 
 which is given by the ratio of B to H is also different. This 
 phenomenon, which is exemplified familiarly by the retention 
 of magnetism in a bar after withdrawal of the magnetising 
 force, has been named by Prof. Ewing hysteresis (from va-repfw, 
 to lag behind). 
 
 The magnetic permeability above defined is a quantity 
 which is in magnetism the analogue of specific inductive 
 capacity in electrostatics, and of the conducting power of a 
 body for heat. It was spoken of by Faraday as the conduct- 
 ing power of a magnetic medium for lines of force (" Exp. Re- 
 searches," Ser. XXVI., 2,797 and 2,846). More recently 
 the reciprocal of \t> has been called the magnetic reluctance. 
 The term magnetic resistance has, however, become so sanc- 
 tified by use that we shall continue to employ it in spite of 
 certain objections which have been urged against it. 
 
 The magnetic resistance of a circuit, composed partly of 
 iron and partly of air, is greater in proportion as the length of 
 the air path is increased in proportion to that of the iron. 
 This fact is strikingly shown in experiments on the magnetic 
 induction produced in closed rings and short bars of soft iron. 
 Thus from some curves given by Prof. Ewing we find that in 
 a certain soft iron ring a magnetising force of 7 C.G.S. units 
 produced an induction of 11,000 C.G.S. units; whereas, in
 
 ELECTRO-MAGNETIC INDUCTION. 41 
 
 the case of an iron rod, of which the length was 50 times the 
 diameter, the same force produced an induction of only 4,000 
 units. In the first case the circuit was a complete iron circuit, 
 and the resistance small. In the second case the magnetic 
 circuit was partly iron and partly air, and therefore the total 
 magnetic resistance was much greater. 
 
 The equation B = ^ H is the expression of the fundamental 
 relation between magnetic force and magnetic induction. 
 For all ordinary non-magnetic materials the value of p and 
 also of its reciprocal //, is taken as unity. Hence, in these 
 cases the magnetic force and magnetic induction have the 
 same value and are in the same direction. In the case of 
 such diamagnetic bodies as bismuth and phosphorus, /* is 
 very slightly less than unity and p very slightly greater, but 
 so little as to be unappreciable except to refined experiments. 
 In speaking of air circuits, or non-magnetic circuits generally, 
 we can speak of lines of force or lines of induction indifferently, 
 as the force and induction have everywhere the same numerical 
 value and same direction, but in dealing with magnetic 
 circuits of permeability greater than unity, the magnetic 
 force and magnetic induction must be carefully distinguished, 
 because they have not the same magnitude and may not have 
 the same direction. In crystalline magnetic bodies the induc- 
 tion might have a very different direction and value to the 
 force, and as wo have seen in the case of soft iron, the induc- 
 tion may be, numerically, two or three thousand times greater 
 than the magnetic force. 
 
 It should be borne in mind, therefore, that in the air-space 
 -outside a magnet or a mass of iron under induction the mag- 
 netic force and magnetic induction have the same direction 
 .and same numerical value, but inside a magnet or mass of 
 iron under induction they must be distinguished. The mag- 
 netic force outside the magnet may be called the magnetic 
 induction through the air, and generally in the non-magnetic 
 material surrounding the magnet the magnetic force and 
 magnetic induction are, for the purposes of measurement, 
 one and the same. In the interior of a mass of iron under 
 induction in a magnetic field, the magnetic force at each 
 point is one compounded of that due to the external or 
 original field and that due to the induced polarity acquired,
 
 42 ELECTRO-MAGNETIC INDUCTION. 
 
 and which acts to produce an opposing magnetic force. Hence 
 the effect of the induced poles on any element in the interior 
 of the iron is to tend to demagnetise it when the external 
 magnetising force is withdrawn. In the inside of a straight 
 uniformly magnetised bar the magnetic force due to the 
 influence of the poles themselves is from the end which points 
 to the north to the end which points to the south, both within 
 the magnet and in the space outside. The magnetic induction, 
 on the other hand, is from the north pole to the south pole 
 outside the magnet, and from the south pole to the north pole 
 inside the magnet. A line of induction followed round, moving 
 always in the positive direction, is found to be a closed loop or 
 endless line. 
 
 It is a fact of fundamental importance that a thin disc of 
 iron or steel, magnetised so that at all points the magnetic 
 axis of each small element of it is in a direction normal 
 to its surface, produces a magnetic field identical Avith 
 that produced by a wire conveying an electric current 
 coinciding in form with the edge of the disc. In short, the 
 magnetic field of a closed circuit conveying a current is 
 identical with that of a magnetic shell filling up the aperture 
 of the circuit. The magnetic field due to electric currents 
 circulating in conductors is, however, of such a nature that 
 each line of induction embraces or surrounds the axis of the 
 circuit once or more. The magnetic field due to permanent 
 or electromagnets is of such a nature that each line of 
 induction passes through the magnet, giving rise to magnetic 
 polarity at the places where it enters and leaves the iron or 
 steel. Every line of induction either surrounds an electrie 
 current or passes through magnetised iron. 
 
 The intensity of magnetisation of any element of a magnet or 
 mass of iron under induction is a term requiring definition. 
 
 The couple required to hold a very small magnet when 
 placed with its axis of magnetisation perpendicularly across 
 the lines of force of a uniform magnetic field in air of unit 
 strength is a numerical measure of the moment of the magnet. 
 The moment of a magnet or of any element of a magnet 
 may be considered numerically to be made up of two factors 
 one its volume, and the other its intensity of magnetisa- 
 tion, or, simply, its magnetisation ; and hence, for a uniformly
 
 ELECTRO-MAGNETIC INDUCTION. 43 
 
 magnetised small linear needle, we may define the intensity of 
 its magnetisation by saying that it is the magnetic moment 
 of unit volume of it. Intensity of magnetisation is, like 
 force and induction, a vector quantity.* In the case of a very 
 long thin wire of soft iron placed along the lines of force of a 
 uniform field the three quantities the magnetic force, the 
 magnetic induction, and the magnetisation are all in align- 
 ment at any point. 
 
 It is important that the full meaning of the phrase " mag- 
 netic force at any point " should clearly be grasped. If there 
 be any uniform magnetic field of strength H , and in it is 
 placed a mass of iron in the shape of an elongated bar, the 
 configuration of this uniform field is disturbed and magnetic 
 polarity is developed in the iron. At any point in the interior 
 of the iron there is a magnetising force, henceforth denoted by 
 the letter H, which is due partly to the original magnetic field 
 H and partly to the induced poles which create a force 
 opposing H . This resultant magnetic force is spoken of as 
 the magnetising force in the iron, and it is the resultant of the 
 external magnetic forces and the internal magnetic forces due 
 to the polarity. If the form of this bar is so chosen that there 
 are no magnetic poles, as in the case of a ring lapped over 
 with an endless solenoid, then the magnetic force in the iron 
 is easily calculable, and it is that due to the external magnetic 
 force alone. If the bar is straight and very long the induced 
 magnetic poles may exert so little effect at the centre of the 
 bar that the induction there and the magnetic force also is 
 that due to the external field alone. 
 
 At each point in the iron the magnetic force H must be 
 thought of as producing magnetisation or magnetic displace- 
 ment, just as in electrostatic phenomena electric force pro- 
 duces in a dielectric electric displacement or electric strain, or 
 just as mechanical stress produces in an elastic body ordinary 
 strain or displacement at every point. This magnetisation is 
 not necessarily in the same direction as the force. 
 
 4. Lines and Tubes of Magnetic Induction.Faraday 
 and Maxwell have raised the conception of a line of magnetic 
 
 * A vector quantity is one which is only precisely fixed when we know its 
 direction as well as magnitude.
 
 44 ELECTRO-MAGNETIC INDUCTION. 
 
 induction from a simply directive notion to one which enables 
 it to be used to convey a quantitative knowledge of the mag- 
 netic field in other words, have enabled lines of induction to 
 be used not only to show the direction of the induction, but 
 also its magnitude in certain units. By this means the mag- 
 netic field can be mapped out into areas and volumes which 
 have a definite dynamical signification. 
 
 In the twenty- eighth series of his " Kesearches on Elec- 
 tricity," Faraday lays stress on the fact stated above that 
 every line of force (induction) is an endless loop ( 3,117, 
 "Exp. Ees.") : "Every line of force must therefore be con- 
 sidered as a closed circuit passing in some part of its course 
 through a magnet, and having an equal amount of force in 
 every part of its course. There exist lines of force within the 
 magnet of the same nature as those without. What is more, 
 they are exactly equal in amount to those without. They 
 have a relation in direction to those without, and are, in fact, 
 continuations of them." 
 
 Let a magnetic field have drawn in it a number of closely 
 contiguous lines of induction. None of these lines can cut 
 each other, because the resultant magnetic induction at any 
 point can have only one definite direction. 
 
 In any region it is possible to describe a surface perpen- 
 dicular to all the lines of induction. Such a surface is called 
 a level surface. 
 
 In the case of a straight infinite current, these level surfaces 
 will be planes radiating out from the axis of the wire, and 
 their traces on a plane perpendicular to the axis of the wire 
 will be a series of radial lines cutting all the circular loops of 
 induction normally. Let A (Fig. 9) represent such a level 
 surface, and let B be another, both cutting the same sheaf of 
 magnetic lines of induction. 
 
 On the level surface A let any unit of area a be taken, and 
 let this area a be projected on to the adjacent level surface B 
 by lines of induction drawn through its boundary. We have 
 then a tubular surface, the ends of which are formed of 
 portions of level surfaces, and the rest of the tubular surface 
 may be conceived to be formed of lines of magnetic induction, 
 supposed to be very closely drawn through the bounding line. 
 Such a geometrical conception is called a tube of induction.
 
 ELECTRO-MAGNETIC INDUCTION. 45 
 
 The characteristic quality of a tube of induction is as follows. 
 If the areas of the sections made by the two level surfaces A 
 and B be called s and s', and if B be the mean magnetic 
 induction over s, and B' that over s', then B s = B' s', or the 
 product of a normal cross -section of tube and mean magnetic 
 induction over that section is constant for all sections of the 
 tube. 
 
 If any level surface be drawn, and on this surface be 
 marked off contiguous small areas such that the magnitude of 
 the area is inversely as the mean value of the magnetic 
 induction over that little area, and if s and B are, as before, 
 the numerical values of any small area and the mean induc- 
 tion over it, then the product B s may be made equal to unity 
 for each of these portions of that level surface. From these 
 
 FJG. 9. 
 
 small areas let tubes of induction be supposed to take their 
 rise, the whole field will be cut up into contiguous tubes of 
 induction. Each of these tubes is called a unit tube of induc- 
 tion. By their mode of description these tubes will have 
 small cross-section at pi aces where the field is strong and 
 widen out in section at places where it is weak, and by the 
 fundamental property of the tubes the value of the mean 
 magnetic induction at any place is inversely as the cross- 
 section of the tubes of induction force at that place. 
 
 From Faraday's point of view, a magnet of any form must 
 be mentally pictured as surrounded with and as having its 
 whole field filled up by a closely packed arrangement of such 
 unit tubes of induction, the tubes being intersected at right 
 angles by the equipotential or level surfaces, and each having
 
 46 ELKi'TRo-MAGNETIV INDUCTION. 
 
 at any point a normal cross-section which is inversely as the 
 magnetic induction at that point. This system of tubes must 
 be supposed to be rigidly attached to the magnet, and to move 
 with it wherever it goes. Furthermore, in accordance with 
 Faraday's conception, each tube is an endless tube, or, as it 
 were, a pipe returning into itself and passing in some part of 
 its course through a magnet or round an electric current. In 
 constructing what may seem to the student to be a highly 
 artificial conception, we are not postulating necessarily any 
 physical existence for these tubes. They should be regarded 
 simply as a device for plotting out the space round a magnet 
 according to a definite rule, and may, in the first place, be 
 regarded as no more than analogous to such subdivisions of 
 .the earth's surface as we make by lines of latitude and longi- 
 
 FIG. 10. 
 
 tude. Having thus divided up a magnetic field into unit 
 tubes of induction, it is simpler in thought to suppose a 
 single line of induction to run down the axis of each tube, and 
 then to mentally disregard the tubular system, and, instead 
 of speaking of a unit tube, to speak of each as a single line 
 of induction. If we imagine a system of induction tubes 
 starting from an equipotential surface and draw any irregular 
 curve on this surface, we shall find that this curve encloses 
 a certain number of tubes or lines of induction (Fig. 10). 
 Bearing in mind that the cross-section s of the tube where it 
 sprouts out from the equipotential surface is inversely as the 
 magnetic induction B at the centre of this cross-section, it is 
 at once evident that the greater the average induction over 
 the area defined so much the more numerous will be the
 
 ELECTRO-MAGNETIC INDUCTION 47 
 
 number of tubes or lines of induction which pass through it. 
 If the cross-section s of each tube should happen to be equal, 
 and there be n tubes passing altogether through an area equal 
 to S, bounded by the black line, then by the very definition of 
 a unit tube 
 
 Ba-1, 
 
 or )i B s = n ; 
 
 but s = S, 
 
 hence B S = n ; 
 
 and the number of tubes passing through any area S on such 
 an equipotential surface in a uniform field is numerically equal 
 to the product of the whole area and of the induction at any 
 point on that area. 
 
 FIG. 11. 
 
 The characteristic quality of a tube of induction is that the 
 flux of induction along it is constant throughout its length. 
 That is to say, the product B s = induction x cross -section is 
 constant, and, since what is true of one tube is true of all, we 
 may say that in a space wholly made up of tubes of magnetic 
 induction the total magnetic induction or flux of induction is 
 the same across all sections of this mass of tubes. 
 
 The lines of induction of a permanent steel magnet are to 
 be thought of as closed loops which pass in their course partly 
 through the steel and partly through the air. The lines of 
 induction of a circuit conveying an electric current are closed
 
 48 ELECTRO-MAGNETIC INDUCTION. 
 
 loops entirely surrounding the axis of the wire. If this circuit 
 is a straight wire, with the return wire at a very great 
 distance, the lines of induction are concentric circles described 
 on a plane perpendicular to the axis of the wire, and having 
 their centre in that axis. 
 
 If a circuit is formed by coiling up into a circular coil a 
 length of insulated wire, the coil having n turns, then to a first 
 degree of approximation we may say that each line of induction 
 forms a closed curve embracing the circuit n times. Thus, if 
 the wire forms a coil of one turn (Fig. 11) each line of induc- 
 tion (represented by the dotted line) is a closed loop em- 
 bracing or linked with the circuit once. If we take a circuit 
 
 of two turns (Fig. 12), then nearly all loops of induction 
 belonging to one single turn embrace not only that turn but 
 the adjacent turn, and if the circuit could be supposed to be 
 opened out straight (Fig. 13), without destroying the loop of 
 induction, it would be found to be twisted twice round that 
 circuit. By similar reasoning, if a loop of induction embraces 
 or is linked with n turns of a conducting circuit, it is in fact 
 the same as linking each loop of induction n times with the 
 single circuit. 
 
 Let a conducting circuit have the form of a helix (Fig. 14), 
 then the lines of induction are closed loops, which embrace
 
 ELECTROMAGNETIC INDUCTION. 49 
 
 some or all the turns of the spiral, and, if the helix have n 
 turns, then each loop of induction, according to its length and 
 
 position, in reality embraces that circuit 1, 2, 3 or n times. 
 
 If there be two circuits, in one or both of which currents are 
 flowing, then each circuit is surrounded by lines or loops of 
 induction, and of those belonging to one circuit some or all are 
 
 FIG. 13. 
 
 linked in as well with the other circuit, so that a certain 
 number of all the loops of induction are common to the two 
 circuits, and are called the loops or lines of mutual induction. 
 It is exceedingly convenient to think of these lines or tubes 
 of induction as linked with various circuits. A line of induc- 
 tion is always linked with an electric circuit or else passes 
 
 through a magnet in some part of its course. If any closed 
 conducting circuit is placed in a field of magnetic induction, it 
 may be thought of as linked with a certain number of unit 
 lines or tubes of induction. If the circuit is moved or the 
 field is changed, the number of linkayes is altered or may be 
 altered. If at any instant N unit tubes or lines of induction 
 are linked with any circuit, and at a very short interval of time
 
 50 
 
 ELECTRO-MA G NET 1C IND UCTION. 
 
 afterwards, say after an interval t, the number of linkages is 
 
 N - N' 
 N', then N - N' is the change in the linkage, and - is the 
 
 rate of change of linkage, and this, by Faraday's law, is the 
 numerical value of the electromotive force set up in the circuit. 
 If during any period of time a circuit is exposed to magnetic 
 induction the rate of change of which varies, then from 
 instant to instant the impressed inductive electromotive force 
 varies and may be represented graphically as follows : Let the 
 straight line X (Fig. 15) be an axis on which lengths are 
 marked off to represent intervals of time, and let ordinates 
 perpendicular to it represent the instantaneous value of the 
 flux of induction through, or lines of induction included by, 
 
 M 
 
 FIG. 15. 
 
 a circuit ; then, if the variation of induction is continuous, it 
 may be represented by a curve drawn between these axes. 
 This curve may be called a curve of induction. If at any 
 point P a tangent P T is drawn to this curve, the trigono- 
 metrical tangent of the angle P T M represents the rate of 
 variation of P M with respect to M, and if P M represents at 
 any instant the induction N, then the slope of the tangent at 
 
 P represents ( , or the rate of change of N with respect to 
 time. 
 
 In the practical application of the above rule it must be 
 borne in mind that if N, or the number of lines or linkages, is 
 measured in units based on the centimetre, gramme, and second 
 system, then the electromotive force is given in the same units.
 
 ELECTRO MAGNETIC INDUCTION. 51 
 
 Since one volt is 10 s C.G.S. electromagnetic units, to get the 
 electromotive force in volts we must divide the time rate of 
 change of induction by 10 8 . Thus, if the change of induction 
 be such that N C.G.S. lines are removed uniformly from the 
 circuit per minute, the electromotive force in volts set up will be 
 
 5. Curves of Magnetisation. One of the most important 
 practical problems in magnetism is to discover the manner 
 in which the induction B varies with the magnetising force 
 H in different cases. Since this variation is often very 
 complicated, it is best represented by a curve of which the 
 ordinates are taken as proportional to the induction and 
 the absciss as proportional to the magnetising force. Such 
 curves are called curves of magnetisation. We shall consider 
 a few special cases and describe the mode of practically 
 determining these curves. The principal instance is the 
 curve of magnetisation of a closed iron ring. Let an iron 
 ring be made of circular cross-section radius a and be circular 
 in form, the radius of the mean circular central axis being 
 E. Let a be very small compared with E. Let such a 
 ring be wound over uniformly with N turns of insulated 
 wire, and let a current of strength A amperes be sent 
 through this wire. The magnetic force creates an induc- 
 tion in the iron, and the lines of induction are circles lying 
 wholly in the iron. Inside the circular solenoid the mag- 
 netic force is nearly the same in value everywhere, and is 
 
 equal to n A, where n is the number of turns of the wire 
 
 N 
 per unit length of the solenoid; or n = - . Hence we 
 
 can calculate the magnetic force acting everywhere on the 
 iron. Let the ring be also wound over with a second 
 insulated coil of wire of N' turns, and let this secondary 
 coil be connected to a ballistic galvanometer or a galvano- 
 meter suitable for measuring quantity of electricity. ' If the 
 current in the primary coil is altered in strength or reversed, 
 the magnetic force undergoes a change, and the induction 
 changes also. Since the lines of induction all pass through, 
 or are linked with, the turns of the secondary coil, this
 
 52 ELECTRO-MAGNETIC INDUCTION. 
 
 change in induction will, by Faraday's law, create a flow 
 of electricity in the secondary circuit. If the current in 
 the primary coil is suddenly changed from one value A amperes 
 to another value A' amperes, a certain quantity of electricity 
 will flow through the secondary coil and galvanometer, and it 
 will cause the galvanometer needle to deflect through an angle 
 6. If k is the galvanometer constant (see Appendix, Note C), 
 then the whole quantity of electricity which flows through 
 
 Q / \\ 
 
 the galvanometer is equal to Q, where Q = A; sin- 
 
 A being the logarithmic decrement of the galvanometer. 
 
 If R' is the total resistance of the secondary coil on the 
 ring, together with that of galvanometer coils and connec- 
 tions, then, as above shown, the total change in the induction 
 through the secondary circuit is equal numerically to R' Q. 
 If B is the mean induction density in the interior of the 
 circular solenoid, S the mean cross-sectional area of the 
 primary coil, and N' the number of turns on the secondary 
 coil, then it is clear that B S N' represents the total induction 
 passing through the secondary circuit, or the number of lines 
 of induction which are linked with the galvanometer circuit. 
 
 Hence if the induction is suppressed by stopping the current 
 the total change in induction is B S N', and this is equal to 
 R' Q. If the current, instead of being simply stopped, is 
 reversed, then the total change in induction is 2 B S N'. 
 Therefore we have 
 
 B S N' = R' Q for stoppage of current ; 
 or 2 B S N' = R' Q for reversal of current. 
 
 By our fundamental equation 
 
 B=//H. 
 
 In the above case the average magnetic force H is equal to 
 _ , in which formula N is the number of primary turns, 
 
 A the primary current in amperes, and I the mean perimeter 
 of the primary solenoid, which last is equal to 2ir R, where R 
 is the mean radius of the primary solenoid. Hence we find 
 that
 
 ELECTRO-MAGNETIC INDUCTION. 53 
 
 according as the primary current is stopped or reversed. 
 Taking the latter as the usual case, we find finally that 
 
 Hence p. is determined in terms of nine quantities, all of 
 which can be easily and exactly measured. If all quantities 
 are measured in centimetres and seconds, then it must be 
 particularly noted that the galvanometer constant k is the 
 number by which the corrected sine of half the angle of throw 
 has to be multiplied in order to obtain the quantity of 
 electricity producing that throw, estimated in absolute C.G.S. 
 electromagnetic measure, which has flowed through the gal- 
 vanometer. Generally speaking, it is most convenient to 
 find k by determining the throw produced by a discharge 
 through the galvanometer of a certain number of microcou- 
 lombs, obtained by charging a condenser of known capacity in 
 microfarads with a certain number of volts. Then it must be 
 remembered that a microcoulomb is 10~ 7 of an electromag- 
 netic unit of quantity in C.G.S. measure. 
 
 By the help of the above formula we can deduce the values 
 of n for the iron ring corresponding to certain values of B or 
 H, and tabulate them. For instance, taking a perfectly new 
 iron ring, we can apply gradually increasing values of H, 
 increasing by small steps, and obtain the corresponding 
 values of B, and then draw curves representing the relation 
 of B and H and of B and /*. Such curves have been given 
 by many observers : Kowland, Hopkinson, Ewing, Shelford 
 Bidwell, and others. For the sake of showing what are the 
 sort of values of B and p. corresponding to certain values of H, 
 a table is given on the next page o'. results of an experiment 
 by Shelford Bidwell on a soft iron ring, and in Figs. 16 and 
 17, page 55, are given diagrams showing the forms of the 
 curves of induction and permeability for such a ring.
 
 5-1 
 
 ELECTRO-MAGNETIC INDUCTION. 
 
 Values of the Permeability and Induction corresponding to 
 various Magnetising Forces for Closed Iron Magnetic Circuits. 
 
 H. 
 
 Magnetising Force. 
 
 B. 
 
 Induction. 
 
 / 
 Permeability. 
 
 0-2 
 
 80 
 
 400 
 
 0-6 
 
 300 
 
 600 
 
 1-0 
 
 1,400 
 
 1,400 
 
 2-0 
 
 4,800 
 
 2,400 
 
 3-9 
 
 7,390 
 
 1,899 
 
 57 
 
 9,240 
 
 1,621 
 
 10-3 
 
 11,550 
 
 1,121 
 
 17-7 
 
 13,630 
 
 770 
 
 22-2 
 
 14,450 
 
 651 
 
 30-2 
 
 15,100 
 
 500 
 
 40 
 
 15,460 
 
 386 
 
 78 
 
 16,880 
 
 216 
 
 115 
 
 17,330 
 
 151 
 
 145 
 
 17,770 
 
 122 
 
 208 
 
 18,470 
 
 89 
 
 293 
 
 18,820 
 
 64 
 
 362 
 
 19,080 
 
 52-7 
 
 427 
 
 19,330 
 
 45 
 
 465 
 
 19,470 
 
 42 
 
 503 
 
 19,480 
 
 38-7 
 
 585 
 
 19,820 
 
 34 
 
 24,500 (Ewing) 
 
 45,300 
 
 1-9 
 
 It will be noticed that the value of the permeability rises 
 very rapidly to a maximum, but that with increasing magnet- 
 ising force it finally diminishes again, and that in very strong 
 magnetic fields iron becarnes scarcely much more permeable 
 than air. Although there is apparently no limit to the 
 induction or number of lines of force which can be forced 
 through iron, yet it seems as if the excess of lines of force 
 generated in the iron, over and above that which would exist 
 if the iron were not there, is limited. Hence, if we consider 
 the strength of the original field as numerically defined by a 
 certain number, the number expressing the induction when a 
 closed circuit of iron is placed in that field is obtained by 
 using a certain multiplier, which becomes less and less as the 
 magnitude of the field increases. 
 
 On looking at these permeability curves it will be seen that 
 the permeability rises to a maximum for a certain value of the
 
 ELECTRO-MAGNETIC INDUCTION. 
 
 induction and then falls again. It has been shown by Lord 
 Rayleigh that for very small magnetic forces soft iron has an 
 
 12,000 
 10,000 
 
 .8,000 
 
 c 
 o 
 
 ^6,000 
 1 
 
 ~4,000 
 
 2,000 
 
 246 
 Magnetising Force. 
 
 FIG. 16. 
 
 Curve of Magnetisation for rising and falling magnetisation in a 
 Soft Iron Ring. 
 
 initial permeability of about 100. Its maximum permeability 
 is about 2,500, and for very great inductions it falls again to 
 something not much greater than unity. Hence the specific 
 
 ^^ 
 
 - ~" 
 
 
 /" 
 
 
 
 
 y 
 
 
 
 B 
 
 
 / 
 
 
 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 ^ 
 
 / 
 
 H 
 
 
 
 3,000 
 
 2.00O 
 
 B 
 
 O 5,000 . 10,000 15,000 20,000 
 
 Induction. 
 
 FIG. 17. 
 
 Permeability Curve for Soft Iron Ring (Rowland). 
 
 magnetic resistance curve is the inverse of the above curves, 
 and the specific magnetic resistance has a minimum value 
 which for a soft iron ring appears to correspond to an indue-
 
 56 ELECTRO-MAGNETIC INDUCTION. 
 
 tion density of about 5,000 C.G.S. (lines per square centi- 
 metre). At less or greater induction densities the specific 
 magnetic resistance is increased. 
 
 Experiments to determine n can only be performed properly 
 on very long bars or rings of iron placed in uniform predeter- 
 mined magnetic fields by lapping over the ring with insulated 
 wire or placing the bar in a helix, so that an electric current 
 traversing this wire generates a field having known values at 
 
 300 
 260 
 
 200 
 
 
 / 
 
 ^-^ 
 
 V 
 
 220^ 
 
 c 
 
 
 
 / 
 
 
 -x\ 
 
 
 
 
 
 // 
 
 j 
 
 
 V 
 
 
 
 
 J5 160 
 
 V 
 
 fcioo 
 
 0. 
 60 
 
 i 
 
 
 
 \ 
 
 \' 
 
 15 C 
 
 
 / 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 B 
 
 
 \ 
 
 
 \ 
 
 2,000 4,000 6,000 
 
 Induction. 
 FIG. 18. 
 Permeability Curves for Nickel (Rowland). 
 
 each point in the interior of the coil. Such experiments have 
 been carried out extensively by various experimentalists, and 
 the results embodied in curves called penneability curves.* 
 The form of these permeability curves is considerably affected 
 
 * Accounts of experiments and investigations on the form of the permea- 
 bility and susceptibility curves for iron and other paramagnetic metals 
 will be found in the following memoirs : Weber, Electrodynamische Maas- 
 bestimmutigen, Bd. III., 26 ; Von Quint us Iciliua, Poggendor/'s Annalen, 
 CXXL, 1864 ; Oberbeck, Pogg. Ann., CXXXV., 1868 ; Riecke, Pogg. Ann., 
 CXLL, 1870 ; Stoletow, Pogg. Ann., Ergbd. V., 1870 ; Rowland, Phil. 
 Mag., Ser. IV., 1873, p. 536, 1874, p. 254 ; Bouty, Comptes Rendus, 1875 ; 
 Froinme, Pogg. Ann., Ergbd. VII., 1875 ; Warburg, Wiedemann's Annalen, 
 XIII., 1881 ; Bidwell, Proc. Roy. Soc. Lond., No. 245, 1886 ; Bidwell, Proc. 
 Roy. Soc. Lond., VoL XLIIL, 1888; Bidwell, Proc. Roy. Soc. Lond., 
 No. 242, 1886 ; Ewing, Trans. Roy. Soc. Lond., Part. II., 1885 ; Hopkinson, 
 Trans. Roy. Soc. Lond., Part II., 1885.
 
 ELECTliO-MAONETIC INDUCTION. 57 
 
 by temperature, and for each magnetic metal there appears 
 to be a temperature at and beyond which it is not much 
 more permeable than air. The permeability of nickel and cobalt 
 varies very much with temperature. In Figs. 17 and 18 are 
 shown the permeability curves for iron and for nickel for two 
 very different temperatures. At about 750C. iron, and at 
 about 400C. nickel, possess a permeability not much 
 greater than air.* In cobalt, permeability appears to be 
 increased up to about 150C. and then diminished. 
 
 In Fig. 19, page 58, is shown the form of the magnetisation 
 curve of a long soft iron rod. It will be seen that the curve 
 is divided roughly into three parts : first, a part in which 
 magnetic induction increases slowly with magnetising force ; 
 second, a part in which it increases very rapidly ; and third, a 
 part when it increases somewhat more slowly again. Generally 
 speaking, it is difficult to force up the induction in a soft iron 
 ring or long rod to a greater value than about 18,000 or 
 20,000 C.G.S. units, but there is no actual physical limit to 
 the amount of induction to be created in iron ; the sole limit 
 seems to be the difficulty of obtaining sufficient magnetising 
 force. If instead of operating on the best soft iron we had 
 selected a hard iron or a steel, it would have been found that 
 the magnetisation curves would not be quite of the same form, 
 but that for a given magnetising force there would be less 
 induction. As soon as the induction reaches the point where 
 the magnetisation curve becomes approximately flat the iron 
 is said to be magnetically saturated. 
 
 If instead of operating on an endless iron circuit or ring we 
 select an iron rod, say 200 diameters long, and wind it over 
 with a magnetising coil, we can, by means of the ballistic 
 galvanometer, determine for it in the above way a magnetisa- 
 tion curve. For if we surround the centre of the rod with a 
 secondary coil, and connect this to the galvanometer, we can, 
 by making changes in the strength of the primary magnetising 
 current, alter the induction through the secondary circuit, and 
 so obtain throws of the galvanometer indicating certain 
 quantities of electricity discharged through it. If the primary 
 helix is very long compared with its diameter we can estimate 
 
 * See Proc. Physical Hoc , London, Vol. IX., p. 187 : Mr. Tomlinson on 
 fi The Temperature at whicli Xiukel begins to Lose its Magnetic Qualities."
 
 58 
 
 ELECTRO-MAGNETIC IND UCTION. 
 
 the strength of its interior field, or the magnetising force, as 
 everywhere approximately equal to 47T/10 times the ampere- 
 turns per unit of length of the helix. Hence we know the 
 
 16,000 
 
 14,000 
 
 I2j000 
 
 10,000 
 
 .0 8,000 
 
 I 6,000 
 
 4,000 
 2,000 
 
 / 
 
 H 
 
 O 2 4 6 8 10 12 14 It 
 
 Magnetising Force of Solenoid in C G S. Units, 
 
 FIG. 19. 
 
 Curve of Magnetisation for rising and falling magnetism in an Iron Rod. 
 Length = 200 diameters (Ewing).* 
 
 * This curve is taken from Prof. J. A. E wing's Paper, " Experimental 
 Researches in Magnetism," Trans. Royal Soc., Part II., 1885, p. 535. 
 The treatise by Prof. Ewing on " Magnetic Induction in Iron and Other 
 Metals," in " The Electrician Series " of Standard Books, will furnish the 
 student with the most complete account of modern magnetic research, 
 and hence it has not been considered necessary to amplify very much this 
 section of the present work. 
 
 See also Dr. J. Hopkinson, " Magnetisation of Iron," Trans. Royal Soc., 
 Part II., 1885, p. 455. 
 
 A very complete summary of recent research in magnetism is to be found 
 in Prof. Chrystal's article, " Magnetism," in the Encyclopaedia Britannica, 
 Ninth Edition. 
 
 See also '' On the Lifting Power of Electro-Magnets and Magnetisation 
 of Iron," Shelford Bidwell, Proc. Royal Soc , June 10, 1886. 
 
 Other references to valuable Papers on the magnetisation of iron are 
 
 Lord Rayleigh, " On the Energy of Magnetised Iron," Phil. May., 
 August, 1886, 'p. 175. 
 
 Lord Rayleigh, " On the Behaviour of Iron and Steel under the Opera- 
 tion of Feeble Magnetic Forces." Phil. Mag., March, 1887, p. 225. 
 
 Ewing and Low, " On the Magnetisation of Iron in Strong Fields '' 
 Proc. Royal Soc., March 24, 1887, Vol. XLIF., p. 200.
 
 ELECTRO-MAGNETIC INDUCTION. 59 
 
 force, and the induction is ascertained when we know the 
 quantity of electricity discharged through the galvanometer 
 on a reversal of the primary current. If the secondary coil 
 makes N' turns round the iron, and if the whole resistance of 
 the secondary circuit, including galvanometer, is equal to R', 
 and if S is the cross-section of the rod, then 2 B S N' is the 
 total change in induction through the secondary circuit 
 obtained on reversing the primary current, and this is equal 
 to R,' Q, where Q is the quantity of electricity discharged 
 through the galvanometer. After finding in this way a series 
 of values of B and the corresponding values of H, we have the 
 means of determining the values of p. for varying values of 
 B. If, as in the above case, the iron rod is not endless, the 
 values of p, so determined will be smaller than the correspond- 
 ing values of p. for an iron ring of the same iron, and will be 
 less as the iron rod is made shorter, because a larger propor- 
 tion of the magnetic circuit is then formed of air and a less 
 portion of iron. 
 
 jected to a cycle of magnetising force in which the force 
 beginning at zero rises up graduaUy to a maximum in one 
 direction, and is then reversed and made a maximum in the 
 other direction, and finally reduced again to zero, we find that 
 the following phenomena exhibit themselves. The induction 
 in the iron and, therefore, its magnetisation has a higher 
 value at all points during the descent of the force than during 
 its ascent. Hence, if a curve is plotted in which horizontal 
 abscissae represent magnetic force and vertical ordinates induc- 
 tion or magnetisation, we obtain a curve of the kind shown in 
 Fig. 20, which is a loop or encloses an area. If at any point 
 in the cycle we stop and reverse the magnetism a small or 
 subsidiary loop (see Fig. 21) is formed on the principal curve. 
 This phenomena is called magnetic hysteresis, because the mag- 
 netism or induction "lags behind" the magnetic force. If 
 the induction B and the magnetic force H are the variables 
 in terms of which the curve is plotted, the curve is called a 
 B H curve of hysteresis. We may next consider the physical 
 meaning of this curve. Consider as before a ring of iron of 
 cross-section S and mean perimeter I wound over with N
 
 60 ELECTRO-MAGNETIC INDUCTION. 
 
 turns of a magnetising coil. Let a current of A amperes be 
 sent through this coil, there are then N A ampere-turns acting 
 
 on it, and the magnetic force operating on the iron is _ . 
 
 At any instant let the difference of potential at the ends of the 
 magnetising coil be V volts, and let the coil have a resistance 
 of K ohms. If the induction in the iron has a value B, the 
 total number of lines of induction linked with the coil is 
 BSN. 
 
 C M><jn<lilinq Fore. H 
 
 Complete Magnetisation Curve for Soft Iron Ring carried from strong 
 positive to strong negative magnetisation. The arrows show the direction 
 of the magnetising operation, and the shaded area the work done due to 
 
 hysteresis. 
 
 If we make a change in the potential and increase the volts 
 to V + 8V, and at the same time increase the current from 
 A to A + 8 A, and the induction from B to B + SB, in the 
 small time 8 1 the following relations between these increments 
 will exist : The time rate of change of the induction is in the
 
 ELECTRO-MAGNETIC INDUCTION. 61 
 
 limit equal to f _, and the time rate of change of the whole 
 number of lines of induction linked with the circuit is 
 
 7 R 
 
 S N _-. Hence this last is numerically equal to the induced 
 electromotive force set up in the circuit by this change ; and by 
 considering the direction of this induction it will be seen that 
 this induced electromotive force is opposed in direction to the 
 
 FIG 2'. 
 
 Magnetisation Curve for Very Soft Iron Ring, showing loops due to 
 hysteresis on the descending branch, and enclosed area due to a complete 
 cycle (Evving). 
 
 impressed electromotive force V producing the current. This 
 induced electromotive force reckoned in volts is equal to 
 
 ^ ^--?. Hence the current A in amperes must be equal at 
 
 any instant to the resultant electromotive force divided by 
 the resistance of the circuit ; or 
 
 V- SN (/B 
 
 10 s dt
 
 62 ELECTRO-MAGNETIC INDUCTION. 
 
 Therefore, multiplying all through by A and d t, we have 
 
 V Kdt = EA?dt + ^ A<7B. 
 But the magnetising force H in the iron at that instant is 
 
 equal to - A. Hence 
 
 . . (18) 
 
 The first term of this equation represents the whole energy 
 in joules given to the ring coil and core in the small time dt ; 
 the second term represents the energy wasted in that time in 
 heating the coppeT magnetising coil ; and the third term must 
 therefore represent the whole energy, measured in joules, 
 absorbed or wasted in the iron core in that same element of 
 time. Accordingly it is easily seen that the integral 
 
 -L f H d B, 
 
 4;r J 
 
 (19) 
 
 taken between any limits, must represent the whole energy 
 measured in ergs, dissipated by a unit of volume (viz., one 
 cubic centimetre) of the iron core in the time limits of the 
 integral. If the time limit is the interval of time occupied in 
 making one complete magnetic cycle, then the above integral 
 will represent the energy dissipated per unit of volume of the 
 iron in this cycle estimated in ergs. But if the diagram of 
 the magnetic cycle is drawn in terms of B and H, the integral 
 
 f H d B taken over the whole limits of the cycle is the value 
 
 of the area enclosed by the induction curve. Finally, there- 
 fore, we reach this rule. If a mass of iron is taken once 
 round a magnetic cycle, and an H B diagram is drawn, 
 showing the relation of induction to magnetising force during 
 the cycle, one centimetre or one unit of length along the hori- 
 zontal being taken as equal to one C.G.S. unit of magnetic 
 force, and one centimetre or unit of length along the vertical 
 being taken for one C.G.S. unit (one line) of induction ; 
 then l/4:r of the area of this closed loop in square units is 
 equal to the energy wasted in such single magnetic cycle 
 measured in ergs. The physical meaning of these loops or 
 enclosed areas in magnetisation curves of complete magnetic
 
 ELECTROMAGNETIC INDUCTION. 63 
 
 cycles was first pointed out by Prof. E. Warburg,* but also 
 independently by Ewing. In the first place, we may remark 
 that however slowly the magnetic cycle may be performed, 
 this waste of energy always takes place, and it is therefore 
 seen to be dependent essentially on the reversal or change of 
 magnetism and not upon the production of local electric 
 currents in the iron. This energy waste is called the hysteresis 
 loss in the iron, and it cannot be got rid of by any amount of 
 lamination or division of the iron. Careful measurements 
 have shown what is the value of this hysteresis loss in iron of 
 various kinds. 
 
 In a Paper entitled " Researches in Magnetism " (Phil. 
 Trans., Part II., 1885), Prof. Ewing has given the values of 
 the energy dissipated in ergs per cubic centimetre, experi- 
 mentally determined for complete magnetic cycles performed 
 on various samples of iron, as follows : 
 
 Energy dissipated in ergs per 
 
 Sample of iron operated cubic centimetre during a coin- 
 
 upon. plete cycle of doubly-reversed 
 
 strong magnetisation. 
 
 Very soft annealed iron 9,300 ergs. 
 
 Less soft annealed iron 16,300 ,, 
 
 Hard drawn steel wire 60,000 ,, 
 
 Annealed steel wire 70,500 ,, 
 
 Same steel, glass hard 7G,000 
 
 Pianoforte steel wire, normal temper. . 116,000 ,, 
 
 Same, annealed 94,000 ,, 
 
 Same, glass hard 117,000 ,, 
 
 If we make one hundred complete cycles of magnetisation 
 per second, the power absorbed per cubic centimetre of metal 
 
 estimated in watts is as follows : 
 
 Power wasted in watts 
 
 per cubic centimetre for 
 
 Cample. 100 ( oj ) cycles per second. 
 
 Very soft annealed iron '093 
 
 Less soft annealed iron 1 63 
 
 Hard drawn steel wire 6 '00 
 
 Annealed steel wire 7 '05 
 
 Same steel, glass hard 7'GO 
 
 Pianoforte steel wire, normal temper ... 11 '600 
 
 Same, annealed 9 '40 
 
 Same, glass hard 11700 _ 
 
 r VViedemann's Annalen, XIII., p. 141, 188 L.
 
 64 ELECTRO-MAGNETIC INDUCTION. 
 
 From the above we can deduce that, roughly speaking, it 
 requires 18 foot-pounds of energy to make a double-reversal of 
 strong magnetisation in a cubic foot of soft iron. The energy 
 so expended can take no other form than that of heat diffused 
 throughout the mass. 
 
 A similar table of experimental results has been given by 
 Dr. J. Hopkinson (Trans. Roy. Soc., Part II., 1885, p. 463), 
 in which Paper the chemical analysis of the samples operated 
 upon is given. The highest value of specific hysteresial dissi- 
 pation was found for Tungsten steel, oil hardened, in which 
 the value of the energy in ergs per cubic centimetre dissipated 
 in a complete magnetic reversal was 216,864. 
 
 Hysteresis is therefore a quality of iron in virtue of which 
 reversal of magnetisation is accompanied by dissipation of 
 energy. The energy so wasted is, of course, converted into 
 heat. This dissipation of energy into heat during magnetisa- 
 tion is something quite apart from any production of heat by 
 eddy (or so-called Foucault) electric currents induced in the 
 mass, and would take place in iron so perfectly divided that no 
 eddy currents could exist. 
 
 One result of Prof. Swing's researches has been to show 
 that if the iron is kept in a state of mechanical vibration 
 hysteresis is greatly diminished, and the value of the energy 
 dissipated in a complete cycle is ?nuch reduced. The removal 
 of strong residual magnetism from soft iron by slight tapping 
 or twisting has also been noticed and commented on by Prof. 
 Hughes.* 
 
 Hysteresis, therefore, is a source of dissipation of energy in 
 the armature of dynamos. For in this case we have a mass of 
 soft iron, viz., the armature core, which has its direction of 
 magnetisation reversed every revolution. Suppose the core 
 has a volume of 9,000 cubic centimetres, and that it makes 15 
 revolutions per second. Taking the specific hysteresis for this 
 sample of iron at 13,356 ergs, we find that the dissipation of 
 energy in ergs per second is equal to 9,000 x 15 x 13,356 
 = 180x10" =180 joules, or a loss of about a quarter of a 
 horse-power-hour. 
 
 * Prof. D. E. Hughes, "On the Cause of Evident Magnetism in Iron," 
 /Voc. Soc. Tel. Engineers, Maj- 24, 1883, p. 3,
 
 ELECTRO-MAGNETIC INDUCTION. 65 
 
 Experiments were at one time made by Joule and others to 
 determine by direct observation the heating effect of magneti- 
 sation upon iron, but in these early experiments it is probable 
 that the results were mostly impure, and qualified largely 
 by the production in the iron of heat by local or eddy electric 
 currents.* 
 
 Since about 10,000 ergs per cubic centimetre are dissipated 
 by a double reversal of strong magnetisation of soft iron, it is 
 not difficult to show that the consequent rise of temperature, 
 even if all the heat is retained in the iron, is 0-000284C., or 
 that some 4,000 reversals would be required to raise the 
 temperature 1C., even provided all the heat generated is 
 retained in the metal. 
 
 If the iron is subjected to very rapid reversals of induction, 
 and if it is in one solid mass, then, in addition to the 
 hysteresis waste of energy, eddy electric currents are set up 
 in the mass of the iron and create heat. In such cases 
 it must be noted that the source of waste of energy in the 
 iron is twofold : first, that due to true magnetic hysteresis, 
 and, second, that due to eddy currents. The last source 
 of waste can be prevented by sufficiently and properly sub- 
 dividing the iron into very thin plates or wires, which are 
 rusted or painted so as to prevent the production to any 
 degree of eddy currents. The first source of waste cannot 
 be prevented by any such lamination. The question has 
 been very much debated and considered whether the true 
 hysteresis loss in iron depends upon the speed at which 
 the magnetic cycle is described whether the dissipation of 
 energy at, say, 100 reversals per second is or is not more 
 than one hundred times that of one slowly performed cycle. 
 
 The matter seems now decided as follows : It appears 
 evident from the researches of J. and B. Hopkinson,f that 
 if the induction density is moderate in amount (for example, 
 not more than 3,000 or 4,000 C.G.S. units) then, whether 
 the reversal of magnetism or cycle is made very slowly 
 
 * See Joule's " Scientific Papers," Vol. I., p. 123, " On the Calorific 
 Effects of Magneto-Electricity," and " On the Mechanical Value of Heat.' 
 Also Phil. Mag., Series 3, Vol. XXJIL, p. 263. 
 
 t The Electrician, September 9, 1892. See also Proc. Roy. Soc., London, 
 April P.O, 1893, Vol. LIII., p. 352.
 
 66 ELECTRO-MAGNETIC INDUCTION. 
 
 or at the rate of a hundred complete cycles per second, the 
 area of the hysteresis curve remains practically unaltered. The 
 same fact has been noted by Messrs. Evershed and Vignoles,* 
 and also by Mr. Steinmetz. We may say, therefore, as the 
 result of the latest work, that the difference between slow 
 cycle and quick cycle hysteresis, once supposed to exist, is 
 found to be non-existent, and that, at any rate for such induc- 
 tions as are used in transformers, we can apply the results 
 of the hysteresis losses obtained by slow cycle methods to 
 the cases of reversals at about a frequency of one hundred 
 or more. 
 
 From the researches of the Messrs. Hopkinson it is also 
 clear that, for higher induction densities, there is a difference 
 between the hysteresis loops for very slow cycles and for 
 rapid ones, and this difference is chiefly in that part of the 
 the curve preceding the maximum induction. As Prof. J. A. 
 Ewing has observed, after sudden changes of magnetising 
 force the induction does not at once attain its full value, 
 but there is a slight increase going on for some seconds. 
 Dr. Hopkinson has remarked that this small difference 
 between the curve as determined by very slow reversals 
 and that as determined by very rapid reversals is a true 
 time effect, the difference being greater between a frequency 
 of 5 per second and 72 per second than between 5 per 
 second and exceedingly slow cycles. There may be, there- 
 fore, a true time lag of magnetism at the higher speeds, 
 but for all such frequencies as are employed in ordinary 
 transformer work, we may take it that the hysteresis loss 
 is constant per cycle and equal to that obtained by slow 
 reversals. 
 
 The reader should carefully note that, if the hysteresis 
 diagrams are taken for a solid iron ring and an equal sized 
 ring made of iron wire in a way afterwards to be described 
 with the wattmeter, when rapidly alternating currents are 
 employed, the area of the diagram will be greater for the 
 solid than for the divided ring. The area of the hysteresis 
 diagram, then, gives us the energy loss due both to eddy 
 currents set up in the iron and also that due to true hysteresis. 
 Hence, in such experiments as above described, the greatest 
 * The Electrician, September 16, 1892,
 
 ELECTRO-MAGNETIC INDUCTION. 67 
 
 care has to be taken to eliminate all eddy current loss before 
 we can draw any conclusions as to hysteresis proper. When 
 a solid mass of iron is magnetised the eddy currents set up in 
 the mass have to die away first before the iron attains its 
 maximum magnetisation, because at every point in the 
 interior of the iron the magnetic force due to the eddy current 
 is in opposition to the external impressed magnetising force. 
 Hence the effect of eddy currents is to delay the rise of 
 magnetism. Over and above this, however, for strong in- 
 ductions there seems to be a slow increase of magnetisation 
 after the magnetising current has become constant. Prof. 
 Ewing says : "I repeatedly observed that when the mag- 
 netising current was applied to long wires of soft iron there 
 was a distinct creeping up of the magnetometer deflection after 
 the current had attained a steady value." 
 
 This time lag appears to be most manifest in the softest 
 iron, and to be especially noticeable near the beginning of the 
 steep part of the magnetisation curve. 
 
 In an investigation on the magnetisation of iron under feeble 
 magnetic forces, Lord Rayleigh has also drawn attention to the 
 fact that the settling down of iron when very soft or annealed 
 into a new magnetic state is far from instantaneous.* He 
 has shown that if the strength of the earth's horizontal 
 magnetic field is called h, for unannealed iron and steel 
 magnetising forces ranging from i h to y^oTy ^ l ca U forth pro- 
 portional magnetisation in other words, the susceptibility is 
 constant over this range, and the value of the corresponding 
 permeability is from 90 to 100, this small proportional 
 magnetisation taking place independently of what may be the 
 actual magnetisation of the iron, provided it is not very near 
 the condition usually called saturation. The moment, how- 
 ever, that the magnetising force is pushed beyond these limits 
 the phenomena of hysteresis and retentiveness make their 
 appearance. 
 
 The subject of hysteresis in iron is by no means yet entirely 
 explored. The chemical and physical states of the iron 
 exercise the greatest influence on its magnetic hysteresis, and 
 high specific electrical resistance seems in general to be an 
 
 * Lord Rayleigh, " On the Behaviour of Iron and Steel Under the 
 Operation of Feeble Magnetic Forces," Phil. Mag., March, 1887, p. 225. 
 
 F2
 
 C8 ELECTRO-MAGNETIC INDUCTION. 
 
 index of large hysteretic power in iron ; and those elements, 
 like manganese, which, when added to iron, increase its specific 
 electrical resistance, have also an effect in increasing its 
 hysteresis waste. 
 
 One important point in connection with magnetic hysteresis 
 is the effect of rise of temperature of the iron upon the 
 hysteresis loss. Experiments made recently by W. Kunz,* 
 are instructive on this point. This experimentalist investi- 
 gated in 1892 and in 1894 the effect of rise of temperature in 
 producing a diminution of magnetic hysteresis in iron, and 
 the following are the results obtained from a long series of 
 observations of this phenomenon : 
 
 Four kinds of iron, two of steel, and one of nickel have been 
 the subject of investigation. Special difficulties occurred in 
 maintaining the wire samples at the high temperature for the 
 required length of time, and in the measurement of these 
 temperatures, and therefore the methods are more particularly 
 described. 
 
 To measure the temperature of the wire, thermo-electric 
 junctions were used, consisting of a platinum wire twisted 
 for about 1-5 cm. of length round a wire of platinum con- 
 taining 10 per cent, of rhodium. Two such couples were 
 used, whose free ends were brought well insulated to a 
 mercury switch, by means of which each couple could be 
 connected to a Deprez-d'Arsonval galvanometer having about 
 200 ohms resistance. The temperature of the junctions at 
 switch and galvanometer was always the same about 20 C. 
 The calibration of these couples up to 300C. was done by 
 means of an accurate mercury thermometer, plunged with one 
 junction of the couple into oil, which was warmed up in 
 large beakers surrounded with asbestos and kept well stirred. 
 The galvanometer deflections corresponding to the tempera 
 tures from 40C to 300C were noted. From 300C upwards, 
 the calibration was done by utilising the known fusing points 
 of certain substances melted by a gas furnace : the junction 
 was placed in the molten material, the flame lowered, and the 
 deflection noted when solidification began. Each couple was 
 thus separately calibrated the outside junctions being kept 
 at a constant temperature by immersion in petroleum at 20C. 
 * See Elektrotechnische Zgitschrift, 1894, No. 14., p. 194,
 
 INDUCTION. 69 
 
 The substances used and temperatures measured, or assumed 
 from Bornstein-Landolt's tables, were as follows : 
 
 Substance Warmed or Melted. 
 
 Temperature or 
 Melting Point. 
 
 Galvanometer 
 Deflection. 
 
 Oil 
 
 Deg. 
 40 
 
 Ifi-fi 
 
 
 73 
 
 27 - 2 
 
 
 104 
 
 38-0 
 
 
 140 
 
 50 .g 
 
 
 200 
 
 77 -fl 
 
 
 300 
 
 125 '4 
 
 KC1O,... 
 
 359 
 
 155 6 
 
 PbCl 
 
 498 
 
 244 '4 
 
 IK 
 
 634 
 
 330 2 
 
 KC1 
 
 734 
 
 393 '6 
 
 NaCl 
 
 772 
 
 417 '8 
 
 Na 2 SO 4 
 
 861 
 
 474*2 
 
 
 
 
 To heat the wire under test a method already employed 
 by Ledeboer was adopted namely, winding an insulated 
 platinum wire round the test wire, and heating it by passing 
 a current through the platinum wire. The iron wire was 
 contained in a porcelain cylinder having a suitable opening to 
 take the thermo-couple, and round this cylinder was wound 
 (non-inductively) the platinum wire. The insulation between 
 the couple and the platinum was tested before each observa- 
 tion. This platinum coil was surrounded by layers of asbestos, 
 among which the wires of the thermo couple were led out. 
 The tube thus formed was placed inside a glass tube and 
 accurately centred by suitable packing. This tube was 
 placed again in another glass tube with asbestos distance 
 pieces. A current of hydrogen passing through the inner 
 tube protected the wires from oxidation. To protect the 
 magnetising coil against the high temperatures, a hollow tube 
 of pure copper was placed between, and a stream of water 
 kept passing through it, and this had to be insulated by 
 asbestos from the glass tubes to prevent their breaking. 
 Observation showed that no heat passed through this jacket. 
 The two couples always agreed in their indications, thus 
 showing that the wire was evenly heated. 
 
 As indicated above, the test wire was a long straight piece ; 
 its magnetic condition was observed by magnetometer, by the 
 "single pole" method. The magnetising coil is placed
 
 70 
 
 ELECTRO-MAGNETIC INDUCTION. 
 
 vertically in the direction east and west from the magneto- 
 meter, and the top end of the magnetised wire is on a level with 
 the instrument. The alteration in the position of the pole due 
 to differences in induction density affect the distance between 
 the magnetometer and wire so little as to be negligible. The 
 vertical component of the earth's field must be taken into 
 account. In magnetising, the cycle was always performed a 
 few times, until the curve became regular. Then a series of 
 observations at varying temperature of a certain cycle was 
 taken, generally with maximum B = 3,590 (about), until at 
 the high temperature (830 C. or so) the magnetism dis- 
 appeared. Then another cycle after cooling. Suitable 
 arrangements were made for compensating for the effect 
 of the magnetising coil itself on the magnetometer, and for 
 demagnetising by reversals of a gradually diminishing current. 
 
 The strength of field in the middle of the solenoid is 
 calculated from the well-known formula, 
 w 47TNC 
 H= ld T' 
 
 where N = number of turns, C = the current in C.G.S. units, 
 and I = the length of the solenoid ; and the induction was 
 calculated from the magnetometer deflection by means of the 
 known value of the horizontal component of the earth's field. 
 
 The following Tables are for a maximum induction density 
 of 3,590, and are the mean of four series : 
 
 Material. 
 
 Temp. 
 
 Hysteresis 
 Loss in Ergs. 
 
 Material. 
 
 Temp. 
 
 Hysteresis 
 Loss in Ergs. 
 
 
 Deg. 
 
 
 
 Deg. 
 
 
 
 20 
 
 2,350 
 
 / 
 
 20 
 
 2,690 
 
 German 
 annealed 
 charcoal 
 
 290 
 470 
 656 
 
 728 
 
 1,600 
 1,204 
 710 
 550 
 
 Swedish J 
 iron \ 
 
 270 
 460 
 650 
 742 
 
 2,080 
 1,550 
 905 
 825 
 
 
 836 
 
 316 | 
 
 812 
 
 712 
 
 
 20 
 
 2,107 
 
 
 20 
 
 Indefinite 
 
 
 20 
 
 3,420 
 
 
 20 
 
 3,100 
 
 Soft 
 
 284 
 
 2,480 
 
 
 275 
 
 2,270 
 
 wrought < 
 iron 
 
 468 
 656 
 744 
 
 1,750 
 821 
 800 
 
 Puddled 
 iron ' 
 
 460 
 560 
 656 
 
 1,730 
 1,310 
 979 
 
 
 20 
 
 900 
 
 
 744 
 
 777 
 
 
 
 |i 
 
 20 
 
 2,090
 
 ELECTRO-MAGNETIC INDUCTION. 
 
 71 
 
 These values show, when plotted as curves, that the equation 
 L = a - b t (where L is the hysteresis loss and t the temperature, 
 and a and b constants) expresses fairly the law. 
 
 Material. 
 
 Temp. 
 
 Hysteresis 
 Loss in Ergs. 
 
 Material. 
 
 Temp. 
 
 Hysteresis 
 Loss in Ergs. 
 
 
 Deg. 
 
 
 
 Deg. 
 
 
 ( 
 
 20 
 
 11,540 
 
 
 20 
 
 9,660 
 
 
 309 i 11,580 
 
 
 309 
 
 9,860 
 
 Hard patent j 
 steel | 
 
 526 
 660 
 790 
 
 6,040 
 2,200 
 1,180 
 
 Patent cast 
 steel 
 
 468 
 560 
 640 
 
 4,950 
 1,985 
 1,614 
 
 !; 
 
 20 
 
 5,230 
 
 
 744 
 
 1,048 
 
 
 
 
 
 20 
 
 4,670 
 
 The above Tables are to be interpreted as follows : Taking 
 a wire of the material named, it was subjected to a magnetic 
 cycle of induction in which the maximum induction reached 
 was 3,590 C.G.S. The wire being taken at a particular 
 temperature, as given in the second column of the tables, 
 a hysteresis loss, diminishing with rise of temperature, was 
 found, the value of which per cycle is given in the third 
 column. 
 
 The two kinds of steel referred to in the last two tables 
 had, for ordinary temperatures, magnetic cycles in shape like 
 a rhombus, altering in shape at about 300, even increasing 
 in area, and between this and 470 changing in form to that 
 of an ordinary iron curve, and decreasing greatly in area. 
 The character of the steel is lost after heating, as shown by 
 the final observations, and it becomes also quite soft. There 
 is, in these cases, no simple relation between hysteresis loss 
 and temperature. 
 
 The following tables relate to charcoal iron : 
 
 B. 
 
 Temp. 
 
 Loss. 
 
 B. 
 
 Temp. 
 
 Loss. 
 
 
 Deg. 
 
 
 
 Deg. 
 
 
 ( 
 
 20 
 
 8,900 
 
 
 20 
 
 21,020 
 
 
 270 
 
 6,690 
 
 
 270 
 
 14,840 
 
 f,200 ( 
 
 468 
 570 
 
 4,660 
 3,340 
 
 
 470 
 570 
 
 9,900 
 7,550 
 
 
 068 
 
 2,270 
 
 
 . 
 
 
 
 
 744 
 
 2,168 
 
 
 
 

 
 72 ELECTRO-MAGNETIC INDUCTION. 
 
 For higher temperatures than 570, B = 14,400 could not be 
 reached. 
 
 The results show that the above law also holds good, 
 generally speaking, for higher values of induction density. 
 
 The corresponding results for steel show that the character- 
 istics described for the lower values of induction density hold 
 also for the higher. In all the above cases a new wire was 
 taken for each series of observations. It is shown, therefore, 
 that when an iron wire is subjected to repeated cycles of 
 temperature and magnetisation the hysteresis loss decreases 
 up to the fourth cycle of temperature, and then becomes 
 uniform ; the results of each temperature cycle being ex- 
 pressible as a straight line, but of different inclination. The 
 steel wire has the first temperature cycle as already described, 
 and the remainder behave like the soft iron. 
 
 In the case of nickel subjected to a cycle of maximum 
 B = 3,590, it was found that the hysteresis fell with the 
 increase of temperature, at first rapidly, and afterwards 
 increasingly slowly : it fell from 11,420 ergs at 20 to 4,700 
 ergs at 288. 
 
 One of the most important results is the fact that repeated 
 cycles of magnetism at a high temperature reduce the 
 hysteresis loss in iron very considerably, and it would appear 
 that this is also the result of one cycle at a very high 
 temperature. 
 
 The above results show that in soft iron a very marked 
 decrease in the hysteresis loss takes place as the temperature 
 of the iron is raised. Further reference to this matter will be 
 made in reference to the losses of energy in transformer 
 working. 
 
 7. The Electromotive Force of Induction. We have in 
 2 enunciated Faraday's law of induction in terms of the 
 variation of a quantity called the flux of induction through 
 the circuit. It is possible to express the fundamental rule in 
 a more elementary manner, and in a way which adapts it to 
 explain every fact yet observed. It is as follows : If any 
 element of a conducting circuit is so placed in a field of 
 magnetic induction that a movement of that element of the 
 conductor or change in the field of induction causes lines of
 
 ELECTRO-MAGNETIC INDUCTION. f3 
 
 induction to intersect it, it creates an electromotive force in 
 the element of which the direction is perpendicular to the 
 plane containing the lines of magnetic force and the direction 
 of motion of the centre of the element. 
 
 This operation is called " cutting lines of magnetic force." 
 We shall allude later to hypotheses which have been con- 
 structed to suggest in some degree an explanation of the 
 nature of this effect. 
 
 The simplest possible case which can be considered is when 
 a short linear element, such as a straight wire, is made to 
 move in a uniform magnetic field, in a direction perpendicular 
 to the plane containing the field lines and the linear con- 
 ductor, the direction of the length of this last being also 
 perpenlicular to the direction of the field lines. 
 
 ( 
 
 
 
 C '''' 
 
 
 7] 
 
 V s' 
 
 .j 
 
 P B 
 
 t 
 i 
 
 v \ L^> 
 
 
 / 
 
 A 
 
 H D 
 
 FIG. 22. 
 
 
 If A B (Fig. 22) is the element of length of the conductor 
 of length L, and if A D represent in magnitude and direction 
 one of the lines of induction of the uniform magnetic field, H, 
 in which it is placed, AB being at right angles to AD, and if 
 AC represent in magnitude and direction a displacement of 
 A B taking place uniformly in one second, so that A B moves 
 uniformly parallel to itself from position A B to position C G 
 in one second, we have then three lines, A B, AC, A D, 
 mutually at right angles, and representing respectively the 
 length L, the velocity V, and the magnetic induction H. 
 
 The result of the motion is to generate in A B an electro- 
 motive force E, numerically equal to the product HLV in 
 consistent units. But since the sides of the parallelopipedon, 
 or solid rectangle, are taken to represent respectively H, L 
 and V, their product E represents the volume of the solid 
 and the magnitude of the electromotive force of induction.
 
 74 ELECTRO-MAGNETIC INDUCTION. 
 
 If the directions of A B, AC and A D are not orthogonal, 
 but inclined, the same still holds good. 
 
 For let the field lines AD (Fig. 23) be supposed to be 
 inclined at an angle 6 with the direction of the length of the 
 conductor A B, and let the direction of motion of A B parallel 
 to itself, represented by A C, be inclined at an angle < with 
 A B. The strength of field estimated perpendicular to A B is 
 H sin 6, and if A C represents the actual velocity of A B, or 
 displacement in one second, then A C sin <, or V sin <, is its 
 velocity in a direction perpendicular to its own length. The 
 magnitude of the induced electromotive force E in A B is then 
 numerically equal to L H sin V sin <, or to H L V sin sin <j> ; 
 but this expression also represents the volume of a doubly 
 skew parallelopipedon or solid rhomboid ; hence, as before, if 
 vectors be drawn representing respectively the length and 
 
 c 
 
 V, 
 
 A Ho 
 
 FIG. 23. 
 
 velocity of a conducting element, and also the field strength 
 in which it is placed, the volume of the solid rhomboid 
 described on these vectors as adjacent sides represents the 
 magnitude of the electromotive force induced in the element. 
 
 The magnitude of this induced electromotive force is not in 
 any way dependent upon the nature of the material of which 
 this conductor is made. Faraday experimentally proved this 
 ("Exp. Ees.," 193-201) by taking a double conductor com- 
 posed of an iron and a copper wire twisted together and united 
 at one end. On passing this double conductor through a 
 magnetic field no induced current was detected in it by a 
 galvanometer. This proved that the electromotive forces set 
 up in each separate conductor were equal and opposite, and 
 hence, since the lengths, field, and velocities were the same, 
 no factor entered into the production of the effect, which 
 depended on the nature of the conductor. From further
 
 ELECTRO-MAGNETIC INDUCTION. 75 
 
 experiments with circuits partly metallic and partly electro- 
 lytic fluids he inferred that in all bodies, whether what are 
 commonly called conductors or non-conductors, or elec- 
 trolytic conductors, identically the same electromotive force 
 is brought into existence by moving the same lengths in 
 the same way in the same magnetic fields. 
 
 When a metallic disc is rotated in a uniform magnetic field 
 so that its axis of rotation is parallel to the direction of the 
 field, there is set up a difference of potential between the 
 centre and the edge. In this case we can tap off a current 
 by an external wire connected to the centre and the edge 
 of the disc. 
 
 We can now show that, starting with the elementary law 
 above stated, as to the magnitude of the induced E.M.F. in 
 an element of a conductor, we can deduce the other principle 
 of the relation of the induced E.M.F. to the rate of change 
 of the induction through the circuit. 
 
 Let A B C D (Fig. 24) be a conducting rectangle, of which 
 the plane is perpendicular to the induction lines of a uniform 
 magnetic field of strength H, the same being shown in plan 
 on the figure ; let the circuit be capable of revolving about an 
 axis in its own plane, and let it be displaced through any 
 angle, 0, as shown in elevation and plan in Fig. 20. If the 
 frame is so displaced it is clear that the sides A C,BD "cut" 
 across lines of magnetic induction, but that the upper and 
 lower sides do not. During this displacement the vertical 
 sides alone will be the seat of electromotive forces. Imagine 
 this frame to revolve round the vertical axis with a uniform 
 angular velocity <a, and at any instant t to have a position 
 such that its plane makes an angle 6 with the plane normal 
 to the lines of force. Let the length of the side A C be 
 L and that of A B be E : the actual velocity of the side 
 
 A C is -, and the strength of the field, in a direction per- 
 pendicular to its length and its direction of motion at that 
 instant, is H sin 6. Hence the electromotive force of induction 
 
 in the side A C is - L H sin 6, and an equal and oppo- 
 sitely directed electromotive force acts in the side B D at the 
 same instant. Hence the total electromotive force acting 

 
 76 ELECTKO-MAGNETIC INDUCTION. 
 
 round the frame is equal to w H B L sin 0. If the area 
 of A B C D is denoted by A we may write the above as 
 w H A sin 6. The angular velocity o> may be expressed as 
 
 7 n 
 
 the time rate of change of 6, or as ; hence the expression 
 for the total electromotive force of induction round the frame 
 
 is HA sin 0', or- 
 d t 
 
 (HAcos^. 
 ^ 
 
 FIG. 24, 
 
 The expression A cos 6 denotes the apparent size of the 
 frame as looked at from a considerable distance along the 
 direction of the lines of induction, and the quantity H A cos 
 is the numerical value of the number of lines of magnetic 
 induction passing through or traversing the frame in its posi- 
 tion when its plane is inclined at an angle 6 to the normal 
 position. We assume that these lines are spaced out according 
 
 * We here suppose the circuit to be foi med of a single loop of wire 
 having a practically negligible self-induction. The above statements would 
 require some modification for a circuit of many turns of wire.
 
 ELECTRO-MAGNETIC INDUCTION. 77 
 
 to the rule proper for such distribution, viz., that the 
 number passing through a unit of area whose plane is taken 
 normal to the direction of these lines is numerically equal to 
 the magnetic induction over that area. 
 
 Writing N for this number of lines so piercing through the 
 frame at any instant, we have, as the expression for the total 
 electromotive force acting round the frame at any instant, the 
 
 quantity - _ ; that is, the electromotive force of induction 
 
 is numerically equal to the rate of change (decrease) of the 
 included lines of induction. It is customary to speak of 
 this induced electromotive force as generated either by the 
 " cutting of lines of force " by the various elements of the 
 conductor or by a change in the number of lines of force 
 piercing through the aperture of the circuit ; but they are 
 merely two different geometrical ways of viewing the same 
 phenomena. The actual results are capable of receiving a 
 physical explanation on the assumption that the act of inter- 
 section of a line of force and a portion of a conducting circuit 
 is productive of an electromotive force. We see that the total 
 electromotive force is the resultant effect due to a summing- 
 up of all the forces acting on each element of the circuit, 
 each elemental E.M.F. being measured by the product of 
 the length of that element, the field strength around it, and 
 its normal velocity in that part of the field. The result is 
 concisely expressed by the number which expresses the time 
 rate of change of the whole number of the lines of induc- 
 tion traversing the circuit. This same may be extended to 
 any circuit of any form moving in any way in any field. 
 
 If a circuit of any form which is traversed by an electric 
 current is placed in a magnetic field due to other neighbouring 
 currents or magnets, there is a flux of induction through that 
 circuit due partly to the current in the conductor and partly 
 to the external field of the other currents or magnets. If 
 there be M lines of induction due to the external field passing 
 through it, and N lines of induction due to its own current, 
 any variation of the external induction, of which the rate of 
 
 change at any instant is represented by - , will produce an 
 impressed electromotive force in such a direction that taking
 
 78 ELECTRO-MAGNETIC INDUCTION. 
 
 lines of induction out of the circuit induces an electromotive 
 force in the clockhandwise ( + ) direction , as seen from that 
 side of the circuit at which the lines enter. When a current 
 is flowing in any conductor, the relation between the direction 
 of the current and that of its own lines of induction is the 
 same as the relation between the thrust and the twist of a 
 corkscrew. Hence, it is evident that, if we consider a circular 
 current (Fig. 25) with the current flowing in it clockhandwise 
 ( + ), as seen from one side, its own lines of induction pass 
 
 through the circuit in the positive direction, or away from 
 the eye. 
 
 Accordingly, a little reflection shows that, if the current in 
 the conducting circuit is made to increase, an opposing 
 electromotive force is created by the increasing induction of 
 the current on its own circuit. The current in the act of 
 increasing crowds its own circuit more full of lines of induc- 
 tion, and creates an electromotive force of induction during 
 the period of this increase equal numerically at any instant to 
 its own rate of increase, and directed in opposition to the 
 impressed external electromotive force which is driving the 
 current.
 
 CHAPTER III. 
 
 THE THEORY OF SIMPLE PERIODIC CURRENTS. 
 
 1. Variable and Steady Flow. In the following pages we 
 shall be chiefly concerned in considering the properties and 
 uses of currents of electricity which are periodic in character ; 
 that is, which are changing in strength from instant to instant 
 in a cyclic or periodic manner. An electric current or an 
 electromotive force may either be steady, in which state it 
 remains uniformly at the same value, or it may be variable, in 
 which case it is changing in value from instant to instant. 
 In this last case we can consider two separate conditions. 
 The current strength or electromotive force may be periodic 
 or non-periodic in value. A non-periodic variable current or 
 electromotive force is one which changes in value from 
 instant to instant accordingly to any assigned law or mode, 
 but in which the same series of values are not regularly 
 repeated. A periodic current or electromotive force is one 
 which runs through a regular cycle of values, returning 
 after a certain period to the same value. It is accordingly 
 said to vary in a cyclic manner, because it changes through 
 a cycle of values. We may take illustrations of these three 
 states from the flow of fluids. A stream of fluid may exist 
 in a steady state ; in this case the motion of each particle 
 of the stream has settled down into a uniform condition 
 as regards velocity. If we imagine a small short tube open 
 at both ends, held anywhere in that flowing fluid, the same 
 volume of fluid would flow through that tube in every unit 
 of time. We may, however, find the fluid in such a condition 
 that the velocity of each particle of the fluid at any point 
 is changing, and the flow is then in a variable condition
 
 80 SIMPLE PERIODIC CURRENTS. 
 
 If that change is of such a character that the motion is 
 regular in its mode of change, then the flow is said to be 
 periodic. Thus, in a non-tidal river the water flows in 
 general uniformly in one direction ; it is in a steady state. 
 At the time of a flood its speed at any point may be 
 rapidly increasing, and in this case its flow is variable. In 
 the case of a tidal river the flow of water is regularly 
 reversed, a cycle of fluid motion is repeated at any point, 
 and the motion is said to be cyclic or periodic in character. 
 
 In considering the motion, either of actual fluids or of 
 electric currents, we can, then, distinguish three states the 
 variable, the periodic, and the steady condition. In the first 
 case the strength or direction of the electric currents or of the 
 fluid velocity is changing at every instant ; in the latter cases 
 the flow has settled down into a permanent state. The 
 questions involved in dealing with the variable or periodic 
 states present rather more difficulties than do problems in 
 steady flow, for the reason that the notions of time and inertia 
 enter into these in a way in which they do not when that flow 
 has reached a steady condition. We shall proceed to examine 
 in an elementary manner some features of electrical flow when 
 variable or periodic. We must, however, prepare the way by 
 considering some purely geometrical properties of certain 
 curves, and also some modes of motion which have special 
 reference to the kind of electric current to be considered sub- 
 sequently. When a mass of water is in motion, a particle of 
 water selected for examination has at any instant a certain 
 velocity in a certain direction. This may be represented 
 graphically by a straight line drawn from that particle 
 representing its velocity in direction and magnitude. 
 Similarly, if electricity is flowing through the mass of a 
 conductor in any manner, it is possible at any point to draw a 
 vector or line representing at that instant the direction and 
 magnitude of the current at the point from which the line is 
 drawn. Lines drawn within the mass of a fluid at any points 
 such that the flow at that instant is along or tangential to 
 these lines are called flow lines. In the first place, let us 
 make the supposition that the flow has reached a steady 
 condition. The flow lines are then fixed. When this is the 
 case each line of flow becomes the actual path of a. fluid
 
 SIMPLE PERIODIC CURRENTS. 81 
 
 particle, and is called a stream line. A. surface may be 
 supposed to be described in the mass of the fluid everywhere 
 perpendicular or orthogonal to the stream lines ; such a 
 surface is called an equipotential or level surface. We may 
 also suppose such a level surface drawn in the mass of a con- 
 ductor through which a current is flowing. Let any area be 
 drawn on the equipotential surface, and let it be divided up 
 into units of area. If the quantity of fluid or of electricity 
 flowing through each unit of area is the same, and if, more- 
 over, it is the same for each unit during each succeeding 
 instant of time, the current is said to be steady and to be 
 uniformly distributed. The quantity flowing per unit of time 
 through any area is the numerical measure of the mean 
 strength of current over that section of the conductor, and 
 the quantity flowing per unit of time through a unit of area 
 is the measure of the mean density of current over that unit 
 of area. If the distribution of current and strength is not 
 uniform, we can only express them at any time and place by 
 calling to aid the language of the differential calculus. If 
 ds be a small area described on an equipotential surface, and 
 if d q be the quantity of electricity which flows in a small 
 time d t through that area d s, and if i is the strength of the 
 current at the centre of that small area at any instant, then 
 in the limit 
 
 2. Current and Electromotive Force Curves. To fix our 
 ideas, let us now suppose the electric flow to take place through 
 a thin cylindrical conductor, such as a wire, in which, at 
 positions sufficiently remote from the ends, the stream lines 
 will be parallel to the axis of the wire and the equipotential 
 surfaces perpendicular to it. Consider any one section, and 
 let the flow across this section be variable both in strength 
 and direction that is to say, let it vary in the quantity of 
 electricity which flows across that section in each succeed- 
 ing instant, and let the flow be first one way and then the 
 other, changing in any manner, however irregular. We can 
 represent graphically the state of things as regards electric 
 flow at that section by means of a curve called a current curve.
 
 82 SIMPLE PERIODIC CURRENTS. 
 
 Take a horizontal line (Fig. 26) to represent the uniform flow 
 of time. At successive instants let ordinates be drawn to this 
 line, representing the strength of current flowing past that 
 section, and let them be drawn above ( + ) or below (-), 
 according as the direction of the flow is to the right or to the 
 left. Thus, if time begins to reckon from O, after the lapse 
 of a time T the current is positive, and is represented by a 
 
 
 
 / 
 
 FJG. 26. 
 
 line T I. After the lapse of a time T' the current is 
 negative, and is represented in strength by a line T' I'. 
 
 This current curve is obviously a single-valued function 
 that is to say, corresponding to a given instant of time the 
 current can only have one value. The curve can never cut 
 itself or double back. 
 
 We may here remind the student of the distinction between 
 single and multiple -valued functions. A single-valued func- 
 
 FIG. 27. 
 
 Single-valued function. 
 
 FIG. 28. 
 Multiple-valued functions. 
 
 tion is one which, when represented graphically by a continuous 
 curve, presents only one value of the ordinate for each value 
 of the abscissa. 
 
 In Fig. 27 is represented graphically a single-valued function, 
 having only one value of the ordinate X Y corresponding to a 
 given value of abscissa X. In Fig. 28 is represented a curve 
 such that there are five different values of the ordinate of the
 
 SIMPLE PERIODIC CURRENTS. 
 
 83 
 
 curve corresponding to one value of the abscissa X. This 
 curve represents a multiple-valued function. 
 
 Amongst single-valued functions, or single ordinate curves, 
 there is one which is particularly important, because it 
 proves to be the constituent element of every single-valued 
 function. This carve is called a simple periodic curve, or simple 
 sine curve, or simple harmonic curve. This curve may be 
 described as follows : Let a circle (say a coach wheel) roll 
 with uniform speed along a straight line, A B : a point P on 
 its circumference will mark out a curve called a cycloid, 
 represented in Fig. 29 by the thick line, A E P B. If the point 
 P be projected at every instant on the vertical diameter of the 
 circle, then the point M will mark out a curve (represented 
 by the dotted curve) as the circle rolls along which has been 
 
 FIG. 29. 
 
 sometimes called " the companion to the cycloid." It is also 
 called a harmonic curve, a sine curve, or a simple periodic 
 curve. Draw a line S N through the centre of the circle 
 and parallel to the base line A B. Let it cut the dotted curve 
 at the point 0. The mathematical student will see that if 
 the point O is taken as origin, and C is called a, and C M 
 called y, then also, if the radius C P of the circle is E, and the 
 angle M P C = P C N is called 6, it is clear that 
 
 and 
 or, 
 
 y = R sin 6. 
 
 = R sin (180 - ). 
 
 If / is the circumference of the circle, then = 2?rR, and, by 
 
 substitution, 
 
 I . 2- 
 
 ^aT^-f 
 
 (21) 
 
 G 2
 
 84 
 
 SIMPLE PERIODIC CURRENTS. 
 
 This last is the equation to the dotted curve E M B, and it 
 is the equation to a simple periodic or sine curve. The 
 quantity I = A B is called the wave length, and R = S E is 
 called the amplitude of the harmonic curve. It will be seen 
 that this simple periodic curve is a smooth wavy curve which 
 has points of maxima above and below the axis C. 
 
 3. Simple and Compound Periodic Curves. If on one 
 
 common axis we draw two simple periodic curves of any wave 
 lengths and any amplitudes, and having any relative position 
 with regard to each other, we may obtain another curve, called 
 a complex periodic curve, by adding together the ordinates of 
 the two simple curves. 
 
 FIG. 39. 
 
 As an example, in Fig. 30 are shown two simple sine curves, 
 represented by the firm lines, of which one has double the 
 wave-length and about two and a-quarter times the amplitude 
 of the other. If these curves are superimposed, and a new 
 curve, represented by the dotted line, formed by adding the 
 ordinates X y v X y z , of a common abscissa, X, into a third, 
 Xt/ 3 , then we obtain, by repeating this at all points, a new 
 curve, which is called a complex periodic curve, because it is 
 compounded of two simple sine carves. The dotted curve is 
 the complex sine curve, and the two firm-line curves are its 
 two components. 
 
 We may in this way add together any number of simple 
 periodic curves and obtain an exceedingly complicated complex 
 periodic curve, which is, however, always, like a simple 
 periodic curve, a single-valued function. It is clear, also,
 
 SIMPLE PERIODIC CURRENTS. 85 
 
 that just as we can compound simple periodic curves into 
 a complex one, so we can resolve a complex single-valued 
 function into a set of simple periodic components, suitably 
 situated with respect to one another. 
 
 4. Fourier's Theorem. One of the most attractive and 
 important of all mathematical discoveries is that of Jean 
 Baptiste Fourier, who in his " Theorie Analytique de la 
 Chaleur," published in 1882, gave a demonstration of the 
 above theorem, viz., that any periodic curve, however com- 
 plex, provided it is a single-valued function, can be resolved 
 into a series of simple periodic curves, of suitable amplitudes 
 and wave-lengths, and be placed in a certain relative position 
 to each other. In mathematical language, any single- valued 
 periodic function can be expressed analytically as a sum of 
 a series of terms the first of which is an arbitrary constant, 
 and each of the following terms is the sine or cosine of an 
 
 \ 
 
 FIG. 31. 
 
 angle multiplied by a constant. Take such a case aa that 
 of a zig-zag line, made up of lines inclined at an angle of 
 GOdeg., like the teeth of a saw (Fig. 31). We can, by Fourier's 
 theorem, express the equation to this periodic line in terms 
 of a series of sine or cosine terms. Thus the equation to 
 the zig-zag line in Fig. 31 is 
 
 = z. \ sin x - - sin 3 x + sin 5 x - &c. 
 
 - 
 
 9 25 
 
 Hence, by adding together the ordinates of a number of sine 
 curves suitably chosen and placed, we can obtain a complex 
 periodic curve which imitates in form any given single valued 
 periodic curve, however complex it may be, provided only that 
 it is periodic, and that the curve does not cut itself. 
 
 This very remarkable theorem has applications in all 
 departments of physics. In acoustics it shows that any
 
 86 SIMPLE PERIODIC CURRENTS. 
 
 continuous sound may be resolved into a series of simple 
 harmonic sounds. In alternating current investigations it 
 demonstrates that any curve of current, however complex, can 
 be resolved into a series of simple periodic currents. If, then, 
 any single function is graphically represented that is to 
 say, any such curve as in Fig. 30 we see that this curve 
 may be described by a point which moves horizontally with 
 a uniform velocity, whilst at the same time it executes in 
 a vertical direction a movement which is the sum of a 
 number of simple harmonic motions superimposed upon 
 one another. The combination of these two rectangular 
 motions causes the point to describe the curve considered. 
 
 In subsequent chapters we shall be examining effects 
 which are due to periodic or fluctuating electric currents. 
 Fourier's theorem gives us, when applied to these cases, 
 a simplification of immense value, in that it enables us to 
 see that, however complicated may be the fluctuation of 
 current in a conductor, it can always be resolved into the 
 sum of a series of simultaneous currents varying in a simple 
 manner, and each of which can be graphically represented 
 by a simple harmonic curve. The general consideration of 
 periodic currents must, then, be preceded by an examina- 
 tion of the elementary theory of electric currents of a 
 periodic character, in which the variation is of the most 
 simple kind. 
 
 Fourier's theorem applies also to many other physical 
 phenomena of great importance. In acoustics it shows, 
 for instance, that however complicated may be the motion 
 of an air particle hi a mass of air through which sound 
 waves are being transmitted, it can be resolved into the 
 sum of a series of motions such as would be produced by 
 the action of tuning forks, each of which gives rise to a 
 motion in the air particles approximately of the nature of a 
 simple harmonic vibration. Helmholtz actually realised this 
 in his synthesis of vowel sounds. 
 
 Physically interpreted, Fourier's theorem means that any 
 variation of motion which can be represented by the changing 
 ordinate of a single-valued periodic curve can be expressed as 
 the sum of a series of simultaneous motions, each one of which 
 is called a simple harmonic, or simple periodic, or simple sine
 
 SIMPLE PERIODIC CURRENTS. 87 
 
 motion. It becomes important, then, to start by examining 
 the simplest form of periodic motion. Suppose a circular 
 disc (Fig. 32), having a pin at its centre, 0, to be pivoted so as 
 to revolve round an eccentric point, C. Let a T bar, moving 
 in guides and having a slot in the cross-piece, be so fixed that 
 the centre pin is constrained to move in the slot. Further- 
 
 FIG. 32. 
 
 FIG. 33. 
 
 more, let the point C round which the disc moves be fixed to 
 some support in the line of the bar AB produced. -If the 
 eccentric is compelled to move round C, the extremity of the 
 bar A will move backwards and forwards with a motion called 
 a simple harmonic motion or a simple periodic motion. 
 
 FIG. 34. 
 
 For it is clear the point (Fig. 33) is compelled to move in 
 a circle round G as a centre, and hence the distance of the 
 point A from C at any instant is the length of the bar A B 
 plus the length B C, which is the projection of C on the line 
 A C. The point B, therefore, executes a simple vibration to 
 and fro along the line A C as moves round, and the point A
 
 88 
 
 SIMPLE PERIODIC CURRENTS. 
 
 imitates the motion of B. If the angle C D is called x and 
 the radius C is a, then the length B C is a sin x, and the 
 displacement of A at any instant from its mean or middle 
 position has the same value. The motion of A is called a 
 simple harmonic motion, and the above eccentric and T bar 
 is a mechanical device for compelling a point to describe a 
 simple harmonic motion (abbreviated into S.H.M.). If such 
 a harmonic motion be executed by point A (Fig. 34), whilst 
 at the same time a strip of paper, S S', is caused to move 
 uniformly in a direction perpendicularly to the line A B, a 
 tracing point fixed to A will describe on the paper a curve of 
 
 FIG. 35. 
 
 which the ordinate A Y is proportional to the sine of the 
 abscissa XY, or the equation to the curve will be of the 
 form y = a sin x, a being some constant quantity. Hence a 
 simple periodic curve is also called a sine curve. 
 
 By combining together two similar pieces of mechanism it 
 is possible to construct a machine which can add together 
 graphically two simple harmonic motions in the same line, 
 but of which the phase angles x and the amplitudes a are 
 different. Machines for doing this have been devised by Lord 
 Kelvin, Mr. Stroh, and others. Apart from complications the 
 general principle is as follows.
 
 SIMPLE PERIODIC CURRENTS. 89 
 
 Let a cord pass over four pulleys (Fig. 85), two of which, 
 F 1 F 2 , are fixed in space, and two, M 1 M 2 , can be made to rise 
 and fall in vertical lines with a simple harmonic motion by 
 being attached to T bars and eccentrics. If the cord has one 
 end, B, fixed, and the other end, A, free, it is easy to see that, 
 if either the pulley M 1 or M 2 rises and falls along a vertical 
 line and the cord is just kept tight, the free end A will be 
 displaced by an amount equal to twice the displacement of 
 M 1 or M 2 , and as M 1 or M 2 moves up anJ down with a S.H.M., 
 the free end of A will also execute similar vibrations. If 
 M 1 and M 2 move together the displacement of A at any instant 
 is equal to the sum of the displacements of M 1 and M 2 . By 
 providing the end A with a tracing point, and moving under 
 it uniformly a sheet of paper in a direction perpendicular to 
 the direction of motion of A, it will describe a curve of which 
 the equation will be of the form 
 
 y = a sin x -t- a sin a/, 
 
 a and a' being the amplitudes and x x the phase angles of the 
 two motions of M 1 and M 2 respectively. This apparatus, or. 
 one of similar principle, has been devised and employed by 
 Lord Kelvin in his researches on the tides. It will be evident 
 from the foregoing explanation that a machine can be con- 
 structed capable of causing a tracing point to move to and fro 
 across a uniformly flowing sheet of paper, with a motion 
 compounded of any number of simple harmonic motions of 
 different amplitude and phase taking place in the same 
 straight line. 
 
 5. Mathematical Sketch of Fourier's Theorem. Without 
 going into a complete proof of Fourier's theorem, for which 
 we must refer the advanced student to mathematical text- 
 books, we propose to indicate to the student how it is prac- 
 tically employed in the analysis of any complex curve into a 
 series of simple harmonic constituents. At a later stage the 
 student will find that this analysis is of use in discussing 
 certain current and electromotive force curves obtained from 
 transformers. 
 
 We start with the assumption, for the propriety of which 
 we must refer the reader to more advanced treatises, that if y
 
 90 SIMPLE PERIODIC CURRENTS. 
 
 is the magnitude of the ordinate of any complex periodic 
 single-valued curve, we can always express y as follows : 
 
 + B 2 cos 2;? t + A 3 sin Sp t + B 3 cos 3 /> t + &c. 
 
 The problem is, given any complex periodic curve, to find the 
 A's and B's in the above equation for its ordinate at any point. 
 To do this we need a preliminary lemma in the integral 
 calculus. It is as follows : 
 
 The integrals, J sin ptsinqtdt, 
 
 and tcosptcosqtdt, 
 
 when integrated between the limits and TT, are equal to 
 zero, if p and q are unequal integers; and equal to J if p 
 and q are equal integers. For, since 
 
 2 sin p t sin q t = cos (p - q) t - cos (p + q) t, 
 and 2 cos p t cos q t = cos (p - q) t + cos (p + q) t ; 
 
 sm(p-q)t 
 
 r 
 therefore, jsmpt S mqtdt= 
 
 sin (p - q) t 
 
 and 
 
 / 
 
 Hence, if p and q are unequal integers, both these inte- 
 grals between the limits t = Q and t = ir are zero. If p = q 
 they both become equal to | . Again, if y is the ordinate of 
 a periodic curve, and if I is the half-wave length, then the 
 
 integral -I yd I represents the mean value of y during half 
 
 U o 
 
 the period ; because it is obvious that, if the mean or average 
 value of y is called M, the area enclosed by the periodic 
 
 curve and the base line between the two limiting ordinates 
 
 n 
 is M /, and this area is also expressed by the integral / y d I. 
 
 Hence the above equality results. From these two simple 
 lemmas it follows that we can easily determine the values of
 
 SIMPLE PERIODIC CURRENTS. 
 
 the constants in the harmonic expansion. Let us assume a 
 .simple case as an example. Let 
 
 V A + A x sin x + A 2 sin 2 x. 
 
 To determine A 2 , multiply all through by sin 2 x and inte- 
 grate between the limits x = and x = IT, 
 
 I y sin 2 x dx = I A sin 2.i- <lx + f A^in a; sin 2 ar<Z.r 
 
 /7T 
 A 9 sin 2 2 a; ^ x. 
 o - 
 
 All the integrals on the right-hand side of the equation 
 
 vanish except the last, which is equal to 2 - . 
 
 2 
 
 Hence 
 
 y sin 2 a; d x. 
 
 In other words, A 2 is equal to f <? the mean value of the 
 product of y and sin 2 x throughout the half period. In 
 
 _^ 
 
 c-^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 ^x 
 
 
 
 
 
 
 
 
 f> 
 
 
 
 
 
 "\ 
 
 \ 
 
 
 
 
 
 ' / 
 
 / 
 
 
 
 
 
 
 N 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 k 
 
 ^ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 4 6 8 10 12 14 16 18 20 22 2< 
 
 Time 
 
 FIG. 36. 
 
 the same way all the other constants may be found. The 
 process of analysing a complex function into its simple 
 harmonic constituents is then reduced to little more than 
 mere arithmetic.
 
 SIMPLE PERIODIC CUEEENTS. 
 
 A single example will make this clear.* There is a certain 
 complex periodic curve, one period of which is represented hi 
 Fig. 36. The problem is to find the simple harmonic or sine 
 curves of which it is composed. Call y the ordinate of the curve. 
 Divide the whole period into twenty-four equal parts. Let T 
 
 be the whole periodic time, and let p stand for ~. Let t be 
 
 any fraction of the periodic time, so that p t is the angular 
 magnitude of the abscissa corresponding to any ordinate y. 
 Since we have divided the period into twenty-four equal parts 
 each of these corresponds to an angular interval of 15. 
 Hence, pt is successively 15, 30, 45, 60, &c. 
 
 Measure from the curve the value of y corresponding to 
 each of these intervals, and tabulate them as follows : 
 
 y pt 
 
 y pt 
 
 y pt 
 
 13-3660 
 
 11-7940 135 
 
 4-8030 270 
 
 14-0355 15 
 
 10-8660 150 
 
 5-9645 285 
 
 14-3300 30 
 
 97060 165 
 
 7-5000 300 
 
 14-3295 45 
 
 8-3660 180 
 
 9-3940 315 
 
 14-1340 60 
 
 6-9645 195 
 
 108660 330 
 
 13-8295 75 
 
 5-6700 210 
 
 12-2940 345 
 
 13-4640 90 
 
 4-6705 225 
 
 13-3660 360 
 
 13 0355 105 
 
 41330 240 
 
 
 12-5000 120 
 
 4-1705 255 
 
 
 Proceed then to make a second table as follows : 
 
 I. 
 
 t 
 
 II. 
 
 y 
 
 III. 
 ft 
 
 IV. 
 
 sinpt 
 
 V. 
 
 y x sinpt 
 
 VI. 
 cospt 
 
 VII. 
 
 y x coap t 
 
 
 
 1 
 
 2 
 3 
 4 
 
 13-3660 
 14-0335 
 14-3300 
 14-3295 
 &c. 
 
 
 
 15 
 30 
 45 
 
 
 
 0-2588 
 0-5000 
 0-7071 
 
 
 3 6324 
 7-1650 
 10-1324 
 
 I 
 
 0-9659 
 0-8660 
 0-7071 
 &c. 
 
 13-3660 
 13-5569 
 124098 
 10-3124 
 
 Similarly in Column VIII. put the values of sin 2 pt ; in 
 Column IX. put the values of y x sin 2 p t ; and in Columns 
 X. and XI. put cos 2 p t and y x cos 2 pt. Then the value of 
 the constant term A + B is the mean or average value of all 
 the 24 numbers in Column II. 
 
 * The example above given is taken almost verbatim from a letter by 
 Prof. John Perry in The Electrician of February 5, 1892, Vol. XXVIII., 
 p. 362.
 
 SIMPLE PERIODIC CURRENTS. 93 
 
 AJ is twice the average of all the 24 numbers in Column V. 
 
 Bj is twice the average of all the 24 numbers in Column VII. 
 
 A 9 is the same for Column IX., and B for Column XI. 
 
 Any number of columns may be calculated corresponding to 
 the multiple angles, 3 p t, 4 fj t, &c., for higher terms of the 
 Fourier series. 
 
 When we have all the sine and cosine terms it is easy to 
 express y in the form 
 
 y = A + B H- v/Aj 2 + B^ 2 sin (pt+6) 
 
 sin (2pt + ff)+ &c., 
 
 by grouping together the sine and cosine terms. 
 
 In the example calculated above it is found that the value 
 of y is approximately 
 
 2/ = 10 + 5 sin (>* + 30) -sin (2^-60), 
 and this shows us that the given periodic carve is made up of 
 two sine curves of amplitudes, 5 and 1 respectively, which 
 differ in phase by 3CK The student will find it to be a useful 
 exercise to take two or three simple periodic curves and add 
 their ordinates into a complex periodic curve, and then by 
 the Fourier analysis to re-discover the simple harmonic 
 constituents again, and see if he can find the amplitudes 
 correctly. 
 
 6. Simple Periodic Currents and Electromotive Forces. 
 Eeturning, then, to electric currents, we may consider how a 
 complex periodic current is made up of simple periodic currents 
 superimposed. It is necessary to examine, in the first place, 
 how a simple periodic current or electromotive force may be 
 generated. Let A B C D (Fig. 37) be a rectangular frame or 
 conductor, able to revolve round a vertical axis, 0', in a 
 uniform magnetic field. The adjacent figure represents the 
 same in plan. If the frame revolve round the axis 0', the 
 total electromotive force acting round the circuit at any 
 instant is numerically equal to the time rate of change of 
 magnetic induction or number of lines of magnetic force 
 passing through the circuit. If H is the field strength in 
 C.G.S. units, I the length of the side A C, and k the length of 
 the side C D, and x the angle which at any instant the plane 
 of the frame makes with a plane drawn at right angles to the
 
 94 SIMPLE PERIODIC CURRENTS. 
 
 lines of the field, then the magnetic induction or number of 
 lines of force through the frame is the product of H, and the 
 apparent size of the frame, as seen along the direction of the 
 lines of force of the field, is equal to H I k cos x. 
 
 If the area of the frame is A square centimetres, the 
 magnetic induction through it is HAcosz. The effective 
 electromotive force acting to produce a current in the circuit 
 
 FIG. 37. 
 
 is numerically equal to the time rate of change (decrease) of 
 the magnetic flux or induction, or to 
 
 dt dt 
 
 This last equation is merely a symbolic statement of the 
 fact that, if such a frame of area A revolve round an axis 
 perpendicular to the lines of force in a, uniform magnetic field,
 
 SIMPLE PERIODIC CURRENTS. 95 
 
 H, with an angular velocity ^f, then the integral electromotive 
 force acting round the frame at any instant corresponding to 
 
 an angular displacement x is H A sin x. 
 
 d t 
 
 If the angular velocity remains constant, the effective 
 electromotive force will be simply proportional at any instant 
 to the sine of the angular displacement of the frame from its 
 initial position. Such a frame produces by its uniform revo- 
 lution a simple periodic variation of electromotive force in its 
 own circuit. If we suppose such a frame to have a closed 
 circuit, then this periodically varying electromotive force will 
 produce in the circuit an electric current which varies in 
 strength very nearly as the sine of the angle of the displace- 
 ment of the frame from its zero position when no lines of 
 force penetrate through its area. Hence, graphically repre- 
 sented, the current varies according to a simple harmonic 
 law, or is a simple sine current. We can then synthesise by 
 
 the superposition of such simple harmonic electric currents 
 any form of variable current, however complicated. Let a 
 series of such sine inductors be joined up on one circuit 
 (Fig. 38), each capable of being regulated as to angular 
 velocity, and imagine these to revolve in magnetic fields of 
 equal strength. These sine inductors are originally set with 
 the plane of their frames at certain different but fixed angles 
 to the planes at right angles to the fields of force in which 
 they revolve, and they must be supposed to maintain these 
 relative positions during their revolution. Accordingly, the 
 effective electromotive force in the whole circuit, when they 
 are all joined up in series and set revolving at fixed speeds, is 
 represented by a function 
 
 e = A sin x + A' sin x + A" sin x" + &c. ; 
 
 and by Fourier's theorem any possible periodic variation of e 
 which, graphically described, is a single-valued function, can
 
 96 SIMPLE PERIODIC CURRENTS. 
 
 be produced by suitable values of the speeds and phase angles 
 of these sine inductors. 
 
 The converse of the above proposition is also true. Let 
 there be any periodic current-generating machine producing 
 in a circuit an electromotive force, and therefore a current 
 varying periodically according to any law. This kind or form 
 of current could be exactly imitated by removing the given 
 machine and substituting a series of sine inductors coupled in 
 series and arranged so as to each produce a simple sine 
 varying E.M.F., the respective sine currents having different 
 phases and amplitudes, but being superimposed upon one 
 another. That is to say, however complicated may be the 
 nature of the periodic current which traverses a circuit, 
 provided the same electric motions are repeated at regular 
 intervals, we may build up this current by suitably super- 
 imposing in the same circuit a number of simple periodic 
 currents of certain amplitude and wave-lengths and fixed 
 difference of phase. 
 
 The above remarks may be taken as an outline of the 
 analysis of any single-valued continuous function into a series 
 of simple harmonic functions. To simplify language, we shall 
 in future speak of a curve whose equation is of the form 
 y = A sin z as a simple periodic curve, and if such curve 
 graphically represents the continuous variation of the flow of 
 electricity past any section of a conductor, or the fluctuation 
 of electromotive force in any circuit, we shall speak of such as 
 a simple periodic current or a simple periodic E.M.F. 
 
 Any other mode of variation of these quantities which, 
 graphically represented, would be a single-valued curve 
 repeating the same form, will be spoken of as a complex 
 periodic curve, current, or E.M.F., and, by the foregoing 
 analysis, a complex periodic function can be analysed into a 
 sum of simple periodic functions. 
 
 7. Description of a Simple Periodic Curve. The follow- 
 ing method affords a very easy means of drawing a simple 
 periodic curve. Take a cylinder or tube of pasteboard (see 
 Fig. 39) and cut it through obliquely with a sharp knife, 
 taking care to make the cut in one plane. The section of this 
 cylindrical tube by an oblique plane will be an ellipse. Slit
 
 SIMPLE PERIODIC CURRENTS. 
 
 97 
 
 the tube open along the line A B and unfold it. Lay it down 
 on another sheet of paper and draw a pencil line guided by 
 the curved edge A Y Y'. Draw a dotted line, X X', so that 
 its vertical distance below the highest point Y on the curve 
 is equal to its vertical distance above the points A and Y', 
 
 FIG. 39. 
 
 or make OY equal to AX. Then move this cardboard 
 template forward through a distance equal to its own width, 
 and draw another piece of curve, repeating the first, and 
 similarly placed (see Fig. 40). 
 
 The resulting curve is a simple periodic or simple sine 
 curve.* The distance X X', equal to the circumference of 
 the tube or to the width of the template, is the wave length. 
 The distance Y of the highest point above the mean line 
 
 " Elements of Dynamics " (Clifford), p. 22.
 
 98 
 
 SIMPLE PERIODIC CURRENTS. 
 
 is the amplitude. If the bottom edge of the template is 
 divided into 360 parts or units, then the distance O'M, 
 measured in such units of the foot of the perpendicular, 
 let fall from any point P on 0' M, is the phase of the point 
 P, measured in degrees. 
 
 It is, perhaps, more convenient to reckon the phase of the 
 point P by the magnitude of the line A N, or the distance 
 of the foot of the perpendicular, let fall from P on X X' from 
 the point A, where the curve crosses the mean line. The 
 phase of the maximum ordinate Y is then 90deg. 
 
 8. The Value of the Mean Ordinate of a Sine Curve. 
 Let Fig. 41 represent the semi-wave of a simple periodic 
 curve ; we shall proceed to prove some geometric proper- 
 ties of such a curve. Considering this curve as bounding 
 an area of which the other including line is the datum line 
 
 FIG. 41. 
 
 X X', we shall first find the value of the mean or average 
 ordinate. Let XX' be divided into equal and very small 
 intervals, such as N N', of which the length is d x, and let 
 X N be called x. Assume as a unit of length the radius of 
 the cylinder of which XX' is the semi-circumference. At 
 each of these small elements raise ordinates, such as P N, to 
 touch the curve. We require to find the mean value of all 
 these equi-spaced ordinates when they are infinitely close. 
 The arithmetic mean value of a number of things is the sum 
 of them divided by their number. If y denote the length of 
 one such ordinate P N, and 2 y the sum of all such ordinates 
 when ruled at n equal and exceedingly small intervals, each 
 of length dx, then the average value of these infinitely 
 numerous ordinates is 
 
 ndx 
 
 ndx
 
 SIMPLE PERIODIC CURRENTS. 99 
 
 but the sum of all such quantities as ydx, or PN . N N' is 
 the sum of all the areas of the little rectangular slips into 
 which these infinitely numerous ordinates divide the area 
 bounded by the curve and X X' , and n d x is the length X X' ; 
 hence we have 
 
 mean ordinate^, 
 length X X' 
 
 The area X Y X' is obtained by integrating the equation to 
 the curve. Calling the maximum ordinate Y, A, and the 
 distance X N, x, the unit being the radius of the cylinder of 
 which X X' is the semi-circumference, we have as the equa- 
 tion to the curve 
 
 y = A sin x, 
 
 and therefore I ydx, or A / sin x d x, 
 
 between the limits and TT, is the value of the area of the 
 curve. But 
 
 A.lsmxdx= -A cos x, 
 
 and this, between the limits x and x = TT, is equal to 2 A. 
 On the same scale, the length 
 
 hence, the average value of the infinitely numerous and equi- 
 
 O A 
 
 spaced ordinates is , or the average ordinate of a simple 
 
 a 
 
 sine curve is equal to - times the maximum ordinate. The 
 value of 
 
 - = 0-6369. 
 
 
 
 Therefore, the average value of the equi-spaced ordinates of a 
 simple periodic curve, or the true mean ordinate, is 0-6869 of 
 the maximum ordinate ; and, if the current or electromotive 
 force varies according to a simple periodic law, the true mean 
 current or the true mean E.M.F. is 0-6369 of the maximum 
 cun-ent or E.M.F. during the phase. 
 
 We have here made use of one simple integration, and it is 
 generally easier to master the elements of the infinitesimal 
 calculus than to construct or follow proofs which aim at 
 
 H2
 
 100 SIMPLE PERIODIC CURRENTS. 
 
 avoiding its use. We shall, however, indicate how the value- 
 of this mean ordinate may be found from first principles. If 
 we call the length of the base line X X' I, and divide it into n 
 equal and very small parts of length 8 x, then n8x=L Erect 
 at each interval an ordinate whose height is y, then the equa- 
 tion to the curve is y = A sin 'j x, where x is, as before, the 
 
 distance X N. The mean value, M, of the ordinate is the sum 
 of all the values of the ordinates divided by their number, or 
 
 is equal to i (y l + */ 2 + y 3 + &e.). 
 M= 1 
 
 The sum of the sine terms in the bracket is known by 
 trigonometry to be equal to 
 
 Hence 
 
 . TT bx 
 sm T2 
 
 which may be otherwise written 
 ~1 &X \ 
 
 ,, A I ~1 /n~8x irSx\ . 
 
 '"aKsnv* (-IT-IT) " 
 
 When n becomes infinite and 8 x becomes zero, n 8 x remains- 
 still equal to I ; hence the above expression in this case- 
 reduces to the following : 
 
 for the value of is 2 when h becomes zero. 
 
 sin-
 
 SIMPLE PERIODIC CURRENTS. 101 
 
 Accordingly the mean value of the ordinates, when they are 
 infinite in number and equi-spaced, is - times the magnitude 
 
 of the maximum ordinate. 
 
 9. The Value of the Mean of the Square of the Ordinates 
 of a Simple Periodic Curve. We require in the next place to 
 find the value of the mean of the square of the ordinates to 
 the same curve, assuming them to be equi-distant and infinite 
 in number. If y v ?/ 2 , &c., are the ordinates, and n the number, 
 "we require to find the value of 
 
 the value of any ordinate being, as above, 
 y = A sin y x. 
 
 If X X' or I is divided into n intervals, each equal to B x, so 
 that n 8 x = I, we have to find the value of 
 
 sin 2 + sin 2 ^ 8 x + sin 2 - 
 
 but, sinco sin 2 # = -fl -cos20Y 
 
 the series in the bracket can be replaced by 
 
 A ALL A A i 
 
 for n terms. Hence the mean value M is 
 
 M = -- -VcosO+ cos- 
 n 2 2;t\ / 
 
 for n terms. 
 
 The cosine series forms a progression of terms which begins 
 with unity, since cosO = l, and passes down through zero 
 to 1, and then up from 1 through zero to unity again, 
 
 for e 
 
 "when n8x l and S x becomes infinitely small.
 
 102 SIMPLE PERIODIC CURRENTS. 
 
 Since the angles are in arithmetic progression, we can pick 
 out from this series pairs of cosine terms such that they are 
 equal in magnitude but opposite in sign, and, when taken 
 pair and pair, cancel each other out. The sum of the cosine 
 series in the bracket is thus equal to zero, and, therefore, 
 
 (22) 
 
 that is, the mean of the values of all the ordinates squared, 
 taken equi-distant and infinite in number, is half the square of 
 the maximum value. 
 
 We have, therefore, this result : If the current in a linear 
 conductor varies in strength and direction in a manner which 
 geometrically would be represented by the ordinate of a simple 
 
 sine curve, the true mean value of the current strength is - 
 
 or 0-637 of its maximum value, and the mean value of the 
 square of the current strength, taken at equal and very small 
 intervals, is half the value of the square of the maximum 
 value. 
 
 Since 1 = 0-637 and = = 0-707, and since the difference 
 V2 
 
 = 0-07, the true mean current is less than the square root of 
 the mean of the squares at each instant by an amount which, 
 is very nearly 10 per cent, of the latter. 
 
 If we proceed by the ordinary rules of the integral calculus, 
 we can find the value of the mean of the squares of the ordi- 
 nates of the sine curve as follows : 
 
 Let y = A sin x 
 
 be the curve ; then 
 
 y 2 = A 2 sin 2 x 
 
 The mean value of the square of the ordinate between tha- 
 limits and 7r-that is, during the half -wave length is 
 
 If-
 
 SIMPLE PERIODIC CURRENTS. 103 
 
 Therefore, M^^T (I~cos2n)dx, 
 
 Hence we reach the same result as above. In order to avoid 
 repeating constantly the clumsy phrase the square root of the 
 mean of the squares of all the equi-spaced ordinates of a curve, we 
 may call this, in speaking, the mean-square value of the 
 ordinate, and express it by the symbol ^mean 2 . Hence, 
 
 ^/mean 2 y stands for the above particular kind of mean of y. 
 
 In practice, in alternating-current work, we hardly ever 
 require to concern ourselves with the true mean of the 
 ordinates of a simple or complex periodic curve. Chiefly 
 we require to know or find the square root of the mean of 
 the squares of the ordinates of a periodic curve taken at 
 equi-distant positions throughout the period. Hence the 
 
 V mean 2 value of the ordinate of a simple periodic curve is 
 equal to the quotient of the maximum ordinate by the Jfi 
 for the mean of the squares of the equi-distant ordinate is 
 
 equal to the value of , as shown above, and hence the 
 
 _ A 
 
 J mean 2 value is -T=. Since N/2 = 1-414 nearly, we see that 
 
 the maximum ordinate of a simple sine curve is VI times 
 the -J mean 2 ordinate. In the practical measurement of alter- 
 nating currents, the value given by the instruments is nearly 
 always the Vmean 2 value of the instantaneous values 
 throughout the period. 
 
 10. Derived Curves. Let the curve in Fig. 42 represent 
 the complete period of a simple periodic curve of which the 
 
 equation is y = A sin x. Let P be any point on the curve. 
 
 ThenPN = i/, OY = A, XX' = J. At P draw a tangent P T to 
 the curve, and let it meet the datum line at T.
 
 104 
 
 SIMPLE PERIODIC CURRENTS. 
 
 We shall call the trigonometrical tangent of the angle P T N, 
 
 P N 
 the slope of the tangent at the point P, hence - r =. = the slope. 
 
 If two points, PP' (Fig. 43), are taken on the curve very 
 near together, and a secant, P' P T, is drawn through them, 
 
 FIG. 42. 
 
 this secant will become a tangent when the points P P' move 
 
 ~p' ivr 
 up into contact. The ratio of f will then, in the limit, 
 
 Jt JVJL 
 
 become the slope of the tangent. If now X N = x - , and 
 
 43. 
 
 X N' = x + , and P M = N N' is 8 x, we have the equations 
 
 and 
 
 The quantity in the bracket is identically the same as 
 2 cos ^ x sin 5 ;
 
 SIMPLE PERIODIC CURRENTS. 
 
 105 
 
 vxrt,)fm 
 
 L I "2 -" 
 
 When 8 x is made infinitely small, the quantity in the square 
 brackets is unity, and we have 
 
 If we plot a curve whose ordinates at any point are the slope 
 -of the primal curve at the corresponding points, the above 
 
 FIG. 44. 
 
 equation shows us three things first, that it is a sine curve 
 or simple periodic curve of the same type as the curve from 
 which it is derived ; second, that its maximum value is 
 
 - times the maximum value of the original ; and third, that its 
 
 zero ordinate corresponds to the maximum one of the original, 
 .and vice versa. 
 
 In Fig. 44 the firm line curve is a curve of sines 
 
 y = A sin ^ x ; 
 
 the dotted line is a curve of sines, whose ordinate QN at 
 any point represents the slope of the tangent at P on the 
 original curve. Accordingly, at Y, where the original curve is 
 -at its maximum, and the slope of its tangent is zero, the
 
 106 SIMPLE PERIODIC CURRENTS. 
 
 derived curve cuts the datum line, or has its phase shifted 
 90deg. backwards relatively to the original- curve. In the 
 language of the differential calculus, the firm line curve is 
 
 the plotting of the curve y = A sin x, and the dotted curve is 
 
 the plotting of -^ as ordinates for the same abscissae. 
 
 We may regard it from another point of view. Let the 
 simple sine curve be supposed to be generated or marked 
 out by a tracing point, P, which moves to and fro along a line 
 P N P' with a simple harmonic motion, whilst the point N 
 moves uniformly along a straight line X X'. (See Fig. 45.) 
 
 FIG. 45. 
 
 Draw as before the dotted curve whose ordinate Q N at any 
 point represents the slope of the firm curve at the correspond- 
 ing point P. Then the magnitude of N Q will represent the 
 rate at which the ordinate P N is increasing or decreasing. 
 For, in this case, distances such as XN, measured along 
 the mean line, are proportional to time, and hence N makes 
 a small movement forward in a small time d t ; there is a 
 corresponding decrease in the ordinate PN, which we may 
 
 denote by dy, and accordingly c -li represents the rate of 
 
 d t 
 
 decrease of P N. If the small forward movement of N causes 
 N to advance through a space d x, d x is proportional to d t, 
 
 as the motion is uniform, and accordingly ( -V- is proportional 
 
 d x 
 
 to ^; hence -- is at any instant graphically represented by 
 d t at 
 
 the slope of the tangent at P that is, by the ordinate Q N. 
 The dotted curve represents, therefore, the ruie of change of the
 
 SIMPLE PERIODIC CURRENTS. 107 
 
 ordinates of the firm curve at that same instant. We shall 
 call the dotted curve the derived curve. 
 
 If the ordinates of the original curve represent the instan- 
 taneous values of a simple periodic current flowing in a con- 
 ductor, then the ordinates of the curve called above the 
 derived curve will represent the rate of change of that current 
 at the corresponding instants. The derived curve is a similar- 
 curve, but shifted backwards by one quarter of a wave length. 
 
 11. Inductance and Inductive Circuits. Before we can 
 proceed to discuss the laws of periodic current flow in circuits 
 of various kinds, we must call attention to some of the funda- 
 mental properties of electric circuits. Every electric circuit in, 
 which a flow of electricity, whether continuous or periodic, 
 can take place possesses three primary qualities, viz., Resistance, 
 Inductance, and Capacity. The resistance of the circuit is a. 
 quality of it, in virtue of which a dissipation of energy takes 
 place when an electric current flows through it. This specific 
 quality is affected by change of temperature and by other 
 alterations of physical condition. In the case of pure metals 
 it has been shown* that, if the metal could be reduced to the 
 absolute zero of temperature, its electrical resistance would 
 vanish. 
 
 It is generally assumed that, apart from the change due to 
 temperature or other altered physical conditions, the electrical 
 resistance of a body is a constant quantity, which is independent 
 of the current flowing through it. It is evident from experience 
 that this is approximately, even if not accurately, the case. It 
 would require very careful and extensive experiments before 
 we should be entitled to say that the resistance of any circuit 
 of any metal, when all corrections have been made for change 
 of volume and temperature, is exactly the same when a thousand 
 amperes are flowing through it as when one-thousandth of an 
 ampere is flowing through it. Still less can we generalise and 
 lay it down as absolutely and universally true. Careful experi- 
 ments made by Prof. Chrystal at the Cavendish Laboratory 
 (B.A. Eeport, 1876) showed that the resistance of a metallic 
 circuit of one ohm is not different for currents of one ampere 
 and for infinitely small currents by as much as 10" 12 part. 
 
 * Dewar and Fleming, fhil. Mag., Sept., 1?93.
 
 103 SIMPLE PERIODIC CURRENTS. 
 
 There is, therefore, a strong prohability that the specific 
 electrical resistance of a body is a quality which is not 
 dependent upon the current flowing through it, but is only 
 aftected by the temperature and physical condition of the 
 body. According to Joule's law the rate of dissipation of 
 energy when a current flows through a conductor is propor- 
 tional to the square of the strength of the current. The total 
 resistance of a circuit may, therefore, be numerically defined 
 by the rate at which energy is dissipated by it when unit 
 current flows in that circuit. In the practical units a circuit 
 which, when traversed by one ampere of current, dissipates 
 energy at the rate of one joule per second, or has a dissipation 
 rate of one watt, is said to have a resistance of one ohni. 
 The energy required to heat one gramme of water one degree 
 centigrade in the neighbourhood of its maximum density is 
 4-2 joules. Since the rate at which energy is being dissipated 
 at any instant in a circuit is measured by the numerical value 
 of the product of the strength of the current flowing in it and 
 the fall in potential down that conductor, it follows that the 
 resistance of the circuit, or of any part of it, is also measured 
 by the ratio between the numerical values of the fall of 
 potential down the circuit or down that part of it and the 
 current strength in that circuit, provided that the inductance 
 of that circuit is negligible. The resistance of a circuit is, 
 therefore, the energy-dissipating quality of it, and the specific 
 resistance of any material is the resistance of one cubic unit of 
 it between opposed faces of the cube. 
 
 In addition to the quality of resistance every circuit possesses 
 also inductance. This quality of a circuit is one in virtue -of 
 which a current of finite value cannot be instantaneously 
 produced even in a circuit of negligible resistance by a finite 
 electromotive force, and when produced cannot be instanta- 
 neously destroyed. On account of the fact that all bodies 
 possess mass, and therefore inertia, a finite force cannot 
 generate a finite velocity in any material body in an in- 
 finitely small time. We see this fact exemplified in every 
 falling body or starting train. A time element due to inertia 
 comes into play which causes the motion of the mass to 
 be acquired gradually, even under the action of a constant 
 finite forc3. Experience shows that in all electric circuits
 
 SIMPLE PERIODIC CURRENTS. tofr 
 
 there is a physical quality present which is related to current-, 
 and electromotive force, just as the mass of a material body is- 
 related to velocity and dynamical force. In virtue of the mass 
 of a body time is required for a finite moving force to generate 
 a finite velocity, and in virtue of inductance of a circuit time 
 is required for a finite electromotive force to generate a finite 
 current. The inductance of the circuit bestows on it a quality 
 which may be called its electrical mass or electrical inertia. The 
 mass of a material body enables it in some way to become the 
 vehicle of energy when in motion, and this energy of motion 
 is called its kinetic energy. This kinetic energy is capable of 
 being removed from the moving body, and the moving body 
 can be brought to rest again only by taking away from it the 
 kinetic energy it possesses as a whole, and transferring that 
 energy to some other body or bodies, or to the molecules of 
 the body itself. In like manner the inductance of a cir- 
 cuit may be said to cause it to be capable of being the 
 vehicle of electrical energy when traversed by an electric 
 current. A current cannot be instantaneously produced in 
 finite value by any electromotive force, and when produced 
 cannot be destroyed except by transforming that energy into 
 some other form. Hence we have a very complete dynamical 
 analogy between material bodies set in motion by what may 
 be called materio-motive force and the flow of electric currents 
 in circuits which possess inductance under the action of 
 electromotive force. 
 
 These qualities may be compared as follows : 
 Motion in matter corresponds to Electric flow in circuits, 
 
 Mass=m ,, Inductance=L, 
 
 Velocity =v ,, Current strength =i, 
 
 Momentum=m, {^^^f^ m men " 
 
 Kinetic energy oH f Electromagnetic energy 
 
 motion = iw v 2 . . . f " j = |L i 2 , 
 
 Rate of change oA ( Bate of change of electro- 
 
 momentum =m^ \ 1 a** 11 ? 1 ? momentum 
 dt) ( = L rf*. 
 
 dt
 
 110 SIMPLE PERIODIC CURRENTS. 
 
 The force acting on a body which is being expended in 
 'making change of momentum is numerically measured at any 
 instant by the rate of change of its momentum existing 
 ;at that instant. So also the electromotive force which is 
 being exerted to produce change of current strength or 
 change of electro -magnetic momentum in a circuit is measured 
 .at any instant by the rate of change of electro-magnetic 
 momentum. 
 
 There is an exact analogy between a heavy body being set 
 in motion against inertia and friction and between an electric 
 current being generated against inductance and resistance. 
 For in the first case one part of the impressed force is being 
 expended to overcome friction and the remainder to accelerate 
 "the mass against inertia, and in the second case one part of 
 4he impressed electromotive force is expended to overcome 
 resistance and the remainder to increase the current against 
 electrical inductance. 
 
 A circuit possessing inductance is called an inductive circuit, 
 and a circuit whose inductance is negligible is called a non- 
 inductive circuit. A truly non-inductive circuit can no more 
 be realised in practice than a mass-less material body. The 
 clear recognition that an electric circuit possessed a quality in 
 virtue of which kinetic energy is associated with it when a 
 current is flowing through it was first reached by Joseph 
 Henry. In 1832 Henry made the observation that if the poles 
 of a single galvanic cell are united by a short thick wire, then 
 on breaking the circuit there is little or no spark ; but if the 
 uniting wire is a very long one, and, better, if it is coiled into 
 a spiral, then there is a considerable spark at the contact on 
 opening the circuit. In 1835 he expanded and continued 
 these observations,* and noticed that if the wire is coiled round 
 an iron core, and thus forms an electro-magnet, the spark 
 and shock at breaking circuit are still more marked. Henry 
 still further elaborated these observations in 1835. t Later 
 still Faraday attacked the same problem, and devoted to its 
 consideration the Ninth Series (1048) of his "Electrical 
 Kesearches." 
 
 * Journal of Franklin Tnstitute, March, 1835, Vol. XV., pp. 169-170. 
 t Phil. May., 1840; see alto Scientific Writings of Joseph Henry, 
 r pp. 87-97.
 
 SIMPLE PERIODIC CURRENTS. Ill 
 
 The chain of experiments which led to this inquiry was 
 apparently started by a question addressed to Faraday by a 
 Mr. Jenkin, one Friday evening, at the Eoyal Institution, as 
 to the reason why a shock was experienced when a circuit 
 containing an electromagnet was broken, the observer retaining 
 in his two hands the ends of the circuit, but no shock was felt 
 if the circuit contained neither magnet nor long coils of wire. 
 Faraday seems speedily to have arranged an organised attack 
 on the subject, and to have returned from his investigation 
 burdened with the spoils of victory in the shape of the 
 following facts : 
 
 1. If a battery circuit is closed by a short thick wire, then, 
 although there may be a very strong current existing in this 
 wire, on breaking contact at any point little or no spark is 
 seen, and if the two ends of the circuit are grasped in the two 
 hands, and the interruption takes place between the hands, 
 then little or no shock is experienced. 
 
 2. If a very long wire is used instead, then, although the 
 absolute strength of the current may be less, yet the spark 
 and shock at interruption are more manifest. 
 
 3. If this length of insulated wire is coiled up into a helix 
 on a pasteboard tube, then, although the length of wire and 
 strength of current are the same, yet the spark and shock are 
 still more marked. 
 
 4. If the above helix has an iron core placed in it, both 
 these effects are yet more exalted. 
 
 5. If the same length of wire is doubled upon itself, being, 
 however, insulated, then the effects nearly vanish, and, whether 
 straight or coiled, this doubled wire with current going up one 
 side and down the other is no better in respect of spark and 
 shock on interruption than a very short wire. 
 
 The first observation which Faraday makes upon the above 
 results is that electricity would seem to circulate with some- 
 thing like momentum or inertia in the wire, and that the 
 greater the length and strength of the current, so much the 
 more power is there to run on and jump over the obstacle 
 presented by the first thin layer of air which is introduced 
 between the contacts as they are separated, giving rise to a 
 spark. He saw, however, at once that, since the form of this 
 circuit is an important factor, the idea of inertia in tfte current
 
 112 SIMPLE PERIODIC CURRENTS. 
 
 itself was fallacious, or else the mere doubling the wire could 
 not nullify all the effects. He did not at that time see that 
 the idea of momentum was exceedingly appropriate, but its- 
 allocation in the electric current itself was wrong. 
 
 The observation, however, which led him to a consistent 
 theory was as follows. A bobbin was prepared, having wound 
 on it two insulated wires, 1 and 2. The ends of 2 being left 
 unconnected, the wire 1 was used to complete a circuit, and 
 gave a spark on interrupting a current traversing it. As we 
 have seen (Chap. I.), Faraday* had three years previously 
 established the fact that the commencement and cessation of 
 a current in one circuit would produce in another circuit, if 
 closed, an inverse or a direct induced electric wave or transi- 
 tory current. Now, on closing the second circuit through a 
 galvanometer or loose contact, and interrupting a steady 
 current flowing in the first circuit, he found that when circuit 
 2 was completed, so that an induced or secondary current 
 could be generated in it, little or no spark happened at thu 
 place of interruption in 1 ; but, if circuit 2 was opened, then 
 the interruption of circuit 1 gave rise to a bright spark at the 
 contact. Faraday therefore inferred that when circuit 2 was 
 closed adjacent to circuit 1, the current in 1 exerted its- 
 full inductive effect in generating secondary currents in 2 - r 
 but that, if circuit 2 was open, then, there being no adjacent 
 conductors, the current in 1 expended its inductive effect in 
 producing induced currents in its own circuit, and this self- 
 vnduction manifested itself by temporarily diminishing the 
 strength of the current at starting and assisting or increasing 
 it momentarily at the interruption. He was thus able, from 
 this point of view, to picture to himself the circuit of 1 as 
 occupied by a steady current, superimposed on which was 
 another current he called the inverse extra current, lasting but 
 a very short time at starting the steady current ; and a direct 
 extra current which flowed on and produced the effects of the 
 spark or shock at the interruption of the circuit. These extra 
 currents, or currents of self-induction, he found could be 
 removed from the circuit itself and exhibited in a neighbouring 
 circuit when that adjacent circuit was closed, and so fitted to 
 be the seat of induced currents due to the mutual induction of 
 * Faraday's " Exp. Res.," 1,090.
 
 SIMPLE PERIODIC CURRENT*. 113 
 
 the primary on this secondary circuit. Faraday then placed 
 this theory under test by requiring it to furnish an explanation 
 of the following experiment : 
 
 M and N (Fig. 46) were two mercury cups* which formed 
 the terminals of three circuits a battery circuit, B, a galvano- 
 meter circuit, G, and a circuit consisting of an electromagnet 
 or helix, C. The needle of the galvanometer was blocked in 
 such a way that the tendency to deflect under the steady 
 current was prevented and the needle kept at zero ; but it 
 was free to deflect in the opposite direction under an oppositely 
 directed current. This being the case, the raising of the 
 battery wires out of the mercury cups was accompanied by a 
 violent " kick " or deflection of the needle in the free direction. 
 
 6 
 
 JL 
 
 ' rr\ 
 
 N I G 
 
 I 
 
 \mmsmj 
 
 FIG. 46. 
 
 The action could clearly be explained by supposing that after 
 the electromotive force of the battery is removed from the 
 coil C, the current in it does not at once stop dead, but runs 
 on like a heavy body and makes a backwash of current 
 through the galvanometer in the direction from M to N. An 
 illustration of the electromagnetic inertia of a coil on inter- 
 rupting the current may be shown in a more modern form, 
 thus : Let E (Fig. 47) be an electromagnet, and let L be an 
 incandescent lamp of which the resistance is very large com- 
 pared with that of E. Let S be a few cells of a storage 
 battery supplying current, and let K be a key. On depressing 
 the key the current flows both in the magnet and in the lamp 
 * Faraday s " Exp. Res.," Vol. L, 1,079.
 
 114 
 
 SIMPLE PERIODIC CURRENTS. 
 
 arranged as a shunt on the magnet. This current, however, 
 is, by assumption, not strong enough to illuminate the lamp. 
 On raising the key and stopping the steady current through 
 the lamp the electric inertia of the coil sends a momentary 
 powerful current through the lamp, which causes it to flash 
 up. Again, if a small shunt-wound dynamo be occupied in 
 supplying current to a few incandescent lamps, and the two 
 hands be employed to raise simultaneously the brushes from 
 the armature, the momentary rush of current from the field- 
 magnet due to this extra current will disagreeably impress the 
 phenomenon upon the mind of the observer if the experiment 
 
 is made with any but a very small dynamo. With a large 
 dynamo this experiment is very dangerous to perform. 
 
 Neither of these experiments is well fitted to illustrate 
 the extra current at the closing of the circuit or the effect 
 of electric inertia on starting the current in a helix. The 
 arrangement most suited to exhibit the whole effect is that of 
 the differential galvanometer as used by Edlund, or that 
 employing Wheatstone's bridge, due to Maxwell. 
 
 In Edlund's arrangement* a differential galvanometer is 
 employed, of which the two coils GX G 2 are so placed and 
 * See Poggendorff's Annalen, 1849.
 
 SIMPLE PERIODIC CURRENTS. 
 
 115 
 
 n-ound that when equal and oppositely -directed currents are 
 sent through them the needle is unaffected. The coils are 
 then connected, as shown in Fig. 48, to a battery, B, an 
 electromagnet or helical coil, L, and a wire, E, of equal 
 v resistance to L, but wound double. The galvanometer coils 
 are so connected to the circuits L and E that when the steady 
 current from the battery flows through the divided circuit 
 the needle remains at zero. On closing the circuit it is then 
 found that the needle makes a sudden deflection in a direction 
 indicating a brief current passing in coil G. 2 , and on breaking 
 the circuit it makes another deflection, indicating a transitory 
 -current passing through Gj. In other words, the balance is 
 
 C, 
 
 FIG. 48. 
 
 destroyed at the instant of breaking and making, but restores 
 itself again when the currents become steady. This experi- 
 ment, therefore, most clearly shows that the electromagnetic 
 helix L, although of exactly the same electrical resistance as the 
 coil E, differs from it in possessing a peculiar quality, which it 
 has in virtue of being in the form of a coil or helix, and to 
 which the name self-induction or inductance has been given. 
 We are able to define this term as follows : The self-induction 
 or inductance of a circuit is, speaking generally, a quality of 
 it in virtue of which a finite and steady electromotive force 
 applied to it cannot at once generate in it the full current due 
 to its resistance, and when the electromotive force is with- 
 drawn time is required for the current strength to fall to zero.
 
 116 
 
 SIMPLE PERIODIC CURRENTS. 
 
 It miist, however, be noticed that not only does the inductance 
 of a circuit depend upon tb.3 geometrical form of the circuit, but 
 it depends upon the magnetic permeability of the region which 
 surrounds the circuit and on the magnetic permeability of the 
 conducting circuit itself. If, in the arrangement with the dif- 
 ferential galvanometer, the steady balance is obtained by using 
 a copper wire helix wound on a cardboard tube and balanced 
 against a non-inductive but equal resistance, it is found that 
 the insertion of a soft iron core into the helix greatly increases 
 the "kick" on making contact, indicating the passage of a 
 greater quantity of electricity through the opposite galvano- 
 meter coil, and therefore a greater delay in the time of 
 establishing the steady balance. 
 
 Maxwell's method of exhibiting the effect of inductance is a 
 preferable arrangement. 
 
 Four conductors are arranged in a rectangle joining the 
 points a, b, c, d, and the diagonals are completed by a galvano- 
 meter and battery (Fig. 49). P, Q and E are non-inductive 
 resistances, and E is an electro-magnetic helix. If E and E 
 are equal in actual resistance and P : Q = E : E, then the 
 permanent closing of the battery circuit does not finally affect 
 the galvanometer indication, and these circuits (battery and 
 galvanometer) are then said to be conjugate circuits. 
 
 When, however, the battery key is first put down the 
 galvanometer receives an impulse in one direction ; when the
 
 SIMPLE PERIODIC CURRENTS. 117 
 
 key is kept down the galvanometer soon returns to zero, or to 
 its original position. On raising the key the needle receives 
 an impulse in the opposite direction. Examination of these 
 impulses shows that if the current enters the quadrangle at d, 
 on closing the key the potential rises at b faster than it does at 
 a, and that on raising the key the potential dies down at b 
 faster than at a; but that, if the "balance" is properly ob- 
 tained, the points a and b reach finally the same potential 
 when the key is kept closed. 
 
 An electromagnetic helix with or without a core of soft 
 iron, behaves itself, therefore, towards an external electro- 
 motive force to which it is submitted as if it had an internal 
 counter-electromotive force which gradually disappears allow- 
 ing the full current due to its resistance to be established in 
 it more or less slowly, and behaves also, at the removal of 
 this external electromotive force, as if a direct internal electro- 
 motive force suddenly made its appearance within it, this 
 also gradually dying away. 
 
 The reader will see, therefore, that every electric circuit can 
 not only dissipate electric energy in virtue of its resistance, 
 but can conserve energy in virtue of its inductance. The 
 resistance is measured by the rate of dissipation of energy 
 which takes place when unit current (one ampere) flows 
 through the circuit, and this rate of dissipation varies as the 
 square of the current strength. The inductance is measured 
 by the electromagnetic momentum associated with the circuit 
 when unit current flows in it. Since, dynamically considered, 
 the rate of change of momentum is a numerical measure of 
 the force producing it, we must define electromagnetic 
 momentum as that quantity the rate of variation of which 
 numerically measures the electromotive force. We have 
 already seen that if lines of magnetic induction (or force) per- 
 forate through and are linked with a circuit, then atiy 
 variation of the number of these lines of induction or linkages 
 gives rise to an induced electromotive force in the circuit 
 equal in numerical magnitude to the rate of change of the 
 included lines of induction. When an electric circuit is re- 
 moved from all other circuits and magnets and is traversed by 
 a current, the turns of this circuit are linked with and include 
 the lines of magnetic induction created by itself. Hence we
 
 118 SIMPLE PERIODIC CUEEEXTS. 
 
 are able to connect the quantity we have called the electro- 
 magnetic momentum with the number of lines of magnetic 
 induction which are linked with the circuit and which are 
 created by the current flowing in that circuit. If a unit cur- 
 rent is flowing in any circuit, there are a certain number of 
 lines of magnetic induction at any instant linked with or 
 perforating that circuit, and the number of these linkages 
 defines the inductance of that circuit. 
 
 12. Electromagnetic Momentum. The justification for the 
 use of the term electromagnetic momentum is as follows : 
 When a heavy body is in motion it possesses at any instant 
 momentum, in virtue of its inertia. Numerically the momentum. 
 of a heavy particle is obtained by taking the product of its 
 mass and its velocity, each measured in appropriate units. 
 The tune rate of change of a body's momentum in any direc- 
 tion is, by the second law of motion, the measure of the force 
 acting upon it in that direction, or, in the notation of the 
 calculus, 
 
 dt 
 
 We have seen that the induced electromotive force in a> 
 circuit depends on the tune rate of change of the magnetic 
 induction through it, and hence the magnetic induction at 
 any instant through a circuit bears the same relation to the 
 induced electromotive force in it that a body's momentum does 
 to the mechanical force acting 011 it. Maxwell has accordingly 
 employed the term electromagnetic momentum to represent the 
 flux of magnetic induction or the number of lines of magnetic 
 induction passing through a circuit, because it is upon the 
 rate of change of this quantity that the induced electromotive 
 force depends. Faraday very early recognised that induction 
 effects depend on a change of some quantity. He makes frequent 
 mention of the electrotonic state, and he spoke of a conductor 
 in a magnetic field, when traversed by lines of induction, as 
 in the electrotonic state, and he considered that when the 
 electrotonic state was either assumed or disappeared its com- 
 mencement or end was marked by the production of the 
 induced electromotive force. Maxwell identified Faraday's 
 electrotonic state with the total induction passing through.
 
 SIMPLE PERIODIC CURRENTS. 119 
 
 the circuit or linked with it. Consider, then, the operations 
 which go on when a conducting circuit say a simple 
 loop of wire is subjected to a steady electromotive force. 
 The instant that force is applied, a current begins to flow in 
 the circuit ; the instant that current begins, lines or rings of 
 induction spread out from the circuit ; and the loop at any 
 instant encloses a certain number of lines of induction which 
 are increasing at that instant at a certain rate. A counter or 
 opposing electromotive force exists in that circuit numerically 
 equal to the time rate of increase of this induction. In circuits 
 which do not enclose or surround iron or other magnetic metal, 
 or which are immersed wholly in a medium of constant perme- 
 ability, the magnetic induction at any point in the neighbour- 
 hood of the circuit is numerically proportional to the strength 
 of the current at that instant flowing in the circuit. This is the 
 fact which lies at the root of the operation of most galvano- 
 meters, viz., that the field at any point in the neighbourhood 
 of the coil is simply proportional to the strength of the cur- 
 rent flowing in the coil. If, then, i represent the strength of 
 the current at any instant in the circuit, and L be a certain 
 constant quantity such that Li represents the induction 
 through the coil or circuit due to the current i in it, then L i 
 is the measure of the electromagnetic momentum of that 
 circuit. This quantity L is a coefficient which, in this case, 
 is dependent only upon the geometrical form of the circuit, 
 and, under the assumption that there is no magnetic material 
 in or near the circuit through which the lines of induction 
 can pass, it is a constant quantity. 
 
 This quantity L is called the constant coefficient of self- 
 induction of the circuit, or, more shortly, the inductance of the 
 circuit. 
 
 The inductance, or the coefficient of self-induction, is thus 
 defined : In the case of circuits conveying electric currents 
 which are wholly made of non-magnetic material and wholly 
 immersed in a medium of constant magnetic permeability, 
 the total magnetic induction through the circuit per unit of 
 current flowing in that circuit when removed from the neigh- 
 bourhood of all other magnets and circuits is the numerical 
 measure of the inductance or of the coefficient of self-induction. 
 Otherwise, the ratio of the numerical values of the electro-
 
 120 SIMPLE PERIODIC CURfiESTS. 
 
 magnetic momentum of such circuit and the current flowing 
 in it when totally removed from all other currents and magnets 
 is the numerical value of the inductance of that circuit. 
 
 13. Electromagnetic Energy. Let us confine our atten- 
 tion first to one circuit of constant inductance or self-induction 
 in which a current is being generated by a constant electro- 
 motive force applied to it. Each increment of strength of the 
 current creates an electromotive force opposing the impressed 
 or external electromotive force. Hence this external electro- 
 motive force has to do work against an opposing force of its 
 own creating all the time the current is rising in strength. 
 When a mechanical force overcomes a resistance through a 
 certain distance, mechanical work is being done, and, accord- 
 ingly, we may ask What is the electromotive force doing all 
 the while it is increasing a current against an opposing electro- 
 motive force ? The answer is, it is doing electrical work. 
 The result of causing a current having a strength i at any 
 instant to flow for a small time, </ 1, against an opposing 
 E.M.F. at any instant equal to e, is that a quantity of work, 
 represented by e i il t, is done in- the time d t . If c is th^ 
 instantaneous value of the opposing electromotive force of 
 self-induction, it is measured at any instant by the rate of 
 
 change of electromagnetic momentum L ?', or by L . 
 
 d t 
 
 Hence the work done in raising the current from a strength 
 i to a strength i+di against the counter-electromotive force of 
 
 self-induction is 'L idt = 'Lidi, and if this is integrated 
 d t 
 
 between limits zero and I, we get the whole quantity of work 
 so done against self-induction alone in bringing up a current 
 from zero to its full value, I, in the conductor, but 
 
 j\idi= 
 
 Exactly in the same way it may be shown that the work 
 done in bestowing a velocity V upon a mass M is measured by 
 the quantity | M V 2 . 
 
 The total work done against the electromotive force of self- 
 induction in creating a current I in a conductor of constant
 
 SIMPLE PERIODIC CURRENTS. 121 
 
 inductance L is, then, numerically equal to half the square of 
 the final current strength, multiplied by the value of the con- 
 stant inductance or coefficient of self-induction. 
 
 The equivalent of this work is found in the magnetic field 
 formed round the conductor, and hence the formation of a 
 magnetic field represents so much energy, measurable in foot- 
 pounds per cubic inch, or in any other similar units, such as 
 ergs or kilogrammetres, per cubic centimetre of field. 
 
 Next let us consider the case of two circuits. Let the con- 
 stant coefficient of self-induction of the first be L, and let it 
 be traversed at any instant by a current i. Let the inductance 
 of the other be N, and let it be traversed by a current i'. Let 
 the coefficient of mutual induction be M. 
 
 The definition of this last quantity is as follows : If both 
 circuits be traversed by unit currents, and if there be no other 
 field than that due to these currents, the number of lines of 
 induction which traverse loth circuits, or are linked with both 
 circuits, is called the constant coefficient of mutual induction. 
 It will be a quantity constant for a given form and position of 
 the two circuits on the assumption that the lines of induction 
 flow in a medium of constant magnetic permeability. Hence, 
 if we consider the work done, d E, in raising the currents 
 i and i' by small increments, d i and d i', in a small time, d t, 
 we find it consists of four parts a part, liidi, representing 
 work done by the current i against its own counter-electro- 
 motive force, and a similar part, Ni' di', for the other circuit, 
 then a portion, ~Midi', representing the work done by the 
 current i in its own circuit against the induced electromotive 
 force, due to the increment of the current i' in the other, and 
 lastly, a similar part, M i' d i, for the second circuit. Hence, 
 we have 
 
 d^ = ltid t + M i d i' + M i' d i+N t' d i'. 
 
 Integrating this between the limits zero and I for one circuit, 
 and zero and I' for the other, we find the whole energy repre- 
 sented by the two currents I and I' flowing in the circuits 
 to be 
 
 MH' + iNr a . . . . (24) 
 
 The electro-kinetic energy is said to be a quadratic function 
 of the currents and the inductances.
 
 122 SIMPLE PERIODIC CURRENTS, 
 
 14. The Unit of Inductance. The Henry. The practical 
 unit of inductance is called one henry. The henry is the 
 unit of inductance which is in consistent relation with the 
 ohrn, the volt, the ampere, the watt, and the joule. A cir- 
 cuit has an inductance of one henry when there are 10* 
 C.G.S. lines of magnetic induction linked with the circuit. 
 or when there are 10 9 linkages of current and magnetic lines 
 of induction, under the condition that one ampere of current 
 traverses the circuit, and that no other lines of induction than 
 those due to itself perforate or are linked with the circuit. 
 If the circuit is a coiled circuit of wire, and the wire makes 
 n turns round a total numher N lines of magnetic force or 
 induction, then there are nN linkages of circuit and induction. 
 Suppose that we have a circular solenoid formed by winding- 
 thin, closely placed, covered wire on a wooden ring of circular 
 cross section. Let the mean cross section of the circular 
 solenoid be S, and let the induction density in the interior of 
 the solenoid be B, when one ampere is sent through the wire 
 windings. Then there are B S lines of induction in all round 
 the interior of the solenoid. Let there be N turns of wire in 
 all on the ring, then there are N S B linkages of current and 
 magnetic lines of force. The inductance of this solenoid, or, 
 its self-induction measured in henrys, is 
 
 If we consider the above circular solenoid or very long 
 straight solenoid to be wound on a wooden or non -magnetic 
 core, the value of the induction B in the interior is numerically 
 the same as that of the magnetic force in the interior, viz., 
 
 - ^- units, where A is the ampere current in the coil, 
 10 L 
 
 N the number of windings, and L the mean length of the 
 coil. Hence the self-induction of such a coil in henrys is 
 
 ^- N 2 , or is proportional to the square of the total 
 
 number of windings N. 
 
 An enormous number of wire windings are, therefore, 
 necessary to obtain any sensible fraction of a henry of in- 
 ductance in a circuit in which the path of the lines of 
 magnetic force is wholly in air, or in some body of unit 
 magnetic permeability.
 
 SIMPLE PERIODIC CUEEENTS. 123 
 
 In the case of such air or non-ferric magnetic circuits the 
 inductance is a constant quantity which depends only on the 
 geometrical form of the circuit. 
 
 The moment, however, that we introduce an iron core we 
 alter the state of affairs. The inductance is then no longer 
 the same for all values of the induction, because the induction 
 varies with the magnetising force, but not proportionately to 
 it. Hence, we cannot speak generally of the inductance of 
 such an electric circuit when linked with an iron, or partly 
 iron, magnetic circuit, except to define its value corresponding 
 to one particular value of the current. We can, however, 
 always refer to the instantaneous value of the inductance when 
 we have occasion to mention a particular value which it has 
 when varying from instant to instant. For very low or very 
 high degrees of magnetisation, however, the inductance of 
 such a circuit will be constant, but very different. 
 
 The following table taken from figures obtained by Mr. A. 
 E. Kennelly and Prof. Ayrton* will furnish the reader with an 
 idea of the approximate magnitude of the inductances of 
 various well-known instruments, measured in henrys and 
 fractions of a henry : 
 
 Garde w voltmeter about 1 microhenry. 
 
 Ordinary telegraph sounder 25 50 millihenrys. 
 
 Astatic mirror galvanometer, about \ o , 
 
 5,000 ohms.. / 2 henrys. 
 
 Mirror speaking galvanometer, 2,250 \ 3-5 henrys 
 
 Single coils of Morse receiver 93 millihenrys. 
 
 Induction coil (giving 2in. spark) \ 
 
 secondary circuit / 
 
 Shunt dynamo (100 volts, 35 amps.)\ n i, 
 
 armature / 
 
 Field magnets of the above dynamo \ 13'6 h 
 
 bCl1 ' 2 ' 5 hmS re 'l 12 millihenrys. 
 
 15. Current Growth in Inductive Circuits. We see, 
 therefore, that when electric energy is spent on a conductor 
 in the production of a current, in addition to the energy 
 taken up in the performance of any chemical or external 
 
 * See The Electrician, Vol. XXVI., p. 290, also pp. 267 and 305.
 
 124 SIMPLE PERIODIC CURRENTS. 
 
 mechanical work, part of it is dissipated as heat by an irre- 
 versible process, and part is associated with the circuit in a 
 recoverable form, and is taken up in the establishment of the 
 energy of the magnetic field, which then exists round the 
 conductor. This last portion of the energy, however, dissipates 
 itself as soon as the impressed electromotive force is with- 
 drawn. 
 
 A mechanical operation analogous to that of starting a 
 current in a wire may be found in the process of starting 
 from rest, or increasing the speed of, a heavy fly-wheel which 
 runs in bearings with friction. On applying a twisting force 
 or torque to the axle of the wheel we get up its speed. To 
 maintain the speed, force has to be continually applied to 
 the wheel, and the work so done against friction is frittered 
 away irreversibly into heat in the bearings. The friction is 
 analogous to the electrical resistance ; it may be called the 
 frictional resistance. 
 
 When the speed of the wheel is constant there is, however, 
 associated with the wheel a certain quantity of energy in a 
 kinetic form measured by ^Iw 2 , where I is the moment of 
 inertia, and w the angular velocity of the wheel. As soon as 
 the maintaining force is withdrawn this accumulated energy 
 dissipates itself in heat by friction, or is utilised in some other 
 way. During the time that the speed of the wheel is being 
 increased, force must be applied to it for two purposes : firstly, 
 to increase its angular momentum, and, secondly, to overcome 
 the friction at the bearings. Suppose that, instead of revolv- 
 ing on bearings with friction, the fly-wheel revolves in a 
 more or less viscous fluid, and that the bearings are truly 
 frictionless ; in such case the frictional resistance to motion 
 would be fluid resistance, and would for low speeds be 
 approximately proportional to the angular velocity. If I is the 
 moment of inertia and w the angular velocity of the wheel at 
 any instant, then it is shown in treatises on dynamics that 
 the product of the moment of inertia and the rate of change of 
 
 the angular velocity at the instant, or I, is the numerical 
 
 el t 
 
 measure of the torque or twisting force acting on the wheel 
 to increase its angular velocity, friction being neglected. If 
 we call the constant frictional coefficient B, so that B w is at
 
 SIMPLE PERIODIC CURRENTS. 125- 
 
 any instant the measure of the force necessary to maintain the 
 motion against friction, the total torsional or twisting force 
 acting on the wheel to maintain its angular velocity against the 
 force of friction, and to increase it against the force of inertia,. 
 
 T-> T> -,d(O 
 
 is F = Bw+I . 
 
 a t 
 
 A precisely similar equation may be found connecting the 
 electromotive force, electric current, electrical resistance, and 
 inductance in the case of current starting in a wire. The above 
 equation gives us a value for the instantaneous angular velo- 
 city, or enables us to find the angular velocity after any tune 
 when F, B, and I are given. When a current of strength i is 
 flowing steadily in a linear conductor, such as the wire under 
 consideration, the energy associated with it in the form of a 
 magnetic field is measured by the quantity ^ L i 2 , where L is 
 the quantity called the inductance of the circuit. Since this 
 quantity L bears to electromagnetic energy a relation similar 
 to that which the moment of inertia of a wheel does to the 
 energy of its rotation, it might be called the coefficient of 
 electromagnetic inertia ; but, as this would be a cumbersome 
 name, it has been called the inductance, or, frequently, the 
 self-induction of the circuit. The numerical product of the 
 moment of inertia and the angular velocity of the wheel is 
 called the angular momentum, and, analogously, the product 
 of the inductance of a circuit and the current flowing at 
 that instant through it is called the electromagnetic 
 momentum. 
 
 The rate at which the angular momentum of a wheel is 
 increasing or diminishing at any instant is a measure of the 
 rotational force, or the couple acting on it at that instant. So 
 also the rate of change of the electromagnetic momentum of 
 a circuit is the measure of the electromotive force acting on 
 it as far as mere change of current strength is concerned, and 
 omitting, for the present, that part of the electromotive force 
 required to overcome the true resistance. We have, then, the 
 following parallel between a fly-wheel, with moment of 
 inertia I, revolving frictionlessly, and having an angular 
 velocity w at any instant, and an electric circuit of inductance 
 L, having a current of strength i flowing in it at any 
 instant :
 
 126 SIMPLE PERIODIC CURRENTS. 
 
 Angular kinetic energy of the wheel, or energy of rotation = ^lu 2 
 Electromagnetic energy of the circuit ........................ = ^Lt 2 
 
 Angular momentum of wheel ................. .................. = I 
 
 Electromagnetic momentum of circuit ..................... = Lt 
 
 Rate of change of angular momentum of wheel = couple"! _ j^" 
 or torsional force causing rotation ....................... / dt 
 
 Rate of change of electromagnetic mornen turn electro- \ _ j^fL* 
 motive force employed in changing current strength/ dt 
 
 The symbol ( = ) must in the above be understood as equivalent to the 
 phrase " is measured by." 
 
 In the electric circuit, over and above the electromotive 
 force which is required to change the electromagnetic 
 momentum, there is an amount required to overcome the 
 frictional resistance of the wire, and which is defined and 
 measured by Ohm's law E = Ri. Hence, at any instant, if E 
 is the impressed electromotive force acting on the circuit, we 
 may divide E into two parts, one part equal to R* by Ohm's 
 law, which is sometimes called the effective electromotive 
 force, and which is that part of the impressed electromotive 
 force which is operating to overcome the true resistance of 
 
 the circuit, and another part equal to L , which is the 
 
 a t 
 
 part operating to change the strength of the current at that 
 instant, producing a small change, d i, in the current strength 
 * in a time d t. Hence, in mathematical language, we have 
 
 (26) 
 
 This is the fundamental equation for varying or periodic 
 currents, when the periodicity is not so rapid as to affect the 
 uniform distribution of the current over the cross section of 
 the wire, and when the electrostatic capacity of the circuit 
 
 may be neglected. The part L is often called the counter- 
 
 dt 
 
 electromotive force of self-induction, and the above equation 
 might be read in words 
 
 Total \ /Electromotive Force"! ^Electromotive Force 
 Impressed 
 Electro- 
 
 motive 
 Force. 
 
 Hrt n 
 
 ' er - h 
 
 ,orV + Ji 
 
 employed in over- employed in chang- 
 < coming resistance, or I + -! ing strength of cur- 
 the E fiective Electro- 1 rent, or the Inductive 
 
 V motive Force. [Electromotive Force.
 
 SIMPLE PERIODIC CUEEENTS. 127 
 
 We might arrive at this fundamental equation otherwise 
 thus : The total rate of expenditure of energy in the circuit is 
 at any instant measured by the product of the current at that 
 instant existing in the wire and the difference of potential 
 between its ends. The energy expended in the circuit is at 
 any instant being partly dissipated at a rate equal to Ei 2 , E 
 being the ohmic resistance and i the current, and partly 
 being stored up in the field at a rate equal to the rate of change 
 of the quantity ^Li 2 . Hence we have : 
 
 Eate of supply! f Rate f dissi -] f Rate of absorption 
 
 of energy \ = 1 patl n of ener ^ \+\ or stora g e of ener y 
 iergy J [as heat J |m the magnetic field, 
 
 and this in symbols is 
 
 Ei = Ei2 + 
 
 or E=E; + L^, . (26) 
 
 dt 
 
 which is our fundamental equation. 
 
 At this stage we must particularly caution the student to 
 note one thing. The quantity L, which is called the induct- 
 ance of the circuit, is a constant and definite numerical 
 quantity for any given form of circuit only as long as this 
 circuit consists of non-magnetic material and is immersed in 
 a non-magnetic medium. If, however, the circuit embraces 
 or is embraced by iron, as in the case of an electromagnet, or 
 is immersed in a medium which is not diamagnetic but 
 magnetic like iron, then it is no longer a constant quantity, 
 but the inductance varies from instant to instant with the 
 strength of the current flowing in the circuit. In this 
 chapter we suppose ourselves dealing only with circuits of 
 constant inductance, and in which the value of L is fixed by 
 the form of the circuit alone. 
 
 16. Equation for Establishment of a Steady Current. We 
 
 return to our discussion of equation (26) (15). When a cur- 
 rent is flowing in a conductor, we may picture it as surrounded 
 by its lines of magnetic induction properly mapped out. 
 That is, so that the number of the lines of induction passing 
 perpendicularly through a small unit of area taken at any
 
 128 
 
 SIMPLE PERIODIC CURRENTS. 
 
 point in the field is equal to the numerical value of the mean 
 strength of the magnetic field over the area. If the circuit has 
 the form of a loop (Fig. 50) lying on a horizontal plane, with 
 the current circulating round it in the opposite direction to 
 that in which rotate the hands of a watch, then the lines of 
 induction must be considered as springing out from the upper 
 surface, and turning outwards and over the conductor, so as 
 to re-enter the loop from the under siirface. The closed 
 circuit is, therefore, linked with a certain number of lines of 
 induction, which, if the circuit is composed of non-magnetic 
 material, are proportional in number to the strength of the 
 current at that instant. Any increase in strength of the 
 
 FIG. 50. 
 
 current causes more lines of induction to grow out from the 
 circuit, and packs the loop fuller of lines of induction. By 
 Faraday's law, any increase of the number of lines of induction 
 traversing or linked with a circuit creates an induced electro- 
 motive force numerically equal to the rate of increase of that 
 number at that instant. Hence, if 100 million lines of in- 
 duction C.G.S. measure are put or inserted at a uniform 
 rate in one second into a circuit, it will create an induced 
 E.M.F ; of one volt in it. If lines of induction are thrust 
 into a circuit, the direction of the current induced is counter 
 clockwise, as seen from that side of the circuit at which they 
 are thrust in (see Fig. 50).
 
 SIMPLE PERIODIC CURRENTS. 129 
 
 Applying this to the case before us, it is easily seen that 
 any increase of current strength in the circuit in Fig. 51 
 crowds the space with more lines of force, and therefore 
 creates in it an electromotive force of self-induction opposed to 
 the impressed electromotive force which is acting to increase 
 the current; and, so long as the current is increasing, this 
 counter E.M.F. is at each instant proportional to the rate of 
 growth of the current strength. 
 
 We can cast our equation (26) 
 
 E = Pu + L 
 
 dt 
 
 into another form, thus : 
 
 __ 
 
 K E d t' 
 E 
 where ^ is the maximum value which the current can attain. 
 
 td 
 
 FIG. 51. Lines of force being crowded into a circuit, inducing a counter- 
 clockwise E.M.F., as seen from the side at which they are put in. 
 
 Let us call this value I. The quantity g, or the ratio of the 
 
 inductance to the resistance of the circuit, is called the time- 
 corn- ant of the circuit ; let this quantity be denoted by T. We 
 
 then have I-*=T^, 
 
 dt 
 
 which, in words, is a statement that if a steady E.M.F. is 
 made to act on any circuit whose time-constant is T, the 
 amount by which at any instant the current falls short of its 
 full value is equal to its rate of growth at that instant, multi- 
 plied by the time-constant. 
 
 K
 
 130 
 
 SIMPLE PERIODIC CURRENTS. 
 
 17. Logarithmic Curves. A curve such that the rate of 
 growth or shrinkage of the ordinate or slope of the curve is 
 proportional to the ordinate itself is called a logarithmic 
 curve. 
 
 Let a curve (Fig. 52) be described by the extremity P of an 
 ordinate, P M, which moves uniformly along X, parallel to 
 itself, and let P M shrink in height at a rate proportional to its 
 height at any instant. The differential equation to such a 
 
 curve is then 
 
 A " 
 ~dt' 
 
 and since <?~ Bz (where <? = the base of Napierian logarithms 
 = 2-71828) is a function which fulfils this condition of having 
 
 FIG. 52. 
 
 a differential coefficient proportional to itself, we can write the 
 
 solution of the above 
 
 i 
 
 y = e A -f a constant, 
 
 for it is at once seen that by differentiating the equation 
 
 i 
 y=e * + a constant, 
 
 t 
 we obtain ^ = _f"\ 
 
 and therefore y = - A -, 
 
 Returning to our equation for the current, we can write, as 
 an equivalent for the equation
 
 SIMPLE PERIODIC CUEBENT8. 
 
 the equation I - i = - T^ffzS, 
 
 dt 
 
 _ 
 T I-i ' 
 
 Integrating this we have as a solution 
 
 - + a constant. 
 
 The constant has to be determined by the condition that. 
 when = 0, i = 0, which gives constant = - log I. Hence the 
 complete solution is 
 
 or I-i=Ie T. 
 
 This last equation expresses the fact that the amount by 
 which the current falls short of its full value, I, at any time, *, 
 after applying the E.M.F., is a fraction of its full value equal 
 to e~ f . When t = 0, or at the instant of closing circuit, I - i = I, 
 or the current * = 0; when = T, I-i = _, or the deficit from 
 
 full current is equal to - x the maximum current. Hence 
 
 ' 
 
 we may define the time -constant of a circuit as the time reckoned 
 from the instant of closing the circuit in which the current 
 
 rises up to a value equal to -!_ of its full value, or to 
 
 about 0-632 of its maximum value. Approximately we may 
 define the time-constant as the time from closing the circuit 
 in which the current rises up to two-thirds of its maximum 
 
 i E 
 value - . 
 
 It 
 
 The rise of current strength in a wire of inductance L and 
 resistance E, when a steady external electromotive force, E, is 
 applied to the circuit, can be represented by a current curve, as 
 shown in Fig. 53. Let OX be a time line on which we mark 
 off time as lengths reckoned from ; let lines drawn vertically 
 to this represent the current strength at any instant in a circuit 
 of time constant T, inductance L, and resistance E ; and let 
 
 K2
 
 132 
 
 SIMPLE PERIODIC CURRENTS. 
 
 Y = I jj represent the maximum current which is finally 
 
 found in the circuit. On applying the electromotive force E 
 to the circuit, the current strength grows up hi the wire as 
 graphically represented by the curve, the law of growth being 
 that the rate of growth at any instant, multiplied by the time- 
 constant, is equal to the difference between the actual current 
 at that instant and the maximum current strength finally 
 attained, or, symbolically, 
 
 i-feT** 
 
 dt 
 
 the solution of the above differential equation being 
 
 or 
 
 i=: 
 
 (27) 
 
 This last equation gives us the value of the current strength 
 at any time t seconds after closing the circuit, in terms of the 
 time-constant, and the maximum current, I, which is finally 
 attained. 
 
 The maximum current, I, would be produced at once in the 
 circuit if its inductance were zero, so that we may finally for- 
 mulate the law of growth of current in a circuit of constant 
 inductance L, resistance E, and no sensible capacity, by saying 
 that the current strength at any instant, added to the rate of growth 
 qftlie cur rent strength at thatlnstant multiplied % the time-constant, 
 is equal to the current which icould exist in the circuit if its in- 
 ductance u-cre zero.
 
 SIMPLE PERIODIC CURRENTS. 133 
 
 18. Instantaneous Value of a Simple Periodic Current. _ 
 
 The application of these principles to the case of simple periodic 
 currents will lead to another important equation. Let there 
 be a circuit which has an inductance L and resistance R, 
 and let a simple periodic electromotive force act upon it ; let 
 the maximum value of this E.M.F. be E, and let p stand 
 
 for 27rtt, where n is the frequency of the oscillation, or _ 
 
 n 
 
 is the duration of one single complete period, p is a quantity 
 of the nature of an angular velocity, and may be called the 
 pulsation. Then, if t is the time which has elapsed from the 
 commencement of the wave of E.M.F. and e is the actual 
 value of the E.M.F. at that instant, 
 
 e = E sin p t. 
 
 In this case the impressed electromotive force varies from 
 instant to instant, passing from zero to a maximum E, then 
 to zero again, and then to a negative maximum - E. Accord- 
 ingly, our fundamental equation for the current strength at 
 any instant is expressed thus : 
 
 rf ( L{ ) +Bf=<?=E swpt. . . . (28) 
 d t 
 
 For, the total rate of expenditure of work on the circuit at 
 any instant when the current has a value i is ei, and this must 
 be equal to the rate at which electrical work is being dissi- 
 pated as heat, or to R & by Joule's law, and to the rate at 
 which work is being stored up in the magnetic field, which is 
 
 Hence ( J L i 2 ) + R t 3 = e i, 
 
 at 
 
 or, L +B=Efflni>t ..... (29) 
 
 dt 
 
 In order to solve this differential equation, and obtain the 
 value of the current i in the circuit at any instant under the 
 periodic electromotive force, we may adopt a well-known 
 algebraic device, and substitute for the value of siupt its 
 equivalent in exponential terms. It is shown in treatises on 
 trigonometry that kff _^ 
 
 "
 
 134 SIMPLE PERIODIC CURRENTS. 
 
 where lc= V^l, and e is now the number 2-71828, which is 
 the base of the Napierian logarithms. 
 
 ke.-TeO 
 Also that cosfl = 6 +e _ ; 
 
 hence cos 6+k sin Q = e k6 . 
 
 These are called the exponential values of the sine and 
 cosine. 
 
 Taking the equation (29), 
 
 L + E i = E sin *, 
 dt 
 
 we divide both sides by L, and, writing T as before for the 
 time-constant , we get 
 
 Multiply both sides by e T (e being here the exponential 
 base, not impressed E.M.F.), and we have 
 7 . t . t T-i * 
 
 ar'+r *-&'** 
 
 The left-hand side of this equation is the complete differential 
 
 
 
 of ie'i, and may be written (*^) and on substituting 
 d t \ / 
 
 the exponential value for aio.pt and putting k for V 1, we 
 have 
 
 i^)..(-i-^)l (30 ) 
 
 The right-hand side of this last equation is the differential 
 with respect to t of 
 
 E 
 
 2fcBT ( 
 
 T~- rjr 
 
 and this last becomes by simplification 
 E ( #* f e
 
 SIMPLE PERIODIC CURRENTS. 135 
 
 Hence, equating both sides of equation (30), when integrated 
 we have 
 
 f- E / ^ ^ \ 
 
 2B/H 1+kpT 1-kpT ) 
 
 Substituting back into sine and cosine terms, and recol- 
 lecting that 
 
 e kp = cos pt -\-lcsinpt, 
 and e~* pt =cospt ksiupt, 
 
 we get finally 
 
 ._ E f shi> 2>T cos p t ) 
 
 This equation gives us a value i for the current at any 
 instant, and at a time t reckoned from the instant when the 
 impressed electromotive force is zero. The value of i is 
 accordingly called the instantaneous value of the periodic 
 current, and the instantaneous value runs through a certain 
 cycle of magnitudes, ranging from zero at one particular 
 instant to a maximum value I at another instant. 
 
 We can, however, put the above equation in a more intel- 
 ligible form. Keplace T by , and let 6 be an angle whose 
 fi 
 
 tangent is equal to t ; 
 
 hence tan 6==pT. 
 
 It follows by an easy transformation that 
 
 R LP 
 
 We have, then, for the value of the current i, the equation 
 i== E (sin p tp T cosp t\ . 
 
 or, by substitution, 
 p
 
 136 SIMPLE PERIODIC CURRENTS. 
 
 This is called the particular solution of the equation 
 
 L +Et=E sinjp t = f, 
 dt 
 
 and it shows us three things : First, that the phase of the 
 current i is retarded behind that of the impressed electro- 
 
 motive force by an angle 6 such that tan Q=p T = _ ^; 
 
 B 
 
 second, that the maximum value of the current is obtained by 
 dividing the maximum value of the electromotive force by a 
 quantity equal to V'lft+p'T?; and, third, that the current 
 curve is a simple periodic curve. The quantity / V/K 2 -fp 2 L 2 
 is called the impedance of the circuit. 
 
 The mathematical student will, however, remember that 
 the complete solution of the equation 
 
 dt 
 
 involves a constant of integration, and this is obtained by 
 adding to the particular solution above obtained the com- 
 plementary function which is obtained by taking the solution 
 of equation (29) when E sin p t = 0. 
 Now, since the solution of 
 
 at 
 
 where C is a constant of integration and e in this last equation 
 is the base of the Napierian logarithms, we have, then, the 
 complete solution of the differential equation 
 
 L +Bi = Esin^ 
 d t 
 
 given by the equation 
 
 -'. . (32) 
 
 The complementary function dies out rapidly as time in- 
 creases at a rate depending on the value of t . Physically, 
 
 the meaning of this is that the current does not settle down 
 into its regular periodic state until a shorter or longer time 
 after the closing of the circuit depending on the value of the 
 
 time -constant . 
 B
 
 SIMPLE PERIODIC CURRENTS. 137 
 
 We shall return again to discuss the complete solution of 
 the above equation (29), and show how to determine the value 
 of the constant C in equation (32). 
 
 We have seen from the explanations on previous pages 
 that the mean-square value of a simple periodic quantity is 
 equal to its maximum value divided by \/2. Hence, if we write 
 Im for the impedance, we can put the equation, giving the 
 instantaneous value of the current produced by a simple 
 periodic impressed electromotive force of maximum value 
 E, operating on a circuit of resistance E, inductance L, 
 with a pulsation p, in the form 
 
 or, if we denote the maximum value of the current during the 
 
 -p 
 
 phase by the letter I, and since I = | , we have 
 
 i = Isin(pt-0). 
 Hence, we may write this in words as follows : 
 
 tJw iji'iximitm value] f the maximum value of the 
 
 -I i> 
 
 of the current ]- = -j impressed electromotive force 
 strength J [ " impedance 
 
 and the mean-square ] f t/w ma<rt -, /mm raZtM 
 
 flafofc? o/ f/te - = 4 j= . 
 
 current strength j (_ ** * 
 
 We see, then, that, in the case of simple periodic electro- 
 motive force, the quantity called the impedance appears to 
 be related to the impressed E.M.F., just as does the resist- 
 ance to the steady E.M.F. in the case of continuous currents, 
 and the above may be called the equivalent of Ohm's law for 
 simple periodic currents. Compare as below 
 
 For steady 1 
 
 or current ~\ (electromotive force 
 
 continuous f strength J = \ ^isTan^e ( Ohmsla w). 
 
 currents J 
 
 For simple^ mean-ware] , IMrtl)I _, ylMW ridue O j 
 periodic or I value oj the I = | , ele ^. omotive foi ^ e 
 
 alternate ( current . 
 
 currents J strength } < impedance
 
 138 
 
 SIMPLE PERIODIC CURRENTS. 
 
 Impedance is a quantity which is measured, like resistance, 
 in ohms, and has for that reason been sometimes, but 
 erroneously, called the virtual resistance. 
 
 19. Geometrical Illustrations. The current equation, 
 expressing the current strength in terms of the impressed 
 electromotive force, the resistance, inductance, and phase 
 angles, which holds good when a circuit of constant induct- 
 ance and no sensible capacity is subjected to moderately 
 great pulsations of electromotive force has been in the 
 previous pages arrived at algebraically from first principles. 
 
 FIG. 54. 
 
 It is, however, possible to elucidate its meaning by geometri- 
 cal methods. Let a circular disc (Fig. 54) be pivoted at 
 the centre 0, and at any point P on the circumference let 
 a plummet line be attached. In front of the circle is a fixed 
 horizontal line XX'. Let the disc move round counter- 
 clockwise at a uniform rate, the time of one revolution 
 being T. As the disc goes round, the length of plummet 
 line P M above X X' fluctuates. Since P M = P sin P M, 
 it follows that, if the magnitude of P M be taken at small 
 equal intervals of time during one revolution, and such 
 heights be plotted off as off-sets at equal distances above 
 and below a datum line, the extremities of these ordinates 
 will lie on a simple periodic or sine curve. In other words,
 
 SIMPLE PERIODIC CURRENTS. 139 
 
 PM grows and shrinks in height in accordance with a 
 simple periodic law. We can, therefore, represent any 
 quantity which fluctuates in magnitude according to a 
 simple sine law of growth by representing it as the pro- 
 jection of a point on the circumference of a circle revolv- 
 ing uniformly, taken on a horizontal or vertical fixed line 
 drawn through the centre. Hence, if P represents the 
 maximum value of an electromotive force fluctuating periodi- 
 cally, P M will represent its various magnitudes during the 
 complete period. The magnitude of P M at any instant 
 is known when we know P, which is called the amplitude, 
 or maximum value, and POM the phase angle of the motion. 
 A diagram, in which the projection on any other line of a 
 radial line revolving round one extremity is made to represent 
 a simple periodic function, is called a clock-diagram. In clock- 
 diagrams radial lines are taken to represent in magnitude the 
 maximum values of the quantities which are to he represented 
 as periodically varying. Any line through the centre may 
 be taken as the line on which projections are taken, and the 
 projections in this line give us the instantaneous values of the 
 periodic quantity whose maximum value is represented by the 
 radius. If different radii are drawn from one centre, repre- 
 senting currents or electromotive forces, then the angular 
 interval between these radii represent the phase difference of 
 these quantities. 
 
 20. Graphic Representation of Periodic Currents. On 
 such a diagram let a radius be drawn to any scale repre- 
 senting by its vertical projection the periodic fluctuation 
 of an impressed electromotive force, varying according to a 
 simple sine law, and acting on a circuit of given inductance 
 and resistance with a fixed periodicity; the problem is to 
 draw on the same diagram another radius, of which the 
 vertical projection shall represent the actual current strength 
 in the circuit at the corresponding instant. The impressed 
 electromotive force at any instant balances, or is equal to, 
 the sum of two others, viz., the effective electromotive force 
 driving the current, which is equal to the product of the 
 ohrnic resistance of the circuit and the current at that 
 instant in it ; and the inductive or counter-electromotive
 
 140 
 
 SIMPLE PERIODIC CURRENTS. 
 
 force, which is equal to the rate of variation of the flux of 
 force or number of lines of force traversing the circuit. The 
 phases, or times of maximum, of these two components are 
 not identical. They differ by 90, since the effective electro- 
 motive force has the same phase as the actual current, and 
 the inductive electromotive force, depending on the rate of 
 variation of the current, comes to a maximum at the instant 
 when the current is zero, or is changing sign. 
 
 By the proposition in 10, these two periodic quantities 
 can therefore be represented by sine curves, one of which 
 is shifted backward relatively to the other, so that the crest 
 of the wave of one coincides with the zero point of the 
 other. We shall first proceed to show that the sum of two 
 
 FIG. 55. 
 
 simple periodic motions of the same periodic time, but 
 different phases and amplitudes, will, when added together, 
 produce a simple periodic motion of the same periodic time. 
 
 Let a parallelogram of cardboard, A B C (Fig. 55), be 
 cut out and pivoted by a pin at the angle 0, so as to turn 
 freely clockhand-wise. Let a vertical line, Y, be drawn 
 through 0, and in any position let the sides A, C, A B 
 be projected on to Y. The projection of lines equal and 
 equally inclined are equal ; hence, since A B is equal and 
 parallel to C, the projection of AB viz., a b is equal to 
 that of OC viz., Oc. But 06 = a + ab always for any 
 position of the card; hence Ob = 0a4-0c. The projection 
 of the diagonal is therefore equal to the sum of the projec-
 
 SIMPLE PERIODIC CURRENTS. 
 
 141 
 
 tions of the adjacent sides. As the card moves uniformly 
 round the magnitudes of the projections fluctuate at each 
 instant, according to a simple periodic law. Hence the 
 sum of the simple periodic motions of which A, C are 
 the amplitudes, and which have a fixed difference of phase 
 represented by the angle A C, is the simple periodic motion 
 represented by B in amplitude and relative phase. If, then, 
 a point be subjected to two simultaneous simple periodic 
 motions of given amplitudes, and of which the phases differ 
 by 90, the actual motion will be represented, as to ampli- 
 tude and phase, by the diagonal of the parallelogram of 
 which these two form the adjacent sides. Returning in 
 thought to electric motion, consider the motion of a particle 
 
 FIG. 56. 
 
 of electricity (if we may be allowed the expression) in the 
 wire subjected to two simultaneous simple periodic motions 
 of unequal amplitude and fixed difference of phase equal 
 to 90. The displacement at any instant due to the two 
 together is equal to the sum of each separately. If the 
 individual motions are represented by the vertical projec- 
 tions of two lines, A, OB, fixed like hands of a toy clock 
 at right angles (Fig. 56), the resultant motion is that indi- 
 cated by the projection of the diagonal C on the same vertical. 
 We have seen (in 10) that, if the variation of a quantity is 
 represented by a simple sine curve, the variation of its rate of 
 change is represented by a sine curve of different amplitude
 
 142 
 
 SIMPLE PERIODIC CURRENTS. 
 
 shifted backward by 90 of phase, or by a quarter of a wave 
 length. It follows from this proposition that if we add 
 together at every instant the motions or the ordinates repre- 
 senting them on a diagram of two simple periodic motions, 
 one of which is the curve representing the rate of change of 
 the ordinate of the other, we shall get a new sine curve, of 
 which the maximum value falls between that of the other 
 two, and of which the amplitude is different, but wave length 
 or periodic time the same. In Fig. 57 the thick line sine 
 curve represents one wave of a simple periodic motion. The 
 fine continuous line is a sine curve of equal wave length, of 
 which the ordinate PM at any point represents or is pro- 
 portional to the rate of change of the ordinate Q M of the 
 
 FIG. 57. 
 
 thick curve at the same instant. Adding together the 
 ordinates of the thick and thin curves, we get a new dotted 
 line sine curve, of which the ordinate R M is equal to Q M 
 + rate of change of Q M. If we substitute for the sine curve 
 diagram a clockhand diagram (Fig. 56), then the projection of 
 OB viz., b corresponds to the ordinate QM of the thick 
 curve; that of A viz., a corresponds to PM, the ordinate 
 of the thin curve ; and that of C, the diagonal of the 
 rectangle A, OB, corresponds to E M, and is the resultant 
 of the motion B, and the rate of change of that motion, viz., 
 OA. If, then, OB represents the amplitude or maximum 
 value of the actual periodic current in a circuit, a line, A, 
 drawn at right angles to B, will represent to a suitable scale 
 the rate of change of that current.
 
 SIMPLE PERIODIC CURRENTS. 143 
 
 We are, then, led to this converse proposition, that we 
 can resolve any simple periodic curve into a pair of component 
 periodic curves of equal periodic time, but of which the maxi- 
 mum value happens for one before and for one after that of 
 the original. 
 
 21. Impressed and Effective Electromotive Forces. If at 
 
 any instant a current of which the instantaneous value is i 
 is flowing in an inductive circuit of which the true resistance 
 is E, the quantity Ei represents the voltage necessary to 
 make this current flow, and this part of the impressed electro- 
 motive force is called the effective electromotive force in the 
 circuit. If the circuit is an inductive circuit, there will be 
 another electromotive force equal in magnitude to L , which 
 
 acts either with or against the total applied or impressed 
 electromotive force. This is called the inductive electromotive 
 force. The electromotive force which is at any instant 
 applied to the circuit is called the impressed electromotive force. 
 The effective electromotive force is always the resultant of 
 the impressed and inductive electromotive forces. Hence, if 
 these last two electromotive forces are represented in a clock 
 diagram in magnitude and relative phase by the two sides 
 of a parallelogram, the effective electromotive force will be 
 represented in magnitude and phase by the diagonal of that 
 parallelogram. A further condition is that, since the inductive 
 electromotive force depends upon the rate of change of the 
 current, it is always at right angles as regards phase with 
 the effective electromotive force. This last is always in 
 step or in synchronism as regards phase with the current. 
 Hence, if we require to draw a clock diagram of electromotive 
 forces for an inductive circuit in which a simple periodic 
 impressed electromotive force is acting, we see that the proper 
 construction is as follows : Take any line, OP (Fig. 58), to 
 represent the magnitude of the maximum value of the 
 impressed electromotive force ; on P describe a semi-circle, 
 M P ; let P be supposed to revolve round the point in 
 the contrary direction to the hands of a watch, and let the 
 projection of OP, at any instant on any line OY drawn 
 through 0, be taken. Then P represents the maximum
 
 144 
 
 SIMPLE PERIODIC CURRENTS. 
 
 value, E, of the impressed electromotive force, and Og, or the 
 projection of P, represents the magnitude of the instan- 
 taneous value of the impressed electromotive force at an 
 instant when P has completed such part of one revolution 
 as is represented by the angle POX. 
 
 Let us suppose P to start from the position X, and 
 let time be reckoned from that instant of starting. Then, 
 if T be the time of one complete revolution, and if t be 
 the time in which P passes through the angle POX, 
 
 i= Lpl COS(pt 9) 
 
 FIG. 58. 
 
 the angle P X is the same fraction of four right angles of 
 2?r that t is of T. Hence the angle POX is, in magni- 
 tude, equal to ^- . For shortness, 2u-/T is written p. There- 
 fore p is a quantity of the nature of an angular velocity. 
 Hence, if the magnitude of q, which is the projection of P, 
 is denoted by e, and if OP is denoted by E, we see that 
 e = 1 E sinpt,
 
 SIMPLE PERIODIC CURRENTS. 145 
 
 or the instantaneous value of the impressed electromotive 
 force runs through a cycle of values represented by the 
 ordinates of a sine curve. 
 
 Next, on P describe a semi-circle, M P, on that side of 
 P which is towards the direction in which P is rotating. 
 Take a point M on this circumference, such that M is to 
 M P in the ratio of LJJ to B, where L is the inductance and K 
 the resistance of the circuit. Through draw K parallel 
 to MP. Produce MO to N, and make ON equal to OM. 
 Draw N K parallel to P and join K. Then, on the same 
 scale on which P represents E, the maximum magnitude of 
 the impressed electromotive force, K will represent the 
 maximum magnitude of El or the effective electromotive 
 force, and N will represent ~Lp I or the maximum magni- 
 tude of the inductive electromotive force. By the geometry 
 of the figure we see that, if the angle P K is called 0, the 
 projection Ok of OK on OY is equal to OK sin(pt 6), 
 and also the projection n of ON on Y is equal to 
 
 N cos (p t - 0} . Hence n = (0 k) . Moreover, K is 
 at 
 
 the resultant of P and N, and N is at right angles to 
 K ; therefore N and K fulfil all the conditions requisite 
 for being the representation of the maximum values of the 
 inductive and effective electromotive forces. For OK is 
 obviously the resultant of P and ON. N is in such 
 a direction that its projection or instantaneous value is 
 numerically determined by the rate of change of the projection 
 or instantaneous value of K ; and we know, by fundamental 
 principles, that the electromotive force of self-induction is 
 determined by the rate of change of the current in this 
 circuit ; that is, by the rate of change of the effective electro- 
 motive force. By considering the relative positions of OP, 
 K, and N, it will be seen that they are in the right 
 directions to represent these three quantities. For, if the 
 system of lines be supposed to revolve round 0, then, when 
 the projection of K is above X that is, when the current 
 in the circuit is increasing the projection of N is negative 
 and is decreasing. This means that the electromotive force 
 of self-induction is in such a direction as to oppose the 
 current, Also, when the effective electromotive force or 
 
 L
 
 146 SIMPLE PERIODIC CURRENTS. 
 
 current is in the same direction, but decreasing, then the 
 inductive electromotive force is positive, or in the same 
 direction as the current, and is increasing. Accordingly, if 
 the magnitude of K is R I, where R is the resistance of the 
 circuit and I is the maximum value of the current, and R I is 
 therefore the maximum value of the effective electromotive 
 force in the circuit, we see that the magnitude of N must 
 be It pi, and that of OP must be A/R 2 + p 2 L 2 I, and this last, 
 we know, is the value of the impressed electromotive force E. 
 Accordingly, on whatever scale P represents the impressed 
 electromotive force E, then OK represents the effective electro- 
 motive force RI, and ON represents the inductive electro- 
 motive force L p I. If we take one Rth part of K, we have 
 the value of the current in the circuit. The angle of lag 6 by 
 
 which the current is behind the impressed electromotive force 
 in phase is an angle, such that 
 
 cos Q = 
 
 Impedance of circuit. ~ -\/R 2 +^y 2 L 2 ' 
 
 The diagram shows us, therefore, not only how to represent 
 the current and impressed electromotive force in an inductive 
 circuit properly as regards phase and magnitude, but tells us 
 practically how the angle of lag should be measured. 
 
 The relation of the impedance and resistance of an inductive 
 circuit may be represented geometrically as follows : Draw a 
 right-angled triangle, ABC (Fig. 59), and take the base A B 
 to represent the resistance of the circuit, and the hypotenuse, 
 AC, to represent the impedance. Then the side BC will 
 represent the magnitude of the quantity p L. This has been 
 called the reactance of the circuit, and since the angle C A B is
 
 SIMPLE PERIODIC CUEEENTS. 147 
 
 -p 
 an angle which has a cosine equal to 2 , we see that 
 
 this angle, which we may call 0, is the angle of lag of current 
 behind electromotive force, and, moreover, that 
 pL__ reactance of circuit 
 ~~ K "resistance of circuit' 
 
 22. The Mean Value of the Power of a Periodic Current. 
 
 Having now seen how the fluctuation of current strength is 
 related to that of the impressed E.M.F. in an inductive circuit 
 under the conditions of a simple sine law of variation, we pass 
 to the consideration of the measurement of the power taken 
 up in or supplied to circuits traversed by periodic currents. 
 
 Let the thin line curve in Fig. 60 represent the curve of 
 impressed electromotive force in an inductive circuit, and the 
 
 FIG. 60. 
 
 thick line the corresponding curve of current. Then at any 
 instant the rate at which energy is being expended on the 
 circuit is equal to the product of the ordinates P M, Q M, 
 which at any point M on the time line represent the electro- 
 motive force and current respectively. The mean rate of 
 expenditure of energy, or the mean power being taken up in the 
 circuit, is then the mean of all such products taken at equal 
 and very near intervals of time during one complete period. 
 This is not by any means identical with the product of the 
 mean current and mean electromotive force. To arrive at an 
 expression for this mean power, we must pave the way by a 
 preliminary proposition on the mean product of two simple 
 periodic quantities. An elegant geometrical method of 
 
 L2
 
 148 
 
 SIMPLE PERIODIC CURRENTS. 
 
 obtaining this has been given by Mr. Blakesley. We shall, 
 however, give here an algebraical proof of this proposition. Let 
 there be two radii OP, OQ (Fig. 61), which revolve in equal 
 periodic times round a common centre 0, separated by a fixed 
 angle, P Q. At equal small intervals of time corresponding 
 to equal angular motions let the projections Op, q of these 
 lines be taken on a vertical line through 0. It is required 
 to find the mean value of the product Op, Oq during one 
 complete period. 
 
 Denote by X the length of P, and by Y the length of Q, 
 and let the angle P Q be ft, and P p be a. (3 is the angle 
 of phase difference, and X and Y are the maximum values of 
 the periodic quantities Op, Oq, which are the vertical projec- 
 tions of P, Q. 
 
 . 61. 
 
 Let Op be denoted by p, and q by q 
 Then p = X cos a, 
 
 and # = Y cos (a+/3); 
 
 and therefore p q = X Y cos a cos (a +/:?). 
 
 Let the pair of radii P, Q be supposed to turn round 
 one complete revolution, proceeding by n small steps, each 
 step increasing the angle a by a very small amount, 8, a and n 
 being a very large number. At each stage let the value of p q 
 be measured as above, then the mean value of the product p q 
 is one ?zth part of the sum of all the n values so taken. Call 
 this mean value of the product M. Then, 
 
 XYjcos /2+cos Sacos(Sa+/3) + cos2Sacos(2Sa-f-/3)x 
 
 n ( .... +COS71 18aCOS(w-l Sa + /3)j 
 
 By trigonometry we have 
 
 cos (n 1 8 a) cos (n 1 8 a+/3) = J cos (2n-l S a+f3) + i cos /3, 
 since cos A+B + cos A B = 2 cos A cos B. 
 
 M
 
 SIMPLE PERIODIC CURRENTS. 149 
 
 Accordingly every term, except the first in the cosine series 
 for M, splits up into the sum of two others, one of which is 
 always | cos /3. Rearranging the terms, we get for the value 
 of M as follows : 
 
 M = f| n cos +icos +cos (2 8 a+0)-f cos (4 3 a 
 
 -fcos 
 
 )) J 
 
 The cosine series in the inner bracket consists of a series of 
 cosines of angles in arithmetic PL egression taken all round the 
 circle. Hence, since the cosine of any angle is numerically 
 equal to that of the cosine of its supplement, but of opposite 
 sign, these cosine terms will cancel each other out pair and 
 pair, when n becomes very great and 8 a very small, and n 8 o 
 equal to 2?r. For when 
 
 n 8 a = ITT, 2 7i-l 8 a+P = 4ir+0, and cos (47T+/3) = cos (3. 
 The first and last terms of the series are in this case identical, 
 and for every term there will exist one of equal magnitude 
 and opposite sign. The sum of the series of cosine terms in 
 the inner bracket is accordingly zero. 
 
 The value of M reduces then to that of the first term, 
 viz. : 
 
 The mean value of the product of two simple harmonic or 
 periodic functions of equal period but different amplitude and 
 phase is equal to half the product of their maximum values, 
 and the cosine of their difference of phase. 
 
 Returning to the consideration of the electrical problem, 
 it is now clear that for simple periodic or sine variation the 
 mean value of the product of the current at any instant and 
 the simultaneous value of the impressed electromotive force in 
 an inductive circuit is obtained by multiplying together half 
 the product of their maximum values, and the cosine of the 
 angle of lag. If i be the current at any instant, and e the 
 impressed E.M.F., I and E being their maximum values, 
 then the mean value of e i during a complete period is 
 
 cos*, ..... (33)
 
 150 
 
 SIMPLE PERIODIC CURRENTS. 
 
 and this is a measure of the mean rate of expenditure of 
 energy on that circuit, or the mean power taken up. It is 
 obvious, then, that if the lag is 90, this mean product is 
 zero, and that no work is done at all. 
 
 When has intermediate values between and 90, the 
 real rate of dissipation or transformation of energy in the 
 
 "P T 
 
 circuit will be intermediate between and zero. In order 
 
 2 
 
 to understand how this can be, and how it is that a circuit may 
 be traversed by a current and yet take up no power, we must 
 examine a little more closely the nature of the phenomena. 
 
 23. Power Curves. Let the periodic curve in Fig. 62 
 represent a sinusoidal variation of electromotive force acting 
 on a circuit which we shall for the moment assume has no 
 
 F.G. 62. Electromotive Force Curve. 
 
 sensible inductance. Let the curve in Fig. 63 represent 
 the corresponding current. The length of each ordinate of the 
 second curve is equal in magnitude to that of the corresponding 
 
 FIG. 63. Current Curve. 
 
 ordinate of the first curve divided by the value of the resist- 
 ance of the circuit. Let the lengths of corresponding ordinates 
 of these two curves be multiplied together, and the product
 
 SIMPLE PEBIODIC CURRENTS. 
 
 151 
 
 set off as the ordinates of a new curve represented by the 
 dotted line in Fig. 64. This dotted curve is, then, the curve 
 of power or activity, and represents the variation of the pro- 
 duct of the current and the electromotive force taken at 
 every instant. 
 
 In multiplying together the ordinates of the first and second 
 curves, we must pay attention to the algebraic sign of each 
 ordinate. Ordinates of each curve drawn above the horizontal 
 
 Ordinates | 
 Products .. 
 
 FIG. 64. 
 
 1-81 3'5 4-05 6'06 676 T'O 
 
 906 1-75 2-47 3-03 338 35 
 
 1-64 6-12 12-25 18-37 22 '86 25'4 
 
 datum line of the curve must be reckoned plus, and ordinates 
 drawn below must be reckoned minus, and in taking the product 
 the algebraic law of signs must be regarded. It will be seen 
 that the dotted line curve consists of a wavy line of two loops 
 lying wholly above the mean datum line. If the area of the 
 two hummocks enclosed by the dotted curve and the horizontal 
 line is indicated or integrated, say, by an Amsler's planimeter,
 
 152 
 
 SIMPLE PERIODIC CURRENTS. 
 
 the area represented by the shaded part so obtained is 
 a measure of the total work done in one complete period of 
 the current oscillation, and, since this area lies wholly above 
 the datum line, it must be reckoned as positive, or as work done 
 by the electromotive force ; in other words, it represents the 
 total energy transformed from electrical energy into heat in 
 one complete period. 
 
 Next let us suppose that the same periodic electromotive 
 force acts upon a circuit having inductance as well as resist- 
 ance, and that therefore, as already shown, the current is 
 retarded in phase behind the electromotive force. Let the 
 thin curve in Fig. 65 represent the periodic impressed electro- 
 
 FIG. 65. Lag 45. 
 
 rr too/ 2 ' 4 ? 3-03 3-3S 3'5 3'38 3'03 2 -47 1 '75 '91 '91 1'75 2'47 
 
 iDdies^ 1-gl 35 4.956-06 6 .7 6 7 6 -7 6 g-06 4 '95 3 "5 l'8l 
 Products ..0 5-5 11-83 17'3 20'48 20'48 17'3 11'83 5'5 318 3 "18 
 
 motive force, and the thick curve the current retarded by 45 
 in phase behind the other. Proceed as before to obtain 
 the power curve by multiplying the heights of corresponding 
 ordinates, the multiplication of the ordinates being shown 
 below the figure. We find that the power curve representing 
 the variation of the activity is a wavy curve, consisting of 
 four sections, two large hummocks above the datum line, 
 which are positive areas, and two small ones below, which are
 
 SIMPLE PERIODIC CURRENTS. 
 
 153 
 
 negative areas. The algebraic sum taken with regard to sign 
 of all these four areas, represented by the shaded parts, is a 
 measure of the total work done in one complete period by the 
 electromotive force. Going one step more, we may imagine 
 that the current lag is 90, and in Fig. 66 we have drawn 
 the power curve in this case, obeying the same instructions. 
 We see that the power curve consists of four loops, two 
 positive and two negative, and that the area of these hum- 
 mocks are equal. Hence, the total area or indicated value 
 in this last case is zero, and the work done in one complete 
 cycle is zero ; hence, the rate of doing work, or the power, is 
 zero. Eeturning to Fig. 64, the first case, it is easy to see 
 
 FIG. 65. Lag 90. 
 
 ' 3 3 ' 38 3 ' 3 2 47 * 73 
 
 rbl 3 . 5 4 .^ ^ 
 
 Products ..0 .613 10 Cl 1223 1061 
 
 6'13 
 
 that the rate of doing work, or the power, is measured by the 
 mean ordinate of the shaded work areas considered as indicator 
 diagrams. Since the dotted curves are perfectly symmetrical, 
 if we draw a line Y Y' at half the height of the maximum 
 ordinate of the dotted curve, it will cut the two hummocks 
 into two parts, and the area of the upper part, or mountain 
 above the line Y Y', would just fill up the valley between the 
 bottom parts of the hummocks. Since the rate of doing work 
 is equal to the work done in one complete cycle divided by the
 
 154 SIMPLE PERIODIC CURRENTS. 
 
 time of duration of that cycle, it obviously follows that this mean 
 ordinate X Y measures the power or rate of doing work and is 
 
 T? T 
 
 equal in magnitude to ; hence this product is a measure of 
 
 2 
 
 the rate at which the electromotive force does work. Eefer- 
 ring to the second case, we see that the mean ordinate X Y is 
 no longer equal to half the maximum ordinate of the positive 
 or upper loop, but is equal to half the difference between the 
 magnitudes of maximum ordinates of the positive and nega- 
 tive loops of the power curve, and is therefore less than 
 
 From the proof given previously we have seen that it 
 
 ' F T 
 is equal to - (cosine of lag). 
 
 In the third case considered, of a lag of 90 degrees, it is 
 easy to see why the resultant rate of doing work is zero. 
 In the first quarter of a stroke the electromotive force propels 
 the current, and this last is in the direction of the E.M.F., 
 but in the second quarter of a stroke the current is negative or 
 opposite to the electromotive force ; in other words, the current 
 is moving against the force and does work against the E.M.F., 
 and the same push and re-push is repeated in the second 
 half of the period. Hence, on the whole, though there is 
 an impressed electromotive force and a current flowing, no 
 resultant work is done and no energy dissipated. 
 
 S 24. The Experimental Measurement of Periodic Currents 
 and Electromotive Forces. At this stage it will be an advan- 
 tage to direct attention to the practical means of measuring the 
 V/rnean square value of periodic currents and electromotive 
 forces, and also the mean value of the power given to an 
 inductive circuit of any kind. There are, amongst others, two 
 instruments especially useful for measuring periodic currents. 
 One of these, the Cardew voltmeter, depends upon the prin- 
 ciple that when a wire traversed by a current, either steady 
 and unidirectional or steadily periodic, is placed in an enclosure 
 the walls of which are approximately at a constant tempera- 
 ture, the wire will itself, after a short time, attain a constant 
 temperature. This constant temperature is reached when 
 there is a state of equilibrium between the rate at which heat
 
 SIMPLE PERIODIC CURRENTS. 155 
 
 is radiated by the wire and the rate at which the walls of the 
 enclosure radiate heat back to it. 
 
 The wire has a definite length corresponding to each tem- 
 perature, and means are provided for measuring this elongation 
 with great accuracy. The total amount of heat generated in 
 the wire per second is dependent upon the rate of generation 
 at each instant. The instantaneous rate of heat development 
 is, by Joule's law, equal in mechanical units to the product of 
 the resistance of the wire and the square of the value of the 
 instantaneous current flowing in it. 
 
 If the wire is traversed by a simple periodic current, and 
 we construct from the current curve diagram another curve 
 whose ordinates are equal to the square of the correspond- 
 ing current ordinates, we have a curve every ordinate of 
 which is proportional to the instantaneous rate of generation 
 of heat in a wire traversed by the periodic current. Since the 
 horizontal line measures time, it is obvious that the whole 
 area of the outer curve, or heat curve, represents the total 
 work done per semi-period by the current in producing heat, 
 and that the same total work would be done by a steady current 
 which had a value equal to the square root of the mean of the 
 squares of all the ordinates of the periodic current curve. 
 
 This square root of the mean of the squares of all the ordi- 
 nates of a simple periodic curve has, however, been shown in 
 9 to be numerically equal to the value of the maximum 
 ordinate of the periodic curve divided by V 2. It follows that 
 the total heat generated per second in the wire is a numerical 
 measure of the -v/mean square value of the current or of half 
 the square of the maximum value of a simple periodic current. 
 A fine wire stretched out in the manner of a Cardew voltmeter 
 wire has a very small inductance, and, when acted upon by a 
 simple periodic electromotive force, the current produced in it 
 is very nearly proportional to this impressed electromotive force. 
 It follows, then, that, when a Cardew voltmeter is subjected 
 to a simple periodic electromotive force, the needle takes a defi- 
 nite position, corresponding to a definite expansion of the wire, 
 which is that which it would take if the wire were subjected 
 
 to a steady electromotive force equal to of the maximum 
 value of the periodic electromotive force.
 
 166 SIMPLE PERIODIC CUEEENTS. 
 
 The Cardew voltmeter is not adapted to measure any but 
 very small currents. The instrument generally employed to 
 measure periodic currents of moderate and large magnitude 
 is some modification of Weber's electro-dynamometer. In the 
 best-known practical form of Siemens there are two coils of 
 wires in series, one fixed and the other movable, and so placed 
 that the currents in the movable coil circuit are traversed at 
 right angles by the lines of force due to those in the fixed 
 coil. When a simple periodic current traverses the coils in 
 series, a force is brought into existence due to the electro- 
 dynamic action, which is proportional to the instantaneous 
 value of the square of the current strength. From instant to 
 instant, however, the current strength varies. If the time of 
 free vibration of the movable coil is very large compared with 
 that of a complete period of the electrical vibrations, and il 
 the movable coil is brought back by a restoring force due to a 
 spring or bifilar suspension or gravity, &c., into a fixed normal 
 position, then, during one complete electrical period, we may 
 consider that the movable portion receives a number of small 
 impulses which are in magnitude represented by the square of 
 the ordinates of the current wave. Hence, the total impulse 
 on the movable coil is equal to the magnitude of the inte- 
 grated area of a sine curve whose ordinates are respectively the 
 squares of those of the current curve, and the mean force on 
 the movable coil will obviously be proportional to the mean 
 ordinate of this force curve. If the movable coil is so heavy 
 that its time of free vibration is very long compared with the 
 time in which the periodic forces on it run through a complete 
 cycle, it will experience a displacement exactly that due to the 
 mean of the forces acting upon it that is, to the square root 
 of the mean of the squares of these instantaneous currents 
 
 or to -, where I is the maximum value of the current during 
 
 A/2 
 
 the period. The periodic force on the movable coil is equiva- 
 lent to a steady force when this periodic force runs through 
 all its values in a time very short compared with the time 
 of free vibration of the coil. Hence, if a simple periodic 
 current has a maximum value I, when it is sent through an 
 electro-dynamometer it will cause a deflection equal to that 
 
 which would be caused by a steady current equal to - .
 
 SIMPLE PERIODIC CURRENTS. 157 
 
 Let us suppose a coil of constant inductance L and resist- 
 ance B to be traversed by a simple periodic current of fre- 
 quency TO ( where 2a-w = p). Let an electro-dynamometer be 
 inserted in series with it, and let a Cardew voltmeter be 
 connected to the extremities of the inductive circuit. 
 
 We have before seen that if E and I are the maximum 
 values during the period of the impressed E.M.F. and current 
 in an inductive circuit, then the power taken up in that 
 
 TT T 
 circuit is equal to -^- cos d, where = angle of lag of current 
 
 behind the E.M.F. But the reading of the Cardew volt- 
 meter when connected to the ends of an inductive circuit is 
 
 p 
 very nearly proportional to , and the dynamometer reading 
 
 v2 
 
 in that circuit is proportional to = ; therefore, the product of 
 v2 
 
 Tjl T 
 
 these readings is proportional to -, and takes no account of 
 
 the difference of phase. The product of the ^/mean square 
 values of the current and of the electromotive force in 
 an inductive circuit is generally called the apparent power 
 or apparent watts given to that circuit, but it is not a measure 
 of the true power given to the circuit. For this reason we 
 can derive no information from the use, in this manner, of 
 these instruments. The observed readings, and hence their 
 product, does not take into account the difference of phase 
 between the current and impressed E.M.F. in the inductive 
 circuit. A very small error, in practice negligible, is also 
 introduced by disregarding the inductance of the wire of the 
 Cardew instrument. On this account, strictly speaking, 
 currents in the wire cannot be taken as accurately propor- 
 tional to potential differences at the extremities, but this is in. 
 ordinary usage a negligible error. 
 
 25. Method of Measuring the True Value of the Power 
 given to an Inductive Circuit. Theory of the Wattmeter. 
 If a current traversing an inductive circuit under a periodic 
 impressed electromotive force is made to pass through another 
 circuit which acts electro-dynamically upon a movable circuit 
 conveying another current proportional in strength to, and
 
 158 
 
 SIMPLE PERIODIC GTJEEENT8. 
 
 agreeing in phase with, the periodic variation of potential 
 difference at the terminals of the inductive circuit, such an 
 arrangement will, if it can be realised, afford a means for 
 obtaining a true numerical measure of the power taken up in 
 the inductive circuit. 
 
 An electro dynamometer having its fixed coil composed of 
 thick wire and its movable coil of fine wire, each circuit 
 being independent, is most usually called a wattmeter. The 
 examination of the circumstances under which the wattmeter 
 can and cannot be used to measure the power expended in a 
 circuit subject to simple periodic electromotive force, leads to 
 some interesting considerations. 
 
 If the thick and thin wire coils of a wattmeter are traversed 
 by two independent steady unidirectional currents, the force 
 
 FIG. 67. 
 
 on the movable coil is at any instant proportional to the pro- 
 duct of the strength of these two currents. If each of these 
 currents are simple periodic currents the force varies with the 
 product of the instantaneous values, and the compound curve 
 formed by taking as ordinates the products of the correspond- 
 ing values of these separate current strengths at each instant 
 is itself a simple periodic curve, provided that the two com- 
 ponent currents have constant amplitudes, equal period, and 
 fixed difference of phase. Let a wattmeter be supposed to be 
 joined up to an inductive circuit (Fig. 67) ; let E and L be 
 the resistance and inductance of this inductive circuit between 
 the points Q Q' ; let the thick wire coil Th of the wattmeter 
 be joined in series with this inductive resistance, and let the
 
 SIMPLE PERIODIC CURRENTS. 
 
 159 
 
 fine wire coil /of the wattmeter of resistance S and inductance 
 N be joined to the points P P' ; let the thick wire coil be 
 of negligible resistance and inductance in comparison with 
 the circuit Q Q'. If a simple periodic electromotive force 
 operates on the double circuit between the points P and P', we 
 shall have a current flowing in B and S. It is required to 
 calculate at any instant the currents in B and S respectively. 
 Consider simply a divided circuit (Fig. 68) in which B and S 
 are the branches. Let x be the current at any instant in B, and 
 y that in S, and let i be the strength of the current in that 
 part of the circuit just before it divides ; in other words, i is 
 the main current, which is divided into x in the inductive 
 resistance and y in the fine wire coil of the wattmeter. Let e 
 be the potential difference between the points P P' at the same 
 instant, and let X, Y, I, and E be the maximum values 
 
 of all these quantities respectively. We assume that i is a 
 simple periodic function of I, and we then write i = I sin p t, 
 where p = 2?r n, n being the frequency. Applying the funda- 
 mental equation of 15 (p. 126) to each circuit, we see that 
 
 dt 
 
 also 
 
 K. 3 ' 1 T X T> XT V 
 
 Accordingly, L + B a; = N -^ 
 
 but, by the principle of continuity, 
 i = x + y 
 
 always, since there can be no accumulation of electricity at 
 P or P' ; 
 
 hence x = iy,
 
 160 SIMPLE PERIODIC CURRENTS. 
 
 and, hence, I/ (t '~ y) +R(t-y) = N^+Sy, 
 dt dt 
 
 or 
 
 at 
 
 But i = I sin ^? t. 
 
 and 
 
 
 or 
 
 dt 
 
 This differential equation is of the type 
 
 where Q is a function of t . The solution of this will be found 
 in " Boole's Differential Equations," p. 38, and it is 
 
 e being here the base of Nap. logs, and not impressed E.M.F. 
 In the case before us P = 
 
 and 
 
 L + .N 
 
 The integrals of e Pt sin p t d t and e Ft cosptdt are required* 
 They are as follows : 
 
 and 
 
 hence it follows that 
 
 RI p (P sin^t-^cosp t) Ijpl e Pt (Pcospt +psin.j)t) 
 = e LTN ~ 
 
 +
 
 SIMPLE PERIODIC CURRENTS. 
 
 Therefore we have by substitution 
 
 _ __ 
 
 (R + S) 2 + (L 
 
 S)L/ , _ 
 
 161 
 
 \ 
 
 Since the original equations are symmetrical in x and y, 
 R and S, L and N, the value for x is given by changing R to 
 S and L to N in the equation for y. 
 
 This equation for y gives us the strength of the current in 
 the fine wire coil, and it shows us that the phase of the 
 currents x and y in the branch circuits differs from that of the 
 main current i by an amount which depends on L, N, R and S. 
 In order to exhibit this in a simple form we may direct atten- 
 tion to a simple trigonometrical transformation. 
 
 FIG. 69. 
 
 Trigonometrical Lemma. The function A sin + B cos 6, 
 where A and B are constants, may otherwise be written 
 
 ^A 2 + B a sin (6 + </>), 
 B 
 A 
 
 Draw any rectangle (Fig. 69) P, Q, and draw a pair of 
 rectangular axes, OX, OY, through 0. Project the points Q, R, P 
 on Y. Then, by geometry, if P X = 6 and P R = <, 
 
 where 
 
 toni-tl. 
 
 = P sin $ + Q cos 6,
 
 162 SIMPLE PERIODIC CURRENTS. 
 
 hence OP sin + Q cos = OR sin (8 + <j>) 
 
 = VO P 2 + Q 2 sin (6 + <). 
 But tan*--|; 
 
 hence A sin 6 + B cos 6 = J&* + B 2 sin (8 + <), . (85) 
 
 where tan < = . 
 
 A 
 
 Returning then to the equation for y, the coefficients of 
 sin 2) t and cos p t in the equation are respectively R (R + S) 
 + L (L + N) f, which represents the A, and (R + S) L p - R 
 (L + N) p, which represents the B, in the above. Squaring 
 each of these expressions, and adding the results, we obtain 
 as a result 
 
 {(B + S) 2 + (L + N) 2 p*} (R 2 + P * L 2 ) ; 
 
 hence we finally arrive by substitution at the equation for y 
 
 
 
 In this form the equation for ?/ shows us that the phase of y 
 is a^at? of that of i, or that the main current lags behind the 
 current in the branch S, provided that S L is greater than R N ; 
 and, since the expression for the current x is perfectly sym- 
 metrical, we can write it down at once, and it is 
 
 (39) 
 
 and it is obvious that, if S L is greater than R N, tan is 
 positive, and tan ff is negative. If SL = RN, then there is 
 no lag, and the branch currents x and y agreein phase with 
 the' main current i. 
 
 The general result is, therefore, this When an impressed 
 electromotive force acts on a circuit which branches into two,
 
 SIMPLE PERIODIC CURRENTS. 163 
 
 having each self but no mutual induction, there is a difference 
 of phase between the currents in the main line and branches ; 
 that is, they do not come to their maximum values at the 
 same instant. The main current lags behind the impressed 
 electromotive force in phase, and the two branch currents 
 respectively lag behind and are pressed ahead of the phase of 
 the main current. 
 
 The question then arises, under what circumstances does 
 the branch current which is in advance in phase of the main 
 current get so much ahead that it comes into consonance with 
 the phase of the impressed electromotive force ? 
 
 To settle this question we shall have to discuss briefly the 
 question of the compound impedance of branch circuits. 
 
 26. Impedance of Branched Circuits. Lord Kayleigh has 
 treated the problem of the impedance of branched circuits 
 under the assumption that any number of circuits are 
 connected in parallel, posessing each self-induction, but 
 having no mutual induction.* 
 
 The problem is : Given the resistance and inductance of each 
 branch, to find the compound resistance and inductance, or 
 equivalent resistance and inductance, of the system for simple 
 periodic currents of given frequency. 
 
 Let K and L be the resistance and inductance of any branch, 
 and p the pulsation = TT n. Let E' and L' be the compound 
 or equivalent resistance and inductance of the system of 
 parallel conductors. 
 
 The solution of the problem given, for which we refer the 
 reader to the original paper, is 
 
 where 
 and 
 
 If we take, as usual, tan 9 = P _ , and write (Im) for impedance, 
 R 
 
 * See Lord Rayleigh " On Forced Harmonic Oscillations of Various 
 Periods " (PhU. May., May, 1886, p. 379). 
 
 Al2
 
 164 SIMPLE PERIODIC CURRENTS. 
 
 where (Im) 2 = R 2 +;/L 2 , we can write the above relations 
 
 Let B"+/> 9 L' 2 be written (IM) 2 . This is the compound or 
 equivalent impedance of the system of parallel conductors. It 
 is obvious that 
 
 hence (IM) 2 = 
 
 Consider the case of a pair of conductors in parallel (Fig. 70) r 
 having resistances R and S and inductances L and N, but no- 
 mutual inductance. 
 Let 
 and 
 and 
 
 then (IM) 2 
 
 The lag e of the maui current just before branching, con- 
 sidered with respect to the impressed electromotive force, will 
 be given by the equation 
 
 hence tan.--
 
 SIMPLE PERIODIC CURRENTS. 
 generally, and in the case considered will be 
 
 165 
 
 tan c 
 
 .hence after reduction 
 tan e = ^~r. 
 
 This is the equation which determines the lag of phase of the 
 current i behind the impressed electromotive force in the main 
 branch before dividing into the branch currents x and y in E 
 .and S respectively. 
 
 Compare this equation with that which determines the 
 angle by which the phase of the branch current y in S is 
 .a/iead of the main current i. It is, as we have seen, 
 
 (SL-RN)p 
 
 In the expressions for tan e and tan 6 put N = 0, and they both 
 become equal to 
 
 This shows that, when N = 0, the current y in the branch S 
 is as much ahead of the main current i as i is behind the im- 
 pressed electromotive force, and hence that y agrees in phase 
 with the impressed E.M.F. acting on the double circuit; 
 in other Words, the current in the branch S is entirely 
 unaffected by being joined in parallel with an inductive 
 circuit R; but if N is not quite zero, then the current in 
 branch S is affected, as regards its lag, by the fact of being
 
 166 SIMPLE PERIODIC CURRENTS. 
 
 joined in parallel with an inductive circuit. The nature of 
 this affection will be dependent on whether SL RN is 
 
 positive or negative that is, whether or is the greater 
 
 R b 
 
 that is, whether the time constant of the R circuit or the 
 
 S circuit is greater. If is greater than , then the current 
 R b 
 
 y in S is ahead of the main current i, but lags behind the im- 
 pressed electromotive force. If - is less than ~, then the 
 R b 
 
 current y hi S lags behind the main current i in phase, and,. 
 a fortiori, behind the impressed electromotive force. 
 
 27. Wattmeter Measurement of Periodic Power. 
 Returning to the wattmeter problem, let one of these 
 divided circuits, viz., the one of resistance R, be a circuit 
 in which it is desired to measure the electrical power. In the 
 ordinary way of using the wattmeter, the fine- wire coil, which 
 we will assume has a resistance S, is placed in parallel with 
 the inductive circuit, the thick- wire coil united in series with 
 the inductive circuit. The main current i is thus divided 
 between the inductive circuit R and the wattmeter fine-wire 
 circuit S. The electro-dynamic action in the wattmeter is 
 then one between a current in S, which we have called y, and 
 one hi the thick- wire circuit, which is the same as that in the 
 inductive circuit R, which we have called x. 
 
 We have above arrived at expressions for the values of ar 
 and y. The question then arises how far the indications given 
 by the instrument, and which are due to the electro-dynamic 
 action of the currents x and y, and proportional to their nume- 
 rical product, are proportional to the real power taken up in 
 the circuit R. 
 
 The current x is the same as the current in R ; hence the 
 error, if any, will result from the current y in S differing in 
 phase or in proportionality from the potential difference 
 between the ends of the circuit R. 
 
 In the ordinary mode of calibrating the wattmeter the 
 instrument would be applied to measure a power hi a non- 
 inductive circuit traversed by a known current, and having 
 a known potential difference at its ends.
 
 SIMPLE PERIODIC CURRENTS. 167 
 
 From this the real watts taken up in the circuit are known 
 and, since the force required to bring back the movable coil 
 to its initial position is proportional to the product of the 
 numerical values of the currents in the fixed and movable 
 coils, we have at once the desired constant of the instrument. 
 
 If a wattmeter so calibrated is applied to measure power 
 in an inductive circuit, there are two causes of error which 
 may or may not neutralise each other, and which may 
 cause the measured watts as determined by the instrument to 
 be greater than, equal to, or less than, the real watts or power 
 taken up in the circuit. 
 
 The first of these causes of error is due to the fact that the 
 fine-wire circuit of the wattmeter always has a sensible induc- 
 tance that is, N is not zero. It may be made very small 
 by arranging the chief part of the wire resistance of the fine- 
 wire circuit as a non-inductive resistance in series with the 
 small inductive resistance which forms the movable coil. It 
 follows that, if E be the maximum potential difference during 
 the period between those points to which the fine-wire circuit 
 is attached, the mean-square ( y'mean' 2 ) value of the current hi 
 
 1 E 
 
 the fine-wire circuit is equal to 7= --when subjected 
 
 to a simple periodic E.M.F. of angular velocity p. This 
 quantity is not proportional merely to E, but depends also 
 on the value of p. One effect of the impedance of the fine- 
 wire circuit is to make the mean-square current in it under 
 periodic E.M.F. less than it would be if produced by a steady 
 E.M.F. equal to the mean-square value of the periodic E.M.F. 
 But, in addition, the impedance causes a lag in phase of the 
 current in the fine-wire circuit behind the phase of the poten- 
 tial difference between its ends. This is the second cause of 
 error, and the effect of this lag is dependent upon the nature, 
 whether inductive or non-inductive, of the circuit E. 
 
 To dissect its action, first let us suppose the circuit R is 
 non-inductive that is, let L be zero. The current a; in it 
 will, therefore, coincide in phase with that of the potential 
 difference at the points of junction. The current hi S, viz., 
 y, will, however, lag in phase behind that of the potential 
 difference at the junction. The effect of this lag in S will be 
 to increase the phase difference between x and y, and to
 
 168 SIMPLE PERIODIC CURRENTS. 
 
 diminish the cosine of this angle of phase difference. Hence, 
 the effect is to diminish the product - cos 8, which measures 
 
 the true mean product of x and y, X and Y being their 
 maximum values and 8 their difference of phase. Since by 
 assumption X agrees in phase with E, any reduction of 
 the above product reduces the instrumental reading, and 
 makes it less than the true-power reading. If, however, 
 we have to deal with a circuit possessing inductance, and 
 in which, therefore, there is a current x, of which the phase 
 lags behind that of the potential difference of the junc- 
 tions, then the lag in the current y in the circuit S, so far 
 from increasing the difference of phase of x and y, may operate 
 to bring them nearer into accordance, and to increase the 
 instrumental reading, and more than make up for the decrease 
 due to the first-named cause of error. 
 
 28. Correcting Factor of a Wattmeter. The action of 
 these two causes of error may be illustrated and explained best 
 by the graphic method by a construction which at the same 
 time shows us how to obtain geometrically the value of 
 the compound resistance and impedance of a branched circuit. 
 
 Describe a circle with centre (Fig. 71), and take any line 
 OA to represent the maximum value of the potential dif- 
 ference between the two points M M' of the divided circuit, of 
 which E. is the resistance of the inductive circuit consisting 
 of the thick wire of the wattmeter in series with the circuit in 
 which the poiver is being measured, and S that of the fine wire 
 of the wattmeter. Then, as before, the vertical projection of 
 A as it revolves represents the periodic variation of this 
 potential difference. On A describe a semi-circle, and set off 
 on A, as a base, two right-angled triangles OCA, B A, 
 of which the sides OB, B A, and 00, C A are in the ratio 
 respectively of the resistance to the reactance of these 
 circuits. Otherwise the angle A B is one whose tangent 
 is p times the time-constant of the S circuit, and A C is one 
 whose tangent is p times the time-constant of the E circuit. 
 Take one S th portion of OB, and set off OY equal to it, 
 then, as in 21 (p. 144), Y represents the maximum value 
 Y of the current in the S circuit.
 
 SIMPLE PERIODIC CURRENTS. 
 
 169 
 
 Similarly, set off X equal to one E th part of C, and 
 ( X represents the maximum current X in E. On X, Y 
 describe a parallelogram Y I X, and draw the diagonal I, 
 and produce it to D. Then I represents the maximum 
 value of the main current I just before division. Join A D ; 
 A D and D will represent the product of the current I and 
 the equivalent reactance and resistance of the two circuits 
 K and S in parallel respectively. 
 
 To prove this last proposition, we must refer again to the 
 paper by Lord Rayleigh on " Forced Harmonic Oscillations 
 -of Various Periods " (Phil. Hag., 1886). 
 
 FIG. 71. 
 
 If E' represents the equivalent resistance of a number of 
 resistances joined in parallel between two points, and L' repre- 
 sents the equivalent inductance of the system, then it is shown 
 in Lord Eayleigh's paper that 
 
 E ' = 'B' and L/ = A 5 ^ B 5 ' 
 
 where
 
 170 SIMPLE PEEIODIC CURRENTS. 
 
 R and L being the resistance and inductance of any branch 
 and the mutual inductance being zero. 
 
 Apply this theorem to the case under consideration, viz, the 
 two inductive resistances (E, L) (S, N) in parallel, and we have 
 
 R S 
 
 ~R 2 +/L 2 
 Effecting the multiplication we have 
 
 R (S 2 +p 2 N 2 ) + S (R 2 +p* L 2 ) 
 
 (b*+>N a )(H a +7>*L s ) 
 L (S 2 + j> 2 N 2 ) + N (R 2 + jo 2 L 2 ) 
 
 
 - 
 
 Turnhig back to Fig. 71, we see from the geometry of 
 the figure that, if the angle B D is as before called 6, 
 BOD = DAB. 
 
 We have then _H = B - A B tan 0. 
 
 cos 
 
 But since A B=p N Y and B = S Y by construction, 
 therefore D = S Y cos 0-p N Y sin 0. 
 
 In 25 we have found the value of tan B to be 
 
 (SL-RN)jp 
 
 ~R(R + S)+L(L + N)/' 
 
 hence, eliminating the sin and cos terms, and substituting for 
 Y the value obtained from equation (36), page 162, we get 
 
 where I is the maximum value of the main current, and Y that 
 of the current hi the S circuit. 
 
 On comparing this value for D with the value above 
 calculated for R' we see that D = R' I.
 
 SIMPLE PERIODIC CURRENTS. 171 
 
 So that, on the same scale on which B and C represent 
 S Y and R X, D represents R' I. Similarly, it may be shown 
 that A D =p L' I. For the angle C D = & = angle CAD, and 
 
 AE *=AC-QC tang'. 
 cos 6 
 
 or AD=pLXcos 0'-RXsiu2'; 
 
 a similar substitution, with help of equation (39), page 162, 
 enables us to see that 
 
 and this is equal to the value found by analysis for p L' I. 
 
 This diagram shows us, then, what is the effect of the 
 inductance of the wattmeter fine-wire circuit, and what must 
 be the correction applied to the readings to get the real power 
 expended in the inductive circuit. 
 
 The actual reading of the wattmeter is proportional to the 
 true mean value of the product of x the current in the induc- 
 tive circuit R and y the current in the fine-wire circuit S ; 
 and this, as previously shown, is equal to half the product oi 
 their maximum values, and the cosine of the difference of 
 
 From Fig. 71 this mean value is therefore 
 QX 2 OY cosine BOG. 
 
 This, however, is not the measure of the power expended hi 
 the R circuit. The true watts are proportional to the mean 
 product of x and a current equal to one S th part of e, having 
 a phase difference equal to the angle C A, viz., that of the 
 angle of lag of the current in R and the potential difference 
 A of its ends. Hence the real power or watts are proportional 
 
 to ? . X . cosine C A, 
 
 S 
 
 since E is the maximum of e, viz., the instantaneous potential 
 difference between the extremities of the branch circuits. 
 
 NowOY is taken as one 8 th part of the effective electro- 
 motive force in the S circuit ; and on the same scale on which
 
 472 SIMPLE PERIODIC CURRENTS. 
 
 O A represents the impressed E.M.F. B represents the effec- 
 tive E.M.F. in that circuit. Hence, in taking the reading of the 
 wattmeter, which is proportional to the quantity 
 
 X - OY cosineBOC, 
 
 2 
 
 as the watts, we are making an error ; the quantity really 
 required is the value of 
 
 ~ X cosine A C, 
 
 2 S 
 
 which is numerically equal to the real power. We see that 
 two errors come in one due to the maximum current in 
 
 f\ -p Tn 1 O A 
 
 the fine-wire circuit being Y or - instead of - or - 
 
 b b b 
 
 and the other due to the phase difference being taken as the 
 angle COB instead of C A. 
 
 To correct the instrumental reading or observed watts to 
 true value or real watts, we have to multiply the observed 
 readings by two factors. 
 
 First, the ratio of - or 
 
 - 
 
 OB b 
 
 which is the correction due to the self-induction of the fine- 
 wire circuit or to the potential part of the wattmeter having 
 a sensible inductance. The second is the ratio of the cosines 
 of the angles C A and C B, or 
 
 cosCOA_ cos CO A _ & 
 cos C B = cos (C A - B A) ~ ' 
 
 But from the diagram 
 
 cosine C A = --_-r. -, 
 
 and cosine B A = 
 
 JB 
 
 TE^L'
 
 SIMPLE PERIODIC CURRENTS. 17? 
 
 Combining these two corrections into a single product, we 
 get as the full correcting factor : 
 
 or 
 
 If we put T s = , where T s is the time-constant of the S,. 
 
 or fine-wire circuit, and T R = _ , where T R is the time-constant 
 
 it 
 of the B circuit, we have 
 
 F- 1+ ? 2T * 2 , (41) 
 
 2 ' 
 
 and the real watts or power taken up in the circuit B is ; 
 obtained by multiplying the observed watts by F. F becomes 
 unity for two cases when L and N are both zero, and also 
 when T S = T R . 
 
 Hence, the ordinary wattmeter, applied as usual to measure 
 the electrical power in a circuit traversed by a simple periodic 
 current, gives absolutely correct readings only in two cases. 
 First, when the fine-wire circuit and the circuit being measured 
 have no inductance ; second, when the fine -wire circuit and the 
 circuit being measured have equal time-constants. 
 
 But if T R is greater than T s , then F is a proper fraction. 
 The wattmeter reads too high, and the real watts are less than 
 the observed. If T K is less than T s , then the observed readings 
 are too low. If T R = T s , then the observed readings are correct. 
 Hence the wattmeter may read too high, too low, or correct. 
 Generally speaking, it reads too high, since the time-constant 
 of the measured circuit will most often be in excess of that of 
 the fine-wire circuit. 
 
 29. Mutual Induction of Two Circuits of Constant In- 
 ductance. As an illustration of the above principles, it is 
 useful to consider the case of the mutual induction of two 
 circuits in one of which a simple periodic electromotive force 
 operates. We suppose two circuits to be so placed relatively to 
 each other that when a change of current occurs in one, which 
 is called the Primary (Pr.), a change of magnetic induction.
 
 174 SIMPLE PERIODIC CURRENTS. 
 
 takes place through the other, called the Secondary (Sec.). 
 We have, then, to regard the primary and secondary as linked 
 together by loops of induction, and the closed lines ol 
 induction, together with the two circuits, must be considered 
 as forming three links of a chain. We shall suppose the self 
 and mutual inductance to be known and to be constant, as 
 also the resistance of each circuit. The primary inductance 
 and resistance will be denoted by L and E, and those of 
 the secondary by N and S, and the mutual inductance by M. 
 The primary circuit is to be subjected to a simple periodic 
 electromotive force of which the maximum value is E, and 
 the result is to generate in the primary circuit a primary 
 current, which, as we have seen, is also a simple periodic 
 quantity, and is to be denoted by its maximum value, Ij. 
 The change of induction through the secondary follows the 
 change of current, and gives rise to an impressed electromotive 
 force in the secondary circuit, which, being represented by the 
 rate of change of the simple periodic induction, is also a 
 simple periodic quantity, and gives rise to a simple periodic 
 current in the secondary, to be denoted by its maximum 
 value I 2 . 
 
 The general description of the phenomena produced in such 
 a system of primary and secondary circuits connected by an 
 air magnetic circuit is a follows : 
 
 1. The application of a simple periodic impressed electro- 
 motive force, E, produces a simple periodic current, I 1} moving 
 under an effective electromotive force, KI U and brings into 
 existence a counter electromotive force of self-induction, 
 which causes the primary current Ij to lag behind E by an 
 angle called the primary lag : . If n is the frequency of the 
 vibrations and 2Trn=p, as before; then, as we have before seen, 
 
 tan #!= * and this counter electromotive force of self- 
 induction is a periodic quantity of which the maximum 
 is Lpln and of which the phase is 90 behind that of the 
 effective electromotive force or current, or is in quadrature 
 with it. 
 
 2. The field round the primary, and therefore the induction 
 through the secondary, is in consonance with the primary 
 current I a j but, since it is also a simple periodic quantity, its
 
 SIMPLE PERIODIC CUEEENTS. 175 
 
 time-rate of change, and therefore the impressed electromotive 
 force in the secondary, is in quadrature with the primary 
 current. Since the induction through the secondary, due to 
 a current ^ in the primary, is M I 1} by the definition of M the 
 maximum value of the rate of change of this induction for a 
 pulsation p is M p I lt 
 
 It is useful to note that in all dealings with simple periodic 
 quantities, if X is the maximum value of a simple periodic 
 quantity which runs through its cycle n, times in a second, the 
 maximum value of its time-rate of change is denoted by p X, 
 where p = 2-!rn. 
 
 If, then, as usual, simple periodic quantities are denoted by 
 the letter signifying their maximum values, prefixing p to any 
 one gives us the value of the maximum of its first differential 
 coefficient with regard to time, or p is here equivalent in 
 
 notation to 
 dt 
 
 3. This secondary impressed electromotive force gives rise 
 to a secondary current, I 2 , moving under an effective secondary 
 electromotive force, S I 2 , and creating a counter electromotive 
 force of self-induction in the secondary, represented by N p I 2 . 
 
 The secondary current lags behind the secondary impressed 
 electromotive force by an angle 0., such that 
 
 4. This secondary current, I 2 , reacts, in its turn, on the 
 primary, and it creates what is called a back electromotive 
 force, or reacting inductive electromotive force, on the primary 
 circuit. The phase of this must be in consonance with that of 
 the electromotive force of self-induction in the secondary, and 
 it is represented by the quantity M I 2 ^>. This is obviously 
 in quadrature with the phase of the secondary current or 
 secondary effective electromotive force. 
 
 5. There is, then, a phase difference between the primary and 
 secondary currents, and also between the primary impressed 
 electromotive force and the primary current. 
 
 The general problem is, then : Given the value of the induc- 
 tances L, M, N, and the resistances R, S, and that of the im- 
 pressed electromotive force E and the frequency n, find from 
 these seven quantities other four, viz., the primary current I Jf
 
 176 SIMPLE PERIODIC CURRENTS. 
 
 the secondary I 2 , and the difference of phase between E, I lf . 
 and I a . 
 
 We shall attack the problem geometrically, as this method 
 exhibits far better than the algebraic method the relation 
 between the various quantities involved. The method adopted 
 is to construct an electromotive force diagram, in which all 
 lines represent on any scale volts ; and moreover, as each of 
 the quantities considered is a periodic quantity, the lines all 
 represent the maximum value of each quantity, and the value 
 at any instant can be obtained by taking the projections of all 
 lines on any straight line through the centre of the diagram, 
 suitably placed. 
 
 Let (Fig. 72) be taken as a centre ; draw any line Q', 
 and on it set off any length, T, which we assume as the 
 magnitude of the maximum of the primary current. All other 
 lines will be in proper proportion to this. Produce T to Q 
 so that Q = R Ii- Q is then the effective electromotive force 
 in the primary circuit. From Q draw Q P at right angles to 
 Q, and set off Q P equal to L p times T or to L p I x ; Q P 
 represents the electromotive force of self-induction in the 
 primary circuit. Join P. 
 
 From draw C at right angles to Q, and set off C' 
 equal to M.p times T. C is then equal to M p I lt and C 
 represents the impressed electromotive force in the secondary 
 circuit. On C describe a semi-circle, and set off B, making 
 
 an angle COB with C such that tan C B = 5L?, or tan 
 
 b 
 
 C B = the ratio of the inductive to the ohmic resistance for 
 the secondary circuit. Join B C. On the same scale on which 
 C represents the impressed electromotive force in the secon- 
 dary circuit, viz., M^I^ OB will then represent the effective 
 electromotive force in the secondary, or will represent S I 2 , and 
 hence, if D is taken equal to one S th part of B, D will 
 represent I 2 , or the secondary current. Next draw a line K 
 perpendicular to B, and therefore parallel to B C, and on it 
 set off a length, K, equal to Np times D or to Mp I 2 . K 
 represents then the back inductive electromotive force of the 
 secondary on the primary. 
 
 The impressed electromotive force which has to be applied 
 to the primary to produce in it the primary current T and
 
 SIMPLE PERIODIC CURRENTS. 
 
 177 
 
 to induce in the secondary the secondary current D has 
 therefore to be equal and opposite to the resultant of three 
 
 
 \ 
 
 M 
 
 Mp', 
 
 / 3 
 
 ' 3 
 t 
 
 a 
 
 s 
 
 / ^ 
 
 >X* 
 
 electromotive forces, or to equilibrate three electromotive 
 forces, viz., the effective electromotive force of the primary
 
 178 SIMPLE PEEIODIG CURRENTS. 
 
 Q, the electromotive force of self-induction in the primary 
 P Q, and the back electromotive force in the primary due to, 
 the inductive effect of the secondary on the primary, viz., K. 
 
 The resultant of Q and Q P is P. If, then, we draw 
 P P' from P, and make it equal and parallel to K, and join 
 O P', P' will be the resultant of Q, Q P, and K ; and 
 hence P' will represent E, or the impressed electromotive 
 force required to be applied to the primary to maintain the 
 currents Ij and I^. 
 
 It is to be understood that in this diagram a unit of length 
 stands for a volt, or unit of electromotive force ; and hence, 
 on that assumption, E represents in volts the impressed 
 E.M.F. that is, the maximum of the simple periodic E.M.F. 
 required to maintain the currents I : la, of which T, D 
 represent the maximum values. The relative phases are 
 indicated by the positions of these lines. To obtain the actual 
 values of the E.M.F. and currents at any instant we have only 
 to take the projections of P', T, and D on any line 
 drawn through suitably placed, and the magnitudes of these 
 projections will give the required quantities. We must then 
 suppose the whole diagram to be enlarged or diminished 
 without distortion until the length of P' is numerically equal 
 to the maximum value in volts of the impressed E.M.F. E, 
 and then T and D, will represent the currents I x and L in 
 magnitude. We may consider the two right-angled triangles 
 Q P, B C as pivoted together at 0, and revolving round ; 
 the fluctuations of the projections of P', T, D on any 
 line will give us the cyclic values of E, I : , and I 2 . We can 
 next obtain some useful relations between these quantities 
 from the geometry of the figure. In the triangle B C, 
 
 Hence I? W f = S 2 L 2 + N 2 f I 2 2 ; 
 
 or ji_x/S*+^N*. 
 
 I 2 ' Mp 
 
 primary current impedance of secondary 
 secondary current" \\.p 
 
 M> might by analogy be called the mutual reactance. 
 
 To obtain the value of \ in terms of E and the inductances 
 and resistances, we project the lines OP' and OP on the
 
 SIMPLE PERIODIC CURRENTS. 179 
 
 -vertical line K, and express the fact that P' or E is in all 
 cases the resultant of K and P. Let the angle P' Q' be 
 called <. cj> is the angle by which the primary current lags 
 behind the total impressed electromotive force. Then COB 
 is B it and T K = C B = 9,, since Q C and K B are both 
 right angles. 
 
 Hence we have by resolution on K 
 E cos (<+ a ) 
 
 but. since L 
 
 we have by substitution 
 
 E cos (* + 0,} = {7== + ^B*+^L' co 3 (6, + &)} I,, 
 
 and therefore a relation established between E and Ij which is 
 known when < is known. 
 
 Since tan 9, =*, and tan 2 =^, 
 
 lii O 
 
 it follows by an easy transformation that 
 RS- 2 LN 
 
 .hence E = a . . 
 
 y {5* _[. p- ]\' 2 cos (<p + tt>) 
 
 To find the value of <, suppose that whilst E and Ij remain 
 the same, the secondary circuit is suppressed. We should 
 then only have an impressed electromotive force, E, creating 
 a current, I 15 and from the diagram and from what has been 
 before explained it is obvious that the effective and self-induc- 
 tive electromotive forces in the circuit would then be repre- 
 sented by Q' and Q' P'. If we denote these by the symbols 
 B/I! and L'^Ii we may properly call E' and L' the equivalent 
 resistance and inductance ; that is to say, these quantities are 
 the resistance and inductance which the primary circuit should 
 have in order that, when there is no secondary circuit, 
 the primary impressed electromotive force may generate in 
 it the same current which it does when the secondary circuit 
 is present and the primary has its natural resistance R and 
 inductance L. We see, then, that the effect of bringing up 
 the secondary and allowing it to be acted upon and react
 
 180 SIMPLE PERIODIC CURRENTS. 
 
 upon the primary is to increase the effective resistance and 
 diminish the effective inductance of the primary ; in other 
 words, the equivalent resistance of the primary circuit is- 
 greater and the equivalent inductance is less by reason oi 
 the presence of the secondary circuit. 
 
 We have then to find the value of cos (< + ft 2 ). 
 Cos (<f> + ft 2 ) = cos < cos ft 2 - sin < sin ft, ; 
 
 R f T '" 
 
 but cos 6 = . sin <i = 
 
 v'R' 2 +p 2 L' 2 
 and cos ft, = ^ sinft 2 - N ^ 
 
 hence cos (</> + ft 2 ) = -^ B'S-L'Njf 
 
 Substituting this in the equation connecting E and I 1} we 
 arrive at 
 
 Es=I (MV + B S - jt) 2 LN) ^B^+y'L'" 
 
 K'S-L'N^ 2 
 
 Returning to Fig. 72, we see from it that P' Q' is parallel 
 to P Q, and hence, if we draw P'V parallel to Q' Q, we have 
 P'Q' = PQ-PY 
 
 = PQ-PP'cosP'PV 
 = PQ-PP'sin6> 2 , 
 
 or 
 
 or , smc e 
 
 we have 
 
 Also, again, Q represents E Ij and Q' on the sama- 
 scale R' I x , and 
 
 Y 
 
 OQ + PP'sinP'PV 
 Q + P P' cos ft, ; 
 
 hence B f ^- 
 
 and ; therefore; R' =
 
 SIMPLE PERIODIC CURRENTS. 181 
 
 These formulae (42) and (43) give us the effective inductance 
 and resistance of the primary circuit as affected by the secon- 
 dary. They were first given by Clerk Maxwell in a paper in 
 the Philosophical Transactions of the Koyal Society in 1865, 
 entitled " A Dynamical Theory of the Electromagnetic Field " 
 {Phil. Trans., 1865, p. 475).* 
 
 If we form from (42) and (43) the function E' S L' N^, we 
 find it to be M' 2 ^? 2 + E S /> 2 LN; and hence, by substitution 
 in the expression already given connecting E and Ii, we arrive 
 finally at the result 
 
 Following thejisual nomenclature, we may call the expres- 
 sion ^/E'' 2 +^ 2 L' 2 the equivalent impedance of the primary 
 circuit, and we have as the final result for the induction coil 
 of constant inductance 
 
 Primary current ^pressed electromotive force 
 streno-th = equivalent impedance of primary 
 circuit 
 
 Secondary current _ -*r primary current 
 
 strength " impedance of secondary circuit* 
 
 The angle of lag of primary current behind impressed 
 E.M.F. = <j>, where tan </>=-??--_, and the angle of lag of the 
 secondary current behind the primary is seen to be 90+ 2 and 
 tan # 2 = i_jP ; hence we have the values and relative phases of 
 
 h 
 
 the currents and the impressed electromotive force. 
 
 In the above equations we are to understand current strengths 
 and electromotive forces to be the maximum values during 
 the period. If ^ and i. 2 be the actual values at any time t, 
 reckoning time from the instant of the zero value of the 
 electromotive force, then, from the principles previously 
 
 * Sec also Lord Rayleigh on " Forced Harmonic Oscillations of Various 
 Periods," Phil. May., May, 1886, p. 375.
 
 182 SIMPLE PERIODIC CURRENTS. 
 
 explained in this chapter, it is obvious that 
 *! = !! sin (pt-$), 
 
 ' P 
 
 or i t = sin (p t - <i) ; 
 
 x/E'- + ^L' a 
 
 and I,-^ 1 ' 
 
 or 
 
 The student will find the above expressions for the primary 
 and secondary currents can be deduced by analytical processes 
 from the simultaneous equations. 
 
 , . , (44) 
 a t a t 
 
 N^L + M^A + S? 2 = 0, . . , , (45) 
 d t d t 
 
 which equations can be established for two circuits by analo- 
 gous methods to that by which in 18 a current equation 
 was arrived at for one circuit, subject to a simple periodic 
 electromotive force.* It is easily seen that if n is very 
 great, or the alternations extremely rapid, then 
 I : _N 
 T 2 ~M' 
 
 If the primary and secondary circuits consist of two equal 
 circuits, so interwound that for these circuits L = M = N, then 
 for very rapid alternations we see that the secondary current 
 I 2 is equal in magnitude, and exactly opposite in phase, to 
 the primary current I 1} and the magnetic fields due to these 
 currents respectively are also equal and opposite in direction 
 at every instant. 
 
 30. The Flow of Simple Periodic Currents into a Con- 
 denser. The electrical capacity of a conductor of any kind is 
 measured by the quantity of electricity required to charge it 
 to unit potential. Two conducting surfaces so arranged as 
 to have constant capacity are generally called a condenser. 
 The most simple and familiar form of this appliance is the 
 Ley den jar, in which two surfaces of tin foil are separated by a 
 
 * See Mascart and Joubert's " Electricity," Vol. L, p. 521.
 
 SIMPLE PERIODIC CURRENTS. 183 
 
 sheet of glass. Condensers for practical purposes are generally 
 formed of sheets of tinfoil, separated by some dielectric, such 
 as paraffined paper, mica, gutta-percha, tissue, or ebonite. If 
 a quantity of electricity, Q, measured in microcoulombs, is 
 given to one of the plates or sets of plates of a condenser, 
 and if the other set are kept at zero potential, the charged 
 plates will be raised to a certain potential, say V volts, above 
 that of the earth. The capacity of the condenser C in 
 microfarads is then such that by definition 
 
 CV = Q. 
 
 If the potential difference of the condenser terminals at any 
 instant is v, and if it is changing, and if q is the charge or 
 quantity of electricity in the condenser at that same instant, 
 then 
 
 C = ; 
 dt dt 
 
 but ?-? is the time rate at which the charge is changing or 
 
 dt 
 
 is the value of the current i at that instant flowing into or 
 out of the condenser. Hence 
 
 -t f^v . 
 
 and C =z. 
 
 dt 
 
 Suppose the condenser is being charged through a circuit 
 whose resistance is E, then i is the current which is flowing 
 through this resistance at the instant considered ; and the fall 
 of potential down the resistance is R i volts. If a constant 
 potential, V, is being applied to the outer extremity of the 
 resistance, so that the total difference of potential between 
 one plate of the condenser and the outer end of the resistance 
 attached to the other plate is V volts, we have at any instant 
 the equation 
 
 v + Ei = V, 
 
 or v+CR ( -=V, 
 
 dt 
 
 or CR'-+t?=V ...... (4G) 
 
 a t
 
 184 SIMPLE PERIODIC CURRENTS. 
 
 This is the differential equation for the potential v of the 
 condenser at any instant, t, when being charged through a 
 resistance, E, by a constant potential, V. The equation 
 (46) is easily integrated as follows : Multiply all through 
 
 by e n ~ c , where e is the base of the Napierian logarithms, viz., 
 2-71828. Then we have, as a result, the equation 
 
 or . 
 
 dt 
 
 Both sides of this last equation can then be integrated. Hence, 
 integrating with respect to time, we obtain the equation 
 
 t_ _t_ 
 v e= V<? RO + a constant, 
 
 The constant C is determined by the condition that when 
 the time * is zero then v is also zero. Hence C= - V, and, 
 therefore, 
 
 ' .... (47) 
 
 The student should compare equations (46) and (47) for the 
 value of the instantaneous potential of a condenser charged 
 by a constant voltage with the equations (26), page 127, 
 and (27), page 132, for the instantaneous value of a current 
 in an inductive circuit acted on by a constant pressure, and it 
 will be seen that they are similarly constructed. The quantity 
 
 CE is called the time-constant of the condenser just as _ 
 
 is called the time-constant of the inductive coil. The greater 
 the condenser time-constant the larger will be the time taken 
 by the condenser potential to arrive at a given fraction of the 
 steady impressed voltage applied to charge it. Practically, 
 the condenser is fully charged in a time equal to eight 
 or ten times its true constant. The reader must note that 
 if the capacity C of the condenser is measured in microfarads, 
 then the resistance E must be measured in megohms to 
 obtain the correct measure of the time-constant C E in seconds.
 
 SIMPLE PERIODIC CURRENTS. 186 
 
 Suppose, for example, that a condenser of one microfarad 
 is being charged through a resistance of one megohm by an 
 impressed voltage of 100 volts. It is required to find the 
 potential difference at the terminals of the condenser at the 
 end of the 1st, 2nd, 3rd, 4th, nth second. Let v lt r 2l r 3 , v n be 
 these potential differences. 
 Then t\ = 100 (1 - e~ l ) = 63 volts nearly, 
 
 where = 2-71828, 
 
 and r 2 = 100 (1 - e~~) = 86 volts nearly, 
 
 Since e~ w is an exceedingly small number, in ten seconds the 
 condenser potential is practically equal to 100 volts. We see, 
 therefore, that in charging condensers through resistances 
 sufficient time must always be allowed for the charge, depend- 
 ing on the value of the quantity C E, and the charge is not 
 practically complete unless contact with the source of im- 
 pressed voltage endures at least for a time equal to 5 C E, or, 
 better still, 10 C E. 
 
 Supposing, in the next place, that the impressed voltage 
 acting on the condenser is periodic in character or alternating, 
 and that the condenser, instead of being charged through a 
 resistance, has its terminals shunted by a resistance. Let 
 the capacity of the condenser be, as before, C microfarads, and 
 let the resistance of the shunting conductor be _ ohms ; that 
 
 is, let K be the shunt conductance. Let the impressed 
 voltage be periodic, and let its frequency be n, and write p 
 for 2-, as usual. Then, as above, the current flowing into 
 the condenser at the instant when its terminal potential 
 
 difference is v is C f . Also the current flowing through the 
 
 a t 
 
 shunt is Kt-. We have, therefore, as the equation for the 
 total current flowing into the shunted condenser, the 
 -expression 
 
 C+Kt-t, 
 
 a t 
 
 where i is the total current flowing into condenser and 
 through the shunt at the instant when the terminal potential 
 difference of the condenser is r. Let time be measured from
 
 186 SIMPLE PERIODIC CURRENTS. 
 
 the instant when the total current denoted by i is zero, then,. 
 if i is a simple periodic current, it may be expressed in terms- 
 of its maximum value by the equation 
 i = Isinpt. 
 
 Hence C + Kv = lsmpt ..... (49) 
 
 at 
 
 On comparing these equations, (48) and (49), for the- 
 instantaneous value of the condenser potential difference 
 when a periodic current is flowing through it with the 
 equations (29), page 133, and (31), page 135, it will be seen 
 that they are structurally the same, and we can at once 
 write down the solution of equation (49) by imitating the 
 solution of (29), writing C instead of L, K instead of R,> 
 instead of t, and I instead of E. Hence the solution of 
 
 C K 
 dt 
 
 sin (p t 6} + a constant, . (50) 
 
 where is an angle such that tan 6 = _^. 
 
 K 
 
 The quantity \/K- + G 2 ^> 2 has not yet received an acknow- 
 ledged name, but it is analogous to the impedance in the 
 case of the current flow in inductive circuits. It has been 
 suggested that this quantity should be called the admittance of 
 the condenser. 
 
 From equation (50) we see that the terminal voltage of the 
 condenser lags behind the charging current in phase ; or 
 otherwise that the current is in advance of the potential 
 difference. If the shunt wire is removed this is equivalent 
 to making K equal to zero, and under these circumstances w& 
 have 
 
 v = _ sin (p t - 90), 
 Gp 
 
 or v = ---cospt ....... (51) 
 
 This shows that in the case of a condenser having a di- 
 electric with no true conducting power, the charging current 
 is exactly 90deg. in advance in phase of the terminal potential 
 difference of the condenser.
 
 SIMPLE PERIODIC CURRENTS. 187 
 
 It is clear, therefore, that as regards producing lag or lead 
 of current capacity acts in the opposite direction to inductance^ 
 and may be considered to be equivalent to a negative inductance. 
 
 It is also evident that if a shunted condenser is placed in 
 series with an inductive circuit a proper relation may be formed 
 between the inductance, resistance, capacity, and conductance, 
 such that the combination of inductive resistance and shunted 
 condenser acts to an impressed simple periodic electromotive 
 force just as if it were totally non-inductive. This annulment 
 of inductance has important applications in practice in tele- 
 graphy, and it is therefore desirable to define these conditions 
 carefully: 
 
 31. A Shunted Condenser in Series with an Inductive 
 Resistance. Let a condenser of capacity C have its terminals 
 closed by a circuit whose conductance is K, and hence its 
 resistance K -1 . Let this shunted condenser be placed in 
 series with an inductive resistance whose inductance is L and 
 resistance E. At any instant let v t be the potential difference 
 of the terminals of the condenser and v those of the inductive 
 resistance when a periodic current having at this same instant 
 a value i is flowing through the system. Let v be the value 
 of the over-all potential difference. Then suppose that v is a 
 simple periodic potential difference, or that 
 
 r = V sin^ t. 
 
 Then the. current i will differ in phase from v, and may be 
 represented by the expression 
 
 Capital letters represent maximum values as usual. We can 
 then form the fundamental equation for the shunted condenser 
 in series with an inductive circuit as follows : We have for 
 the shunted condenser terminal potential i\ the equation 
 
 C^ + Kr^-i, (52) 
 
 and for the inductive resistance the equation 
 
 L'-+Rf = * a , (53) 
 
 and also t'i + 1- 4 = v, (54)
 
 188 SIMPLE PERIODIC CUBRENT8. 
 
 since the over-all potential v is the sum of the separate 
 potential steps vi and v t . Eliminate Vi and v a from these 
 equations, and we get 
 
 + K V , . (55) 
 also, since v = V sin p t, 
 
 we have C d = Cp Vcospt, 
 
 at 
 
 and K v = K V siup t. 
 
 Hence we have 
 
 C ^ + K v = JR 2 + Cy V sin (p t-0). . (56) 
 
 This last follows at once from the lemma on page 161. 
 Then, since i = I sm (pt-<f>), 
 
 we obtain = I p cos (p t - <), 
 
 and ^=-Ij^sm(pt-^), 
 
 and by substitution of these last values in equation (55), we 
 arrive at the equation 
 
 (K K + 1 - C Lp>) I sin (p t - ) + (C R^ + K "Lp) I cos (pt-<$ 
 
 0) . . (57) 
 
 dt 
 
 = v/K* + CV 2 V sin (^ - 0). . . (58) 
 
 Since it is shown on page 161 that any function of the form 
 A sin 6 + B cos can always be expressed in the form 
 
 -r> 
 
 . ^/A' 2 + B 2 sin (0 + a) , where tan a = -r-, 
 
 A 
 
 it follows at once that the maximum value of the current I ia 
 given by the equation 
 
 = _ 
 
 (K R + 1 - C L /") + (C B/> + K L p)' 2 ' ' 
 
 and it is not difficult to show that this last equation (59) may 
 be written in the form 
 
 P _ 
 
 "
 
 SIMPLE PERIODIC CUEEENTS. 189- 
 
 Let E 1 stand for 
 and L 1 for 
 
 If the condenser and shunt are removed it is clear that the 
 maximum value of the current I will be given by 
 
 Hence the effect of adding the shunted condenser is as if 
 the resistance of the inductive circuit were increased bv an 
 
 T7- 
 
 amount equal to , and the inductance of the 
 
 K 2 + C 2 > 2 
 
 inductive circuit diminished by an amount equal to 
 C 
 
 Hence, if the quantity _ - _ - is equal to L, which is 
 
 the inductance of the circuit, then the total effective induct- 
 ance of the circuit will vanish, and the inductive resistance 
 and shunted condenser together will act as if there were 
 no resultant inductance at all. This condition is fulfilled. 
 when we have 
 
 which equation gives us the required magnitude of the capacity 
 to annul the inductance. Uniting - instead of K, so that r is- 
 the resistance of the shunt, we have 
 L 
 
 (60) 
 
 C(l-CLj> 2 ) 
 
 as the final equation, which tells us what must be the 
 resistance of the shunt to be put across the terminals of a 
 condenser of capacity C in order that the combination may 
 just annul the inductance of a resistance in series with it 
 whose resistance is E and inductance L. If L is measured in. 
 henries, C must be in farads, and E and r be given in ohms. 
 
 It will be seen that the annulment is only exact for one- 
 particular frequency, and that change of frequency means 
 a change in the value of r requisite to neutralise the induct-
 
 190 
 
 SIMPLE PERIODIC CURRENTS, 
 
 ance. In telegraphic work it is usual to employ a shunted 
 condenser to annul more or less completely the self-induction 
 of relay magnets. 
 
 32. Representation of Periodic Currents by Polar Diagrams. 
 Two methods of graphically representing simple periodic 
 currents or electromotive forces, viz., clock diagrams and wave 
 diagrams, have been hitherto used. Each of these methods 
 has some peculiar advantages of its own. There is, however, 
 a third method which has sometimes especial utility, and this 
 is by a polar diagram. 
 
 Let a straight line P (Fig. 73) revolve round one of its 
 .extremities 0, and let the angle of inclination 6 which the 
 
 FIG. 73. 
 
 revolving line makes with another fixed straight line OA 
 passing through this centre of revolution be called the angle 
 of displacement of the revolving line. Let any point P be 
 taken on the revolving line, and let the distance of this point 
 from the centre be denoted by r. The length r is called the 
 radius vector. Let r increase or diminish as P revolves, 
 but so that the length P = r varies according to any law 
 connecting it with the angle of displacement A P. The 
 extremity P of the line P will then trace out a curve called 
 a polar diagram. 
 
 Suppose that P runs through a cycle of values beginning 
 with a zero value when = 0, and, after reaching a maximum
 
 SIMPLE PERIODIC CURRENTS. 191 
 
 value, becoming zero again after the line P has completed 
 one half revolution, or when 6 = 7r, the polar diagram will be a 
 closed curve passing through the origin. Since the radius 
 P runs through a cycle of values from zero to a maximum, 
 and returns to zero again during a revolution of the radius 
 through two right angles, or an angle TT, the radius r may 
 suitably represent the value of any periodic quantity which 
 completes a cycle of values in the time represented by one 
 complete half revolution of the radius vector. Suppose, then, 
 that P varies as sin 0, where 6 is the displacement phase 
 angle that is, let 
 
 r=Rsin0. 
 
 It is then clear that in this case the polar curve is a circle. 
 For if we draw the line B through perpendicular to A, 
 and draw PB at right angles to OP, since AOP = and 
 OP = r, and since by supposition r=R sin 0, it follows that 
 -the angle P B is always equal to 0, and that, therefore, the 
 length B is a constant length equal to E. In other words, 
 the locus of the point P is a circle passing through 0, P, and B. 
 Hence the circle is the curve which represents in a polar 
 diagram a simple periodic quantity, and if the length P is 
 taken to represent the magnitude of a simple periodic current 
 
 or electromotive force at any instant corresponding to a phase 
 angle A P, the circle passing through will be the polar 
 
 curve representing this simple periodic current or electro- 
 motive force. 
 
 If the current or electromotive force is periodic, but not 
 simply periodic, then the polar curve representing it will be a 
 unicursal curve passing through the origin 0, but will not be 
 a circle. The polar diagram of an alternating current or 
 electromotive force enables us very easily to determine the 
 square root of the mean-square value of the periodic quantity 
 represented by it. It is more difficult to do this with the 
 ordinary wave-curve diagram. 
 
 Suppose, for example, that in Fig. 74 we have a wave 
 diagram drawn representing a half wave of an alternating 
 electromotive force which is not a simple sine curve. The 
 ordinates represented by the dotted lines are proportional 
 to the instantaneous values of the periodic quantity taken 
 every 20deg. If we wish to find the Vrnean 2 value of the
 
 192 
 
 SIMPLE PERIODIC CURRENTS. 
 
 ordinates of this curve there is no other way of doing this- 
 than by drawing a number of numerous equidistant vertical 
 ordinates, measuring their lengths, squaring these magnitudes,, 
 and taking the square root of the mean of these squares. This- 
 is a troublesome arithmetical process, and in proportion as the 
 wave curve is more irregular or complex, so much the more 
 numerous must be the ordinates to obtain a correct result for 
 the mean-square value. It is, however, a much easier process 
 if the periodic quantity is represented by a polar curve. Let 
 Fig. 75 represent the same periodic function as in Fig. 74,. 
 drawn in a polar form. The dotted radii are proportional to 
 the instantaneous values of the periodic quantity, and are 
 
 FIG. 74. 
 
 placed at angular intervals of lOdeg. Let any radius P (see 
 Figs. 75 and 76) be denoted by r, and let the corresponding 
 phase angle P A be- denoted by the letter 6. If we consider 
 the radius to move forward by a small angle d 6 and to increase 
 in length by a small amount dr, then it is obvious that the 
 small increment of area d A swept out by the radius r is equal 
 to | r 2 d 6. To obtain the whole area, and included by the polar 
 curve P Q E, we have to integrate this quantity ^r*d6 from 
 to TT, or to obtain the integral
 
 SIMPLE PERIODIC CURRENTS. 
 
 193 
 
 On the base line drawn through the polar centre let a 
 semicircle A C B (see Fig. 75) be described equal in area to 
 the area included by the polar curve OPQ. Let OA = R 
 be the radius of this semicircle. Then 
 
 R = 
 
 The expression on the right hand side of this last equation 
 is obviously the square root of the mean of the squares of the 
 instantaneous values of the periodic quantity r. Hence, we 
 can obtain at once the /s/mean^ value of a periodic quantity 
 current or electromotive force as follows. On any straight 
 
 FIG. 75. 
 
 line describe a semicircle and draw radii of this semicircle at 
 angular intervals equal to those phase intervals for which the 
 instantaneous values of the current or electromotive force are 
 observed. Set off on these radii lengths equal to the respec- 
 tive values of these instantaneous quantities and join the 
 extremities of all these lines so set off by a curve. This curve 
 is the polar diagram, and it is a closed curve. Take the area 
 included by the polar diagram with an Amsler's or other plani- 
 meter, and from a table of areas of circles find the circle whose 
 area is double that of the polar curve. Then the radius of 
 this circle is the square root of the mean of the squares of the 
 periodic quantity. If, for instance, we have a periodic current 
 
 o
 
 194 SIMPLE PERIODIC CURRENTS. 
 
 curve plotted down in polar form, this operation will give 
 us the dynamometer value of the current, or the value which 
 would be read on a dynamometer. 
 
 The polar curve, therefore, lends itself very easily to the 
 determination of the mean -square value of a periodic current 
 or electromotive force, of which the instantaneous values are 
 known throughout a semi-period, whether those values are 
 ia'ven at equidistant intervals of time or not. The reader 
 w^, therefore, note that if we plot down a periodic quantity 
 ii rectangular co-ordinates (Fig. 74) or wave form, and find 
 the rectangle A P P' B of equivalent area to the half wave, 
 the altitude AP of this rectangle gives us the true mean 
 ordinate of the periodic curve ; but if we plot down the 
 periodic quantity to a polar diayram (Fig. 75) and find the 
 
 FIG. 76. 
 
 semicircle A C B of equivalent area, the radius A of this 
 aemicircle gives us the square root of the mean-square value of 
 the periodic quantity represented by the radii of the polar curve. 
 
 33. Initial Conditions on starting Current Flow in a Cir- 
 cuit having Resistance and Inductance. It has been shown 
 in the foregoing sections that if an impressed electromotive 
 force of simple periodic kind acts upon a circuit having in- 
 ductance, the resulting current is a simple periodic current, 
 but lags behind the impressed electromotive force in phase. 
 These, however, are the conditions when the resulting current 
 has become steady. At the instant of closing the circuit there 
 are peculiar conditions of augmentation of the current which 
 are called initial conditions, and which have very important
 
 SIMPLE PERIODIC CUHEENTS. 195 
 
 consequences in practice. It will be advisable, therefore, to 
 examine a little more closely how these are produced, and 
 what results may be expected at the instant of starting or 
 stopping the current in such a circuit. To do this we will, 
 in the first place, examine more carefully the solution of the 
 fundamental equations for current creation in an inductive 
 circuit. It has been shown that the differential equation for 
 current at any instant in the circuit of constant inductance 
 L and resistance R under an impressed simple periodic electro- 
 motive force v V sin p t is 
 
 . . (61) 
 
 at 
 
 To solve this equation completely, we differentiate it twice, 
 and eliminate thereby the term V sin p t, thus obtaining the 
 equation 
 
 L+B+ ^ L+ ^ Et ' =0 - (62) 
 
 A differential equation of this type is called a linear differ- 
 ential equation of the third order. It is shown in treatises 
 on differential equations that its solution depends on the 
 solution of a cubic equation called the auxiliary equation. 
 The auxiliary equation in this case is 
 
 Lw 3 + Bm 2 +^ 2 Lm+j9 2 E = 0. . . . (63) 
 This cubic equation can be split up into two factors and 
 be written 
 
 and hence the roots of the cubic equation (63) are 
 DI= + v 1 p 
 
 For a linear equation of the type of (62), the roots of the 
 auxiliary cubic being a and J -1(3, the solution is known 
 io be of the form 
 
 i = A e a * + B sin /3 + B' cos t. 
 The solution of the differential equation (62) is, then, given by 
 
 Rt 
 
 . . (64) 
 o2
 
 196 SIMPLE PERIODIC CURRENTS. 
 
 By the Trigonometrical Lemma on page 161 we can write, 
 instead of the second and third terms on the right hand side 
 of (64), the single term v/B 2 + B' 2 sin (p t-<f>), where >/B 2 + B' 2 
 is obviously the maximum value of the current call it I 
 when the initial state is over, and <f> is the angle by which 
 the current, when steady, lags behind the electromotive force 
 in phase. Hence (64) may be written, 
 
 * 
 t = Ae L +Isin(pt-<f>). . . . (65) 
 
 To find the constant quantity A, we note that at the instant 
 when the circuit is closed the current has necessarily a value 
 zero. Let this closing of the current happen at a time t' 
 reckoned from the instant when the electromotive force is 
 zero. Then at the instant of closing the circuit we have 
 
 Q = ke~ Lt + Isin(pt'-<t>), 
 
 R 
 
 or A= - 
 
 Hence, substituting this value of A in equation (65), we obtain 
 
 -4)4 ^ ', . . (66) 
 and this is the complete solution of the differential equation 
 (61) for the current in the circuit. 
 
 We note that the expression for the current at any instant 
 in the inductive circuit is made up of two terms ; the first 
 term, I sin (p t </>), is a simple sine function, and represents by 
 itself a simple periodic current having a maximum value I. 
 
 - t) 
 The second term, I sin (pt'-$)e~*- , is an exponential 
 
 function having a maximum value I sin (pt' - </>), when t' = t r 
 and this term represents, therefore, a logarithmic curve begin- 
 ning with the value I sin (p t' - <j>) and dying away gradually 
 to zero. 
 
 Hence the resulting current curve consists of these two 
 curves superimposed upon one another, a periodic curve and 
 a logarithmic or diminishing curve. 
 
 In Fig. 77 are shown two such curves ; curve 1 being the 
 sine curve, curve 2 the logarithmic curve, and the resultant 
 curve 3, represented by the dotted line, which is obtained by 
 adding together the ordinates of the sine curve and the 
 logarithmic curve. It will be seen that the effect of the
 
 SIMPLE PERIODIC CURRENTS. 197 
 
 superposition is to make the resultant curve lopsided with 
 respect to the curve axis for a certain period, but beyond that 
 time it is sensibly symmetrically situated with respect to the 
 time axis. Hence, during this initial period, the maximum 
 values of the current in opposite directions are not the same. 
 At the instant when t = t' the value of the current is zero. 
 It is obvious that, when pt' = 9Q + (f), sin (pt'-$) = \, and 
 that then the logarithmic curve begins with its greatest value ; 
 but that, when pt' <}>, then the logarithmic curve has no 
 existence at all. When pt' = QQ + <f> that is if the circuit 
 
 is closed at an instant t = -L? reckoning from the instant 
 
 P 
 
 when the impressed electromotive force is zero the disturbance 
 of the uniformity of the periodic curve of current is the 
 
 FIG. 77. 
 
 greatest possible. At the same time the maximum value of 
 the current in the negative direction can never be greater 
 than 2 1 where I is the maximum value of the steadily periodic 
 current. For the maximum value of the logarithmic curve at 
 the instant of closing the circuit is I, and at that instant 
 i = ; hence the value at which the periodic component of the 
 current must begin will be + 1. At the time when the periodic 
 part has reached a maximum of - 1, which happens after half 
 a period, the ordinate of the logarithmic curve will have fallen 
 to something less than I by an amount depending on the rate 
 
 T> 
 
 of decrease, which in turn depends upon the value of =- or the 
 reciprocal of the time-constant of the circuit.
 
 198 SIMPLE PERIODIC CURRENTS. 
 
 Consider the case when the circuit is closed at a time t 
 reckoned from the zero of electromotive force such that 
 
 T 
 If T is the complete periodic time, then at a time - after 
 
 the instant of closing the value of the periodic part in the 
 expression for the current is 
 
 = 1 sin (2) t' -<f> + ir) 
 = -I 
 
 since t' - & = 90. 
 
 m 
 
 The value for the exponential part at this instant '+ is 
 
 _ K T 
 
 - 1 sin (p t' - 4>) e L 2 
 
 and hence the total value of that current at that instant is- 
 
 RT 
 
 since pt' <f> = 90. 
 
 The greatest value which the time constant ^ can have is 
 
 Ti K 
 
 infinity. Hence, when _ = 0, or approximately zero, e ^"2=1^ 
 
 Jj 
 
 and the quantity in the bracket in the last equation may 
 approach to 2 but can never exceed it. This shows us that, 
 with an inductive circuit of very large time constant, the value 
 of the current after half a period from the instant of closing 
 the circuit may be something a little less than twice the value 
 of its periodic steady maximum, provided that the circuit is 
 closed at the instant when the electromotive force has a value 
 e such that 
 
 e = E sin (90 + #), 
 or e = E cos <, 
 
 where tan <f> = -=^. 
 H 
 
 This amounts to saying that the circuit must be closed at 
 the instant of zero electromotive force.
 
 SIMPLE PERIODIC CURRENTS. 199 
 
 Hence we see that during the initial period there may be a 
 greatly increased mean-square value of the current, and thus 
 the production of a current-rush on closing the circuit. 
 
 When therefore a circuit of constant inductance is switched 
 on to a source of steadily periodic electromotive force at the 
 instant when the electromotive force has a value corresponding 
 to the maximum value of the current, when the variable state 
 is over, we find that; before the current settles down into its 
 steady swing, lagging behind the electromotive force in phase, 
 there is a period of disturbance during which the current has 
 greater maximum values in one direction than in the other, 
 and the current virtually consists of a rapidly evanescent 
 unidirectional current superposed upon the normal periodic 
 current which ultimately survives. If however the circuit is 
 closed at the instant when the electromotive force is passing 
 through a value which corresponds to that at which the 
 current has its zero value, when the variable period is passed, 
 then there is no period of disturbance, but the current begins 
 at once in its normal manner lagging behind the electromotive 
 force and having constant maximum values + 1 and I alter- 
 nately. This phenomenon of current rushes into inductive 
 circuits such as transformers will be treated at length in a 
 later chapter, and the attention of the reader is merely at this 
 point directed to the general nature of the effects taking place 
 in a circuit of constant inductance when suddenly switched on 
 to a source of simple periodic electromotive force. 
 
 34. Initial Conditions in Circuits having Capacity, 
 Inductance and Resistance. It is somewhat more difficult to 
 discuss completely the conditions which arise at the instant of 
 connecting to a source of periodic electromotive force a circuit 
 having not only inductance and resistance but also capacity. 
 Generally speaking, they may be described as consisting of a 
 variable period and a succeeding steady period. Wte shall in 
 outline indicate how these conditions respectively arise. Sup- 
 pose, in the first place, that a condenser of capacity C is con- 
 nected through an inductive resistance of inductance L and 
 resistance R to a source of periodic electromotive force of which 
 the value at any instant is represented by 
 i/ = V sin pt.
 
 200 SIMPLE PERIODIC CURRENTS. 
 
 Let i be the instantaneous value of the current flowing into 
 the circuit, and let v be the value at the same instant of the 
 potential difference between the terminals of the condenser, 
 and V! the fall of potential down the inductive resistance. 
 Then we have the following fundamental equation connecting 
 the current and potential : 
 
 For the resistance L + E i = v lt , (67) 
 
 dt 
 
 for the condenser C = i, (68) 
 
 d t 
 
 and for the total fall of potential 
 
 *'-+* (69) 
 
 Hence, from (67) and (68), 
 
 i/l' 
 
 dt 
 and eliminating i by the help of (68), we get 
 
 .... (70) 
 
 It t U I 
 
 This is the differential equation defining the value of the 
 condenser terminal potential in terms of the constants and 
 the time. To solve (70) we must eliminate V sin^ t. This 
 is done by differentiating (70) twice and eliminating V sinpt 
 "between the original equation (70) and the twice differentiated 
 equation. As a result we reach the equation 
 
 The solution of this linear differential equation of the fourth 
 order depends on the solution of the biquadratic 
 
 L C m* + C E ;/t 3 + ( 1 + C L j9*) m 2 + C E p* 
 and this last splits up into two factors 
 
 (m 2 + 2 r) (C L m 2 + C B m + 1) - 0. 
 Hence the roots of this biquadratic are, 
 
 m = v^n: P
 
 SIMPLE PERIODIC CURRENTS. 201 
 
 It is shown in treatises on differential equations that the 
 solution of equation (71) is then 
 
 A -D .--* - B 
 
 v=A.smpt + B cosp t + A e ^smqt + ^e ticosqt, . (72) 
 
 where A, B, A', B' are constants and 
 
 o= Alj^Cll 2 
 
 v -TCI?-' 
 
 The solution for v may obviously be written 
 
 ~ ^ ' sin (9 1 -<'). (73) 
 
 This equation for the value of the condenser potential v 
 shows us that the variation of v is made up of two parts. 
 First a simple periodic part V A 2 + B' 2 sin (p t - <f>), which may 
 be written V sin (p t-<f>), and which indicates a simple periodic 
 "variation of v differing in phase from the impressed electro- 
 motive force v' by an angle </>. The other term of the solution 
 indicates a superposed periodic variation, gradually decreasing 
 in amplitude as time increases, and dying out as the exponent 
 
 T> 
 
 t increases with time. If, for the sake of brevity, we write 
 C' for ^A /:! + B' 2 , we can put the solution for v in the form 
 
 <j>). . (74) 
 
 Two constants have therefore to be determined, viz., C' and <', 
 in order that we may completely solve the problem. 
 
 Since the quantity of electricity in the condenser at any 
 instant is numerically equal to the product of the capacity 
 and potential, if we multiply (74) all through by C, the con- 
 denser capacity, we have an expression for the charge x in the 
 condenser at any instant. Since this charge is null at the 
 instant of closing the circuit, if the circuit is closed at the 
 instant t', reckoning time from the instant when the impressed 
 electromotive force is zero, we have 
 
 = V sin (p t' - <f>) + C' e ~ 2/ sin (q t' - <') 
 as an equation to determine the constant C'. 
 
 Therefore c,'= -*? - <. . . . (75) 
 
 sin (q t' - <f>)
 
 202 SIMPLE PERIODIC CURRENTS. 
 
 No sufficient advantage is to be obtained by working out 
 the rather complicated algebraical expressions in terms of 
 C, L, B, and p, for the constant $', but we can indicate gene- 
 rally what the equations teach. They show us that if such a 
 condenser in series with an inductive circuit is 'switched on to 
 a source of periodic impressed electromotive force, before the 
 oscillations of the condenser potential settle down into a 
 regular state, there is a variable period in which a second 
 set of oscillations of gradually diminishing amplitude are 
 superimposed on the steady set, and the second set of oscil- 
 lations have a quite different frequency and initial amplitude to 
 the steady set which ultimately survive. In the initial period 
 the superposition of the two sets of oscillations may increase 
 the instantaneous value of the condenser terminal potential 
 difference to a value greater, and perhaps much greater, than 
 it would have if there were no such additional oscillations. If 
 the circuit is closed at an instant t' such that pt' = <f>, viz., at 
 an interval after the impressed electromotive force has passed 
 its zero value equal to the final permanent difference of phase 
 of condenser and impressed electromotive force, then, since 
 this value makes the constant C' zero, we see that there are 
 no superposed vibrations at all. On the other hand, if t' is so 
 chosen that p t' = 90 + <, then the disturbing effect is greatest. 
 It is clear, therefore, that, in switching on a condenser to an 
 alternating current circuit through an inductive resistance, 
 initial effects of abnormal rise of condenser voltage may result. 
 We shall see later on that these are practically very important 
 matters in dealing with alternating current systems of supply, 
 and that caution must always be used in switching on a con- 
 denser to such a circuit. 
 
 35. Complex Periodic Functions. Before leaving the 
 subject of periodic currents and electromotive forces it is 
 desirable to explain some properties of the trigonometrical 
 expressions or series by which such functions can be 
 represented as the sum of a series of simple periodic con- 
 stituents or terms. It has already been explained that the 
 value of an ordinate y of any single valued function can be 
 expressed by Fourier's method as follows : 
 
 y = Y + Y! sin (/> * + &) + Y 2 sin (2 p t + >) + , &c. ; (76)
 
 SIMPLE PERIODIC CURRENTS. 203 
 
 or, taking advantage of the trigonometrical equality 
 
 A sin B + B cos 6 = V A 2 + B 2 sin (6 + <), where tan <f> = T , 
 
 A. 
 
 we can write the above expression for ?/ 
 
 y = Y + y-L sinpt + z eosp t + y 2 sin 2 pt + ^ cos 2^> + &c. (77) 
 
 The coefficients Y 1} Y 2 , &c., in the series in (76) are the 
 amplitudes of the simple sine curves or harmonic constituents 
 whose added ordinates together build up the function y. 
 
 If the periodic quantity represented is a wave curve 
 symmetrical with respect to the axis of time, and repeating 
 the same form continually, then the constant term Y is 
 absent. 
 
 We can therefore express the instantaneous value e of any 
 periodic electromotive force by the expression 
 
 = E 1 sin^ + F 1 cos^e + E 2 sin2^* + F 2 cos2^ + ,&c (78) 
 
 and the instantaneous value i of any periodic current by the 
 series 
 
 : cos^e + I 2 sin 2^ + J 2 cos 2pt + , &c. (79) 
 
 In the series (78) the amptitude of the first harmonic is 
 l/Ej' + Fi* and that of the second V^ + F^, and so on. 
 
 We have already seen that if we take the mean or average 
 value of sin 2 B over one half-period, or from to TT, the value 
 of the mean is - ; but that the mean value of such a product 
 
 as sin d cos B or sin B sin 2 B over half a period is zero. 
 
 Accordingly, if we square the series in (78) or (79) we find 
 that the values for e 2 and for i partly involve terms like 
 sm 2 pt, sin 2 2jz, &c., and partly terms like siupt sinZpt. 
 Hence, if we integrate the value of #dtm ffdt over half a 
 period and divide the result by IT, or take the definite integrals 
 
 - PV 8 d t, - f "V* d t, which is equivalent to finding the mean- 
 square values of e and /, we find by the above theorem that 
 
 -| Vrft =T + T + + +>&c - ' (80> 
 
 7T / n _ _ Jfi 36
 
 204 SIMPLE PERIODIC CURRENTS. 
 
 and, similarly, by multiplying (78) and (79) and taking the 
 mean value of the products from p t = to p t = TT, we obtain 
 
 VA + TVu.ZA+V. + .to. (.) 
 
 The ordinary alternate current ammeter or dynamometer 
 measures the V'mean 2 value of the current i, or the quantity 
 
 _ | i 2 d t ; and the ordinary alternating voltmeter, such as 
 7r / o _ 
 
 an electrostatic voltmeter, measures the V'mean 2 value of the 
 
 electromotive force e, or the quantity */ - (*V 2 d t, whatever be 
 
 V ""Jo 
 the force of the curves of e and i. A wattmeter in proper form 
 
 reads the true mean product of e, i or - (* eidt when currents 
 
 ^J 
 
 respectively proportional to e and i traverse its two coils. For 
 
 shortness, let us write e for *J - ("" e 2 d , and i' for the similar 
 V *J o 
 
 function of i z , and (e' i') for - I* eidt ; then 
 */ 
 
 +, &c., 
 , &c., 
 
 2 (e i') = E! I x + F! Ji + , &C. 
 
 We see therefore that twice the V mean 2 value of e or i is 
 equal to the sum of the squares of the coefficients of the sine 
 and cosine terms, or to the sum of the squares of the 
 amplitude of the harmonic constituents. Also that the twice 
 mean product of two periodic functions taken over a half 
 period at similar instants is equal to the sum of the products 
 of the coefficients of similar sine or cosine terms taken in 
 pairs from each expansion. Suppose that e and i represent 
 the instantaneous values of the impressed electromotive force 
 and current of any circuit, we may ask whether the product 
 of the V mean' 2 value of e and the Vmean 2 value of i is equal 
 to, greater, or less than the mean of the product of e and i t 
 
 T eidt.
 
 SIMPLE PERIODIC CURRENTS. 20c 
 
 We see that the square of the expression for 2 (e 1 1')' is made 
 up of terms like E^ 2 , and F^J^, and also of terms like 
 2E 1 I 1 F 1 J 1 . 
 
 The product of the expressions for e and i' consists 
 obviously of terms like Ej 2 !^, F^Jj 2 , and also of terms like 
 Ffli 2 and E, 2 ^ 2 . 
 
 Hence, the question whether the product of ' and i' or 
 e' x. i' is or is not greater than (e i') will depend upon the 
 relative collected magnitude of terms like F/Ij 2 + E 1 2 J x 2 in the 
 one series, and of terms like 2 E^FjJ^ If F : : E^:^;!!, 
 
 or if J^ 1 = J- 1 , then E x 2 J x 2 + F x 2 1 2 = 2 E x F : I, J lf and if the same 
 
 &i i 
 
 proportion holds good for the coefficients of the terms in 
 sin 2 p t, &c., then we see that e' xi' = (e'i'), or the product of 
 the ,yinean 2 values e and i is equal to the mean product of e 
 and /. If the above proportionality does not hold good, then, 
 since generally a'-' + fc 2 is greater than 2 a b, it is not difficult 
 to see that e x i' is greater than (e' i') or the product of 
 
 is greater than the mean value- /** i d t. 
 -J o 
 
 Hence it follows that, in this last case, the product of the 
 amperes and volts as read on alternating current instruments 
 is greater than the true value of the power as read on a watt- 
 meter. The mean value - feidt is called generally the 
 
 7 o 
 true watts or poicer given to the circuit, and the product 
 
 - f* e z dtx. \ - f^ftdt is called the apparent watts or 
 
 TTJ TTJ 
 
 power given to the circuit. The apparent watts are equal 
 under some conditions to the true watts. This is the 
 case when the circuit is non-inductive, and when the 
 different harmonic constituents of the current and electro- 
 motive force are in step or in synchronism with each other. 
 Under these conditions the angle of lag of the electromotive 
 force harmonics is equal to the lag of the current harmonics 
 of the same degree, and this is expressed by relations of the
 
 206 SIMPLE PERIODIC CURRENTS. 
 
 form Fj : Ej : : Ji * Ij holding good. Under other conditions, the 
 apparent watts are greater than the true watts. The ratio of 
 the true power or watts taken up in any circuit to the apparent 
 power or watts is called the power factor of the circuit, and the 
 power factor (P.F.) is therefore given for any alternating current 
 circuit by the ratio of the wattmeter reading to the product of 
 the ammeter reading and voltmeter reading for that circuit.
 
 CHAPTER IV. 
 
 MUTUAL AND SELF INDUCTION. 
 
 1. Researches of Prof. Joseph Henry. We have already, 
 in the first chapter, made a brief allusion to the share taken by 
 Joseph Henry in the fundamental discovery of the induction of 
 
 FIG. 78. 
 
 electric currents. A full account of his labours in this field is 
 to be found in the collected " Scientific Writings of Joseph 
 Henry," republished by the Smithsonian Institution. It 
 will be of advantage to consider at this stage some of his chief 
 investigations. 
 
 The principal pieces of apparatus used by Henry in his 
 experiments on the induction of electric currents consisted of 
 
 * See also the Philosophical Magazine, Vol. XVI., 3rd Ser., 1840, and 
 Transactions of the American Philosophical Society. Vol. VI., 1838, pp. 
 303-337.
 
 208 MUTUAL AND SELF INDUCTION. 
 
 a number of flat coils of copper strip or band, which were 
 designated by the names Coil No. 1, Coil No. 2, &c., also 
 several long bobbins -of wire, and these, to distinguish them 
 from the ribands, were called Helix No. 1, Helix No. 2, &c. 
 
 His description of these coils and helices is as follows : Coil 
 No. 1 was formed of thirteen pounds of copper strip one inch 
 and a-half wide and ninety-three feet long ; it was well 
 covered with two coatings of silk, and was generally used in 
 the form represented in Fig. 78, which is that of a flat spiral 
 sixteen inches in diameter. It was, however, sometimes 
 formed into a ring of larger diameter, as is shown in Fig. 79. 
 
 FIG. 79. 
 
 Coil No. 2 was also formed of copper strip of the same width 
 and thickness as coil No. 1. It was, however, only sixty feet 
 long. Its form is shown at b in Fig. 78. The opening at the 
 centre was sufficient to admit helix No. 1. Coils No. 8, 4, 5, 
 6, were all about sixty feet long, and of copper strip of the 
 same thickness, but of half the width of coil No. 1. 
 
 Helix No. 1 consisted of sixteen hundred and sixty yards of 
 copper wire ^th of an inch in diameter ; No. 2 of nine 
 hundred and ninety yards, and No. 3 of three hundred and 
 fifty yards of the same wire. These helices were wound on 
 bobbins of such size as to fit into each other, thus forming 
 one long helix of three thousand yards, or, by using them 
 separately and in different combinations, seven helices of 
 different lengths. The wire was covered with cotton thread
 
 MUTUAL AND SELF INDUCTION. 209 
 
 saturated with bees' wax, and between each stratum of spires 
 a coating of silk was interposed. 
 
 Helix No. 4, shown at a, Fig. 79, was formed of five hundred 
 and forty-six yards of wire Jg-th of an inch in diameter, the 
 several spires of which were insulated by a coating of cement. 
 
 Helix No. 5 consisted of fifteen hundred yards of silvered 
 copper wire, j4 T th of an inch in diameter, covered with cotton, 
 and of the form of helix No. 4. 
 
 In addition, a long spool of copper wire covered with cotton, 
 T ^th of an inch in diameter and five miles long, was provided. 
 It was wound on a small axis of iron, and formed a solid 
 cylinder of wire eighteen inches long and thirteen in diameter. 
 
 For determining the direction of the induced currents Henry 
 employed a magnetising spiral, which consisted of about thirty 
 spires of copper wire in the form of a cylinder, and so small as 
 just to admit a sewing needle into the axis. 
 
 Also a small iron horseshoe is frequently referred to, which 
 was formed of a piece of soft iron about three inches long and 
 f ths of an inch thick ; each leg was surrounded with about 
 five feet of copper bell wire. This length was so small that 
 only a current of considerable strength could develop sensible 
 magnetism in the iron. This horseshoe was used for indicat- 
 ing the existence of such a current. The battery which was 
 used was a simple copper-zinc cylinder battery, having about 
 If square feet of zinc surface. In some experiments a series of 
 cells was used, but most experiments were performed with one 
 or two cells of the above kind. For interrupting the circuit of 
 the conductor Henry employed the simple device of scraping 
 one end of the conductor along a rasp held in contact with 
 the battery terminal. 
 
 Provided with this apparatus, Henry entered on a pre- 
 liminary series of experiments on the induction of electric 
 currents, and in 1838 published an account of his investiga- 
 tions on the phenomenon which had been' previously named 
 by Faraday electro-dynamic induction. The fact which seems 
 to have chiefly attracted the attention of the numerous investi- 
 gators who rapidly entered the region of research opened out 
 by Faraday's discovery of the mutual induction of electric 
 circuits and the production of electric currents in conducting 
 circuits by the variation of the magnetic induction linked with
 
 210 MUTUAL AND SELF INDUCTION. 
 
 them, and by Henry's discovery of the self-induction of electric 
 circuits, seems to have been the possibility of obtaining from 
 a single cell of a galvanic battery effects such as spark and 
 shock. These effects connected what was then known as voltaic 
 electricity with the then more familiar effects of electrification by 
 friction.* Henry took up the train of investigation at this point, 
 and proceeded to employ the above described helices and coils in 
 an investigation of the facts of the self- and mutual-induction 
 of electric circuits. His mode of operating was to close the 
 battery circuit by dipping the ends of a coil or helix into two 
 mercury cups connected with the terminal plates, and then to 
 break the circuit by lifting out one end from its mercury cup, 
 the hands being at the same time in contact with the battery 
 terminal and the end of the conductor which is being raised. 
 In this way the extra current, or electro-magnetic discharge of 
 the coil, passed through the operator's body. 
 
 When the electromotive force was small, as in the case of a 
 thermopile or a large single cell, and the circuit taken was the 
 flat riband coil No. 1, ninety-three feet long, it was found to 
 give brilliant snaps at the surface of the mercury when contact 
 was broken, but the shocks were very feeble, and could only 
 be felt in the fingers or through the tongue. The induced 
 current in a short coil, which thus produced deflagration but 
 not shocks, he called, for distinction, one of quantity. 
 
 When the length of the coil was increased, the battery 
 being the same, the deflagrating power decreased, while the 
 intensity of the shock continually increased. With five- 
 riband coils in series, making an aggregate length of three 
 hundred feet, and a small battery the deflagration was less 
 than with coil No. 1, but the shocks were more intense. 
 
 There appeared to be, however, a limit to this increase of 
 intensity of the shock, and this took place when the increased 
 resistance or diminished conduction of the lengthened coil 
 began to counteract the influence of the increasing length of 
 the current. The following experiment illustrated this fact. 
 
 A coil of copper wire ^th of an inch in diameter was 
 increased in length by successive additions of about thirty- 
 
 * For a more extended description of the historical order of discoveries 
 in connection with the induction coil the reader is referred to the First 
 Chapter in the Second Volume of this treatise.
 
 MUTUAL AND SELF INDUCTION. 211 
 
 two feet at a time. After the first two lengths, or sixty-four 
 feet, the brilliancy of the spark began to decline, but the 
 shocks continually increased in intensity until a length of 
 five hundred and seventy feet was obtained, when the shocks 
 also began to decline. This was, then, the proper length to 
 produce the maximum effect with a single battery and a wire 
 of the above diameter. With a battery of sixty cells (Cruick- 
 shank's trough), having plates four inches square, scarcely 
 any shock could be obtained when the coil formed a part of 
 the circuit. If the length of the coil was increased, then the 
 inductive effect became very apparent. 
 
 When the current from ten cells of the above-mentioned 
 trough was passed through the large spool of copper wire, the 
 induced shock was too severe to be taken through the body. 
 Again, when a small battery of twenty-five cells having plates 
 one inch square, which alone would give but a very feeble 
 shock, was used with helix No. 1, an intense shock was 
 received from the induction when the contact was broken. 
 Also a slight shock in this arrangement was given when the 
 contact was formed, but it was very feeble in comparison with 
 the other. The spark, however, with the long wire and 
 compound battery was not as brilliant as with the single 
 battery and short riband coil. 
 
 When the shock was produced from a long wire, as in the 
 last experiments, the size of the plates of the battery might 
 be very much reduced without a corresponding reduction in 
 the intensity of the shock. A small battery was made, 
 formed of six pieces of copper bell wire one inch and a- half 
 long and an equal number of pieces of zinc of the same 
 size. When the current from this was passed through a coil 
 consisting of five miles of wire, the shock was given at once 
 to twenty-six persons joining hands. 
 
 With the same coil, and the single battery used in the 
 former experiments, no shock, or at most a very feeble one, 
 could be obtained. 
 
 The induced current in these last experiments he called 
 one of considerable intensity and small quantity. 
 
 2. Mutual Induction. Henry then passed on to consider 
 the mutual induction of two circuits. Coil No. 1 (see c, 
 
 p2
 
 212 MUTUAL AND SELF INDUCTION. 
 
 Fig. 80) -\vas arranged to receive the current from a small 
 battery of a single cell, and coil No. 2, b, was placed over it 
 with a plate of glass between to secure perfect insulation. 
 As often as the current in No. 1 circuit was interrupted, a 
 powerful secondary current was induced in No. 2. When the 
 ends of the secondary were joined to a magnetising spiral, the 
 enclosed needle became strongly magnetic. Also when the 
 ends of the second coil were attached to a small water 
 decomposing apparatus, a stream of gas was given off at 
 each pole ; and when the secondary current was passed 
 through the wires of the iron horse-shoe, magnetism was 
 developed. The shock, however, from the secondary coil 
 was very feeble, and scarcely felt above the fingers. This 
 secondary current had, therefore, the properties of one of 
 moderate intensity but considerable quantity (to use the 
 
 FIG. 80. 
 
 terms then employed) when developed by the current in one 
 flat riband coil acting on another flat riband coil. 
 
 Coil No. 1, remaining as before a longer coil, formed by 
 uniting Nos. 3, 4, and 5, was substituted for No. 2. With this 
 arrangement as a secondary circuit the magnetising power of 
 the current and the brilliancy of the spark at breaking 
 contact was less than before, but the shocks were more 
 powerful in other words, the intensity of the secondary 
 induced current was increased, whilst its quantity was 
 decreased: 
 
 A compound helix, formed by uniting Nos. 1 and 2 helices, 
 and therefore containing two thousand six hundred and fifty 
 yards of wire, was next placed on coil No. 1. The weight of 
 this helix happened to be precisely the same as that of coil
 
 MUTUAL AND SELF INDUCTION. 213 
 
 No. 2, and hence the different effects of the same quantity of 
 metal (as secondary circuit) in the two forms of a long and 
 short conductor could be compared. With this arrangement 
 the magnetising effects with the apparatus above-mentioned 
 disappeared. The sparks were much smaller and the decom- 
 position less than with the short coil, but the shock was 
 almost too intense to be received with impunity except 
 through the fingers of one hand. The secondary current 
 in this case was one of small quantity but of great intensity. 
 
 The following experiment is important in establishing the 
 fact of a limit to the increase of the intensity of the shock as 
 well as to the power of decomposition with a wire of given 
 diameter. 
 
 Helix No. 5, consisting of a wire T | T th of an inch in 
 diameter, was placed on coil No. 2, and its length increased to 
 about seven hundred yards. With this extent of wire neither 
 decomposition nor magnetism could be obtained, but shocks 
 were given of a peculiarly pungent nature. The wire of the 
 helix was further increased to about fifteen hundred yards ; the 
 shock was now found to be scarcely perceptible in the fingers. 
 
 As a counterpart to the last experiment, coil No. 1 was 
 formed into a ring of sufficient internal diameter to admit the 
 great spool of wire, and, with the whole length of this (five 
 miles), the shock was found so intense as to be felt at the 
 shoulder when passed only through the forefinger and thumb. 
 Sparks and decomposition were also produced, and needles 
 rendered magnetic. The wire of this spool was T \th of an inch 
 in diameter ; and Henry noted therefore from this experiment 
 that, by increasing the diameter of the wire, its length might 
 also be increased with increased effect of shock. 
 
 The previous experiments were repeated, using a battery of 
 sixty cells (Cruickshank's trough). When the current from 
 this was passed through the riband coil No. 1, no indication, or 
 a very feeble one, was given of a secondary current in any of 
 the coils or helices arranged as in the preceding experiments; 
 but when the long helix No. 1 was placed as a primary, instead 
 of coil No. 1, a powerful inductive action was produced on each 
 of the circuits used as before. 
 
 First, helices Nos. 2 and 3 were united into one coil and placed 
 within helix No. 1, which still conducted the battery current.
 
 214 MUTUAL AND SELF INDUCTION. 
 
 With this disposition a secondary current was produced, which 
 gave intense shocks but feeble decomposition and no magnetism 
 in the soft iron horse-shoe. It was therefore one of intensity, 
 and was produced by a battery current also of intensity. 
 Instead of the helix used in the last experiment for receiving 
 the induction (secondary), one of the coils, No. 3 (copper riband), 
 was now placed on helix No. 1, the battery remaining as 
 before. With this arrangement the induced current gave no 
 shocks, but it magnetised the small horse-shoe, and when the 
 ends of the coil were rubbed together produced bright sparks. 
 It had, therefore, the properties of a current of quantity, and 
 it was produced by the induction of a current from a battery 
 of intensity.* 
 
 This experiment was considered of so much importance that 
 it was varied and repeated many times, but always with the 
 same result ; and it therefore established the fact that an inten- 
 sity current could induce one of quantity ; and by the preceding 
 experiments the converse has also been shown, that a quantity 
 current could induce one of intensity. 
 
 3. Induction at a Distance. In the experiments on mutual 
 induction detailed above, the primary circuit was separated 
 from the secondary only by a pane of glass, but the action was 
 so energetic that an obvious experiment was to investigate the 
 effect of distance on the mutual induction. For this purpose 
 coil No. 1 was formed into a ring of about two feet in diameter 
 (see Fig. 79), and helix No. 4 placed as shown. When the helix 
 was at the distance of about sixteen inches from the middle of 
 the plane of the ring, shocks could be perceived, through the 
 tongue, and these rapidly increased in intensity as the helix 
 was lowered, and when it reached the plane of the ring they 
 were quite severe. The effect, however, was still greater when 
 the helix was moved from the centre to the inner circumference, 
 
 * This last experiment is very interesting, as showing that in 1838 Prof. 
 Henry had already realised that which used to be called the reverse use of 
 the induction coil. He had employed a current flowing in a fine wire of 
 many turns and moving under a high electromotive force, to induce a cur- 
 rent of greater strength in a secondary circuit, consisting of a lesser number 
 of turns of copper riband, and moving under a less electromotive force. In 
 other words, he had constructed what we should now call a step-down 
 transformer. Note Henry's explicit statement in the following paragraph.
 
 MUTUAL AND SELF IND UCTION. 215- 
 
 as at c, but when it was placed without the ring, in contact 
 with the outer circumference at b, the shocks were very slight, 
 and when placed within, but with its axis at right angles to 
 that of the ring, not the least effect could be observed. Coil 
 No. 1 remaining as before (the primary) helix No. 1, which was 
 nine inches in diameter, was substituted for the small helix in 
 the last experiment, and with this the effect at a distance waa 
 much increased. When coil No. 2 was added to coil No. 1, and 
 the currents from two small batteries sent through, these 
 shocks were distinctly perceptible through the tongue when the 
 distance of the planes of the coil and the three helices united 
 as one was increased to thirty-six inches. The action at a dis- 
 tance was still further increased by coiling the long wire of the 
 large spool into the form of a ring of four feet in diameter, and 
 placing parallel to this another ring formed of the four ribands 
 of coils Nos. 1, 2, 3, 4. When a current from a single cell 
 having thirty-five feet of zinc surface, was passed through the 
 riband conductor, shocks through the tongue were felt when 
 the rings were separated to a distance of four feet. In another 
 experiment, to illustrate induction across a distance, Henry 
 (Phil. Mag., Vol. XVIII., 1841, 3rd Series, p. 592) joined all 
 his coils, so as to form a single conductor of about 400 feet in 
 length, and this was rolled into a ring of five and a-half feet 
 in diameter and suspended vertically against a door. On the 
 other side of the door, and opposite to the coil, was placed a 
 helix formed of upwards of a mile of copper wire one sixteenth 
 of an inch in thickness and wound in a hoop of four feet in 
 diameter. With this arrangement, and with a battery of one 
 hundred and forty-seven square feet of zinc surface divided into 
 eight elements, shocks were perceptible on the tongue when the 
 two conductors were separated by a distance of seven feet, and 
 at a distance of between three and four feet the shocks were 
 quite severe. 
 
 In the fifty years which have elapsed since Henry performed 
 the classical experiments described above, the progress of know- 
 ledge has placed in our hands an appliance vastly more delicate 
 than physiological shock for detecting induction at a distance, 
 viz., the articulating telephone receiver. Aided by this, it has 
 recently been found possible to find indications of the mutual 
 induction between conductors separated by miles instead of feet.
 
 216 MUTUAL AND SELF INDUCTION. 
 
 Along the Gray's Inn-road, London, the English Post-office 
 service placed a line of iron pipes buried underground carrying 
 many telegraph wires. The United Telephone Company placed 
 a line of open wires along the same route over the housetops, 
 situated 80ft. from the underground wires. Considerable dis- 
 turbances were experienced on the telephone circuit, and even 
 Morse signals were read, which were said to be caused by the 
 Continuous and parallel telegraph circuits. A very careful series 
 of experiments,* extending over some period, proved unmis- 
 takably that this was so, and that the well-known pattering 
 disturbances due to induction are experienced at a much 
 greater distance than was anticipated. 
 
 Experiments conducted on the Newcastle Town Moor 
 extended the area of the disturbance to a distance of 3,000ft., 
 while effects were detected on parallel lines of telegraph 
 between Durham and Darlington at a distance of 10J miles. 
 But the greatest distance experimented upon was between the 
 east and west coast of the Border, when two lines of wire 
 40 miles apart were affected the one by the other, sounds 
 produced at Newcastle on the Jedburgh line being distinctly 
 heard at Gretna on a parallel line, though no wires connected 
 the two places. 
 
 Distinct conversation has been held by telephone through 
 the air without any wire through a distance of one quarter of 
 a mile, and this distance can probably be much exceeded. 
 
 Effects are not confined to the air, for submarine cables 
 half a mile apart in the sea disturb each other. It may well 
 be doubted whether the inductive effects above described as 
 taking place over very large distances above mentioned are 
 not complicated by current leakage, but it has been abundantly 
 established that inductive effects can be produced and detected 
 between circuits separated by great distances. 
 
 Practical application of current induction across large air 
 spaces has been made in the methods of carrying on tele- 
 graphic communication with railway trains when in motion. 
 There are two methods by which this has been accomplished. 
 (1) The magneto-induction method, which was devised by 
 
 * Mr. W. H. Preece on "Induction between Wires and Wires" (The 
 Electrician, Vol. XVII., 1836, p. 410 ; British Association Report, Birming- 
 ham, 1886).
 
 MUTUAL AND SELF INDUCTION. 217 
 
 Mr. L. J. Phelps and was tried about the year 1885 on a line 
 about 15 miles long between Haarlem Eiver and New 
 Eochelle Junction, in the United States. In the other 
 system (2) the principle involved is that of electrostatic 
 induction, and, after having been suggested in a more or less 
 imperfect form by Mr. W. Wiley Smith in 1881 (U.S. 
 Patent No. 247,127), has been worked out in great detail by 
 Messrs. Edison and Gilliland. 
 
 In the magneto-induction system a telegraphic car attached 
 to the train carries a great circuit of wire wound on a frame 
 extending the whole length of the car, and so placed that one 
 side of the windings is as near the track as possible and one 
 side as high above as the height of the car will permit. 
 Between the rails is laid down a fixed insulator conductor, 
 and the fluctuations of a current in this last induce currents 
 in the lower side of the large coil carried on the car. The 
 secondary current so induced is detected by a telephone and 
 by suitable interruptions. A Morse code of audible signals 
 can be transmitted from the fixed conductor to the moving 
 train. The signals are thus made to jump over the air space, 
 and continuous communications can be kept up between a 
 station or stations in connection with the fixed conductor and 
 a person in the moving telegraph car. 
 
 Mr. Phelps used a conductor of No. 12 (A.G.) insulated 
 wire, which was placed in a kind of small wooden trough 
 mounted on blocks attached to the sleepers. The car con- 
 taining the telegraphing instruments carried beneath its floors, 
 about Tin. above the rail level, a 2in. iron pipe, in which 
 was a rubbe: tube holding about 90 convolutions of No. 14 
 (A.G.) copper wire, so connected as to form a continuous 
 circuit about a mile and a-half long, and presenting something 
 like three-quarters of a mile of wire parallel to the primary 
 line wire mounted between the metals. The instrument, 
 consisted of a delicate polarised relay as a receiving instru- 
 ment, which acted as a sounder, and a " buzzer," or rapid 
 current-breaker, for transmitting signals by means of the 
 Morse key, which were received at the station in a telephone. 
 This arrangement was so far a practical success that 
 Mr. Phelps was encouraged to proceed ; but meantime it was 
 discovered that the patent above referred to had already been
 
 218 MUTUAL AND SELF INDUCTION. 
 
 issued, while Edison and Gilliland had also been working on 
 similar lines. In Wiley Smith's specification no mention is 
 made of a " buzzer," which turns out to be an important 
 feature in the invention ; but the practical success of the 
 experiments made is due to a combination of the devices of 
 Phelps, Edison, and Gilliland. The latest system is an 
 improvement on that of Phelps, briefly described above, in 
 that it dispenses with the insulated line wire laid between the 
 metals, and uses ordinary telegraph wire strung on what 
 are known as short poles alongside the permanent way. 
 The line wire is, in fact, stretched on poles about 16ft. high 
 and at an average distance of 8ft. from the rails. On the 
 Lehigh Valley Railroad, U.S., from Perth Junction to Easton, 
 interesting experiments of this kind were made in 1887. As 
 a rule the roof of the car, usually sheathed with metal, is 
 available for securing the necessary electrical condition, but, 
 where a metal covering is absent, all that is necessary is to 
 attach a wire or rod to the roof and another to some portion 
 of the metallic or rolling part of the coach in order to obtain 
 " earth." The instruments are inserted in this circuit, and 
 comprise a 12-cell chromic acid battery (the cells being 2in. 
 wide by 4in. deep), which is closed on an induction coil having 
 a primary of about 3 ohms and a secondary of about 500 
 ohms, and provided with an ordinary vibrating make-and- 
 break. The messages are sent by means of a Morse key 
 placed in the secondary circuit, this key being of the double- 
 pointed kind with extra contact. The receiving telephone has 
 a resistance of about 1,000 ohms ; but Mr. Phelps states that, 
 even when wound so as to have a resistance of 10,000 ohms, 
 the sound is quite clear, so high is the electromotive force of 
 the induction on the roof. The car-roofs are frequently of 
 metal usually painted tin plates, sheet zinc, or galvanised 
 iron, and these answer admirably as inductive receivers ; but 
 where the roofs are of wood, covered with painted canvas, an 
 iron or brass rod or tube, |in. in diameter, is carried along 
 under the eaves throughout the length of the train. The 
 metallic roof or the rod is connected by a wire to the secondary 
 of an induction coil, while the primary of the coil is connected 
 to the front contact of the double-pointed key, and through 
 that with the battery. Opposite the core of the coil is the
 
 MUTUAL AND SELF INDUCTION. 219 
 
 " buzzer," which transmits a series of impulses to the lin& 
 whenever the key is worked. The extra contact, which is 
 placed on the upper surface of the front contact of the key, 
 closes the secondary circuit, and allows the charges to be sent 
 into the roof, while, when the key is on the back contact, the 
 secondary and primary coils are cut out, and the charge from 
 the roof then passes direct to the key and through the telephone 
 to earth, which, as a rule, is made by connecting wires from 
 the coil and the telephone to one of the axle-boxes. The coil 
 and the key, with suitable connections, are mounted on a 
 board which is large enough to contain a telegraph form 
 besides, and the telephone is attached by flexible connections, 
 and is, when in use, strapped to the operator's head. The 
 battery is put up in a case with a handle, so that the whole 
 apparatus can be carried from one end of a train to the other. 
 The arrangements at the terminal and other stations on the 
 line, so far as induction telegraphy is concerned, are practically 
 identical with those in the railway coach ; but, in addition, 
 they have a duplex Morse equipment, by which ordinary 
 messages can be sent by the dot-and-dash system. 
 
 Of late years interesting experiments have been made under 
 the direction of Mr. W. H. Preece in carrying on telegraphic 
 communication across considerable distances by means of 
 induction between parallel circuits. 
 
 4. Induced Currents of Higher Orders. In 1888 Henry 
 made a further remarkable discovery, viz., that secondary 
 currents, though only of momentary duration, could in their 
 turn induce other induced currents in neighbouring conductors ; 
 and these he called tertiary and currents of higher orders. 
 
 A primary current was passed through coil No. 1, while 
 coil No. 2 was placed over it to receive the secondary current, 
 and the ends of this last coil joined to a third coil, No. 3. 
 By this disposition the secondary current passed through No. 3, 
 and sinca this was at a distance (see Fig. 81), and beyond the 
 influence of the primary, its separate induction could be 
 rendered manifest by the effects on helix No. 1, arranged 
 as a secondary circuit to this third coil. When the handles 
 a ft of the last helix d were grasped a powerful shock was 
 received, proving the induction of a tertiary current in the last
 
 220 
 
 MUTUAL AND SELF INDUCTION. 
 
 coil. By a similar more extended arrangement of inducing 
 coils (shown in Fig. 82) shocks were received from currents 
 of a fourth and fifth order; and with a more powerful 
 primary current and additional coils a still greater number 
 of successive inductions might be obtained. Henry thus 
 established by decisive experiments that, in a properly placed 
 series of connected coils, a primary current could give birth 
 to secondary currents, and these last to tertiary currents, and 
 
 
 FIG. 81. 
 
 so on, a whole family of induced currents arising from the 
 starting or stopping of the primary current. 
 
 It was found that with a small battery a shock could be 
 given from the current of the third order to 25 persons 
 joining hands ; also shocks perceptible in the arms were 
 obtained from a current of the fifth order. 
 
 FIG. 82. 
 
 When the long helix is placed over a secondary current 
 generated in a short coil, and which is one of quantity, a 
 tertiary current of intensity was obtained capable of producing 
 shocks. When the intensity current of the last experiment 
 was passed through a second helix, and another flat riband 
 coil placed over this (see Fig. 82 A), a quantity current was 
 again produced. Therefore, in the case of these currents of 
 higher orders also a quantity current could be induced from 
 one of intensity, and the converse.
 
 MUTUAL AND SELF INDUCTION. 221 
 
 The arrangement in Fig. 82 shows these different results 
 produced at once. The induction from coil No. 3 to helix 
 No. 1 produces an intensity current, and from helix No. 2 to 
 coil No. 4 a quantity current. 
 
 The next stage in Henry's inquiry had reference to the 
 direction of these induced currents. Knowing that a current 
 on starting in a conductor induces an inverse or oppositely- 
 directed induced current in a neighbouring secondary circuit, 
 and a direct or like directed induced current on stopping, it 
 was clear that each tertiary current must consist, in its 
 simplest form, of two oppositely directed currents succeeding 
 each other instantaneously ; for at the " make " or "break" 
 of the primary the secondary circuit is traversed by a brief 
 secondary current in "opposite" or "like" direction. We 
 shall speak of these as the inverse and direct secondary currents 
 produced on closing or opening the primary circuit. 
 
 FIG. E2.v. 
 
 Each of these secondary currents rises to a maximum and 
 then sinks to zero again. If there is a tertiary circuit present, 
 then during the rise of the secondary current to its maximum 
 it is developing an inverse tertiary and during its decrease to 
 zero a direct tertiary current. Since, as we shall see, the 
 duration of the secondary current is a very small fraction of a 
 second, these two component tertiary currents must succeed 
 each other at an excessively short interval of time. Physio- 
 logically their separate effect is, so to speak, united, and 
 they make themselves felt as one shock. Henry adopted 
 the method of employing a magnetising spiral containing a 
 sewing needle as a means of analysing the nature of these 
 induced currents of higher order. By inserting such a 
 spiral in the circuit of the successive conductors and noting 
 the direction of the magnetisation of the steel needle he 
 arrived at the conclusion that there exists an alternation in
 
 222 MUTUAL AND SELF INDUCTION. 
 
 the direction of the currents of the several orders, and 
 that the directions of the several induced currents could 
 he expressed by saying that at the " make " of the primary 
 we get an inverse secondary, a direct tertiary, and inverse 
 quarternary current, and so on ; or, symbolically : 
 Primary current started stopped 
 
 Secondary ,, inverse direct - 
 
 Tertiary ,, direct + inverse -j- 
 
 Quarternary inverse direct 
 
 d-c. dc. 
 
 The use of a magnetising spiral as a means of determining 
 the direction of an induced current is, however, liable to 
 lead to serious errors in drawing conclusions as to direction 
 of currents, and the above experiment cannot be regarded 
 as an exhaustive examination of the whole phenomena of 
 induced currents of higher orders. Before entering into a 
 more detailed discussion of the exact nature of the effects 
 which here present themselves, it will be of assistance to 
 gather together the principal observations on the induction 
 of transient electric currents. 
 
 5. Inductive Effects Produced by Transient Electric Cur- 
 rents. It was an obvious inference, from all the foregoing 
 facts, that Leyden jar discharges, or the transient currents 
 formed by discharging charged condensers, should in like 
 manner be able to give rise to a family of induced currents in 
 suitably-placed circuits. Henry thus opened up a new field of 
 research, which was diligently cultivated by Marianini, Abria, 
 Matteucci, Eeiss, Verdet, and many other physicists. Henry's 
 first experiment was as follows : A hollow glass cylinder (see 
 Fig. 83) of about six inches hi diameter was prepared with a 
 narrow riband of tinfoil about thirty feet long pasted spirally 
 around the outside, and a riband of the same length pasted on 
 the inside, so that the corresponding spires of the two were 
 directly opposite each other. The ends of the inner spiral 
 passed out of the cylinder through a glass tube to prevent 
 direct communication between the two circuits. When the 
 ends of the inner riband were joined by the magnetising spiral 
 containing a sewing needle and a discharge from a half-gallon 
 jar sent through the outer riband, the needle was strongly
 
 MUTUAL AND SELF INDUCTION. 223 
 
 magnetised in such a manner as to indicate an induced current 
 through the inner riband in the same direction as that of the 
 current of the jar. If, instead of using the magnetising spiral, 
 the ends of the inner riband were brought near together, a 
 small spark was detected at the instant of sending a jar dis- 
 charge through the outer conductor. Experiments were next 
 made in reference to the production of induced currents of 
 different orders by electric discharges. For this purpose a 
 series of glass cylinders with tinfoil spirals pasted on them was 
 prepared and joined up so that the inner spiral of one cylinder 
 was in connection with the outer spiral of another. When a 
 discharge was passed through the outer riband of the first 
 cylinder it produced an induced secondary discharge circulating 
 in the inner spiral of the first and the outer spiral of the second 
 cylinder. This in turn generated a tertiary current, and so 
 
 FIG. 83. 
 
 forth. Each of these discharges was a brief wave of current, 
 and by the use of the magnetising spiral in each circuit an 
 attempt was made to determine the direction of the discharge. 
 Here, however, an anomaly presented itself. By the use of 
 this magnetising spiral it appeared that the induced discharges 
 were all in the same direction. Leyden jar discharges were 
 then passed through the first member of the series of coils and 
 helices used in the experiments on galvanic currents, and here 
 the directions of the induced discharges in the several con- 
 ductors were found to alternate. After various experiments 
 Henry considered that he had found the solution of this 
 anomaly in the different distances, of the inducing and induc- 
 tive circuits. As an experiment illustrating this he gives the 
 following: Two narrow strips of tinfoil about twelve feet long
 
 224 MUTUAL AND SELF INDUCTION. 
 
 were stretched parallel to each other, and separated by thin 
 plates of mica to the distanca of about ^th of an inch. When 
 a discharge from a half-gallon jar was passed through one of 
 these an induced current in the same direction was obtained 
 from the other. When the ribands were separated to a distance 
 of about ith of an inch, no induced current, as evidenced by 
 the absence of effect in the magnetising spiral, could be 
 obtained. When the circuits were still further separated the 
 induced current reappeared, but in the opposite direction to the 
 primary discharge. The distance at which the induced dis- 
 charge changes direction appears, according to Henry, to be 
 dependent on a number of circumstances, such as the capacity 
 and charge of the jar and the length and thickness of the 
 wires. 
 
 With a battery of eight half-gallon jars and parallel wires 
 of about ten feet long the change in direction did not take 
 place until the wires were separated by twelve or fifteen inches. 
 The currents of all the higher orders were found to change 
 sign with a change in the distance between the inducing and 
 inductive circuit. 
 
 One interesting experiment was made by Henry to illustrate 
 the inductive effect of jar discharges across considerable dis- 
 tances. In this case a primary circuit was formed consisting 
 of an insulated wire eighty feet long. Around this, and 
 separated from it by a distance of about twelve feet, was 
 another circuit consisting of a wire one hundred and twenty- 
 feet long. When the discharge from thirty large Leyden jars 
 was sent through the primary wire an induced discharge was 
 obtained in the other sufficiently strong to magnetise to 
 saturation a small needle placed in a magnetising spiral 
 interpolated in the secondary circuit. We may, however, 
 remark here, once for all, that all these experiments directed 
 to determine the direction of induced discharges in which the 
 magnetising power of the discharge is made use of for this 
 purpose are difficult to interpret, and too much reliance must 
 not be placed on the conclusions thus drawn. Leaving out of 
 account for the moment all consideration of what are called 
 electric oscillations, to which we shall allude subsequently, we 
 may say that, if two discharges are passed through a magnetis- 
 ing spiral, the discharges being oppositely directed and of equal
 
 MUTUAL AND SELF INDUCTION. 225 
 
 quantity but different durations, the resulting direction of 
 magnetisation will be dependent upon several conflicting 
 elements. Speaking generally, the intensity of magnetisation 
 is determined by the relative magnitude of the maximum 
 current strength during the discharge, and of two discharges 
 having equal quantity the one lasting the shortest time would 
 rise to the highest current strength during the period of the 
 discharge, and exercise the greatest magnetising force. Even 
 then it would not be safe to draw too pronounced a con- 
 clusion from the direction of magnetisation as to the relative 
 magnitudes of the maxima of two alternate discharge currents 
 rapidly succeeding each other, for, as Abria pointed out long 
 ago,* the demagnetisation of a steel needle requires a less 
 magnetising force than that necessary to magnetise it in the 
 first instance, and hence the final results are complicated by 
 the relative order of imposition as well as the relative maximum 
 magnitude of the magnetising discharge currents. One fact 
 which has to be borne in mind in attempting to interpret 
 these results of Henry is that the magnetising current whose 
 direction we are seeking to determine acts by induction also 
 on the mass of the needle or iron in the testing magnetising 
 coil, and generates in its mass induced currents circulating 
 round its surface. Under the head of Magnetic Screening 
 in a later section we shall examine the circumstances under 
 which such currents induced in a metallic mass shield to a 
 greater or less extent conducting circuits lying beyond them 
 from inductive effects. Meanwhile we may say that the 
 effect of a very sudden discharge in one direction in the 
 magnetising coil is to induce eddy currents in the surface 
 of the needle which shield the inner and deeper portions of 
 the steel from the magnetising action, and the resulting 
 magnetisation is chiefly superficial. If, however, the dis- 
 charge is prolonged or dragged out whilst retaining the same 
 electric quantity, the shielding action will not be so pronounced, 
 and the magnetisation will penetrate deeper down into the 
 mass of the steel. Accordingly two equal discharges, i.e., 
 discharges of equal quantity, may produce a greater or less 
 magnetic moment in the steel, according as the duration 
 of the same is greater or less, a very sudden discharge 
 
 * Abria, Ann. dc Chcm. et dc Phys., [3] Vol. I., p. 429, 1844. 
 
 Q
 
 226 MUTUAL AND SELF INDUCTION. 
 
 Laving much less magnetic-moment -producing power than 
 the same quantity more dragged out. We may in general 
 also say that the magnetising power of a discharge current is 
 determined by the value of the maximum current strength 
 during the discharge, and hence of two equal quantity dis- 
 charges, the one which lasts the shorter time, and which has, 
 therefore, the greatest maximum value, will, if the discharges 
 are approximately equal in duration, produce the greatest 
 magnetising effect. 
 
 The tertiary currents, produced by ordinary galvanic currents, 
 and the secondary currents, produced by Ley den jar discharges, 
 consist, as we have seen, in their simplest form of a double 
 discharge or flow, one part inverse, or oppositely directed to its 
 inducing parent current, and the succeeding part direct, or 
 similarly directed, the two component currents of the total 
 discharge having equal quantity but different durations. In 
 general the first portion, or the inverse current, is that which 
 has the greatest maximum value and the shortest duration, the 
 second half, or the direct current, being more dragged out in 
 tune ; and, for a reason to be stated further on, the approxima- 
 tion of the induced and inducing circuits exaggerates this 
 difference, or increases the maximum value of the inverse 
 current at the expense of its duration. The explanation which 
 may be offered, however, of the phenomena of the magnetisa- 
 tion of steel by tertiary currents, or by the secondary currents 
 due to Leyden jar discharges, is as follows : When the induced 
 and inducing circuits are not very near to each other, and when 
 the inducing current reaches its maximum not very suddenly, 
 the two induced currents are not very different in duration, but 
 the first or inverse current has, of the two, a rather greater 
 maximum and less duration. It follows that a magnetisa- 
 tion is produced in the needle, which, on the whole, is in the 
 direction produced by the inverse current, and the inference 
 from the direction of magnetisation is that the induced and 
 inducing currents are in the opposite direction. If, however, 
 the inducing current reaches its maximum value very suddenly, 
 as it does if the circuits are very close, then the first half, or 
 the inversed induced current, is so brief in its duration that the 
 magnetisation of the needle due to it is very superficial. On 
 the other hand, the magnetisation due to the rather more pro-
 
 MUTUAL AND SELF INDUCTION. 227 
 
 longed direct current is more diffused through the needle, and 
 the resultant magnetisation found on testing the needle is that 
 apparently due to the direct current, and the inference from 
 the resulting magnetism of the needle would be that the 
 induced and inducing currents are in the same direction. By 
 some such explanation as the above we may reconcile these 
 experimental results of Henry with known facts, but it is 
 evident, since the resulting magnetisation of the needle is an 
 effect determined by the relative maximum values of the two 
 portions of the total induced current, and by their duration, as 
 well as by their order of superposition, that considerable caution 
 is necessary in attempting to interpret the results of experi- 
 ments made with a magnetising helix. Henry was followed 
 in the same field of investigation by Abria, Marianini, 
 Eeiss, and Matteucci. Matteucci endeavoured to determine 
 the direction of the induced discharges by employing a pro- 
 cess founded upon the experiment of the pierced card, in 
 which the hole made by the spark on a piece of paper or a card 
 is always nearer to the negative electrode. By means of this 
 process, combined with the employment of the galvanometer, 
 Matteucci considered that the inductive discharges are deter- 
 mined by the following law: If the inducing and induced 
 circuit are both closed, the induced discharge is in the opposite 
 direction to the inducing discharge. If, however, the induced 
 circuit is interrupted at any point so that there is a spark, the 
 induced discharge is in the same direction as the inducing. 
 Abandoning these methods above described, M. Verdet* em- 
 ployed another, which depends upon the polarisation of elec- 
 trodes in dilute sulphuric acid. 
 
 From more recent knowledge we may state the facts with 
 regard to the action of alternate currents upon a dilute sul- 
 phuric acid voltameter as follows! : 
 
 If a current of electricity consisting of alternate short fluxes 
 of current of opposite sign is passed through a voltameter having 
 platinum electrodes, and if these electrodes are large, there is no 
 visible decomposition, but if the electrodes are reduced in size 
 below a certain limit visible decomposition begins. For every 
 
 * Verdet, An*, de Chem. et de Phys., [3] Vol. XXIX., p. 501, 1850. 
 t See a Paper by MM. Maneuvrier and J. Chappuis, abstracted in The 
 Electrician, June 29, 1888, Vol. XXL, p. 237. 
 
 Q2
 
 228 MUTUAL AND SELF INDUCTION. 
 
 current there is a certain size of electrode, above which gas is not 
 visibly evolved, and for every given size of electrode there is a 
 current below which gas is not apparently liberated. When 
 the conditions are suitable for the liberation of gas, the gases 
 collected at both electrodes have the same composition. If the 
 quantities of electricity passing in each alternate and oppositely- 
 directed flux are equal, then the electrodes are not sensibly 
 polarised. If, however, the quantities are not equal, then there 
 is, on the whole, a greater flow of current in one direction than 
 in the other, and the electrodes exhibit the state known as 
 polarisation, and yield a reverse current when connected with 
 the galvanometer. Verdet, in his experiments, made use of 
 flat spirals, the wires of which were insulated from each other 
 with great care by silk and a layer of gum-lac varnish. The 
 primary spiral was made of copper wire fths of an inch in dia- 
 meter and 92 feet in length, forming 24 spirals. The secondary 
 circuit consisted of three spirals of wire -^th of an inch in dia- 
 meter and 157 feet in length, making 95 turns. The inducing 
 discharge was supplied from a Leyden jar battery of nine large 
 jars. The induced discharge was sent through a voltameter 
 having small platinum electrodes, and which could be connected 
 with a delicate galvanometer for detecting polarisation of the 
 electrodes immediately after the discharge. Verdet's experi- 
 ments led him to recognise that when the induced circuit is 
 continuous, and not interrupted anywhere except by the 
 insertion of the voltameter, no traces of polarisation are 
 obtained except by very powerful discharges. This indicates 
 that the induced discharge consists of a double current of two 
 oppositely- directed and equal quantities of electricity. In the 
 case of very powerful discharges there was a slight galvano- 
 metric deflection, indicating a preponderating secondary dis- 
 charge in the same direction as the primary. If the induced 
 or secondary circuit is interrupted at one point, so that the 
 discharge has to pass as a spark at that place, then very per- 
 ceptible polarisation of the electrodes presents itself, and the 
 direction of this is such as to indicate a predominant induced 
 current passing in the same direction as the primary. 
 
 To sum up. It follows from all the numerous researches on 
 induced discharges that this is a very complex phenomenon, 
 and is influenced by a large number of conditional circum-
 
 MUTUAL AND SELF INDUCTION. 229 
 
 stances, and also by the very mode employed for determining 
 it. It may be, however, taken as proved that an induced dis- 
 charge, produced either as a secondary discharge by a transi-' 
 tory primary, such as the discharge from a Leyden jar, or a 
 tertiary current produced by induction by a secondary current 
 of very brief duration, is, hi its simplest form, a wave of electric 
 current, consisting of two short fluxes or currents in opposite 
 directions, and succeeding each other immediately. This 
 Poggendorff* holds to be shown by the action of such tertiary 
 or higher order currents on a galvanometer. If these currents 
 are led through a galvanometer of which the arrangement is 
 such that the magnetic axis of the needle is accurately at right 
 angles to the direction of the magnetic axis of the coil, then 
 no deflection of the needle is observed, or at most a very slight 
 one. If, however, the needle makes an angle with the plane 
 of the coils, then these induction currents cause a deviation of 
 the needle. This effect (die doppehinnige Ablenkung) arises 
 from the fact that the magnetism of the needle is not rigid, 
 and that the alternate twisting couples to which the needle is 
 subjected are not equal, by reason of the fact that one of the 
 halves of the complete induced current say the direct half 
 increases the magnetic moment of the needle, and hence 
 increases slightly the deflecting couple in the direction tending 
 to increase the deviation of the needle ; the other half say 
 the inverse part of the induced current tends to reduce the 
 moment of the needle, and hence to subject it to a smaller 
 reverse couple. Hence it follows that, if discharges of equal 
 quantity and opposite sign succeed each other through a 
 galvanometer when the needle is accurately in the plane of the 
 coils, Little or no deviation is observed ; but if the coils are 
 turned so that the needle makes an angle with them, then 
 these alternate currents' will affect the needle and increase the 
 ;,angle of deflection. 
 
 This behaviour towards a galvanometer, and the action on a 
 voltameter of liberating mixed gases of equal composition at 
 each pole, prove that each induced current of the third and 
 higher orders consists of two oppositely-directed discharges, 
 produced by the operation of two successive electromotive 
 impulses of opposite sign and very brief duration acting upon 
 * Poggendorff, Pogg. Ann,, Bd. XLV., 1838, p. 353.
 
 230 
 
 MUTUAL AND SELF INDUCTION. 
 
 the circuit. The quantities of these discharges are equal ; but 
 the durations are different, and hence the maximum value of 
 the current strength during the opposite discharges may be 
 very different. 
 
 This may be illustrated graphically thus : 
 
 Let the curve a P b Q c (Fig. 84) be a current curve represent- 
 ing two waves of current of opposite sign succeeding each 
 other. Let the horizontal line a c be a time line, and vertical 
 ordinates represent instantaneous current strengths. Then the 
 shaded areas will represent the quantity in each discharge. 
 Let these shaded areas be equal, then the diagram represents 
 two discharges of equal quantity succeeding each other in 
 opposite directions, but having different maximum current 
 
 FIG. 84. 
 
 strengths I and I'. The duration of the first discharge is 
 represented by a b, and that of the second by b c. This diagram 
 represents the conditions in the simplest case of tertiary 
 current. If the instantaneous value of the current at any 
 time is called i, then the whole quantity of the discharge will 
 be represented by the shaded area and by the integral / i d t 
 between proper limits. 
 
 We may classify the effects of induced discharges or currents 
 in the following way : 
 
 (1) Those effects dependent upon I i d t, or upon the whole 
 quantity of the discharge. These are the gahanometric and 
 the electro-chemical effects. If a discharge is passed through a.
 
 MUTUAL AND SELF INDUCTION. 231 
 
 galvanometer, the duration of which is very small compared 
 with the time of free oscillation of the needle, the galvanometer 
 needle experiences a "throw" such that the sine of half the 
 angle of deflection is proportional to the whole quantity of the 
 discharge. Also in a voltameter, by Faraday's law, the whole 
 quantity of the electrolyte broken up is proportional to the 
 quantity of electricity which has passed through it. 
 
 (2) Those effects dependent upon I i-dt, or upon the average 
 of the square of the strength of the current at every instant 
 during the discharge. These are the heating and the electro- 
 dynamic effects. By Joule's law, at every instant the rate of 
 dissipation of energy is proportional to the square of the current 
 strength, and hence the whole heat generated by the discharge 
 is proportional to the integral above. Similarly, if the dis- 
 charge passes through a circuit, part of which is movable and 
 can react upon a fixed part, so that attraction or repulsion may 
 take place between them, the force is dependent at any instant 
 on the square of the current strength, and hence the whole 
 effect or average force upon the same integral. 
 
 (3) We have, lastly, effects dependent chiefly upon the 
 maximum ordinate I, or upon the rate of change of the cur- 
 rent that is, upon the steepness of the slope of the current 
 curve. These are the physiological, telephonic, luminous, and 
 magnetic 'effects. 
 
 The physiological effect of a discharge in giving a shock 
 appears to depend in great part upon the suddenness with 
 which the maximum current strength is reached. Of two dis- 
 charges which reached equal maxima, that which arrived at it 
 in the shortest time would be the most effective in producing 
 shocks. The value of the maximum current strength is also 
 important. Two induced currents of equal quantity but different 
 durations cause a greater shock in proportion to their lesser 
 duration. The telephone in this respect resembles the animal 
 body. It is affected more by the rate of change of the current 
 strength than by the absolute current strength at any instant. 
 
 The magnetic effect depends, as has been shown by Lord 
 Rayleigh,* upon the maximum current strength during the 
 
 * See Phil. Mag., Ser. 4, Vol. XXXVIIL, 1869, p. 8 : The Hon. J. W 
 Strutt (Lord Rayleigh), " On some Electromagnetic Phenomena." Also 
 Phil. Mag., Ser. 4, Vol. XXXIX., 1870, p. 431.
 
 232 MUTUAL AND SELF INDUCTION. 
 
 discharge, or upon the initial current strengMi, in those cases 
 in which the current dies gradually away. In the two Papers 
 referred to below it is shown by direct experiment that, since 
 the time required for the permanent magnetisation of steel is 
 small compared with the duration of induced currents generally, 
 the amount of acquired magnetism depends essentially on the 
 initial or maximum current strength during a transitory 
 current, without regard to the time for which it lasts. It is, 
 then, not difficult to understand that the effort to settle by 
 experiment with a magnetising coil the direction of induced 
 discharges may lead to very conflicting results, and, in any 
 case, it is hardly competent to do more than indicate the 
 direction in which the maximum current flow takes place 
 during the discharge. 
 
 The spark effects are also included in this category. The 
 air or other dielectric is broken down when the difference of 
 potentials between the two discharging points reaches a 
 certain magnitude, and in the case of a varying electric 
 pressure the question whether a spark will pass or not is 
 evidently determined by the maximum magnitude of that 
 quantity.* 
 
 It is evident from the above considerations that the complete 
 analysis of the effects and phenomena of induced currents of 
 the higher orders, and of those of secondary currents due to 
 discharges from condensers, requires a knowledge of the form 
 of the current curve in each case. We proceed to consider the 
 problem of the theory of induced currents in some of its simpler 
 aspects. 
 
 6. Elementary Theory of the Mutual Induction of Two 
 Circuits. Aiming rather at the elucidation of principles than 
 very copious treatment, we shall consider in the next place the 
 problem of the mutual induction of two circuits in its simplest 
 form. Let there be two bobbins of wire in suitable positions 
 for producing mutual induction and without iron cores. Let 
 the constant inductance of the first or primary coil be denoted 
 by L and its resistance E, and the similar quantities for the 
 
 * See Bertin, " Notes on Electrodyuamic Induction," Ann. de Chimie, 
 4th Sen, Vol. XXII., April, 1871, p. 486.
 
 MUTUAL AND SELF INDUCTION. 233 
 
 second or secondary coil be N and S, and let M be the co- 
 efficient of mutual induction.* 
 
 Let there be a source of constant electromotive force, E, 
 which can be applied or withdrawn from the primary circuit. 
 We shall denote by x the strength of the current in the 
 primary at any time t after closing the primary circuit by 
 applying the battery to it. Also we shall denote by y the 
 current in the secondary circuit at any time reckoned from 
 the same zero. 
 
 If, then, at any instant the currents are x and y, the follow- 
 ing state of things exists in the circuits. 
 
 The electromotive force E is the impressed force on the 
 primary circuit. 
 
 That part of the impressed electromotive force producing 
 the current x is E. That part employed in overcoming the 
 
 counter-electromotive force of self-induction is L , and the 
 
 dt 
 
 counter-electromotive force of mutual induction due to the 
 
 current y at that instant in the secondary circuit is - M _^. 
 
 dt 
 
 Hence the relation which at any instant holds good between 
 these quantities is 
 
 The above equation is an expression of the fact that the 
 external impressed electromotive force at any instant is equal 
 to the internal electromotive forces and the effective electro- 
 motive force driving the current. 
 
 Similarly, for the secondary circuit we have an induced 
 electromotive force due to the induction of the primary on 
 
 the secondary equal to M and a counter-electromotive force 
 
 
 a t 
 
 of self-induction N ^. 
 a t 
 
 Hence N+M 
 
 d t d t 
 
 since there is no external impressed electromotive force. The 
 complete solution of the problem of finding the currents x 
 
 * Continental writers often call L and N" the potentials of the bobbins 
 on themselves, and M the potential of one bobbin on the other.
 
 234 MUTUAL AND SELF INDUCTION. 
 
 and y at any instant is obtained by the solution of these 
 simultaneous differential equations 
 
 dt 
 
 at dt 
 
 As our object is to illustrate principles rather than mathe- 
 matical methods, we shall simplify the problem by supposing 
 that the two circuits are similar in every respect. This makes 
 R = S and L = N, and the equations become 
 
 L^+M^+Bs-E, .... (83) 
 ..... (84) 
 
 Bearing in mind that the inductance L is, in ordinary 
 parlance, the "number of lines of force" which are linked 
 with the primary circuit when unit current flows in its own 
 circuit, and that M signifies the number of lines of force which 
 are common to both, or linked in with both circuits, when 
 unit current flows in each, we see that M can never be greater 
 than L, but that under all circumstances we must have 
 
 M <or = L, 
 
 also M < or = N ; 
 
 hence M 2 <or = LN, 
 
 or LN M 2 always a positive quantity, and the maximum 
 value which the co-efficient of mutual inductance M can have 
 is x/L N, or the square root of the product of the self- induc- 
 tances of the separate circuits. 
 
 In order to separate the differentials- in (83) and (84) we 
 differentiate each equation with respect to t, and obtain
 
 MUTUAL AND SELF INDUCTION. 235 
 
 Multiply (85) by L, (86) by - M, and (83) by E, and then 
 adding the three equations together we obtain 
 #x , 2LE dx . R 2 EE 
 
 , v 
 L a -M*d* L'-M 3 * ' ' 
 
 and a similar elimination gives us 
 
 We have now separated the differentials in x and ?/, and the 
 solution of these equations depends, as is well known,* upon 
 the solution of an auxiliary quadratic equation 
 
 the solution of which is 
 
 R 
 
 m= - - , or- 
 
 + M' L-M 
 
 Hence the general solution of (83) and (84) is 
 
 -Ar*-i+B/ 1 -+! f . . . (89) 
 
 and y = A.'e L +* +BV L - M , (90) 
 
 where A, B, A', B' are constants of integration to be determine J 
 from the circumstances of the flow. To do this, however, a 
 preliminary discussion is necessary. Let us suppose that the 
 primary current is fully established, and has a steady value I r 
 and hence that M I lines of induction penetrate through the 
 secondary circuit. This quantity is then the electromagnetic 
 momentum of the secondary circuit, because when the current 
 ill the primary is steady there is no current in the secondary 
 circuit. 
 
 Let us now suppose that the primary circuit is broken, and 
 that the circumstances of the " break " are such that all these 
 MI lines of induction are removed at a uniform rate in a 
 small time B t from the secondary circuit. 
 
 During this time 8 1 an electromotive force will operate upon 
 
 the secondary circuit equal in magnitude to ~^T, or to the rate 
 
 of decrease of the included lines of force. We have seen in 
 
 * See Boole's " Differential Equations," p. 192, 2nd Edition.
 
 236 MUTUAL AND SELF INDUCTION. 
 
 Chap. III. that when an electromotive force E acts on a circuit 
 of inductance L and resistance E that the current i at any time 
 after the commencement of the application of the electro- 
 motive force is given by the equation 
 
 In the case' considered the inductance and resistance of 
 the secondary circuit are L and E, and the impressed electro- 
 
 motive force applied during a time 8 t is -yr- . Hence, at the 
 end of the interval of time St, the value of the secondary 
 current is given by the equation 
 
 This gives us the value of the inverse induced current at 
 the instant of breaking the primary. Expand the above 
 expression by the exponential theorem, and it becomes 
 
 i M MBSt M E 2 8 t 2 
 
 _ _ 
 I~T~L'l-2 + L 3 l-2-3' 
 
 At the instant when the removal of lines of force or the ces- 
 sation of the induction through the secondary takes place the 
 impressed electromotive force ceases and the secondary current 
 begins to die away. If we suppose the "break" of the primary 
 to be very sudden, 8 1 becomes practically zero, and we have 
 
 that is to say, the secondary current starts with a value equal 
 
 to of that of the steady primary. ' 
 
 Li 
 
 The state of things in the secondary circuit immediately 
 after the break of the primary is, then, this : The electromotive 
 impulse due to stoppage of the primary has generated a current 
 
 of initial value I in the secondary, but there is no impressed 
 
 L 
 
 electromotive force in the secondary circuit. If at any instant 
 after the break the current in the secondary circuit is i, the 
 law of decay of this current is expressed by the equation : 1 
 
 dt
 
 MUTUAL AND SELF INDUCTION. 
 The solution of this is 
 
 237 
 
 and the constant C is found from the condition that when 
 
 t = i = I. Hence we have 
 L 
 
 (91) 
 
 This gives us the value of the direct or "break" induced' 
 current in the secondary at any instant after the break of the 
 primary. Graphically, this may be represented by a curve, 
 such as that in Fig. 85. During the time T in which the 
 primary is being broken the induced electromotive force is 
 
 FIG. 85. 
 
 creating an induced current, the rising strength of which is 
 represented by the rise P. The time occupied by the break 
 8 t is T. As T is diminished in value, the magnitude of 
 the maximum ordinate P T approximates to I, and this is 
 
 the initial value of the inverse secondary current when the 
 break is very sudden. After the break the current decays 
 away along a path represented by P Q, and becomes zero only 
 after an infinite time. 
 
 The whole quantity of the induced current is obtained by 
 integrating equation (91) with respect to the time from zero to- 
 infinity, thus : 
 
 M T -?' ,. MI
 
 238 MUTUAL AND SELF INDUCTION. 
 
 We see, then, that both the maximum value and whole 
 quantity of the direct secondary current are proportional to 
 the coefficient of mutual induction and to the strength of 
 the primary current, and, moreover, that the whole quantity of 
 electricity set in motion in a secondary circuit of total resis- 
 tance R by suddenly removing from it M I lines of force is 
 equal to the quotient of number of lines removed by the total 
 resistance of the secondary circuit. 
 
 If the induced current is sent through a galvanometer the 
 
 indications are proportional to the magnitude of - If, how- 
 
 R 
 
 ever, the induced current is employed to magnetise steel 
 needles, the magnetisation acquired is dependent upon the 
 
 magnitude of -, and is therefore greater in proportion as 
 
 the coefficient of self-induction of the secondary circuit is less. 
 Lord Rayleigh has pointed this out,* and shown by experi- 
 ment that, within certain limits, the magnetising effect of the 
 ^break-induced current on steel needles is greater the smaller 
 the number of turns of which the secondary consists, the 
 opposite being, of course, true of the galvanometer. The 
 galvanometer takes account of the total quantity of the induced 
 current ; whilst the magnetising power depends mainly on the 
 magnitude of the current at the first moment of its formation, 
 without regard to the time which it takes to subside. 
 
 Returning to the equations (89) and (90), we can now find 
 the constants of integration, counting the time from the instant 
 of " make " of the primary. It is obvious that when t = 0, y = 
 and x = 0, and that the whole quantity of the make-induced 
 
 /QO 
 y d t, must be equal to the whole quantity of the 
 D 
 
 break-induced current, which we have seen is equal to . 
 
 R 
 In (90) put t = 0, / = ; we get 
 
 A' + B' = 0, or B'=-A'. 
 
 / Rt Rt \ 
 
 Hence, y = A' ^~?B--e=sY 
 
 and 
 
 *Phil. May., Ser. 4, Vol. XXXIX., 1870, p. 429.
 
 MUTUAL AND SELF INDUCTION. 239 
 
 Hence the whole quantity of the "make "-induced current is 
 
 - p , and this must be equal to Mi which is the whole 
 ti K 
 
 quantity of the " break " current. Hence A' = - _. 
 
 2 
 
 Therefore we get for the instantaneous strength of the 
 " make " .secondary current 
 
 i ^ 5 (92) 
 
 Again, in (89) put t = 0, x = Q, 
 and we get A + B + I = 0, 
 
 or B=-(I+A); 
 
 and by substitution in (89) 
 
 3 = A e~L+ai-(A + I) e~Z=*+I. 
 From this equation we can find the value of A by substitut- 
 
 ing the value of -^derived from equation (92), and derived 
 
 dt dt 
 
 from the above in the original differential equation (83), and we 
 
 find A= -_. Hence we arrive at the equation for the value 
 
 of the primary current at any instant, and it is 
 
 nt\ 
 ^ . . . (93) 
 
 This gives the law according to which the primary current 
 grows up in its circuit. If M = 0, that is, if there is no secon- 
 dary circuit ; then 
 
 ( 
 i- 
 
 which is the ordinary law of current growth. If M = L, which 
 is the greatest possible value of M, then 
 
 Hence it is obvious that the presence of the secondary circuit 
 hastens the rise of the primary current and operates on it to 
 reduce its inductance. 
 
 On making the primary we get a " make " or inverse 
 secondary current according to the law of growth expressed by 
 the equation
 
 240 
 
 MUTUAL AND SELF INDUCTION. 
 
 and we see that under the circumstances assumed the " make ' 
 secondary starts from an initial value zero, rises up to a maxi- 
 mum, and then decays aWay again. To find the time of reach- 
 ing maximum, equate _ to zero, and we find 
 
 f _L2-M 2 
 
 ~TKM 
 
 and this function increases as M decreases. So that the more 
 nearly M is equal to L the sootier does the secondary reach 
 its maximum. It is not difficult to show that when M = L the 
 
 above value for ' becomes zero, and when M = ' = -. 
 
 ^,-M=L (nearly) 
 
 O ( nearly J 
 
 FIG. 86. 
 
 Curves representing roughly the current value of the make-induced current for 
 different and increasing values of M. 
 
 If, then, we trace a series of curves (Fig. 86) representing 
 the values of y, or the make-induced current at each instant 
 
 for various and increasing values of , as the coils are moved 
 
 M 
 
 further apart, we find a series of curves with decreasing maxima, 
 but the maxima happening later as M decreases. 
 
 Lastly, on breaking the primary current we have a break- 
 induced current in the same direction as the primary, which at
 
 MUTUAL AND SELF INDUCTION. 241 
 
 any instant after the " break " is decaying away according to 
 the law 
 
 .-*!.-* 
 
 L 
 
 If the break was absolutely instantaneous, the induced current 
 would start with a finite value equal to __ of that of the primary, 
 
 but as no form of break entirely eliminates sparking, the rise of 
 the direct secondary current is a gradual one. Also we have 
 another element of disturbance which enters into the case. 
 The self-induction of the primary creates direct electromotive 
 force in its own circuit at the instant when the induction 
 through the primary due to its own current vanishes. When 
 the primary is broken either at a mercury cup or at a platinum 
 point the fusion and volatilisation of metal which takes place 
 keeps open for a little time a conductive path through which 
 flows the extra current due to the self-induction of the primary. 
 As will be explained later, the decay of the current on breaking 
 a circuit may often be by a series of oscillations or diminishing 
 periodic currents. 
 
 This direct extra current in the primary will have its effect 
 in introducing a very short inverse-induced current, which 
 will precede the main direct-induced current due to the decay 
 of the primary current. In any event it will introduce an 
 electrical oscillation tending to render the growth of the direct 
 secondary current a gradual matter. It is an interesting case 
 to examine the relative maximum values and duration of the 
 two induced currents under an assumption very nearly realised 
 when the primary and secondary are wound together on the 
 same bobbin, viz., when M = L. In this case the values of y 
 and z become 
 
 I - R < 
 
 The maximum of the direct currents ("break") is I, and 
 that of the inverse (or "make") is -. If we wish to know at 
 the end of what times * and t' the strengths of the two induced 
 currents y and z are reduced to of that of the primary we
 
 242 MUTUAL AND SELF INDUCTION 
 
 obtain by substitution of for y and z in the two above equa- 
 
 tions the following : 
 
 1 _?.' 
 
 _=<? "IT for the direct-induced current, 
 
 in 
 
 1 "| R ' 
 
 and _ = _g~2L for the inverse-induced current. 
 
 m 2 
 
 and therefore _ = 2 l - 
 
 We see that t' is always greater than t, and that, in propor- 
 tion as m increases, t' tends towards a limit 2 t, or the inverse 
 current has a duration about double that of the direct secondary. 
 We shall now see how this theory is confirmed by experiment. 
 
 7. Comparison of Theory and Experiment. Masson and 
 
 Breguet carried out a series of experimental researches on induced 
 currents which illustrate and confirm the foregoing theory. 
 The principal part of their apparatus was a commutator keyed 
 on a revolving shaft, which enabled them to separate the 
 direct and inverse-induced currents. Two brass wheels were 
 keyed on one shaft, but insulated from it, and the wheels 
 had depressions cut in their periphery which were filled up 
 with ivory. These wheels could be shifted relatively to 
 each other, and were insulated from each other and from 
 the shaft (see Fig. 87). Two springs pressed against the edge 
 of the wheels, and two against the hub of the wheel. The 
 whole arrangement served as a means to break and make one 
 circuit, and at the same time to control a second circuit so that 
 it was broken at the time when the first was made, and made 
 at the time when the first was broken, or vice versa. One of 
 these wheels was inserted in the circuit of a primary coil and 
 battery, and the other in the circuit of a secondary coil and 
 galvanometer. On rotating the wheel at a certain fixed speed 
 the series of "break" and " make "-induced currents are 
 separated out ; all one set are stopped out and all the other 
 are sent through the galvanometer. In this way it was shown 
 that the quantities of the induced currents were equal, but 
 very different in maximum magnitude, and hence in duration, 
 the break-induced currents being greatly superior in making 
 sparks.
 
 MUTUAL AND SELF INDUCTION. 
 
 243 
 
 Lenz* wound a spiral of wire on the soft iron armature of a 
 magnet and connected the ends of the wire to a ballistic gal- 
 vanometer. He detached the armature suddenly, and observed 
 the throw of the galvanometer. If denotes the angle of 
 deflection and x the number of windings, he found that the 
 
 1 /9 
 
 product - sin _ was a constant quantity, which shows that 
 x A 
 
 cceteris paribus, the quantity of electricity set in motion was in 
 proportion to the number of lines of induction withdrawn 
 from the circuit. He also established experimentally, in con- 
 firmation of Faraday, that the electromotive force of induction 
 was independent of the width, thickness or material of the 
 
 FIG. 87. 
 
 wire windings,! and by other experimentalists also the fact 
 has been established that the electromotive force is indepen- 
 dent of everything except the form of the conductor and the 
 nature of the change it experiences in relation to the magnetic 
 induction through it. FeliciJ carried out an extensive series 
 of experiments on induction, using a form of induction 
 balance. 
 
 * Lenz, Poggendorff's Annalen, Bd. XXXI., 1835, p. 385. 
 
 t See Faraday, " Exp. Researches," Ser. II., 193, et seq. ; also for 
 Electrolytic Circuits, see L. Hermann, Pogg. Ann., 1871, p. 586. 
 
 t Felici, Nvovo Cimento, Vol. IX., 1859, p. 345, also Ann. de Chimic [3], 
 Vol. XXXIV., 1852, p. 64.
 
 244 
 
 MUTUAL AND SELF INDUCTION. 
 
 In this apparatus a secondary circuit, consisting of two coils, 
 is arranged in series with a galvanometer. These coils are so 
 far apart as not to influence one another. In contiguity to 
 each secondary coil is a primary coil, and the primaries are 
 wound in opposite directions. The primaries are in circuit 
 with a battery and a key. The circuits can be so arranged, by 
 adjusting the distances of the coils, that the induction of the 
 primaries on their respective secondaries balance each other, 
 and the galvanometer indicates no current, however strong may 
 be the primary current. If three pairs of coils (see Fig. 88) are 
 thus taken and balanced, two and two, so that the induction 
 of A on a is equal to that of B on b and C on c, then, if we con- 
 nect the primary A in series with B and C in parallel, so that 
 
 Fio. 88. 
 
 the current divides between them in the ratio of their resis- 
 tances, and connect the secondaries with a galvanometer, all in 
 series, so that the current in a is opposed to that in b and in c, 
 then no induced current is detected when the battery circuit is 
 made and broken. This proves that the quantity of the induc- 
 tion current is proportional to the strength of the primary 
 current. 
 
 If a primary and secondary coil are taken in fixed positions 
 and the " throw " of a galvanometer observed when a definite 
 steady electromotive force E is applied to the primary, then, if 
 the position of battery and galvanometer are reversed, the 
 application of the same electromotive force E to the secondary
 
 MUTUAL AND SELF INDUCTION. 245 
 
 will give the same " throw" on the galvanometer now attached 
 to the primary circuit, provided that the galvanometer and 
 battery either have equal internal resistance or that their 
 resistance is negligible in comparison with that of the coils. 
 Hence we may assert that the induction of a circuit A upon B is 
 the same as that of B upon A. For, if the resistances are R and 
 S, then we have seen that the total quantity Q of the secondary 
 
 current is t -, where I is the steady value of the primary 
 
 and M is the mutual inductance; but I = , hence Q = . 
 
 R SR * 
 
 If, then, the positions of battery and galvanometer are reversed, 
 
 M F 
 
 we get a quantity of induced current equal to , which is 
 
 R S 
 
 the same as before. For any two coils it is possible to find a 
 number of relative positions in which the interruption of a 
 current in one produces no induced current in the other. In 
 such cases the coils are said to be conjugate to each other. It 
 is manifest that when in these positions the lines of induction 
 produced by one coil do not pass through the other. It is 
 possible to use one coil in this way to explore the field of 
 another. 
 
 Let P be a primary coil and S be a small flat secondary coil, 
 both being shown in section in Fig. 89. Then, if S is placed 
 in a position conjugate to P, it will be found possible to move 
 the coil S along a certain line ABC, maintaining the flat face 
 of the coil always tangent to that line and so that in all these 
 positions P and S are conjugate. It is evident that such a line 
 is a line of induction of the coil P. 
 
 When one coil is in a conjugate position to the another, as 
 far as regards inductive action they may be considered to be at 
 an infinite distance apart. It follows, therefore, that if a coil 
 is moved suddenly from a conjugate position to one not conju- 
 gate in the field of a primary traversed by a steady current, 
 and then the primary current is stopped at the instant of arriv- 
 ing at the second position, a galvanometer in the second circuit 
 will have its needle jerked from one position of rest to another 
 of rest, because the interruption of the current takes out 
 of the circuit of the second coil just as many lines of induction 
 due to the first coil as the motion from one position to the
 
 246 
 
 MUTUAL AND SELF INDUCTION. 
 
 other put in. A series of well-devised experiments on the 
 conjugate positions of two coils has been carried out by Mr. 
 W. Grant.* 
 
 An elaborate investigation into the duration of induced 
 currents was made by Blaserna.f 
 
 A commutator was constructed which consisted of two insu- 
 lating cylinders keyed on one shaft and having on part of their 
 surface brass coverings cut into steps (see Fig. 90). These 
 cylinders were capable of being set in any relative position to 
 each other on the shaft. The shaft could be revolved at a high 
 rate of speed, and its velocity ascertained by a siren plate 
 attached to the axis. This siren plate consisted of a disc 
 pierced with holes against which was directed a jet of air. 
 
 From the pitch of the musical note given out, when ascertained 
 by comparison with standard tuning forks, the speed could be 
 determined. Two springs pressed against the hubs of these 
 cylinders and two against the surfaces of these cylinders, and a 
 current entering by the hub was conducted to the brass coating 
 and escaped by the other spring, if the cylinder was in such a 
 position that this last spring was pressing on the metal part. 
 The apparatus, therefore, formed a device by which each 
 
 * See Proc. Physical Soc., London, Vol. III., p. 121 ; also Proc. Physical 
 Soc. London, VoL IV., p. 361. 
 
 t Blaserna, " Sul sviluppo e la durata delle Correnti d'induzione," Gwrnale 
 di Science Naturali, Vol. VI. (Palermo, 1870).
 
 MUTUAL AND SELF INDUCTION. 247 
 
 pair of springs might be brought into electrical contact for 
 a definite portion of the time of a revolution of the cylin- 
 ders and be insulated also for a given time, each pair of 
 springs being in connection relatively to the other in a deter- 
 mined manner for a determined time. In the circuit of the 
 one cylinder and pair of springs m M was placed a battery 
 primary coil and tangent galvanometer, and in the circuit 
 of the other pair a secondary coil and sensitive galvano- 
 meter. This being prepared, the primary coil P and the 
 secondary S were placed a given distance apart. On revolving 
 the commutator it periodically interrupts the primary current, 
 the time during which the primary current is kept on depend- 
 ing upon the position of the spring M on its cylinder. The 
 other cylinder can be so set as to collect either the direct or 
 
 C dd 
 
 FIG. 90. 
 
 inverse secondary currents, and send them in series through the 
 sensitive galvanometer, the time during which this secondary 
 circuit is closed being capable of regulation by the adjustment 
 of the spring M t . In his experiments Blaserna first investi- 
 gated the duration of each of the induced currents. The 
 interrupters were so arranged relatively to one another that, 
 whilst the primary circuit was made and broken, the secondary 
 circuit was not closed until a small time after " making" the 
 primary, and then broken again before the primary was broken. 
 By adjusting the secondary interrupter a position could be 
 found in which the galvanometer just showed no current. The 
 interval between the closing of the primary and the opening of 
 the secondary was then the interval occupied by the secondary
 
 248 
 
 MUTUAL AND SELF INDUCTION. 
 
 current, and this was the duration of the " make "-induced 
 current. Blaserna found that the " make " secondary (inverse) 
 lasts a longer time than the "break" current (direct). For 
 the coils used the times were 
 
 Inverse secondary lasts '000485 second. 
 Direct secondary lasts -000275 second. 
 
 He next proceeded to obtain the curve of each current, and 
 to determine the time of arrival at- a maximum. 
 
 The secondary interrupter was so set that the secondary 
 circuit was closed just before the primary, and opened after at 
 a certain definite interval of time. The galvanometer thus 
 
 FIG. 91. 
 
 received a current which was made up of repeated doses of the 
 whole quantity of the induced current up to a certain fraction. 
 Knowing the speed of the commutator and the coefficient of the 
 galvanometer, the value of the whole quantity of the induced 
 current, extending over a certain fraction of its whole duration, 
 was known ; and from those observations, repeated at regular 
 progressive intervals during the whole period of the current, 
 the value of the ordinates of the current curve can be obtained. 
 For, if the curve (Fig. 91) A P P' B (upper figure) represents the 
 variation of current during a time A B, so that P X = y repre-
 
 MUTUAL AND SELF INDUCTION. 249 
 
 sents the current strength at a time X, and P' X' represents the 
 current strength after a very small interval of time, X X' = d t 
 then the area P P' X' X = y d t represents the quantity of elec- 
 tricity which has passed in the time XX'. Call this dQ. 
 
 Hence dQ = ydt, or y = ^. 
 
 dt 
 
 Suppose another curveA'P'R (lower curve) is drawn on an 
 equal abscissa A'B', such that its ordinate at every point 
 represents the whole area of the upper curve up to the 
 corresponding point that is to say, the lower curve is a curve 
 such that its ordinate P' X' is proportional to the area A P X 
 of the upper curve, AX (upper curve) being equal to A'X' 
 (lower curve), when the time interval d t becomes very small. 
 It is easily seen that if the area A P X (upper curve) is called 
 Q, and the ordinate P X is called y, that the tangent of the 
 angle P'YX' (lower curve) which the geometrical tangent 
 drawn at P' makes with the axis A' B' , and which is repre- 
 sented by -?, is proportional to the ordinate PX. Hence the 
 dt 
 
 upper curve is a derived curve of the lower, and, if we are 
 given a curve like the lower curve, the ordinates of which 
 represent the whole quantity of electricity which has from a 
 given epoch flowed past a point, we can, by drawing a curve 
 whose ordinates represent the slope of the first curve, obtain a 
 second curve, which is a curve of current. In this way it is 
 possible to describe the current curve, and to determine its 
 form and position of maximum. 
 
 Blaserna found that the greater the distance apart of the 
 primary and secondary in other words, the less the mutual 
 inductance the less was the maximum value of the secondary 
 current, and the greater the delay in the appearance of that 
 maximum. This is in accordance with the above elementary 
 theory. In the case of the "break," or direct secondary 
 current, he found the delay in establishing the maximum not 
 BO great, and the maximum ordinate was greater though the 
 total duration of the current was less. He established by direct 
 experiment the equality of the quantity of the two induced 
 currents. When the coils were very near together the induced 
 current at starting established itself by a series of electrical 
 oscillations.
 
 250 MUTUAL AND SELF INDUCTION. 
 
 By the help of the same apparatus Blaserna investigated 
 the rise of a current in a coil when the same is placed suddenly 
 in connection with a constant source of electromotive force. 
 For the "make" extra current only one of the revolving 
 interrupters was used, and the circuit was completed by the 
 means of a battery, galvanometer, and coil. When the com- 
 mutator was revolved it first started the current and then 
 after an interval cut it off again, and the effect on the 
 galvanometer is due to the sum of all these small quantities 
 of electricity so cut off and integrated whilst the current is in 
 process of increasing. As the duration of the time of contact 
 was increased the galvanometer deflection increased (speed of 
 revolution remaining constant), but when the time of contact 
 was long enough to fully establish the current, then increase 
 
 B 
 
 FJG. 92. 
 
 of speed of rotation did not increase the galvanometer deflec- 
 tion. By this apparatus the fact was established that the 
 primary current established itself in its coil by a series of 
 oscillations, or short alternating currents. 
 
 Similarly, on breaking the circuit the course of the current 
 was investigated. For this purpose one revolving interrupter, 
 I, was inserted in the circuit of a battery, B, and coil, C, and 
 from the ends of the coil (see Fig. 92) other ,wires were brought 
 and led through the galvanometer G, and other interrupter I', 
 arranged as a shunt on the coil. The break in the battery 
 circuit at p was so arranged that each time the current was 
 fully established before being broken again. The break in the
 
 MUTUAL AND SELF INDUCTION. 251 
 
 galvanometer or shunt circuit was so arranged relatively to 
 the other that the shunt circuit was closed a little before the 
 battery circuit was broken, and then opened at a definite 
 interval afterwards. In this way there was a little flow of 
 current through the galvanometer due to the steady current, 
 but this could be estimated and allowed for. On plotting out 
 a current curve from the quantity curve it was found that the 
 current decayed away on interrupting the circuit by a series of 
 oscillations which followed each other much quicker than those 
 on the establishment of it, and the whole duration of the extra 
 current at " break," or the time of falling from steady current 
 to practical zero, was less than the time required to f ally estab- 
 lish the current. It was found that the first oscillation, on 
 beginning to interrupt the steady current, had a much greater 
 amplitude than any of those on starting the current. 
 
 The duration of an oscillation was perhaps three or four ten- 
 thousandths of a second, and about 50 to 100 oscillations pro- 
 bably happened before the current became steady ; hence the 
 whole duration of the variable period, or of the extra current, 
 was about two to three-hundredths of a second. Very roughly, 
 the nature of the oscillatory character of the current at the 
 make and break may graphically be represented by the curve 
 in diagram Fig. 93.* 
 
 Blaserna drew from his observations the deduction that there 
 is an interval of delay in the starting of the secondary currents, 
 and that a small but measurable time elapses between the 
 instant of making or breaking the primary circuit and the 
 beginning of the secondary current. From this he made a 
 calculation as to the velocity of electromagnetic induction, and 
 he also stated that the interposition of dielectric substances 
 such as glass or shellac between the coils reduced the so- 
 calculated velocity. 
 
 Bernstein (Pogg. Ann., Bd. CXLIL, 1871, p. 72) repeated 
 these observations of Blaserna, but did not confirm these last 
 results. He found that the first oscillation always began at 
 the instant of breaking or making the primary circuit, and he 
 
 * In The Electrician for June 1, 1888, a curve is given by Mr. F. Higgins, 
 showing the rise of current in the magnets of type-printing telegraphs, and 
 the oscillatory character of the current at starting is well marked. Mr. 
 Higgins's curve gives the results of actual observations.
 
 252 
 
 MUTUAL AND SELF INDUCTION. 
 
 found no effect produced by the interposition of dielectric 
 media. 
 
 Helmholtz has carefully examined these results of Blaserna 
 and criticised them.* He remarks that Blaserna used for his 
 coils flat spirals of wire with many turns, and also he used the 
 current from several Bunsen cells to create the primary current. 
 Not only do the spirals act like a condenser, giving the whole 
 apparatus a sensible electrostatic capacity, but the use of a 
 battery of high electromotive force causes a considerable spark 
 at the break, which spark has a very sensible and rather 
 irregular duration. Also in Blaserna's experiments, the two 
 circuits were placed at various distances apart. If a current 
 
 FIG. 93. 
 
 is started in a primary coil the effect of the induced current 
 created in the secondary by its re-acfcion on the primary is to 
 hasten the rise of the primary current, and at the break to 
 accelerate its decay. As the secondary circuit is moved further 
 off this effect is less marked. Hence, the rise and fall of the 
 primary is more gradual and the arrival of the secondary 
 current at its maximum value is more delayed. From this 
 results, then, an apparent retardation of the time of the arrival 
 of the maximum of the induced current. 
 
 * Helmholtz, " On the Velocity of the Propagation of Electrodynaraic 
 Effects," Phil. May., Ser. 4, VoL XLIL, 1871, p. 232.
 
 MUTUAL AND SELF INDUCTION. 053 
 
 Helmholtz conducted a series of experiments by means of 
 his pendulum chronoscope. A heavy iron pendulum P (see 
 Fig. 94), 'the lower end of which carried two plates of agate, 
 could be made to execute one swing and then be caught by a 
 detent. These plates of agate in the course of the swing were 
 caused to strike against and tip over two little levers I, I'. One 
 of these levers was fixed, and the other could be moved 
 forward so as to separate the blows. One was made to break 
 the circuit of a primary coil Pi; when tipped over, and the 
 
 Se, 
 
 FIG. 94. 
 
 other by its movement separated a connection between a con- 
 denser and the ends of a secondary coil, Sec, attached to it. 
 
 These being arranged, the fall of the pendulum executed 
 these two "breaks" successively, separated by an interval of 
 time capable of being calculated from the known motion of 
 the pendulum. The two circuits were placed 170 centimetres 
 apart. The primary consisted of 12 turns of thick wire, and 
 the secondary of 560 turns of fine wire, The current was sent
 
 254 MUTUAL AND SELF INDUCTION. 
 
 from one Daniell cell. The two ends of the secondary were 
 connected to the two plates of the condenser, and when the 
 pendulum fell it broke the primary current and started in the 
 secondary circuit an oscillatory current reverberating to and 
 fro in the secondary wire, the condenser acting as a resonator. 
 At a definite interval after rupture of the primary, the 
 condenser was separated and examined by a quadrant electro- 
 meter. The charge in the condenser showed the phase of the 
 electrical oscillation existing at the instant of such separation. 
 In one case Helmholtz observed 35 oscillations in J^th of a 
 second. In order to discover if any retardation took place 
 with increased distance of the coil, it was necessary to fix 
 attention upon some phase in the oscillations. The successive 
 zero points of the current were very sharply defined, and 
 suitable for this purpose. Helmholtz found that alteration of 
 the distance between the primary and secondary coils made 
 no perceptible difference in the position of the zero points, 
 and that, as far as the apparatus he was using could detect, 
 the velocity of the electro -magnetic impulse must be greater 
 than 195 miles per second. He pointed out in this Paper 
 that the commencement of the secondary current is not a 
 sharply marked thing. The spark which takes place at break 
 of the primary lasts an appreciable time, and all this time the 
 primary is dying gradually, and the induced current therefore 
 is increasing. The period of duration of the break spark may 
 be something like y-giy^th to ^thn^k f a second, and is, 
 therefore, a large fraction of the duration of a single electrical 
 oscillation, which amounted to about -^Vyth of a second. 
 The duration of the break spark can be found by observation 
 of the time which elapses from beginning of break up to the 
 first zero point of the secondary current oscillations, as com- 
 pared with the mean value of the duration of an oscillation. 
 The interval up to the first zero point is the duration of the 
 break spark plus the time of half a complete oscillation. The 
 duration of the spark is never constant, and depends a good 
 deal on the amount of platinum thrown off from the contacts 
 each time. The average duration of the spark in Helmholtz's 
 experiments was found to be about one-tenth of the whole 
 period of an oscillation. Helmholtz also noticed in some 
 earlier observations evidence of electrical oscillations set up in
 
 MUTUAL AND SELF INDUCTION. 255 
 
 a flat spiral, one end of which was insulated. In this case 
 some 45 oscillations were detected in the space of ^th 
 of a second. Henry also noticed that the time of subsidence 
 of the current, when the circuit is broken by means of a 
 surface of mercury, is very small, and probably does not 
 much exceed the ten-thousandth part of a second. It has, 
 however, a quite appreciable duration, for Henry found 
 that the spark at ending presents the appearance of a band 
 of light of considerable length when viewed in a mirror 
 revolving at the rate of six hundred revolutions per second. 
 
 Bernstein, with the aid of a contact break somewhat 
 different from that used by Blaserna, also examined the 
 duration of the oscillations set up in a secondary coil. He 
 found that the duration of the first oscillation at breaking 
 primary was longer than that of the subsequent ones. The 
 mean duration when using a single Grove cell in the primary 
 circuit was '0005 second, and when using a Daniell cell only 
 0001 second. We shall return later to consider more recent 
 researches on these electrical oscillations in inductive circuits 
 and point out that they can only occur when some part of the 
 circuit possesses sensible electrical capacity. In the case 
 of a coil or bobbin of wire we have not only resistance and 
 inductance, but measurable capacity present in the conductor. 
 
 8. Magnetic Screening and the Action of Metallic 
 Masses in Induction Coils. At one stage* of his investi- 
 gations Henry made the important discovery that, if a 
 primary and secondary coil are separated by a metallic sheet, 
 a notable decrease takes place in the intensity of the shock 
 taken from the secondary circuit when a sudden discharge is 
 passed through the primary, or continuous current started or 
 stopped in the primary circuits. A thick copper plate was found 
 more effective than a thin one in thus preventing the inductive 
 effect of the primary upon the secondary coil. If a radial slit 
 was cut in a circular metallic plate the annulling effect was 
 altogether stopped. If the two edges of the gap (see Fig. 95) 
 were furnished with wires leading to a magnetising spiral, 
 Henry found he could in this way make evident the existence 
 in the plate of a current induced by the action of the primary. 
 * ^hoTMag., Vol. XVI.7l840, p. 257.
 
 256 
 
 MUTUAL AND SELF INDUCTION. 
 
 A flat coil of insulated wire was substituted for the metal 
 plate, and it was found that the screening action of this coil 
 was only sensible when the two ends were joined so as to com- 
 plete the circuit. This action, by which the induction of a 
 primary coil on a secondary is prevented by the interposition 
 of a metallic plate, cylinder, or closed circuit of insulated wire, 
 is called magnetic screening. The elementary explanation of 
 this effect is not difficult to arrive at. Suppose a small con- 
 ducting circuit of resistance B to be placed in a magnetic field 
 so that it is traversed normally by N lines of magnetic induction. 
 Let the constant coefficient of self-induction of this circuit be 
 L. If, then, in any small time d t a variation of the lines of 
 induction traversing this circuit takes place, the impressed 
 
 FIG. 95. 
 
 d N 
 
 electromotive force on that circuit will be represented by 
 
 d t ' 
 
 and if at that instant the current in the circuit is i, by the 
 principles laid down in the last chapter the current equation 
 will be 
 
 ~dt' 
 --(Li + NUR-0. 
 
 Suppose the conductivity of this circuit to be perfect, and E 
 therefore zero, we have, by integration of the above, equation, 
 the result 
 
 L i + N = const. ;
 
 MUTUAL AND SELF INDUCTION. 257 
 
 in other words, the lines of induction L i, linked to the circuit 
 at any instant due to the induced current generated in it, are 
 opposite in direction to those whose variation is producing 
 the current, and together with them make up a constant 
 number. Hence, if the variation of N is such as to take 
 lines of induction out of the circuit, the action of the 
 current thereby induced is to add or increase them in the 
 circuit at an equal rate. If we suppose our circuit to be 
 a perfectly conducting metal plate, and just behind this metal 
 plate there is another small closed circuit, then any variation 
 of lines of induction passing through this plate will not take 
 effect in producing any induced current in the small circuit, 
 because the inductive action of the current induced in the plate 
 nullifies, as far as the small circuit is concerned, any vari- 
 ation of the external field. It is clear that these conclusions 
 would apply to any surface of finite extent which possessed 
 perfect conductivity ; the induced currents which any vari- 
 ation of the external field would produce in this surface 
 would always be such that the induction through each portion 
 would be kept constant in other words, that the perpendicular 
 component of the magnetic induction at each point on the 
 surface would retain a fixed value. It follows that a closed 
 surface of zero resistance is a complete screen for all points in 
 the interior against the effects of variation of the field on 
 conductors on the outside of the surface ; these effects reduce 
 to the production of surface currents in the shielding conductor, 
 which keep the resultant field in the interior constant or at 
 zero. 
 
 Faraday describes (" Exp. Researches," Vol. L, 1720 et seq.) 
 an experiment which at first sight seems to disprove the fact 
 of magnetic screening. He placed a flat copper wire spiral, 
 which was in connection with a battery and key, between two 
 other flat spirals which were respectively connected with the 
 two coils of a differential galvanometer. The coils were so 
 joined up that the inductive effect of a break and make of the 
 battery circuit produced no movement of the galvanometer 
 needle because it was subjected to two equal and opposite 
 impulses from the two coils. When an exact balance was 
 obtained a flat plate of copper, nearly three-quarters of an inch 
 thick, was interposed between the primary spiral and one of 
 
 s
 
 258 MUTUAL AND SELF INDUCTION. 
 
 the secondaries. The galvanometer needle was not, however, 
 any more affected than if the copper was absent. To under- 
 stand this we must bear in mind that the break or make of the 
 primary current produces in the copper a secondary current, 
 but as the effect of the primary coil on the secondary coil on 
 that side is balanced by the other one we may regard the 
 secondary coil next the copper plate as free to receive any 
 inductive effect it can from the eddy current induced in the 
 copper block. This secondary current induced in the copper 
 generates a tertiary current in the secondary spiral, and this 
 tertiary current consists, as we have seen, of a double short 
 flux of electricity equal in quantity and opposite in sign. The 
 galvanometer is then traversed by two small equal quantities 
 of electricity in opposite directions, and as this does not 
 sensibly affect a not very sensitive galvanometer no movement 
 of the needle is seen. If, however, instead of the differential 
 galvanometer, Faraday had used a differential telephone, he 
 would have found distinct evidence of a screening action. 
 Again, suppose that, instead of a simple make or break, 
 Faraday had employed a steadily periodic or alternate cur- 
 rent in the primary, this would have set up a steady periodic 
 secondary current of equal frequency in the copper plate, and 
 this again would have set up in the secondary coil on that side 
 a steadily periodic tertiary current of equal period, and this 
 might have been detected by the use of a sensitive differential 
 electro-dynamometer or a soft iron needle galvanometer. 
 
 Henry found that a sheet of tinfoil afforded a very small 
 amount of screening for shock, but a thick sheet of copper a 
 very considerable one in the case of induction by battery cur- 
 rents, and in the case of induction by Ley den jar discharges 
 the same phenomenon was apparent. In the case of an iron 
 screen there is an additional effect, due to the fact that the 
 iron, by its small magnetic resistance, conducts away the lines 
 of induction somewhat through its mass, and prevents them 
 from extending to the space on the other side. In this case 
 also a considerable thickness of metal is necessary to bring 
 about the effect of annulment. When we are limited to the 
 use, as we are in practice, of materials whose conductivity is 
 far from being perfect, it is found that a thin screen of metal 
 hardly affords any sensible protection from inductive effect.
 
 MUTUAL AND SELF INDUCTION. 259 
 
 In other words, the field on the other side of the screen is " 
 very far from constant. This has been well demonstrated 
 in certain investigations by Prof. D. E. Hughes in carrying 
 on some highly valuable experimental researches into the 
 means of preventing induction upon lateral telegraph wires.* 
 It has many times been proposed to annul mutual induction 
 between telegraph and telephone wires by covering them 
 over with thin metal covering, which covering is kept " to 
 earth." It is now known, and well exemplified in Prof. 
 Hughes's experiments, that this shielding affords no protec- 
 tion when the covering is not very thick and when the rate ol 
 change of the currents is not very rapid. A gutta-percha 
 wire was enclosed in ten coverings of tinfoil, and such 
 arrangement was not found to afford protection to induction, 
 as detected by a telephonic wire stretched alongside. Even 
 when twenty coatings of thin charcoal iron were put round 
 the wire, not only was there found to be a very sensible per- 
 manent field outside the iron, but changes of field were made 
 manifest also. It is not to be taken that these experiments 
 disprove the fact of magnetic screening, but only that the 
 low conductivity of the envelopes used is ineffective at the 
 speed of current change employed to render visible the effect 
 of magnetic screening. It is different, however, if the induc- 
 tive effects are being produced by a very rapid rate of change 
 of field. For suppose that a small circuit, as before, is placed 
 in a uniform field, and is traversed by q lines of induction due 
 to this external field. Suppose q varies according to a simple 
 periodic law, so that q = Q cosp t, where p = 2?r n t n being the 
 frequency of the alternations. Then we have 
 
 - 
 
 but - is the value of the impressed electromotive force in 
 
 d t 
 
 the circuit, and if we call the current at any instant i, then, 
 by the principles in Chap. III., we have 
 
 sm(pt-6), 
 
 * See a Paper by Prof. Hughes " On Lateral Induction in Telegraph 
 Wires," read before the Society of Telegraph Engineers, March 12, 1879. 
 Published in The Electrician, March 22, 1879. 
 
 s 2
 
 260 MUTUAL AND SELF INDUCTION. 
 
 in which R is the resistance and L the inductance of the- 
 circuit, and 
 
 Suppose that R is very small compared with L^, whichi 
 is the case when n or the frequency of alternation is made 
 very great, then R vanishes compared with ~Lp, and if we call 
 i' the value towards which i approximates in this case, we 
 have 
 
 
 
 - cosp t, 
 L 
 
 and l = 
 
 d t, L 
 
 . 
 dt dt 
 
 Hence L^'--li, 
 
 dt dt 
 
 or L i' + q = constant. 
 
 Hence the field due to the current in the circuit, together 
 with the external field, is a constant quantity, and we get the- 
 condition of perfect shielding. We may sum up the fore- 
 going by saying that, if a screen of absolutely no electrical 
 resistance is interposed between a primary and secondary coil, it- 
 effects a perfect magnetic screening, whatever may be its thick- 
 ness. If, on the other hand, the screen has a finite conductivity, 
 then the screening will be very imperfect, unless a very great. 
 thickness of material is used, and the above will be true when 
 the change of field or the change of primary current is a simple- 
 "make" and "break" or a slowly periodic change. When,. 
 however, the change of current in the primary is very rapidly 
 periodic, then the screening effects of even imperfect conductors 
 will make themselves felt, and a comparatively thin screen of 
 metal will effect a nearly perfect shielding for induction. This 
 theory is strikingly confirmed by some very beautiful experi- 
 ments of Mr. Willoughby Smith, which are described in the- 
 Journal of the Society of Telegraph Engineers (November 8, 
 1883, Vol. XII., p. 458),* and entitled " Experiments on Volta- 
 
 * See also The Electrician, November 17, 1883, p. 18.
 
 MUTUAL AND SELF INDUCTION. 261 
 
 Electric Induction." Mr. Willoughby Smith's apparatus con- 
 sisted of two flat coils A and B (see Fig. 96), placed a certain 
 distance apart. One of these was a primary coil connected with 
 -a battery, and the other was connected with a sensitive galvano- 
 meter. In the circuit of both were current reversers, which 
 reversed the galvanometer and battery alternately, and hence 
 made the opposite induced currents both affect the galvano- 
 meter in the same direction. This being arranged, the commu- 
 tator was started so as to reverse the currents very slowly, and 
 a sheet of copper interposed between the spirals. Under these 
 circumstances the interposition of the copper produced but 
 little effect. If, however, the commutator was driven at a very 
 rapid rate the copper plate caused a marked diminution in the 
 galvanometric deflection, and this diminution was greater in 
 proportion as the speed was greater. In the original Paper 
 
 FIG. 96. 
 
 a curve is given (Fig. 97) which shows the decrease in the 
 galvanometer deflection, expressed as a percentage of the 
 original undiminished deflection, corresponding to various 
 speeds of reversal. It will be seen that the less the conduc- 
 tivity of the metal the greater must be the speed in order that 
 the magnetic screening may approach perfection. Iron, of 
 course, occupies an exceptional position. It cuts off, even at 
 very low speed reversals, a large portion of the field, not by a 
 true screening action, but by conducting away the lines of 
 magnetic force and preventing their access to the secondary 
 <j}il. It will be seen that at any given speed the order in 
 which the metals reduce the deflection is the order of their 
 electric conductivity, and that as far as the diagram goes the 
 lines all (except iron) slope upward, indicating that at very
 
 -202 
 
 MUTUAL AND SELF INDUCTION. 
 
 "high speeds the screening of even the worst conductors will 
 approach perfection. It would no doubt be found that, if the 
 
 -telephone were used as a detector, the magnetic screening of a 
 
 "copper plate or thin tinfoil sheet would become very manifest 
 for high notes when not in anyway marked or distinguishable 
 
 -for notes or sounds of low frequency of vibration.* 
 
 As far back as 1840 Dove had made experiments! on the 
 
 effect of the introduction of cores of various materials into the 
 primary circuit of an induction coil. His apparatus consisted 
 of two similar primary bobbins wound on tubes of non-metallic 
 substance and connected in series (Fig. 98). Over each primary 
 bobbin was wound a secondary circuit, and these secondary 
 circuits were connected in series, but so that the induction of 
 
 90 
 
 80 
 
 I 70 
 s 53 
 
 r 
 
 40 
 30 
 20 
 IO 
 
 
 1 ^^ 
 
 
 Copper 
 Zinc 
 
 Tin 
 Iron 
 
 Lfead 
 
 \ /^ 
 
 
 ^*^~ 
 
 \ / 
 
 ^~^~~ 
 
 
 JL- 3 
 
 
 
 
 
 
 ^s^ 
 
 / 
 
 / 
 
 ^s' 
 
 
 / 
 
 / 
 
 ^^ 
 
 
 
 / 2 
 
 ^^> 
 
 / / 
 
 7 / ^5 
 
 
 
 
 
 100 00 1000 1500 200C! 
 
 Rsversals Per Minute 
 
 FIG. 97. 
 
 the two primary bobbins operated in opposite directions and 
 nullified on the whole secondary circuit each other's effect. 
 Exact neutralisation was obtained by adjusting one of the 
 secondaries. When this was the case, various cores of iron 
 rods of different kinds were inserted in one primary bobbin, 
 and it was found that the induction balance was destroyed. 
 
 * The above explanation of the cause of the difference between the 
 screening of the different metals is not that given by the distinguished 
 investigator, but it is the explanation which to the author seems most in 
 accordance with known principles. 
 
 t Dove, Poggendorffs Annalen, Vol. XLIX., 1840.
 
 MUTUAL AND SELF INDUCTION. 
 
 263 
 
 By inserting iron wires of a certain size in the other core, 
 balance could be again obtained, but not simultaneously, as 
 estimated by the galvanometer and by the shock. Thus, with 
 a bar of forged iron, 110 wires had to be inserted in the other 
 coil to obtain an equilibrium, as estimated by the galvano- 
 meter; but, as far as could be judged by the shock, 15 wires 
 
 were sufficient. With regard to different kinds of iron, experi- 
 ment shows that if we class them according to galvanometrie 
 effect we obtain a different series to that at which we arrive 
 when classifying them in the order in which they create 
 sensation by shock. Thus grey rough cast-iron is the kind
 
 264 MUTUAL AND SELF INDUCTION. 
 
 which approached nearest to bundles of soft iron wire in 
 respect of increasing the shock. Enclosing iron wires in a 
 brass tube reduced the action of the wires in disturbing the 
 indictive balance and rendered them very little better than a 
 bar of solid iron. When the primary current was a discharge 
 from a Leyden jar, Dove found that the physiological effect 
 (shock) of the secondary current, as estimated, was reduced by 
 the introduction into the primary bobbin of non-magnetic 
 conducting cores ; in other words, the introduction of a core of 
 non-magnetic but highly conducting material into the primary 
 bobbin reduced the power of a primary discharge to create a 
 secondary discharge. These last results may be obtained in a 
 more modern form by the substitution of a Bell telephone to 
 detect the tertiary currents generated by the metal core. 
 
 Let a Bell telephone be connected in series with the 
 secondary coil of a small induction coil, of which the primary 
 is wound on a hollow bobbin and the frames are wholly 
 of wood or non-metallic substance. A convenient form is 
 that known as Du Bois-Beymond's sliding coils. Let an 
 interrupter in the primary circuit make and break the circuit 
 rapidly. This being so, the telephone emits a steady rattle 
 or hum. If a massive copper rod is introduced into the 
 primary bobbin as a core, the telephonic rattle is more 
 or less suppressed ; if a core of soft iron wire is introduced 
 the noise is increased ; if a core of solid iron or steel is 
 used the noise may be increased, but not so much as when 
 the divided iron is used. The explanation of the exalting 
 effect of the soft iron wire is simple. The presence of the 
 iron reduces the reluctance of the magnetic circuit. More lines 
 of induction therefore flow through the secondary circuit, and 
 hence the strength of the secondary current is increased, and the 
 mean rate of change of induction through it is also increased. 
 The diminishing effect of the copper core is explicable in the 
 light of the knowledge that in such a conducting core the 
 primary current generates induced currents, and these in their 
 turn re-act upon the secondary circuit, inducing in it a tertiary 
 current. The directions of the currents induced by the primary 
 in the solid core and in the secondary circuit are the same. 
 The direction, however, of the first half of the tertiary current 
 developed in the secondary by the current in the copper core is
 
 MUTUAL AND SELF INDUCTION. 265 
 
 opposite to the direction of the current developed in the secon- 
 dary by the action of the primary. Hence it results that the 
 current in the secondary circuit is more or less wiped out by 
 the opposing inductions due to the primary circuit and the 
 currents induced in the copper core. Otherwise the operation 
 might be regarded thus : Suppose the primary circuit to be 
 traversed by a periodic current creating a simple periodic flux 
 of induction through the copper core. As we have seen, under 
 the head of magnetic screening, this variation of induction 
 would induce currents in the copper core, which would them- 
 selves generate a flux of induction which would, if the con- 
 ductivity of the core were perfect, or the rapidity of change 
 of induction infinite, be exactly equal and opposite at each 
 instant to the flux of induction producing those currents. 
 
 If the conductivity is not quite perfect, or the rate of 
 -variation not very great, yet nevertheless the direction of 
 
 FIG. 99. 
 
 the field of magnetic force inside the copper, due to the cur- 
 rents induced in its mass, will more or less oppose the field of 
 force at every instant which is by its fluctuations generating 
 those currents. If the thick line III in Fig. 99 represents the 
 sinusoidal or simple periodic change of induction or magnetic 
 field in the interior of the copper, due to the primary helix, 
 :and if the dotted line 22 represents roughly the changing 
 field due to the eddy currents generated in the core, which are 
 nearly 180 behind the primary in phase, the integral or sum 
 of both superimposed fields represented by 3 3 at any instant 
 is less than the original one due to the primary alone at the 
 corresponding instant. Also the mean rate of cliamje of the 
 resultant field is less, and the secondary circuit experiences at 
 every instant a less inductive electromotive force. The same 
 reasoning which we have employed in the case of magnetic
 
 266 MUTUAL AND SELF INDUCTION. 
 
 shielding applies here, and the differences in the reducing 
 effect of cores of various rnetals would be found to be less 
 at high speeds of alternation than at low. In some small 
 induction coils used for medical purposes the strength of the 
 secondary current is graduated by drawing in or out of the 
 primary coil a copper tube which slips over the bundle of fine 
 iron wires used as a core. The rationale of the action of this 
 copper tube in so operating is in a general way to be found 
 in the principles laid down above. 
 
 When Henry obtained possession of the " Experimental 
 Researches " of Faraday, as detailed in the fourteenth series of 
 his " Experimental Researches," he was exercised in his mind 
 to reconcile the results obtained by Faraday on the interposi- 
 tion of metallic screens between inducing and induced circuits 
 with his own. Faraday had found that when the galvanometer 
 was used as a current finder " it makes not the least differ- 
 ence " whether the space between the primary and secondary 
 coils was air, sulphur, shellac, or such conducting bodies as 
 copper and other non-magnetic metals. On the other hand, 
 Henry found that a shock from a secondary coil which, 
 would paralyse the arms was so much reduced by the inter- 
 position of a metallic plate as hardly to be sensible on the 
 tongue. Here was evidently something to be explained, and 
 in a long memoir (Phil. Mag., Series 3, Vol. XVHL, 1841, 
 p. 492 ; also Transactions of the American Philosophical 
 Society, Vol. VHL, 1840) Henry examined this and other 
 matters. He first verified Faraday's experience by attaching 
 the ends of a secondary coil to a galvanometer and bringing 
 up suddenly towards it a permanent magnet, or a coil 
 traversed by a steady current. The swing of the galva- 
 nometer was found to be quite unaffected in extent by the 
 interposition of a plate of copper. Again, in place of the 
 copper plate, a closed metallic conductor (an endless coil) was 
 employed, but whether the circuit of this coil was open or 
 closed it made not the slightest difference on the galvano- 
 meter deflection. 
 
 Forty feet of copper wire, covered with silk, were wound on 
 a short cylinder of stiff paper, and into this was inserted a 
 hollow cylinder of sheet copper, and into this again a rod 
 of soft iron. When the latter was rendered magnetic, by
 
 MUTUAL AND SELF INDUCTION. 267 
 
 suddenly bringing in contact with its two ends the different 
 poles of two magnets, a current was generated in the wire, 
 but the strength of this current, as measured in the galvano- 
 meter, was the same whether the copper cylinder was present 
 or was removed. Henry then noticed that there was one 
 element of difference between the indications of a galvano- 
 meter and that of the magnetising spiral. If the two 
 secondary currents at "break" and "make" of a primary 
 were sent through a magnetising spiral and through a gal- 
 vanometer, the arrangement might be such that the induced 
 current at "make" of the primary was unable to give any 
 sensible magnetisation to the steel needle enclosed in the 
 spiral, but at " break" was able to magnetise it to saturation. 
 Nevertheless, in both cases the "throw" of the galvanometer 
 was the same. Similarly with the degree of shock felt, the 
 galvanometer indications being alike for the inverse and 
 direct induced current ; yet that induced current gave the 
 greatest shock which was able to produce the greatest magne- 
 tisation. The explanation of these facts became clear as soon 
 as it was seen that the deflections of the galvanometer 
 depended upon the whole quantity of the discharge, and must 
 necessarily be alike for the inverse and for the direct current, 
 but that the magnetising effect and the physiological shock 
 depended upon the maximum value of the instantaneous 
 discharge current, and might therefore be very different for 
 the two induced currents. It was then evident that any 
 -actions by which this maximum value of an induced current 
 was decreased, whilst its duration was increased and total 
 quantity left unaltered, would result in rendering this current 
 less easily detectable by shock or magnetisation, but make no 
 difference in its effect on a galvanometer. Aided by this 
 thought, he repeated Faraday's experiment with the balanced 
 coils referred to in 8 ("Experimental Eesearches," Vol. L, 
 1,790 et seq.). A galvanometer was provided having two 
 equal wires of the same length and thickness wound on the 
 same frame, and also a double magnetising spiral was pre- 
 pared by winding two equal wires round the same piece of 
 hollow straw. Coil No. 1, connected with a battery, was 
 supported perpendicularly on the table, and coils Nos. 3 and 4 
 were placed parallel, one on each side, and each coil connected
 
 268 MUTUAL AND SELF INDUCTION. 
 
 in series with one coil of the differential galvanometer and 
 with one spiral of the magnetising helix. The two outside 
 coils were then adjusted so that when the battery circuit was 
 made and broken, and the current started and stopped in the 
 middle coil, no indication was given by the galvanometer, and 
 no magnetisation produced in a steel needle placed in the 
 double helix. A thick zinc plate was then introduced between 
 the primary coil and one of the secondaries, and it was found 
 that the needle of the galvanometer still remained stationary 
 on making and breaking the primary current, but that the 
 steel needle in the spiral became powerfully magnetic. This 
 indicated that the two secondary currents, whilst still equal 
 in total quantity, had been so affected that one had a less 
 maximum value than the other, and hence a differential 
 magnetising action was produced. A similar effect was 
 observed when a galvanometer and magnetising spiral were 
 together introduced into the secondary circuit of a single 
 primary and secondary circuit. The interposition of a metal 
 sheet considerably reduced the magnetising power or the 
 shock, but left the galvanometer deflection unaltered. In 
 order to increase the number of facts, this last experiment 
 was varied by the exchange of a soft iron needle for the hard 
 steel needle in the magnetising coil, the metal screen being 
 interposed in each case, and it was found that whereas the 
 metal screen cut off almost entirely the power of the secondary 
 current to magnetise hard steel, it could yet slightly magnetise 
 the soft iron. A screen of cast iron half an inch thick, how- 
 ever, not only neutralised the power to magnetise hard steel, 
 but reduced the deflection of the galvanometer as well. The 
 general explanation of the foregoing facts, as due to Henry, is 
 as follows : The secondary current, as we have seen, is a 
 brief discharge, which rises very suddenly to its maximum 
 value and then fades gradually away. The current curve 
 of the secondary current, due to the rupture of a primary 
 circuit, may be represented by the thick firm line in Fig. 100. 
 If a metal screen is interposed between the primary and the 
 secondary circuit the screen gets a similar secondary current 
 generated in it, and this last again acts by induction to gene- 
 rate a tertiary current in the secondary circuit. This tertiary 
 current consists of two portions first, an inverse part opposite
 
 MUTUAL AND SELF INDUCTION. 269> 
 
 in direction to the secondary current in the screen, and, secondly, 
 a succeeding direct current. Let the current curve of this 
 tortiary current in the secondary circuit be represented by the 
 fine firm line in Fig. 100. The total quantities of electricity 
 flowing in each part of the two portions of the tertiary current 
 are equal. The resultant effect, then, of the action of the 
 primary current when interrupted is to cause in the secondary 
 circuit the true secondary current, which is an unidirectional 
 flux (thick curve), and a superimposed tertiary current, which 
 is a bi-directional flux, its algebraic total of quantity being, 
 zero. 
 
 FIG. 100, 
 
 If we add together at each instant the ordinates of the two- 
 current curves we get a resultant curve (dotted line) which 
 represents the actual current curve in the secondary circuit. 
 The total area (electric quantity) enclosed between the hori- 
 zontal line and the dotted curve must be equal to the total 
 area enclosed between the thick firm line and the horizontal, 
 because we have added and substracted equal areas ; but the 
 maximum ordinate of the dotted curve will be less than that of
 
 270 MUTUAL AND SELF INDUCTION. 
 
 the thick firm line curve, and the form of the curve will be 
 very different also. It is, then, clear that the superposition 
 of a complete tertiary current, which is of itself but very little 
 able to affect a galvanometer on a secondary current which 
 gives a definite galvanometer indication, is not able to alter 
 that galvanometer deflection, depending as it does 011 the total 
 quantity of the discharge. The magnetising power and shock, 
 however, depend upon the maximum value or suddenness 
 with which the induced current rises to its maximum value, 
 and this factor is very much affected by the overlaying of 
 a secondary current by a tertiary. We see, then, that the 
 experiences of Faraday and Henry may be completely recon- 
 ciled, and that the detection of magnetic screening depends 
 upon the nature of the detecting instrument in the secondary 
 circuit. 
 
 The practical outcome of much of the foregoing discussion 
 of magnetic screening is that the use of lead-covered cable for 
 the conveyance of periodic currents of the usual frequency (60 
 or 100 alternations per second) is of HO advantage in respect 
 of prevention of inductive disturbance in neighbouring tele- 
 phone wires. Not only is the lead too poor a conductor, 
 but the frequency of alternation is too small to render the 
 magnetic screening effective. The only effective method of 
 annulling the inductive disturbance is to carry the periodic 
 current along a conductor which lies in the axis of, and is 
 insulated from, a concentric enclosing tube or sheath, which 
 acts as a return. This return must be itself insulated from 
 the earth, and the condition to be fulfilled is that at any 
 instant, and at any section the algebraic sum of the currents 
 in the core and sheath must be zero ; reckoning current in 
 one direction positive, and in the other negative. 
 
 The whole question of magnetic screening has been worked 
 out .mathematically by several mathematicians, and besides 
 the section in Clerk-Maxwell's Treatise (Vol. H. 654, 2nd 
 Ed.), the advanced student may be referred to memoirs by 
 Prof. Charles Niven " On the Induction of Electric Currents 
 in Infinite Plates and Spherical Shells " (Phil. Trans. Roy. 
 Soc., 1881, p. 307), and also to Prof. H. Lamb " On Elec- 
 trical Motions in a Spherical Conductor" (Phil. Trans. Eoy. 
 Soc., 1883, p. 519).
 
 MUTUAL AXL> SELF INDUCTION. 271 
 
 9. Reaction of a Closed Secondary Circuit on the Pri- 
 mary. If a Bell telephone is placed in series with a coil 
 of many turns of fine wire wound on a hollow bobbin, and if 
 both are placed in series with the secondary circuit of a small 
 induction coil, the strength of the secondary current can be so 
 adjusted that the telephone emits a low murmur or rattle. 
 This being the case, let a solid bar of copper be introduced 
 into the bobbin of fine wire, and it will be found that the 
 noise of the telephone is increased. If a bundle of fine iron 
 wires is substituted for the copper rod it will, on the other 
 hand, reduce the noise or stop it altogether. The explanation 
 of this effect is to be found in the reaction which a closed 
 secondary circuit has upon its primary in changing the 
 resultant impedance of the primary. We have shown in 
 Chapter in. (p. 180), that the re-active effect of the 
 secondary is to increase the resistance and reduce the 
 inductance of the primary circuit, and we have deduced 
 two formulae given by Maxwell for the value of the equivalent 
 resistance E' and the equivalent inductance I/ of a primary 
 coil of resistance B and inductance L in the presence of a 
 secondary coil of resistance S and inductance N, the mag- 
 netic circuit having a constant resistance, and the mutual 
 inductance being M. Hence, the equivalent impedance of 
 the primary coil in presence of the secondary is -/E^+p 2 !/ 2 , 
 and that which we may call its isolated or intrinsic impedance 
 is equal to Vw + p* L 2 . For brevity we may write the symbol 
 Im for VK'+^L* and Im' for VB' a +# a L' a , also Itn 2 for 
 V s 2 + j9 2 K 2 . The question then arises, which is the greater 
 Im' or Im ? To discover this, take for E' and L' the values 
 given on page 180, and we have 
 
 T , T 
 and L =L- *L 
 
 s-+ 
 
 Forming from these the function E' 2 + p 2 L' 2 , we have
 
 272 MUTUAL AND SELF INDUCTION. 
 
 If S = x, or the secondary circuit is open, the right-hand; 
 side of the above equation is zero, and we find that the 
 impedance of the primary circuit is not altered by the pre- 
 sence of the open secondary, as of course it should not be. 
 
 If S is not infinite, that is if the secondary circuit is closed, 
 then the above equation shows us that, if the quantity 2 R & 
 is greater than the quantity jo 2 (2 L N - M 2 ), then Im' is greater 
 than Im, or the impedance of the primary circuit is increased by 
 closing the secondary. But if 2 R S is less than p* (2 L N - W) , 
 then Im' is less than Im, or the impedance of the primary is- 
 decreased by closing the secondary circuit. 
 
 If <*! stands for ^-, and o. 2 for J -^, and also if /? stands for 
 
 K JN 
 
 M 
 , , , it is not difficult to show* that to make Im' greater 
 
 than Im we must have 
 
 % oa less than - - . 
 
 When the secondary circuit has a certain critical value it is 
 possible to show experimentally that above this value closing 
 the secondary circuit increases the primary impedance, but 
 below this value closing the secondary circuit decreases the 
 primary impedance. 
 
 The following experiment was made in the laboratory of 
 Prof. Elihu Thomson t : A small induction coil had its 
 primary circuit arranged in series with nine incandescent 
 lamps joined in parallel, thus exciting it with an alternating 
 current of about ten amperes. When the secondary circuit 
 was closed by means of a vacuum tube of high resistance, a 
 marked fall occurred in the candle-power of the lamps used 
 as a resistance in the primary circuit. The impedance of the 
 
 * See Mr. E. C. Rimington, "On the Behaviour of an Air Core Transformer 
 when the Frequency is Below a Certain Critical Value," Proc. Physical 
 Soc., London, October 27, 1893. 
 
 t See The Electrician, Vol. XXXII., p. 225.
 
 MUTUAL AND SELF INDUCTION. 273 
 
 primary was thus increased. When the secondary circuit 
 was closed through a water resistance, the lamps brightened 
 up, thus showing that the primary impedance was decreased. 
 
 Hence the closing of the secondary circuit does not always 
 decrease the primary impedance. Mr. Rimington (loc. cit.) 
 quotes an experiment with an air core transformer or induction 
 coil consisting of two circuits without iron core, in which 
 closing the secondary circuit had the effect of decreasing the 
 primary current by about 3 per cent., thus showing an increased 
 primary impedance. Generally speaking, however, the closing 
 of the secondary circuit so that the total secondary circuit 
 resistance is small has the effect of decreasing the primary 
 circuit impedance. 
 
 Hence also holding a conductor or conducting circuit of low 
 resistance near a primary coil has the effect of decreasing the 
 impedance of that coil and therefore increasing the flow of 
 primary current through it under the influence of a constant 
 impressed primary electromotive force. 
 
 The explanation of our experiment with the induction coil 
 and the copper rod is now simple. The introduction of the 
 copper rod into the fine wire helix is equivalent to approxi- 
 mating to a primary coil a closed secondary circuit. The 
 impedance of the fine wire circuit to the alternating current 
 from the secondary circuit of the induction coil is hence 
 reduced ; it gets more current, and the telephone is made to 
 emit a louder sound. If, however, a core of divided fine iron 
 wire is introduced into the fine wire helix, the result is simply 
 to increase the impedance of that circuit, and therefore to 
 reduce the current actuating the telephone. When considering 
 in particular the theory of the induction transformer as applied 
 to electric distribution we shall see the above principles have 
 important practical bearings. 
 
 In a Paper recording some experimental results on the 
 self-induction and resistance of compound conductors* Lord 
 Eayleigh has given some comparisons of the results of theory 
 and experiment on Maxwell's formulas above alluded to. By 
 the use of a resistance and inductance bridge, very similar to 
 one designed by Prof. Hughes, the measurements of the 
 inductance and resistance of a circuit can be made separately 
 * See Phil. Mag., December, 1886, p. 469.
 
 274 MUTUAL AND SELF INDUCTION. 
 
 with ease. A pair of wires was wound on one bobbin ; 
 each wire had a resistance of nearly ! ohm, and a diameter 
 of -037in. Each coil consisted of nine double convolutions. 
 In certain arbitrary units the resistance of one of these copper 
 wires to steady currents was 1-75, and its inductance 11-2. 
 These values were obtained when the other coil was on open 
 circuit. On closing the unused coil, the resistance of the first 
 rose to 2-67 and its inductance fell to 4-7. 
 To compare this with the theory. 
 
 The formulae are E' = E + 
 
 B*- 
 
 M 2 N 
 
 L'-L- 
 
 Now E = S = 1-75 x -0492 xlO 9 absolute C.-G.-S. units of 
 resistance, 
 
 and L = N = ll-2 x 1553 centimetres, 
 
 M = 11 X 1553 centimetres, 
 
 and p = 27r n = 2 x 3-1415 x 1050. 
 
 The periodic current used had a frequency of 1050 per second; 
 hence 
 
 Therefore E fc =E (1 + -6) = 1-6R, 
 
 and L'- = L (l-'6)= -4L; 
 
 but 1-6 x 1-75 = 2-8 = E', 
 
 and 4xll-2 = 4-5=L'. 
 
 These calculated values compare very favourably with the 
 observed values, viz. : 
 
 E' = 2-67, L'= 4-7, 
 
 and experimentally confirm the truth of Maxwell's formulae 
 for the increased resistance and diminished inductance of a 
 circuit when placed near a closed secondary circuit. 
 
 10. Hughes's Induction Balance and Sonometer. In 1879 
 Prof. Hughes constructed and described a very perfect induc- 
 tion balance, with which he was able to conduct researches of 
 an exceedingly interesting character. In order to have a 
 perfect induction balance he found it necessary to make all the
 
 MUTUAL AND SELF INDUCTION. 
 
 275 
 
 four coils exactly similar.* Four boxwood bobbins (see Fig. 101) 
 are each wound over with 100 metres of No. 32 copper wire. 
 These coils are arranged in pairs at a considerable distance 
 apart, so that the coefficient of mutual induction between the 
 separated pairs is negligible. Two of the coils, A andB, are joined 
 in series with each other and with a battery and interrupter I, 
 and the other two coils, C and D, are employed respectively as 
 secondary coils to these two. These secondary coils are in series 
 with each other and with a telephone receiver T, and are so 
 joined up that the direction of the induction of A on C is oppo- 
 site to that of B on D. One pair of coils is placed hi a fixed 
 position, and the other pair can be slightly moved to or from 
 
 FIG. 101. 
 
 each other by means of a micrometer screw. The coils are first 
 adjusted so that the inductions are equal and opposite, and on 
 listening at the telephone the opposing secondary currents 
 produce at best but a very slight sound, which can be perfectly 
 abolished by adjusting the distance of one pair of coils. When 
 this is the case, if we insert in the opening of the bobbin of one 
 of the primary coils a disc or piece of metal d, the balance is 
 destroyed, and we hear sounds more or less intense. In order 
 to get some comparative measurements, Prof. Hughes designed 
 
 * " On an Induction Current Balance." By Prof. D. E. Hughes. Proc. 
 Koy. Soc., No. 196, May 5, 1879. 
 
 T 2
 
 276 
 
 MUTUAL AND SELF INDUCTION. 
 
 a companion instrument, called a sonometer. In this instrument 
 a pair of primary coils are, as before (see Fig.. 102), joined in 
 series with each other and with a battery. The coils are fixed 
 at the extremities of a bar. Between these primary coils slides 
 a single secondary coil, and the primary coils are so wound that 
 their inductions on this secondary coil are equal and opposite. 
 When this secondary coil is exactly between the two primary 
 coils, a telephone placed in series with the secondary coil gives 
 out no sound when the primary current is rapidly interrupted. 
 If, however, the secondary coil is slid from one primary and 
 towards the other, the differential action creates an induced 
 current detected by the telephone. By reading off on the bar 
 the extent of displacement necessary to create in the telephone 
 a sound of a certain magnitude an arbitrary reading can. 
 
 FIG. 102. 
 
 be obtained corresponding to every different value of the 
 secondary current. A switch is provided, by means of which 
 the same telephone can be shifted rapidly from the induction 
 balance secondary circuit to the sonometer secondary circuit. 
 The experiments first performed consisted in placing within 
 one primary coil of the induction balance certain equal-sized 
 discs of different metals, and then so arranging the sono- 
 meter secondary coil that the noise in the telephone produced 
 by the current in the secondary of the sonometer was judged 
 by the ear to be equal to the sound produced in the telephone 
 when it was shifted to the secondary circuit of the induction 
 balance, and in which the inductive balance had been broken
 
 MUTUAL AND SELF INDUCTION. 277 
 
 down by the insertion of the disc of metal. Discs of various 
 metals the size and shape of an English shilling were made, 
 and, when inserted in the induction coil, the sonometer bar 
 readings, reckoned from the centre or absolute zero of sound 
 given in certain arbitrary degrees, were as follows : 
 
 Silver (chemically pure) 125 ! German Sitver 50 
 
 Gold 117 ! Iron (pure) 40 
 
 Silver coin 115 | Copper (alluy) 40 
 
 Aluminium 112 j Lead 38 
 
 Copper 100 Antimony """ 35 
 
 Zinc 80 Mercury 30 
 
 Bronze 76 i Bismuth 10 
 
 Tin 74 !Zinc(alloy) 6 
 
 Iron (ordinary) 52 | Carbon 2 
 
 This list does not agree in order entirely with that of any 
 of the lists of electrical conductivity. In some degree it 
 evidently has reference to conductivity, because, roughly 
 speaking, the best conductors come at the top and the worst 
 at the bottom ; but whilst it is headed by silver, which has the 
 highest conductivity per unit of volume, we find aluminium, 
 which has the highest conductivity per unit of mass, occupying 
 a position above that of copper. The disturbing effect of 
 the metal on the inductive balance is not, however, simply 
 proportional either to the conductivity per unit of mass or per 
 unit of volume. In more recent experiments a graduated zinc 
 wedge pushed in more or less between one pair of coils of the 
 induction balance was employed to obtain comparative numbers 
 representing the disturbance produced when discs of various 
 metals are inserted in the other coil. The elementary theory 
 of the induction balance is of course contained in all that has 
 gone before in this and the last chapter. It is, generally 
 speaking, dependent for its action on effects similar to those 
 producing magnetic shielding. If the discs are slit so as to 
 prevent circumferential electric currents in their mass, their 
 action in disturbing the inductive balance is mitigated or 
 annulled. If the metal disc is replaced by a copper coil with 
 open extremities no effect is observed on the inductive balance. 
 If the ends of the coil are joined, the coil behaves as if it were 
 a metallic disc and causes loud sounds in the telephone. The 
 effect due to the iron disc is a mixed one. It in part acts like 
 any other metal disc, but it differs from them in one respect. 
 If any non-magnetic disc is placed edgeways in the centre of
 
 278 MUTUAL AND SELF INDUCTION. 
 
 the primary bobbin it has a diminished effect in disturbing the 
 balance ; in the case of iron the disturbance is increased by 
 turning the disc edgeways. In order to have before us a 
 typically simple case, imagine an induction balance made of 
 two very long primary helices and each embraced near the 
 centre by a small secondary coil. Let the primary coils be 
 traversed by a simple periodic current. We have then in the 
 interior of the primary coil a uniform magnetic field varying 
 synchronously with the primary current in a simple periodic 
 manner, and the rate of change of the magnetic field at any 
 instant will be a measure of the electromotive force acting in 
 the secondary circuit. Suppose into one primary helix is 
 inserted a thin copper tube ; this will form a closed secondary 
 circuit, and secondary periodic currents will be induced in it, 
 flowing round the cylinder in directions parallel to the turns 
 of the primary helix. As this copper cylinder possesses a very 
 sensible time constant, the phase of these secondary currents 
 in the copper cylinder will be nearly opposite to that of the 
 primary current. The resultant magnetic field in the interior 
 of the cylinder is therefore that due to the resultant of these 
 two simple periodic currents which are nearly opposed in 
 phase. Hence the absolute magnitude of the interior field 
 and its rate of variation will be less than if the copper 
 cylinder was removed. It results, therefore, that the induction 
 through the secondary helix and the electromotive force 
 impressed on it will be diminished by the presence in the 
 primary coil of this copper cylinder. The diagram given on 
 page 177, showing a geometrical construction for the 
 magnitudes of the primary and secondary currents in an 
 induction coil without iron, shows us why the primary and 
 secondary currents are thus more or less opposite in phase. 
 Since, in a general way, the higher the conductivity of the 
 tube or disc introduced into the primary the greater the time 
 constant, and the greater the lag in phase of the currents 
 induced in this metallic circuit behind the phase of the 
 inducing primary, it follows that the resultant interior field 
 acting to produce inductive electromotive force in the 
 secondary helix will be diminished by the introduction of 
 discs of very high conductivity more than by discs of very 
 low conductivity.
 
 MUTUAL AND SELF INDUCTION. 279 
 
 From the principles discussed under the head of magnetic 
 shielding it would appear that the differences between various 
 metals inserted as discs in the induction balance would be less 
 marked at very high speeds of interruption than at very low 
 ones. With respect to the action of iron, two effects have to be 
 considered which are the results of very different actions. The 
 introduction of the iron into the primary coil reduces the 
 magnetic resistance of the circuit of induction of that coil, and 
 this cause, if it operated alone, would destroy the inductive 
 balance by raising the inductive electromotive force in that 
 secondary circuit corresponding to the primary into which the 
 iron is introduced; but the iron disc, like every other disc, 
 gets circumferential induced currents created in it, and these, 
 if they acted alone, would destroy the inductive balance by 
 lowering the inductive electromotive force in that secondary 
 coil. 
 
 These two effects conflict, and it is an interesting confirmation 
 of theory to find that Prof. Hughes says it is possible to intro- 
 duce into one primary coil of the induction balance a disc of 
 iron and some soft iron wires in such positions that these 
 opposite actions nullify each other, and, though each mass of 
 iron separately would destroy the induction balance, yet the 
 two together being introduced complete silence in the tele- 
 phone is the result. The sensibility of the induction balance 
 to minute differences of electric conductivity and magnetic 
 permeability is very remarkable. If into one coil of a carefully- 
 adjusted balance we place a good sovereign, or shilling, and 
 into the other a bad one, the telephone detects the base coin 
 with unerring certainty by the loud noise given out. In the 
 same way, if two pieces of soft iron are introduced into the two 
 primary coils, and a balance is obtained, the mere magnetisa- 
 tion of one of them will be at once detected, because that 
 magnetised piece becomes thereby less permeable, and destroys 
 the balance. We may present the rough general theory of the 
 induction balance in another way. Let the " coin " be simply 
 regarded as a closed circuit, between which and the primary 
 circuit surrounding it there is a certain coefficient of mutual 
 induction. The two primary coils forming one primary 
 circuit have, on the whole, no action on the two secondary 
 coils forming one secondary circuit, and we may therefore
 
 280 MUTUAL AND SELF INDUCTION. 
 
 consider the primary circuit as if it were in a position 
 conjugate to the secondary. The coin, however, is acted upon 
 inductively by the primary circuit, and the eddy currents or 
 secondary currents generated in it react on the secondary 
 circuit, causing in it tertiary currents, which affect the 
 telephone. Looking at it from this point of view, we might 
 construct an induction balance thus. Let A (see Fig. 103) 
 be a single primary coil, and B a secondary coil, having a 
 telephone in series with it. Place the coil B in a position 
 conjugate to A that is, with its axis at right angles to that 
 of A. Then let variation of current in A produce no current 
 in B. Now hold a sheet of copper anywhere, say at C, and 
 the telephone will be caused to sound. For A, though it 
 
 FIG. 103. 
 
 cannot affect B inductively directly, yet it can produce a 
 secondary current in C held at a non-conjugate position, and 
 these secondary currents in C will create other tertiary 
 currants in B. The experiment thus appears to indicate a 
 sort of reflection of inductive power.* 
 
 This was experimentally shown by Mr. Willoughby Smith 
 in his Paper on " Volta-Electric Induction " (see Journal of 
 Society of Telegraph-Engineers, Vol. XII., page 465). 
 
 * The full theory of the induction balance has been given by Prof. Oliver 
 Lodge. See Proc. Phys. Soc. London, Vol. III., p. 187. Also in the same 
 volume is a Note by Prof. J. H. Poynting " On the Graduation of the 
 Sonometer."
 
 MUTUAL AND SELF-INDUCTION. 281 
 
 An interesting experiment due to Mr. Willoughby Smith is 
 to employ a simple Bell telephone receiver, unconnected with 
 any circuit, as an induction finder. If a coil of wire is traversed 
 by an electric current, either rapidly intermittent or alternating, 
 then a Bell telephone held anywhere in the magnetic field 
 emits a sound. The pulsating field disturbs the magnetism 
 of the telephone magnet, and enables us, therefore, to detect 
 rapid electromagnetic disturbances at the place where it is held. 
 It is obvious, then, that the induction balance, combined with 
 a telephone, is an apparatus of extreme sensitiveness. It 
 can render evident the smallest differences of weight, nature, 
 degree of purity or temperature of two conductors of identical 
 dimensions, such as two coins placed in identical conditions 
 in respect of the two systems of coils. 
 
 It enables us to detect very small masses of metal in a badly 
 conducting body, and may be employed with much advantage in 
 verifying the insulation of the different windings of a coil, the 
 ends of which are open. At the same time, however, it lends 
 itself better to qualitative than to quantitative work, as it is 
 difficult to interpret rigorously the results obtained.* 
 
 11. The Transmission of Kapidly Intermittent or Alter- 
 nating Currents through Conductors. Some experiments by 
 Prof. Hughes in 1886 on the self-induction of metallic wires 
 were the means of directing the general attention more closely 
 than before to the nature of the propagation of electric 
 currents of high frequency through metallic conductors, and 
 although mathematical writers, particularly Maxwell and 
 Oliver Heaviside, had previously considered the problem 
 theoretically, the experimental results drew the attention 
 of many to this question to whom the more recondite 
 mathematical investigations were unknown. Prof. Hughes's 
 
 * For further information on the use and theory of the induction balance 
 the student may consult, with advantage, Mascart and Joubert's " Elec- 
 tricity," Vol. II., 986 ; also Hughes, Phil. Mag. [5], Vol. II., p. 50, 1879. 
 On the differential telephone, see Chrystal, Phil. Trans. Roy. Soc. Edin., 
 Vol. XXIX., p. 609, 1880. 0. Lodge, Proc. Physical Soc. London, 
 Vol. III., p. 187, on intermittent currents and the theory of the induction 
 balance.
 
 282 MUTUAL AND SELF INDUCTION. 
 
 experiments* on the self-induction of metallic wires were 
 made with a combined resistance and induction bridge of 
 somewhat novel form. Suppose that a quadrilateral arrange- 
 ment be formed of four conductors P, Q, E, S, only one 
 of which, P, has any sensible self-induction, and let the 
 diagonals be completed by a telephone T, and battery B, 
 with interrupter I. In the first place, let the resistance- 
 balance be obtained for steady currents. This can be 
 achieved by placing the telephone with the interrupter as a 
 conjugate circuit to the battery (see Fig. 104), and altering one 
 resistance, say, 11, until a balance is obtained. By a suitable 
 
 FIG. 104. 
 
 adjustment of the four resistances complete silence can be 
 obtained in the telephone. 
 
 Next, let the interrupter be removed to the battery circuit,. 
 all the other arrangements remaining the same (see Fig. 105). 
 It will be found that the balance is destroyed, and that no 
 mere change in the value of the resistance E will enable a per- 
 fect balance to be obtained. The reason for this is that, on 
 
 * These experiments formed the subject of Prof. Hughes's Inaugural 
 Discourse to the Society of Telegraph-Engineers on the occasion of his 
 election to the office of President. See Journal of the Society of Telegraph 
 Engineers, January 28, 1886, " The Self-induction of an Electric Current 
 in Relation to the Nature and Form of its Conductor."
 
 MUTUAL AND SELF INDUCTION. 283 
 
 closing the battery circuit, the inductance of P introduces a 
 counter electromotive force into P and the potential rises at 
 c faster than at d, and on breaking the circuit the potential at 
 c dies down faster than at d ; and hence at each make and 
 break the telephone is subjected to an alternate flux of current 
 which causes it to emit a sound. Supposing that an attempt 
 is made to get rid of this sound by shifting the point c so as to 
 alter E, the steady balance will be destroyed, and the telephone 
 will be traversed by a current during the time when all the 
 currents have become steady ; but no such change in the value 
 of B will prevent a variation of current taking place through 
 
 the telephone during the complete period from the first instant 
 when the battery circuit is closed to the instant when it is 
 opened again. 
 
 The only way in which a balance can be obtained in this last 
 arrangement is by introducing into the telephone circuit an elec- 
 tromotive force which shall be capable of being made to balance 
 at every instant the inductive electromotive force due to the 
 inductance of P. Prof. Hughes does this very ingeniously by 
 introducing a pair of mutually inductive coils into the battery
 
 284 
 
 MUTUAL AND SELF INDUCTION. 
 
 and telephone circuits, and the final arrangement is as shown 
 in Fig. 106. M! and M 2 are a pair of coils, one of which, M 2 , 
 is in the battery circuit and is fixed, and the other, M lf is 
 in the telephone circuit, and can be placed so that, whilst its 
 centre coincides with that of M 2 , its axis makes any required 
 angle with that of M lt In this way the mutual inductance 
 between M a and M 2 can be varied from zero when the coil axes 
 are at right angles to a definite maximum value when they are 
 co-linear. 
 
 It is found that, when the coils M : M 2 are in certain 
 positions, the inductive electromotive force set up in the 
 
 a 
 
 -- \ 
 
 FIG. 106. 
 
 telephone circuit by the induction of M x on M 2 can be made 
 to neutralise the electromotive force of self-induction due to the 
 inductance of P, when, in addition, a certain value is given to 
 the resistance E. Under these circumstances the bridge can 
 be balanced and the telephone completely silenced, both when 
 the interrupter is in the battery circuit and also in the tele- 
 phone circuit ; in other words, the bridge can be balanced both 
 for steady and for variable currents. 
 
 In the arrangement adopted by Prof. Hughes the resist- 
 ances Q, E, and S, were sections of one and the same fine
 
 MUTUAL AND SELF INDUCTION. 
 
 285 
 
 German silver, 1 metre long, and having a total resistance of 
 4 ohms (see Fig. 107). The ends of this wire were joined ta 
 
 61 
 
 FIG. 107. 
 
 the conductor P under investigation, and the rest of the 
 apparatus was arranged as described. 
 
 In order to investigate the relation between the resistances 
 and inductances which holds good when the bridge is balanced for 
 
 FIG. 108. 
 
 steady and also for variable currents, a diagram must be drawn 
 (Fig. 108) representing the network of conductors. Then call
 
 286 MUTUAL AND SELF INDUCTION. 
 
 the current at any instant in the inductive branch P, x, 
 that in the branch Q, y, and that in the telephone circuit z. 
 The current in the branch battery is then x + y. Let L be the 
 inductance of P, and M the mutual inductance of the coils placed 
 in the circuits B and T, and let all the other circuits, Q, E, 
 and S, have no sensible inductance. Let e be the electromotive 
 force of the battery at any instant t. Then the currents in the 
 various branches at that instant are as follows : 
 In the branch P the current is x 
 R ,, x-vz 
 S y-z 
 
 Q y 
 
 B x+y 
 T z 
 
 Applying Kirchhoff's corollaries to each of the three meshes 
 of the network, we have three equations, viz., 
 
 -L-jf---!*^ (94) 
 s-U-- - (95) 
 
 T*-SF^=-My^ (96) 
 
 and these three equations enable us to find at any time 
 t the current in any branch.* If we suppose the bridge 
 to be balanced for variable currents, then z is zero, and on 
 making this limitation we find the above equations reduce to 
 the two, 
 
 Qy-Ps-L^f-Ba-Sy, . - (97) 
 
 and -M^I-M^RZ-ST/. . . . (98) 
 
 Furthermore, let us assume that the currents vary according 
 to a simple periodic law. In this case, if X is the maximum 
 value of x, then we can write 
 
 * The general method of finding the current equations for any network 
 is given in Maxwell's "Treatise on Electricity," 2nd Edition, Vol. II., 
 755. Also see " Problems on Networks of Conductors," by J. A. Fleming, 
 Phil. Mag., September, 1E85, Vol. XX., p. 221 ; or Proceedings Phys. Soc., 
 Lond., 1885.
 
 MUTUAL AND SELF INDUCTION. 287 
 
 where p as usual = 2- n, n being the frequency of the alterna. 
 
 dx 
 tion. Hence ^ =p!cospt, 
 
 cPx 
 and -75 = 
 
 Adopting the fluxional notation, it is convenient to write x for 
 7-7 and x for -r^- Hence, for simple periodic variation of a 
 current a;, we always have the condition 
 x = p* x. 
 
 If we differentiate with respect to t the two equations (97) 
 and (98), and eliminate x by the help of the equation x = -p z x, 
 we obtain two other equations, which, together with the original 
 two (97) and (98), give us the necessary four equations for elimi- 
 nating the four variables x, y, x, y. We have thus, 
 
 Qy-Px-~Lx = Ea;-Sy. . . . (99) 
 
 -Mac -My = Ra?-Sy. . . . (100) 
 
 = Ex-Sy. . . . (101) 
 
 =Rx-Sy. . . - (102) 
 
 The student who has mastered the elements of determinant 
 analysis will recognise that the variables x, y, x, y can be 
 eliminated from these equations, and the relation which must 
 always hold good between the constants can be found by 
 equating to zero the determinant of these four equations. We 
 have then 
 
 -L, 0, -(P + B), (Q + S) 
 
 -M, -M, B, S 
 
 -(P + R), (Q + S), Lp 2 , 
 
 -E, S, M/> 2 , Mp 2 =0. 
 
 This determinant writes oat into the sum of three terms, 
 viz. :
 
 288 MUTUAL AND SELF INDUCTION. 
 
 This long equation reduces to the simpler form 
 
 -(QB-SP)2] = 0. 
 
 In order that the sum of the two left hand terms in the 
 above equation may always he zero, each factor in the square 
 brackets must be separately zero, and it will be seen that each 
 of these factors equated to zero are equivalent to the two- 
 equations : 
 
 QB-SP = ML/, ..... (103) 
 and M(P + Q-f-B + S) = SL ..... (104)* 
 
 These equations express the relation which holds good 
 between the resistances of the branches and the self and mutual 
 induction coefficients of a Hughes bridge when the bridge is 
 balanced for variable currents. 
 
 It will be seen that the ordinary relation of the resistances 
 for steady balance, viz., P;Q = B;S is departed from, and that 
 we have for the resistance of branch P, when traversed by 
 variable currents, the value 
 
 p _QB-MLp_QB 
 
 . S iS S 
 
 and for the inductance of branch P under these circumstances, 
 the value 
 
 L _M(P + Q + B + S) < m t (10G) 
 
 S 
 In some of his experiments Prof. Hughes interpreted his 
 
 O T? 
 
 results on the assumption that P was always equal to 5L- , and 
 
 b 
 
 L was equal to M ; but the complete investigation shows that 
 this is not the case. A very full theoretical and practical 
 examination of the induction bridge has been given by Prof. 
 H. F. Weber, for which the student is referred to the pages of 
 the Electrical Review, Vol. XVIII., p. 321, 1886, and VoL XIX., 
 p. 30, 1886.1 
 
 * These equations were given by Lord Rayleigh in the discussion on 
 Prof. Hughes's Paper. See also Lord Rayleigh " On the Self-induction 
 and Resistance of Compound Conductors," Phil. Mag., Dec. 1886, p. 471. 
 Equivalent equations have been also arrived at by Prof. H. F. Weber and 
 Mi-. Oliver Heaviside. 
 
 t See also Mr. Oliver Heaviside in the Phil. Mag., August, 18B6.
 
 MUTUAL AND SELF INDUCTION. 239 
 
 The whole method of the construction and use of the induc- 
 tion bridge has been the subject of elaborate examination by 
 Lord Kayleigh in a Paper on the self-induction and resistance 
 of compound conductors (Phil. May., December, 1886), from 
 which we shall quote freely in what follows. Lord Bayleigh 
 discarded the tooth-wheel interrupter, as it does not give a 
 regular variation of current corresponding in period to the 
 passage of a tooth ; and he substituted a harmonium reed, 
 the vibrating tongue of which made contact once during each 
 period with the slightly-rounded end of a brass or iron 
 wire advanced exactly to the required position by means 
 of a screw cut upon it. Blown with a regulated wind, such 
 reeds are capable of giving interruptions of current up to 
 about 2,000 per second. The one usually employed had a 
 frequency of 1,050 vibrations per second. The induction 
 compensator consisted of two circular coils, one of which 
 was fixed and the other movable round an axis, so placed 
 that the flat circular coils could be placed either with their 
 planes coincident or at right angles. If the inner coil is very 
 small compared with the other, and the coils are placed with 
 centres coincident and axes inclined at any angle, 6, and if M 
 be the maximum mutual inductance and M the inductance in 
 any position, 6, then 
 
 M = M cos 0. 
 
 This law is, however, not followed when the coils are sensibly 
 of the same size. In this case Lord Eayleigh has shown that 
 the mutual induction is very approximately proportional to the 
 angle between the axes of the coils for a range between 40 and 
 140. In the actual experiments the mutual inductance of the 
 coils was determined for each degree of angular displacement 
 of the axes by comparing it with the calculated coefficient 
 between two wires, Avound in measured grooves, cut in a cylinder, 
 and it was found that every degree of movement of the movable 
 coil, when the axes were not far removed from perpendicu- 
 larity, was equal to 776*3 centimetres of mutual induction, the 
 maximum when = being 56,100 centimetres. The first 
 experiment described in the Paper referred to is one on the 
 self-induction and resistance of a coil of copper wire. In the 
 bridge used the resistances Q + R + S were together equal to 
 4'00 ohms. Resistances were, however, measured in scale 
 
 u
 
 290 MUTUAL AND SELF INDUCTION. 
 
 divisions of the bridge wire, each one equal to 2-04 x 10 s 
 centimetres per second. The copper coil being balanced on 
 the bridge, it was found that the readings of the three 
 resistances and of M were as follows : 
 
 Q = 610, R=190, 8 = 1,160, 
 
 M = 36 = 36 x 776 centimetres, 
 
 and the frequency n of the vibrations = 1,050. Hence p = 2-7r 
 x 1,050. Taking the equations (103) and (104) on page 288, 
 and eliminating L, we have for the value of P, the equation 
 
 p /M 2 (Q + B + S)-) 
 Q.E *- S.Q.B 
 
 t| 13? 
 
 Substituting the values above, we find 
 
 O T? 
 P = -876 -FT- = 87 -5 scale divisions. 
 
 b 
 
 This gives the value of the real resistance of P for the periodic 
 currents used ; and we see that if we neglected the peculiarity of 
 the bridge, and simply assumed the ordinary law, that the resis- 
 tance of P was equal to Q R-i-S, we should make an error 
 of some 12 per cent. On actually balancing the bridge for 
 steady currents the resistance of P was found to be 87'3 scale 
 divisions, thus indicating that for this copper coil at the 
 frequency employed the resistance to variable currents was 
 the same as to steady ones. 
 
 On inserting a solid copper rod into the aperture of the coil 
 and measuring again the resistance and self-induction, it was 
 found that the values of the reading were Q = 660, R = 190, 
 M = 295, instead of as before, Q = 610, M = 36. Hence the 
 introduction of another closed secondary circuit (viz., the 
 copper rod) increased the real resistance and diminished the 
 real self-induction in accordance with the principles explained 
 on page 180, at which place we demonstrated Maxwell's 
 equations for the increased resistance and diminished self- 
 induction of a primary circuit when in contiguity to a closed 
 secondary circuit. 
 
 The next example selected was that of a soft iron wire, 160 
 centimetres long and 3-3mm. dia. Here, with the variable
 
 MUTUAL AND SELF INDUCTION. 291 
 
 currents from the reed interrupter of the same period as 
 before, a balance was obtained for 
 
 Q = 178, R = 190, 8 = 1,592, M = 8 x 776 centimetres, 
 from which we find 
 
 P = -985 %? = 20-93 scale divisions. 
 
 D 
 
 The resistance of the same wire to steady currents was 
 
 P 0= 100 * 190 = 11-88 scale divisions. 
 1,6/0 
 
 Hence the effective resistance to variable currents having a 
 frequency of 1,050 was 1'84 times the resistance to steady 
 currents. We have presented to us here the phenomena 
 characteristic of the behaviour of conductors to electric 
 currents rapidly intermittent or reversed. The real resistance 
 of the conductor is increased. This is not to be confused 
 with the fact that for intermittent currents the impedance 
 (\/R--f2>2L2) measured in ohms is greater than the ohmic 
 resistance (E) ; but it is to be understood as a real increase 
 in the rate at which energy is dissipated per unit of current. 
 It is now well understood that such increase of resistance 
 is due to the fact that the current density for rapidly 
 periodic currents is not uniform over the cross-section of 
 the wire, but is greatest along the outer layers of the wire. 
 Hence, under rapidly periodic currents the inner portions 
 of a conducting wire are never reached by the current, and, 
 as far as current carrying duty is concerned, might as well 
 he away. This difference may be graphically represented 
 thus : Let relative density of current or quantity passing per 
 second through unit of cross-section of a conductor per unit 
 of time be represented, like relative density of population, by 
 degree of density of shading. Then the flow of a steady 
 current through the section of a wire might be represented 
 as in Fig. 109; and the flow of current over the cross- 
 section when the current is rapidly periodic might be repre- 
 sented as in Fig. 110. 
 
 We must consider that the current in beginning in a 
 conductor starts its flow first on the outside, and soaks or 
 penetrates inwards into the deeper layers by degrees. We see 
 that, in consequence of this, if the current is reversed in sign, 
 
 u 2
 
 292 MUTUAL AND SELF INDUCTION. 
 
 or rapidly intermitted, it will not have time to soak or diffuse 
 very far into the mass of the conductor before it is, so to- 
 speak, re-called, and its operations will be confined to- 
 the outer layers. This is a rather broad way of stating 
 modern views on the modus operandi of current flow- 
 According to these views the current in a wire is not 
 established by a process analogous to starting a flow of 
 water in a pipe by a push applied one end, but it is put 
 into the wire at all points of its surface by energy absorbed 
 from the surrounding dielectric. Other things being equal,, 
 the rate at which this equalisation of current across the cross- 
 section of the conductor goes on will be a function of the 
 magnetic permeability of the material. The current in, 
 flowing along a magnetisable circuit magnetises it circularly. 
 This magnetisation involves work, and the impressed electro- 
 
 FIG. 109. FIG. 110. 
 
 motive force which is increasing the current has to do work r 
 not only against that which may be called the formal 
 inductance of the circuit, or against that part of the counter 
 electromotive force of induction which depends on the form of 
 the circuit, but has to create this circular magnetisation. 
 
 By keeping to the outer layers of the conductor the periodic 
 current avoids magnetising the deeper layers of the material. 
 Proof will be given later in describing the remarkable investi- 
 gations of Hertz that this description of the mode of establish- 
 ment of a current is one supported by experimental facts. 
 We are thus able to offer a consistent theory of the real 
 increase of resistance which we find for rapidly periodic 
 currents. The inner core or central portion of the conductor 
 is not used by the current, and, so far as conducting it goes,.
 
 MUTUAL AND SELF INDUCTION. *^93 
 
 might as well be absent ; hence the solid conductor does no 
 more, or not much more, in the way of carrying the current 
 than a hollow or tubular conductor would do : and, accord- 
 ingly, the real or ohmic resistance of the conductor for such 
 variable currents is greater than it is for steady currents. 
 
 Another way of regarding this inequality of current distribu- 
 tion over the cross-section of a wire is as follows : The counter 
 electromotive force arising from self-induction is greater at 
 the axis or central portion of the wire than it is near the 
 surface. If we consider the whole current flowing across any 
 section of the conductor as made up of little streamlets of 
 currents flowing parallel to each other, the central streamlets 
 or filaments of current experience more opposition in reaching 
 full magnitude than do the outer ones, because of the mutual 
 induction with those surrounding them. The current there- 
 fore arrives at its maximum value at the surface of the con- 
 ductor before it does at the deeper or central portions. If 
 the current is periodic or transitory the central streamlets or 
 current filaments are always greatly inferior in strength to 
 those at the surface. There is reason, then, to believe that a 
 sudden rush of current, very brief in duration, such as the dis- 
 charge from a Leyden jar or condenser, moves chiefly along 
 the surface of a discharging wire, and the same statement 
 holds good for very rapid pulsatory or alternate currents. 
 Although it may be said that the general principles governing 
 the behaviour of alternating current flow as conductors were 
 virtually given by Maxwell,* they have been subsequently 
 chiefly developed mathematically by Mr. Oliver Heaviside and 
 Lord Rayleigh, and were brought to the notice of practical 
 electricians principally by the experiments of Prof. Hughes 
 previously mentioned. 
 
 This increase of the resistance proper of a wire for rapidly 
 periodic currents is one of the most striking of the results of 
 Prof. Hughes's researches. The full mathematical develop- 
 ment of the problem, even for comparatively simple cases, 
 leads to some very complex mathematical expressions. Lord 
 
 * Maxwell's " Electricity," Vol. II., 689-690. In this paragraph it is 
 shown that the counter electromotive force of self-induction at any point 
 in a conductor is a function not only of the time but of the position of the 
 point considered, and varies over the cross-section of the conductor.
 
 294 MUTUAL AND SELF INDUCTION. 
 
 Eayleigh has, however, treated with great fulness* one or 
 two cases of practical importance. If E and L are the true 
 ohmic resistance and inductance of a cylindrical straight wire 
 of length Z and magnetic permeability p to steady currents or 
 currents of very slow alternations, and if an alternating 
 current of simple periodic form and frequency n is sent 
 through it, then the resistance is increased to E 1 and the 
 inductance diminished to L 1 in such wise that if p = 2?r n, as 
 usual, we have 
 
 A being some constant depending on the position of the return 
 wire. 
 
 These formulas express the fact that the resistance is in- 
 creased and the inductance diminished in proportion as the 
 frequency of alternation gradually increases from zero to 
 infinity. 
 
 At slow rates of alternation the chief opponent with which 
 the impressed electromotive force has, so to speak, to contend 
 is the ohmic resistance ; and the distribution of current across 
 the cross-section of the conductor under these conditions is 
 such as to make that resistance a minimum, and this is known 
 to be so when the distribution is a uniform distribution. The 
 current is then taking the greatest advantage of the conductor, 
 and the heat generated and dissipated per unit of time is less 
 under these conditions than if the same total current were 
 distributed in any other way over the cross-section of the 
 conductor. This last statement can be easily proved. Let the 
 cross-section of the conductor, supposed to be a cylindrical wire, 
 be divided into two equal zones by a circular line. Let the 
 resistance per unit of length of the conductor be r for each 
 portion corresponding to the outer and inner zone. Call the 
 outer portion the sheath and the inner the core of the con- 
 ductor for brevity. If a total quantity of current, x, flows 
 through the conductor, then the rate of dissipation of energy 
 
 * "On the Self-induction and Eesistance of Straight Conductors,' 
 Phil. May., May, 1886, p. 382.
 
 MUTUAL AND SELF INDUCTION. 
 
 as heat is^- for each portion per unit of length, or If 2 for the 
 
 whole conductor, on the assumption that the current is equally 
 divided between the sheath and the core. 
 
 If, however, we suppose the total current, x, to be distributed 
 so that a portion, y, travels by the sheath, and the remainder, 
 z, travels by the core, then the heat generated per unit of 
 length per unit of time is rf for the sheath and rz 2 for the 
 core. Hence, for the equi-distribution of current, the energy 
 dissipation is 1 ^_ = r (y+ z ) t anc 1 f or the unequi-distribution it 
 
 is r^+z 2 ). Which, then, is greater, r ^ /+8 ) 8 or r(i/ a +z 8 )? 
 Consider the following inequalities : 
 
 (y - z) 2 is greater than 1 (y - z) 2 , 
 or ?/ 2 + z 2 - 2 t/z is greater than- (y-zf\ 
 
 Hence, y* + z 2 - 2 y z is greater than - (y + zf - 2 // z. 
 Adding 2 >j z to both sides, we have 
 
 y z + z 2 is greater than ~(y + zf. 
 2 
 
 Accordingly it follows that 
 
 r y* + r z 2 is greater than - fa + zf, 
 
 or r y 2 + r z 2 is greater than - a; 2 ; 
 
 2 
 
 that is to say, the rate of energy dissipation is greater for the 
 assumed unequal distribution than for the distribution in 
 which the current is equal in density over the cross-section of 
 the conductor. The same kind of proof may be extended 
 to any other arbitrary distribution of current over the cross- 
 section, and the reasoning will lead to the conclusion that 
 the equi- dense distribution is that which causes the least rate 
 of dissipation of energy per unit of current.
 
 296 MUTUAL AND SELF INDUCTION. 
 
 For slew alternations, therefore, the current adopts that mode 
 of distributing itself over the cross-section of the conductor 
 which makes the rate of energy dissipation a minimum. 
 On the other hand, for rapid alternations the current meets 
 with its greatest obstacle from the counter electromotive force 
 of self-induction, and it accordingly distributes itself over the 
 cross-section of the conductor, so as to get as much to the 
 outside as possible, and thus avoids, in the case of magnetic 
 conductors, magnetising the inner layers or portions of the 
 conductor. The endeavour is to make the self-induction a 
 minimum irrespective of resistance. This is only an instance 
 of the broad, general principle that behaviour of current for 
 very rapid pulsations, or alternations, is determined by the 
 inductances rather than the resistances, whereas for steady or 
 slowly periodic currents the behaviour is governed by resistance 
 rather than by self-induction. 
 
 In order to see under what conditions the alteration of resis- 
 tance and self-induction becomes sensible, we have to examine 
 
 the value of the term ^ ^ in the above-given series for 
 
 E 1 . We will first take the case of an iron wire 0-4 centimetre, 
 say, 0-16 inch diameter (No. 8 B.W.G.). The specific resis- 
 tance of iron in C.G.S. measure is about 10 4 ; so that 
 B_ 10 4 = 10 6 
 
 I TTXO'04 477* 
 
 ^ 2 =4;r 2 w 2 , n being the frequency. 
 
 Let us take w=100, so that there are supposed to be 100 
 complete alternations per second. The value of //, is more 
 difficult to assign. For small degrees of magnetisation and 
 solid iron, we may, perhaps, take /A = 300; 
 
 then L P'Vf- 1 4- 2 nVZ 2 _ 
 
 12 E 2 12 E 2 10 10 
 
 If *=300, ?i = 100, ^^OxlO 8 , and 1- =0-47 
 
 = 0'5 nearly. 
 
 Accordingly, for this case W = E (1+0-47) nearly, or the 
 resistance is increased to about half as much again. 
 
 If n = 1,000 we should find E 1 = 48 E, or the resistance would 
 be increased nearly fifty times.
 
 MUTUAL AND SELF INDUCTION. 297 
 
 Consider next the case of copper. The specific resistance is 
 1,640 C.G.S. units. If a be the radius of the wire in centi- 
 metres, then we have 
 
 _ 
 
 12 ~~W~ IT (i,640) 2 10* 
 
 If, as before, ?& = 100, this fraction becomes equal to 0'12a 4 . 
 'This shows that for a diameter of one centimetre we should 
 
 have B^K (1+0-12); 
 
 and hence for diameters of one centimetre and upwards the 
 resistance of round copper rods becomes very sensibly increased 
 for alternating currents of a frequency about 100 and up- 
 wards. The practical conclusions of importance in electrical 
 engineering from the above investigation are these : First, 
 copper rods or conductors should be used, and not iron, for 
 transmitting alternate or intermittent electric currents having 
 .a moderate frequency, say of 100 to 1,000 per second ; secondly, 
 to avoid, as far as possible, the increase of resistance due to 
 the current keeping to the outer portions of the conductor, the 
 conductor should be in the form of a thin strip, or better, a 
 tube having walls thin in proportion to the radius. It is to be 
 noted that mere stranding of the conductor, or building it up 
 of separate insulated conductors joined in parallel, will not 
 prevent this augmentation of resistance, unless the stranding is 
 of such a kind that portions of the cable which at one point of 
 its length form the inner parts or heart of the cable at another 
 part of its length form the outside. 
 
 The object to be achieved is to construct some kind of 
 stranding by which all portions of the conductor are brought as 
 near as possible to the dielectric, so that the energy arriving 
 from the dielectric finds all parts of the mass of the cable, 
 both surface and interior, equally accessible. In order to 
 avoid external inductive disturbance, the proper form to give 
 to a cable intended to convey rapidly intermittent or alternate 
 currents is a couple of rather thin concentric tubes of copper 
 well insulated from each other, and both insulated from the 
 earth, of which one forms the lead and the other the return. 
 By this device the metal will be most economically employed. 
 An equivalent device used in practice is a concentric cable,
 
 298 
 
 MUTUAL AND SELF INDUCTION. 
 
 which consists of a central core of stranded copper cable 
 covered with insulation, and then plaited over with a sheath 
 of other copper wires which form the return conductor. 
 
 In a further experiment, Lord Eayleigh (loc. cit.} examined 
 the resistance of an iron wire of hard Swedish iron 10-03 
 metres long and 1-6 millimetre in diameter. In arbitrary units 
 the resistance of the wire to steady currents was 10-4 units 
 or 0-51 ohm, and to currents of 1,050 complete alternations 
 per second its resistance was 12-1 units, or 0-595 ohm, which 
 is an increase of about 20 per cent. In the case of a stouter 
 wire, 18-34 metres long and 3-3 millimetres in diameter, the 
 
 FIG. 111. 
 
 resistance to steady currents was 4-7 units, and the resistance 
 to the interrupted currents of the above-mentioned frequency 
 was 8-9 units, or nearly double. This illustrates the fact that, 
 for a given frequency of alternation, the ratio in which the 
 resistance is increased is greater the greater the diameter of 
 the conductor, assuming it to be a round solid rod. 
 
 Lord Eayleigh found it more convenient in many researches 
 to slightly alter the arrangement of the induction balance as 
 described by Prof. Hughes,"and to make it as follows (Fig. 111).
 
 MUTUAL AND SELF INDUCTION. 292 
 
 Two arms of a quadrilateral, E and S, consist of equal resist- 
 ances of German-silver wire, wound double, so as to have 
 negligible inductance. One arm, Q, consists of a coil having 
 inductance and resistance greater than that of any conductor! 
 P, to be placed in the fourth arm. B and I are a battery and' 
 an interrupter, T is a telephone in the "bridge," and rr 1 is 
 a German-silver wire of appropriate resistance, along which 
 slides the contact of the bridge. The arm P includes a pair 
 of coils joined in series, and which act upon each other by 
 mutual induction, so that the resulting self-induction of 
 the two coils in series can be varied within certain limits by 
 turning one coil round within the other. For the resulting 
 self-induction of such a pair of coils used in this manner 
 may be regarded as made up of the component self-inductions 
 of each coil taken separately and of twice the positive or 
 negative mutual self-induction, depending upon which faces 
 of the coils are presented to each other. It is possible, then, 
 within certain limits to vary the inductance of the branch 
 PC, and to vary also the resistance of the branches Q and 
 P C by shifting the contact of the telephone along r ? J . 
 
 The condition for obtaining a true balance when the current 
 is periodically interrupted is that the resistances and induc- 
 tances of the branches Q and P C shall be separately equal. 
 Suppose a balance has been obtained without the use of P, in 
 which the resultant self-induction of C is made to balance the 
 inductance of Q, and the resistance of C + r 1 is made to be equal 
 to that of Q + r. Let, now, any conductor, P, be inserted as in 
 the figure ; the telephone contact will have to be shifted, and 
 also the inductance of C will have to be changed to re-obtain a 
 balance. The inductance of P is measured by the amount by 
 which that of C has to be reduced on inserting P, and the 
 resistance of P is measured by twice the resistance of that 
 length of the German- silver wire r?- 1 by which the telephone 
 contact point has to be shifted to regain the balance. This 
 method of employing the induction balance separates out at 
 once the real resistance of P from its effective induction. 
 
 With the aid of this balance an interesting experiment was 
 made, showing the effect of a closed secondary circuit on the 
 resistance and inductance of the primary. The frequency was 
 again, as usual, 1,050 per second. A coil was prepared of
 
 300 MUTUAL AXD SELF INDUCTION. 
 
 two copper wires, wound side by side on one bobbin. The 
 diameter of each wire was about 0-OSin., and the length of 
 each wire 318in. There were 20 (double) turns, so that the 
 mean diameter of the coil, wound as compactly as possible, 
 was about oin., and the resistance of each wire was 0-05 ohm. 
 
 The coefficient of mutual induction of the two wires was 
 determined by comparison of the self-induction L of one wire 
 with that of the two wires connected oppositely in series, viz., 
 (2 L - 2 M). In this way it appeared that 
 
 M = 48-1 = 43-1 x 1,553 centimetres. 
 
 Observation showed that closing of the circuit of one wire 
 reduced the self-induction of the other from 44-4 to 3-4. The 
 resistance to steady currents was 0-92 (arbitrary units). The 
 resistance to the periodic currents was 0-97 with the secondary 
 circuit open, and 1-74 with the secondary circuit closed. 
 Hence, L = 44-4x 1,553 centimetres, and 
 
 E = 0-97 x 0-0492 x 10 9 centimetres per second. 
 
 From Maxwell's formulae, page 180, we get 
 
 * 10" x 1-951 _ 
 
 B* +/>*L a 10 7 x 0-023 + 10 17 x 2-071 
 Hence, L^L (1-0-932), 
 
 where L 1 is the decreased inductance. Hence, 
 
 L! = 0-068 L, 
 or L! = 0-068x44-4 = 3, 
 
 and the observed value is 3-4, which is in very tolerable 
 agreement. 
 
 Again, the steady resistance with secondary open is 0-92, 
 and hence the resistance R 1 with secondary closed is 
 
 R! = 1-932x0-92 = 1-77; 
 
 and observation gives the value 1-74. We see, then, that 
 observations with this bridge confirm, with a considerable 
 degree of accuracy, the deductions from the theory of simple 
 periodic currents, that the closing of a secondary circuit 
 increases the resistance and diminishes both the inductance 
 and the impedance of an adjacent primary circuit. 
 
 From a practical point of view the most important difference 
 between the conduction cf steady electric currents and rapidly
 
 MUTUAL AND SELF INDUCTION. 301 
 
 periodic currents is that of the locale of the currents in the 
 conductor and the consequent rise in the ohmic resistance of 
 the conductor as a whole when employed with such periodic 
 currents. Prof. Hughes called attention in 1883 to this great 
 difference in the resistance of an electrical conductor if mea- 
 sured during the variable instead of the stable condition of the 
 current.* 
 
 In experiments with his induction bridge Prof. Hughes was 
 able to assure himself that the resistance of an iron telegraph 
 wire of the usual size was more than three times greater for 
 rapid periodic currents of about 100 per second than for steady- 
 currents. The full elucidation of the propagation of currents 
 in conductors under periodic electromotive force is not to be 
 attempted without following out some very elaborate mathe- 
 matical analysis. The subject has received its most complete 
 treatment perhaps in the published writings of Mr. Oliver 
 Heavisidef and all that can be attempted here is to give a 
 slight sketch of the views which are now very generally 
 held on this subject. 
 
 Consider a long level tank or canal full of liquid. There 
 are, amongst others, two ways in which we might suppose 
 this liquid to be set in motion. A paddle or the hand 
 might be placed in the liquid, and by giving the liquid 
 bodily a push it might be made to move forward ; or we 
 might suppose some body floating on the surface, such as a 
 plank of wood to be dragged along the surface. The friction 
 between the plank and the layer of water beneath it would 
 then cause the subjacent layer of liquid to move with the 
 plank, and the motion of this layer would be gradually com- 
 municated to the other and deeper-lying layers by reason 
 of the viscosity of the fluid. Or take the case of a basin 
 containing water. The liquid might be set in rotation by 
 stirring it with a paddle or the hand, but it might also 
 be set in rotation by twisting the basin rapidly. In this las: 
 case the rotation of the basin would be communicated by 
 friction to the water in contact with its sides, and then handed 
 
 * Discussion on a Paper by Mr. \V. H. Preece on " Electrical Conductors," 
 Proceedings In&t. Civil Engineers, Vol. LXXY., 1883. 
 
 t "Electromagnetic Theory," by Oliver Heaviside in The Electrician 
 Series, and " Electrical Papers," published by Messrs. Maemillan.
 
 302 MUTUAL AND SELF INDUCTION. 
 
 on from layer to layer of the water by internal fluid friction. 
 Thus the twist or spin of the basin would be gradually propa- 
 gated inwards from the circumference to the centre. Imagine 
 the whole mass of the liquid divided up into very thin con- 
 centric shells, like the coats of an onion. If the liquid were 
 a perfect fluid there would be no friction between these layers, 
 but since every liquid possesses some degree of viscosity or 
 internal fluid friction, the sliding of one layer of fluid over 
 another gradually causes the second layer to partake of the 
 motion of the first. Hence, when the rotation of the basin 
 commences the friction between its sides and the first layer 
 of fluid starts that gradually in motion ; this motion is then 
 transmitted to the second layer, and so on, until the whole mass 
 of the liquid possesses an equal angular velocity round the 
 axis of rotation. The greater the fluid friction or viscosity the 
 more rapid will be this equalisation of the angular velocity of 
 all parts of the fluid, and so a rotating vessel full of tar would 
 arrive at a stationary condition as regards angular velocity 
 sooner than one filled with a limpid liquid as alcohol or ether. 
 Just as the angular velocity diffuses inwards from the circum- 
 ference to the centre in the case of such a revolving basin of 
 liquid, so, according to modern views, does the current diffuse 
 inwards from the circumference to the axis of the electric con- 
 ductor, The student who has been accustomed to think of a 
 current as produced in a conductor by a sort of push given to 
 it in the conductor such conception being based on a rough 
 working hypothesis of a hydrodynamic nature will perhaps 
 have some difficulty in discarding this notion and realising that 
 the current in a wire may perhaps be generated in it by an 
 action taking place at all parts of the surface of the u-ire which 
 gradually soaks or diffuses into the conductor out of the sur- 
 rounding dielectric, but he will find that this new hypothesis 
 serves to establish a mode of viewing the induction phenomena 
 which makes various experimental results much more easily cor- 
 related. It was well demonstrated by the experiments of Prof. 
 Hughes and others that a flat sheet or strip of metal has a less 
 self-induction than a round wire of equal cross-sectional area. 
 On the present hypothesis, this is explained by saying that the 
 flat strip offers a greater absorption surface to the dielectric ; the 
 current therefore soaks in more quickly to the centre and arrives
 
 MUTUAL AND SELF INDUCTION. 303 
 
 at a uniform distribution over the cross section very soon in 
 other words, the variable state is sooner over, and we express 
 this fact by saying that the self-induction is small. Again, if the 
 electromotive force is oscillatory or rapidly periodic, we see at 
 once that the current has not time to penetrate right into the 
 core of the conductor before its sign or direction is reversed. 
 It has hardly started on its journey inwards, soaking from sur- 
 face to centre, before it is recalled ; hence the flow of a current 
 when very rapidly periodic is confined to the surface of the 
 conductor, the real or ohmic resistance is increased, and the 
 self-induction is diminished. 
 
 Lord Kelvin has shown (Bath British Association Meeting, 
 1888) that for alternating currents of a frequency equal to about 
 150 complete alternations per second, the depth to which the 
 currents penetrate into the substance of the copper is about 
 three millimetres, so that portions of the conductor beyond 
 this distance from the surface are almost useless for conduction. 
 The practical moral of this is that the proper form for a con- 
 ductor for alternate currents is either a flat sheet of copper or 
 a copper tube, in which, for the above frequency, the thickness 
 of material is not more than one-quarter of an inch. It is 
 useful in this connection to note a few facts with regard to 
 cables as used for alternating currents. A seven strand cable 
 has an overall diameter of three times one strand. A nine- 
 teen strand cable has an over-all diameter of five times one 
 strand. A No. 12 wire (S.W.G.) has a diameter of 0-109 inch. 
 Hence a 19/12 cable has a diameter of 0-5 inch, and a cross- 
 sectional area of 0-1615 square inch. At a current density of 
 600 amperes per square inch this cable will carry 100 amperes, 
 and it has a resistance of one- sixth of an ohm per 1,000 yards. 
 For alternating currents, therefore, of about 100 frequency, 
 a 19/12 stranded cable is about the largest size that should be 
 employed. For alternating currents of 100 frequency, beyond 
 about 100 amperes, a form of cable must be employed in 
 which the thickness of the conductor is not at any part 
 greater than about one quarter of an inch ; and this is only 
 to be achieved by the employment of concentric tubes or 
 concentric stranded cables in which the core or central strand 
 is not of greater thickness than half an inch, and which con- 
 dition necessitates, therefore, that when above a certain cross-
 
 304 MUTUAL AND SELF INDUCTION'. 
 
 sectional area the central conductors should also be of tubular 
 form. One of the advantages to be gained by the employment 
 of alternating currents of low frequency is that the limiting 
 diameter of the conductors is much larger for low than for 
 high frequency. To return to our illustration of the twisting 
 basin of fluid. Suppose the action on the vessel consists in 
 rapidly twisting it through a small angle, first one way and 
 then the other, the liquid in the interior would be subjected to 
 a strain which would consist in the various concentric layers 
 of the liquid sliding backwards and forwards over each other. 
 The interior of the liquid would be thrown into stationary 
 waves, in which the nature of the wave motion consisted hi 
 each particle of water being displaced first one way and then 
 another along an arc of a circle described on a horizontal 
 plane, with its centre in the axis of rotation. The more rapid 
 the motion the greater would be the rate of decrease in the 
 amplitude of each wave in passing from the circumference to 
 the centre of the vessel ; in other words, for very rapid oscilla- 
 tions the bulk of the water in the centre of the basin would 
 remain nearly at rest. 
 
 Every experiment as yet made on the self-induction or change 
 of self-induction in conductors is consistent with the above 
 hypothesis. It shows, for instance, why a conductor composed 
 of thin insulated wires or thin insulated strips has a less self- 
 induction than a solid conductor of equal cross-section. Prof. 
 Hughes says* : " We can reduce the self-induction of a 
 current upon itself to a mere fraction of its previous value 
 by simply separating the contiguous portions of a current 
 from each other, the results proving that a comparatively small 
 separation, such as is obtained by employing ribbon conductor? 
 in place of a wire of the samo weight, reduces the self-induction 
 80 per cent, in iron and 85 per cent, in copper, and if we still 
 divide the current by cutting the ribbon into several strips 
 (separating the strips at least 1 centimetre from each other), 
 then the combined but separated strips show a still greater 
 reduction, being 94 per cent, in iron and 75 per cent, in copper." 
 
 These, and many other experiments of a similar sort, indicate 
 that we may regard the inductance of a conductor as an effect 
 which is due to the fact that the current takes time to pene- 
 * Inaugural Address, Journal Soc. Tfl. ng., 1886.
 
 MUTUAL AND SELF INDUCTION. 305 
 
 trate into the conductor, and that a reduction of the time 
 required to arrive at an equal current density in all parts 
 of the conductor can be effected by any change of form 
 which brings the inner parts of the conductor nearer the 
 surface, or makes them more get-at-able from the dielectric, 
 The better the conductor the slower is the rate of equalisation 
 of current density over its cross-section in other words, the 
 less rapid is the rate of diffusion of the current inwards from 
 circumference to centre; and the " time constant " of the 
 circuit, or the time in which, under the operation of a constant 
 electromotive force, the current will rise to a definite fraction 
 of its maximum value, is a quantity proportional to the con- 
 ductivity of the circuit, and to another factor (the formal 
 inductance), which may be considered as expressing the 
 accessibility of the conductor as regards geometrical form to 
 the entrance of the current into it ; and finally, in the case of 
 magnetic conductors, to a quantity (the permeability) deter- 
 mined by the capacity of the conductor to utilise part of this 
 incoming energy in producing magnetisation of its substance. 
 
 We are indebted to a Paper read before the Austrian 
 Academy by Prof. Stefan for a simple and intelligible analogy 
 helping to comprehension of the electrical distribution of 
 current in a conductor. Imagine a cylinder or cylindrical 
 wire heated throughout to a uniform temperature ; let it be 
 suddenly brought into a chamber where the temperature is 
 higher. The outer layers of the cylinder will rise first in 
 temperature, and gradually convey the heat to the successive 
 interior layers. Precisely the same order of phenomena 
 occurs if an E.M.F. is suddenly set up between the ends of 
 the wire or cylinder. The current during the variable state 
 passes first through the outer layers alone, and gradually 
 penetrates the inner layers. When the external E.M.F. 
 is suddenly removed, the action, of ceasing in the current 
 resembles the cooling of the cylinder. The current ceases 
 first, or, rather, most quickly, in the outer layers. 
 
 Now, let us imagine the cylinder transferred to and fro 
 from a very hot place to a cool one. It is easy to see that 
 waves of heat will pass in and out radially, and also that the 
 condition at any instant will depend largely upon the rate of 
 transference. 
 
 x
 
 306 MUTUAL AND SELF INDUCTION. 
 
 When the rate of motion is sufficiently slow, the waves of 
 heat passing any given point in the radius of the wire follow 
 exactly with the periodic changes of position. The amplitudes 
 of these variations have values which decrease from the sur- 
 face inwards. "When the rate of change is increased, the 
 amplitude of the waves gets shorter and shorter, and at an 
 infinite velocity of transference the wire would acquire an 
 equable temperature throughout. In the electrical analogue 
 the rate of transference corresponds to the inverse of the 
 periodic time of an alternating current. The heat conducting 
 power of the material corresponds to electrical resistance. 
 
 Prof. Stefan gives some numerical illustrations which are 
 useful. If an alternating current have a frequency of 250 per 
 second and is passed through an iron wire of 4mm. diameter, 
 the amplitude of the waves of current density is about twenty- 
 five times greater upon the surface than at the axis of the wire. 
 For double the number of vibrations per second the external 
 amplitude becomes only six times as great. The difference of 
 phase is one-third the duration of the vibration in the first 
 case and one-half in the second. The latter statement implies 
 that the external current is at a given moment actually in the 
 reverse direction to the internal current. 
 
 For non-magnetic wires the difference is not nearly so 
 marked, and it decreases as the specific resistance increases. 
 For a copper wire of 4inm. diameter, with a periodic time of 
 one 500th second, the difference between the current density 
 at the surface and at the centre is only 14 per cent. If, how- 
 ever, the copper wire Jbe increased to 20rnm. diameter, then 
 we should get the same difference as in the particular iron 
 wire quoted. 
 
 . It is obvious that this non-homogeneous distribution of 
 current must increase dissipation of energy, which is, of 
 course, proportionate in each transverse section to the square 
 of the current strength, at that spot. In the case of the iron 
 wire quoted,. the increase of resistance is 48 per cent, at the 
 250 per second frequency, and 100 per cent, at the higher 
 speed. As the frequency of alternation is increased, the 
 resultant self-induction of the circuit is lessened, but although 
 the true resistance is increased, the impedance may be 
 diminished on the whole.
 
 MUTUAL AND SELF INDUCTION. 307 
 
 12. Electromagnetic Repulsions. The effects of self and 
 mutual induction in conducting circuits are well illustrated in 
 studying the dynamical actions taking place between con- 
 ductors conveying currents and other circuits. On the 2nd 
 day of October, 1820, Ampere presented to the Royal Academy 
 of Sciences in Paris an important memoir, in which he summed 
 up the results of his own and Arago's investigations in the 
 then new science of electro-magnetism, and crowned that labour 
 by the announcement of his great discovery of forces of 
 attraction or repulsion existing between conductors conveying 
 electric currents.* Kespecting that achievement, when deve- 
 loped in its experimental and mathematical completeness, 
 Clerk Maxwell speaks of it as "one of the most brilliant in 
 the history of physical science." Our wonder at what was 
 then accomplished is increased when we remember that hardly 
 more than two months before that date John Christian Oersted 
 had startled the scientific world by the announcement of the 
 discovery of the magnetic qualities of the space near a current- 
 traversed conductor. Oersted named the actions around the 
 conductor, which we now refer to as the magnetic field, the 
 electric conflict, and in his first Paper, f in describing the 
 newly-observed facts, he says : " It is sufficiently evident that 
 the electric conflict is not confined to the conductor, but is 
 dispersed pretty widely in the circumjacent space." " We 
 may likewise collect," he adds, " that this conflict performs 
 circles round the wire, for without this condition it seems 
 impossible that one part of the wire when placed below the 
 magnetic needle should drive its pole to the east and when 
 placed above it to the west." These words are taken from the 
 original paper, which stimulated the philosophic thought of 
 
 * Me"moire pre"sente a 1' Academic Royale des Sciences le 2 Octobre, 
 1820, ou se trouve compris le re'sume' de ce qui avait ete lu a la meme 
 Academic lea 18 me eb 25 me Septembre, 1820, sur les effets des courants 
 Electrique, par M. Ampere. See Vol. XV. Annales de Chimie, 1820. 
 
 t In the Annals of Philosophy for October, 1820, VoL XVI., p. 274, is 
 to be found an English translation of Oersted's original Latin essay, dated 
 July 21, 1820, describing his immortal discovery. This Paper is entitled 
 "Experiments on the Effects of a Current of Electricity on the Magnetic 
 Needle," by John Christian Oersted, Knight of the Order of Danneborg, 
 Professor of Natural Philosophy in Copenhagen. 
 
 x2
 
 308 MUTUAL AND SELF INDUCTION. 
 
 Ampere, and finally led him to the valuable discovery of 
 the electro-dynamic actions between conductors conveying 
 currents. 
 
 Eeferring the student to text -books on Physics for the 
 complete statement of Ampere's work, we may describe briefly 
 some illustrations of the interactions of two circuits traversed 
 by currents in the same or opposite directions. Holding a 
 circular coil traversed by a continuous electric current near to 
 a similar circuit free to move, we find that when the circuits 
 are parallel to each other there is an attractive force between 
 them if the currents in adjacent parts of the circuits flow in 
 the same direction, and a repulsive effect if they flow in the 
 opposite. This is the electro-dynamic action discovered by 
 Ampere and utilised in the construction of instruments for the 
 measurement of electric currents in practical work. If one 
 conducting coil, such as that of an electro-magnet, is traversed 
 by an alternating current, and the other is simply a closed 
 circuit or coil placed a little distance off, but in its field, it has 
 been previously explained that the closed circuit becomes the 
 seat of an alternating induced current, which, if the inducing 
 current is sufficiently powerful, can be made to render itself 
 evident by illuminating a small incandescent lamp placed in 
 the secondary circuit.* We notice, however, that in perform- 
 ing the experiment the secondary circuit must be so placed 
 that the magnetic induction of the primary coil perforates 
 through the secondary circuit. If the secondary circuit is 
 held in such a position that the reversal of direction of the 
 primary current causes no reversal of direction of the magnetic 
 field traversing the secondary circuit, because it is not linked 
 with any of the lines of induction of the primary, the secondary 
 circuit is no longer the seat of any induced current. 
 
 * The experiments described in the following paragraphs can be shown 
 with an alternating current magnet, having a core formed of a bundle of 
 fine iron wire about 3in. in diameter and 12in. long, excited by an alternat- 
 ing-current dynamo, giving a current at an electromotive force of about 
 100 volts. A small shelf around the core a little above the middle serves 
 as a support for rings, &c., to be projected. The performance of these 
 experiments on a scale suitable for large audiences requires from 10 to 
 15 horse-power at least, and can hardly be shown well unless the alternator 
 can provide a current of 100 amperes at 100 volts available at the moment 
 of maximum demand.
 
 MUTUAL AND SELF INDUCTION. 309 
 
 This electromagnetic induction thus taking place across 
 space is not stopped by the interposition of a non-conducting 
 screen. The magnetic induction passes freely through a 
 deal board or a plate of glass, but if we interpose a thick sheet 
 of copper (Fig. 112) we thereby screen the secondary circuit 
 from the inductive action of the primary. The rapid heating 
 of this copper screen makes us aware that the secondary 
 currents are induced in the copper sheet in the form of eddy 
 currents, and it therefore screens the secondary circuit, as 
 already explained, because the inductive action of these eddy 
 
 FIG. 112. Copper Plate interposed between a Primary and a Secondary 
 Coil and shielding the Secondary from Induction. 
 
 currents on the side remote from the magnet is exactly equal 
 and opposite in inductive effect to that of the primary circuit 
 on the secondary coil. 
 
 If a continuous current is sent through the coils of an 
 electro-magnet, and magnetises its iron core very powerfully, 
 it is found to be impossible to strike the pole of the magnet a 
 sharp blow by means of a sheet of copper. Holding a sheet 
 of copper over such a magnetic pole, and exciting the magnet, 
 the hand holding the copper sheet feels a repulsive action at
 
 310 MUTUAL AND SELF INDUCTION. 
 
 the moment when the current is put on and an attractive 
 action when it is cut off. If we try to slap the magnet pole 
 sharply with the copper sheet, it is found that this repulsive 
 force prevents anything like such a sharp blow being given to- 
 the pole when the current is on as can be given when the 
 current is off. Moreover, when a very powerful electro- 
 magnet is employed, it is found that a disc of copper let fall 
 over the pole does not fall down sharply and quickly on to it 
 when the current is flowing through the coils of the magnet, 
 but settles down softly and slowly as if falling through some 
 viscous fluid. The correct explanation of these facts is to be 
 found in the statement that the motion of the conductor 
 towards the magnetic pole causes eddy electric currents to be 
 generated in it by electro-magnetic induction, and that these, 
 being in the opposite direction to the exciting current of the 
 magnet, cause a repulsive force to exist between the inducing 
 and secondary circuits, which creates the apparent resistance 
 we feel. 
 
 In order to exhibit the stress brought into existence between 
 an electro-magnet and a metal sheet held near it when induced 
 currents are set up in the disc, we may arrange the following 
 experiment : Over the pole of a powerful electro-magnet 
 we balance a small disc of copper, the size of a penny, 
 carried on one end of a delicately-balanced bar. A mirror 
 attached to that bar serves to reflect on to a screen a ray 
 of light indicating the smallest motion of the copper disc. 
 On magnetising the magnet the copper is suddenly repelled, 
 but comes to rest again immediately in its initial position. 
 When the magnet is demagnetised the copper experiences a 
 momentary attraction. Or we may illustrate the same action 
 in another way. Consider, for instance, a ring of copper 
 hanging in front of the pole of an electro-magnet (see Fig. 113), 
 having the plane of the ring perpendicular to the lines of 
 magnetic force proceeding out from the pole. Let the 
 magnet be an electro-magnet, and let the pole be suddenly 
 made a north or marked pole. Lines of magnetic force are 
 thrust into the aperture of the ring. This magnetic flux, in 
 accordance with a well-known law, generates an inductive 
 electromotive force, which causes a transient current to flow 
 round the ring in a counter-clockwise direction, as looked at
 
 MUTUAL AND SELF INDUCTION. 311 
 
 from the north magnetic pole. The ring becomes virtually a 
 magnetic shell, having a north pole facing the north pole of 
 the exciting magnet. By the fundamental laws of action 
 between currents and magnets established by Ampere, the 
 ring experiences a slight repulsive force, due to the electro- 
 dynamic action between the current in the ring and the 
 magnetic pole. The generation of the momentary induced 
 current in the ring is accompanied by an electro-dynamic 
 impulse tending to thrust it away from the pole. 
 
 Suppose, next, that the electro-magnet is demagnetised. 
 The ring has generated in it a reverse induced current flowing 
 in the direction the hands of a clock move when looked 
 at from the magnetic pole. This is also accompanied by an 
 electro-dynamic attraction of the ring towards the pole, but 
 
 FIG. 113. Copper Ring hung in the Field of an Electro-magnet, and 
 Repelled or Attracted when the Current is put on or cut off. 
 
 which is much more feeble than the previous repulsion. These 
 attractions and repulsions are obviously due to the Amperian 
 stress set up between the magnet and the metal by reason 
 of the induced currents set up in the latter. It has been 
 pointed out by Prof. S. P. Thompson that Ampere himself 
 probably observed an effect of this kind (Proc. Phys. Soc. of 
 London, Vol. XIII., p. 493, "Note on a Neglected Experiment 
 of Ampere."). Impulsive effects of this nature have been also 
 studied by Prof. Vernon Boys (see Proc. Phys. Soc. of London, 
 Vol. VI., p. 218, " A Magneto -electric Phenomenon "). 
 
 Let us in the next place consider a circuit, say a closed 
 conducting ring, suspended in front of the pole of an electro- 
 magnet, and let the coils of the electro -magnet be trans-
 
 312 
 
 MUTUAL AND SELF INDUCTION. 
 
 versed by an alternating current of electricity (Fig. 113). 
 The magnetic field of the magnet is then an alternating field. 
 We shall suppose it to vary in strength according to a 
 simple periodic law. The closed circuit is therefore subjected 
 to an inductive action, and we know that the induced electro- 
 motive force in that circuit is at any instant proportional 
 to the rate of change of the magnetic field in which it is 
 immersed. If, therefore, the variation in strength of that field 
 is represented geometrically by the ordinates of a periodic 
 curve, the varying electromotive force acting in the ring circuit 
 is represented by the ordinates of another such curve of equal 
 
 FIG. 114. Diagram showing the Equality of the Attractive and Repulsive 
 Impulses in a Non-inductive Circuit when held in an Alternating Magnetic 
 Field. 
 
 wave length, shifted a quarter of a wave length behind the 
 first. In the diagram (see Fig. 114) the variation of the in- 
 duced magnetic field, and the induced electromotive force in 
 the circuit, are represented as usual by two harmonic curves. 
 This induced electromotive force creates an induced current 
 flowing backwards and forwards in the ring, and we shall, 
 in the first instance, suppose that this current flows in 
 exact synchronism with its electromotive force. The 
 induced current and the inducing magnetic field may there- 
 fore be represented as to relative phase and strength by the 
 curves in the diagram (Fig. 114). The dynamical action,
 
 MUTUAL AND SELF INDUCTION. 313 
 
 or the force which the ring experiences, is at any instant 
 proportional to the product of the strength of the magnetic 
 field in which the ring is immersed and to the strength of the 
 induced current created in it. If we multiply together the 
 numerical values of the ordinates of these two curves at any 
 and every point on the horizontal line, and set up a new 
 ordinate at that point representing this product, the extremi- 
 ties of these last ordinates define a curve, which is a curve 
 representing the force acting on the secondary circuit ; and it 
 is seen from the diagram (Fig. 114) to be a wavy curve having 
 a wave length equal to half that of the first two curves. More- 
 over, the whole area enclosed between the outline of this force 
 curve and the horizontal line represents to a certain scale the 
 time integral of that force, or the impulse acting on the secon- 
 dary circuit, and the theory shows us that, under the assump- 
 tions made, the secondary circuit so acted upon experiences in 
 each period of the current four impulses, two positive or repul- 
 sive, and two negative or attractive. Hence, it follows that 
 such an ideal conducting circuit held in front of an alter- 
 nating electro-magnet should experience a rapid alternate series 
 of equal pushes and pulls, or of little impulses to and from the 
 magnet. These equal and opposite impulses in quick succession 
 would neutralise one another, and our supposed circuit would 
 not, on the whole, be subject to any resultant force. 
 
 When we present a real conducting circuit to the pole of an 
 electro-magnet traversed by a powerful alternating current, 
 we find that under the actual circumstances there is a powerful 
 repulsive action between the pole and the circuit. With a 
 powerful alternating current electro-magnet striking effects 
 of repulsion may be thus shown. 
 
 If we hold a copper ring over the pole of a powerful vertical 
 alternating electro-magnet, we find at once that there is a 
 perceptible and strong repulsion. Letting the ring go, it 
 jumps up into the air, impelled so to do by the electro-magnetic 
 repulsion acting upon it (Fig. 116). All good conducting rings 
 will execute this gymnastic feat, and rings of copper and 
 aluminium are found to be most nimble of all. Eings of zinc 
 and brass are sluggish, and a ring of lead will not jump at all. 
 Prof. Elihu Thomson was the first to call attention to this 
 strong repulsive action between conducting rings and an
 
 314 
 
 MUTUAL AND SELF INDUCTION. 
 
 alternating electro-magnet. He has thus described his first 
 notice of these effects : " In 1884, while preparing for the 
 International Electrical Exhibition at Philadelphia, we had 
 occasion to construct a large electro-magnet, the cores of which 
 were about Gin. in diameter and about 20in. long. They 
 were made of bundles of iron rod about j^-in. in diameter. 
 When complete the magnet was energised by a current from 
 a continuous-current dynamo, and it exhibited the usual 
 powerful magnetic effects. It was found also that a disc of 
 sheet copper of about ^in. in thickness and lOin. in diameter, 
 if dropped flat against a pole of the magnet, would settle down 
 softly upon it, being retarded by the development of currents 
 
 FIG. 115. 
 
 in the disc, due to its movement in a strong magnetic field, 
 and which currents were of opposite direction to those in the 
 coils of the magnet. In fact, it was impossible to strike the 
 magnet pole a sharp blow with the disc, even when the 
 attempt was made by holding one edge of the disc hi the hand 
 and bringing it down forcibly towards the magnet. In 
 attempting to raise the disc quickly off the pole a similar 
 but opposite action of resistance to movement took place, 
 showing the development of currents in the same direction as 
 those in the coils of the magnet, and which current, of course, 
 would cause attraction as a result. The experiment could be 
 tried in another way. Holding the sheet of copper by one
 
 MUTUAL AND SELF INDUCTION. 
 
 315 
 
 edge, just over the magnet pole (see Fig. 115), the current in the 
 magnet coils was cut off by shunting them. At that moment 
 was felt an attraction of the disc, or a dip towards the pole. 
 On starting the current the plate experienced a powerful re- 
 pulsion." The question may then be asked : Why is it the 
 metal rings are always repelled by the alternating magnet ? 
 The explanation is not difficult to find. The real ring possesses 
 a quality, called its inductance, of which we took no account 
 in our examination of the case a moment ago. As a con- 
 
 FIG. 116. Aluminium Kiug projected from the pole of an Alternating 
 Electro-magnet, and floating over the pole when restrained by three 
 strings. 
 
 sequence of this inductance we have seen that the current 
 induced lags in phase behind the inducing electromotive force. 
 We have then to correct the diagram considered just now, 
 to make it fit in with the facts of nature, and we must repre- 
 sent the periodic curve which stands for the fluctuations of 
 the induced current in the ring as shifted backwards or 
 lagging behind the curve which represents the electromotive 
 fore.) in the circuit brought into existence by the fluctuating
 
 316 MUTUAL AND SELF INDUCTION. 
 
 magnetic field. Making this change (Fig. 117), and forming, 
 as before, a force curve to represent the impulses on the ring, we 
 then find that, owing to the "lag" of the secondary current, 
 one set of the impulses, namely, the positive or repulsive 
 impulses, has been enlarged at the expense of the negative or 
 attractive impulses. Theory, therefore, points out that, as a 
 consequence of the self-induction of the ring, the balance 
 between the attractive impulses and the repulsive impulses is 
 upset, and that the latter predominate. The real ring behaves 
 therefore, very differently to the ideal non-inductive ring. 
 The real ring is strongly repelled, because the resultant action 
 
 /"N 
 
 FIG. 117. Diagram showing the Inequality of the Attractive and Repul- 
 sive Impulses in the case of an Inductive Circuit when held in an Alter- 
 nating Magnetic Field. 
 
 of all the impulses is to produce, on the whole, an electro- 
 magnetic repulsion. This repulsion is evidence of the self- 
 induction or inductance of the circuit exposed to the magnetic 
 field, and it forms a new way of detecting it. But although 
 this is part of the truth, it is not the whole truth. The lag of 
 the induced current in the ring, and hence the predominance of 
 the repulsive impulses, depends on the conductivity of the 
 material of which the ring or circuit is made ; and the better
 
 MUTUAL AND SELF INDUCTION. 317 
 
 this conductivity the greater is that repulsion, because both 
 the induced current and the lag are thereby increased. Hence 
 it comes to pass that there are two factors involved in making 
 this repulsive effect, the conductance of the ring or disc and 
 its inductance. For equal conductivities, the greater the self- 
 induction the greater the repulsion. For equal self-inductions, 
 the greater the conductivity of the circuit so much the more 
 repulsive effect will be produced. 
 
 We can show the effect of the relative conductivity of discs 
 of equal size, and therefore of equal self-induction, by weighing 
 similar discs of various metals over an alternating pole. If we 
 take discs of copper, zinc, and brass of equal form and size, 
 and weigh these discs on the scale pan of a balance placed 
 over the pole of an alternating current magnet, the scale pan 
 being made of a good non-conductor, we can measure the 
 electro-magnetic repulsion on the disc by the loss in weight it 
 experiences.* 
 
 The same result can be illustrated by placing over the pole 
 of our alternating magnet a paper tube. Taking one of the 
 copper rings, and first exciting the magnet, we let the ring 
 drop down the tube. It falls as if on an invisible cushion that 
 buoys it up, and it remains floating in the air. If rings of 
 different metals and equal size are placed on the tube, they 
 float at different levels like various specific-gravity beads in a 
 liquid. The greater the conductivity of the ring the greater 
 is the repulsion on it in any given part of the alternating 
 field, and hence the highly conducting rings will be sustained 
 in a weaker field than the feebler conducting ring, assuming 
 the rings to have about equal weights. Moreover, we are able 
 to show by another experiment the fact that these rings are 
 traversed, when so held, by powerful electric currents. If we 
 press down the copper ring upon the zinc or brass ring floating 
 beneath it, the rings are attracted together and the copper ring 
 holds up the zinc. This is obviously because the rings are all 
 traversed by induced currents circulating in the same direction. 
 
 It is, of course, an immediately obvious corollary, from all 
 that has just been said, that any cutting of a ring or disc which 
 
 * Experiments of this kind have been made by M. Borgman. See 
 Comptes Rendus, No. 16, April 21, 1890, p. 849, and also February 3, 1890 V 
 Vol. CX., p. 233.
 
 318 MUTUAL AND SELF INDUCTION. 
 
 hinders the flow of the induced currents causes the whole of 
 the repulsion effects to vanish. We illustrate this by causing 
 a ring of copper wire to jump off the pole, and then cutting it 
 with pliers, find it has ceased to be capable of giving signs of 
 life. When the metallic masses or circuits which are pre- 
 sented to the alternating magnetic pole are of very low 
 resistance the electro-magnetic repulsion may become very 
 powerful, many pounds of thrust or push being produced by 
 apparatus of quite moderate size. It is, in fact, quite startling 
 to hold over the pole of a very powerful alternating magnet a 
 very thick plate of high conductivity copper. It would greatly 
 surprise anyone not acquainted with these principles to be told 
 that a massive copper ring weighing eight or ten pounds could 
 
 FIG. 118. Copper Ring "floating" in air over the pole of an Alternating 
 Current Electro-magnet, \vhen restrained by strings. 
 
 be made to float in the air, but it is possible to show this 
 easily. The ring needs to be tethered by light strings 
 (Fig. 118) to prevent it from being thrown off laterally, 
 although these strings in no way support its weight. 
 
 One of the most beautiful of Prof. Elihu Thomson's experi- 
 ments exhibits this effect of electro-magnetic repulsion on a 
 closed coil, which is buoyed up in water by a small incandes- 
 cent lamp in circuit therewith. In a glass vase is floated 
 a little glow-lamp like a balloon (Fig. 119). The car con- 
 sists of a coil of insulated wire, and the ends of this coil
 
 MUTUAL AND SELF INDUCTION. 319 
 
 are connected with the lamp. The whole arrangement is 
 accurately adjusted to just, or only just, float in water. 
 Placing the vase over an alternating magnetic pole, the 
 magnetic induction creates a current in the coil which lights 
 the lamp, and, moreover, the electro-magnetic repulsion on 
 the coil causes the lamp and coil to rise upward in the water. 
 There is also another class of actions namely, deflections 
 and rotations produced by electro-magnetic repulsion on 
 highly conducting discs or rings. If the conducting ring 
 or disc which is presented to the alternating magnet is con- 
 strained by being fixed to an axis around which it can rotate, 
 
 FIG. 119. Incandescent Lamp and Secondary Coil floating in water and 
 Repelled by an Alternating Current Electro-magnet placed beneath. 
 
 the action may reduce to a deflective force. On presenting a 
 flat suspended disc to the pole, the disc is prevented by its 
 constraint from being repelled bodily ; so it sets its plane 
 parallel to the lines of magnetic induction, and places itself in 
 a position such that the induced currents in it are reduced to 
 a minimum. On this principle, before becoming acquainted 
 with Prof. Elihu Thomson's original work, the author devised,
 
 320 MUTUAL AND SELF INDUCTION. 
 
 a little copper disc galvanometer for detecting small alter- 
 nating currents. 
 
 This deflection by an alternating current of a copper disc 
 suspended -within a coil with its plane inclined to the plane of 
 the coils was, in March, 1887, noticed independently by the 
 author, who subsequently described a copper disc galvanoscope 
 for alternating currents based on this fact (see The Electrician, 
 May 6, 1887). He did not at the time know how thoroughly 
 Prof. Thomson had explored the phenomena, but the sub- 
 stantial explanation of the facts as above given had already 
 occurred to him. 
 
 More interesting than the deflective actions are those which 
 result in the production of continuous rotation in highly 
 
 FIG. 120. Alternating Electro-magnet with Shaded Poles, causing a- 
 Copper Disc placed between the Jaws to revolve. 
 
 conducting bodies placed in an alternating field. We employ 
 for this purpose an electromagnet having a laminated iron 
 core (see Fig. 120), the ends of the iron circuit being provided 
 with copper bars, which embrace and cover portions of the 
 polar terminations of the magnet. When the magnet ia 
 excited by a periodic current, these secondary circuits become 
 the seat of powerful induced secondary currents. Taking in 
 hand a large copper disc pivoted at the centre and held in a 
 fork, we hold this wheel so that part of the disc is inserted 
 between the jaws of the electro-magnet. Immediately, rapid
 
 MUTUAL ANP SELF INDUCTION. 321 
 
 rotation is produced. The reason is not far to seek. The 
 alternating field creates induced currents, both in the closed 
 coils and in the neighbouring portions of the disc ; and the con- 
 ductors in which they flow are therefore drawn together. If 
 the polar coils are so placed as to partly shield the poles 
 these attractive actions act unsymmetrically on the disc and 
 pull it continuously round. The action is, perhaps, better 
 illustrated by a simpler experiment. If we hold a pivoted 
 copper disc (Fig. 121) symmetrically over an alternating 
 pole, the action of the pole is one of pure repulsion on the 
 disc, and it causes no rotation in it. When a copper sheet 
 is so placed as to shield or " shade," as Prof. Thomson calls 
 
 FIG. 121. Revolution of a " Shaded " Copper Plate held over the Pole of 
 an Alternating Current Electromagnet. 
 
 it, part of the magnetic pole, currents are induced both in 
 the fixed plate and in the movable one. The fixed disc 
 shields part of the other from the induction of the pole, and 
 I e i '-e causes the induced currents in that plate and disc to be 
 so located that they are in positions to cause continual attrac- 
 tion between the conductors and to continuously pull round 
 the movable disc into fresh positions, so creating regular 
 rotation. 
 
 This principle of " shading " a portion of a conductor from 
 the inductive action of the pole, and so causing the eddy 
 currents in it to be located in a portion of its service and tc 
 cause attraction between that conductor and the shading 
 conductor, is capable of being exhibited in various ways.
 
 322 MUTUAL AND SELF INDUCTION. 
 
 We place on a copper plate a light, hollow, copper ball (see 
 Fig. 122), and support it in a little depression in a copper 
 plate. Holding the arrangement over the alternating magnet, 
 the ball begins to spin round rapidly when the magnet is 
 excited. This rotation is caused by the continual attraction 
 of the eddy currents induced in the fixed plate and in that 
 part of the ball which is not shielded from the pole by the 
 plate. We may vary the experiment, and exhibit many more 
 or less curious and amusing illustrations of it. If we float 
 these copper balls in water (Fig. 123), and place the glass 
 bowl containing them over the alternating pole, the interposi- 
 
 Fio. 122. Light, Hollow, Copper Ball standing in a depression on th 
 edge of a Copper Plate, and set in rotation when held over the Pole of an 
 Alternating Electro-magnet. 
 
 tion of a copper sheet between the pole and the balls causes the 
 latter to begin to spin in a highly energetic manner.* 
 
 Amongst other illustrations of the principles above described 
 Prof. Elihu Thomson invented a novel form of electro- 
 magnetic gyroscope (Fig. 124). Over the alternating magnet 
 a gyroscope of the usual form is suspended. The wheel of the 
 gyroscope is made of iron, and the tyre of the wheel is a thick 
 copper band. Immediately the iron core is magnetised, the 
 
 * For a mathematical discussion of these electro-magnetic rotations, see 
 Phif.. Trans. Royal Soc., Vol. CLXXXIIU., 1892, p. 279, Mr. G. T. 
 Walker on "Repulsion and Rotation produced by Alternating Electric 
 Currents."
 
 MUTUAL AND SELF INDUCTION. 
 
 323 
 
 gyroscope begins to rotate with great rapidity over the pole. 
 In this case the unsymmetrical disposition of the eddy currents 
 in the copper band around the wheel is sufficient by itself to 
 cause the rotation to occur. The phenomenon, however, which 
 lies at the bottom of all these effects is that the self-induction 
 of the secondary circuit causes the eddy currents to be delayed 
 in phase behind the magnetising field, and hence to persist 
 into the period of reversal of that field, and so produce the 
 repulsion between the primary conducting circuit and that 
 part of the secondary conducting circuit in which the eddy 
 currents are set up. 
 
 Fro. 123. Hollow Copper Ball floating in water over an Alternating 
 Current Electro-magnet, and caused to revolve by the interposition of a 
 " shading " copper plate. 
 
 One more experiment in this part of the subject may be 
 referred to. Eeturning to the use of the electro-magnet, 
 in which the iron circuit is all but complete, we find that, 
 when a highly-conducting disc is put between the closely 
 approximated half -shielded jaws of this electro-magnet, and an 
 alternating current employed to excite it, the conducting disc 
 is held up in the air gap by reason of the attraction set up 
 between the currents induced in the disc and the shielding 
 polar plates. If, however, the disc has a relatively poor con-
 
 324 MUTUAL AND SELF INDUCTION. 
 
 ductivity the attraction is not nearly so marked. A good or 
 bad silver coin can thus be discriminated, because the good 
 silver coin has sufficient conductivity to be the seat of power- 
 ful induced currents, but the bad coin has not. 
 
 Closely akin to the foregoing, but more difficult to explain, 
 are the rotations in copper and iron discs which can be 
 caused by the approximation to them of a laminated iron bar 
 alternately magnetised. These actions have been carefully 
 studied by Prof. Elihu Thomson, and applied by him and 
 others in many practical devices. Across the top of an 
 electro-magnet is placed a long bar of laminated iron with the 
 plane of the lamination vertical (Fig. 125). This bar is 
 
 FIG. 124. Electromagnetic Gyroscope Revolving over the Pole of an 
 Alternating Current Electro-magnet. 
 
 surrounded at intervals by copper bands, which form small 
 closed secondary circuits upon it. If we excite the magnet 
 and hold near the bar an iron disc capable of free rotation, it 
 begins to rotate rapidly. Not only can this be done with a 
 laminated bar throttled by conducting circuits, but even a solid 
 bar of hard steel will serve the same purpose, and a couple of 
 steel files placed across the poles can cause rapid rotation in 
 pivoted discs of copper or of iron held with their edges close 
 to the bars so alternately magnetised. 
 
 To understand the operations which produce this rotation, 
 we have to return to some elementary principles. Consider
 
 MUTUAL AND SELF INDUCTION. 
 
 325 
 
 a conducting ring held in front of the electro-magnet as in 
 Fig. 113. Let a sudden flux of magnetic induction be made 
 through the ring aperture, that is, in common parlance, let 
 " lines of force " be thrust through the opening of the ring. 
 If these lines proceed out from a north pole, they will create 
 an electromotive force in the ring in such a direction as to 
 make a current flow counter-clockwise round the ring as seen 
 from the pole. This current in the ring itself creates a 
 magnetic field round the ring, and a consideration of the 
 direction of the current -will show that in the central aperture 
 of the ring the dire :tion of the inducing field and the field due 
 to the induced current are in opposite directions. The effect of 
 
 C 
 
 'FiG. 125. Alternating Current Magnet with Laminated Iron Bar across its 
 pole causing Revolution of two Iron Discs held near its extremities. 
 
 this opposition is to retard the formation of the field due to 
 the magnet in the aperture of the ring. In other words, 
 the current induced by the internal field causes the lines of 
 induction due to the external pole to be, as it were, momen- 
 tarily thrust out laterally, and resisted in their endeavour to 
 pass through the aperture of the ring. 
 
 If we consider a bar of iron surrounded with a copper 
 band, and imagine that this bar is suddenly acted upon by 
 a magnetising force at one end, the result of the current 
 induced in the ring will be that for a brief time the
 
 326 MUTUAL AND SELF INDUCTION. 
 
 magnetic induction in the bar will be caused to leak 
 out laterally, and go round the copper ring on the out- 
 side, its passage through the ring being resisted. If, then, 
 we throttle a magnetic circuit, such as a laminated iron 
 bar, with copper coils closed upon themselves, and place a mag- 
 netising coil at one end, the closed conducting circuits hinder 
 the rise of magnetic induction in the bar ; or, in other words, 
 give it what may be called magnetic self-induction. If the 
 source of magnetism is a rapidly-reversed pole, the consequences 
 of this delay or " lag " in the induction is that a series of alter- 
 nating magnetic poles are always travelling with retarded speed 
 up the bar, and these may be considered to be represented 
 by tufts of lines of magnetic induction which spring out from 
 and move laterally up the bar. If the bar is not laminated and 
 not throttled, the eddy currents set up in the mass of the bar 
 itself act in the same way, and operate to resist the rise of 
 induction in the bar and to delay the propagation of mag- 
 netism along it. Hence we must think of such a throttled 
 bar, when embraced by a magnetising coil at one end, as sur- 
 rounded by laterally moving bunches of lines of magnetic 
 induction, which move up the bar. Each reversal of current 
 in the magnetising coil calls into existence a fresh mag- 
 netic pole at the one end of the bar, which is, as it were, 
 pushed along the bar to make room for the pole of opposite 
 name, which appears the next instant behind it. When an iron 
 disc is held near such a laminated and throttled bar, these 
 laterally moving lines of force induce poles in the disc which 
 travel after the inducing poles, and hence the disc is continually 
 pulled round. If the disc is a copper disc, the laterally moving 
 lines of magnetic force induce eddy currents in the disc, and 
 these, by the principle already explained, create a repulsion 
 between the pole and that part of the disc in which the eddy 
 currents are set up, causing revolution of the disc. 
 
 An interesting application of the above principle has been 
 made in the meter of Messrs. Borel, Wright and Ferranti 
 for measuring alternating currents. It consists of a pair of 
 vertical electro-magnets (see Fig. 126), with laminated iron 
 cores, and each magnet bears at the top a curved horn of 
 laminated iron which is throttled by copper rings. These 
 curved horns, springing from the magnets, embrace and
 
 328 MUTUAL AND SELF INDUCTION. 
 
 nearly touch a light iron-rimmed -wheel, free to turn in the 
 centre. The actions just explained drive the wheel round, 
 when the magnet coils are traversed by an alternating current. 
 The iron wheel can* es on its shaft a set of mica vanes, which 
 retard the wheel by air friction. Under the opposing influ- 
 ences of this retardation and the electro-magnetic rotation 
 forces, the wheel takes a certain speed corresponding to 
 different current strengths in the magnetic coils, and hence 
 the total number of revolutions of the wheel in a given time, 
 as recorded by a counter, serves to determine the total 
 quantity of alternating current which has passed through 
 the meter. 
 
 The rotation of iron discs can be shown also by means of a 
 badly-designed transformer. If a closed laminated iron ring 
 
 FIG. 127. Magnetic Leakage across a Throttled Iron Ring, causing rotation 
 of an Iron Disc placed near the Secondary Coil. 
 
 (Fig. 127) is wound with a couple of conducting circuits, such 
 an arrangement constitutes a transformer. If these two 
 circuits are wound on opposite sides of the iron ring, the 
 previous explanations show that the arrangement will be 
 productive of great magnetic leakage across the iron circuit. 
 In designing transformers for practical work,' one condition 
 amongst others which must be held in view is to so arrange 
 the conductive and magnetic circuits that a great magnetic 
 leakage of lines of force across the air does not take place. 
 If, however, this leakage exists, it indicates that the secondary 
 circuit is not getting the full benefit of the induction created 
 by the primary. To detect it we have merely to hold near the
 
 MUTUAL AND SELF INDUCTION. 329 
 
 iron circuit a little balanced or pivoted iron disc, and if it is 
 set in rapid rotation it indicates that there are laterally- 
 moving lines of magnetic force outside the iron, which have 
 escaped from the iron in consequence of the back-magneto- 
 force of the secondary circuit. 
 
 The above described phenomena have been utilized in the 
 construction of measuring instruments of various kinds, and 
 the effects due to the magnetic leakage of magnetic fields will 
 be found to have applications which will be considered more 
 carefully in discussing the action of transformers. 
 
 of the effects described in the present chapter will have dis- 
 closed to the careful reader that there is a complete symmetry 
 between the two fundamental quantities, electric current 
 and magnetic induction. Let the diagrams in Fig. 128 
 represent a circuit of iron (magnetic circuit) linked with 
 a circuit of copper (conductive circuit) and let the iron circuit 
 have wound on it a magnetising coil capable of imposing a 
 magneto-motive force (M.M.F.) on it, whilst the conductive 
 circuit has a source of electromotive force (E.M.F.), say a 
 battery, introduced into it. Suppose, then, that this E.M.F. is 
 suddenly introduced in the conductive circuit, the linking of 
 this circuit with the iron circuit bestows considerable self- 
 induction on the conductive circuit, and this operates to delay 
 the rise of the current strength in the conductive circuit when 
 the E.M.F. is suddenly applied. If the two circuits were 
 plunged into a good conducting liquid medium, the action of 
 the iron circuit would be to cause a leakage of current across 
 the conductive circuit. Quite similarly, we find that if a 
 magnetomotive force is suddenly applied to the iron circuit, 
 the induced current set up in the conductive circuit opposes 
 the growth of the induction in the magnetic circuit, and, as 
 air is not an insulator for magnetic induction, it causes a 
 leakage of magnetic induction across the circuit. Hence the 
 growth of electric current in the conductive circuit is hindered 
 by linking it with an iron circuit, and the growth of magnetic 
 induction in an iron circuit is hindered by linking it with a 
 copper circuit. This is only one instance of the fact that the 
 laws of current establishment in conductive circuits are similar 

 
 a^U MUTUAL AND SELF INDUCTION. 
 
 to the laws of establishment of magnetic induction in magnetic 
 circuits. 
 
 The growth of current from surface to centre of con- 
 ductors has been described in sections of this chapter. The 
 gradual soaking in, or growth of the magnetic induction, from 
 the surface to the centre of the iron cores of electro-magnets, 
 when a sudden external magnetising force is applied, has been 
 experimentally examined with great skill by Dr. J. Hopkinson 
 and Mr. E. Wilson, and the reader is referred for a full account 
 
 Electric Ci 
 
 Magnetic Circuit. 
 
 Copper 
 
 Iron. 
 
 FIG. 128. Diagrams illustrating the Symmetry in relation between 
 Electromotive Force and Electric Current, and Magnetomotive Force 
 and Magnetic Induction. 
 
 of their work to The Electrician, Vol. XXXIV., 1895, p. 510. 
 Very briefly it may be said that these experiments consisted in 
 showing experimentally that in the case of an electro-magnet 
 with a very large solid iron core the magnetic induction in 
 the iron is not established instantaneously at its full value 
 at all points in the iron when the magnetising force is
 
 MUTUAL AND SELF INDUCTION. 331 
 
 applied, but it begins at the surface of the core and slowly 
 works inwards to the centre. In an entirely similar manner 
 we know that in a conductor of large cross-section the 
 actions involved in the production of a current in the 
 conductor establish the current first at the surface of the 
 conductor, and the central portions of the conductor are 
 only reached by it after a certain finite time. We shall 
 return in the next chapter to the discussion of some of these 
 theoretical Questions.
 
 CHAPTEE Y. 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 1. Electromagnetic Theory. In the matter so far before 
 the reader attention has been directed to the chief facts ot 
 electromagnetic induction without any inquiry into the 
 possible mechanism by which this may be effected. Attention 
 may at this stage be directed to modern views of the subject, 
 which have been the outcome of the work of Faraday and 
 Maxwell and all their illustrious followers in this field of 
 study. The cardinal principle of these methods of viewing 
 the phenomena is the denial of action at a distance. That is 
 to say, if at any point in a field we find a force due to a 
 current flowing in some conductor, this force cannot be 
 regarded as appearing there without anything happening in 
 the interspace, but must be the consequence of successive 
 changes in closely contiguous places, and not the result of 
 operations at a distance without intermediate machinery. 
 Whenever we find an electromagnetic effect taking place 
 at any locality we are directed therefore by these notions 
 to look for its antecedents or consequences at the adjacent 
 places, and the apparent phenomenon is not to be regarded as 
 the whole of it, but to be taken as a portion of the whole 
 of the effects which are produced in every part of the region 
 or medium. The finite velocity of light, and the impossibility 
 of accounting for its propagation on any other hypothesis than 
 that of actual transmission of something across space, or the 
 propagation of a state of stress and strain or periodic change 
 of some kind through a medium, led to attempts to settle 
 between the rival hypotheses by crucial experiment, with the 
 result that the vast bulk of the accumulated evidence decides
 
 DYNAMICAL THEORY OF INDUCTION, 333 
 
 in favour of the existence in space of a medium which has 
 properties not possessed by the ordinary atomic matter, hut 
 which may certainly be called a material substance in the 
 sense that it can be the recipient or vehicle of energy. The 
 study of the phenomena of liyht indicates that along the path 
 of a ray there are certain changes which are periodically 
 repeated, such that at portions of the medium separated by a 
 distance called a wave-length, changes of a similar kind are 
 being coincidently effected. The application of mathematical 
 analysis to optical phenomena has led to the conclusion that 
 we can offer a tolerably satisfactory account of them by 
 making the supposition that there exists such a universally 
 diffused ether or medium in which these changes go on. At 
 this point, however, the profound difficulties of the subject 
 begin. To offer a complete account of the phenomena of light, 
 and to deduce all the observed effects from a fundamental 
 principle, we have to construct a hypothesis as to the structure 
 of this ether and the nature of the periodic changes which 
 constitute the wave motion in it. The periodic changes which 
 in the case of sound and fluid waves are known to exist 
 suggested that in the case of the ether the periodic changes 
 are motions of the parts of the ether relatively to one another, 
 and that these motions are the result of displacements taking 
 place under certain stresses. We cannot even attempt here a 
 sketch, however brief, of the various hypotheses which have 
 formed as to the sort of motions which may occur. On one 
 assumption the ether has been regarded as capable of having 
 displacements or deformations made in it against internal 
 forces, resisting these changes similar to the shearing strains 
 and stresses in solids. From this point of view, now some- 
 times called the elastic solid theory of light, we may picture 
 this ether to ourselves as a distortable but incompressible 
 jelly-like solid, which exists everywhere and penetrates into 
 the interior of all material bodies. As long as the hypothesis 
 of a universal ether was demanded merely to correlate the 
 observed phenomena of light a limited order of facts had 
 alone to be considered ; but the conception that electric and 
 magnetic effects also required a similar assumption, increased 
 the difficulties to be dealt with. The mind of Faraday con- 
 tinually turned to the thought that the medium assumed
 
 334 DYNAMICAL THEORY OF INDUCTION. 
 
 in both these regions of phenomena might be the same ; and 
 his great disciple, Maxwell, was led more definitely to formu- 
 late a similar conception. If an ether or medium is demanded 
 as a fundamental cause of two or more classes of facts, then it 
 is certainly unscientific to fill space several times over with 
 ethers of different kinds until the attempt has been made 
 to ascertain if one alone cannot be found to fulfil all the 
 required functions. Maxwell was led therefore to the con- 
 clusion that both luminous, electric, and electromagnetic 
 phenomena might be explained by the supposition of one 
 single medium capable of certain internal changes, and 
 possessing certain mechanical properties, and he thus avoided 
 the unscientific process of thought of postulating two different 
 ethers by boldly adopting the hypothesis that the medium 
 on which electric effects and optical phenomena depend for 
 their existence is one and the same. We shall see later how 
 this supposition has been supported. 
 
 One important element in Maxwell s electric theory is his 
 conception of electric displacement. When an electromotive 
 force acts upon any part of a dielectric which is uniform and 
 non-crystalline it is assumed that at all points along the line 
 of electrostatic induction there is an electric displacement, as 
 Maxwell calls it. The theory does not tell us what is the 
 physical nature of this displacement. We may, in the first 
 place, merely for the purpose of illustration, suppose that the 
 unknown something which we call electricity is moved along 
 a line of induction, and that on the removal of the electric 
 force it returns to its original position, and that a dielectric 
 or insulator is a material in which the electricity, when dis- 
 placed by the application of an electrostatic stress or force, 
 resists this displacement in virtue of an electric elasticity. The 
 apparent charge on conductors, according to this view, is the 
 electricity displaced out of, or into, the dielectric, and positive 
 charge or electrification may be regarded as the possession of 
 an excess which is extruded from the dielectric on to the con- 
 ductor, and negative as a deficit when the conductor gives up 
 some to the dielectric. 
 
 Maxwell's next principle is that change in electric displace- 
 ment is an electric current whilst the change lasts. He calls 
 this a displacement cutrent, to distinguish it from a current in
 
 DYNAMICAL THEORY OF INDUCTION. 335 
 
 conductors called the conduction current. The displacement 
 current is supposed to have, however, all the properties of an 
 electric current. Conducting bodies must he regarded as 
 those in wh'ch there is no elastic force resisting displacement, 
 or, in other words, have no electric elasticity, and in which, 
 therefore, electric displacement can go on continuously. The 
 existence of a current of conduction is recognised by two 
 co-existing effects first, the dissipation of energy into heat ; 
 and second, the existence of magnetic force the direction of 
 which is along closed lines described around the line of the 
 current. The displacement current in dielectrics, which takes 
 place at the instant of applying or changing the electric force, 
 is also considered to be accompanied by magnetic force. In 
 other words, we must consider the displacement current which 
 takes place in a dielectric when electrostatic force acts on it 
 as a very brief conduction current, and as originating a system 
 of lines of magnetic induction surrounding it, just as a con- 
 ducting wire is so surrounded, by its loops or closed lines of 
 magnetic induction. Conversely, when lines of magnetic 
 force penetrate through an insulator or dielectric, any change 
 in the density of these lines creates eddy displacement cur- 
 rents in the mass. If the lines penetrate through a conductor 
 they produce, under similar circumstances, eddy currents of 
 conduction, whose energy is ultimately frittered down into 
 heat. Also, if a conductor is moved across a magnetic field 
 so that it " cuts " lines of induction we have seen that if the 
 conductor is a portion of a closed circuit it has a current of 
 conduction produced in it. Similarly, if a dielectric body is 
 moved in a magnetic field in a like manner it has during the 
 continuance of the motion a displacement current produced in 
 it. Since a dielectric circuit is always a closed circuit, a dis- 
 placement current, or the production of electric displacement 
 in it is always the result of any change in the magnetic 
 field in its interior. For the purpose of obtaining a rough 
 illustrative working model of the actions going on in a 
 dielectric submitted to the action of electric force, it is 
 necessary to fall back on some material, hypothesis of elec- 
 tricity that is, we must conceive of electricity as a something 
 which can be displaced relatively to the molecules of the 
 dielectric, and that it resists this displacement, and that
 
 336 DYNAMICAL THEOfiY OF INDUCTION. 
 
 when this displacement is made under the action of electric 
 stress the removal of this stress causes a disappearance of the 
 displacement. Dr. Lodge has suggested a form of apparatus 
 serving as a rough working model of this dielectric action, 
 iA which buttons sliding along a rod, and held in certain 
 positions by elastic strings, represent the electric particles 
 capable of elastic displacement.* 
 
 We may quant tatively define electric displacement by saying 
 that in a homogeneous non-crystalline dielectric, if a plane be 
 drawn perpendicular to the line of action of the resultant 
 electric force, then under th.3 operation of this electric force the 
 quantity of electricity displaced normally across a unit of area 
 of this plane is called the electric displacement. This displace- 
 
 FIG. 129. 
 
 ment is of the nature of an elastic strain, and is removed when 
 the electric force is removed. Let us fix our ideas by imagin- 
 ing a sphere immersed in a dielectric medium to receive an 
 electric charge of quantity Q. Suppose this sphere to be sur- 
 rounded by a concentric spherical shell (Fig. 129) also immersed 
 in the dielectric. On giving the central sphere a charge + Q 
 we know that on the inside surface of the insulated concentric 
 shell will appear an inductive charge - Q of equal quantity 
 and opposite sign, and a charge -f Q on the outside surface. 
 
 * See Dr. 0. J. Lodge "On a Model Illustrating Mechanically the 
 Passage of Electricity through Metab and Dielectrics," Phil. Mag. t 
 November, 1876.
 
 DYNAMICAL THEORY OF INDUCTION. 337 
 
 Let this spherical shell be very thin and be placed at a distance 
 > from the central sphere, supposed to be very small. The 
 electric force due to the central charge Q at the surface of the 
 
 concentric shell is represented by . , and this force exerts a 
 
 displacing action on the electricity of the shell, causing posi- 
 tive electricity to be displaced outwards or in the direction of 
 the force and negative electricity to be displaced inwards or 
 against the force. 
 
 The quantity K which appears in the above expression for 
 the magnitude of the electric force is called the dielectric con- 
 stant, or the specific inductive capacity, according to Faraday, 
 of the medium. If the dielectric is air or other gas, K is very 
 nearly unity, and the law of the force becomes the ordinary 
 Newtonian law, viz., force varies as quantity divided by square 
 of distance that is, the electric force at any point due to a 
 small quantity, Q, collected on a sphere is numerically equal 
 
 to - , where r is the distance of the point from the centre of 
 
 the sphere. The quantity K is assumed to have a value of 
 unity in the case of a vacuum, and varies for known dielectrics 
 from a little above unity for dry air up to a value of 6 to 10 for 
 glass. In the case of metals and conducting bodies we may 
 consider K to be infinitely great, and generally K is a number 
 which expresses the ratio of the displacement in the given 
 dielectric to the displacement which would take place under the 
 same electric force if the dielectric was removed and a vacuum 
 left in its place. The whole quantity of electricity displaced out- 
 wards through the conducting shell is + Q, and since the radius 
 of this spherical shell is r, its surface is 4?r r 2 and the quantity 
 displaced through unit of area of this shell in the direction of 
 
 the force is JSL. This quantity, then, Maxwell calls the 
 47T r* 
 
 electric displacement, and denotes by the symbol D. 
 
 The electric force or resultant electric intensity at all points 
 
 over the spherical shell is , and this quantity Maxwell calls 
 Kr 2 
 
 the electromotive intensity at that point, and denotes it by E 
 We may also speak of D as the electric strain and E as the 
 electric stress by an extension of usual mechanical language.
 
 338 DYNAMICAL THEORY OF INDUCTION. 
 
 The quotient of a stress by its corresponding strain is, in 
 mechanics, called the coefficient of elasticity corresponding to 
 that stress. For instance, the quotient of stretching force by 
 longitudinal extension in the case of solid rods subjected to 
 extending forces is called Young's Modulus of Elasticity, or 
 the longitudinal elasticity. By a similar use of language the 
 quotient electric stress by electric strain may be called the 
 electric elasticity, and we have 
 Q 
 
 -- = - = tJie electric elasticity. 
 
 Q 
 
 47T7-2 
 
 Hence the series of numbers obtained by dividing the 
 number 4 x 3'1416 by the specific inductive capacities give a 
 series of numbers which are the electric elasticities of these 
 substances.* 
 
 2. Displacement Currents and Displacement Waves. 
 Maxwell's second fundamental conception, as we have men- 
 tioned, is that a displacement of electricity whilst it is taking place 
 is an electric current. That is to say, the variation of displace- 
 ment, whether of increase or decrease, is a movement of elec- 
 tricity which is in effect an electric current. A dielectric must, 
 however, be considered as a body which does not permit any 
 but a very transient electric current passing through it. If 
 continuous electric force is applied to it the dielectric is soon 
 .strained to its utmost extent, and no more current or flow or 
 displacement takes place through it until the sign or direction of 
 the electric force is reversed. A dielectric may be considered 
 to be pervious to very rapidly reversed periodic currents, but 
 very opaque or impervious to continuous currents. This is 
 familiarly illustrated by the fact that a condenser inserted in a 
 telephonic circuit does not stop telephonic communication, but 
 .does stop continuous currents. If D be at any instant the 
 displacement at any point in a dielectric, and if D varies with 
 
 * The occurence of this 4^r in electric and magnetic equations is an 
 objection from some points of view. Mr. Oliver Heaviside has discussed 
 the subject fully in his writings in The Electrician, and proposed a system 
 of rational units in which it is suppressed.
 
 DYNAMICAL THEORY OF INDUCTION. 339 
 
 the time so that - is its time rate of variation, then ( , or 
 
 d t dt 
 
 as it may be best written in Newtonian fashion D, is the dis- 
 placement current, or rate of change of displacement. If at 
 any point in a dielectric rapid changes of displacement are 
 taking place, these variations of electric displacement are in 
 effect electric currents producing magnetic induction in the 
 surrounding portions of the dielectric. When we come to 
 discuss the investigations of Hertz we shall see that this view 
 receives support from experimental research. An electric 
 displacement taking place all along a certain plane is equi- 
 valent to a current sheet, and an electric displacement taking 
 place along a certain line is a linear current. Electrostatically 
 speaking, lines of electric displacement are lines of electro- 
 static induction, and these lines, when the displacement is 
 changing, become lines along which electric current flow is 
 taking place. The denial of action at a distance involves the 
 assumption that the only portions of a dielectric which can act 
 directly upon each other are those which are in immediate 
 contact or are contiguous. 
 
 3- Maxwell's Theory of Molecular Vortices. Given a 
 medium possessing certain mechanical qualities, such as elas- 
 ticity, a definite density, a capability of relative displacement 
 of its parts, we may ask, is it possible to imagine a structure 
 which will mechanically account for the effects we have to 
 consider in electrical phenomena ? A full discussion of ether 
 theories is not possible here, but it may be of assistance to the 
 student to place before him a general account of one such 
 attempt to construct a mental imagery of its mechanism. 
 We should always remember, however, that even if we are 
 able to imagine a mechanism capable of producing even all 
 the effects we find in Nature in any region of fact, it does 
 not in the least follow that the real state of affairs agrees with 
 our conception of it. Maxwell put forward his theory of 
 Molecular Vortices in the Philosophical Magazine for 1861 and 
 1862. A general account of this theory has been given in the 
 " Life of James Clerk Maxwell," and as the limits of such an 
 elementary treatise as the present one preclude any detailed 
 account of the mathematical portion of this theory, we shall 
 
 z2
 
 340 DYNAMICAL THEORY OF INDUCTION. 
 
 borrow the language of the authors of the above-mentioned 
 work* hi describing it. Maxwell supposes that any medium 
 which can serve as the vehicle of electro-magnetic energy 
 consists of a vast number of very small bodies called cells, 
 capable of rotation, which we may consider to be spherical,. 
 or nearly so, when in their normal position. When magnetic 
 force is transmitted by the medium or acts through it, these 
 cells are supposed to be set in rotation with a velocity propor- 
 tional to the intensity of the magnetic force, and the direction 
 of rotation is related to the direction of the force in the same 
 manner as the twist and thrust of a right-handed screw- We 
 have thus all the magnetic field filled with molecular vortices, as 
 Maxwell calls them, all rotating round the lines offerees as axes. 
 These cells as they revolve tend to flatten out like revolving 
 spheres of fluid, and to become oblate spheroids ; they thus con- 
 tract along the lines of force and expand at right angles, creating 
 a tension along the lines of force, and a pressure at right angles 
 to them. These cells are supposed to be elastic spheres closely 
 packed together and incapable of separating from each other. 
 If any line of cells is set rotating the contraction of each cell 
 along its axis of revolution must set up a tension or pull along 
 that line, it behaves like a filament of muscular tissue, and 
 contracts in length and swells out or increases in thickness. If 
 several adjacent lines of cells or vortices are all set revolving; 
 in the same direction, the swelling out of each line causes them 
 to press on each other ; hence there is a lateral pressure and a 
 longitudinal tension. In any space filled with these cells so- 
 revolving the lines of tension or axes of revolution of the cells 
 will take up certain positions, depending on the necessity exist- 
 ing for the stresses to adjust themselves to equilibrium, and 
 Maxwell has shown mathematically that such a system of 
 cells in tension and pressure is a system which will behave- 
 in a manner similar to that in which we find actual lines of 
 magnetic force do, and that the behaviour of magnetic poles 
 to each other can be explained fully by the assumption of a_ 
 
 * "Life of James Clerk Maxwell." By Lewis Campbell and William. 
 Garnett. 1st Edition. 1882. The general description of Maxwell's views 
 in the above-mentioned work is due to Prof. Garnett, and in the annexed 
 paragraphs the expository account of this theory is taken in part almost- 
 verbatim from the pages of this book.
 
 DYNAMICAL THEORY OF INDUCTION. 341 
 
 tendency on the part of the lines of force between them to 
 contract like elastic threads along their length, and to push 
 one another apart when laid parallel and proceeding in the 
 same direction. To account for the transmission of rotation 
 from one cell to another in the same direction, and from one 
 line of cells or vortex to the next, Maxwell supposed that 
 there exists bet-ween the cells a number of extremely minute 
 spherical bodies which can roll without sliding in contact 
 with the vortex cells. These bodies serve the same purpose 
 as "idle wheels" in machinery, which coming between a 
 driving wheel and a following wheel serve to cause both to 
 turn in the same direction. These minute spheres Maxwell 
 supposed to constitute electricity. We shall speak of them 
 collectively as the electric matter. These electric particles 
 are furthermore supposed to be free to move in conductors ; 
 but in dielectrics they are tethered to one spot, or rather into 
 one molecule of the substance, and can only be displaced a 
 little way against an elastic resilience, which brings them 
 back to their original position when the displacing force is 
 withdrawn. Furthermore, we must assume that both cells 
 and particles are very small, compared with the molecules of 
 matter. The passage of electric particles from molecule to 
 molecule in conductors, however, sets up molecular vibration, 
 or generates heat. Something of the nature of friction must, 
 therefore, be also postulated to account for the fact that the 
 electric particles, when set moving in a conductor, give up 
 energy to the molecules, and the energy is in them dissipated 
 in the form of heat. That there is some kind of rotation 
 going on along the lines of magnetic force has been held by 
 Maxwell to be indicated by the behaviour of a ray of polarised 
 light when passing through a dielectric along a line of 
 magnetic force, and he states* that Faraday's discovery of the 
 magnetic rotation of the plane of polarised light furnishes 
 complete dynamical evidence that wherever magnetic force 
 exists there is matter small portions of which are rotating 
 ! about axes parallel to the direction of that force. The further 
 , assumption is made that the cells are composed of an elastic 
 ; material, and that they can be distorted or squeezed slightly, 
 I returning again in virtue of their resistance to their original 
 * Article "Faraday," Encyclopaedia ritannica, 9th Edition.
 
 342 DYNAMICAL THEORY OF INDUCTION. 
 
 form. In order to obtain a clear conception of (he inter- 
 relation of the idle wheels or the electric particles and the 
 revolving cells or lines of induction, we may construct a 
 mechanical illustration of one element of the mechanism as 
 it is supposed to exist in the dielectric. 
 
 Consider A and B (see Fig. 130) to he two wheels of india- 
 rubber, and that C is another small wheel lying between A 
 and B and transmitting motion from one to the other. Let C 
 be tethered to a fixed point, D, by an elastic spring, and let C 
 be at the same time capable of rotation round its centre. 
 Suppose A is set in rotation, clock-hand wise, whilst B is held 
 fast, and that the wheel C cannot slip on A, the result will be 
 to drag down C to the position of C 1 , stretching the spring and 
 displacing C. Let B be then set free ; the wheel C continues 
 to roll on A, and transmits its rotation to B. Owing to the 
 
 FIG. 130. 
 
 assumed elasticity of the discs A and B, the wheel C can be 
 drawn down between them, and yet within the limits of its 
 displacement equally transmit the rotation of A to B without 
 slip. The same action of a preliminary displacement of C and 
 subsequent rotation of B will take place if the wheel B possesses 
 inertia that is, if we assume it to be a heavy wheel which 
 cannot in virtue of its mass be set rolling with finite speed 
 in an infinitely short time. 
 
 If, then, we suppose a long row of such wheels with inter- 
 mediate displaceable idle wheels, the main wheels being heavy 
 bodies, the result of causing the first wheel to rotate would be 
 to propagate along the line a successive displacement of the 
 idle wheels, and to set the main wheels successively in rota-
 
 DYNAMICAL THEORY OF INDUCTION. 343 
 
 tion. Translating these mechanical concepts into their elec- 
 trical equivalents, Maxwell considers that the heavy wheels 
 are the analogues of the molecular vortices or lines of force, 
 and that their density is determined by what we call the 
 magnetic permeability of the medium ; the elastically dis- 
 placeable idle wheels are the electricity in the dielectric ; and 
 that when a line of force is brought into existence in a 
 dielectric, or, in other words, when a line of cells is set 
 rotating, this action propagates itself outwards, producing 
 successive displacements of the electric particles, or generates 
 a displacement wave, and is accompanied by the successive 
 appearance of rotation in the cells, or by the propagation of a 
 wave of electromagnetic force. 
 
 The velocity of propagation of this wave will depend on the 
 elastic forces restraining displacement, and on the inertia of 
 the revolving vortices. We have seen that the elasticity of 
 
 the dielectric is expressed by the quantity 7r> where K is the 
 
 K 
 
 specific inductive capacity. We shall see later on that the 
 electromagnetic density of the medium is expressed by 4ir/i, 
 where //, is the magnetic permeability. 
 
 The velocity of propagation of a disturbance through an 
 elastic medium is numerically equal to the quotient of the 
 square root of its effective elasticity e, by the square root of 
 
 its density d, or by v. 
 v d 
 
 If, then, for the electromagnetic medium e = and 
 
 1 
 
 d = 4:r p., we have v = , , or the velocity of lateral propaga- 
 
 tion of a wave of electric displacement or of magnetic force in 
 a medium is numerically equal to the square root of the 
 reciprocal of the product of its specific inductive capacity and 
 its magnetic permeability. Such a mechanical hypothesis 
 shows us how the spin of one line of vortices results in pro- 
 ducing displacement of the idle wheels or electricity along 
 lines which are circles described round the initial vortex as 
 axis, and in propagating outwards the vortex spin or mag- 
 netic force with a finite velocity from one line of molecular 
 vortices to another.
 
 344 DYNAMICAL THEORY OF I 
 
 By the aid of the ideas which were discussed in the last 
 section we are enabled to arrive at a mechanical conception 
 which helps us to connect together observed facts, and which, 
 even if not a real representation of what is taking place, is 
 at least a working model, which may assist us to correlate 
 the actions taking place when an electric current is started 
 in a wire. 
 
 An electric current on this hypothesis is a flow or pro- 
 gression of the electric particles which are free to move 
 forward in a conductor, and which only can move steadily 
 forward, owing to their incompressibility, when the circuit in 
 which they flow is a complete circuit. Suppose a thin con- 
 ductor bent into the form of a very large circle, and that an 
 electromotive force urges a procession of electric particles 
 round it. As these particles go forward they cause the electric 
 cells next them to rotate, and the motion of this line of cells 
 embracing the line of current will be just like that which 
 would take place if a bracelet of spherical beads strung on an 
 elastic thread were rolled along a round rod which it closely 
 embraces. Each bead would turn over and over, rolling on 
 the rod, and the motion of the whole bracelet would be like 
 that of a tightly-fitting india-rubber umbrella ring pushed 
 along a round ruler. The progression of the electric particles 
 would start circular vortex rings revolving round the line of 
 motion. This corresponds to the fact that a linear current 
 creates a magnetic field composed of circular embracing lines 
 of forces. The first or adjacent line of vortices would, by the 
 intervention of the idle wheels, set in rotation another set of 
 cells lying on a concentric line, and cause them to rotate in the 
 same manner as the first ones. Also, it would cause a back- 
 ward displacement of the intermediate idle wheels, if we con- 
 sider that only the central line of electric particles are conduct- 
 ing matter, and that the next and all succeeding rows are in 
 a dielectric. The starting of the progressive movement of 
 the line of electric particles in the conductor will result 
 in an elastic displacement in the opposite direction of all 
 surrounding electric particles in the dielectric along lines 
 parallel to the line of current ; and also in setting up a 
 system of molecular vortices composed of revolving cells, the 
 axes of these vortices being co-axial circles described round
 
 DYNAMICAL THEORY OF INDUCTION. 345 
 
 the line of flow, the rotations and displacements being propa- 
 gated out laterally from the line of current. In consequence 
 of the fact that the revolving cells are supposed to possess 
 inertia or mass, and that all the mechanism is supposed to be 
 rigidly connected together, a steady force applied to set the 
 central line of electric particles in motion will not be able to 
 produce in them the full velocity until time has elapsed suffi- 
 cient to allow the inertia of the connected mechanism to be 
 overcome. We are thus able mechanically to imitate the 
 phenomena of self-induction of the circuit and the gradual rise 
 of current strength in an inductive curcuit under the operation 
 of a steady impressed electromotive force, and to deduce it as 
 a consequence of the fundamental hypothesis. 
 
 Oar theory, then, points out that a current should rise 
 gradually in strength, and also that the embracing lines of 
 
 B 
 
 magnetic force must be considered to come into existence 
 successively as the rotation is taken up in ever-widening 
 circles by the molecular vortices successively receiving motion 
 of rotation. Also, on withdrawing the impressed electromotive 
 force the inertia of the mechanism tends to make it run on 
 for a little and the electric particles, which by their motion 
 started the vortices, are now themselves urged forward for a 
 little in the same direction, and this constitues the extra 
 current at "break." 
 
 ' Let us next endeavour to see what ought to happen on the 
 supposition that there are two conducting circuits in the field, 
 both forming closed circuits, and to one of which an impressed 
 electromotive force can be applied. LetV 1 ,V 2 ,V s ,&c. (Fig. 131),
 
 346 DYNAMICAL THEORY OF INDUCTION. 
 
 represent the sectional view of a series of vortex lines of electric 
 cells, and let I 1? I 2 , 1 3 , &c., be the idle wheels or electric particles. 
 Let the row of electric particles I a be supposed to be lying 
 inside a conducting circuit, A, represented by the dotted lines, 
 and by our fundamental supposition, the particles I a are quite 
 free to move along the conductor, and to rotate on their axes. 
 Let there be another conductor, B, placed parallel to A, and 
 let I 5 be the electric particles in it. The space C between is 
 supposed to be occupied by a dielectric, and in it the electric 
 particles can only be displaced elastically from a fixed position. 
 We may regard these idle wheels I 2 1 3 1 4 as tethered by springs- 
 to one spot. Such being the mechanism, imagine that the row 
 of particles I x is urged forward in a downward direction. As 
 the row of particles pass between the cells V x V 2 they will set 
 them in rotation in opposite directions. Owing to the inertia 
 of the vortices thi first effect of the rotation of V 2 will be 
 to cause I 2 to roll over V 3 and be displaced in an upward 
 direction ; its disph cement is resisted by the elastic force of 
 the spring. The rotation of 1^, however, sets V 3 in rotation, 
 and after a short interval V 3 is rotating at the same speed and 
 in the same direction as Y 2 . I 2 then ceases to be displaced, 
 because the action of V 2 on I 2 , and the reaction of V 3 on 1^ 
 simply amount to a couple or twist on I.,. The same sort of 
 action results in a gradual handing on of the rotation from 
 vortex to vortex, and a propagation of displacement from 
 one idle wheel to the other. When the motion reaches the 
 conductor B, the first result is to cause a displacement of 
 the electric particles upwards, the rotation of V 5 not being 
 instantly acquired by V 6 . This amounts to a current in the 
 upward or opposite direction. As soon, however, as the vortex 
 V 6 has accepted the full speed of rotation, then the forces 
 on the electric particles I 5 amount only to twists, and not 
 to forces of displacement ; hence the particles I 5 cease to 
 experience any forca impelling them forward, and come to rest 
 in virtue of the fact that the conductor offers a resistance to 
 their motion. They fritter down their energy of motion into 
 heat, and come to rest. Hence the induction current in the 
 conductor after a short flow ceases, and the vortex spin becomes 
 equal in the vortices on either side of it. Suppose now that 
 the impressed force in the circuit A is withdrawn, the electric-
 
 DYNAMICAL THEORY OF INDUCTION. 347 
 
 particles in the A circuit are driven forward for a short time 
 by the energy stored up in the adjacent vortices ; these last, 
 however, give up one by one their energy to the circuit A, 
 where it is dissipated as heat. This surrender of velocity is 
 propagated outwards until at the surface of the circuit B the 
 state of things finally is, that when the vortex V 5 has come 
 nearly to rest, the motion of V 6 still continues. The energy of 
 V 6 and of vortices beyond expends itself in moving forward the 
 electric particles in circuit B in the same direction as that in 
 which the current in A was travelling originally in other 
 words, part of the energy of the field is spent in making a 
 transitory current in B as well as in A in the same direction. 
 It follows, therefore, that there is a less induction current in 
 A at breaking circuit when a closed circuit B is present 
 than if B were not there that is to say the presence of a 
 closed secondary circuit B diminishes the self-induction of 
 the primary circuit, as is known to be the case. We see, 
 therefore, that the theory is so far in accordance with observed 
 facts. 
 
 The theory must, however, be taken for no more than it is 
 worth, viz. : an attempt to construct a mechanical system 
 which shall act in the manner in which we find electro-magnetic 
 fields and circuits do act. The true mechanism may be 
 very different; the one described * has at least the utility 
 that it shows a way in which the observed effects might be 
 produced. The various dynamical elements in the supposed 
 mechanism have their equivalents in the recognised electrical 
 and electro-magnetic qualities. The angular velocity of the cells 
 or vortices around their axis represents the intensity of the mag- 
 netic force, or the strength of the magnetic field. The angular 
 momentum of the vortices represents the magnetic induction, 
 hence the mass of each cell, or the density of the medium, is the 
 analogue of the magnetic permeability. This is greater in 
 paramagnetic substances than in air or vacuum, and greatest of 
 all in iron ; in fact, so exceptional is it in iron that Maxwell 
 supposed the particles of the iron themselves to take part in 
 the vortex action. Hence, the energy of a magnetic field is 
 greater if that field contain iron, and accordingly the presence 
 of iron in a core immensely increases the vortex energy for a 
 given vortex velocity, that is, it increases the inductance of
 
 348 DYNAMICAL THEORY OF INDUCTION. 
 
 the circuit. The energy associated with any revolving cell or 
 vortex is proportional to the product of its velocity and 
 momentum, or the product of the magnetic force, and the 
 magnetic induction estimated in the same direction is a 
 measure of the energy per unit of volume existing in that 
 portion of the field. The " number of lines of force " passing 
 through any circuit is on this theory to be identified with the 
 whole momentum of the molecular vortices linked with that 
 circuit. If any circuit is traversed by lines of force or linked 
 with lines of molecular vortices, and the cause creating this 
 field is removed, say, by withdrawing the magnet or repressing 
 the electric current creating it, the vortices give up their energy 
 gradually to this secondary circuit, and it appears there as 
 energy of motion of the electric particles or as an electric cur- 
 rent. When one system of bodies iu motion sets another set in 
 motion by mutual action and reaction, and there is no loss of 
 energy by anything like friction or imperfect elasticity, then 
 the momentum gained by one must be equal to thai lost by 
 the other, and the rate of gain of momentum of the one 
 system is at any instant equal to the rate of loss of momentum 
 by the other. Hence, if the vortices lose momentum their 
 rate of loss of momentum that is, the rate of withdrawal of 
 lines of induction from the circuit, must he equal to the rate 
 of gain of momentum of, or to the force acting on, the electric 
 particles which are absorbing the momentum. Hence we see 
 that the impressed electromotive force in the circuit must be 
 equal to the rate of withdrawal of lines of induction, and the 
 theory conducts us to Faraday's law of induction, as a 
 necessary dynamical consequence of our fundamental assump- 
 tion. Maxwell has extended the theory of molecular vortices 
 to the explanation of electrostatic phenomena, with which we 
 are not, however, here directly concerned. We have seen 
 that the theory is capable of affording an explanation on 
 mechanical principles, of self-induction, mutual induction, and 
 the law of electro-magnetic induction. In order to complete 
 the theory as far as regards the phenomena of magnetism, it 
 is necessary to suppose that the particles of magneti sable 
 metals, such as iron, are set in rotation by the molecular 
 vortices which traverse them, and that an increase of speed of 
 these vortices does not increase proportionallv the rotation of
 
 DYNAMICAL THEORY OF INDUCTION. 349 
 
 the iron molecules. These last behave like wheels slung 
 loooely on a shaft, between which shaft and the wheel there 
 is friction decreasing as the speed of rotation of the shaft 
 increases. If, then, the wheel experiences a constant fric- 
 tional resistance from external causes, indefinite increase of 
 speed of the shaft would accelerate the wheel's rotational 
 velocity up to a certain point, and the wheel would then cease 
 to rotate. This supposition would enable us to make our 
 theory agree with the fact that increase of magnetic force 
 does not increase indefinitely the magnetic induction through 
 iron, but brings is up to a point at which, approximately 
 speaking, the induction remains stationary. To sum up, we- 
 may say that the hypothesis of molecular vortices is an 
 endeavour to imagine a mechanism capable of accounting for 
 electro-magnetic induction on dynamical principles, and on 
 the assumption that the energy of a magnetic field is energy 
 stored up in a medium in virtue of a particular kind of 
 rotation of its parts. 
 
 This medium consists of portions capable of elastic displace- 
 ment when we consider parts of it lying in dielectrics or 
 capable of progressive movement when in conductors, and 
 these portions constitute what we call electricity. Other por- 
 tions are capable of rotation round closed axes of rotation, and 
 these constitute what we call " lines of force." The medium 
 possesses, therefore, an elastic resilience, and the reciprocal 
 of this quality, or its freedom of yielding to electromotive 
 force, is recognised as the specific inductive capacity. The 
 medium possesses also density, and we call this its magnetic 
 permeability, or magnetic inductance. The mass of unit of 
 length of the vortices is equal for all vortices, whether in 
 vacuum, air, or non-magnetic bodies, but in iron the vortices 
 are loaded by the adhesion to them of the molecules of the 
 metal, and the density is increased, and hence the permea- 
 bility ; but for very great angular velocities that is, for great, 
 magnetic forces the adhesion of the molecules and vortices 
 must be supposed to cease, and the permeability approximates 
 to unity. The magnetic force at any point in a field is the 
 angular velocity of the vortex motion at that point, and the 
 magnetic induction is the angular momentum. Magnetic 
 attraction and repulsion is due to the tension set up along a
 
 350 DYNAMICAL THEORY OF INDUCTION. 
 
 vortex line by the polar contraction and equatorial expansion 
 of the vortex cells. At places where there is magnetic 
 polarity or free magnetism there is a discontinuity in the 
 angular velocity of the vortices within and without the iron. 
 Self-induction is the result of the inertia of the molecular 
 vortices, whereby motion set up in them cannot be generated 
 or checked instantaneously. Mutual induction, or the pro- 
 duction of induction currents, is due to the fact that differences 
 in the angular velocity of adjacent vortex filaments or cells 
 cause a displacement of the electric particles or idle wheels. 
 Finally, electromotive force is the force causing displacement 
 of the electric particles, and electric currents consist in con- 
 tinuous or periodic movements of these electric particles. 
 Electric currents always produce magnetic fields because 
 there is nothing of the nature of slip between the particles 
 and cells, and, therefore, any progressive movement of the 
 first sets up rotation in the second, and conversely differential 
 rotations or spins of the cells or vortices sets up displacement 
 of the electric particles, causing either electric strain in a 
 dielectric or electric current in a conductor. 
 
 4. Comparison of Theory and Experiment. The test of 
 any physical theory is its power to predict new phenomena 
 .as well as to interpret ascertained experimental results. The 
 theory of molecular vortices leads to the conclusion that 
 electro-magnetic induction must be propagated through the 
 medium with a finite velocity, and that in dielectrics of unit 
 permeability the velocity of propagation is inversely as the 
 square root of the specific inductive capacity. In the 
 -dynamical theory of light it is shown that the ratio of 
 the velocity of light in vacuo to its velocity in any given 
 transparent medium is a constant quantity for each definite 
 wave length, and is called the index of refraction of that body 
 for that wave length, and is denoted in physical optics by the 
 symbol p. Hence, the velocity of light of definite wave-length 
 is inversely as the refractive index for that wave-length. The 
 refractive index for very long wave-lengths can be calculated 
 from observed values of /* for definite rays, and hence numbers 
 obtained representing the relative velocity of these undulations 
 in various transparent bodies. The values of the dielectric
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 351 
 
 constants, or reciprocal of the electric elasticities, of various 
 transparent and semi-transparent bodies have also been deter- 
 mined, and it has been found that for a large group of bodies 
 there is a tolerably close agreement between the values of 
 the square root of the dielectric constant and the index of 
 refraction /tx for very long waves, as shown by the selection 
 from the results of some experimental determinations given 
 in Table A. 
 
 Table A. 
 
 K 
 
 (Dielectric Constant). 
 
 -v/K 
 
 fJ.-X> 
 
 (Refrac- 
 tive Index). 
 
 Authority. 
 
 Reference. 
 
 Sulphur 3'84 
 
 1-96 
 
 2-041 
 
 
 
 Colophonium.. .. 2'55 
 Paraffin 2'32 
 
 1-59 
 1-52 
 
 1-54 I 
 1-54 
 
 Boltzmann 
 
 / Pof/^.Ann.,CLl., 
 \ Ib74, p. 482. 
 
 Pure rubber 2-12 
 
 1-45 
 
 1-50 
 
 Schiller 
 
 / Pony. Ann., 
 \ CLII.,p.535. 
 
 Oil of turpentine 2'21 
 Petroleum 2'037 
 
 1-49 
 1-43 
 
 1-461 
 T46 I 
 
 Silow 
 
 ( Por/g. Ann., 
 I CLVI., 1875, 
 
 Benzine 2198 
 
 1-48 
 
 1-43 j 
 
 
 1 p. 395. 
 
 Petroleum spirit T92 
 Petroleum oil ...2'07 
 Ozokerite 2'13 
 
 1-38 
 1-44 
 i-46 
 
 1-38 \ 
 1-44 
 1-44 f 
 
 J. Hopkinson 
 
 (Trans. Roy. Soc, 
 1877, 1878 and 
 
 Turpentine 2'23 
 
 1-49 
 
 1-4&J 
 
 
 1881. 
 
 For some other dielectrics, such as glass and the vegetable 
 and animal oils, the agreement is not by any means so close 
 but for gases, as determined by Boltzmann (Pogy. Ann., CLL, 
 1875, p. 403), there is a fair coincidence. (See Table B.) 
 
 Table B. 
 
 Gas. 
 
 Air 
 
 ! r00059 
 
 1-000295 1-000294 
 
 
 1-000946 
 
 1-000473 1-000 149 
 
 
 1 1-000264 
 
 1-000132 1-000138 
 
 Carbonic oxide 
 
 | 1-OOC69C 
 
 1-000345 1-000340 
 
 
 i 1-000994 
 
 1-000497 1-000503 
 
 
 1 1-001312 
 
 1-000656 1-000678 
 
 Marsh gas 
 
 ! 1-000944 
 
 1-000472 1-000443 
 
 
 
 
 The gases are taken at 0C. and 760 millimetres pressure. 
 Accordingly, we can say that, for a large group of dielectrics, of 
 which the magnetic permeability is unity, and hence the 
 velocity of propagation of an electro-magnetic impulse proper-
 
 352 DYNAMICAL THEORY OF INDUCTION. 
 
 tional to the square root of the electric elasticity or to the 
 reciprocal of the square root of the dielectric constant, we 
 do find a fair agreement between these numbers and the 
 numbers representing the refractive indices or the relative 
 velocities of propagation of very long waves or disturbances in 
 the ethereal medium postulated to account for the phenomena 
 of light. The imperfect agreement between the values of the 
 refractive index for long wave-lengths and the square root of 
 the dielectric constant for some other bodies shows that the 
 theory is only approximately in agreement with fact, and 
 that the results obtained by the methods adopted for deter- 
 mining the dielectric constant are perhaps impure, and do- 
 not give the true value of the electric elasticity. When we 
 consider that the displacements which constitute the light 
 wave motion of the luminiferous ether are changed some 
 billions of times per second, it is seen to be highly prob- 
 able that measurements of the specific inductive capa- 
 city in which the electric stresses are only reversed tens or 
 hundreds of times in a second may be rendered impure or 
 mixed owing to the presence of effects due to an imperfect 
 electric elasticity introduced by the superposition of electric 
 conduction or of electrolytic transport upon the true or elastic 
 displacement effect. In fact those bodies, such as glass and 
 the vegetable oils, which exhibit the greatest discrepancy, are 
 those in which the chemical composition indicates a possibility 
 of electrolysis. There may be an electro displacement in such 
 electrolisable bodies over and above the true electrostatic 
 displacement which is engendered by a molecular change in 
 the body, which change results in actual decomposition when 
 the electric force reaches a certain limit. Put broadly, it 
 may amount to this, that the true electric displacement 
 is a displacement of electricity within the molecule, but that 
 in electrolisable bodies electric stress sets up a strain of 
 the molecule itself which, within certain limits, is an elastic- 
 strain, and disappears with the removal of the stress, but 
 that beyond these limits molecular disruption takes place. 
 In these cases the displacement measured in taking the specific 
 inductive capacity is the true or dielectric displacement plus 
 a displacement due to strain of the molecule, and the result 
 would be to make K appear too great, and, in fact, for glass.
 
 DYNAMICAL THEORY OF INDUCTION. 353 
 
 and certain oils the values in Table C have been obtained, 
 which in all cases are such that \/K" exceeds the value of 
 IJL^ , or the refractive index, for very long waves of light.* 
 
 Table C. 
 
 Substance. 
 
 K 
 
 x/K 
 
 V x (approx.) 
 
 Glass, extra dense flint 
 light flint . 
 
 9-896 
 6-72 
 
 31 
 
 2-59 
 
 1 
 
 
 6-96 
 
 2'63 
 
 } 1-5 to 1-6 
 
 
 8-45 
 
 2'90 
 
 J 
 
 Castor oil ... 
 
 4-78 
 
 2-18 
 
 T46 
 
 
 3-02 
 
 1-73 
 
 1'46 
 
 Olive oil 
 
 3-16 
 
 1-77 
 
 1'46 
 
 
 3-07 
 
 1'75 
 
 1'45 
 
 
 
 
 
 J. Klemencic (abstract in the Journal of the Society of Tele- 
 graph Engineers, 1886, p. 108) has experimented also on the 
 specific inductive capacity of gases and vapours, and given a 
 table (see Table D) in which he compares \/K with /* (refrac- 
 tive index) of these same bodies. It is seen that the agree- 
 ment of vK and /* is very close for the simple gases, but 
 that a marked difference exists in the case of more complicated 
 molecules. 
 
 Table D. 
 
 Gas. 
 
 v/K 
 
 Boltzmann. 
 
 N/K 
 Klemencic. 
 
 H 
 Refractive 
 index. 
 
 Air 
 
 1-000295 
 1-000132 
 
 1-000293 
 1-000132 
 
 1-000295 
 1-000139 
 
 
 1-000473 
 
 1-000492 
 
 1-000454 
 
 Carbonic oxide 
 
 1-000345 
 
 1-000347 
 
 1-000335 
 
 Nitrous oxide 
 
 1-000497 
 
 1-000579 
 
 TOO 516 
 
 
 1-000656 
 
 1-000729 
 
 1-000720 
 
 Marsh gas 
 Carbonic bisulphide .. 
 
 1-000472 
 
 1-000476 
 1 001450 
 
 1-000442 
 1-001478 
 
 
 
 1-OC477 
 
 1-000703 
 
 Ether 
 
 
 1-00372 
 
 1-00154 
 
 Ethyl chloride 
 
 
 1-00776 
 
 1-001174 
 
 
 
 1-00773 
 
 1-00122 
 
 
 
 
 
 The specific inductive capacity of a vacuum is taken as unity, and Boltz- 
 mann's values are given for comparison. 
 
 * See Dr. J. Hopkinson, Phil. Trans. Royal Society, Vol. CLXXIL, 1881, 
 p. 372. 
 
 A A
 
 S54 DYNAMICAL TBEOKY OF INDUCTION. 
 
 5. Velocity of Propagation of an Electromagnetic Dis- 
 turbance. There is another line of experimental enquiry 
 which leads to an important relation between electric and 
 optic phenomena. This is the comparison of electrostatic and 
 electromagnetic measurements. If two very small spheres 
 are electrostatically charged and placed with their centres 
 at a unit of distance apart, the stress between them may 
 be mechanically measured. If the conductors are equally 
 charged with opposite kinds of electricity, and the stress 
 when at a unit of distance in air is one unit, the electric 
 quantities are said to be unit electrostatic quantities. If 
 such unit quantities are discharged through a conductor at 
 the rate of one discharge per second, the resulting flow or 
 current is called an electrostatic unit of current. 
 
 In the above definition we suppose the dielectric to be a 
 vacuum or some substance such as air, of which the dielectric 
 constant does not differ sensibly from unity. If q and q l be 
 two quantities measured electrostatically, and then be placed 
 on small conductors separated by a distance r in a dielectric of 
 constant K, the dynamical force between them will be nu- 
 
 numerically equal to < ~. i ', and if g = q l , then the force is ^-^. 
 
 Hence, if r is always taken equal to unity, the real quantity of 
 electricity producing by its action on another equal quantity a 
 unit of force will vary as the square root of K when the experi- 
 ment is performed in various dielectrics. In other words, the 
 absolute magnitude of the electrostatic unit of quantity, and 
 therefore also of the current, will vary as the square root of the 
 specific inductive capacity of the medium in which the charges 
 exist. There is another mode in which a unit of current may be 
 defined, and this depends on the definition of a unit magnetic 
 pole. If two magnetic poles of equal strength, 7??, are placed at 
 a distance r apart in a magnetic medium of permeability p, the 
 
 stress or force between them will be numerically equal to ^, 
 
 ^7-2 
 
 in which expression it is seen that m and p. appear as quantities 
 analogous to q and K in the electrostatic analogue. Hence, when 
 r is unity, we see that to produce a unit stress between the 
 poles m the pole strength must vary as the square root of /*, or 
 the absolute magnitude of the unit magnetic pole varies directly
 
 DYNAMICAL THEORY OF INDUCTION. 355 
 
 as the square root of the magnetic inductive capacity of the 
 medium in which the experiment is performed, the absolute 
 unit magnetic pole being denned as a pole which at a unit of 
 distance acts on another like pole with a unit of force in a 
 magnetic medium, assumed to be vacuum, or some standard 
 substance of unit permeability. 
 
 Since an electric current produces a magnetic force, it may be 
 defined as to magnitude by agreeing that the unit of current is 
 to be one which, when flowing in a circular circuit of unit radius, 
 acts for every unit of length of that circuit with a unit of force 
 on a unit magnetic pole placed at the centre of that circle. 
 The magnitude of the force on the magnetic pole is proportional 
 to the product of the strength of the pole and the strength of 
 the current. Hence, if the magnitude of the unit pole is varied 
 the magnitude of the unit of current will vary inversely as the 
 magnitude of the strength of magnetic pole which is taken as 
 the unit pole. When the medium is varied, the magnitude of 
 the unit magnetic pole, or of the pole which fulfils the condi- 
 tion of acting on another equal pole at a unit of distance with 
 a unit of force varies directly as the square root of the permea- 
 bility of the medium. It follows, then, that the magnitude of 
 the electro-magnetic unit of current varies inversely as the square 
 root of the magnetic permeability of the medium in which the 
 experiment is made. 
 
 We have, then, that the electrostatic unit of current is a 
 quantity which varies directly as the square root of the electro- 
 static inductive capacity of the medium, or as \/K, and the 
 electromagnetic unit of current is another unit of current 
 which varies inversely as the square root of the magnetic induc- 
 tive capacity of the medium, or as >//*. The electrostatic unit 
 of current represents a much smaller quantity of electricity 
 per second than the electro-magnetic in other words the 
 value of the ratio of the magnitude of the unit electro-magnetic 
 current based on the definition of a unit magnetic pole, to the 
 magnitude of the unit electrostatic current, based on the 
 definition of a unit of electrostatic quantity, is an integer 
 number, and a large one. This ratio of the two units of 
 current varies when the fundamental inductive capacities of 
 the medium is changed, but so that the ratio of the electro- 
 magnetic to electrostatic unit varies inversely as the square 
 
 AA2
 
 356 DYNAMICAL THEORY OF INDUCTION. 
 
 root of the product of K and p. If C, n is the magnitude of the 
 electro-magnetic unit of current, and C g is that of the electro- 
 
 static unit for the standard dielectric, in which K = 1 and /* = !, 
 
 r\ 
 then, when the dielectric is. changed, -^ is changed in the ratio 
 
 Cg 
 
 of 1; /K p. Let E wac denote the value of the ratio for 
 vacuum or for a standard dielectric, of which K = 1 and p. = 1, 
 and E OT denote its value for any other medium of which the 
 dielectric constant is K and the magnetic constant /x, then 
 
 We have next to consider what is the physical meaning of 
 this ratio of the electro-magnetic and electrostatic units. 
 
 The degree in which one quantity is greater or less than 
 another, or to put it more precisely, that amount of stretching 
 or squeezing which must be applied to the latter in order to 
 produce the former, is called the ratio of the two quantities.* 
 The ratio of two physical quantities is therefore the expres- 
 sion of the operation which must be performed on the one to 
 make it the physical equivalent to the other. What operation 
 must be performed on an electrostatically measured unit of 
 electricity to make it the equivalent in every way of an electro- 
 magnetically measured unit of electricity? The reply is, it 
 must be set in motion with a definite velocity. The electric 
 current produces a magnetic field. The electro-magnetic mea- 
 sure of current is obtained by defining the field by stating its 
 dynamical effect on a defined magnetic pole, and the unit of 
 electric quantity measured electro -magnetically is the quantity 
 conveyed by the unit current so measured in a unit of time. 
 If we imagine a circular or other conductor conveying a unit 
 (electro-magnetic) current to have stretched alongside of it 
 another closely adjacent conductor of like form, each unit of 
 length of which is charged electrostatically with a unit (electro- 
 static) of electric quantity, we might submit the following 
 question : The current flowing in the first named conductor 
 transmits a unit (electro-magnetic) quantity of electricity across 
 each section of it per unit of time : with what velocity must 
 electricity in the second conductor be set flowing in order that 
 
 * W. K. Clifford, " The Common Sense of the Exact Sciences," p. 99.
 
 DYNAMICAL THEORY OF INDUCTION. 357 
 
 there may be an equality in the quantities flowing past any 
 sections in each of the conductors, as evidenced by equality in 
 the magnetic fields produced by the first-named current and 
 the moving electric charge ? This velocity is evidently a 
 concrete velocity, which depends on the very nature of the 
 qualities of the medium which determine magnetic and 
 electrostatic attraction, and this velocity may be called the 
 ratio of the magnitude of the electro-magnetic to the electro- 
 static unit of quantity. This velocity is evidently one which 
 is determined by the nature of the medium, and not by the 
 particular units of length, time, and mass selected for use in 
 the measurements. This comparison assumes that a moving 
 electrostatic charge is in effect the equivalent of an electric 
 current. This has been put to the test of experiment by Prof. 
 Bowland.* A rigid gilt ebonite disc was fixed to an axis, and 
 could be rotated between two gilt glass discs. One member 
 of a very delicate astatic system of magnetic needles was 
 placed near the disc and shielded from electrostatic disturbance. 
 On charging the gilt ebonite disc and setting it in rapid 
 rotation it was found to affect the magnetic needle whilst 
 rotating just as a current of electricity would have done if 
 flowing in a circular conductor coinciding in form with the 
 periphery of the disc. Since 1876 Prof. Rowland has again 
 in the United States repeated the experiment and confirmed 
 the general result. There is, therefore, experimental founda- 
 tion for the view that a static charge of electricity conveyed 
 on a moving body creates a magnetic field whilst it is in 
 movement. This kind of electric current, in which a static 
 charge is bodily moved on a conductor, is called a convection 
 current. The experiment of comparing the magnitudes of an 
 electrostatic and an electro-magnetic unit of electric quantity 
 as above defined was first made by Profs. Weber and 
 Kohlrausch, and the value of that ratio for a medium such as 
 air, in which approximately we have K and u. both equal to 
 unity, gave as a result a velocity very nearly identical with the 
 velocity of light. Since that time very many experimentalists 
 have determined the value of this ratio, which is denoted by 
 
 * See Phil. Mag., 1876, Vol. II., Fifth Series, p. 233 : Dr. Helmholtz, 
 ' On the Electro-Magnetic Action of Electric Convection." These experi- 
 ments of Prof. Rowland were carried out a,t Berlin,
 
 358 DYNAMICAL THEORY OF INDUCTION. 
 
 the symbol " v." The names and the results of the observa- 
 tions made by some of the principal observers are set out in 
 the Table on opposite page. 
 
 One of the best determinations of the velocity of light is 
 that made by Prof. Newcomb, at Washington, in 1882. The 
 method employed was the revolving mirror method of Foucault, 
 the distance between the revolving and fixed mirror being in 
 one portion of the experiments 2,550 metres, and in the other 
 portion 3,720 metres. The resulting velocity of light in vacua 
 is 2-99860 x 10 10 centimetres per second. 
 
 The following results of other observations are abstracted 
 from Prof. Everett's book, " Units and Physical Constants," 
 2nd edition : 
 
 Observe, *"****"' 
 
 Michelson, at Naval Academy, 1879 .................. 2'99910 x 10 10 
 
 Michelson, at Cleveland, 1882 ...................... 2'99853 x 10 l 
 
 Newcomb, at Washington, 1882 (best results) ... 2*99860 x 10 W 
 Newcomb (other results) .............................. 2'99810 x 10 l 
 
 Foucault, at Paris, 1862 ................................ 2'98COO x 10i 
 
 Cornu, at Paris, 1874 .................................. 2 98500 x 10 10 
 
 Cornu, at Paris, 1878 .................................. 3'004 x 10 10 
 
 Last result discussed by Listing ..................... 2'9999 x 10i 
 
 Young and Forbes, 1880-81 .................. : ........ 3'01382xl0 10 
 
 Earlier observations gave as follows : 
 
 Koemer's method, by Jupiter's satellites ............ 3 '000 x 10 10 
 
 Bradley's method, by stellar aberration ............ 2'977 x 10 10 
 
 Fizeau ..................................................... 3'142 xlO*> 
 
 The general result of the best determinations is that the 
 velocity of light is very close to 3-000 x 10 10 centimetres per 
 second, or nearly one thousand million feet per second. 
 
 We have, therefore, the following facts : The velocity V TO 
 of light of definite wave length in any medium is connected 
 with the velocity Y p of the same ray in vacuo by an equation 
 
 P- 
 
 where /* is the refractive index of that medium for the par- 
 ticular wave length considered, and also that the velocity V is 
 very nearly 3 x 10 10 centimetres per second. Also we find that 
 the ratio of the electro-magnetic to the electrostatic unit of 
 electric quantity or current in any dielectric and magnetic
 
 DYNAMICAL THEORY OF INDUCTION. 359 
 
 1! 
 
 SJ 
 
 x 8 
 
 s! 
 
 8 
 
 da a 
 
 I 
 
 I 
 
 IB 
 
 i 
 ij 
 
 if. 
 
 ts - 
 
 I* 
 
 
 ig 
 
 * 
 
 P 
 
 EH 02
 
 360 . DYNAMICAL THEORY OF INDUCTION. 
 
 medium E m is connected with the same ratio measured in vacuo 
 R, by an equation 
 
 E - 
 
 "" 
 
 where K is the dielectric constant and fi the magnetic per- 
 meability.* Experiment has also indicated that within narrow 
 limits, taking best results, E B and V have the same value, 
 namely, 8 x 10 10 centimetres per second, and that ^/K has the 
 same value as p (refractive index) for media, for which p, (per- 
 meability) has the value unity. We are led, therefore, to infer 
 that this close relationship is not a matter of accident, but 
 that it indicates a very intimate connection between electricity 
 and light, and that the hypothesis that light is a disturbance 
 propagated through an elastic medium may be supplemented 
 with some considerable show of reason by the hypothesis that 
 electro-magnetic phenomena are the result of actions taking 
 place in identically the same medium or ether. There are no 
 transparent media for which the magnetic permeability differs 
 by more than a very small quantity from unity, and hence the 
 approximate identity of the values of the ratio of the units 
 compared in air with the value of the velocity of light waves of 
 very long wave-length ; and the approximate identity for true 
 dielectrics of the value of the refractive index and of the square 
 root of the dielectric constant furnishes a test of the proba- 
 bility of the truth of the electro-magnetic theory of light. 
 Maxwell's mathematical method of arriving at this theory 
 consisted hi forming certain equations expressing the velocity of 
 propagation of vector potential, and noticing that these equations 
 were mathematically of the same form as those which determine 
 the velocity of propagation of a disturbance through an elastic 
 medium. The physical meaning of this term, vector potential, 
 may be arrived at as follows : 
 
 Suppose a regiment of soldiers to set off marching down a 
 street, the ranks being well spaced out. At any place in the 
 street let two lines be drawn across the street parallel to 
 each other and a few yards apart. Let two observers take 
 
 * It is unfortunate that usage has consecrated the same Greek letter /* 
 for refractivity in optics and magnetic inductivity in electro-magnetics. In 
 some respects it would be an advantage in electro-optics if these quantities 
 were differently symbolised.
 
 DYNAMICAL THEORY OF INDUCTION. 361 
 
 note of how many soldiers cross each line. At any instant 
 the total number of soldiers which are contained between the 
 two lines is equal to the difference between the numbers which 
 have crossed each line respectively. However irregular the 
 movement may be, the total number of soldiers at any instant 
 in the area or the product of the area, and the number of 
 soldiers per unit of area within the boundary, will be equal to the 
 number obtained by reckoning the algebraic sum of the soldiers 
 which have from the beginning of the time crossed the whole 
 boundary line, calling those numbers jiositive when soldiers 
 have stepped into the area and negative when they have stepped 
 out of it. We have here a simple example of the way in which 
 a line integral may be the equivalent of a surface inter/ml. If 
 the area be irregular in shape and contain A square yards, and 
 if the perimeter be I linear yards, then if % n z , &c., are the 
 number of men which have stepped across each yard length 
 of the boundary, and if N x N 2 , &c., are the number of men 
 in respective square yards within the area at any instant, then 
 Ni + N 2 + , &c., to A terms or 2N is called a surface integral 
 and will be equal to n 1 + n.i + , &c., to I terms, which is a line 
 integral, provided that each n is reckoned positive when men 
 step in, and negative when men step out of the area over each 
 yard of the boundary. The algebraic sum of all the stepping 
 over the boundary all the way round the area is equal to the 
 sum of the men per square yard all over the area. We have 
 here given an illustration of an important proposition in 
 mathematical physics, viz., that a surface integral, or the sum- 
 mation of a certain quantity over an area, can be replaced by 
 a line integral, or the summation of another relative quantity 
 all along the boundary line of that area. We proceed to 
 illustrate it from an electrical point of view. 
 
 Let C (Fig. 182) be the circular cross-section of an infinite 
 straight wire conveying a current C. Eound C describe a 
 circle of radius r. The magnetic force at p is known to be 
 
 equal to units, and is directed along the circumference 
 of the circle ; the line integral of the magnetic force along the 
 dotted line is equal to _ x27rr = 47rC, and the surface 
 integral of the current through the area enclosed by the dotted
 
 362 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 circle is C. Hence we have generally that the line integral of 
 the magnetic force is equal to 4?r times the surface integral of 
 the current. This proposition is generally true, and it is easy 
 to show that if A be any area (see Fig. 133) traversed normally 
 by a current, such that the current density is u over any element 
 
 FIG. 132. 
 
 of area d s, then the integral of u d s all over the area, or \uds, 
 
 is equal to the line integral of the magnetic force taken along 
 the boundary line. The mathematical operation of taking a 
 line integral has been called by Maxwell curling, and we express 
 
 FIG. 133. 
 
 the above proposition by saying that 4:r times the total current 
 through the area is equal to the curl of the magnetic force 
 round it. On the theory that lines of magnetic force do not 
 spring suddenly into existence in a field, but are propagated 
 onwards from point to point in ihe field, it is possible to show
 
 DYNAMICAL THEORY OF INDUCTION. 363 
 
 that just as the current is the curl of the magnetic force so the 
 magnetic force is the curl of another quantity called the vector 
 potential. 
 
 Let A B (Fig. 134) be a portion of a straight conductor in 
 which a current can be started. Let x x, y y' be two lines drawn 
 a unit of distance apart, parallel to each other and at right 
 angles to the conductor. These lines bound a strip of plane space 
 taken in the plane of the current. Draw any two transverse 
 lines a b, c d, parallel to the conductor and separated by a small 
 distance. We know that when a current is started in the con- 
 ductor the lines of magnetic force F will be circles formed round 
 A B as axis, and having their planes perpendicular to the 
 plane x x', y y'. Let us now assume that if a current is 
 suddenly started in the conductor A B the magnetic force is 
 
 B 
 
 propagated outwards from the conductor with a finite velocity 
 v. In other words, each circular line of force must be con- 
 sidered to expand outwards like a circular ripple on the surface 
 of water. When once the field has arrived everywhere at its 
 normal value the magnetic force at a distance r from the wire 
 
 20 
 
 is , where C is the value of the current, and we shall sup- 
 pose, as usual, that the magnetic field is indicated as to value 
 by the density of the lines of force, or that the number per 
 square centimetre traversing normally the plane x x', y y' is at 
 any point proportional or numerically equal to the magnetic 
 force at that point. If, then, we neglect for the moment all 
 effect of self-induction, and suppose the current in the wire to 
 rise up instantaneously to its full value, we may yet regard the
 
 364 DYNAMICAL THEORY OF INDUCTION. 
 
 circular lines of force as expanding outwards with a certain 
 velocity of enlargement, and attaining or taking up their final 
 positions after a short interval of time. If we represent the 
 intersections of these rings of force on the plane of x x y y by 
 dots, these dots will march forward like the soldiers in the pre- 
 vious illustration. The total number of lines of force which at 
 any instant are found traversing the area a b d c is equal numeri- 
 cally to the difference in the number between those which from 
 the beginning of the epoch have intersected or cut through the 
 Jine a b and those which have cut through c d. In other words, 
 the surface integral of the magnetic force over abed may be 
 represented by, or is equal to, the line integral round abed 
 of a certain quantity called the vector potential, which, physi- 
 cally interpreted, is the total number of lines of force which 
 have cut through a unit element of the boundary in the process 
 of expansion or propagation outwards. This term vector poten- 
 tial is justified as follows : If F be the total number of lines 
 of force per unit of length of a ft which have cut through a b 
 from the instant of beginning the current, and if the small 
 distance b d is called Sa*, the length x b being called x, then by 
 Taylor's theorem (Diff. Calc.), the number which have cut 
 
 dF 
 through unit of length of c d is F -3 8 x, and hence the 
 
 difference between F and this last quantity is -j-, 8x, and this 
 last when multiplied by 8 y, which we may take for the length 
 
 T TTt 
 
 of a ft or c d that is -j-^ 8x8 y is the total number of lines of 
 
 force included in the area abed. If we call the induction 
 through this area B that is to say, the number of lines of 
 force per square centimetre is B it follows that the number 
 through ab c d is B 8x8y. Hence, equating the two values, 
 
 we have -^ 8 x 8 y = B 8 x 8 y, 
 
 *I-R 
 
 dx 
 
 Hence, the mean magnetic force over the small area is numeri- 
 cally equal to the space variation of a certain quantity F. In 
 electrostatics the electric force X at any point in the electric
 
 DYNAMICAL THEORY OF INDUCTION. 365 
 
 field is the space variation of a certain quantity V, called the 
 electrostatic or scalar potential that is to say, 
 
 ~~^T = X; 
 
 and accordingly by anology that quantity F whose space varia- 
 tion gives the magnetic force under the circumstances considered 
 above is called the vector potential of the current. From Ampere's 
 investigations it is known that the magnetic force due to an 
 element of a current C of length 8 s at a distance r from this 
 
 fl 
 
 element, has the value , and is along a line at right angles 
 
 to the plane containing & s and r. The space variation of - 
 
 /i o r 
 
 is ; hence the vector potential of an element of current at 
 
 any point is proportional to the length of that element divided 
 by its distance from that point. 
 
 In electrostatic phenomena we obtain the static potential at 
 any point due to any charge Q by taking each element q of the 
 charge, and dividing the magnitude of this element of charge 
 by its distance from the point at which the potential is required, 
 
 and taking the sum 2 3. of all such quotients. In electrostatics 
 
 the potential at a point is a scalar or directionless quantity, 
 and the summation is merely an algebraic sum ; but in dealing 
 c* /? 
 
 with currents the quotients are vectors, or directed quan- 
 
 r 
 
 tities, and have to be added together according to the laws 
 for the addition of vector quantities just as forces and velocities 
 are added. Hence the potential of a current at any point is a 
 vector or directed quantity. The lines of vector potential of 
 a straight current are lines described in space parallel to the 
 current, and the lines of vector potential of a circular current 
 are circles described on planes parallel to the plane of the cur- 
 rent. Returning to the simple case of a straight current, let 
 us suppose that a unit of length is described somewhere parallel 
 to the current, and that on starting the current suddenly cir- 
 cular lines of magnetic force are propagated outwards with a 
 velocity V ; these lines will, as they expand, cut perpendicularly 
 through the element of length just as the expanding ripples on 
 water due to a stone dropped into it would "cut through" a
 
 366 DYNAMICAL THEORY OF INDUCTION. 
 
 stick held perpendicularly in the water a little way from the 
 place where the "splash " was made. Suppose that after N 
 lines of force have cut through the element of length this little 
 line is made to move forward parallel to itself, so that there is 
 no further increase in the number of lines of force which after- 
 wards cut through it, it is evident that it must move with the 
 velocity of propagation of the expanding rings of force. But the 
 number expressing the number of lines of force which have cut 
 through the element of length already is the value of the vector 
 potential at that point where the element is at that instant ; 
 hence the velocity of propagation of the vector potential is 
 the velocity of propagation of an electro-magnetic disturbance. 
 Maxwell's general mathematical method of investigating the 
 propagation of an electro-magnetic disturbance consisted in 
 forming equations expressing the change of the value of the 
 vector potential of a current or system of currents at any point 
 in the field, and deducing equations which mathematically are 
 of the same type as those which express the propagation of a 
 disturbance through an elastic solid or fluid, and his result 
 was that the velocity of propagation of the vector potential 
 through a medium of electrostatic and magnetic inductivities 
 
 K and p. was equal to , or to (K/*)-i. 
 
 N/ K M 
 The complete proof of the above proposition as given by 
 
 Maxwell in all its generality requires some elaborate analysis, 
 but is is not difficult to give a simple illustration by treating a 
 reduced case, and which will exhibit the principles of the more 
 complete problem. 
 
 Let an infinite straight conductor be supposed situated in a 
 dielectric medium of specific inductive capacity (electrostatic 
 inductivity) K and of permeability (magnetic inductivity) p.. 
 We proceed to investigate the velocity of lateral propagation of 
 electro-magnetic induction on the supposition that if a current 
 is instantaneously started at its full value in the conductor, 
 supposing this possible, the magnetic force travels outwards 
 laterally from the conductor in all directions with a velocity v. 
 This amounts to the supposition that the circular lines of 
 magnetic force surrounding the conductor swell out or expand 
 outwards from the surface of the conductor, so that the radius 
 of any determinate circular line of force increases or grows
 
 DYNAMICAL THEORY OF INDUCTION. 367 
 
 with a velocity v. It must be borne in mind that the 
 magnetic force at any point in the field at any instant is 
 
 defined by the density or concentration of the lines of force 
 
 that is, by the number passing normally through a unit 
 of area. If we complicate the problem by supposing the 
 strength of the current in the conductor to gradually increase, 
 then the concentration of the lines at any point must be 
 supposed to increase gradually, but the rate of increase of 
 concentration that is, of the force is a different thing from 
 the rate of outward movement of the lines of force. 
 
 We might in imagination suppose each line of force to be 
 labelled so as to recognise it. All the lines travel outward 
 from the conductor at the same rate, but some go out farther 
 than others. The first ones shed off expand out to reach 
 
 FIG. 135. 
 
 positions in the most distant portions of the field, and the 
 succeeding ones reach intermediate positions, and as the 
 current strength grows up fresh arrivals or deliveries of lines 
 of force happen which pack the space fuller, and increase the 
 concentration at all points of the field, at a rate depending on 
 the rate of growth of the current. 
 
 Let C (Fig. 135) be a portion of the straight conductor. 
 In the plane of C take any little rectangular area abed, with 
 side a c equal to unit of length, and side a b equal to 8 x, 8 x 
 being a very small quantity compared with the distance between 
 OC and ac, that is, let the distance Oc = x and Od = x + 8x, 
 and let the distance Sx be the distance by which the radius 
 of any circular line of force of the conductor OC increases
 
 368 DYNAMICAL THEORY OF INDUCTION. 
 
 in a small time 8 1. At any instant the number of lines 
 of force which pass normally through the small area abed 
 is equal to the difference between the number which have 
 "cut" across ac and those which have cut across Id in 
 consequence of our supposition as to the outward growth or 
 expansion of the circular lines of force. Let F be the total 
 number of lines of force due to the current in C which have 
 from the beginning of the current flow " cut across " a c, then, 
 by the principles of the Differential Calculus, the number 
 which have cut across bd is represented by the quantity 
 
 7 -[7\ 
 
 F - 8 x, and the number existing in, or perforating through, 
 
 dx ,p 
 
 the area abed is the difference between F and F- 8x, or 
 
 ,p 
 equal to 8 x. Let B stand for the induction through 
 
 unit of area of the rectangle abed, or to the number of lines of 
 force per unit of area, then the total number of lines of force 
 through abed is represented also by B 8 x, since the area of 
 a b c d is 8x square units, a c = b d being unity. 
 
 Hence, H =B ( 109 ) 
 
 or the induction is represented by the space rate of change of 
 the vector potential of the current at that point in the direc- 
 tion of x. In this case let it be borne in mind that the vector 
 potential signifies the number of lines of force which have from 
 the beginning of the epoch cut through unit length taken 
 parallel to the current. Again, since by supposition each line 
 of force moves outwards parallel to itself through a distance 
 
 8x in a time 8t, is the velocity of propagation v of the 
 o t 
 
 electro-magnetic disturbance or of the vector potential. The 
 rate of " cutting across " a c at any instant is represented by 
 
 7 T71 
 
 ; hence the number of lines of force added to the area in 
 d t 
 
 a time 8t must be 8t, and this must be equal to the accu- 
 dt 
 
 mulation of the lines in a b c d in the same time in the area 
 a b c d.
 
 DYNAMICAL THEORY OF INDUCTION. 300 
 If in a small time interval the rate of cutting across a c is 
 
 J TJ1 
 
 _, then the rate at which " cutting" is taking place across 
 a length b d, removed by a distance 8 x, is 
 
 ~dt d~x \ dt) X ' 
 
 and the rate at which accumulation of lines of induction is 
 going on in the area is 
 
 dx \dt 
 
 Hence, since B is the induction per unit of area and the area 
 of abed is 8x square units, the rate of increase of induction 
 through a b c d is 
 
 Accordingly we have 
 
 <B.)---* 
 
 rf V ; dx\dt 
 
 or since 8 # is constant, 
 
 rfB = __^ 
 dt d\dt' 
 _ d fdF\dt 
 dt\dt) dae 
 dE dx = _d 2 F dt . 
 or ' ~dx ~di~ ~dt? d~x ' 
 
 but =v = velocity of propagation of the impulse. Hence, 
 dt 
 
 (no) 
 
 dx 
 
 T T71 
 
 or, generally, sinco B = , 
 
 weha ve *| + ^|I.O .... (HI) 
 
 d t* d x 2 
 
 as the equation of motion of the vector potential. This 
 equation, which is a reduced case of the general one, is of 
 the same type as that obtained in the theory of sound for the
 
 370 DYNAMICAL THEORY OF INDUCTION. 
 
 propagation of an impulse along a tube or canal. In the case 
 of sound the symbol F would be the velocity potential.* In the 
 electro-magnelic problem the F is the rector potential. It might 
 perhaps be more expressively called the induction potential. 
 
 The rate of cutting, or the value of , also expresses the 
 
 dt 
 
 electromotive force acting along the unit of length a c in the 
 dielectric. On Maxwell's hypothesis this electromotive force in 
 the dielectric acting parallel to the current in the conductor 
 produces a displacement in the dielectric, such that if E is the 
 electromotive force we have as above 
 
 dt K 
 
 where D is the displacement through unit of area ; hence, 
 
 -# < 112 > 
 
 and - is the rate of displacement or the displacement current 
 dt 
 
 flowing through unit of area taken perpendicularly to the cur- 
 rent in C at the point considered. Let this displacement 
 
 current be denoted by u. We have then that r = -^> K 
 
 rfr Iv 
 
 being the dielectric constant of the medium. 
 
 Consider now a small parallelopipedon (Fig. 136) or solid 
 rectangle described in the dielectric, of which the sides are 
 respectively ac = l, cd = Sx, c e = 8y. 
 
 The effect of the cutting across of this solid rectangle by 
 expanding lines of induction will be to generate in it a displace- 
 ment current such that the total displacement current parallel 
 to a c and through c d fe will be u d x d y. By a previous theorem 
 the line integral of magnetic force round any line is equal to 
 4?r times the surface integral of the current through the area 
 bounded by that line, and this is true whether the magnetic force 
 be produced by that current, or whether it is a current produced 
 by a certain changing magnetic force. Apply the theorem to 
 the small rectangle bounded by the lines c efd. The surface in- 
 tegral of the current through c efd is u d x d y. The magnetic 
 
 * See Besant's " Hydromechanics," p. 251 (Third Edition).
 
 DYNAMICAL THEORY OF INDUCTION. 
 B 
 
 371 
 
 force along c e is -, where B is the induction at c and //, is the 
 
 magnetic permeability of the medium, since by a fundamental 
 theorem the magnetic induction B at any place is equal to p. 
 times the magnetic force at that point. The magnetic force 
 
 along df removed by a distance 8 x from ce is -(B- 8x] 
 
 p\ dx /' 
 
 and there is no magnetic force along c d and ef, for these sides 
 are perpendicular to the direction of the magnetic force of the 
 
 FIG. 136. 
 
 current in C. Hence, the line integral of magnetic force round 
 cefdis 
 
 hence, 
 
 4;r u 8 x 8 y = _ 8 x 8 y, 
 d x 
 
 
 (113) 
 
 Accordingly, in the equations (112) and (113) above, we have 
 
 ^F dE . 
 obtained values for the quantities -^ and -^ in terms of 
 
 the permanent constants of the medium ; and by substitution
 
 372 DYNAMICAL THEORY OF INDUCTION. 
 
 of these values in equation (111) above, we see that the square 
 of the velocity of propagation of the vector potential is 
 d-F 4:r M 
 
 *-*JL. * 
 
 d B 4?r p. u K fj.' 
 
 V IT' 
 
 that is, the velocity of propagation of the magnetic force is the 
 square root of the reciprocal of the product of the magnetic 
 and electrostatic inductive constants of the medium. We have 
 above proved that the ratio of the electro-magnetic to the 
 electrostatic units of electric current is expressed by the same 
 quantity, and indicated that accurate experiment shows this 
 ratio to be numerically the same as the velocity of light. 
 
 Hence, the velocity of an electro-magnetic disturbance or 
 magnetic force is the same as the velocity of light, and the 
 conclusion is urged upon us with great force that the medium 
 concerned in both phenomena is the same. 
 
 6. Electrical Oscillations. A survey of the phenomena of 
 electric current induction would be very incomplete if it did 
 not contain some reference to the subject of electrical oscilla- 
 tions. Recent researches have endowed this department of 
 electrical investigation with fresh interest. We proceed to 
 consider the manner in which electrical oscillations may arise. 
 If a material body is subjected to elastic constraint, and is dis- 
 turbed from a position of equilibrium, it returns when set free 
 to its original position. If that body is endowed with mass, 
 and hence possesses the quality of inertia, its motion of return 
 to its position of equilibrium will, under certain circumstances, 
 carry it beyond that point and set up oscillations, which decay 
 gradually away. Two illustrations of this readily present 
 themselves, one a mechanical and the other a pneumatical 
 example. The first case is that of a pendulum or straight 
 spring. Let this pendulum or spring be deflected from its 
 position or condition of equilibrium and held in constraint. Next 
 let it be set free the elastic or restoring forces urge it back 
 again to its first position. In virtue of its mass it will acquire
 
 DYNAMICAL THEORY OF INDUCTION. 373 
 
 a certain momentum, and on reaching the position of equili- 
 brium this momentum may carry it past this point, and the 
 acquired kinetic energy will then be expended in making a 
 displacement against the elastic forces. If there is nothing of 
 the nature of friction present to fritter away the work expended 
 on the body in making the first displacement, then the energy 
 would remain associated with it for ever, being alternately 
 potential and kinetic, and the oscillations continue with undi- 
 minished amplitude. If the spring or pendulum vibrates in a 
 viscous fluid, then a frictional retardation will be experienced, 
 and in so far as this is present the energy is gradually dissi- 
 pated, and the oscillations decay away, becoming gradually less 
 and less in amplitude. It may so happen that the work done 
 against frictional resistance during the first quarter of a com- 
 plete oscillation in starting to return from the position of 
 greatest displacement is just equal to the work done in origin- 
 ally making the displacement. When this is the case the 
 whole energy is dissipated by the time the deflected or dis- 
 placed body reaches its original position of rest, and there are 
 then no oscillations. Accordingly a pendulum or spring may 
 be set in a viscous fluid of such a kind that the frictional 
 resistance is just sufficient to secure that when the body 
 is disturbed and then set free it returns to its original 
 position without ever passing it ; in other words, there are 
 no oscillations. Another illustration of oscillatory and non- 
 oscillatory establishment of equilibrium is as follows : Sup- 
 pose there be two large vessels, or reservoirs, connected by a 
 pipe, closed or able to be closed in the middle by a stop- 
 cock. Let one of these vessels, A, be exhausted of its air, and 
 let the other, B, have air in it at the atmospheric, or a greater 
 than the atmospheric pressure. First, let the connecting pipe 
 be supposed to be long and narrow ; on opening the stopcock 
 air will rush over from B into A, and the flow of air will con- 
 tinue uniformly in the pipe in one direction until the pressure 
 in A and B is equalised. Second, let the connecting pipe be 
 very short and large, so that little tubular friction is offered to 
 the flow of air. Under these circumstances the result of open- 
 ing the tap would be that a rush of air would take place, which 
 would be succeeded by a series of oscillations of the air in the 
 tube. The air, in fact, rebounds from side to side, and the
 
 374 DYNAMICAL THEORY OF INDUCTION. 
 
 equilibrium is only finally established after a series of gradually 
 diminishing oscillations or backward or forward currents of 
 air in the tube. This establishment of equilibrium or pressure 
 by oscillatory movement takes place when the resistance to the 
 flow is small. That this is no fanciful description is proved 
 by the experience of MM. Clement and Desormes in their 
 experiments to determine the ratio of the specific heats of gases. 
 In these experiments a large glass vessel had a partial vacuum 
 made in it. A stopcock was then quickly opened and closed, 
 and the pressure of the air determined after a short time. 
 These experiments were repeated by MM. Gay Lussac and 
 ^Yelter. See Journal de Physique, LXXXIX., 1819, 428, and 
 Ann. de Ch. et de Phys. [1], XIX., 1821, 436. 
 
 M. Cazin (Ann. de Ch. et de Phys. [3] LXVL, 1862, 206) 
 first pointed out a source of error which resulted from these air 
 oscillations, and showed that the final pressure depended upon 
 the phase of the oscillation at which the stopcock is closed. 
 
 These examples are sufficient to indicate that when a material 
 system of bodies having inertia is displaced against elastic forces 
 which compel it to return, if free, to a definite position, whilst 
 at the same time its motion is resisted by actions of the nature 
 of frictional resistance which dissipate its energy, we have a 
 resulting motion which may be oscillatory or non-oscillatory, 
 according to the relation of the constants of the system. Under 
 certain conditions as to mass, or inertia and friction, we have 
 oscillations dying gradually away. Under other conditions we 
 have a gradual return to the original position without ever 
 passing it. The motion is then said to be perfectly dead-beat. 
 "We shall investigate presently the conditions which must hold 
 good, and the relation between the inertia factor, in virtue of 
 which the moving system possesses kinetic energy, and the 
 resistance factor, in virtue of which the energy bestowed upon 
 the system at its first displacement is frittered away into heat, 
 in order that the motion may be vibratory or dead-beat. 
 
 When a condenser or Ley den jar is discharged through a 
 conductor, the potential energy runs down in the form of an 
 electric current. In this case we have a similar state of things 
 to that existing when a bent spring is released. This trans- 
 formation of the potential energy may take place either by a 
 vibratory current, that is, by a series of electrical oscillations or
 
 DYNAMICAL THEORY OF INDUCTION. 375 
 
 by a uni-directional discharge. It is highly probable that Prof. 
 Joseph Henry, as far back as 1842, was the first to recognise 
 that the discharge of a condenser might be of an oscillatory 
 character. It is remarked by him* that "The discharge, 
 whatever may be its nature, is not correctly represented by 
 a single transfer of imponderable fluid from one side of the 
 jar to the other ; the phenomena require us to admit the 
 existence of a principal discharge in one direction and then 
 several reflex actions backward and forward, each more feeble 
 than the preceding, until equilibrium is attained. All the 
 facts are shown to be in accordance with this hypothesis, and 
 a ready explanation is afforded by it of a number of phenomena 
 which are to be found described in the older works on electricity, 
 but which have until this time remained unexplained." A little 
 later on in the Paper he gives an explanation of the reversal 
 of polarity of the needles by the oscillatory discharge. In his 
 celebrated Essay, " Erhaltung der Kraft " (Berlin, 1847), 
 Helmholtz alluded also to such a possible form of electric 
 discharge in the following words : " We assume that the dis- 
 charge (of a jar) is not a simple motion of the electricity in one 
 direction, but a backward and forward motion between the 
 coatings in oscillation, which become continually smaller until 
 the entire vis viva is destroyed by the sum of the resistances." 
 He adds : " The notion that the discharge consists of alter- 
 nately opposed currents is also favoured by the phenomena 
 observed by Wollaston while attempting to decompose water 
 by electric shocks, that both descriptions of gases are evolved 
 at both electrodes." The investigation which, however, marks 
 an epoch in this subject is the Paper by Lord Kelvin (then Sir 
 William Thomson) in the June number of the Philosophical 
 Magazine for 1853, on " Transient Electric Currents." In 
 this Paper the author discusses, first, the equations which 
 determine these currents at any instant when a condenser or 
 Leyden jar is discharged through a conductor. The dis- 
 charging conductor is supposed to have self-induction, or as 
 
 * " The Scientific Writings of the late Prof. Joseph Henry." Washing- 
 ton : 1886. Vol. I. This statement of Prof. Henry had attention directed 
 to it by Mr. A. D. Raine in 2 he Electrician of November 2, 1888, p. 831. 
 It had been previously mentioned, however, in the sketch of the life of 
 Prof. Joseph Henry, given in the Encyclopedia Brittanica, Ninth Edition.
 
 376 DYNAMICAL THEORY OF INDUCTION. 
 
 Lord Kelvin then called it, " electro-dynamic capacity," and 
 also to have ohmic resistance, which is constant, and indepen- 
 dent of the rate of discharge. On these two assumptions he 
 builds up an equation which mathematically contains the whole 
 theory, as follows : 
 
 If C is the electrostatic capacity of the jar or condenser, and 
 E the ohmic resistance, and L the constant inductance of the 
 discharging conductor ; and if q is the electric quantity hi the 
 jar, and v the potential difference of its coatings at any instant 
 t, then by the definition of electric capacity we have 
 
 q = C v , 
 
 and _? = i = the current at that instant in the conductor, which 
 di 
 
 is equal by Ohm's law to ^. By the principle of conservation 
 of energy the rate at which electro-magnetic energy is being 
 
 taken up by the conductor, viz., (A L i 2 ), together with the 
 a t 
 
 rate at which energy is being dissipated as heat in the con- 
 ductor, viz., Ei 2 (by Joule's law), must be equal to the rate of 
 decay of the energy contained in the jar, or to 
 
 -<*)--;(*) 
 
 tit a t \ \jj 
 
 __ (I f , Q \ f? 
 
 Hence - ( L 1 _ 
 
 Cdt~ J Jt 
 but t = '-^, or the current is the rate of loss of charge, there- 
 
 The value of q, or the charge in the jar at any instant, is 
 given by the solution of this equation. 
 Let us write the equation in the form 
 
 In order to solve this equation we may proceed as follows : 
 The charge g in the jar begins by possessing a certain initial
 
 DYNAMICAL THEORY OF INDUCTION. 377 
 
 value, and ends by being zero. Let us assume that q can be 
 expressed as a function of the time t in the form q=A.e mt , 
 where A is a constant and e is the base of the Naperian 
 logarithms, and m is also a certain function determined by 
 the capacity, resistance, and inductance of the system. For 
 it is clear that by a suitable value for A and m the func- 
 tion A e mt may be made to express the mode in which the 
 charge q dies away with increase of the time t. The problem 
 is reduced, then, to finding A and m. The solution of nearly 
 every differential equation is by a process of happy guessing ; 
 there is generally no systematic or direct method of obtaining 
 the required result. Take, then, the expression g^Ae"", 
 form the first and second differential coefficients, and sub- 
 stitute these results in the original equation, and we arrive 
 at the expression 
 
 Hence, the value A e mt assumed for q will satisfy the equa- 
 tion (115) ; that is, when substituted for q in the original 
 expression, render it zero, provided that m is such a quantity 
 that m a + a ?/i-|- 6 = 0. The two roots of this last quadratic 
 equation are obtained by a simple solution, and they are 
 
 Two cases then arise, first, when is greater than b that 
 
 is, when J?l- is greater than JL , or - greater than _ . In 
 
 4 J_r Li O 4 Jj 
 
 this case the roots of the quadratic are real, and if we call 
 them m^ and m 2 we can say that the solution of the dif- 
 ferential equation is 
 
 (116) 
 
 where A and B are constants determined by the initial circum- 
 stances of the discharge, and w : and m. 2 are equal respectively 
 
 to-" + A? -6 and--- /--b. This solution for the 
 
 2V 4 'J V 4 
 
 value of q is called an exponential solution, and it indicates 
 that under these circumstances when the inductance, resistance
 
 378 DYNAMICAL THEOEY OF INDUCTION. 
 
 and capacity are of such magnitudes that E is greater than 
 / __ , the quantity q dies away regularly, diminishing with 
 
 the time in a continuous manner. In this case the discharge 
 of the jar is always in one direction, and the current or rate of 
 
 decay ( - ) of the charge is also always in one direction. 
 \ d t/ 
 
 If, however, E is less than / , then (^.-6) is a nega- 
 f\J C \4 / 
 
 tive quantity, and the square root of it is an imaginary one, 
 and the roots of the quadratic m' 2 + am + b = Q are unreal. 
 It is shown in treatises on algebra that a quadratic equation 
 has either two real or two imaginary roots, and when this last 
 is the case the roots of the quadratic can always be expressed 
 in the form a + /3 V - 1. 
 
 Accordingly, the solution of the original equation (115) 
 under these circumstances is of the form 
 
 a -^i*. . (117) 
 
 By a simple transformation, based on the employment of 
 the exponential values of the sine and cosine, as given on 
 page 106, this solution can be thrown into the form 
 
 q = e at (Pcoapt + P l smpt) . . . (118) 
 where P and P 1 are constants, and 
 
 a = -=--?., and/?- lb-?L= /I 
 2 2L V 4 Y LC 
 
 The general result is then that the equation 
 
 L(V 
 
 has two solutions one, called the dead beat case which applies 
 
 when E is greater than . /. , and is of an exponential form, 
 
 > C 
 
 and indicates that the charge q dies away regularly with lapse 
 of time, and the discharge current is uni-directional ; the 
 other, called the oscillatory case, which applies when E is less 
 
 than */ > contains sine and cosine terms, and indicates a 
 periodically changing discharge decreasing by a series of
 
 DYNAMICAL THEORY OF INDUCTION. 379 
 
 oscillations, in which case the charge on each plate of 
 the condenser is first positive and then negative, but 
 at the same time always decreasing; or, in other words, 
 is a periodic variation superimposed on a steadily decreasing 
 variation, the currents or rates of discharge following 
 the same distinction. These two modes of discharge, or 
 
 Time 
 
 Curve representing the Discharge of a Condenser through a large .Resistance. 
 Discharge Uni-directional and Continuous. 
 
 Curve representing the Discharge of a Condenser through a Small Resistance. 
 
 Discharge is Periodic and Alternate. Maxima gradually diminishing in Geometric 
 
 Progression. 
 
 FIG. 137. 
 
 solutions of the differential equation, are best indicated 
 graphically by the two curves in Fig. 137, in which the 
 upper curve represents the gradual decrease, according to an 
 exponential law, which is indicated as the proper solution of 
 the equation, when the value of B or the resistance of the
 
 resistance E is less than */ -. When B has such a 
 
 380 DYNAMICAL THEORY OF INDUCTION. 
 
 discharging circuit is greater than */ r , and the lower one 
 
 V G 
 
 the oscillatory discharge, which is indicated by the trigono- 
 metrical solution of the differential equation, when the 
 
 '4L 
 __ C" 
 
 value that R=\/- the discharge is just non-oscillatory. 
 
 ' G 
 
 We find, then, that according to Lord Kelvin, analysis 
 indicates that for a certain relation between the resistance 
 and inductance of the discharge circuit and of the capacity of 
 the jar the discharge is a simple current in one direction 
 or an oscillatory but decreasing current, according as E 
 
 is greater or less than */ . If the discharge is oscillatory, 
 
 then the electrical oscillations are isochronous, and the 
 periodic time of a complete oscillation is 
 
 rp_ 2?T 
 
 V LC~H7 
 for in the second solution (118), 
 
 q = e a *(P cos (3 1 + Q sin t), 
 we see that at intervals of time equal to ^ the sine and cosine 
 
 terms have the same values, since sin = sm/3u + 5Y and 
 
 the same for the cosine. Hence, the trigonometrical factor in 
 the value for q periodically repeats itself in value at intervals 
 
 of time equal to - , and is zero at times when tan y8 t = - -. 
 
 ft Q 
 
 Hence the complete periodic time of the oscillation is 
 
 ft 
 and the frequency of the oscillations, or number in one second, 
 
 18 tI.\J f-y.-* TV
 
 DYNAMICAL THEORY OF INDUCTION. 381 
 
 Accordingly, when E= /y/_ there are no oscillations in one 
 
 second, or the motion is just non-oscillatory, or dead beat. In 
 the case of the uni-directional discharge the values of the in- 
 stantaneous current in the discharge circuit can be represented 
 as we have seen by the ordinates of an exponential curve, and in 
 the case of the oscillatory discharge by those of a periodic curve 
 whose successive maxima descend in geometric progression as 
 the time increases in arithmetic progression. During equal 
 intervals of time the whole quantities which pass decrease 
 also in geometric progression, and the zero points, or instants 
 of reversals of sign of current, are uniformly separated. 
 
 The foregoing predictions of analysis have been confirmed 
 by the experiments of Feddersen, Paalzow, Bernstein, Blaserna, 
 Helmholtz, Schiller and Eood. Lord Kelvin in his original 
 Paper pointed out and suggested the application of Wheat- 
 stone's mirror in the examination of the discharge. In 
 Feddersen's experiments the spark from a Leyden jar battery 
 was taken between two brass balls placed in front of a revolv- 
 ing mirror. The discharge was passed through a high resist- 
 ance. The image of the spark was viewed by a telescope. 
 Under these circumstances the image of the spark was drawn 
 out when the mirror revolved into a continuous band of light 
 in a direction perpendicular to that of the discharge.* When 
 the resistance was gradually reduced a point was reached at 
 which the image was broken up into a series of separated 
 strips, each strip corresponding to a discharge. This showed 
 that the discharge was intermittent. 
 
 In Paalzow's experiments a similar discharge from a Leyden 
 battery was passed through a resistance coil and through a 
 vacuum tube, and the image of the discharge in the vacuum 
 tube viewed in a revolving mirror. As before, with a small 
 resistance the image consisted of a number of separate images, 
 each of which corresponded to a discharge, and a bluish light 
 showed itself at both poles of the vacuum tube. When the 
 
 * An experimental research of a very complete character on the duration 
 and nature of the discharge of a Leyden jar is described by Prof. Ogden 
 Rood in the American Journal of Science and Arts for September, 1869 ; 
 January, 1871; September, 1871 ; October, 1872; November, 1872; March. 
 1873.
 
 382 DYNAMICAL THEORY OF INDUCTION. 
 
 resistance was increased the bluish light showed itself only at 
 one pole. In the former case a magnet held outside the tube 
 split the discharge into tico lines of light, showing that it con- 
 sisted of currents travelling in both directions ; but in the last 
 case the magnet did not divide the discharge. This sufficiently 
 indicated that with a low resistance the discharge was oscillatory 
 and alternate, and not uniform or uni- directional. 
 
 Feddersen found that the critical resistance at which the 
 discharge just becomes oscillatory varies inversely as the square 
 root of the capacity of the battery, which is in agreement with 
 the predictions of theory. 
 
 A good account of the researches of these experimentalists is 
 given in Wiedemann's Galvanismus, Part II, 800, et seq* 
 
 We can cast the expressions for the charge at any instant 
 left in the condenser into more convenient forms. First, 
 consider the dead-beat case (equation 116) is 
 
 where m l and m. 2 are the real roots of the quadratic equation 
 
 * For the sake of readers wishing to pursue the subject we give here a 
 few references, to original Papers, in which are included some collected 
 by Mr. Tunzelraann in a series of articles on Electrical Oscillations in 
 The Electrician of September 14, 1 888, and succeeding numbers. 
 
 Feddersen, PoggendorfFs Annalen, Vol. CIIL, p. 69, 1858 ; Vol. CVIIL, 
 p. 497, 1859; Vol. CXIL, p. 452, 1861 ; Vol. CXIIL, p. 437, 1861 ; Vol. 
 CXV., p. 336, 1862 ; Vol. CXVI., p. 132, 1862. 
 
 Paalzow, Pogg. Ann., Vol. CXIL, p. 537, 1861 ; Vol. CXVIIL, p. 178, 1863. 
 
 Bernstein, Pogg. Ann., Vol. CXLII., p. 54, 1871. 
 
 Heltuholtz, Monatsberichte der Berl. Akad., 1874. 
 
 Kirchoff Gesammdte Abhandlungen, p. 168, containing remarks and 
 criticisms of Feddersen's results. 
 
 Von Oettingen, Pogg. Ann., Vol. CXV., p. 115, 1862; also Jubelbaud, 
 p. 269, 1874. 
 
 L. Lorenz, Wiedemann's Annalen, Vol. VII., p. 161, 1879. 
 
 Schiller, Pogg. Ann. Vol. CLIL, p. 535, 1872. 
 
 Mouton, These, Paris, 1876, Journal de Physique, Vol. VI., pp. 5 and 
 46, 1876. 
 
 Kolacek, Beiblatter en Wiedemann's Annalen, VoL VII., p. 541, 1883. 
 
 Oleareky, Verhandlungen der Academic von Kralcau, Vol. VII., p. 141, 
 1882. 
 
 Oberbeck, Wiedemann's Annalen, Vol. XVII., pp. 8161,040, 1882; Vol. 
 XIX., pp. 213 and 265, 1883. 
 
 Bichat et Blondlot, Comptes Rcndus, VoL XCIV., p. 1,590, 1882.
 
 DYNAMICAL THEORY OF INDUCTION. 383 
 
 and as a = - and 6 = __, we have m, = - _ + /_?! _. l 
 
 L CL' 2L V4L 2 CL' 
 
 which we will write as - a + ft, and similarly, m 2 is -a- f3. 
 
 The constants A and B are determined by the condition that 
 when t = the charge q is the original charge Q ; 
 
 hence Q = A + B, (119) 
 
 and since the current i at any instant is the rate of loss of 
 charge, or - _?, we have i = - - = - A m^ e m i i - B m 2 e m z t , 
 
 when t = 0, i = Q. 
 
 Hence A.m 1 + 'Bm t = (120) 
 
 From these two equations (119) and (120) A and B are deter- 
 mined in terms of m l and m 2 , or of a and (3, and we find 
 
 Let the quantity be called T 1 and let . be called 
 
 a p a + p 
 
 T 2 , then it is easily seen that A = TI Q, and B = - T ^ Q, 
 
 and the equation for q may be written 
 
 The ratio of the potential v of the condenser at any instant 
 to its original potential V is the same as that of q to Q. 
 
 The two quantities T x and T 2 are such that their sum is 
 equal to C E and their product to C L statements easily 
 verified by taking the values of T! and T 2 in terms of a and /?, 
 
 T? / T?2 i 
 
 and recollecting that a stands for , and ft for * / __ - _.--. 
 
 A Lt v 4 LJ (J Jj 
 
 Hence also the current * at any instant is given by the 
 equation 
 
 (122) 
 
 These two equations (121) and (122) contain the complete 
 solution of the discharge in the dead-beat case, giving the 
 current, potential and quantity at any instant reckoned from 
 the moment of closing the circuit of the condenser.
 
 384 DYNAMICAL THEORY OF INDUCTION. 
 
 Suppose that the discharging circuit possesses no inductance, 
 then L = 0, and the equation reduces to 
 
 In the above expression the product E C, or the product of the 
 resistance of the discharging circuit and the capacity of conden- 
 ser, is a quantity of the dimensions of a time, and is called the 
 time constant of the condenser. It represents the time in which 
 
 the charge of the condenser falls to -th part of its original value 
 ( being 2-71828). Let E C be denoted by T. Then if we begin 
 with a charge Q, in a time T the charge left is . In a time 
 
 2 T it is 5, and in a time n T it is 9. Now, since (? = (2-71828) 3 , 
 
 6 6 
 
 or nearly 20, and e 4 is nearly 54, it follows that in time 7 T 
 only one-thousandth of the original charge remains, and in a 
 time 21 T only one thousand millionth ; so that in a period of 
 time equal to 5 or 6 times the length of the time constant the 
 condenser is practically discharged. If the discharging circuit 
 possesses inductance then in the dead-beat case there are two 
 time constants of unequal importance. These are the quantities 
 we have called T x and T 2 above. T : is the larger of the two. 
 The rapidity of decay of the charge with an inductive dis- 
 charger depends chiefly on T : . For if we refer again to equa- 
 tion 121, we see that q will become zero when the quantity 
 
 -L - t 
 
 in the bracket, viz., the function {T^ Ti-T 2 e 1's}, becomes 
 
 zero. 
 
 Starting with given values of Tj and T 2 depending on the 
 values of L, C, and E, and knowing that T! is greater than 
 T 2 , the function starts with a value equal to T x - T 2 when t = 0, 
 and as t increases without limit both exponentials tail away 
 down to zero ; but since T : is greater than T 2 , the first expo- 
 
 _ t 
 nential,viz., e TJ, is longer getting down to practical zero 
 
 _^ 
 
 than the other. Hence, the evanescence of e T I practically 
 determines the time of discharge of the condenser, and we 
 may call T : the principal time constant of the system.
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 385 
 
 If we call the expression __ 2 A, then bearing in mind that 
 
 1 
 
 ^ 
 
 where a = anc 
 
 /E^_J_ 
 V 4r,a ni.' 
 
 we can express T t and T 2 in terms of A and C E or T, and we 
 have by simple substitution 
 
 T,.. 2TA 
 
 and 
 
 1- Vx-4A' 
 2TA 
 
 l+Vl^4A' 
 and the product T x T 2 = T 2 A. 
 
 Hence, if a horizontal line is taken, on which the values of 
 A are set off (see Fig. 138), and values for T t and T a plotted 
 
 FIG. 138. 
 
 off vertically, the locus of the extremities of these ordinates is 
 a parabola. In the figure, lengths along 1 represent values 
 of A, and the corresponding values of T : and T 2 define a 
 parabola P M 0, such that P = T = C E, and the ordinates of 
 the upper portion P M of the curve are the values of T v and 
 those of M are those of T 2 . The value of A = is the abscissa 
 
 A, for which T 1 = T 2 , for when 
 
 then (3 = 0, and in 
 
 cc
 
 386 DYNAMICAL THEOEY OF INDUCTION. 
 
 this case T 1 = T 2 , and T l has its minimum value. For this 
 particular value of A, which is just the value when the dis- 
 charge ceases to be dead-beat, and becomes oscillatory that 
 
 is, when - = TTT or -TT^> = \ ^ ne ^ me constants have equal 
 4 L 2 C L C K 
 
 values, and TJ. becomes a minimum. Hence, for this particular 
 value of the inductance the time of discharge of the condenser 
 is a minimum, and less, therefore, than the time of discharge 
 when the discharge circuit has no inductance.* 
 
 Turning next to the case when the inductance of the dig- 
 
 charge circuit is such that A is greater than , or when _L. is 
 
 ^ 4L 2 
 
 less than - , we have to consider the periodic function which 
 CL 
 
 then applies. 
 
 Referring to equation 118 for the value of q in terms of t we 
 have = e 
 
 where a = - _^ as before, but /3 now stands for / __ - J*L 
 From the conditions that q = Q when t = 0, and that when 
 
 t = 0, t = ^f = 0, we find that P = Q and P : = Q |. 
 Hence, q = Qe~>*< |cos/3 + 
 
 ^- 
 On the convention that y is such an angle that 
 
 we can write the above expression 
 
 sin 7 
 
 and t = l?= Q e 
 
 dt /2LC 
 
 Hence, we see that the expression for the currents and for the 
 remanent quantity of electricity at any time t consists of a 
 periodic part, which is a sine function, and a decreasing part, 
 
 * This appears to have been first noticed by Dr. W. E. Sumpner (Phil. 
 Mag., June, 1&77), and discussed by Prof. Oliver Lodge in an interesting 
 paper in The Electrician for May 18, 1888, p. 59, from which article some 
 portion of the above paragraph and figures have been taken.
 
 DYNAMICAL THEORY OF INDUCTION. 387 
 
 which is an exponential function, and that the rate of decay 
 of the maxima of the waves is determined by the value of 
 
 p Q T 
 
 __; in other words, _ is the time constant for the oscillatory 
 form of discharge. 
 
 This is expressible as 2 T A. in our notation, and is, hence, 
 simply proportional to A. In Fig. 138 the variation of the 
 time constant T 3 , or _ , for oscillatory discharge is represented 
 
 by the straight line M Q. 
 
 The really important part of the time constant curve is the 
 part P M Q, consisting of a bit of a parabola and a straight 
 line, and having a minimum ordinate corresponding to A = ^. 
 
 The current at different times for the two cases A = and 
 X = % are plotted in Figs. 139 and 140. 
 
 For A = I r we have T 3 = T, since T 3 = 2 T A. In other words, 
 
 the time of discharge of the condenser when 2 = | is the 
 
 C B 
 
 same as when L = 0, and just double that when A = ^ ; and in 
 this last case the rate of discharge is a maximum. Hence, so 
 far from reducing the rate of discharge, a little self-induction 
 in the discharge circuit is a positive help to the condenser in 
 getting rid of its charge. Dr. Sumpner* has pointed out that 
 since a lightning discharge resembles that of a condenser, a 
 little inductance in a lightning rod may assist matters instead 
 of blocking the way of the discharge. 
 
 A pendulum swinging in treacle was long ago suggested by 
 Lord Eayleigh as a mechanical analogue to the Leyden jar dis- 
 charge. Dr. Lodgef has pointed out that we may make the 
 analogy exact by considering a loaded spring bent aside or 
 compressed in a resisting medium in such way that gravity is 
 not concerned in the motion and then let go. 
 
 The pliability of the spring corresponds to the capacity of 
 the condenser, its displacement to the electric charge. The 
 load or inertia corresponds to the self-induction of the circuit ; 
 the viscosity of the fluid to its resistance. If the viscosity 
 friction be supposed to vary accurately as the speed, then the 
 equation of motion is 
 
 m^-Rv = Rx, 
 
 dt ' 
 
 Loc. cit. t See The Electrician, May 18, 1888, p. 41. 
 
 Od2
 
 388 DYNAMICAL THEORY OF INDUCTION. 
 
 where x is the displacement and v the velocity = - . Writing 
 
 i ilt 
 
 L for m, and _ for C, and x for Q, we have the condenser 
 
 \ 
 
 FIG. 139. 
 
 Curve I. represents the strength of the discharge current of a condenser in a circuit 
 of no self-induction. T =S R. This curve corresponds to the point P in Fig. 138. 
 
 Curve II. represents the strength of the discharge current ot the same condenser 
 in a circuit of the same resistance, but with self-induction enough just to bring the 
 discharge to the verge of oscillation, this being the condition which effects complete 
 discharge in the shortest time possible. This curve corresponds to the point M in 
 
 Charge 
 in Jar 
 
 o TO TO i zero 2 TO Time 
 
 FIG. 140. 
 
 Curve I. shows the charge remaining in the jar at any time, the circuit being 
 practically devoid of self-induction. 
 
 Curve II. shows the same thing for L=J S R2 that is, for the quickest discharge 
 possible. At first Curve I has the advantage, but at a time 1-26 K 8 the second 
 curve overtakes it and discharges the jar more rapidly. 
 
 equation (115); the two are seen to be the same, and every- 
 thing we have said of the electrical problem applies to the 
 mechanical one.
 
 DYNAMICAL THEORY OF INDUCTION. 389 
 
 It is obvious mechanically that if the resistance is moderate 
 and the mass considerable, the recoil of the spring will be 
 accompanied by oscillations, and that with great resistance and 
 small inertia the motion will be a slow sliding back without 
 oscillation ; and there must exist between the strength of 
 the spring, the mass of its load, and the viscosity resistance 
 of the medium some definite relation which shall constrain 
 the recoil to be dead beat, just returning to the original 
 position of equilibrium without overshooting the mark. This 
 relation is now seen to be 
 
 and under these circumstances the recovery of the spring is 
 effected in the shortest possible time. 
 
 In addition to the experimental researches of Blaserna, to 
 which reference has been made at page 246 et seq., very 
 extensive experiments have been made by Bernstein* and by 
 Moutonf on the subject of electrical oscillations in the case of 
 induced currents. Bernstein's experiments were made with a 
 revolving wheel interrupter, which closed a primary circuit, and 
 for a very short time, at a determinable period after the closure 
 of the primary, put the secondary circuit in series with a delicate 
 ballistic galvanometer. In this way the state of the secondary 
 circuit could be investigated at various instants of time after 
 closing or opening the primary circuit, and the general results 
 of Blaserna were confirmed. In Mouton's experiments a rather 
 different form of commutator (see Jamin's " Cours de Physique," 
 Vol. IV., p. 201, third edition) was employed to break a, 
 primary circuit and to examine with a quadrant electrometer 
 the electrical state of the terminals of an open secondary 
 circuit at various instants afterwards. Mouton found that a 
 potential difference declared itself at less than one four-mil- 
 lionth of a second after rupture of the primary, and that this 
 potential difference died away with decreasing amplitude by 
 rapidly reversing sign, thus indicating the existence of electrical 
 oscillations set up in the open secondary circuit. The duration 
 of the first semi-oscillation was greater than that of succeeding 
 ones. In the case of a secondary circuit of 13,860 turns he 
 
 * Pogg. Ann., Vol. CXLIL, p. 54, 1871. 
 
 t "Etude Experimental BUT lea Phdnom&nea d'Induction Electrodyna- 
 mique." The'se de Doctorat, 1876.
 
 390 DYNAMICAL THEORY OF INDUCTION. 
 
 found that the first semi-oscillation had a duration of 110 
 millionths of a second, and the succeeding ones about 77 
 millionths of a second, and he was able to count about 30 
 complete oscillations. 
 
 7. The Function of the Condenser in an Induction Coil. 
 
 Fizeau appears to have been the first* to suggest that the 
 action of an induction coil employed for raising the electro- 
 motive force of a current would be increased by the employ- 
 ment of a condenser. Its mode of use is as follows : Let P 
 be a primary circuit which takes current from a few cells of a 
 battery, and let I be an interrupter in the primary circuit, either 
 automatically worked by the magnetisation and demagnetisation 
 of the iron core or by any other means. Let S be a secondary 
 circuit of many more turns and high resistance. Under these 
 circumstances each break of the primary current is accompanied 
 by the production of an electromotive force in the secondary cir- 
 cuit capable of producing a discharge across an air space in the 
 secondary circuit. This electromotive force in the secondary is 
 increased by any action tending to increase the suddenness of 
 the stoppage of the primary current, and decreased by anything 
 promoting a spark at the points of rupture of the primary circuit. 
 Fizeau found that if a condenser, formed of alternate sheets of 
 tinfoil and mica or paraffined paper in such fashion as to form 
 a Ley den jar, has its two opposite coatings connected with the 
 two extremities between which the rupture of the primary 
 circuit takes place, then the electromotive force in the secon- 
 dary circuit under these circumstances is increased. In 
 most current text-books this action is explained by saying 
 that the extra current in the primary circuit, instead of 
 being expended in making a spark at the contact points, 
 darts into the condenser and hastens the decay of the primary 
 current. This explanation as generally given is, however, very 
 imperfect. A more complete examination of the nature of the 
 condenser action has been given by Lord Kayleigh (Phil. Mag., 
 Vol. XXXIX., 1870, p. 428, et seq.). In the experiments there 
 detailed a sewing needle was submitted to the magnetising action 
 of an induced secondary current produced by the " break " of 
 the current in a primary circuit. In some previous experiments 
 * Comptes Rendus, Vol. XXXVL, p. 418, 1853.
 
 DYNAMICAL THEORY OF INDUCTION. 391 
 
 by the same writer (PJiil. Mag., July, 1869, p. 9) it had been 
 shown that the magnetising effect of the secondary current 
 was, cet. par., proportional to the initial strength of the in- 
 duced current, and that this initial strength was proportional 
 to the quotient of M by N, or to the value of the ratio of the 
 coefficient of mutual induction to the coefficient of self-induc- 
 tion of the secondary circuit. It was then found that the mag- 
 netising effect of the secondary current was greatly increased 
 by connecting the plates of a condenser respectively to the 
 two points between which the break of the primary circuit 
 occurred. The complete investigation of the values of the 
 induced and primary currents would under these conditions be a 
 good deal more complicated than the investigation of the more 
 simple case of the discharge of a condenser through a single 
 inductive circuit. We are here, however, only concerned with 
 the first part of the electrical motion, the manner in which the 
 currents wear down under the action of the resistances being 
 of subordinate importance. It appears that when the electrical 
 motion is decidedly of the oscillatory type the first few oscil- 
 lations will take place almost uninfluenced by resistance, and 
 on this supposition the calculation (following Lord Eayleigh) 
 becomes remarkably simple. 
 
 Let L, M and N be the primary, mutual and secondary in- 
 ductance, and let i and i' be the primary and secondary current 
 strengths at any instant, and q and q' the quantities of elec- 
 tricity which have flowed through these circuits from the 
 instant of beginning to reckon the time t, 
 
 then <*_-, and ^-f; 
 
 dt dt 
 
 and if we neglect resistance effects, as we can do at the instant 
 after "breaking" the primary circuit, and call C the capacity 
 of the condenser bridging across the " break " of the primary 
 circuit, the equations giving the values of the primary and 
 secondary current i and i' at the instant after breaking the 
 primary circuit are 
 
 L^ + M^'+^O. . . . (123) 
 dt dt (j 
 
 M^ + N^l' =0. ... (124) 
 
 dt dt
 
 392 DYNAMICAL THEORY OF INDUCTION. 
 
 Eliminating i' we have 
 
 (125) may be written 
 
 A differential equation of this type always indicates an 
 oscillatory motion. For, consider the simple periodic function 
 
 x = A sin p t, where p = _^, T being the periodic time of the 
 motion, we have _5=^ A cos p t, and * = - p* A sin p t ; 
 
 (It (i t 
 
 hence, 3 ^ + p z x = Q, and therefore x = A sin pt is a particular 
 
 solution of this equation. 
 
 In the above differential equation p is seen to be TT times 
 the frequency of the oscillation. 
 
 Accordingly, equation (126) indicates an oscillation of the 
 primary current, of which the periodic time is equal to 
 
 and this is the periodicity of the electric oscillation set up in 
 the primary at the first instant after "break." 
 
 Equation (124) gives by integration the connection between 
 i and i', and it is 
 
 Mi + N t" = constant, . . . (127) 
 
 which shows that the currents in the primary and secondary 
 oscillate synchronously, the maximum of the one coinciding 
 with the minimum of the other. Since i' is zero at the 
 instant of " break," the constant in equation (127) must be 
 equal to M I, where I is the current strength in the primary 
 just before " breaking " primary circuit. 
 Accordingly, we have 
 
 ./ JV1 /T \ 
 
 .=-(!-*), 
 
 so that when, after half an oscillation of the primary, i becomes 
 equal to - 1, we have 
 
 i' = 2I (128)
 
 DYNAMICAL THEORY OF INDUCTION. 393 
 
 This equation gives us the initial value of the secondary 
 current i' in terms of the value of the primary current just 
 before the " break " when the condenser is used. Comparing 
 equation (128) with the results on page 236, where it is shown 
 that, if the condenser is not used across the " break " of the 
 primary, the initial value of the secondary current under the 
 
 assumption of a perfectly sudden break is equal to I, 
 
 we see that the value of the secondary current just imme- 
 diately after the break of the primary, is double that which 
 is there deduced as the value when the primary is simply 
 suddenly stopped without the intervention of the condenser. 
 Stripped of symbolism, what the above amounts to is this : if 
 a condenser is inserted across the " break points " of a primary 
 circuit, then on breaking the primary circuit, the primary 
 current continues to run on into the condenser for a short time ; 
 it then rebounds, and is reversed in sign, retaining initially its 
 full strength. The electromotive force set up in the secondary 
 circuit is then the result of a stoppage of a primary current and 
 its immediate reversal in direction, and this is equivalent to the 
 removal of a certain number of lines of induction from the 
 secondary circuit, and their immediate insertion into it in the 
 opposite direction. Hence, when a condenser is so employed, 
 the inductive electromotive force in the secondary must be just 
 double that which it would be if there were no such rebound 
 of the primary. The condenser acts by setting up electrical 
 oscillations, and it does away with the spark, or largely dimi- 
 nislies it, in virtue of the fact that the condenser acts at the 
 moment of " break" as if it were a shunt circuit of negative 
 self-induction, only with this difference that instead of dissi- 
 pating energy like a conducting circuit it returns it again to 
 the primary circuit in the form of a reversed current, and 
 increases the total change of induction through the secondary 
 circuit in the short interval of time immediately succeeding 
 the " break." 
 
 Since the sparking distance of the secondary current depends 
 on the initial electromotive force in the secondary that is, on 
 the maximum of the electromotive force we see that the con- 
 denser so applied can greatly increase the sparking distance 
 of the secondary discharge.
 
 394 DYNAMICAL THEORY OF INDUCTION. 
 
 The action is essentially a phenomenon of resonance. The 
 condenser causes an elastic recoil in the current and enables 
 the electro-kinetic energy of the steady primary current to he 
 utilised in producing secondary electromotive force rather than 
 suffer dissipation in the form of a contact spark. In order to 
 he efficient in quenching spark the capacity of the condenser 
 must be great enough to take the full primary current, or 
 to receive charge at a rate equal to the delivery of the full 
 primary current for a time during which the contact or break 
 points are separating to a distance too great to permit of much 
 sparking jumping across. There is a certain capacity of 
 condenser suitable for any given coil which produces the most 
 beneficial result in quenching contact spark and lengthening 
 secondary spark. The required capacity is best determined by 
 trial, since the experimental data necessary to furnish the 
 means to calculate it would be probably more difficult to obtain, 
 owing to the fact that it will be determined by several variables, 
 viz., the effective resistance and inductance of the primary 
 circuit, the rate of breaking, and probably also by the ampli- 
 tude of movement of the " break points." If the primary coil 
 of an induction coil is traversed by an alternating current then 
 the condenser as ordinarily used becomes superfluous. It will 
 be remembered that the late Mr. Spottiswoode obtained 
 secondary sparks of great magnitude from his large coil by so 
 using the alternating current of a De Meritens machine. 
 
 If a condenser is discharged through a circuit of which the 
 resistance is so small that it may be neglected in numerical 
 comparisons, then the equation of discharge is 
 
 where the symbols have the same signification as before. As 
 above explained, this indicates that the discharge is oscillatory, 
 and that the time of a complete oscillation is 2?r ^/ITC. 
 
 In describing the experiments of Blaserna we saw that the 
 frequency of the electrical oscillation set up in circuits on 
 starting and stopping currents in them could be reckoned by 
 tens of thousands per second. In the case of Leyden jars dis- 
 charged through very short circuits, the frequency may rise to 
 numbers reckoned by millions per second. Since the frequency
 
 DYNAMICAL THEORY OF INDUCTION. 395 
 
 of luminous vibrations falls between 400 and 700 billions 
 per second, these condenser oscillations fall in frequency in the 
 gap between the acoustic and luminous vibrations. 
 
 It is of interest here to note that since these electrical 
 oscillations in a circuit are creating pulsatory electrical dis- 
 turbances, which spread out from the wire laterally, the 
 wire in which the electrical oscillations are going on is 
 virtually emitting "light," although not such light as can 
 affect our eyes. The ether waves in the case of these elec- 
 trical disturbances are too long to be eye-affecting. If the 
 velocity of a wave disturbance is V, and the wave length is A, 
 and the frequency of the oscillations corresponding to this 
 wave length is n, then V = n X, for the wave motion travels 
 over the length of one wave in the time of one complete oscil- 
 lation. In the case of ether disturbances we have seen that 
 V is 3 x 10 10 centimetres per second, or 186,000 miles per 
 second. Hence when the frequency of the electrical oscilla- 
 tions is known, the wave length of the lateral disturbance 
 emitted can be found. According to Dr. Lodge, a microfarad 
 condenser discharging through a good conducting coil having 
 an inductance of one henry gives a current alternating 160 times 
 in a second, and emits ether waves about 1,200 miles long. 
 A gallon Ley den jar (capacity about 0-003 microfarad) dis- 
 charging through a stout wire suspended round an ordinary 
 sized room emits ether waves between three and four hundred 
 yards in length, its current alternating at the rate of about one 
 million per second. A pint Ley den jar sparking through an 
 ordinary pair of discharging tongs gives a current of 15 million 
 alternations per second, with ether waves some 20 yards in 
 length. An ordinary electrostatic charge on a sphere two feet 
 hi diameter, if disturbed in any way, will surge to and fro at 
 the rate of 300 million vibrations per second, emitting ether 
 waves a yard long. Electric charges on bodies of atomic 
 dimensions, if able to oscillate at all, would vibrate thou- 
 sands of billions of times a second, and produce ultra-violet 
 light. 
 
 The ordinary use of a condenser with an induction coil shows 
 how it can be employed to neutralise the effect of self-induction 
 in a circuit. We have considered on page 187, "31, of Chapter 
 IV., the case of a condenser having its terminals shunted by a
 
 396 DYNAMICAL THEORY OF INDUCTION. 
 
 resistance, and the combination placed in series with an 
 inductive circuit and there shown that capacity can neutralize 
 self-induction. We may also consider the case of a condenser 
 in parallel with an inductive circuit as another similar problem. 
 Let L B (Fig. 140) be an inductive circuit, and let the ter- 
 minals a ft be closed by a condenser C of capacity C. Let L be 
 the inductance and B the resistance of the coil. Let i be the 
 value at any instant of a simple periodic current sent through 
 the relay and condenser hi parallel, and let i v i 2 be the simul- 
 taneous current strengths at that instant in the condenser 
 circuit and the coil circuit. As the potential difference of the 
 points a and b oscillates, an ebb and flow of current is produced 
 in the condenser circuit ; the condenser, in fact, is charged and 
 discharged by the periodic current ; also a periodic current is 
 produced in the inductive circuit L E. The current in L B 
 
 LR 
 
 lags in phase behind the impressed electromotive force or 
 potential difference of the points a b, and the current flowing 
 into the condenser lags 90deg. in phase behind the same im- 
 pressed electromotive force. From this it results that the 
 mean current through the inductive circuit may, under some 
 circumstances, be greater when the condenser is joined up to 
 its ends than when it is not so joined ; its effective self-induction 
 is thereby lessened, and it acts as if it had experienced a 
 diminution of self-induction. The condition most favourable 
 for producing this result may be investigated as follows : 
 
 Let v be the potential difference of the points a and b at the 
 instant when the current in the undivided circuit is i and that 
 in the branches is i, and ^. We then have, by the principle 
 of continuity, 
 
 (129)
 
 DYNAMICAL THEORY OF INDUCTION. 397 
 also t, = C ^, (130) 
 
 and L+Bv=, .... (181) 
 
 and we may take the original current before division to be 
 simply periodic, and to be expressed by 
 
 i = Iswpt ..... (132) 
 where I is its maximum value. 
 
 Then by elimination of v and t"i and i from the above four 
 equations we arrive easily at the equation 
 
 OL^ + CB^ + t.-Isinpt. . . (133) 
 
 Now, since i z must be a simple periodic current lagging in 
 phase behind that of the undivided current i, we may take t' 2 
 to be of the form 
 
 4 = I 2 sin(pi-<9) ...... (134) 
 
 I 2 being the maximum value of t' 2 , and 9 its phase lag be- 
 hind Ij. 
 
 Hence, by differentiation of (134) and substitution in (133) 
 we arrive at 
 
 (1 - CLp 8 ) I 2 sin (pt - 6) + C E P I 2 cos (pt- 0) = I sin;? t, (135) 
 which by the lemma on page 161 may be written 
 
 IV (1 - C L p*)* + C 2 BV (sin^ t - + <) = I sin p t. (136) 
 
 Both sides of this last equation are the expressions for the 
 same thing, viz., the value of i, and hence, equating the 
 coefficients, we have 
 
 This gives us the value of the ratio of the maximum or mean 
 values of the strengths of the undivided current and the 
 current in the inductive circuit. If we differentiate the 
 right-hand side of (137) with respect to C, and apply the 
 usual criterion to ascertain whether we have a maximum 
 or minimum value, we find that the expression on the right- 
 hand side of (137) has a minimum value when 
 
 C- L . (138)
 
 393 DYNAMICAL THEORY OF INDUCTION. 
 
 In other words, if the capacity of the condenser is so chosen as 
 to have a capacity equal numerically to the quotient of the 
 inductance hy the impedance of the coil, then, under these 
 circumstances, the mean strength of the current in the coil 
 circuit will be greater than the mean strength of the current 
 before subdivision ; and it is easily seen, by substituting in 
 equation (137) the value of C given by (138), which makes the 
 ratio of current strength a minimum, that with this value of 
 the capacity the strength of the current in the inductive coil 
 is to the strength of the current before division in the ratio of 
 the impedance to the resistance of the inductive circuit. 
 
 The expression (138) gives the value of the condenser capacity 
 which will produce the required result of minimising the self- 
 induction of a relay of resistance E and inductance L when 
 applied to it. Another problem of a like kind, but not so 
 practically useful, is the investigation of the behaviour of a 
 condenser when joined in series with an inductive coil and 
 traversed by a simple periodic current. Let a condenser of 
 capacity C be joined in series with an inductive circuit of 
 resistance E and inductance L, and let a simple periodic 
 current of frequency n be sent through the two in series. It 
 is not difficult to show that, if we take p for 2?r n, as usual, 
 and if the capacity and inductance are so related to the 
 
 frequency of oscillation that p = . , then, under these 
 
 v L C 
 
 circumstances, the condenser just annuls the self-induction 
 of the coil, and the two together permit the passage of the 
 same current which would traverse the coil in virtue of its 
 resistance E, assuming it to have no inductance. This is 
 easily proved as follows : Let L be the inductance and E the 
 resistance of the inductive circuit, and C the capacity of the 
 condenser in series with it. Let v = V sin p t be the potential 
 difference at the instant t, measured over the condenser and 
 inductive resistance, and let v : and v 2 be the fall of potential 
 down the inductive circuit and condenser respectively. Then 
 v = v 1 + v z ; (139) 
 
 also, L^ + Bi-i*, (140) 
 
 and
 
 DYNAMICAL THEORY OF INDUCTION. 399 
 
 where i is the value of the current flowing in the circuit at 
 the instant when the potential difference between the ends of 
 the whole circuit is v. 
 
 Hence by substitution we have 
 
 *=Vsinj>*. . . (142) 
 
 The current being also simply periodic must have a value i 
 expressed by the equation, 
 
 t = Isin(p*-0), .... (143) 
 
 since it will differ in phase by an angle from the potential 
 difference v. 
 
 Accordingly, we find that by differentiating (143) and 
 substituting the values in (142) we arrive at the equation 
 
 (1 - C L /) sin (p t - 0} + E C p cos (pt-6} = cos p t. 
 
 The maximum value of the current, viz., I, is therefore 
 given by the expression 
 
 If then 1 = C Li> 2 or p = , we see that the above equation 
 
 vCL 
 
 y 
 
 reduces to I = ^, 
 
 Jtt 
 
 and the whole circuit of coil and condenser is equivalent to a 
 simple non-inductive circuit of resistance R, in other words 
 the inductance is annuled. Hence a certain relation between 
 the inductance, capacity, and frequency, causes the inductance 
 to be neutralised by the capacity, and the whole circuit to be 
 effectively non-inductive. 
 
 8- Impulsive Discharges and Relation of Inductance 
 thereto. If between the ends of a conductor a difference of 
 potential is created which is brought about slowly, the result 
 shows itself in a current in the conductor, and the resulting 
 current is determined as to strength by the mode of variation 
 of the potential and by the capacity as well as by the induct-
 
 400 DYNAMICAL THEORY OF INDUCTION. 
 
 ance and ohmie resistance of the conductor. If, however, the 
 difference of potential is created with great suddenness, the 
 resulting electric flow is less determined by the true resist- 
 ance, and more by the inductance of the conductor. In this 
 case we have the phenomena of impulsive discharges. We have a 
 mechanical analogy in the case of impulses or sudden blows 
 given to heavy bodies, which well illustrates how strikingly 
 force phenomena may be altered when for steady or slowly 
 varying forces we substitute exceedingly brief impulses or 
 blows. If an explosive, such as gun-cotton, is laid on a stone 
 slab in open air, and simply ignited, it burns away with com- 
 parative slowness ; the slab is uninjured, and the evolved gases 
 simply push the air away to make room for themselves. But 
 it is well known that by means of detonators the same explo- 
 sive can be fired with enormously greater rapidity, and in this 
 case the blow or impulse given to the air is so sudden that it 
 has not time to be pushed away, and in virtue of its inertia its 
 incapacity of receiving a' finite velocity in an infinitely small 
 time bestows on it an inertia resistance, which causes nearly the 
 whole of the effect of the explosion to take effect downwards 
 on the slab, and this last is shattered. The inductance oi 
 conductors introduces a series of phenomena which are the 
 electrical analogues of the above mechanical experiment. We 
 have seen that the counter electromotive force of self-induc- 
 tion is proportional to the rate of change of another quantity, 
 called the electro-kinetic momentum, and this quantity phy- 
 sically interpreted is the total flux of induction or number 
 of lines of induction enclosed by the conducting circuit at 
 that instant. A conductor of sensible inductance can no more 
 have a current of finite magnitude created in it instantaneously 
 than a body of sensible mass can have a finite velocity in- 
 stantaneously given to it. In both cases there is an immense 
 resistance to very sudden change of condition. A very loose 
 plug of snow or earth stuffed into the muzzle of a loaded gun 
 will cause it to burst when fired, since the inertia resistance 
 of the plug to very sudden motion is exceedingly large, 
 though the frictional resistance may be small. Accordingly, 
 the study of the behaviour of conductors under exceedingly 
 sudden electric blows or electromotive impulses leads us to 
 consider some very interesting effects. We shall best eluci-
 
 DYNAMICAL THEORY OF INDUCTION. 401 
 
 date these effects by describing some interesting and suggestive 
 experiments due to Dr. Oliver Lodge.* 
 
 His first experiment is called the experiment of the alternative 
 path. The two terminal knobs of a Voss or Wimshurst electrical 
 machine (see Fig. 141) are connected to the two inside coatings 
 of a pair of Ley den jars. The two outside coatings are con- 
 nected to the balls of another discharger, B, and the terminals 
 of this discharger are short-circuited by a metal wire, indicated 
 by the dotted line. The Leyden jars stand on a badly insu- 
 lating wooden base. On turning the handle of the electrical 
 machine the inside coatings receive equal and opposite elec- 
 trical charges, and there is an induced charge on the outer 
 
 L 
 
 Fio. 141. 
 
 coating of each, which, in the language of the old school of 
 electricians, was called the " bound " charge. When the differ- 
 ence of potentials of the inner coatings reaches a certain value 
 the air space at A is cracked, and a spark passes, discharging 
 the inner coatings of the jars. At that instant the charges of 
 the outer coatings are set " free," or, in modern language, the 
 potential of one rises and that of the other falls. The effect of 
 this is that whereas before the spark passed at A the balls at B 
 
 * The account of these experiments is taken from the report of Dr. 
 Lodge's Mann Lectures before the Society of Arts. These suggestive 
 lectures were reprinted in The Electrician, entitled " Protection of Build- 
 ings from Lightning," Vol. XXL, pp. 234, 273, 302. 
 
 D D
 
 402 DYNAMICAL THEORY OF INDUCTION, 
 
 were at equal potential, on a spark at A happening the balls 
 at B are instantaneously brought to a very great difference 
 of potential. It might be thought that since the balls are 
 short-circuited by a metallic wire this difference of potential 
 will expend itself on making a current in the wire. On the 
 contrary, very little of the discharge may take place through 
 the wire. A spark passes at B, or, in other words, the 
 discharge passes in great part across the exceedingly highly 
 resisting air space at B, rather than take the circuit of the 
 metallic wire of very low resistance, so that although there is 
 a divided circuit open to the discharge, one branch of which 
 measures hundreds of thousands of ohms or megohms and 
 the other only a small fraction of an ohm, it nearly all goes by 
 the route of higher resistance. The explanation of this is that 
 when the balls at B are thrown with great suddenness into 
 
 FIG. 142. 
 
 opposite electrical states the counter electromotive force of 
 self-induction of the circuit of metal L makes it virtually 
 non-conducting. The electromotive impulse meets with 
 such resistance owing to the electro-magnetic inertia of the 
 circuit that it rebounds and cracks through the air. In order 
 that it shall do this, however, the distance of this air gap 
 at B has to be less than a certain amount. There is a certain 
 critical distance of the knobs B for less than which the dis- 
 charge always jumps across B, and for greater than which the 
 discharge keeps mainly to the metallic circuit. Even if the 
 short-circuiting metal is a thick rod, still when B is not great 
 the discharge chooses the air-gap path. The phenomenon here 
 presented has had fresh interest and attention called to it by 
 Dr. Lodge, but it has really been long a familiar one, though 
 its explanation has not stood out hitherto so sharply as it does
 
 DYNAMICAL THEORY OF INDUCTION'. 403 
 
 Faraday was acquainted with it, and showed that if a charged 
 Leyden jar is discharged by means of a wire crossed or bent so 
 that there was a loop (Fig. 142), the wires at a nearly but not 
 quite touching, then when the spark happened at b, a spark 
 took place also at a, showing that some at least of the discharge 
 jumped across a instead of pursuing the course of the metal 
 loop. 
 
 The same fact lies at the base of the action of all lightning 
 arresters placed on telegraph or telephone instruments. Mr. 
 C. F. Varley, we believe, first suggested that the coils of the 
 single-needle instrument might be protected from damage by 
 lightning by twisting together the earth and line wires where 
 they leave the case, the theory being that although ordinary 
 
 FIG. 143. 
 
 currents were not short-circuited by reason of the cotton 
 covering of the wires, yet lightning discharges would meet 
 with such resistance in the inductive coils that they would 
 jump across the knot from wire to wire rather than pass 
 round and damage the coils. In the same way the ordinary 
 comb protector is supposed to act. Between the line wire 
 L (see Fig. 143) and the electromagnetic instrument, relay, 
 telephone, &c., is placed a metal comb, which has its points in 
 opposition to another comb in connection with the earth, and 
 the other terminal of the electromagnetic instrument is also 
 "to earth. ' ' An incoming current has then two paths open to it 
 to get to earth, one of comparatively low resistance through the 
 instrument, and one of enormously high resistance across the 
 
 DD 2
 
 404 DYNAMICAL THEORY OF INDUCTION. 
 
 air-gap between the comb points. Ordinary currents, steady or 
 periodic, pass entirely through the metallic circuit. Very violent 
 electric impulses, such as a lightning discharge, meet with an 
 enormous inductive opposition in the electromagnetic instru- 
 ment owing to the inability of an inductive circuit to respond 
 to an electromotive impulse instantaneously. Hence the air- 
 gap is cracked, and the discharge passes across the combs and 
 the instrument may be saved. Evidence exists, however, 
 pointing to the fact that the protection afforded by these con- 
 trivances is very far from complete. We are not here concerned 
 with their efficiency as practical devices, but only with them 
 as illustrating the principle of the alternative path and the 
 behaviour of inductive circuits to impulsive discharges. That 
 these devices are insufficient has been fully demonstrated by 
 many experimentalists.* 
 
 In these experiments of the alternative path it was found by 
 Dr. Lodge that the critical distance at which the discharge just 
 prefers to jump the air-gap was greater for a thick copper rod 
 40 feet long (No. 1 B.W.G.) than for an iron wire (No. 27 
 B.W.G.) of 33-3 ohms resistance, indicating a less inductive 
 inertia on the part of the iron ; but this fact is only true for 
 the particular circumstances of the experiment. A very clear 
 difference was established between copper rod and tape, using 
 conductors of the same length and weight. The tape has an 
 advantage in permitting more easily the passage of sudden 
 electric discharges. A controversy on the relative suitability of 
 rod and tape for lightning conductors dates from the time of 
 Faraday and Sir W. Snow Harris, and a possible explanation 
 of the reasons for preferring one rather than the other presents 
 itself when we consider the matter in the light of those con- 
 siderations which induce us to think that an electric current 
 begins always at the surface of conductor, and takes a certain 
 tune to diffuse or soak into the mass of the metal. It is not 
 cross -section but surface which is here concerned ; and, other 
 things being equal, the conductor which offers the greatest 
 surface to the dielectric is able to drain the energy out of the 
 
 * For an account of some interesting experiments by Prof. Hughes and 
 Prof. Guillemin on " Lightning Protectors " see The Electrician, Vol. XXI., 
 p. 301, July 13, 1888. It was found that a protector consisting of two 
 opposed flat plates was better than a comb or opposed ooint.
 
 DYNAMICAL THEORY OF INDUCTION. 405 
 
 dielectric most quickly and dissipate it as heat in the con- 
 ductor. We have referred to this on a previous page (see ante, 
 p. 252), and it will be mentioned again in connection with some 
 views of Prof. Poynting.* With respect to the apparent supe- 
 riority of iron, it would naturally have been supposed that, 
 since the magnetic permeability of iron bestows upon it greater 
 inductance, it would form a less suitable conductor for dis- 
 charging electric energy with great suddenness. Owing to the 
 fact that the current only penetrates just into the skin of the 
 conductor, there is but little of the mass of the iron magnetised, 
 even if these instantaneous discharges are capable of magne- 
 tising iron. This last fact has been thought to be due to an 
 actual time lag of magnetisation, viz., that magnetising force 
 required to endure for a sensible time in order to produce 
 magnetisation, but recent views tend in the direction of con- 
 sidering the apparent lag as a consequence of the fact that the 
 eddy currents produced in the surface layers of the metal by 
 the discharge shield the inner and deeper layers from inductive 
 influence, as described under the head of Magnetic Screening. 
 In any event the final result is the same ; the electromotive 
 impulses, or sudden rushes of electricity, do not magnetise the 
 iron, and hence do not find in it any greater self- inductive oppo- 
 sition than they would find in a non-magnetic but otherwise 
 
 * For some special remarks on the self-induction of wires of various cross- 
 sections see Mr. Oliver Heaviside in the Phil. Mag., January, 1887, p. 11 : 
 " The magnetic energy per unit of length of a circuit is \ L i 2 , where i is 
 the current in the wire and L the inductance per unit of length. As regards 
 the diminution of L in general by spreading out the current in a strip 
 instead of concentrating it in a wire, that is a matter of elementary reason- 
 ing founded on the general structure of L. If we draw apart currents, 
 keeping the currents constant, thus doing work against their mutual 
 attraction, we dimmish their energy at the same time by the amount of 
 work done against their attraction. Thus the quantity L i" of a circuit 
 is the amount of work that must be done to take the current to pieces, 
 so to speak that is, to separate all its filamentary elements of currents to 
 an infinite distance. If wires are taken, each of a unit of length and of 
 the same total cross-sectional area, but of different forms of cross-section, 
 round, square, elliptical, equilateral triangle, narrow rectangle, &c., the 
 ratio of their inductances is the same as the ratio of their torsional rigidities. 
 Thus the narrow strip has the least torsional rigidity, and the circular- 
 sectioned wire the greatest, and this is true also for their relative self- 
 inductions."
 
 406 DYNAMICAL THEORY OF INDUCTION. 
 
 similar conductor. Dr. Lodge's further researches seem to 
 show that there is a real advantage in using iron for lightning 
 conductors over copper, and that its greater specific resistance 
 and higher fusing point enable an iron rod or tape to get rid 
 safely of an amount of electric energy stored up in a dielectric 
 which would not be the case if it were copper. This point is 
 further elucidated by some other experiments of Dr. Lodge. 
 Two tinfoil conductors were prepared of approximately equal 
 resistance and length. One of these was formed into a spiral, 
 each layer being insulated with paraffin paper, and wound on 
 a glass tube. The other was made into a zig-zag or non- 
 inductive resistance. These conductors were then employed as 
 alternative paths, as in the former experiment with the copper 
 wire. In the case when the tin-foil zig-zag was employed to 
 short-circuit the jars it was not possible to get a B spark (see 
 Fig- 141) until the distance of the A balls was shortened to 0-6 
 (tenths of inch). When the tinfoil spiral was used the critical 
 spark distance at B rose to 6-4. When the iron wire bundle 
 was inserted in the tube it did not in any perceptible degree 
 increase this distance. The length of the sparking distance at 
 A was 7'3, and when no alternative path was used at all to 
 connect the jars the critical distance of the B balls, at which 
 sparks sometimes passed and sometimes failed, was 11-1. Here, 
 then, we have the non-magnetisability of iron by sudden dis- 
 charges illustrated. Dr. Lodge has called attention to the 
 fact that a " choking " coil having a core of divided iron and 
 wound over with many turns of wire does not add to the 
 apparent self-induction of a circuit discharging a Leyden jar. 
 It may even diminish it when the discharge is oscillatory and 
 of sufficient frequency, although the oscillations may be as few 
 as 500 per second. This experiment shows, as we know from 
 other facts, that eddy currents are set up even in a core of 
 finely-divided iron, and that these eddy currents, under suffi- 
 ciently rapid alternations, are confined to the surface of the 
 core, and moreover, since they are as regards phase nearly in 
 opposition to that of the current in the coil, they actually tend 
 to diminish the total flux of induction through the coil, and 
 hence diminish the self-induction of the circuit. 
 
 The inductive opposition to electric discharge presented by 
 even a short length of conductor, when the difference of poten-
 
 DYNAMICAL THEORY OF INDUCTION. 407 
 
 tial between the ends is made very suddenly, is seen in the 
 tendency under such circumstances to side flash. If a conduc- 
 tor, say, a straight rod of copper, has one end to earth, and 
 somewhere very near its side is the end of another conductor 
 also " to earth," then if the free end of the first conductor is 
 suddenly exalted in potential the impulsive rush of electricity 
 meeting with such an obstacle in the inductance of the conduc- 
 tor, spits or flashes out laterally and sparks to the other con- 
 ductor. No conductor is able to prevent side flash altogether 
 unless it has practically no inductance. As long as a conduc- 
 tor must be straight (like a lightning conductor) so long will 
 there be a tendency to side flash. This is illustrated by the 
 following experiment. A massive conductor has (Fig. 144) a 
 very fine wire stretched alongside and air gaps in this bye- 
 path left by bringing the ends of the fine wire very near to the 
 sides of the large conductor. On sending an impulsive rush of 
 electricity through the large conductor little sparks are seen at 
 a and b, showing that some of the discharge has left the thick 
 
 FIG. 144. 
 
 conductor and travelled along the fine wire, even although it 
 had to leap across an air gap. If the bare hands are applied to 
 the ends of an open spiral of very stout copper wire, one end of 
 which is connected to a "good earth," shocks will be felt when 
 a Leyden jar is discharged through the copper. In this case 
 the human body forms the bye-path, and the experiment indi- 
 cates that the law of division of steady currents or slow dis- 
 charges between conductors in parallel, viz., a division in the 
 ratio of their conductivities, does not hold good for impulsive 
 discharge, and that the relative inductance of the circuits 
 has more influence in the latter case in determining what 
 happens. 
 
 The distinction between the resulting discharges due to a 
 steady electromotive force or strain and that due to an elec- 
 tromotive impulse or impulsive rush of electricity has been 
 illustrated by some further experiments by Dr. Lodge on the
 
 408 DYNAMICAL THEORY OF INDUCTION. 
 
 behaviour of model lightning conductors when subjected to 
 the action of these two modes of discharge. Two tin plates 
 are placed horizontally and insulated, and these are supposed 
 to represent the earth and a thunder- cloud. These plates are 
 connected, as in Fig. 145, to an electrical machine, and by work- 
 ing the handle are brought up to a steady potential difference. 
 On the lower plate are placed little rods of various heights, 
 sharp, or having knobs, and these represent lightning con- 
 ductors. At a certain potential difference the electric strain 
 set up in the air exceeds the limit which the dielectric can 
 sustain, and it breaks down, giving rise to a spark. A discharge 
 then takes place towards one or other of the mimic lightning 
 conductors. In one experiment three conductors were used 
 one with a large knob, 0-9in. less in height than the distance 
 between the plates, the second with a small knob, 2in. less in 
 height, and a sharp short point. The point even when very 
 
 FIG. 145. 
 
 low prevents discharge altogether. It may be too low to be 
 effective, or it may be insufficient to cope with the supply of 
 electricity if that is supplied very fast, but it acts to prevent 
 discharge. If the point is removed or covered up we then find 
 that the discharge takes place, when the potential difference 
 of the plates is made great enough, to the small knob by 
 preference, and it does so even when the stem of the short 
 knob is lower than that of the large knob. In other words, 
 when the stems are the same height the small knob protects 
 the large one, and it does this until lowered in height to about 
 two inches less than the other; when this is the case both 
 knobs are struck indifferently. And it does this even when a 
 resistance of one megohm is interposed in the stem of the 
 smaller knob. The state of things is, however, very much 
 altered if in place of bringing up the two plates gradually to a 
 sparking potential difference they are very suddenly thrown
 
 DYNAMICAL THEORY OF INDUCTION. 409 
 
 into opposite electrical conditions by connecting them to the 
 electrical machine as shown in Fig. 146. 
 
 The jars charge up as they stand on the same wooden table, 
 and when the potential rises to sparking amount they discharge 
 at A, and a violent electric rush then takes place between the 
 two plates, and the conductors between are struck. If the same 
 three kinds of conductors are used, and they be adjusted until 
 they are all about equally struck, we find that the smaller and 
 shorter-stemmed knob no longer protects the larger one, and 
 the sharp point no longer protects either ; all three, large ball, 
 small ball and point, are liable to be struck equally if at the 
 same height, and if they differ in height the highest is most 
 likely to be struck, no matter what it is. Points are, then, no 
 protection against these impulsive rushes of electricity. The 
 special virtue of a point in the case of the slower-timed dis- 
 charges is that it prepares the path of the discharge to itself, 
 
 FIG. 146. 
 
 for in this case the path is pre-arranged by induction. If one 
 of the conductors has a large resistance say a liquid megohm 
 inserted in it then this one is no longer struck ; it ceases to 
 protect the other conductors even if higher than them, and 
 even if it be so raised in height that it touches the top plate, 
 thus connecting the plates by a bad conductor, the two other 
 conductors get struck with apparently the same ease as before. 
 This indicates that a lightning conductor with a bad earth can- 
 not protect well against discharges of the nature of a sudden 
 rush. Mr. Wimshurst has, however, shown reason for consider- 
 ing that in this experiment the electrical state of plates, as 
 regards sign of electrification, may be of importance. The 
 question how far the point protects from the impulsive rush is 
 not altogether cleared up. It is still sub judice.
 
 410 DYNAMICAL THEORY OF INDUCTION. 
 
 In performing the first experiment of the alternative path 
 (Fig. 141) it was noticed that the B spark was longer than the A 
 spark. Plainly this indicates that the discharge at A sets up 
 electrical oscillations. The manner in which this is brought 
 about is as follows : On the commencement of the discharge 
 the air-space is intensely heated, and its conductivity so far 
 increased that the conditions as to the relation of inductance, 
 resistance and capacity of the discharger and condenser are ful- 
 filled, and the discharge takes the oscillatory form. If a couple 
 of long leads are attached to the A discharger (Fig. 147), the 
 farther ends being insulated, and a discharger B bridged across 
 at BH B.J or B 3 , then it is found that at every discharge at A a 
 spark can be obtained at B, and for a certain length of A spark 
 the B spark will be longer at B 3 than at the nearer positions. 
 Evidently what happens is that the electrical oscillation across 
 
 FIG. 147. 
 
 the A discharge intervals sets up violent surgings to and fro 
 in the open circuit wires, just like water in a long trough 
 when it is tilted, and the recoil at the insulated ends, combined 
 with the inductance of these leads, produces a cross flash at 
 B. It is, in fact, a case of resonance ; the long open circuit 
 leads act like resonators to the oscillating discharge across A, 
 and the nearer the length of the leads approaches to half a 
 wave length or to some multiple of half a wave length the 
 more perfect will be the resonance and the greater the recoil 
 at the open ends, and hence the greater the spark at B 3 . 
 
 If the experiraent is tried in the dark, the B discharger 
 being removed, it is seen that the leads glow at the ends with 
 a vivid brush light at the moment when the jars are dis- 
 charged. When the proper length of open circuit lead has
 
 DYNAMICAL THEORY OF INDUCTION. 411 
 
 been found which resonates best in accord with the jars used 
 as dischargers, then the whole of the effects described can be 
 made to disappear by connecting a very small Leyden jar to 
 the ends of the wires. The increase of static capacity thus 
 given to the leads reduces their potential below sparking 
 point. Arranging the jar so as to leave an air-space between 
 it and one of the wires, a spark passes into it at each A 
 spark ; but the jar is not in the least charged afterwards, 
 proving that the spark is a double one, first in and then out 
 of the jar, a real recoil of the reflected pulse. Hence, also, we 
 see that the brush visible in the dark is the same on each wire, 
 and one is not able to say that one brush is positive and the 
 other negative, for each is both. 
 
 A curious experiment illustrating the electrical surgings or 
 oscillations set up in a conductor which is suddenly discharged 
 at one end is as follows : Attach one end of a long wire to one 
 knob of a "Wirnshurst machine, and connect the other pole to 
 
 FIG. 148. 
 
 earth. The wire is otherwise insulated, and now forms one 
 coating of a condenser of which the other is the walls of the 
 room. The wire is bent round so that its free end nearly touches 
 its initial end (see Fig. 148). Under these circumstances one 
 would naturally say that a spark at B was absurd, and yet 
 it is found that even if the wire is a stout copper wire a 
 spark happens at B when one is produced at A. This B spark 
 is caused by an electrical oscillation in the wire. The wire is, 
 as it were, pumped full of electricity by the machine, and 
 when the spark happens at A a release is given at that end for 
 one brief instant. Then ensues a rebound of the electricity, 
 and the pressure rises at the free end to sparking amount. 
 The whole effect is just analogous to the effect of suddenly 
 opening and closing a tap on a high-pressure water service a 
 concussion is heard in the tap on shutting, and if one could see 
 the water it would be found that it rebounds, and a reflected
 
 412 DYNAMICAL THEORY OF INDUCTION. 
 
 wave is set up in the pipe, which, if the pipe is not strong 
 enough, will burst it at some weak point. The practical moral 
 of this is that any large conductor suddenly discharged has set 
 up in it violent electrical surgings, which may cause it to spit 
 off discharges at other points, and these sparks may be as long 
 as the principal spark. 
 
 Another way of making these electrical surgings conspicuous 
 is by their effect in causing a Leyden jar to overflow, i.e., 
 to spark round its edge. A jar does this when its coat- 
 ings are very suddenly raised to a great potential difference. 
 Fig. 149 shows the arrangement. The inside of the jar is made 
 to communicate direct to one machine pole, and the outer 
 coating, through the intervention of a long wire, to the other 
 pole. 
 
 When a spark happens at A, and the length of the wire L is 
 sufficiently great, the jar sparks over its edge. The explanation 
 
 FIG. 149. 
 
 of this is as follows : Whilst the handle of the machine is being 
 turned the potential difference of the j ar coatings increases. At 
 a certain limit the air in the A space breaks down, and, being 
 lieated, becomes for a moment a very good conductor ; there is, 
 therefore, a rush of electricity out of the inner coating and into 
 the outer coating, but the spark at A ceasing, this outflow from 
 the jar is suddenly stopped and rebounds, whilst at the same 
 time the inductance of the wire L causes a rush to continue into 
 the jar. The rebound of the flow when the rush through the air 
 space is suddenly stopped causes the potential difference of the 
 coatings to rise to a point at which they spark over the edge of 
 the glass. In an example given by Dr. Lodge the jar was a 
 one gallon jar, with glass fully three inches above the tinfoil. 
 L was a thick No. 1 copper wire circuit round a room. The jar
 
 DYNAMICAL THEORY OF INDUCTION. 413 
 
 ovei-flows every time a spark happens at A, even though the 
 length of this spark is only 0-64in. If the long lead L is short- 
 circuited, then the jar refuses to overflow until the A spark has 
 been increased to l-Tin. The higher potential difference needed 
 to cause overflow or rebound in the case with a short circuit is 
 illustrative of the fact that a little self-induction in the dis- 
 charging circuit bestows momentum on the flow and assists in 
 making a back splash. 
 
 A hydraulic analogue to the above might be found in con- 
 sidering the case of a liquid flowing steadily along a trough or 
 canal. If an obstruction was suddenly created, as by closing a 
 valve or sluice, the liquid would rebound and a wave would be 
 created ; and, as in the case of the hydraulic ram, the rebound of 
 the liquid against a closed valve might be made to lift some of 
 it to a higher level than that from which it originally fell. In 
 the electrical case, the rebound is made to raise the jar coatings 
 to a greater potential difference than that which existed at the 
 instant when the jar commenced to discharge. 
 
 9. Theory of Experiments on the Alternative Path. We 
 may proceed, following Dr. Lodge,* and quoting freely from 
 him in what follows, to examine a little more in detail the 
 electrical oscillations set up in an open circuit by Leyden jar 
 discharges. These stationary electrical oscillations in linear 
 conductors resemble those which can be set up in a cord fixed 
 at one end, or in a trough of liquid, by suitably-timed im- 
 pulses. As we have seen, if a jar discharges at A (see Fig. 150) 
 in the ordinary way, simultaneously an even longer spark may 
 be obtained at B, at the far end of two long open circuit 
 leads. Or if the B ends of the wire are too far apart to allow 
 of a spark, the wires glow and spit off brushes every tune 
 a discharge occurs at A. The theory of the effect seems 
 to be that oscillations occur in the A circuit with a period 
 T = 2?r ,/LC, where L is the inductance of the A circuit and 
 C the capacity of the jar. These oscillations disturb the 
 surrounding medium, and send out radiations of the precise 
 nature of light, only too long in wave length to affect the 
 
 * See Phil. Mag., August, 1838; also The Electrician, August 10, 1888, 
 p. 435.
 
 414 DYNAMICAL THEORY OF INDUCTION. 
 
 retina of our eyes. The velocity of these electro-magnetic 
 impulses is, as we have seen, equal to v, where 
 
 so the wave length of the oscillations is 
 
 Now is the electro-magnetic measure of inductance, and 
 P K 
 
 is the electrostatic measure of capacity, //, being the magnetic 
 permeability, and K the electrostatic inductivity of the medium 
 surrounding the wire. 
 
 Each of these quantities is of the dimensions of a length, and 
 the wave length of the radiation is 2?r times their geometric 
 mean. We may look upon it, then, that the magnetic field 
 due to the oscillatory current in the A circuit, which circuit 
 
 FIG. 150. 
 
 consists partly of metal wires, partly of the dielectric of the 
 jar, and partly of the heated air in the spark space, acts 
 inductively upon the other or B circuit which is adjacent to 
 it, and has, in fact, the jar dielectric as a common boundary. 
 The pulsating field induces oscillatory currents in the open B 
 circuit. These electric pulses rush along the surface of the 
 wires with a certain amount of dissipation, and are reflected at 
 the distant end, producing a recoil kick or impulse tending to 
 break down the dielectric in the air gap B with production of 
 a spark. These currents continue to oscillate to and fro unti 
 damped out of existence by the resistance of the wires. The 
 best effect in the way of spark at B is observed when the 
 length of each wire is such that the time occupied by an elec- 
 tric pulse in travelling along the wires and back again is equal 
 to the time of a complete oscillation in the A circuit ; that is,
 
 DYNAMICAL THEORY OF INDUCTION. 415 
 
 when the length of the open circuit wires is equal to half a 
 wave length or to some multiple of half a wave length. The 
 natural period of oscillation in the long wires will then agree 
 with the oscillation period of the discharging circuit and the 
 oscillations in the open circuit wires, and the field due to the 
 oscillations in the A circuit will vibrate in unison like a 
 column of air in a pipe resonating a tuning fork, or like a 
 string vibrating when attached to the tongue of a reed. 
 
 The elementary theory of the open circuit oscillations is as 
 follows : 
 
 Let li and i\ be the inductance' and resistance of the straight 
 wires per unit of length, as affected ~by the periodicity, and let c x 
 be the capacity per unit of length. It has been shown by 
 Lord Eayleigh (Phil. Mag., May, 1886) that with very rapid 
 oscillations owing to the circumferential distribution of the 
 current the inductance and resistance have values different from 
 the steady current values, and when the frequency of the oscil- 
 lations is very great the resistance r : per unit of length is the 
 geometric mean, of its ordinary value r and ^ p /*, where /^ is 
 the magnetic permeability of the material of the conductor, or 
 r i = *J%P Po?', P being, as usual, 2?r n, n being the number of 
 complete oscillations per second. 
 
 And again, when n is very great, the inductance Z x per unit 
 
 of length is equal to a constant plus ^, or 
 
 I being the induction for slowly fluctuating currents. 
 
 In the case of the two parallel wires we have for the slope 
 
 of the potential -^J- along them the usual equations, 
 
 4S'--g ..... (U4) 
 
 i being the instantaneous current in the section of the length 
 lying at a distance #*from the origin ; and also for the accumu- 
 lation of charge in this element dx of the length we have the 
 equation 
 
 -11 = 1 ..... (145) 
 dt (jdx
 
 416 DYNAMICAL THEORY OF INDUCTION. 
 
 The elimination of i between these equations gives us a 
 differential equation for V, and shows that stationary waves 
 of current are set up in finite wires of suitable length under 
 the action of an alternating electromotive force. The solution 
 of the equation for a long wire when i\ is small and p is very 
 large is 
 
 mi 
 
 V ~ "i* cosy* (*---), 
 
 \ ?ij/ 
 
 where w, = 1- and w t = . -. 
 
 2 L v -Uj O| 
 
 The velocity of propagation of the wave is therefore 7^ and the 
 
 wave length is n r 
 
 P 
 
 For two parallel wires, as in the Ley den jar case, we have 
 each wire 
 
 i= ppo r > 
 
 r being the ordinary resistance. And again, as Lord Kayleigh 
 has shown (Phil. May., May, 1886), we have 
 
 b being the distance between the parallel wires and a the 
 radius of either, and /* the magnetic permeability of the 
 material of the conductors. 
 
 For immensely quick oscillations the second term is zero. 
 Also, the capacity G : of the wires per unit of length is, by a 
 known theorem, 
 
 0,- 
 
 " 
 
 hence 
 
 and the velocity of the pulse along the wires is the same as in 
 the dielectric round them. In other words, the electric pulses 
 set up in the wires rush to and fro with a velocity equal to 
 that with which the electro-magnetic impulse is propagated 
 through the dielectric round them. Hence, we have here a 
 means of determining experimentally the wave length of a given 
 discharging circuit. Either vary the size of the A circuit or
 
 DYNAMICAL THEORY OF INDUCTION. 417 
 
 adjust the length of the B wires until the recoil spark B is as 
 long as possible. Then measure, and see whether the length 
 of each wire is not equal to 
 
 A small condenser can be made having an electrostatic 
 capacity of, say, two or three centimetres, and if such a coated 
 pane be made to discharge over its edge, the discharged circuit 
 will have an electro-magnetic inductance of a few centimetres. 
 Under these circumstances the electrical oscillations would be 
 at the rate of a thousand million a second, and the wave length 
 of the electro-magnetic disturbance radiated would be about 20 
 to 30 centimetres. 
 
 If a conductor as small as an atom could have its electrical 
 charge disturbed in the same way, oscillations would be set up 
 of the frequency of light waves and electro-magnetic disturb- 
 ances of light wave length radiated ; and it seems probable 
 that this is just what light waves are, viz., electro-magnetic 
 disturbances propagated through the ether and due to electric 
 oscillatious set up in the atomic charge. 
 
 10. Impulsive Impedance. In the experiments of the 
 "alternative path," as described by Dr. Lodge, the main result 
 is very briefly summed up by saying that when a sudden dis- 
 charge had to pass through a conductor it was found that iron 
 and copper acted about equally well, and indeed iron sometimes 
 exhibited a little superiority, and that the thickness of the 
 conductor and its ordinary conductivity mattered very little 
 indeed. We are led by this to see that the impedance which a 
 conductor offers to a sudden discharge, and which may be 
 called its impulsive impedance, is something quite different 
 from its ordinary or ohmic resistance, or even its impedance, 
 defined as Vft 2 +p' 2 L' 2 , to slowly periodic or oscillatory cur- 
 rents. As already mentioned, the resistance of a conductor 
 to very rapidly changing currents is expressed by E u where 
 
 E being the resistance to steady current, /* the permeability 
 of the material of the conductor and I its length, and p = 2ir
 
 418 DYNAMIC AL THEORY OF INDUCTION. 
 
 times the frequency of the oscillation. Also the corresponding 
 inductance L x is 
 
 L.-L+?.. 
 
 where L is a constant depending, on the size and form of the 
 circuit, but only in a small degree upon its thickness. Hence, 
 forming the function VR^+y-L^, and calling this Im 1 , 'we 
 have 
 
 im 
 
 where 
 
 In the case of enormously rapid oscillations the value 
 practically reduces to p L, and hence the impulsive impedance 
 varies in simple proportion to the frequency, and depends on 
 the form and size of the circuit, but not at all on its specific 
 resistance, magnetic permeability, or diameter. 
 
 All this is borne out by experiment. In some of his 
 experiments Dr. Lodge found the impedance of a No. 2 wire 
 of two and a-half metres length bent into a circle to be 180 
 ohms at twelve million oscillations per second, and for a 
 No. 40 wire the impedance was only 300 ohms, although the 
 ohmic resistances of these wires were respectively -004 ohms 
 and 2'6 ohms. At three million oscillations per second, or 
 at one-fourth the frequency, the impedances of the same 
 circuits were 43 ohms and 78 ohms. At one-quarter million 
 oscillations per second the impedances are reduced to 
 four and six ohms respectively for the thick rod and fine wire. 
 Hence, for frequencies of a million per second and upwards, 
 such as occur in jar discharges, and perhaps in lightning, the 
 impedance of all reasonably conducting circuits is the same, 
 and independent of conductivity and permeability, and hardly 
 affected greatly by enormous changes in diameter. 
 
 11. Hertz's Kesearches on the Propagation of Electro- 
 magnetic Induction. The classical researches of Hertz on 
 electrical oscillations and the propagation of electro-magnetic 
 induction through space form an epoch in the history of
 
 DYNAMICAL THEORY OF INDUCTION. 419 
 
 electrical science. These investigations have been well 
 described by Dr. Lodge in his book on " The "Work of 
 Hertz,"* and the reader is referred to this for an account 
 of the chief work of Hertz and his followers. There is 
 therefore no need to enter here at very great length into 
 an account of these discoveries ; but a very excellent 
 abstract of Hertz's work has been given by Mr. G. W. 
 de Tunzelmann.f 
 
 Preliminary Experiments. It is known that if in the second- 
 ary circuit of an induction coil there be inserted, in addition to 
 the ordinary air space across which sparks pass, a Eiess spark 
 micrometer, with its poles joined by a long wire, the discharge 
 will pass across the air space of the micrometer in preference 
 to following the path of least resistance through the wire, 
 provided this air spaca does not exceed a certain limit ; and it 
 is upon this principle that lightning protectors for telegraph 
 lines are constructed. It might be expected that the sparks 
 could be made to disappear by diminishing the length and 
 resistance of the connecting wire ; but Hertz found that though 
 the length of the sparks could be diminished in this way, it is 
 almost impossible to get rid of them entirely, and they can 
 still be observed when the balls of the micrometer are con- 
 nectsd by a thick copper wire only a few centimetres in length. 
 
 This shows that there must be variations in the potential 
 measurable in hundreds of volts in a portion of the circuit 
 only a few centimetres in length, and it also gives an indirect 
 proof of the enormous rapidity of the discharge ; for the differ- 
 ence of potential between the micrometer knobs can only be 
 due to self-induction in the connecting wire. Now the time 
 occupied by variations in the potential of one of the knobs 
 must be of the same order as that in which these variations 
 can be transmitted through a short length of a good conductor 
 to the second knob. The resistance of the wire connecting 
 the knobs is found to be without sensible effect on the results. 
 
 * Published by " The Electrician " Printing and Publishing Company, 
 Limited. 
 
 t This section originally appeared as a series of articles in the pages of 
 The Electrician, in Vol. XXI., pp. 587, 625, 663, 696, 725, 757, 788 (1888). 
 The writer felt it would be difficult to make a more complete digest of 
 Hertz's work than is contained in these excellent articles, and, by the kind 
 permission of their author, he is allowed to reproduce them in these pages. 
 
 EE2
 
 420 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 In Fig. 151, A is an induction coil and B a discharger. The 
 wire connecting the knobs 1 and 2 of the spark micrometer M 
 consists of a rectangle, half a metre in length, of copper wire 
 two millimetres in diameter. This rectangle is connected with 
 the secondary circuit of the coil in the manner shown in the 
 diagram, and, when the coil is in action, sparks, sometimes 
 several millimetres in length, are seen to pass between the 
 knobs 1 and 2, showing that there are violent electrical oscil- 
 lations not only in the secondary circuit itself, but in any 
 conductor in contact with it. This experiment shows even 
 more clearly than the previous one that the rapidity of the 
 oscillations is comparable with the velocity of transmission of 
 electrical disturbances through the copper wire, which, accord- 
 
 FIG. 151. 
 
 ing to all the evidence at our disposal, is nearly equal to the 
 velocity of light. 
 
 In order to obtain micrometer sparks some millimetres in 
 length a powerful induction coil is required, and the one used 
 by Hertz was 52 centimetres in length and 20 centimetres in- 
 diameter, provided with a mercury contact breaker, and excited 
 by six large Bunsen cells. The discharger terminals consisted 
 of brass knobs three centimetres in diameter. The experiments 
 showed that the phenomenon depends to a very great extent 
 on the nature of the sparks at the discharger, the micrometer 
 sparks being found to be much weaker when the discharge in-
 
 DNTAM1CAL THEORY OF INDUCTION. 421 
 
 the secondary circuit took place between two points or between 
 a point and a plate than when knobs were used. The micro- 
 meters sparks were also found to be greatly enfeebled when the 
 secondary discharge took place in a rarefied gas, and also when 
 the sparks in the secondary were less than half a centimetre 
 in length ; while, on the other hand, if they exceeded 1| centi- 
 metres the sparks could no longer be observed between the 
 micrometer knobs. The length of secondary spark which was 
 found to give the best results, and which was therefore em- 
 ployed in the further observations, was about three-quarters 
 of a centimetre. 
 
 Very slight differences in the nature of the secondary sparks 
 were found to have great effect on those of the micrometer, 
 : and Hertz states that after some practice he was able to deter- 
 mine at once from the sound and appearance of the secondary 
 spark whether it was of a kind to give the most powerful 
 effects at the micrometer. The sparks which gave the best 
 results were of a brilliant white colour, only slightly jagged, 
 and accompanied by a sharp crack. 
 
 The influence of the spark is readily shown by increasing 
 the distance between the discharger knobs beyond the striking 
 distance, when the micrometer sparks disappear entirely, 
 although the variations of potential are now greater than 
 before. The length of the micrometer circuit has naturally 
 an important influence on the length of the spark, as the 
 greater its length the greater will be the retardation of the 
 electrical wave in its passage through it from one knob of the 
 micrometer to the other. 
 
 The material, the resistance, and the diameter of the wire 
 of which the micrometer circuit is formed have very little 
 influence on the spark. The potential variations cannot, 
 therefore, be due to the resistance ; and this was to be 
 expected, for the rate of propagation of an electrical disturb- 
 ance along a conductor depends mainly on its capacity and 
 coefficient of self-induction, and only to a very small extent 
 on its resistance. The length of the wire connecting the 
 micrometer circuit with the secondary circuit of the coil is 
 also found to have very little influence, provided it does not 
 exceed a few metres in length. The electrical disturbances 
 must therefore traverse it without undergoing any appreciable
 
 422 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 change. The position of the point of the micrometer circuit 
 which is joined to the secondary circuit is, on the other hand, 
 of the greatest importance, as would be expected, for, if the 
 point is placed symmetrically with respect to the two micro- 
 meter knobs, the variations of potential will reach the latter 
 in the same phase, and there will be no effect, as is verified 
 by observation. If the two branches of the micrometer circuit 
 on each side of the point of contact of the connection with the 
 secondary are not symmetrical the spark cannot be made to 
 disappear entirely; but a minimum effect is obtained when 
 the point of contact is about half-way between the micrometer 
 knobs. This point may be called the null point. 
 
 Fig. 152 shows the arrangement employed, e being the null 
 .point of the rectangular circuit, which is 125 centimetres long 
 
 M 
 
 12 
 
 FIG. 152. 
 
 by 80 centimetres broad. When the point of contact is at ! a 
 or b sparks of from three to four millimetres in length are 
 observed ; when it is at e no sparks are seen, but they can be 
 made to reappear by shifting the point of contact a fetor 
 centimetres to the right or left of the null point. It should 
 be noted that sparks only a few hundredths of a millimetre in 
 length can be observed. If, when the point of contract is at , 
 another conductor is placed in contact with one of the micro- 
 meter knobs, the sparks reappear. 
 
 Now, the addition of this conductor cannot produce any 
 alteration in the time taken by the disturbances proceeding
 
 DYNAMICAL THEORY OF INDUCTION. 423 
 
 from e to reach the knobs, and therefore the phenomenon can- 
 not be clue simply to single waves in the direction c a and d b 
 respectively, but must be due to repeated reflection of the 
 waves until a condition of stationary vibration is attained, 
 and the addition of the conductor to one of the knobs must 
 diminish or prevent the reflection of the waves from that ter- 
 minal. It must be assumed, then, that definite oscillations 
 are set up in the micrometer circuit just as an elastic bar is 
 thrown into definite vibrations by blows from a hammer. If 
 this assumption is correct, the condition for the disappearance 
 of- the sparks at M will be that the vibration periods of the two 
 branches e 1 and e 2 shall be equal'. These periods are deter- 
 mined by the products of the coefficients of self-induction of 
 these conductors into the capacities of their terminals, and are 
 practically independent of their resistances. 
 
 In confirmation of this it is found that if, when the point of 
 contact is at e and the sparks have been made to reappear 
 by connecting a conductor with one of the knobs, this con- 
 ductor is replaced by one of greater capacity, the sparking is 
 greatly increased. If a conductor of equal capacity is con- 
 nected with the other micrometer knob, the sparks disappear 
 again ; the effect of the first conductor can also be counter- 
 acted by shifting the point of contact towards it, thereby 
 diminishing the self-induction in that branch. The conclusions 
 were further confirmed by the results obtained when coils of 
 copper wire were inserted into one or other and then into 
 both of the branches of the micrometer circuit. 
 
 Hertz supposed that as the self-induction of iron wires is, for 
 slow alternations, from eight to ten times that of copper wires, 
 therefore a short iron wire would balance a long copper one ; 
 but this was not found to be the case, and he concludes that, 
 owing to the great rapidity of the alternations, the magnetism 
 of the iron is unable to follow them, and therefore has no effect; 
 on the self-induction.* 
 
 * In a note in Wiedemann's Annalcn, Vol. XXXI., p. 543, Dr. HertZ 
 stated that since the publication of his Paper in the same volume he had 
 found that Von Bezold had published a Paper, in 1870 (Poggendorff 8 
 Annalcn, Vol. CXL., p. 541), in which he had arrived by a different method 
 of experimenting at similar results and conclusions as those given by him 
 under the head of Preliminary Experiments.
 
 424 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 Inditction Phenomena in Open Circuits. In order to test 
 more fully his conclusion that the sparks obtained in the 
 last experiments described were .due to self-induction, Hertz 
 placed a rectangle of copper wire with sides 10 and 20 centi- 
 metres in length respectively, broken by a short air space, 
 with one of its sides parallel and close to various portions of 
 the secondary circuit of the coil and of the micrometer circuit, 
 with solid dielectrics interposed to obviate the possibility of 
 sparking across, and he found that sparking in this rectangle 
 invariably accompanied the discharges of the induction coil, 
 the longest sparks being obtained when a side of the rectangle 
 was close to the discharger. 
 
 CXZZEDO 
 
 FIG. 153. 
 
 A copper wire, igh (Fig. 153), was next attached to the 
 flischarger, and a side of the micrometer circuit, which was 
 supported on an insulating stand, was placed parallel to a 
 portion of this wire, as shown in the diagram. The sparks at 
 M were then found to be extremely feeble until a conductor, 
 C, was attached to the free end, h, of the copper wire, when 
 they increased to one or two millimetres in length. That the 
 action of C was not an electrostatic one was shown by its pro- 
 ducing no effect when attached at g instead of at h. When
 
 DYNAMICAL THEORY OF INDUCTION. 425 
 
 the knobs of the discharger B were so far separated that no 
 .sparking took place there, the sparks at M were also found 
 to disappear, showing that these were due to the sudden dis- 
 charges and not to the charging current. The sparks at the 
 discharger which produced the most effect at the micrometer 
 were of the same character as those described under the head 
 of Preliminary Experiments. Sparks were also found to occur 
 between the micrometer circuit and insulated conductors in its 
 vicinity. The sparks became much shorter when conductors 
 of large capacity were attached to the micrometer knobs, or 
 when these were touched by the hand, showing that the 
 quantity of electricity in motion was too small to charge 
 these conductors to a similarly high potential. Joining the 
 micrometer knobs by a wet thread did not perceptibly diminish 
 the strength of the sparks. The effects in the micrometer 
 circuit were not of sufficient strength to produce any sensation 
 when it was touched or the circuit completed through the body. 
 In order to obtain further confirmation of the oscillatory 
 nature of the current in the circuit k i h g (Fig. 153), the con- 
 ductor C was again attached to h, and the micrometer knobs 
 drawn apart until sparks only passed singly. A second con- 
 ductor, C', as nearly as. possible similar to C, was then attached 
 to k, when a stream of sparks was immediately observed, and 
 it continued when the knobs were drawn still further apart. 
 This effect could not be ascribed to a direct action of the 
 portion of circuit i k, for in this case the action of the portion 
 of circuit g h would be weakened, and it must therefore have 
 consisted in C' acting on the discharging current of C a result 
 which would be quite incomprehensible unless the current 
 in g h were of an oscillatory character. 
 
 : Since an oscillatory motion between C and C' is essential for 
 the production of powerful inductive effects, it will not be 
 sufficient for the spark to occur in an exceedingly short time, 
 'but the resistance must at the same time not exceed certain 
 .limits. The inductive effects will therefore be excessively small 
 if the induction coil included in the circuit C C' is replaced by 
 an electrical machine alternately charging and discharging 
 itself, or if too small an induction coil is used, or, again, if the 
 air space between the discharger knobs is too great, as in all 
 these cases the motion ceases to be oscillatory.
 
 426 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 The reason that the discharge of a powerful induction coil 
 gives rise to oscillatory motion is that, firstly, it charges the 
 terminals C and C' to a high potential ; secondly, it produces 
 a sudden spark in the intervening circuit ; and thirdly, as soon 
 as the discharge begins the resistance of the air space is so 
 much reduced as to allow of oscillatory motion being set up. 
 If the terminal conductors are of a very large capacity for 
 example, if the terminals are in connection with a battery the 
 current of discharge may indefinitely reduce the resistance of 
 the air space, but when the terminal conductors are of small 
 
 . capacity this must be done by a separate discharge, and there- 
 fore, under the conditions of Hertz's experiments, an induction 
 
 . coil was absolutely essential for the production of the oscilla- 
 tions. 
 
 FIG. 154. 
 
 As the induced sparks in the experiment last described were 
 several millimatres in length, Hertz modified it by using the 
 arrangement shown in Fig. 154, and greatly increasing the 
 distance between the micrometer circuit and the secondary 
 circuit of the induction coil. The terminal conductors C and C'' 
 were three metres apart, and the wire between them was of 
 copper, 2 millimetres in diameter, with the discharger B at its 
 centre. 
 
 The micrometer circuit consisted, as in the preceding experi- 
 ments, of a rectangle 80 centimetres broad by 120 centimetres 
 long. With the nearest side cf the micrometer circuit at a
 
 DYNAMICAL THEORY OF INDUCTION. 427 
 
 distance of half a millimetre from C B C', sparks two milli- 
 metres in length were obtained at M, and though the length of 
 the sparks decreased rapidly as the distance of the micrometer 
 circuit was increased, a continuous stream of sparks was still 
 obtained at a distance of one and a-half metres. The interven- 
 tion of the observer's body between the micrometer circuit and 
 the wire C B C' produced no visible effect on the stream of 
 sparks at M. That the effect was really due to the rectilinear 
 conductor C B (7 was proved by the fact that when one or other, 
 or both, halves of this conductor were removed, the sparks at M 
 ceased. The same effect was produced by drawing the knobs 
 of the discharger B apart until sparks ceased to pass, showing 
 that the effect was not due to the electrostatic potential differ- 
 ence of C and C', as this would be increased by separating the 
 discharger knobs beyond sparking distance. 
 
 The closed micrometer circuit was then replaced by a straight 
 copper wire, slightly shorter than the distance C C', placed 
 parallel to C B C' and at a distance of GO centimetres from it. 
 This wire terminated in knobs, 10 centimetres in diameter, 
 attached to insulating supports, and the spark micrometer 
 divided it into two equal parts. Under these circumstances 
 sparks were obtained at the micrometer as before. 
 
 With the rectilinear open micrometer circuit sparks were 
 still observed at the micrometer when the discharger knobs 
 of the secondary coil circuit were separated beyond sparking 
 distance. This was, of course, due simply to electrostatic 
 induction, and shows that the oscillatory current in C C' was 
 superposed upon the ordinary discharges. The electrostatic 
 action could be got rid of by joining the micrometer knobs by 
 means of a damp thread. The conductivity of this thread was 
 therefore sufficient to afford a passage to the comparatively 
 slow alternations of the coil discharge, but was not sufficient to 
 provide a passage for the immeasurably more rapid alternations 
 of the oscillatory current. Considerable sparking took place at 
 the micrometer when its distance from C B C' was 1'2 metre, 
 and faint sparks were distinguishable up to 3 metres. At these 
 distances it was not necessary to use the damp thread to get 
 rid of the electrostatic action, as, owing to its diminishing more 
 rapidly with increase of distance than the effect of the current 
 induction, it was no longer able to produce sparks in the micro-
 
 423 D YNA MICAL THEOR Y OF IND UCTION. 
 
 meter, as was proved by separating the discharger knobs beyond 
 sparking distance, when sparks could no longer be perceived at 
 the micrometer. 
 
 Resonance Phenomena. In order to determine whether the 
 oscillations were of the nature of a regular vibration, Hertz 
 availed himself of the principle of resonance. According to 
 this principle, an oscillatory current of definite period would, 
 other conditions being the same, exert a much greater inductive 
 effect upon one of equal period than upon one differing even 
 slightly from it.* 
 
 If, then, two circuits are taken having as nearly as possible 
 equal vibration periods, the effect of one upon the other will 
 be diminished by altering either the capacity or the coefficient 
 of self-induction of one of them, as a change in either of them 
 would alter the period of vibration of the circuit. 
 
 This was carried out by means of an arrangement very simi- 
 lar to that of Fig. 154. The conductor C C' was replaced by a 
 .straight copper wire 2-6 metres in length and 5 millimetres in 
 diameter, divided into two equal parts as before by a discharger. 
 The discharger knobs were attached directly to the secondary 
 terminals of the induction coil. Two hollow zinc spheres, 30 
 centimetres . in diameter, were made to slide on the wire, one 
 on each side of the discharger, and since, electrically speaking, 
 these formed the terminals of the conductor, its length could 
 be varied by altering their position. The micrometer circuit 
 was chosen of such dimensions as to have, if the author's 
 hypothesis were correct, a slightly shorter vibration period than 
 that of C C'. It was formed of a square, with sides 75 centi- 
 metres in length, of copper wire 2 millimetres in diameter, and 
 it was placed with its nearest side parallel to CB C' and at a 
 distance of 30 centimetres from it. The sparking distance at 
 the micrometer was then found to be 0-9 millimetre. When 
 the terminals of the micrometer circuit was placed in contact 
 with two metal spheres 8 centimetres in diameter, supported on 
 insulating stands, the sparking distance could be increased up 
 to 2-5 millimetres. When these were replaced by much larger 
 spheres the sparking distance was diminished to a small 
 fraction of a aaillimetre. Similar results were obtained on 
 .connecting the micrometer terminals with the plates of a 
 
 * See Oberbeck, Wiederuanu's Annalen, Vol. XXVI., p, 245, 1835.
 
 DYNAMICAL IHEORY OF INDUCTION. 
 
 429 
 
 Kohlrausch condenser. When the plates were far apart the 
 increase of capacity increased the sparking distance, but when 
 the plates were brought close together the sparking distances- 
 again fell to a very small value. 
 
 The simplest method of adjusting the capacity of the micro- 
 meter circuit is to suspend to its ends two parallel wires the 
 distance and lengths of which are capable of variation. By this 
 means the author succeeded in increasing the sparking distance 
 up to three millimetres, after which it diminished when th& 
 wires were either lengthened or shortened. The decrease of the 
 sparking distance on increasing the capacity was naturally to 
 be expected ; but it would be difficult to understand, except on 
 the principle of resonance, why a decrease of the capacity 
 should have the same effect. 
 
 FIG. 155. Curve showing relation between length of side of rectangle 
 (taken as abscissa) and maximum sparking distance (taken as ordinate), the- 
 sides consisting of straight wires of varying lengths. 
 
 The experiments were then varied by diminishing the capa- 
 city of the circuit C B C' so as to shorten its period of oscil- 
 lation, and the results confirmed those previously obtained \. 
 and a series of experiments in which the lengths and capacities 
 of the circuits were varied in different ways showed conclu- 
 sively that the maximum effect does not depend on the con- 
 ditions of either one of the two circuits, but on the existence of 
 the proper relation between them. 
 
 When the two circuits were brought very close together, and 
 the discharger knobs separated by an interval of 7 millimetres, 
 sparks were obtained at the micrometer, which were also
 
 430 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 7 millimetres in length, when the two circuits had been care- 
 fully adjusted to have the same period. The induced E.M.F's 
 must in this case have attained nearly as high a value as the 
 inducing ones. 
 
 To show the effect of varying the coefficient of self-induction, 
 a series of rectangles, abed (Fig. 154), were taken, having a 
 constant breadth, a b, but a length, a c, continually increasing 
 from 10 centimetres up to 250 centimetres : it was found that 
 the maximum effect was obtained with a length of 1-8 metre. 
 The quantitative results of these experiments are shown in 
 Fig. 155, in which the abscissae of the curve are the double 
 lengths of the rectangles, and the ordinates represent the cor. 
 responding maximum sparking distances. The sparking dis- 
 tances could not be determined with great exactness, but the 
 
 FIG. 156. Curve showing relation between length of side of rectangle 
 (taken as abscissa) and maximum sparking distance (taken as ordinate), the 
 sides consisting of spirals gradually drawn out. 
 
 errors were not sufficient to mask the general nature of the 
 result. 
 
 In a second series of experiments the sides a c and b d were 
 formed of loose coils of wire which were gradually pulled out, 
 and the result is shown in Fig. 156. It will be seen that the 
 maximum sparking distance was attained for .a somewhat 
 greater length of side, which is explained by the fact that in 
 the latter experiments the self-induction only was increased by 
 increase of length, while in the former series the capacity was 
 increased as well. Varying the resistance of the micrometer
 
 DYNAMICAL THEORY OF INDUCTION. 431 
 
 circuit by using copper and German silver wires of various 
 diameters was found to have no effect on the period of oscilla- 
 tion, and extremely little on the sparking distance. 
 
 When the wire c d was surrounded by an iron tube, oi 
 when it was replaced by an iron wire, no perceptible effect 
 was obtained, confirming the conclusion previously arrived 
 at that the magnetism of the iron is unable to follow such 
 rapid oscillations, and therefore exerts no appreciable 
 effect. 
 
 It is only proper, however, to interpolate at this point the 
 remark that other observers do not endorse entirely this 
 statement of Hertz. We may especially draw attention to 
 the work of Prof. J. Trowbridge and of Mr. C. E. St. 
 John* on the propagation of electrical oscillations on iron 
 wires. The experimental results obtained by these investi- 
 gators may be summed up as follows : 
 
 1. The magnetic permeability of iron wires exercises an 
 Important influence upon the decay of electrical oscillations 
 of high frequency. The influence is so great that the oscilla- 
 tions may be reduced to half an oscillation on a circuit of 
 suitable self-induction and capacity for producing them. 
 
 2. Currents of high frequency such as are produced in 
 Leyden jar discharges therefore magnetise iron. 
 
 3. The self-induction of iron circuits is sensibly greater 
 than that of similar copper circuits under rapid electrical 
 oscillations 115 x 10 6 reversals per second. 
 
 4. This increase in self-induction produces a shortening of 
 the wave-length. 
 
 o. The permeability of annealed iron under the atove rate 
 of alternation is about 885. 
 
 For full information as to the methods of obtaining these 
 results we must refer the reader to the original Papers. 
 
 Nodes. The vibrations in the micrometer circuit which have 
 been considered are the simplest ones possible, but not the only 
 ones. While the potential at the ends alternates between two 
 fixed limits, that at the central portion of the circuit retains a 
 constant mean value. The electrical vibration, therefore, has 
 
 * See Phil. May., December, 1891, Mr. J. Trowbridge on " Damping of 
 Electrical Oscillations on Iron Wires ; " and Phil. Mag., Xovember, 1894, 
 Mr. C. E. St. John on " Wave-Lengths of Electricity on Iron Wires. '
 
 432 DYNAMICAL THEORY OF INDUCTION. 
 
 a node at the centre, and this will be the only nodal point. Its- 
 existence may be proved by placing a small insulated sphere 
 close to various portions of the micrometer circuit while sparks 
 are passing at the discharger of the coil, when it will be found 
 that if the sphere is placed close to the centre of the circuit 
 the sparking will be very slight, increasing as the sphere is 
 moved further away. The sparking cannot, however, be 
 entirely got rid of, and there is a better way of determining 
 the existence and position of the node. After adjusting the 
 two circuits to unison, and drawing the micrometer terminals 
 so far apart that sparks can only be made to pass by means of 
 resonant action, let different parts of the circuit be touched 
 by a conductor of some capacity, when it will be found that 
 the sparks disappear, owing to interference with the resonant 
 action, except when the point of contact is at the centre of the- 
 circuit. Hertz then endeavoured to produce a vibration 
 with two nodes, and for this purpose he modified the apparatus 
 previously used by adding to the micrometer circuit a second 
 rectangle, ef g h, exactly similar to the first (as shown in Fig. 
 157), and joining the points of the circuit near the terminals 
 by wires 1 3 and 2 4, as shown in the diagram. 
 
 The whole system then formed a closed metallic circuit, the 
 fundamental vibration of which would have two nodes. Since 
 the period of this vibration would necessarily agree closely 
 with that of each half of the circuit, and, therefore, with that 
 of the circuit C C', it was to be expected that the vibration 
 would have a pair of loops at the junctions 1 3 and 2 4, and a 
 pair of nodes at the middle points of c d and g h. The vibra- 
 tions were determined by measuring the sparking distance 
 between the micrometer terminals 1 and 2. It was found that, 
 contrary to what was expected, the addition of the second 
 rectangle diminished this sparking distance from about three 
 millimetres to about one millimetre. The existence of 
 resonant action between the circuit C C' and the micrometer 
 circuit was, however, fully demonstrated, for any alteration 
 in the circuit ef g h, whether it consisted in increasing or in 
 decreasing its length, diminished the sparking distance. It 
 was also found that much weaker sparking took place between 
 c d or g h and an insulated sphere than between a e or bf and 
 the same sphere, showing that the nodes were in c d and g h f
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 433 
 
 as expected. Further, when the sphere was made to touch 
 c d or g h it had no effect on the sparking distance of 1 and 2 ; 
 but when the point of contact was at any other portion of the 
 circuit the sparking distance was diminished, showing that 
 these nodes did really helong to the vibration, the resonant 
 action of which increased this sparking distance. 
 
 The wire joining the points 2 and 4 was then removed. 
 As the strength of the induced oscillatory current should be 
 zero at these points, the removal ought not to disturb the 
 vibrations, and this was shown experimentally to be the case, 
 the resonant effects and the position of the nodes remaining 
 unchanged. The vibration with two nodal points was, of 
 
 -0- 
 
 )- 
 
 FIG. 157. 
 
 course, not the fundamental vibration of the circuit, which 
 consisted of a vibration with a node between a and e, and for 
 which the highest values of the potential were at the points 
 2 and 4. 
 
 When these spheres forming the terminals at these points 
 were brought close together slight sparking was found to take 
 place between them, which was attributed to the excitation, 
 though only to a small extent, of the fundamental vibration. 
 This explanation was confirmed in the following manner : 
 The sparks between I and 2 were broken off, leaving only the
 
 434 DYNAMICAL THEORY OF INDUCTION. 
 
 sparks between 2 and 4, which measured the intensity of the 
 fundamental vibration. The period of vibration of the circuit 
 C C' was then increased by drawing it out to its full length, 
 and thereby increasing its capacity, when it was observed that 
 the sparking gradually increased to a maximum, and then 
 began to diminish again. The maximum value must evidently 
 occur when the period of vibration of the circuit C C' is the 
 same as that of the fundamental vibration of the micrometer 
 circuit, and it was shown that when the sparking distance 
 between 2 and 4 had its maximum value the sparks corre- 
 sponded to a vibration with only one nodal point, for the 
 sparks ceased when the previously existing nodes were 
 touched by a conductor, and the only point where contact 
 could take place without effect on the sparking was between 
 a and e. These results show that it is possible to excite at 
 mil in the same conductor either the fundamental vibration 
 or its first overtone, to use the language of acoustics. 
 
 Hertz appeared to consider it very doubtful whether it was 
 possible to get higher overtones of electrical vibration, the 
 difficulty of obtaining such lying not only in the method of 
 observation, but also in the nature of the oscillations them- 
 selves. The intensity of these is found to vary considerably 
 during a series of discharges from the coil even when all the 
 circumstances are maintained as constant as possible, and the 
 comparative feebleness of the resonant effects shows that there 
 must be a considerable amount of damping. There are, more- 
 over, many secondary phenomena which seem to indicate that 
 irregular vibrations are superposed upon the regular ones, as 
 would be expected in complex systems of conductors. If, 
 therefore, we wish to compare electrical oscillations from a 
 mathematical point of view with those of acoustics, we must 
 seek our analogy in the high notes intermixed with irregular 
 vibrations, obtained, say, by striking a wooden rod with a 
 liammer rather than in the comparatively slow harmonic 
 vibration of tuning forks or strings ; and in the case of vibra- 
 tions of the former class we have to be contented even in the 
 study of acoustics with little more than indications of such 
 phenomena as resonance and nodal points. 
 
 Referring to the conditions to be fulfilled in order to obtain 
 the best results, Hertz noted a fact of very considerable interest
 
 DYNAMICAL THEORY OF INDUCTION. 435 
 
 and novelty, namely, that the spark from the discharger 
 should always be visible from the micrometer, as, when 
 this was not the case, though the phenomena observed were 
 of the same character, the sparking distance was invariably 
 diminished. This effect of the light from the spark of an 
 induction coil in increasing the sparking distance in a 
 secondary circuit has been fully described by Dr. Lodge in 
 his book on the work of Hertz, and he has pointed out that 
 the same effect is produced by light from burning magnesium 
 wire or other sources rich in the ultra-violet rays. 
 
 Theory of the Experiments. The theories of electrical oscilla- 
 tions which have been developed by Lord Kelvin, von 
 Helmholtz, and Kirchoff have been shown* to hold good for 
 the open circuit oscillations of induction apparatus, as weh 1 as 
 for the oscillatory Leyden jar discharge ; and it is of interest 
 to inquire whether the observed results are of the same order 
 as those indicated by theory. 
 
 Hertz considers, in the first place, the vibration period. 
 Let T be the period of a single or half vibration proper to the 
 conductor exciting the micrometer circuit ; L its coefficient of 
 self-induction in absolute electromagnetic measure, expressed, 
 therefore, in centimetres ; C the capacity of one of its terminals 
 in electrostatic measure, and therefore also expressed in centi- 
 metres ; and v the velocity of light in centimetre-seconds ; 
 then, if the resistance of the conductor is small, 
 
 v 
 
 In the case of the resonance experiments, the capacity C was 
 approximately the radius of the sphere forming the terminal, 
 so thr.t C = 7'5 centimetres.! The coefficient of self-induction 
 
 * Lorentz, Wiedemann's Annalen, Vol. VII., p. 161, 1879. 
 
 t In Hertz's original Paper the capacity of the spherical terminal ball 
 was taken as 15 units. M. Poincare first drew attention to the fact that 
 the capacity C in the above formula denotes the amount of electricity 
 which exists at one end of an oscillating conductor when the difference of 
 potential between the two ends is equal to unity. Hence, if the spheres 
 are far apart, the difference of potential between each of them and sur- 
 rounding space is +|. Therefore the charge on the sphere is formed by 
 dividing its capacity, i.e., its radius in centimetres, by 2. Hence, C in the 
 
 FF2
 
 436 DYNAMICAL THEORY OF INDUCTION. 
 
 was that of a wire of length 1 = 150 centimetres and diameter 
 d = 1/2 centimetre. 
 
 According to Neumann's formula, 
 
 Lf [cos , j , 
 = I I ds ds t 
 
 which gives in the case considered 
 
 L = 2 7 (log ^L - 0-75^ = 1,902 centimetres. 
 \ d / 
 
 As, however, it is not quite certain that Neumann's formula 
 is applicable to an open circuit, it is better to use von Helm- 
 holtz's more general formula, containing an undetermined 
 constant k, according to which 
 
 Putting Jc = l, this reduces to Neumann's formula; for & = 0< 
 it reduces to that of Maxwell, and for A- = 1 to Weber's. The 
 greatest difference in the values of L obtained by giving these 
 different values to k would not exceed a sixth of its mean 
 value, and therefore, for the purposes of the present approxi- 
 mation, it is enough to assume that k is not a large positive 
 or negative number; for if the number 1,902 does not give 
 the correct value of the coefficient for the wire 150 centimetres 
 in length, it will give the value corresponding to a conductor 
 not differing greatly from it in length. 
 
 Taking L = 1,902 centimetres, we have TT VC L = 531 centi- 
 metres, which represents the distance traversed by light during 
 the oscillation, or, according to Maxwell's theory, the length 
 of an electromagnetic ether wave. The value of T is then 
 found to be 1-26 hundred-millionths of a second, which is of 
 the same order as the observed results. 
 
 The ratio of damping is then considered. In order that 
 oscillations may be possible, the resistance of the open circuit 
 must be less than 2 v \/L/C. Fcr the exciting circuit used this 
 gives 676 ohms as the upper limit of resistance. If the actual 
 resistance, r, is sensibly below this limit, the ratio of damping 
 
 will be e*t. The amplitude will therefore be reduced in the 
 ratio 1:2-71 in 
 
 2L_2j,- /L 67G 215 
 rT"rV C"r".r
 
 D YNA MICA L THEOE Y OF IND UCTION. 437 
 
 oscillations. We have, unfortunately, no means of deter- 
 mining the resistance of the air space traversed by the spark, 
 but as the resistance of a strong electric arc is never less than 
 a few ohms we shall be justified in assuming this as the 
 minimum limit. From this it would follow that the number 
 of oscillations due to a single impulse must be reckoned ill 
 tens, and not in hundreds or thousands, which is in accordance 
 with the character of the experimental results, and agrees with 
 results observed in the case of the oscillatory Leyden jar 
 discharge. In the case of closed metallic circuits, on the 
 other hand, theory indicates that the number of oscillations 
 before equilibrium is attained must be reckoned by thousands. 
 Hertz compares, lastly, the order of the inductive actions of 
 these oscillations according to theory with that of the effects 
 actually observed. To do this it must be noted that the 
 maximum E.M.F. induced by the oscillation in its own circuit 
 is approximately equal to the maximum potential difference 
 at its extremities ; for if there were no damping these quanti- 
 ties would be identical, since at any moment the potential 
 difference at the extremities and the E.M.F. of induction 
 would be in equilibrium. In the experiments under con- 
 sideration the potential difference at the extremities was such 
 as to give a spark 7 to 8 millimetres in length, which must 
 therefore represent the maximum inductive action excited in 
 its own circuit by the oscillation. Again, at any instant the 
 induced E.M.F. in the micrometer circuit must be to that 
 in the exciting conductor in the same ratio as that of the 
 coefficient of mutual induction M of the two circuits to the 
 coefficient of self-induction L of the exciting circuit. The 
 value of M for the case considered is easily calculated from the 
 ordinary formulae, and it is found to lie between one-ninth 
 and one-twelth of L. This would only give sparks of from 
 to f millimetre in length, so that according to theory visible 
 sparks ought in any case to be obtained ; but, on the other 
 hand, sparks several millimetres in length, as were obtained 
 in the experiments previously described, can only be explained 
 on the assumption that the successive inductive actions pro- 
 duce an accumulative effect ; so that theory indicates the 
 necessity of the existence of the resonant effects actually 
 observed.
 
 438 DYNAMICAL THEORY OF INDUCTION. 
 
 Hertz was at first inclined to suppose that as the micro- 
 meter circuit was only broken by the extremely short air space 
 limited by the maximum sparking distance under the condi- 
 tions of the experiment, it might therefore be treated as a 
 closed circuit, and only the total induction considered. The 
 ordinary methods of electro-dynamics give the means of com- 
 pletely determining the total inductive effect of a current 
 element on a closed circuit, and would, therefore, in this case 
 have sufficed for the investigation of the phenomena observed. 
 He found, however, that the treatment of the micrometer 
 circuit as a closed circuit led to incorrect results, so that it, 
 as well as the primary, had to be treated as an open circuit, 
 and therefore a knowledge of the total induction was insuffi- 
 cient, and it became necessary to consider the value both of 
 the E.M.F. of induction and of the electrostatic E.M.F. due to 
 the charged extremities of the exciting circuit at each point of 
 the micrometer circuit. 
 
 The investigations to which these considerations led are 
 described by Hertz in a Paper, " On the Action of a Rectilineal- 
 Electrical Oscillation upon a Circuit in its Vicinity," published 
 in Wiedemann's Annalen, Vol. XXXIV., p. 155, 1883. 
 
 In what follows the exciting circuit will be spoken of as the 
 primary and the micrometer circuit as the secondary. Hertz 
 points out that the reason that electrostatic effect cannot be 
 neglected is to be found in the extreme rapidity with which 
 the electrostatic forces change their sign. If the electrostatic 
 alternations in the primary were comparatively slow they 
 might attain a very high intensity without giving rise to a 
 spark in the secondary, since the electrostatic distribution on 
 the secondary would vary so as to remain in equilibrium with 
 the external E.M.F. This, however, is impossible, because the- 
 variations in direction follow each other too rapidly for the 
 distribution to foUow them. 
 
 In the present investigations the primary circuit consisted of 
 a straight copper wire 5 millimetres in diameter, carrying at its 
 extremities hollow zinc spheres 30 centimetres in diameter- 
 The centres of the spheres were one metre apart, and at the 
 middle of the wire was an air space f centimetre in length. 
 The wire was placed in a horizontal position, and the observa- 
 tions were all made at points near to the horizontal plane
 
 DYNAMICAL THEORY OF INDUCTION. 439 
 
 through it, which, however, did not of course affect their 
 generality, as the same effects would necessarily be produced 
 in any plane through the horizontal wire. The secondary 
 circuit consisted of a circle of 35 centimetres radius, of copper 
 wire 2 millimetres in diameter, the circle being broken by 
 an air space capable of variation by means of a micrometer 
 screw.* 
 
 The circular form was selected for the secondary circuit 
 because the former investigations had shown that the sparking 
 distance was not the same at all points of the secondary, even 
 when the conductor as a whole remained unchanged in posi- 
 tion, and with a circular circuit it was easier to bring the air 
 space to any part than if any other form had been used. To 
 attain this object the circle was made movable about an axis 
 passing through its centre perpendicular to its plane. 
 
 The circuits of the dimensions stated were very nearly in 
 unison, and they were further adjusted by means of little 
 strips of metal soldered to the extremities, and varied in 
 length until the maximum sparking distance was obtained. 
 
 We shall follow Hertz in first considering the subject 
 theoretically, and then examining how far the experimental 
 results are in accordance with the theoretical conclusions. It 
 will be assumed that the E.M.F. at every point is a simple 
 harmonic function of the time, but that it does not undergo 
 reversal in direction, and it will further be assumed that the 
 oscillations are at any given moment everywhere in the same 
 phase. This will certainly be the case in the immediate neigh- 
 bourhood of the primary, and for the present we shall confine 
 our attention to such points. Let s be the distance of a point 
 measured along the circuit from the air space of the secondary, 
 and F the component E.M.F. at that point along the circular 
 arc d s. Then F is a function of s, which assumes its original 
 value after passing once round the circle of circumference S. 
 It may, therefore, be expanded in the form 
 
 JTTS B'sin 27rs + 
 
 "s^ a IT 
 
 * This small circular detector circuit may be called an electro-magnetic 
 eye, because it enables us to see the electro-magnetic disturbance and to 
 localise it.
 
 440 DYNAMICAL THEORY OF INDUCTION. 
 
 The higher terms of the series may he neglected, as the only 
 result of so doing will be that the approximate theory will 
 give an absolute disappearance of sparks where really the 
 disappearance is not quite complete, and indeed the experi- 
 ments are not delicate enough to enable us to compare their 
 results with theory beyond a first approximation. 
 
 The force A acts in the same direction and is of constant 
 amount at all points of the circle, and therefore it must be 
 independent of the electrostatic E.M.F., as the integral of the 
 latter round the circle is zero. A, then, represents the total 
 E.M.F. of induction, which is measured by the rate of varia- 
 tion of the number of magnetic lines of force which pass 
 through the circle. If the electro-magnetic field containing 
 the circle is assumed to be uniform, A will therefore be 
 proportional to the component of the magnetic induction 
 perpendicular to the plane of the secondary. It will therefore 
 vanish when the direction of the magnetic induction lies in 
 the plane of the secondary. A will consist of an oscillation, 
 the intensity of which is independent of the position of the 
 air space in the circle, and the corresponding spaiking dis- 
 tance will be called a. 
 
 The term B' sin '\ can have no effect in exciting the funda- 
 mental vibration of the secondary, since it is symmetrical on 
 opposite sides of the air space. 
 o_ 
 
 The term B cos *~~ will give a force acting in the same 
 
 fc> 
 
 direction in the two quadrants opposed to the air space, and 
 will excite the fundamental vibration. In the two quadrants 
 adjacent to the air space it will give a force in the opposite 
 direction, but its effect will be less than that of the former 
 one ; for the current is zero at the extremities of the circuit, 
 and therefore the electricity cannot move so freely as near the 
 centre. This corresponds to the fact that if a string fastened 
 at each end has its central portion and ends acted on respec- 
 tively by oppositely-directed forces, its motion will be that 
 due to the force at the central portion, which will excite the 
 fundamental vibration if its oscillations are in unison with the 
 latter. The intensity of the vibration will be proportional to 
 B. Let E be the total E.M.F. in the uniform field of the
 
 DYNAMICAL THEOET OF INDUCTION. 441 
 
 secondary, cf> the angle between its direction and the plane of 
 the latter, and 6 the angle which its projection on this plane 
 makes with the radius drawn to the air space. Then we shall 
 have, approximately, 
 
 S 
 and, therefore, B = - E cos < sin 0. 
 
 B, therefore, is a function simply of the total E.M.F. due 
 both to the electrostatic and electro -dynamic actions. It will 
 vanish when $ = 90 that is to say, when the total E.M.F. is 
 perpendicular to the plane of the circle, whatever be the posi- 
 tion of the air space on the circle. B will also vanish when 
 = that is to say, when the projection of the E.M.F. on 
 the plane of the circle coincides with the radius through the 
 air space. If the position of the air space on the circle is 
 varied, the angle B will vary, and, therefore, also the intensity 
 of the vibration and the sparking distance. The sparking 
 distance corresponding to the second term of the expansion 
 for F can therefore be represented approximately by a formula 
 of the form (3 sin 9. 
 
 Now, the oscillations giving rise to sparks of lengths a and 
 ,/3 sin respectively are in the same phase. The resulting 
 oscillations will therefore be in the same phase, and their 
 amplitudes must be added together. The sparking distance 
 being approximately proportional to the maximum total 
 amplitude may therefore also be obtained by adding the 
 sparking distances due to the two oscillations respectively. 
 The sparking distance will therefore be given as a function of 
 the position of the air space on the secondary circuit by the 
 expression a + (3 sin B. Since the direction of the oscillation 
 in the air space does not come into consideration, we are con- 
 cerned only with the absolute value of this expression and not 
 with its sign. The determination of the absolute values of 
 the quantities a and would involve elaborate theoretical 
 investigations, and is, moreover, unnecessary for the explanation 
 of the experimental results. 
 
 Experiments with the Secondary Circuit in a Vertical Plane. 
 When the circle forming the secondary circuit was placed 
 with its plane vertical, anywhere in the neighbourhood of the 
 .primary, the following results were obtained.
 
 442 DYNAMICAL THEORY OF INDUCTION. 
 
 The sparks disappeared for two positions of the air space, 
 separated by ISOdeg., namely, those in which it lay in the 
 horizontal plane through the primary ; but in every other 
 position sparks of greater or less length were observed. 
 
 From this it followed that the value of a must have been 
 constantly zero, and that 6 was zero when the air space was in 
 the horizontal plane through the primary. 
 
 The electromagnetic lines of force must therefore have been 
 perpendicular to this horizontal plane, and therefore consisted 
 of circles with their centres on the primary ; while the electro- 
 static lines of force must have been entirely in the horizontal 
 plane, and therefore this system of lines of force consisted of 
 curves lying in planes passing through the primary. Both of 
 these results are in agreement with theory. 
 
 When the air space was at its greatest distance from the 
 plane the sparking distance attained a maximum value of 
 from 2 to 3 millimetres. The sparks were shown to be due to 
 the fundamental vibration by slightly varying the secondary 
 so as to throw it out of unison with the primary when the 
 sparking distance was diminished, which would not have been 
 the case if the sparks had been due to overtones. Moreover, 
 the sparks disappeared when the secondary was cut at its 
 points of intersection with the horizontal plane through the 
 primary, though these would be nodal points for the first 
 overtone. 
 
 When the air space was kept at its greatest possible distance 
 from the horizontal plane through the primary, and turned 
 about a vertical axis, the sparking distance attained two 
 maxima at the points for which < = 0, and almost dis- 
 appeared at the points for which < = 90. 
 
 The lower half of Fig. 158 shows the different positions of 
 minimum sparking. A A' is the primary conductor, and the 
 lines in n represent the projections of the secondary circuit on 
 the horizontal plane. The arrows perpendicular to these give 
 the direction of the resultant lines of force. As this did not 
 anywhere vanish in passing from the sphere A to the sphere 
 A', it could not change its sign. 
 
 The diagram brings out the two following points : 
 
 1. The distribution of the resultant E.M.F. in the vicinity 
 of the rectilinear vibration is very similar to that of the electro-
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 443 
 
 static E.M.F. due to the action of its two extremities. It 
 should be specially noted that near the centre of the primary 
 the direction is that of the electrostaiic E.M.F., showing tha"t 
 it is more powerful than the electro-dynamic, as required by 
 theory. 
 
 2. The lines of force deviate more rapidly from the line 
 A A' than the electrostatic lines, though thia is not so evident 
 on the reduced scale of the diagram as in the author's original 
 drawings on a much larger scale. 
 
 It is due to the components of the electrostatic E.M.F. 
 parallel to A A' being weakened by the E.M.F. of induction, 
 while the parpendicular components remained unaffected. 
 
 FIG. 158. 
 
 Experiments with the Secondary Circuit in a Horizontal Plane. 
 The results obtained when the plane of the secondary was 
 horizontal can best be explained by reference to the upper half 
 of the diagram in Fig 158. 
 
 In the position I., with the centre of the circle in the line 
 A A' produced, the sparks disappeared when the air space 
 occupied either of the positions b l or b\, while two equal maxima 
 of the sparking distance were obtained at % and a' 1} the length 
 of the spark in these positions being 2-5 millimetres. Both 
 these results are in accordance with theory. 
 
 In the position II. the circle is cut by the electro-magnetic 
 lines of force, and therefore a does not vanish. It will, how-
 
 444 DYNAMICAL THEORY OF INDUCTION. 
 
 ever, be small, and we should expect that the expression a + /? 
 sin would have two unequal maxima, /3 + a and (3 -a, both 
 for = 90, and having the line joining them perpendicular 
 to the resultant E.M.F., and between these two maxima we 
 should expect two points of no sparking near to the smaller 
 maximum. This was confirmed by the observations. 
 
 The maximum sparking distances were 3 - 5 millimetres at 
 # 2 and 2 millimetres at a' 2 . Now, with the air space at o 2 , 
 the sphere A being positive, the resultant E.M.F. in the oppo- 
 site portion of the circle will repel positive electricity from A, 
 and therefore tend to make it flow round the circle clockwise. 
 Between the two spheres the electrostatic E.M.F. acts from A 
 towards A', and the opposite E.M.F. of induction in the neigh- 
 bourhood of the primary acts from A' to A, parallel to the 
 former, and, acting more strongly on the nearer than on the 
 further portion of the secondary, tends to cause a current in 
 same direction as that due to the former, namely, in a clock- 
 wise direction. Thus the resultant E.M.F. is the sum of the 
 two as required by theory, and in the same way it is easily 
 seen that when the air space is at a' a the resultant E.M.F. is 
 equal tc their difference. 
 
 As the position III. is gradually approached the maximum 
 disappears, and the single maximum sparking distance a s was 
 found to be four millimetres in length, having opposite to it a 
 point of disappearance ' 3 . In this case clearly a = /?, and the 
 sparking distance is given by the expression a(l + sin 0). The 
 line a 3 a' & is again perpendicular to the resultant E.M.F. 
 
 As the circle approaches further towards the centre of A A' f 
 a will become greater than (3, and the expression a + ft sin 
 will not vanish for any value of 6, but will have a maximum 
 a + /3 and a minimum a -(3; and in the experiments it was 
 found that the sparks never entirely disappeared, but varied 
 between a maximum and a minimum, as indicated by theory. 
 
 In the position IV. a maximum sparking distance of 5-5 
 millimetres was observed at 4 , and a minimum of 1-5 milli- 
 metre at o' 4 . 
 
 In the position V. there was a maximum sparking distance 
 of 6 millimetres at a 5 , and a minimum of 2-5 millimetres at a' t . 
 In these experiments the air space should be screened off from 
 the primary in the latter positions as well as in the earlier
 
 DYNAMICAL THEORY OF INDUCTION. 445 
 
 ones, in which it is unavoidable, as otherwise the results would 
 not be comparable. 
 
 In passing from the position III. to the position V. the line 
 a a' rapidly turned from its position of parallelism to the 
 primary circuit into a position perpendicular to it. In the 
 latter positions the sparking was essentially due to the induc- 
 tive action, and therefore Hertz was justified in his previous 
 experiments in assuming the effect in these positions to be due 
 to induction. 
 
 Even in these positions, however, the sparking is not totally 
 independent of electrostatic action, except when the air space 
 is half-way between the maximum and minimum positions, and 
 therefore & sin 6 = 0. 
 
 Other Positions of the Secondary Circuit. Hertz made 
 numerous observations with the secondary circuit in other 
 positions, but in no case were any phenomena observed which 
 were not completely in accordance with theory. As an 
 
 example of these consider the following experiment : 
 
 The secondary was first placed in the horizontal plane in the 
 position V. (Fig. 158), and the air space was in the position a. 
 relatively to the primary. The circle was then turned about a 
 horizontal axis through its centre and parallel to the primary, 
 so as to raise the air space above the horizontal plane. During 
 this rotation 9 remained equal to 90deg., and the value of /8 
 remained nearly constant, but a varied approximately in the 
 same ratio as cos W, "*& being the angle between the plane of the 
 circle and the horizontal, for a is proportional to the number of 
 magnetic lines of force passing through the circle. Let a be 
 the value of a in the initial position, then in the other postions 
 its value would be o<, cos "*?, and therefore the sparking dis- 
 tance should be given by the expression a n cos ^F + /3, in which 
 a was known to be greater than (3. This was confirmed by 
 observation, for it was found that as the air space increased 
 its height above the horizontal plane the sparking distance 
 diminished from 6 millimetres down to 2 millimetres, its value 
 when the air space was at its greatest distance above the hori- 
 zontal plane. During the rotation through the next quadrant 
 the sparking distance diminished almost to zero, and then 
 increased to the smaller maximum of 2-5 millimetres, which it 
 attained when the circle had turned through ISOdeg., and was
 
 446 DYNAMICAL THEORY OF INDUCTION. 
 
 therefore again horizontal. Similar results were obtained in 
 the opposite order as the circle was rotated from 108deg. to 
 360deg. When the circle was kept with the air space at its 
 maximum height above the horizontal plane, and then raised 
 or lowered bodily without rotation, the sparking distance was 
 found to diminish in the former case and to increase in the 
 latter results completely in accordance with theory. 
 
 Forces at Greater Distances. Experiments with the secondary 
 at greater distances from the primary are of great importance, 
 as the distribution of E.M.F. in the field of an open circuit is 
 very different according to different theories of electro-dynamic 
 action, and the results may, therefore, serve to eliminate some 
 of them as untenable. In making these experiments, how- 
 ever, an unexpected difficulty was encountered, as it was found 
 that at distances of from 1 to 1'5 metre from the primary, the 
 maximum and minimum, except in certain positions, became 
 indistinctly defined ; but when the distance was increased to 
 upwards of two metres, though the sparks were then very 
 small, the maximum and minimum were found to be very 
 sharply marked when the sparks were observed in the dark. 
 The positions of maximum and minimum were found to occur 
 with the circle in planes at right angles to each other. At 
 considerable distances the sparking diminished very slowly as 
 the distance was increased. Hertz was not able to determine 
 an upper limit to the distance at which sensible effects took 
 place, but, in a room 14 metres by 12, sparks were distinctly 
 observed when the primary was placed in one corner of the 
 room, wherever the secondary was placed. When, however, 
 the primary was slightly displaced no effects could be observed, 
 even when the secondary was brought considerably nearer. 
 The interposition of solid screens between the two circuits 
 greatly diminished the effect. 
 
 Hertz mapped out the distribution of force throughout the 
 room by means of chalk lines on the floor, putting stars at the 
 points where the direction of the E.M.F. became indeter- 
 minate. A portion of the diagram obtained in this manner is 
 shown on a reduced scale in Fig. 159, with respect to which 
 the following points are noteworthy : 
 
 1. At distances beyond three metres the E.M.F. is every- 
 where parallel to the primary oscillation. Within this region,
 
 DYNAMICAL THEORY OF INDUCTION. 447 
 
 therefore, the electrostatic E.M.F. is negligible in comparison 
 with the E.M.F. of induction. Now all the theories of the 
 mutual action of current elements agree in giving an E.M.F. 
 of induction inversely proportional to the distance ; while the 
 electrostatic E.M.F., being due to the differential action of 
 the two extremities of the primary, is approximately inversely 
 proportional to the cube of the distance. Some of these 
 theories, however, are not in accordance with the experi- 
 mental result that the effect diminishes much more rapidly 
 in the direction of the primary oscillation than in a direc- 
 tion at right angles to it, induced sparks being observed at 
 a distance exceeding 12 metres in the latter direction, while 
 they disappeared at a distance of about four metres in the 
 former direction. 
 
 FIG. 159. 
 
 2. That, as already proved, for distances less than one 
 metre the distribution of E.M.F. is practically that of the 
 electrostatic E.M.F. 
 
 8. There are two straight lines at all points of which the 
 direction of the E.M.F. is determinate, namely, the line in 
 which the primary oscillation takes place, and the perpendicular 
 to the primary through its middle point. Along the latter the 
 E.M.F. does not vanish at any point : the sparking diminishes 
 gradually as the distance is increased. This, again, 1? incon- 
 sistent with some of the theories of mutual action of current 
 elements, according to which it should vanish at a certain 
 definite distance. A very important result of the investigation 
 is the demonstration of the existence of regions within which
 
 448 DYNAMICAL THEORY OF INDUCTION. 
 
 the direction of the E.M.F. becomes indeterminate. These 
 regions form two rings encircling the primary circuit. Since 
 the E.M.F. within them acts very nearly equally in every 
 direction, it must assume different directions in succession, for, 
 of course, it cannot act in different directions simultaneously. 
 
 The observations, therefore, lead to the conclusion that 
 within these regions the magnitude of the E.M.F. remains 
 very nearly constant, while its direction varies through all 
 the points of the compass at each oscillation. Hertz stated 
 that he was unable to explain this result, as also the 
 existence of overtones, by means of the simplified theory in 
 which the higher terms of the expansion of F are neglected, 
 and he considers that no theory of simple action at a distance 
 is capable of explaining it. If, however, the electrostatic 
 E.M.F. and the E.M.F. of induction are propagated through 
 space with unequal velocities, it admits of very simple explana- 
 tion. For within these annular regions the two E.M.F.s are 
 at right angles and of the same order of magnitude ; they will, 
 therefore, in consequence of the distance traversed, differ in 
 phase, and the direction of the resultant will turn through all 
 the points of the compass at each oscillation. 
 
 This phenomenon appeared to him to be the first indication 
 which had been observed of a finite rate of propagation through 
 space of electrical actions, for, if there is a difference in the 
 rate of propagation of the electrostatic and electro-dynamic 
 E.M.F., one at least of them must be definite. 
 
 At the end of the Paper in which the preceding experiments 
 are described, Hertz describes some observations which he 
 made' on the conditions at the primary sparking point 
 which affect the production of sparks in the secondary circuit. 
 He found that illuminating the primary spark diminished 
 its power of exciting rapid oscillations, the sparks in the 
 secondary being observed to cease when a piece of magnesium 
 wire was burnt or an arc lamp lighted near the primary 
 point. The observed effect on the primary sparks is that 
 they are no longer accompanied by a sharp crackling sound 
 as before. The effect of a second discharge is especially note- 
 worthy, and it was found that the secondary sparks could be 
 made to disappear by bringing an insulated conductor close 
 to the opposed surfaces of the spheres forming the terminals
 
 DNYAM1CAL THEORY OF INDUCTION. 449 
 
 at the primary air space, even when no visible sparking took 
 place between the latter and the insulated conductor. The 
 secondary sparking could also be stopped by placing a fine 
 point close to the primary air space, or by touching one of 
 the opposed surfaces of the terminals with a piece of sealing- 
 wax, glass, or mica. Hertz states that further experiments 
 led him to conclude that, even in these cases, the effect is due 
 to light too feeble to be perceived by the eye, arising from a 
 side discharge. He points out that these effects afford another 
 example of the effects of light on electric discharges, which 
 have been observed by E. Wiedemann, H. Hebert, and W. 
 Hallwachs. 
 
 Hertz's next Paper in order of publication in Wiedemann's 
 Annalen is "On Some Induction Phenomena Arising from 
 Electrical Actions in Dielectrics " (Vol. XXXIV., p. 273), and 
 contains an account of some researches which were under- 
 taken with a view of obtaining direct experimental confirma- 
 tion of the assumption involved in the most suggestive theory 
 of electrical actions, viz., that of Faraday and Maxwell, that 
 the well-known electrostatic phenomena observed in dielectrics 
 are accompanied by corresponding electro-dynamic actions. 
 The method of observation consisted in placing a secondary 
 conductor adjusted to unison, as regards electrical oscillations, 
 with the primary, as near as possible to the former, and in 
 such a relative position that the sparks in the primary pro- 
 duced no sparking in the secondary. As the equilibrium 
 could be disturbed and sparking induced in the secondary by 
 the approach of conductors, it formed a kind of induction 
 balance ; but the point of special interest in connection with 
 it was that a similar effect was produced when the conductors 
 were replaced by insulators, provided the latter were of com- 
 paratively large size. The observed rapidity of the oscillations 
 induced in the dielectrics showed that the quantities of elec- 
 tricity in motion under the influence of dielectric polarisation 
 were of the same order of magnitude as in the case of metallic 
 conductors. 
 
 The apparatus employed is shown diagrammatically in 
 Fig. 160, and was supported on a light wooden framework, not 
 shown in the illustration. The primary conductor consisted 
 of two brass plates, A A', with sides 40 centimetres in length, 
 
 oo
 
 450 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 joined by a copper wire 70 centimetres long and half a centi- 
 metre in diameter, containing an air space of three-quarters of 
 a centimetre, with terminals formed of polished brass spheres. 
 When placed in connection with a powerful induction coil, 
 oscillations are set up, the period of which, determined by the 
 dimensions of the primary, can be determined to within a 
 hundred-millionth of a second. The secondary conductor con- 
 sisted of a circle, 35 centimetres in radius, of copper wire two 
 millimetres in diameter, containing an air space, the length 
 of which could be varied by means of a screw from a few 
 
 FIG. 160. 
 
 hundredths of a millimetre up to several millimetres. The 
 dimensions stated were such as to bring the two conductors 
 into unison, and secondary sparks up to six or seven milli- 
 metres in length could be obtained. 
 
 The circle was movable about an axis through its centre 
 perpendicular to its plane, to enable the position of the air 
 space to be varied. The axis was fixed in the position m n in 
 the plane of A and A', and half-way between them. The 
 centre of the circle was at a distance of 12 centimetres from 
 the nearest points of A and A'.
 
 DYNAMICAL THEORY OF INDUCTION. 451 
 
 When / was in either of the positions a or a' lying in the 
 plane of A A' no sparking occurred in the secondary, while 
 maximum sparking took place at b and V 90deg. from the 
 former positions. The E.M.F. giving rise to the secondary 
 sparks is, as in previous experiments, partly electrostatic and 
 partly electro-magnetic, and the former being the greater will 
 determine the sign of the resultant E.M.F. The oscillations 
 must, for the reason previously explained, be considered as 
 produced in the part of the secondary most remote from the 
 air space. Assuming the E.M.F. and the amplitude of the 
 resulting oscillation to be positive when/ is in the position b', 
 they will both be negative when /is at b. 
 
 When the circle was slightly lowered in its own plane the 
 sparking distance was increased at b' and diminished at b, and 
 the null points lay at a certain distance below a and a'. The 
 electrostatic E.M.F. is scarcely affected by such a displacement, 
 but the integral of the E.M.F. of induction taken round the 
 circle is no longer zero, and therefore gives rise to an oscil- 
 lation which will be of positive sign whatever be the position 
 of /, for the direction of the resultant E.M.E\ of induction 
 is opposite to that of the electrostatic E.M.F. in the upper 
 half of the circle, and coincides with it in the lower half, where 
 the electrostatic E.M.F. has been assumed to be positive. Since 
 the new oscillation so produced is in the same phase as the 
 previously existing one, their amplitudes must be added 
 io give the resultant amplitude, which explains the pheno- 
 mena. 
 
 Effects of tJie Approach of Conductors. In making these 
 observations it was found necessary to remove all conductors to 
 a considerable distance from the apparatus, in order to obtain 
 a complete disappearance of sparking at the points a and a'. 
 Even the neighbourhood of the observer was sufficient to set 
 up sparking when the air space / was in either of these posi- 
 tions, and the sparks had therefore to be observed from a 
 distance. The conductors used for the experiments were of 
 the form shown at C (Fig. 160), and consisted of thin metal 
 foil. The objects kept in view in selecting the material and 
 dimensions were to obtain a conductor which would give a 
 moderately large effect and having an oscillation period less 
 than that of the primary. 
 
 GG 2
 
 452 DYNAMICAL THEORY OF INDUCTION. 
 
 When the conductor C was brought near to A A', it was 
 found that the sparking distance decreased at b and increased 
 at b', and the null points were displaced upwards that is, in 
 the direction of C. 
 
 From the results of experiments already described it is 
 evident that the effect of displacing A A' upwards would be 
 the same, qualitatively, as that of a current in the same 
 direction as that in A A' directly above it. The effect pro- 
 duced by the approach of C was the reversa of this, and could 
 be explained by an inductive action, supposing there were a 
 current in C in the opposite direction to that in A A', which 
 is exactly what must occur ; for the electrostatic E.M.F. would 
 give rise to such a current, and since the oscillations in C are 
 more rapid than those of the E.M.F., the current must be in 
 the same phase as the inducing E.M.F. The truth of this 
 explanation was confirmed by the following experiments. 
 The horizontal plates of the conductor C being left in the same 
 position as before, the vertical plate was removed, and succes- 
 sively replaced by wires of increasing length and fineness, in 
 order to lengthen the oscillation period of C. The effect of 
 this was to displace the null points more and more in an 
 upward direction, while at the same time they became less 
 sharply defined, a minimum sparking taking the place of the 
 previous absolute disappearance. The sparking distance at the 
 highest point had previously been much less than at the lowest 
 point, but after the disappearance of the null points it began 
 to increase. At a certain stage the sparking distance at the 
 two positions became equal, and then no definite minimum 
 points could be found, but sparking took place freely at all 
 positions of/. Beyond this stage the sparking distance at the 
 lowest point diminished, and very soon two minimum points 
 made their appearance close to it, not clearly defined at first, 
 but gradually becoming more distinct, and at the same time 
 approaching the points a a' , with which they ultimately coin- 
 cided, when the minimum points again became absolute null 
 points. These results are in agreement with the conclusion 
 drawn from the former observations, for as the oscillation 
 period of C approaches that of A A' the intensity of the 
 current in the former increases, but a difference of phase 
 arises between it and the existing E.M.F. When the two-
 
 DYNAMICAL THEORY OF INDUCTION. 453 
 
 are in unison the current in C attains its maximum, and, 
 as in other cases of resonance, the difference of phase gives 
 rise to a slightly clamped oscillation, having a period of about 
 a quarter that of the original one, which makes any inter- 
 ference between the oscillations excited in the circle B by A A' 
 and C respectively impossible. These conditions clearly corre- 
 spond to the stage at which the sparking distances at b and 
 V were equal. When the oscillation period of C becomes 
 decidedly greater than that of A A', the amplitude of the 
 oscillation in the former will again diminish, so that the 
 difference in phase between it and the exciting E.M.F. will 
 approach half of the original period. The current in C will 
 therefore always be in the same direction as that in A A', so 
 that interference between the two oscillations excited in B will 
 again become possible, and the effect of C will then be opposite 
 to its original effect. When the conductor C was made to 
 approach A A' the sparks in B became much smaller, which 
 is explained by the fact that its effect will be to increase the 
 oscillation period of A A', and therefore to throw it out of 
 unison with B. 
 
 Effects of the Approach of Dielectrics. A very rough estimate 
 shows that when a dielectric of large mass is brought near 
 to the apparatus the quantities of electricity set in motion by 
 dielectric polarisation are at least as large as in metallic wires 
 or thin rods. If, therefore, the action of the apparatus were 
 unaffected by the approach of such masses, it would show that, 
 in contradiction to the theories of Faraday and Maxwell, 
 no electro -dynamic actions are called into play by means of 
 dielectric polarisation, or as Maxwell calls it, electric displace- 
 ment. The experiments, however, showed an effect similar to 
 that which would be produced if the dielectric were replaced 
 by a conductor with a very small oscillation period. In the 
 first experiment made, the mass of dielectric consisted of a 
 pile of books, 1-5 metre long, O p o metre broad, and 1 metre 
 high, placed under the plates A A'. Its effect was to displace 
 the null points through about lOdeg. towards the pile. A 
 block of asphalte (D, in Fig. 160), weighing 800 kilogrammes, 
 and measuring 1-4 metre in length, 0-4 metre in breadth, and 
 0-6 metre in height, was then used in place of the books, the 
 plates being allowed to rest upon it.
 
 454 DYNAMICAL THEORY OF INDUCTION. 
 
 The following results were then obtained : 
 
 1. The spark at the highest point of the circle -was now 
 decidedly stronger than that at the lowest point, which was 
 nearer to the asphalte. 
 
 2. The null points were displaced through about 23deg. 
 downwards that is, in the direction of the block and at 
 the same time were transformed into mere points of mini- 
 mum sparking, a complete disappearance being no longer 
 obtainable. 
 
 3. When the plates A A' rested on the asphalte block the 
 oscillation period of the primary was increased, as shown by 
 the fact that the period of B had to be slightly increased in 
 order to obtain the maximum sparking distance. 
 
 4. When the apparatus was moved gradually away from the 
 block its action steadily diminished without changing its 
 character. 
 
 5. The action of the block could be compensated by bringing 
 the conductor C over the plates A A' while they rested on the 
 block, the null points being brought back to a and a' when C 
 was at a height of 11 centimetres above the plates. When the 
 upper surface of the asphalte was 5 centimetres below the 
 plates, compensation was obtained when C was placed at a 
 height of 17 centimetres above them, showing that the action 
 of the dielectric was of the order of magnitude which had been 
 anticipated. 
 
 The asphalte contained about 5 per cent, of aluminium 
 and iron compounds, 40 per cent, of calcium compounds, and 
 17 per cent, of quartz sand. In order to make sure that the 
 observed effects were not due to the conductivity of some of 
 these substances a number of further experiments were 
 made. 
 
 In the first place, the asphalte was replaced by a mass of 
 the same dimensions of the so-called artificial pitch prepared 
 from coal, and effects of a similar kind were observed, but 
 slightly weaker, the greatest displacement of the null points 
 amounting to 19deg. Unfortunately this pitch contains free 
 carbon, the amount of which it is difficult to determine, and 
 this would have some conductivity. 
 
 The experiments were then repeated with a conductor, C, of 
 half the linear dimensions of the former one, and smaller blocks
 
 DYNAMICAL THEORY OF INDUCTION. 455 
 
 of various substances, on account of the great cost of obtaining 
 large blocks of pure materials. The substances used were 
 asphalte, coal-pitch, paper, wood, sandstone, sulphur, paraffin, 
 and also a fluid dielectric, namely, petroleum. With the 
 smaller apparatus it was not possible to obtain quantitative 
 results of the same accuracy as before, but the effects were of 
 an exactly similar character, and left little room for doubt of 
 the reality of the action of the dielectric. 
 
 The results might possibly be supposed to be due to a change 
 in the distribution of the electrostatic E.M.F. in the neighbour- 
 hood of the dielectric, but, in the first place, Hertz stated 
 that he was unable to explain the details of the observations 
 on this hypothesis, and in the second place it is disproved by 
 the following experiment : 
 
 The smaller apparatus was placed with the line r s on the 
 upper near corner of one of the large blocks, in which position 
 the dielectric was bounded by the plane of the plates A A' and 
 the perpendicular plane through r s, both of which are equi- 
 potential surfaces, so that if the action were electrostatic no 
 effect should be produced by the dielectric. It was found, 
 however, to produce the same effect as in other positions. It 
 might also be supposed that the effects were due to a slight 
 conductivity, but this could hardly be the case with such 
 good insulators as sulphur and paraffin. Suppose, moreover, 
 that the conductivity of the dielectric is sufficient to discharge 
 the plate A in the ten-thousanth of a second, but not much 
 more rapidly; then, during one oscillation, the plates would 
 loose only the ten-thousandth part of their charge, and the 
 conduction current in the substance experimented on would 
 not exceed the ten-thousandth part of the primary current in 
 A A', so that the effect would be quite insensible. 
 
 It is thus shown in the experiments described above that 
 when variable electrical forces act in the interior of dielec- 
 trics of specific inductive capacity not equal to unity the 
 corresponding electric displacements produce electro-dynamic 
 effects. In a Paper, " On the Velocity of Propagation of Electro- 
 Dynamic Actions," in Wiedeniann's Annalen, Vol. XXXIV., 
 p. 551, Hertz showed that similar actions take place in the 
 air, which proves, as was previously pointed out, that electro- 
 dynamic action must be propagated with a finite velocity.
 
 456 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 The method of investigation was to excite electrical oscilla- 
 tions in a rectilinear conductor in the same manner as in 
 former experiments, and then to produce effects in a secondary 
 conductor by exciting electrical oscillations in it by means of 
 those in the rectilinear conductor, and at the same time by the 
 primary conductor acting through the intervening space. This 
 distance was gradually increased, when it was found that the 
 phase of the vibrations at a distance from the primary lagged 
 behind those in its immediate neighbourhood, showing that 
 the action is propagated with a finite velocity which was found 
 to be greater than the velocity of propagation of electrical 
 waves in wires in the ratio of about 45 to 28, so that the former 
 is of the same order as the velocity of light. Hertz was 
 unable to obtain any evidence with respect to the velocity 
 of propagation of electrostatic actions. 
 
 FIG. 161. 
 
 The primary conductor A A' (Fig. 161) consisted of a pair 
 of square brass plates with sides 40 centimetres in length, con- 
 nected by a copper wire 60 centimetres in length, at the middle 
 point of which was an air space, across which sparks were 
 made to pass by means of powerful discharges from the induc- 
 tion coil J. The conductor was fixed at a height of 1*5 metre 
 above the base -plate of the coil, with its plates vertical, and 
 the connecting wire horizontal. A straight line, rs, drawn 
 horizontally through the air space of the primary, and perpen- 
 dicular to the direction of the primary oscillation, will be called 
 " the base-line ; " and a point in this, situated at a distance of 
 45 centimetres from the air space, will be referred to as " the 
 null point."
 
 D YNA MICAL THEOE Y OF IND UC TWN. 457 
 
 The experiments were made in a large lecture-room, with 
 nothing near the base-line for a distance of 12 metres from 
 the primary conductor. The room was darkened during the 
 experiments. 
 
 The secondary conductor consisted either of a circular wire, 
 C, of 35 centimetres radius, or of a square of wire, B, with 
 sides 60 centimetres long. The primary and secondary air 
 spaces were both capable of adjustment by means of micro- 
 meter screws. Both the secondary conductors were in unison 
 with the primary, the (half) vibration period of each being one 
 hundred- millionth of a second, as calculated from the capacity 
 and coefficient of self-induction. It is doubtful whether the 
 ordinary theory of electrical oscillations would lead to accurate 
 results under the conditions of these experiments, but as it 
 gives correct numerical results in the case of Leyden jar dis- 
 charges, it may be expected to be correct as far as the order 
 of the results is concerned. When the centre of the secondary 
 lies in the base-line, and its plane coincides with the vertical 
 plane through the base-line, no sparks are observed in the 
 secondary, the E.M.F. being everywhere perpendicular to the 
 direction of the secondary. This will be referred to as " the 
 first principal position" of the secondary. When the plane of 
 the secondary is vertical and perpendicular to the base-line, the 
 centre still lying in the base-line, the secondary will be said to 
 be in its " second principal position." Sparking then occurs 
 in the secondary when its air space is either above or below 
 the horizontal plane through the base-line, but not when it is 
 in this plane. As the distance from the primary was increased, 
 the sparking distance was observed to decrease, rapidly at first, 
 but ultimately very slowly. Sparks were observed throughout 
 the whole distance of 12 metres available for the experiments. 
 The sparking in this position is due essentially to the E.M.F. 
 produced in the portion of the secondary remote from the air 
 space. The total E.M.F. is partly electrostatic and partly 
 electro-dynamic, and the experiments show beyond the possi- 
 bility of doubt that the former is greater, and therefore deter- 
 mines the direction of the total E.M.F. close to the primary, 
 while at greater distances it is the electro-dynamic E.M.F. 
 which is the greater. 
 
 The plane of the secondary was then turned into the hori-
 
 458 DYNAMICAL THEORY OF INDUCTION. 
 
 zontal, its centre still lying in the base-line. This may be 
 called "the third principal position." When the centre of the 
 circular secondary conductor was kept fixed at the null point, 
 and the air space was made to travel round the circle, vigorous 
 sparking was observed in all positions. The sparking distance 
 attained its maximum length of about six millimetres when its 
 air space was nearest to that of the primary, and its minimum 
 length of about three millimetres when the distance between 
 the two air spaces was greatest. If the secondary had been 
 influenced by the electrostatic force, sparking would only be 
 expected when the air space was close to the base-line, and a 
 cassation of sparks in the intermediate positions. The direction 
 of the oscillation would, moreover, be determined by the direc- 
 tion of the E.M.F. in the portion of the secondary furthest 
 from the air space. There is, however, superposed upon the 
 electrostatically excited oscillation a second oscillation, due to 
 the E.M.F. of induction, which produces a considerable effect, 
 since its integral round the circle (considered as a closed circuit) 
 does not vanish; and the direction of this integral E.M.F. is 
 independent of the position of the air space, opposing the 
 electrostatic E.M.F. in the portion of the secondary next to 
 A A', and assisting it in the portion furthest from A A', as 
 explained previously. 
 
 The electrostatic and electro-dynamic E.M.F.s, therefore, 
 act in the same direction when the air space is turned towards 
 the primary conductors, and hi opposite directions when the 
 air space is turned away from the primary. In the latter 
 position it is the E.M.F. of induction which is the more 
 powerful, as is shown by the fact that there is no disap- 
 pearance of sparking in any position of the air space, for when 
 this is 90deg. to the right or left of the base-line it coincides 
 with a node with respect to the electrostatic E.M.F. In these 
 positions the inductive action in the neighbourhood of the 
 primary can be observed independently of the electrostatic 
 action. 
 
 Waves in Rectilinear Wires. In order to produce in a wire- 
 by means of the primary oscillations a series of advancing 
 waves of the character required for these experiments, the 
 following arrangements were made : Behind the plate A was 
 placed a plate, P, of equal size. A copper wire one millimetre
 
 
 DYNAMICAL THEORY OF INDUCTION. 45 
 
 in diameter connected P to the point M of the base-line. 
 From M the wire was continued in a curve about a metre in 
 length to the point N, situated about 30 centimetres above the 
 air space, and was then further continued in a straight line 
 parallel to the base-line for such a distance as to obviate all 
 danger of disturbance from reflected waves. In the present 
 series of experiments the wire passed through a window, and 
 after being carried to a distance of about 60 metres, was put 
 to earth, and a special series of experiments showed that this 
 length was sufficient. When a wire, bent so as to form a 
 nearly closed circuit with a small air space, was brought near 
 to this straight wire, a series of fine sparks was seen to- 
 accompany the discharges of the induction coil. Their 
 intensity could be varied by varying the distance between the 
 plates P and A. The waves in the rectilinear wire were of the 
 same period as that of the primary oscillations, as was proved 
 by their being shown to be in unison with each of the two 
 secondary conductors previously described. The existence of 
 stationary waves showed that the waves in the rectilinear wire 
 were of a steady character in space as well as in time. The 
 nodal points were determined in the following manner : The 
 further end of the wire was left free, and the secondary con- 
 ductor was brought near to it in such a position that the wire 
 lay in its plane, and had the air space turned towards it. As 
 the secondary was moved along the wire, points of no sparking 
 were observed to recur periodically. The distance from the 
 point n to the first of these was measured, and the length of 
 the wire made equal to a multiple of this distance. The 
 experiments were then repeated, and it was found that the 
 nodal points occurred at approximately equal intervals along 
 the ware. 
 
 The nodes could also be distinguished from the loops in 
 other ways. The secondary conductor was brought near to 
 the wire, with its plane perpendicular to it, and with its air 
 space neither directed completely towards the wire nor com- 
 pletely away from it, but in an intermediate position, so 
 as to produce E.M.F.s perpendicular to the wire. Sparks 
 were then observed at the nodes, while they disappeared 
 at the loops. When sparks were taken from the rectilinear 
 wire by means of an insulated conductor, they were found
 
 460 DYNAMICAL THEORY OF INDUCTION. 
 
 to be stronger at the nodes than at the loops ; the difference, 
 however, was small, and was, indeed, scarcely distinguishable 
 unless the position of the nodes and loops was previously 
 known. The reason that this and other similar methods do 
 not give a well-defined result lies in the fact that irregular 
 oscillations are superposed upon the waves considered ; the 
 regular waves, however, can be picked out by means of the 
 secondary, just as definite notes are picked out by means of a 
 Helmholtz resonator. If the wire is severed at a node, no 
 effect is produced upon the waves in the portion of wire next 
 to the origin ; but if the severed portion of wire is left in its 
 place the waves continue to be propagated through it, though 
 with somewhat diminished strength. 
 
 The possibility of measuring the wave-lengths leads to 
 various applications. If the copper wire hitherto used is 
 replaced by one of different diameter, or by a wire of some 
 other metal, the nodal points retain their position unchanged. 
 It follows from this that the velocity of propagation in a wire 
 has a definite value independent of its dimensions and material. 
 Hertz states that even iron wires offer no exception to this, 
 showing that the magnetic susceptibility of iron does not play 
 any part in the case of such rapid motions. This conclusion 
 is not, however, confirmed by the researches of Prof. J. 
 Trowbridge, and investigations, referred to on page 431, show 
 that the magnetisability of the iron does exert an influence 
 sensible though small. It would be interesting to investi- 
 gate the behaviour of electrolytes in this respect. In their 
 case we should expect a smaller velocity of propagation, 
 because the electrical motions are accompanied by motions of 
 the molecules carrying the electric charges. It was found 
 that no propagation of the waves took place through a tube 
 10 millimetres in diameter, filled with a solution of sulphate 
 of copper ; but this may have been due to the resistance being 
 too high. By the measurement of wave-lengths the relative 
 vibration periods of different primary conductors can be deter- 
 mined, and it therefore becomes possible to compare in this 
 manner the vibration periods of plates, spheres, ellipsoids, &c. 
 
 In the experiments made by Hertz, nodes were very dis- 
 tinctly produced when the wire was severed at a distance of 
 .either 8 metres or 5-5 metres from the null point of the base-
 
 DYNAMICAL THEORY OF INDUCTION. 461 
 
 line. In the first case the nodes occurred at distances from the 
 null point of - 0-2 metre, 2-3 metres, 5*1 metres, and 8 metres, 
 and in the latter case at distances of -0-1 metre, 2-8 metres, 
 and 5-5 metres. It appears, therefore, that the (half) wave- 
 length in a free wire cannot differ much from 2*8 metres. The 
 fact that the wave-lengths nearest to P were somewhat smaller 
 was to be expected from the influence of the plates and of the 
 curvature of the wire. This wave-length, with a period of one 
 hundred-millionth of a second, gives 280,000 kilometres per 
 second for the velocity of propagation of electrical waves in 
 wires. Fizeau and Gounelle (Poggendorffs Annalen, Vol. 
 LXXX., p. 158, 1850) obtained for the velocity in iron wires 
 100,000 kilometres per second, and 180,000 in copper wires. 
 W. Siemens (Poggendorff's Annalen, Vol. CLVIL, p. 309, 
 1876), by the aid of Leyden jar discharges, obtained a velocity 
 of from 200,000 to 260,000 kilometres per second in iron 
 wires. Hertz's result is very nearly the same as the velocity 
 of light. Space will not allow us to fully discuss the causes 
 which led to certain discrepancies in Hertz's earlier results. 
 Suffice it to say that he subsequently found that the velocity 
 of propagation of an electromagnetic disturbance along a wire 
 was the same as in free space, viz., the velocity of light. The 
 apparent difference between the velocity of long and short 
 waves was afterwards explained by Hertz himself, and the 
 causes of this were made clear by the experiments conducted 
 in the large hall of the Rhone waterworks by MM. Sarasin 
 and de la Rive. From these experiments it became clear 
 that the interference due to surrounding objects was the cause 
 of the apparent difference between the velocities of long and 
 short waves, but that in a sufficiently large space this 
 difference disappeared, and the velocity of both long and 
 short electromagnetic waves was the same. The reader may 
 consult with advantage on this point the notes and text of 
 the full translation of Hertz's electrical Papers made by Mr. 
 D. E. Jones.* 
 
 Interference of tlie Direct Actions with those transmitted 
 through the Wire. If the square circuit B is placed at the 
 null point in the second principal position, with the air space 
 
 * " Electric Waves." Authorised English translation of Hertz's Papers, 
 by D. E. Jones.
 
 462 DYNAMICAL THEORY OF INDUCTION. 
 
 at its highest point, it will be unaffected by the waves in tha 
 wire, but the direct action when in this position was found to 
 produce sparks 2 millimetres in length. B was then turned 
 about a vertical axis into the first principal position, in which 
 there would be no direct action of the primary oscillation, but 
 the waves in the wire gave rise to sparks, and by bringing P 
 near enough to A a sparking distance of 2 millimetres could 
 be obtained. In the intermediate positions sparks were pro- 
 duced in both these ways, and it would therefore be possible 
 to get a difference of phase, such that one should either 
 increase or diminish the effect of the other. Phenomena of 
 this nature were, indeed, observed. When the plane of B was 
 in such a position that the normal drawn towards A A' was 
 directed away from that side of the primary conductor on 
 which P was placed, there was more sparking than even in 
 ths principal position ; but if the normal were directed 
 towards P the sparks disappeared, and only reappeared when 
 the air space was made smaller. When the air space was at 
 the lowest point of B, the other conditions remaining the 
 same, the sparks disappeared when the normal was turned 
 away from P. Further variations of the experiment gave 
 results in accordance with these. 
 
 It is easily seen that these phenomena were exactly what 
 were to be expected. To fix the ideas, suppose the air space 
 to be at the highest point, and the normal directed towards P, 
 as in Fig. 161. Consider what happens at the moment that 
 the plate A has its greatest positive charge. The electrostatic, 
 and therefore the total, E.M.F. is directed from A towards A'. 
 The oscillation to which this gives rise in B is determined by 
 the direction of the E.M.F. in the lower portion of B. There- 
 fore positive electricity will flow towards A' in the lower 
 portion, and away from A' in the upper portion. 
 
 Consider next the action of the waves. As long as A is 
 positively charged, positive electricity will flow from the plate 
 P. This current is at the moment considered at its maximum 
 value at the middle point of the first half wave-length. A 
 quarter of a wave-length further from the origin that is to 
 say, in the neighbourhood of the null point it first changes 
 its direction. The E.M.F. of induction will here, therefore, 
 impel positive electricity towards the origin. A current will
 
 DYNAMICAL THEORY OF INDUCTION. 463 
 
 therefore flow round B towards A' in the upper portion and 
 away from A' in the lower portion. The electrostatic and 
 electro -dynamic E.M.F.s are therefore in opposite phases and 
 oppose each other's action. If the secondary circuit is rotated 
 through OOdeg., through the first principal position, the direct 
 action changes its sign, but not so the action of the waves, so 
 that they now tend to strengthen each other. The same 
 reasoning holds when the air space is at the lowest point of B. 
 
 Greater lengths of wire were then included between m and n, 
 and it was found that the interference became gradually less 
 marked, until within a length of 2-5 metres it disappeared 
 entirely, the sparks being of equal length whether the normal 
 were directed towards or away from P. When the length of 
 wire between m and n was further increased, the distinction 
 between the different quadrants reappeared, and with a length 
 of 4 metres the disappearance of the sparks was fairly sharp. 
 The disappearance, however, then took place (with the air space 
 at the highest point) when the normal was directed away from 
 P, the opposite direction to that in which the disappearance pre- 
 viously took place. With a still further increase in the length 
 of the wire the interference reappeared, and returned to its 
 original direction with a length of 6 metres. These phenomena 
 are clearly to be explained by the retardation of the waves in 
 the wire, and show that here again the direction of motion in 
 the advancing waves changes its signs at intervals of about 
 2-8 metres. 
 
 To obtain interference phenomena with the secondary circuit 
 C in the third principal position, the rectilinear wire must be 
 removed from its original position and placed in the horizontal 
 plane through C either on the side of the plate A or of the 
 plate A'. Practically it is sufficient to stretch the wire 
 loosely, and to fix it by means of an insulated clamp on each 
 side of C alternately. It was found that when the wire was 
 on the same side as the plate P the waves in it diminished 
 the previous sparking, and when on the opposite side the 
 sparking was increased, both results being unaffected by the 
 position of the air space in the secondary circuit. Now it has 
 been already pointed out that at the moment when the plate A 
 has its maximum positive charge, and at which, therefore, the 
 primary current begins to flow from A, the current at the first
 
 464 DYNAMICAL THEORY OF INDUCTION. 
 
 node of the rectilinear wire begins to flow away from the 
 origin. The two currents, therefore, flow round C in the 
 same direction when C lies between the rectilinear wire and A, 
 and in opposite directions when the wire and A are on the 
 same side of C. The fact that the position of the air space is 
 indifferent confirms the conclusion formerly arrived at that 
 the direction of oscillation is that due to the electro-dynamic 
 E.M.F. These interferences are also changed in direction 
 when the wire m n, I metre in length, is replaced by a wire 4 
 metres in length. 
 
 Hertz also succeeded in obtaining interference phenomena 
 when the centre of the secondary circuit was not in the base- 
 line, but these results were of no special importance, except 
 that they confirmed the previous conclusions. 
 
 Interference Phenomena at Various Distances. Interference 
 may be produced with the secondary at greater distances than 
 that of the null point ; but care must then be taken that the 
 action of the waves in the wire is of about the same magnitude 
 as the direct action of the primary circuit through the air. 
 This can be effected by increasing the distance between P 
 and A. 
 
 Now, if the velocity of propagation of the electro-dynamic 
 disturbances through the air is infinite, the interference will 
 change its sign at every half- wave length in the wire that is 
 to say, at intervals of about 2-8 metres. If the velocities of 
 propagation through the air and through the wire are equal, 
 the interference will be in the same direction at all distances. 
 Finally, if the velocity of propagation through the air is finite, 
 but different from the velocity in the wire, the interference will 
 change in sign at intervals greater than 2-8 metres. 
 
 The interferences first investigated were those which occurred 
 when the secondary circuit was rotated from the first into the 
 second principal position, the air space being at the highest 
 point. The distance of the secondary from the null point was 
 increased by half-metre stages from up to 8 metres, and 
 at each of these positions an observation was made of the 
 effects of directing the normal towards and away from P 
 respectively. The points at which no difference in the 
 sparking was observed in the two positions of the normal 
 are marked in Table I. Those in which the sparking
 
 DYNAMICAL THEORY OF INDUCTION. 465 
 
 was least, showing the existence of interference, when the 
 normal was directed towards P, are marked + , and those in 
 which the sparking was least when the normal was directed 
 away from P are marked . The experiments were repeated 
 with different lengths of wire m n, varying by steps of half a 
 metre from 1 metre up to 6 metres. The first horizontal line 
 in the table gives the distance, in metres, of the centre of the 
 secondary circuit from the null point, while the first vertical 
 line gives the lengths of the wire m n, also in metres. 
 
 Table I. 
 
 
 
 
 
 1 
 
 1 
 
 
 2 
 
 
 3 
 
 
 4 
 
 
 5 
 
 
 6 
 
 
 7 
 
 
 8 
 
 100 
 150 
 200 
 250 
 
 300 
 ?50 
 
 nr\n 
 
 1 1 00+ + 
 
 
 
 
 
 i 
 
 - 
 
 
 
 
 
 - + + O O O 1 
 
 
 
 f 
 
 n 
 
 
 
 
 
 
 
 
 
 00 + +OOC 
 
 
 
 
 
 
 
 0+000 1 
 
 
 
 
 
 f 
 
 
 
 
 
 1 | 00+ + 
 
 1 I 000 + 
 
 450 
 500 
 550 
 600 
 
 
 
 
 
 
 
 i- + + + + 
 
 
 
 
 
 o 
 
 h +00 1 
 
 
 
 
 
 
 
 
 
 
 1 000 1 
 
 I +001 
 
 + 001 
 
 
 
 
 
 
 An inspection of the table shows, in the first place, that the 
 changes of sign take place at longer intervals than 2 - 8 metres ; 
 and, in the second place, that the change of phase is more 
 rapid in the neighbourhood of the origin than at a distance 
 from it. As a variation in the velocity of propagation is very 
 unlikely, this is probably due to the fact indicated by theory 
 that the electrostatic E.M.F., which is more powerful than the 
 electro-dynamic E.M.F. in the neighbourhood of the primary 
 oscillation, has a greater velocity of propagation than the 
 latter. 
 
 In order to obtain a definite proof of the existence of similar 
 phenomena at greater distances, Hertz continued the observa- 
 tions, in the case of three of the lengths m n, up to a distance 
 of 12 metres, and the result is given in Table II. 
 
 If we make the assumption that at the greater distance it is 
 only the E.M.F. of induction which produces any effect, the 
 experiments would show that the interference of the waves
 
 466 
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 exsited by the E.M.F. of induction with the original wares in 
 the wire changes its sign only at intervals of about 7 metres. 
 
 Table II. 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 | 12 
 
 100 
 
 + 
 
 
 
 _ 
 
 _ 
 
 
 
 
 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 260 
 
 
 
 
 _ 
 
 
 
 + 
 
 + 
 
 
 
 
 
 
 
 
 
 
 - - 
 
 400 
 
 
 
 
 + 
 
 + 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 Table TIL 
 
 . 
 
 
 
 1 2 
 
 3 
 
 4 
 
 ICO 
 
 _ 
 
 _ 
 
 _ 
 
 _ 
 
 
 
 150 
 
 - 
 
 - 
 
 
 
 
 
 
 
 200 
 
 
 
 
 
 
 
 + 
 
 4- 
 
 250 
 
 
 
 + 
 
 -t- 
 
 + 
 
 + 
 
 300 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 350 
 
 + 
 
 + 
 
 + 
 
 + 
 
 
 
 400 
 
 + 
 
 + 
 
 + 
 
 -I- 
 
 
 
 450 
 
 + 
 
 + 
 
 + 
 
 
 
 
 
 500 
 
 + 
 
 + 
 
 
 
 
 
 
 
 550 
 
 + 
 
 
 
 
 
 
 
 _ 
 
 600 
 
 
 
 
 
 
 
 
 
 
 
 In order to investigate the E.M.F. of induction close to the 
 primary oscillation, where the results are of special importance, 
 Hertz made use of the interferences which were obtained 
 when the secondary circuit was in the third principal position, 
 and the air space was rotated through 90deg. from the base-line. 
 The direction of the interference at the null point, which has 
 already been considered, was taken as negative, the interference 
 being considered positive when it was produced by the passage 
 of waves on the side of C remote from P, which make the signs 
 correspond with those of the previous experiments. It must be 
 borne in mind that the direction of the resultant E.M.F. at the 
 null point is opposed to that of the E.M.F. of induction, and 
 therefore the first table would have begun with a negative sign 
 if the electrostatic E.M.F. could have been eliminated. The 
 present experiments showed that up to a distance of 8 metres 
 interference continued to occur, and always of the same sign as 
 at the null point. It was unfortunately impossible to extend 
 these observations to a greater distance than 4 metres on
 
 DYNAMICAL THEORY OF INDUCTION. 467 
 
 Account of the feebleness of the sparks, but the results obtained 
 were sufficient to give distinct evidence of a finite velocity of 
 propagation of the E.M.F. of induction. These observations, 
 like the former ones, were repeated with various lengths of 
 the wire mn in order to exhibit the variation in phase, and 
 the results obtained are given in Table III., which shows 
 that, as the distance increases, the phase of the interference 
 changes in such a manner that a reversal of sign takes place 
 at intervals of from 7 to 8 metres. This result is further con- 
 firmed by comparing the results of Table III. with the results 
 for greater distances given in Table II., for in the former 
 series the effect of the electrostatic E.M.F. is eliminated, 
 owing to the special position of the secondary circuit, while 
 in the latter it becomes insensible at the greater distances 
 owing to its rapid decrease with increasing distance. We 
 should therefore expect the results given in the first table for 
 distances beyond 4 metres to follow without a break the 
 results given in Table III. for distances up to 4 metres. This 
 was found to be the case, as is evident from inspection of 
 Tables II. and III. 
 
 To show this more clearly, the signs of the interference of 
 the waves, due to the electro-dynamic E.M.F., with the waves 
 in the wire are collected together in Table IV., the first four 
 columns of which are taken from Table III., and the remain- 
 ing columns from Table II. 
 
 Table IV. 
 
 _ 
 
 
 
 ^ 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ICO 
 
 _ 
 
 _ 
 
 _ 
 
 _ 
 
 
 
 
 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 
 
 250 
 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 
 
 
 
 
 
 
 
 _ 
 
 
 
 400 
 
 + 
 
 + 
 
 + 
 
 + 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 From the results given in this table Hertz drew the 
 following conclusions : 
 
 1. The interference does not change its sign at intervals of 
 2-8 metres. The electro-dynamic actions are therefore not 
 propagated with an infinite velocity. 
 
 2. The interference is not in the same phase at all points. 
 Therefore the electro-dynamic actions are not propagated 
 through air with the same velocity as electric waves in wires. 
 
 HH2
 
 468 DYNAMICAL THEORY OF INDUCTION. 
 
 3. A gradual retardation of the waves in the wire has the 
 effect of displacing a given phase of the interference towards 
 the origin of the waves. The velocity of propagation through 
 the air is therefore greater than through a wire. 
 
 4. The sign of the interference is reversed at intervals of 
 7-5 metres, and therefore in traversing this distance an electro- 
 dynamic wave gains one length of the waves in the wire. 
 
 Thus, while the former travels 7*5 metres, the latter travels 
 7*5 - 2-8 = 4-7 metres, and therefore the ratio of the velocities 
 is 75 I 47, which gives for the half wave-length of the electro- 
 dynamic action 2-8 x 75/47 = 4-5 metres. Since this distance 
 is traversed in 1-4 hundred-millionth of a second, the absolute 
 velocity of propagation through the air must be 320,000 kilo- 
 metres per second. This result can only be considered reliable 
 as far as its order is concerned ; but its true value can hardly 
 exceed half as much again, or be less than two-thirds of this 
 amount. In order to obtain a more accurate determination of 
 the true value it will be necessary to determine the velocity of 
 electric waves in wires with greater exactness. 
 
 It does not necessarily follow from the fact that in the imme- 
 diate neighbourhood of the primary oscillation the interference 
 changes its sign after an interval of 2-8 metres, that the velocity 
 of propagation of the electrostatic action is infinite, for such 
 a conclusion would rest upon a single change of sign, which 
 might, moreover, be explained independently of any change of 
 phase, by a change in the sign of the amplitude of the resultant 
 force at a certain distance from the primary oscillation. Quite 
 independently, however, of any knowledge of the velocity of 
 propagation of electrostatic actions, there exist definite proofs 
 that the rates of propagation of electrostatic and electro dynamic 
 E.M.F.s are unequal 
 
 In the first place, the total force does not vanish at any 
 point on the base line. Now, near the primary the electro- 
 static E.M.F. is the greater, while the electro-dynamic E.M.F. 
 is the greater at greater distances. There must, therefore, be 
 some point at which they are equal, and since they do not 
 balance they must take different times to reach this point. 
 
 In the second place, the existence of points at which the 
 direction of the resultant E.M.F. becomes indeterminate does 
 not seem capable of explanation, except on the supposition.
 
 DYNAMICAL THEORY OF INDUCTION. 469 
 
 that the electrostatic and electro-dynamic components perpen- 
 dicular to each other are in appreciably different phases, and, 
 therefore, do not compound into a rectilinear oscillation in a 
 fixed direction. The fact that the two components of the 
 resultant are propagated with different velocities is of con- 
 siderable importance, in that it gives an independent proof 
 that one of them at any rate must have a finite velocity of 
 propagation. 
 
 Further researches of Hertz on electrical oscillations, of 
 which accounts have been published, are to be found described 
 in a Paper, "On Electro-Dynamic Waves in Air, and their 
 Reflection," in Wiedemann's Annalen, Vol. XXXIV., p. 609. 
 The author had been endeavouring to find a more striking and 
 direct proof of the finite velocity of propagation of electro- 
 dynamic waves than those which he had hitherto given ; for, 
 though these are quite sufficient to establish the fact, they can 
 only be properly appreciated by one who has obtained a grasp 
 of the results of the entire series of researches. 
 
 In many of the experiments which have been described, 
 Hertz had noticed the appearance of sparks at points in the 
 secondary conductor where it was clear from geometrical con- 
 siderations that they could not be due to direct action, and it 
 was observed that this occurred chiefly in the neighbourhood 
 of solid obstacles. It was found, moreover, that in most 
 positions of a secondary conductor the feeble sparks produced 
 at a great distance from the primary became considerably 
 stronger in the vicinity of a solid wall, but disappeared with 
 considerable suddenness quite close to the wall. The most 
 obvious explanation of these experiments was that the waves 
 of inductive action were reflected from the wall and interfered 
 with the direct waves, especially as it was found that the 
 phenomena became more distinct when the circumstances 
 were such as to favour reflection to the greatest possible 
 extent. Hertz therefore determined upon a thorough investi- 
 gation of the phenomena. 
 
 The experiments were made in the Physical Lecture Theatre, 
 which is 15 metres in length, 14 metres in width, and 6 metres 
 in height. Two rows of iron columns, running parallel to the 
 sides of the room, would collectively act almost like a solid 
 wall towards electro-dynamic action, so that the available
 
 470 DYNAMICAL THEORY OF INDUCTION. 
 
 width of the room was only 8-6 metres. All pendant gas- 
 fittings were removed, and the room left empty, with the- 
 exception of wooden tables and forms, which would not exert 
 any appreciable disturbing effect. The end wall, from which 
 the waves were to be reflected, was of solid sandstone, with 
 two doors in it, and the numerous gas pipes attached to it 
 gave it, to a certain extent, the character of a conducting 
 surface, and this was increased by fastening to it a sheet of 
 zinc four metres high and two metres broad, connected by 
 wires to the gas pipes and a neighbouring water pipe. Special 
 care was taken to provide an escape for the electricity at the 
 upper and lower extremities of the zinc plate, where a certain 
 accumulation of electricity was to be expected. 
 
 The primary conductor was the same that was employed in 
 the experiments described on page 456, Fig. 161, and was placed 
 at a distance of 13 metres from the zinc plate, and, therefore } 
 two metres from the wall at the other end of the room. The 
 conducting wire was placed vertically, so that the E.M.F.s to 
 be considered increased and diminished in a vertical direction. 
 The centre of the primary conductor was 2-5 metres above the 
 floor of the room, which left a clear space for the observations 
 above the tables and benches. The point of intersection of the 
 reflecting surface with the perpendicular from the centre of the 
 primary conductor will be called " the point of incidence," and 
 the experiments were limited to the neighbourhood of this 
 point, as the investigation of waves striking the wall at a con- 
 siderable angle would be complicated by the differences in their 
 polarisation. The plane of vibration was therefore parallel to 
 the reflecting surface, and the plane of the waves was perpen- 
 dicular to it, and passed through the point of incidence. 
 
 The secondary conductor consisted of the circle of 35 centi- 
 metres radius, which has been already described. It was 
 movable about an axis through its centre perpendicular to its 
 plane, and the axis itself was movable in a horizontal plane 
 about a vertical axis. In most of the experiments the secon- 
 dary conductor was held in the hand by its insulating wooden 
 support, as this was the most convenient way of bringing it into 
 the various positions required. The results of these experi- 
 ments, however, had to be checked by observations made 
 with the observer at a greater distance from the secondary, as
 
 DYNAMICAL THEOEY OF INDUCTION. 471 
 
 the neighbourhood of his body exerted a slight influence upon 
 the phenomena. The sparks were distinct enough to be 
 observed at a distance of several metres when the room was 
 darkened, but when the room remained light they were 
 practically invisible even when the observer was quite close to 
 the secondary. 
 
 When the centre of the secondary was placed in the line of 
 incidence, and with its plane in the plane of vibration, and the 
 air space was turned first towards the reflecting wall and then 
 away from it s a considerable difference was generally observed 
 in the strength of the sparks in the two positions. At a 
 distance of about 0-8 metre from the wall the sparks were 
 much stronger when the air space was directed towards the 
 wall, and its length could be adjusted so that, while there was 
 
 tear 
 
 ,_t I--'!* -^ j 
 
 W\ * c @ 
 
 FIG. 162. 
 
 a steady stream of sparks when in this position, they disap- 
 peared entirely when the air space was directed away from the 
 wall. These phenomena were reversed at a distance of 3 metres, 
 and recurred, as in the first case, at a distance of 5-5 metres. 
 At a distance of 8 metres the sparks were stronger when the 
 air space was turned away from the wall, as at the distance of 
 3 metres, but the difference was not so well marked. When 
 the distance was increased beyond 8 metres no further reversal 
 took place, owing to the increase in the direct effect of the 
 primary oscillation and the complicated distribution of the 
 E.M.F. in its neighbourhood. 
 
 The positions L, II., III. and IV. (Fig. 162) of the secondary 
 circle are those in which the sparks were strongest, the distance
 
 472 DYNAMICAL THEORY OF INDUCTION. 
 
 from the wall being shown by the horizontal scale at the foot. 
 When the secondary circle was in the positions V., VI. , and 
 VII., the sparks were equally strong in both positions of the 
 air space, and quite close to the wall the difference between 
 the sparking in the two positions again diminished. Therefore 
 the points A, B, C, D in the diagram may in a certain sense be 
 regarded as nodes. The distance between two of these points 
 must not, however, be taken as the half wave-length, for if 
 all the electrical motions changed their directions on passing 
 through one of these points the phenomena observed in the 
 secondary circuit would be repeated without variation, since 
 the direction of oscillation in the air space is indifferent. 
 
 The conclusion to be drawn from the experiments is that in 
 passing any one of these points part of the action is reversed, 
 while another part is not. The experimental results, however, 
 warrant the assumption that twice the distance between two 
 of these points is equal to the half wave-length, and when this 
 assumption is made the phenomena can be fully explained. 
 
 For suppose a wave of E.M.F., with oscillations in a vertical 
 direction, to impinge upon the wall, and to be reflected with 
 only slightly diminished intensity, thus giving rise to stationary 
 waves. If the wall were a perfect conductor, a node would 
 necessarily be formed in its surface, for at the boundary and 
 in the interior of a perfect conductor the E.M.F. must be 
 infinitely small. The wall cannot, however, be considered as 
 a perfect conductor, for it was not metallic throughout, and 
 the portion which was metallic was not of any great extent. 
 The E.M.F. would therefore have a finite value at its surface, 
 and would be in the direction of the impinging waves. The 
 node, which in the case of perfect conductivity would occur at 
 the surface of the wall, would, therefore, actually be situated 
 a little behind it, as shown at A in the diagram. If, then, 
 twice the distance A B that is to say, the distance A C is 
 half the wave-length, the steady waves will be as represented 
 by the continuous lines in Fig. 162. The E.M.F.s acting on 
 each side of the circles, in the positions L, II., III., and IV., 
 will, therefore, at a given moment be represented in magnitude 
 and direction by the arrows on each side of them in the 
 diagram. If, therefore, in the neighbourhood of a node, the 
 air space is turned towards the node, the strongest E.M.F. in
 
 DYNAMICAL THEORY OF INDUCTION. 473 
 
 the circle will act under more favourable conditions against a 
 weaker one under less favourable conditions. If, however, the 
 air space is turned away from the node, the stronger E.M.F. 
 acts under less favourable conditions against a weaker one 
 under more favourable conditions. In the latter case the 
 resultant action must be less than in the former, whichever 
 of the two E.M.F.s has the greater effect, which explains the 
 change of sign of the phenomenon at each quarter wave- 
 length. 
 
 This explanation is further confirmed by the consideration 
 that, if it is the true one, the change of sign at the points B 
 and D must take place in quite a different manner from that 
 of the point C. The E.M.F.s acting on the secondary circle, 
 an the positions V., VI., and VII., are shown by the corre- 
 sponding arrows, and it is clear that in the positions B and 
 D, if the air space is turned from one side to the other, the 
 vibration will change its direction round the circle, and there- 
 fore the sparking must, during the rotation, vanish either once 
 or an uneven number of times. In the position C, however, 
 the direction of vibration remains unaltered, and therefore the 
 sparks must disappear an even number of times, or not at all. 
 
 The experiments showed that at B and D the sparking dimi- 
 nished as the air space receded from a, vanished at the highest 
 point, and again attained its original value at the point ft. At 
 C, on the other hand, the sparking continued throughout the 
 rotation, being a little stronger at the highest and lowest 
 points. If, then, there is any change of sign in the position 
 C, it must occur with very much smaller displacements than 
 in the other positions, so that in any case there is a distinction 
 such as is required between this and the other two cases. 
 
 Another very direct proof of the truth of Hertz's repre- 
 sentation of the nature of the waves was obtained. If the 
 secondary circle lies in the plane of the waves instead of in 
 the plane of vibration, the E.M.F. must be equal at all points 
 of the circle, and for a given position of the air space the 
 sparking must be directly proportional to its intensity. When 
 the experiment was made, it was found, as expected, that at 
 all distances the sparking vanished at the highest and lowest 
 points of the circle, and attained a maximum value at the 
 points in the horizontal plane through the point of incidence.
 
 474 DYNAMICAL THEORY OF INDUCTION. 
 
 The air space was then placed at such a point and close to 
 the wall, and was then moved slowly away from the wall, 
 when it was found that, while there was no sparking quite 
 close to the metal plate, it began at a very small distance 
 from it, rapidly increased, reached a maximum at the point 
 B, and then diminished again. At C the sparking again 
 became excessively feeble and increased as the circle was 
 moved still further away. The sparking continued steadily 
 to increase after this, as the motion of the circle was con- 
 tinued in the same direction, owing, as before, to the direct 
 action of the primary oscillation. 
 
 The curves shown by the continuous lines in Fig. 162 were 
 obtained from the results of these experiments, the ordinates 
 representing the intensity of the sparks at the distances repre- 
 sented by the corresponding abscissa?. 
 
 The existence in the electrical waves of nodes at A and C, 
 and of loops at B and D, is fully established by the experi- 
 ments which have been described; but in another sense the 
 points B and D may be regarded as nodes, for they are the 
 nodal points of a stationary wave of magnetic induction which, 
 according to theory, accompanies the electrical wave and lags 
 a quarter wave-length behind it. 
 
 This can easily be shown to follow from the experiments, for 
 when the secondary circle is placed in the plane of vibration 
 with the air space at its highest point, there will be no spark- 
 ing if the E.M.F. is uniform throughout the space occupied by 
 the secondary. This can only take place if the E.M.F. varies 
 from point to point of the circle, and if its integral round the 
 circle differs from zero. This integral is proportional to the 
 number of magnetic lines of force passing backwards and for- 
 wards across the circle, and the intensity of the sparks may be 
 considered as giving a measure of the magnetic induction, 
 which is perpendicular to the plane of the circle. Now, in this 
 position vigorous sparking was observed close to the wall, 
 diminishing rapidly to zero as the point B was approached, 
 then increasing to a maximum at C, falling to a well-marked 
 minimum at D, and finally increasing continuously as the 
 secondary approached still nearer to the primary. If the 
 intensities of these sparks are taken as ordinates, positive and 
 negative, and the distances from the wall as abscissa?, the
 
 DYNAMICAL THEORY OF INDUCTION. 475 
 
 curve shown by the dotted lines in Fig. 162 is obtained, which 
 therefore represents the magnetic waves. 
 
 The phenomena observed in the first series of experiments 
 described above may therefore be regarded as due to the 
 resultant electric and magnetic actions. The former changes 
 sign at A and C, the latter at B and D, so that at each of these 
 points one part of the action changes sign, while the other does 
 not, and therefore the resultant action which is their product 
 must change sign at each of these points, as was found to be 
 the case. 
 
 When the secondary circle was in the plane of vibration the 
 sparking in the vicinity of the wall was observed to be a maxi- 
 mum on the side towards the wall and a minimum at the 
 opposite side, and as the circle was turned from one position to 
 the other there was found to be no point at which the sparks 
 disappeared. As the distance from the wall was increased, the 
 sparks on the remote side gradually became weaker, and 
 vanished at a distance of 1-08 metre from the wall. When 
 the circle was carried further in the same direction the sparks 
 appeared again on the side remote from the wall, but were 
 always weaker than on the side next to it ; the sparking, how- 
 ever, no longer passed from a maximum to a minimum merely, 
 but vanished during the rotation once in the upper and once 
 in the lower half of the circle. The two null points gradually 
 receded from their original coincident positions until at the 
 point B they occurred at the highest and lowest points of the 
 circle. As the circle was moved further in the same direction 
 the null points passed over to the side next to the wall, and 
 approached each other again, until, when the centre was at a 
 distance of 2-35 metres from the wall, the two null points were 
 again coincident. B must be exactly half-way between this 
 point and the similar point previously observed, which gives 
 1 -72 metre as the distance of B from the wall a result which 
 agrees, within a few centimetres, with that obtained by direct 
 observation. Moving further in the direction of C, the spark- 
 ing at different points of the circle became more nearly equal, 
 until at C it was exactly so. In this position there was no null 
 point, and as the distance was further increased the phenomena 
 recurred in the same order as before. 
 
 Hertz found that the position of C could be determined
 
 476 DYNAMICAL THEORY OF INDUCTION. 
 
 within a few centimetres, the determinations of its distance 
 from the wall varying from 4-10 to 4-15 metres; he gives its 
 most probable value as 4-12 metres. The point B could not be 
 observed with any exactness, the direct determinations varying 
 from 6 to 7-5 metres as its distance from the wall. It could, 
 however, be determined indirectly, for the distance between B 
 and C being found to be 2*4 metres, taking this as the true 
 value, A must have been 0-68 metre behind the surface of the 
 wall, and G-52 metres in front of it. The half wave-length 
 would be 4-8 metres, and by an indirect method it was found 
 to be 4-5 metres, so that the two results agree fairly well. 
 Taking the mean of these as the true value, and the velocity 
 of light as the velocity of propagation, gives as the vibration 
 period of the apparatus 1-55 hundred-millionth of a second, 
 instead of 1-4 hundred-millionth, which was the theoretically 
 calculated value. 
 
 A second series of experiments was made with smaller 
 apparatus, and though the measurements could not be made 
 with as much exactness as those already described, the results 
 showed clearly that the position of the nodes depends only on 
 the dimensions of the conductors, and not on the material of 
 the wall. 
 
 Hertz states that after some practice he succeeded in obtain- 
 ing indications of reflections from each of the walls. He was 
 also able to obtain distinct evidence of reflection from one of 
 the iron columns in the room, and of the existence of electro- 
 dynamic shadows on the side of the column remote from the 
 primary. 
 
 In the preceding experiment the secondary conductor was 
 always placed between the wall and the primary conductor 
 that is to say, in a space in which the direct and reflected 
 rays were travelling in opposite directions, and gave rise to 
 stationary waves by their interference. 
 
 He next placed the primary conductor between the wall and 
 the secondary, so that the latter was in a space in which the 
 direct and reflected waves were travelling in the same direction. 
 This would necessarily give rise to a resultant wave, the inten- 
 sity of which would depend on the difference in phase of the 
 two interfering waves. In order to obtain distinct results it was 
 necessary that tne two waves should be of approximately equal
 
 DYNAMICAL THEORY OF INDUCTION. 477 
 
 intensities, and therefore the distance of the primary from the 
 wall had to be small in comparison with the extent of the 
 latter, and also in comparison with its distance from the 
 secondary. 
 
 To fulfil these conditions the secondary was placed at a dis- 
 tance of 14 metres from the reflecting wall, and, therefore, 
 ahout 1 metre from the opposite one, with its plane in the plane 
 of vibration, and its air space directed towards the nearest wall, 
 in order to make the conditions as favourable as possible for 
 the production of sparks. The primary was placed parallel to 
 its former position, and at a perpendicular distance of about 
 80 centimetres from the centre of the reflecting metallic plate. 
 The sparks observed in the secondary were then very feeble, 
 and the air space was increased until they disappeared. The 
 primary conductor was then gradually moved away from the 
 wall, when isolated sparks were soon observed in the secondary, 
 passing into a continuous stream when the primary was between 
 1-5 and 2 metres from the wall that is, at the point B. This 
 might have been supposed to be due to the decrease in the 
 distance between the two conductors, except that as the primary 
 conductor was moved still further from the wall the sparking 
 again diminished, and disappeared when the primary was at 
 the point C. After passing this point the sparking continually 
 increased as the primary approached nearer to the secondary. 
 These experiments were found to be easy to repeat with 
 smaller apparatus, and the results obtained confirmed the 
 former conclusion that the position of the nodes depends only 
 on the dimensions of the conductor, and not on the material 
 of the reflecting wall. 
 
 Hertz points out that these phenomena are exactly analogous 
 to the acoustical experiment of approaching a vibrating tuning- 
 fork to a wall when the sound is weakened in certain positions 
 and strengthened in others, and also to the optical phenomena 
 illustrated in Lloyd's form of Fresnel's mirror experiments ; 
 and as these are accepted as arguments tending to prove that 
 sound and light are due to vibration, his investigations give a 
 strong support to the theory that the propagation of electro- 
 magnetic induction also takes place by means of waves excited 
 in a medium. They therefore afford a confirmation of the 
 Faraday-Maxwell theory of electrical action.
 
 478 D YNA M1CAL THEOR Y OF IND UCTION, 
 
 12- Further Researches on Electro-Magnetic Radiation. 
 When once the experimental proof had been given that the 
 result of electrical oscillations in a conductor is to propagate 
 out into surrounding space radiations which are in all respects 
 of the same nature as light, except in that they cannot affect 
 the eye, it became evident that a new and vast field of 
 investigation had been opened, and one in which it would 
 be possible to produce the electro-magnetic analogues of all 
 the more familiar optical phenomena. 
 
 The reflection, refraction, dispersion, and polarisation of light 
 waves are well-known optical phenomena. We can perform 
 analogous experiments with rays of dark heat which differ only 
 from light rays in having a greater wave-length, and in being 
 thereby unable to affect the optic nerve. In performing these 
 experiments with dark heat or non-luminous radiation we have 
 to make use of the thermopile as a perceiver of the ray. The 
 electro-magnetic radiation scattered from a conductor in which 
 electric oscillations are set up differs again from light and dark 
 heat in having a still longer wave-length. In performing 
 experiments with electro-magnetic radiation we have seen that 
 Hertz's invention of the electro-magnetic resonator put us in 
 possession of an apparatus which is the exact equivalent of a 
 thermopile, or the human eye, as a ray localiser ; and more 
 recent researches have shown us how to construct a large 
 number of forms of receiver of even more sensitive character, 
 by means of which we can detect this electro-magnetic radiation. 
 
 In these electro-magneto-optic experiments of Hertz, the 
 source of radiation is a divided metallic cylinder about one 
 inch in diameter and twelve inches long. This is divided in 
 halves, and the two parts separated by a small distance. 
 They are respectively attached to the ends of the secondary 
 coil of a small induction coil. When the coil is put in action, 
 electrical oscillations are set up in these cylinders which 
 result in the outward propagation of ethereal undulations of 
 about two feet in wave-length and having a frequency of 
 about five hundred millions a second. 
 
 In order to see these waves, Hertz employed a resonator 
 consisting of a metallic circuit having a small spark interval. 
 With these simple appliances he has been able to show the 
 reflection of the electro -magnetic waves from plane surfaces,
 
 DYNAMICAL THEORY OF INDUCTION. 479 
 
 and the concentration of radiation by parabolic mirrors of 
 sheet zinc, repeating in fact. the old experiment of the con- 
 jugate mirrors. The radiation from this source could, he 
 found, be gathered up by one parabolic mirror, reflected to a 
 second and concentrated again to a focus. Another achieve- 
 ment was the refraction of the rays by a great prism of pitch. 
 Placed in the path of an electro-magnetic ray, he found that 
 this pitch prism refracted it through an angle of 22deg., and 
 that the material had a refractive index of 1-7 for these long 
 waves. Again, it was found that metallic sheets were opaque 
 to this radiation, but that it passed through such non- 
 conductors as dry wood, and that a laboratory door, although 
 opaque to light, is transparent to this ultra-ultra red or 
 electro-magnetic radiation. 
 
 The reader may be referred to Dr. Lodge's book on the 
 "Work of Hertz, and some of his Successors" for a full 
 account of the experimental proofs that electro-magnetic 
 radiation and that radiation we commonly call light are one 
 in essential nature, although differing in wave-length. These 
 experiments are akin to the acoustic ones in which air waves, 
 too short to be audible, are generated ; and in place of the 
 ear, now useless, a sensitive flame is employed to find or 
 indicate the waves, and inform us of the presence or absence 
 of aerial wave motion. In the same way all well-known 
 optical effects can be reproduced with ether radiation too long 
 in wave-length to affect the eye, but capable of acting on a 
 proper receiver. 
 
 It is a necessary corollary of Maxwell's electro-magnetic 
 theory of light that good conducting bodies should be opaque 
 a,nd good insulators transparent. As a matter of fact, for dis- 
 turbances of the period of light many good insulators, such as 
 ebonite, are opaque, even in very thin sheets, and conversely 
 gold, silver, and platinum are semi-transparent when in very 
 thin sheets. It must be borne in mind, however, that the 
 frequency of light oscillations falls between 400 and 700 
 million-million oscillations per second, or are of the order of 
 5 x 10 14 . 
 
 We cannot .by any of Hertz's methods produce electrical 
 oscillations so rapid as this. Hence, since conductivity and 
 insulating power of materials have generally been determined
 
 480 DYNAMICAL THEORY OF INDUCTION. 
 
 with reference either to steady currents or to moderately 
 great oscillations, we cannot institute a comparison between 
 these qualities as possessed by any given substance and 
 opacity or transparency for the much greater frequency of 
 luminous electro-magnetic waves. It has been shown that 
 ebonite is very transparent to long waves of dark heat,* and 
 hence there is no difficulty in understanding that it is trans- 
 parent to the longer waves produced by electrical oscillations 
 set up in moderately small conductors, whilst it is opaque to 
 the very short ones of light. Also the transparency of thin 
 metallic sheets to light is an indication of imperfect conducti- 
 bility. We have seen that an infinitely perfect conductor is 
 a perfect magnetic screen, and accordingly we should expect 
 that the more perfect the conductivity of a metal the greater 
 would be its opacity even in very thin films. It is well known 
 that cooling copper increases its conductivity. Wroblewski 
 showed (Comptes Rendus, Vol. 01., July, 1885, p. 160) that by 
 cooling copper to -200C., or to the temperature of the 
 solidification of nitrogen, its conductivity is increased to 
 about nine times its value at 0C. These experiments have 
 been greatly extended by Dewar and Fleming (Phil. Mar/., 
 September, 1893), who have shown that perfectly pure metals 
 have most probably no electrical resistance at the absolute 
 zero of temperature. It would be interesting to know if the 
 opacity of a very thin film of copper is increased by cooling to 
 - 200 to any perceptible extent. With respect to electrolytes 
 some interesting experiments have been made by Prof. J. J. 
 Thomson.! In these experiments electrical oscillations of 
 about 10 8 per second in frequency were established in a 
 primary circuit by means of an induction coil. These alter- 
 nating currents were caused to induce secondary oscillations 
 in a neighbouring parallel circuit of appropriate size. The 
 secondary circuit oscillations were rendered visible by minute 
 sparks at a break in that circuit. The interposition of a thin 
 sheet of tinfoil or of the thinnest sheet of Dutch metal or 
 
 * See note on " The Index of Refraction of Ebonite," by Profs. Ayrton 
 and Perry, Proceedings of the Physical Society, London, Vol. IV., p. 345. 
 
 t " On the Resistance of Electrolytes to the Passage of Rapidly Alter- 
 nating Currents," Proceedings of the Physical Society, London, Vol. XLV., 
 No. 276, 1889.
 
 DYNAMICAL THEORY OF INDUCTION. 481 
 
 gold-leaf supported on glass at once stopped completely the 
 secondary sparks. This is a very interesting confirmation of 
 the theory of magnetic screening laid down on p. 255 of 
 Chap. IV. We have seen that for moderately rapid alterna- 
 tions the conductivity of tinfoil is not sufficient to make it 
 opaque to electro-magnetic radiations, hut for disturbances of 
 a frequency equal to about 10 8 the tinfoil affords a perfect 
 screening, or, in other words, is opaque. 
 
 With regard to the gold-leaf, Prof. Thomson remarks that 
 he has not been able to get any leaf thin enough to be trans- 
 parent to oscillations of this rate. On inserting a sheet of 
 ebonite between the primary and secondary circuit, it was 
 found to produce no effect on the sparking, indicating that 
 ebonite, although opaque to ordinary light, is transparent to 
 ether disturbances of the rate here employed. A thin layer of 
 transparent electrolyte was then used as a screen, and it was 
 found that whilst a very thin layer produced little or no effect, 
 a depth of three to four millimetres of dilute sulphuric acid 
 was sufficient to stop the sparking. Experiment showed that 
 the conductivity of various electrolytic solutions was about 
 the same for currents reversed 120 times a second as for 
 currents reversed 100 million times a second. As, however, 
 these electrolytes are transparent, they must, according to the 
 electro-magnetic theory, be insulators for currents reversed 
 about 10 15 times per second, and the molecular processes on 
 which electrolytic conduction depends must occupy a time 
 between one-hundred millionth and one-thousand billionth of 
 a second. Space will not permit further reference to this 
 exceedingly promising department of future research more 
 than to say that if the electro-magnetic theory of light is true 
 it will be able to furnish an electrical explanation, not only of 
 the simpler optical phenomena, but of such complex phenomena 
 as those embraced in the sciences of spectroscopy and photo- 
 graphy. 
 
 13. Propagation of Electro -Magnetic Energy. In our 
 exposition of the various electro-magnetic phenomena we have 
 directed attention to the facts which make it evident that even 
 in the simple phenomenon of the propagation of an electric 
 current in a wire we must divest ourselves of the idea that the 
 
 ii
 
 482 DYNAMICAL THEOEY OF INDUCTION. 
 
 so-called flow of current is analogous to the movement of a 
 material fluid in a pipe. It is true that there are effects in the 
 case of the electric current which correspond to the inertia and 
 resistance effects in the case of water flow ; but the progress of 
 knowledge has indicated that what we are in the habit of calling 
 the electric current is as much outside the wire as in it, and 
 that we must release ourselves from the trammels of any ideas 
 which cause us to concentrate attention exclusively or mainly 
 on the actions in the conductor. In fact, at the absolute 
 zero of temperature there would be no dissipation of energy 
 in the conductor at all, if it were a pure metal, and all the 
 processes would be confined to the medium. "We are indebted 
 to, amongst others, Prof. Poynting for an enlargement of our 
 views on the nature of electric current propagation, and in two 
 valuable memoirs these matters have been discussed by him.* 
 He says : A space containing electric currents may be 
 regarded as a field where energy is transformed at certain 
 points into the electric and magnetic form by means of 
 batteries, dynamos, thermopiles, &c., and in other parts of 
 the field this energy is being again transformed into heat, 
 work done by electro-magnetic forces, or any other form 
 yielded by currents. Formerly a current was regarded as 
 something travelling in the conductor, and the energy which 
 appeared at any part of the circuit was supposed to be 
 conveyed thither through the conductor by the current. But 
 the existence of induced currents and electro-magnetic actions 
 has led us to look on the medium surrounding the conductor 
 as playing a very important part in the development of the 
 phenomena. If we believe in the continuity of the motion 
 of energy, we are forced to conclude that the surrounding 
 medium is capable of containing energy, and that it is capable 
 of being transferred from point to point. We are thus led to 
 consider the problem how does the energy connected with an 
 electric current pass from point to point, by what paths does it 
 
 * " On the Transfer of Energy in the Electro-Magnetic Field," by Prof. 
 J. H. Poynting, Philosophical Transactions of the Royal Society, 1884, 
 Part II., VoL CLXXY., p. 343. Also " On the Connection between Electric 
 Current and the Electric and Magnetic Inductions in the Surrounding 
 Field," by Prof. J. H. Poynting, Philosophical Transactions of the Royal 
 : Society, 1835, Part II., Vol. CLXXVL, p. 277.
 
 DYNAMICAL THEORY OF INDUCTION. 483 
 
 travel, and according to what laws? Let us put a specific 
 case. Suppose a dynamo at one spot generates an electric 
 current, which is made to operate an electric motor at a dis- 
 tant place. We have here in the first place an absorption of 
 energy from the prime motor into the dynamo. We find the 
 whole space between and around the conducting wires mag- 
 netised and the seat of electro-magnetic energy. We have, 
 further, a re-transformation of energy in the motor. The 
 question which presents itself for solution is to decide how the 
 -energy taken up by the dynamo is transmitted to the motor. 
 
 Briefly stated, the general tendency of recent views is 
 that this energy is conveyed through the electro-magnetic 
 medium, or ether, and that the function of the wire is to 
 localise the direction or concentrate the flow in a particular 
 path, and is at the same time also a sink or place in which 
 energy is dissipated. A consideration of the whole phe- 
 nomenon has enabled Prof. Poynting to formulate an im- 
 portant law, as follows: At any point in the magnetic field 
 of conductors conveying currents the energy moves perpendicularly 
 to the plane containing the lines of electric force and the lines of 
 magnetic force, and the amount crossing a unit of area of this plane, 
 per second is equal to the product of the intensities of the two forces 
 multiiilied by the sine of the angle between them and dicided by 4ir. 
 Tf E denote the electric force, or force on a very small body 
 charged with a unit of positive electricity, and H denote the 
 magnetic' force, or force on a unit magnetic pole, and if at any 
 point in the electro-magnetic field these forces are inclined at 
 an angle 0, then there is a flow of energy e at this point in a 
 direction perpendicular to the planes of E and H, and equal 
 per second to the value of 
 
 j. E H sin 
 
 "1^ 
 
 The full proof of this law is given in the first of the Papers 
 mentioned on the preceding page. 
 
 Prof. Poynting has here introduced the important notion of 
 a flow of energy. We may remark in passing that this notion 
 does no violence to previous notions of energy. Energy, like 
 matter, is conserved that is, it is unalterable in total amount ; 
 and if in any circumscribed space some form of energy makes 
 its appearance, then we know that either an equal quantity 
 
 n2
 
 484 DYNAMICAL THEORY OF INDUCTION. 
 
 must Lave passed into that space from outside, or that an 
 equivalent quantity of some other form already in the 
 enclosure must have been transformed. If energy disappears 
 at one point and reappears at an adjacent point in equal 
 amount, we can with perfect propriety speak of it as having 
 been transferred from one point to the other, although we are 
 unable to identify the respective portions of it as we can in 
 the case of the movement of matter. Applying this view 
 to the simple phenomena of a battery producing heat in a 
 conducting wire, the notion to be grasped is that the potential 
 energy of the chemical combinations in the battery causes 
 energy to be radiated out along certain lines, the means 
 of conveyance being the electro -magnetic medium ; this 
 energy flows into the wire at all points, and is there 
 re-transformed into heat or light. A simple illustration 
 of Poyn ting's law is to consider the case of a section of a 
 straight conductor traversed, as we usually say, by a current. 
 Let the conductor be a right cylinder, or round wire, of 
 length I, radius r, and let E be the electric force at any 
 point in the wire, and H the magnetic force at the surface ; 
 also let V be the potential difference between the ends, C the 
 steady current, and B the total ohmic resistance. Consider 
 the energy flowing in on this section of the wire through its- 
 surface. It is equal per second to the area of the surface, 
 
 multiplied by ^5, or to 2 ^ EH . 
 
 Now JT r H is the line integral of the magnetic force taken- 
 round the wire following the circular surface, and this, as pre- 
 viously shown (p. 25) is equal to 4?r C. Also we have the 
 potential difference at the ends of the cylinder equal to the line- 
 integral of the electric force, or to ZE. Since, then, 2;rrH 
 = 4jr C and E I = V, we get, by substitution in the value of 
 the energy sent per second into the section of the wire, viz.,. 
 
 27r '' * H E , the equivalent C V. But by Ohm's law C E = V ; 
 
 hence the energy absorbed per second by the conductor is C 2 R, 
 and we know by Joule's law that this is the measure of the 
 energy dissipated per second in the wire as heat. We see, 
 then, that the energy dissipated in each section of the con- 
 ductor is absorbed into it from the dielectric, and the rate of
 
 DYNAMICAL THEORY OF INDUCTION. 485 
 
 ihis supply can be calculated by Poynting's law for each 
 element of the surface. None of the energy of a current travels 
 along the wire, but enters into it from the surrounding non- 
 conductor, and as soon as it enters it begins to be transformed 
 into heat, the amount crossing successive layers of the wire 
 decreasing till, by the time the centre, where there is no magnetic 
 force, is reached, it has all been transformed into heat. 
 
 In the Paper another simple case treated is that of a con- 
 denser discharged by a wire. In this case, before the discharge, 
 we know that the energy resides in the dielectric between the 
 plates. If the plates are connected by an external wire, accord- 
 ing to these views the energy is transferred outwards, along the 
 electrostatic equipotential surfaces, and moves on to the wire, 
 and is there converted into heat. According to this hypothesis, 
 we must suppose the lines or tubes of electrostatic induction 
 running from plate to plate to move outwards as the dielectric 
 strain lessens and, whilst still keeping their ends on the plates, 
 finally to converge in on the wire and be there broken up and 
 their energy dissipated as heat. At the same time the wire 
 .acquires transient magnetic qualities. This means that some 
 part of the energy of the expanding lines of electrostatic induc- 
 tion is converted into magnetic energy. The magnetic energy 
 is contained in ring-shaped tubes of magnetic force, which 
 .expand out from between the plates and then contract in upon 
 some other part of the circuit. 
 
 The whole history of the discharge may be divided into 
 three parts. First, a time when the energy associated with 
 the system is nearly all electrostatic and is represented by the 
 energy of the lines or tubes of electrostatic induction running 
 from plate to plate ; second, a period when the discharge is at 
 its maximum, when the energy exists partly as energy asso- 
 ciated with lines of electrostatic induction expanding outwards, 
 and partly in the form of closed rings or tubes of magnetic 
 force expanding from and then contracting back on the wire ; 
 .and lastly, a period when nearly all the energy has been 
 .absorbed or buried in the wire, and has there been dissipated 
 in the form of heat, which is radiated out again as energy of 
 -dark or luminous radiation. The function of the discharging 
 wire is to localise the place of dissipation, and also to localise 
 .the place where the magnetic field shall be most intense ; and
 
 486 DYNAMICAL THEORY OF INDUCTION. 
 
 all that observation is able to tell us about a conductor which 
 is conveying that which we call an electric current is that ifc 
 is a place where heat is being generated, and near which 
 there is a magnetic field. These conceptions lead us to fresh 
 views of very familiar phenomena. Suppose we are sending 
 a current of electricity through a submarine cable by a 
 battery, say, with zinc to earth, and suppose the sheath is- 
 everywhere at zero potential, then the wire will be everywhere 
 at a higher potential than the sheath, and the level surfaces 
 will pass through the insulating material to the points where 
 they cut the wire. The energy which maintains the current 
 and which works the receiver at the distant end travels through 
 the insulating material, the core serving as a means to allow 
 the energy to get into motion or to be continually propagated, 
 The energy absorbed by the core is, however, transformed 
 into heat and radiated again as dark heat. 
 
 In the case of an arc or glow-lamp worked by an alter- 
 nating current, we have to consider that the energy which 
 moves in on the carbon is returned again, with no other 
 change than that of a shortened wave-length, and the carbon 
 filament performs the same kind of change on the electro- 
 magnetic radiation as is performed when we heat a bit of 
 platinum foil to vivid incandescence in a focus of dark 
 heat. If we adopt the electro-magnetic theory of light, it 
 moves out again still as electro-magnetic energy, but in a 
 different form, with a definite velocity and intermittent 
 in type. We have, then, in the case of the electric light 
 this curious result that energy moves in upon the arc or 
 filament from the surrounding medium, there to be con- 
 verted into a form in which it is sent out again, and which, 
 though the same in kind, is now able to affect our senses. 
 A current passing through a seat of electromotive force is 
 therefore a place of divergence of energy from the conducting 
 circuit into the medium, and this energy travels away and is 
 converged and transformed by the rest of the circuit. From 
 this aspect the function of the copper conducting wire fades 
 into insignificance in interest in comparison with the function 
 of the dielectric. When we see an electric tramcar, or motor, 
 or lamp, worked from a distant dynamo, these notions invite 
 us to consider the whole of that energy, even if it be thousands-
 
 DYNAMICAL THEORY OF INDUCTION. 487 
 
 of horse-power per hour, as conveyed through the electro- 
 magnetic medium, and the conductor as a kind of exhaust 
 valve, which permits energy to be continually supplied to the 
 dielectric. 
 
 Consider, for instance, the simple case of an alternating- 
 current dynamo connected to an incandescent lamp by con- 
 ducting leads. We have in this case a closed conducting loop, 
 consisting partly of the armature wire, partly of the leads, and 
 lastly of the lamp filament. The action of the dynamo when 
 at work consists in alternately inserting into and withdrawing 
 tubes of magnetic induction from a portion of this enclosed 
 area or loop The insertion of this induction causes an 
 electro-magnetic disturbance which travels away through the 
 enclosed dielectric in the form of an ether displacement in 
 its most generalised sense. In reaching the surface of the 
 enclosing conductor this wave begins to soak into it, the 
 electro-magnetic energy at the same time dissipating itself in 
 it in the form of heat. By a suitable arrangement of the 
 resistances and surfaces of various portions of the circuit, 
 we are able to localise the principal place of transformation, 
 and to control its rate so as to compel this transformation 
 of energy to take place at a certain rate in a limited portion 
 of the conductor. Energy is then sent out again in a 
 radiant form partly in the form of ether waves capable of 
 exciting the retina of the eye, but very largely in the form 
 of dark heat. The ether, or electro-magnetic medium, is, 
 therefore, the vehicle by which the energy is carried to 
 the lamp and conveyed away from it in an altered form ; 
 and, whatever be the translating device employed, the ether 
 is the seat of the hidden operations, which are really the 
 fundamental ones, and the visible apparatus is only the con- 
 trivance by which the nature of the energy transformation 
 is determined and its place defined. 
 
 These views are the outcome of that half-century of 
 scientific thought which dates from the period of Faraday's 
 conception of an electro-magnetic medium. We can with- 
 out hesitation predict that the ideas which have thus 
 guided to so much discovery are destined to conduct to 
 further revelations of the nature of the unseen mechanism 
 which lies behind the apparent actions taking placa in the
 
 488 DYNAMICAL THEORY OF INDUCTION. 
 
 electromagnetic field, and of which these actions are the 
 result. 
 
 14. Propagation of Currents along Conductors. We 
 have in a previous section enunciated the modern ideas on 
 this subject, according to which the propagation of a current 
 in a conductor depends upon actions taking place in the 
 dielectric, and various illustrations were given in explanation 
 of this process. We owe to Hertz, however, an absolute 
 experimental demonstration of the correctness of the opinions 
 on this matter, which had previously, from a mathematical 
 standpoint, been put forward by Oliver Heaviside, Poynting, 
 and Lodge. Hertz placed on record his experimental 
 demonstrations in a remarkably interesting Paper which we 
 shall quote here almost verbatim.* 
 
 Dealing with the deductions made by the above writers 
 from Maxwell's equations Hertz described (loc. cit.) his 
 experiments in confirmation of their views, and makes the 
 following remarks : 
 
 " Mathematical investigation points to the conclusion that 
 the electric force which determines the current is in no wise 
 propagated in the wire itself, but under all circumstances enters 
 the wire from without and spreads itself in the metal com- 
 paratively slowly, and according to laws similar to those 
 governing the changes of temperature in a conductor of heat. 
 If the forces in the neighbourhood of the wire are continually 
 altering in direction, the effect of these forces will only enter 
 to a small depth into the metal ; the more slowly the changes 
 take place, so much deeper will the effect penetrate ; and if, 
 finally, the changes follow one another infinitely slowly, the 
 force has time to fill the whole interior of the wire with uni- 
 form intensity." 
 
 The first and most important question with regard to 
 this theory is, whether it agrees with fact. Since, in the 
 experiments which Hertz carried out on the propagation 
 of electric force, he made use of electric waves in wires 
 which were of extraordinarily short period, it was con- 
 
 * This Paper was translated from Wiedemann's Annalen, XXXVII., 
 p. 395, July, 1839, by Dr. J. L. Howard for the Phil. Mag. of August, 1889, 
 and by kind permission of the translator is given here.
 
 DYNAMICAL THEOET OF INDUCTION. 489 
 
 Tenient to prove, by means of these, the accuracy of the 
 inferences drawn. In fact, the theory was proved by the 
 experiments which will now be described ; and it will be 
 found that these few experiments suffice to confirm in the 
 highest degree the view of Messrs. Heaviside and Poynting. 
 Analogous experiments, with similar results, but with quite 
 different apparatus, had already been made by Dr. 0. J. 
 Lodge,* chiefly in the interest of the theory of lightning- 
 conductors. Up to what point the conclusions which were 
 drawn by Dr. Lodge in this direction from his experiments 
 are just must depend in the first place on the velocity with 
 which the alterations of the electrical conditions really follow 
 each other in the case of lightning. The apparatus and 
 methods which are here mentioned are those which Hertz 
 described in full in previous memoirs.! The waves used 
 were such as had in wires a distance of nearly three metres 
 between the nodes. 
 
 " If a primary conductor acts through space upon a secondary 
 ^conductor, it cannot be doubted that the effect penetrates the 
 latter from without. For it can be regarded as established 
 that the effect is propagated in space from point to point ; there- 
 iore it will be forced to meet first of all the outer boundary of 
 the body before it can act upon the interior of it. But a 
 closed metallic envelope is shown to be quite opaque to this 
 effect. If we place the secondary conductor having a spark 
 gap in such a favourable position near the primary one that 
 we obtain sparks 5mm. to 6mm. long, and surround it with a 
 closed box made of zinc plate, the smallest trace of sparking 
 can no longer be perceived. The sparks similarly vanish if 
 we entirely surround the primary conductor with a metallic 
 box. It is well known that, with relatively slow variations of 
 current, the integral force of induction is in no way altered 
 by a metallic screen. This is, at the first glance, contradic. 
 tory to the present experiments. However, the contradiction 
 .is only an apparent one, and is explained by considering the 
 duration of the effects. In a similar manner, a screen which 
 conducts heat badly protects its interior completely from 
 
 * Lodge, Journ. Soc. of Arts, May, 1888 ; Phil. Mag. [5], XXVI., p. 217 
 <1888). 
 
 t Hertz, Wied. Ann., XXXIV., p. 551 (1888).
 
 490 DYNAMICAL THEORY OF INDUCTION. 
 
 rapid changes of the outside temperature, less from slow 
 changes, and not at all from a continuous raising or lowering 
 of the temperature. The thinner the screen is the more rapid 
 are the variations of the outside temperature which can be- 
 felt in its interior. In this case the electrical action must 
 plainly penetrate into the interior, if we only diminish suffi- 
 ciently the thickness of the metal a box covered with tinfoil, 
 protected completely, and even a box of gilt paper, if care were 
 taken that the edges of the separate pieces of paper were in 
 metallic contact. In this instance the thickness of the con- 
 ducting metal was estimated to be barely ^mm. Hertz then 
 fitted the protecting envelope as closely as possible round the 
 secondary conductor. For this purpose its spark-gap was- 
 widened to about 20mm., and, in order to detect electrical 
 disturbances in it, an auxiliaiy spark-gap was added exactly 
 opposite to the one ordinarily used. The sparks in this latter 
 were not so long as in the ordinary spark-gap, since the effect 
 of resonance was now wanting, but they were still very bril- 
 liant. After this preparation the conductor was completely 
 enclosed hi a tubular conducting envelope as thin as possible, 
 which did not touch it, but was as near it as possible ; and in 
 the neighbourhood of the auxiliary spark-gap (in order to be 
 able to use it) the envelope contained a wire-gauze window. 
 Between the poles of this envelope brilliant sparks were pro- 
 duced, just as previously in the secondary conductor itself ; but 
 in the enclosed conductor not the slightest electrical move- 
 ment could be recognised. The result of the experiment is not 
 affected if the envelope touches the conductor at a few points ; 
 the insulation of the two from each other is not necessary in 
 order to make the experiment succeed, but only to give it the 
 force of a proof. Clearly we can imagine the envelope to be 
 drawn more closely round the conductor than is possible in the 
 experiment ; indeed, we can make it coincide with the outer- 
 most layer of the conductor. Although, then, the electrical 
 disturbances on the surface of our conductor are so powerful 
 that they give sparks 5mm. to 6mm. long, yet at -^min. 
 beneath the surface there exists such perfect freedom from dis- 
 turbance that it is not possible to obtain the smallest sparks. 
 We are brought, therefore, to the conclusion that what we call 
 an induced current in the secondary conductor is a phenomenon,
 
 DYNAMICAL THEORY OF INDUCTION. 491 
 
 which is the result of actions taking place in the surrounding 
 dielectric. 
 
 11 One might grant that this is the state of affairs when the 
 electric disturbance is conveyed through a dielectric, but main- 
 tain that it is another thing if the disturbance, as one usually 
 says, has been propagated in a conductor. If we place near 
 one of the end plates of our primary conductor a eonducting- 
 plate, and fasten to it a long straight wire, we have already 
 seen in the previous experiments how the effect of the primary 
 oscillation can be conveyed to great distances by the help of 
 this wire. The usual theory is that a wave travels along the 
 wire in this case. But we can show in the following manner 
 that all the alterations are confined to the space outside and 
 on the surface of the wire, and that its interior knows nothing 
 of the wave passing over it. A piece about 4 metres long was 
 removed from the wire conductor and replaced by two strips 
 of zinc plate 4 metres long and 10cm. broad, which were laid 
 flat one above the other, with their ends permanently connected 
 together. Between the strips along their middle line, and 
 therefore almost entirely surrounded by their metal, was laid 
 along the whole 4 metres length a copper wire covered with 
 gutta-percha. It was immaterial for the experiments whether 
 the outer ends of this wire were in metallic connection with 
 or insulated from the strips ; however, the ends were mostly 
 soldered to the zinc strips. The copper wire was cut through 
 in the middle, and its ends were carried, twisted round each 
 other, outside the space between the strips to a fine spark-gap, 
 which permitted the detection of any electrical disturbance 
 taking place in the wire. When waves of the greatest possible 
 intensity were sent through the whole arrangement, there was 
 nevertheless not the slightest effect observable in the spark- 
 gap. But if the copper wire was then displaced anywhere a 
 few decimetres from its position, so that it projected just a 
 little beyond the space between the strips, sparks immediately 
 began to pass. The sparks were the more intense according 
 to the length of copper wire extending beyond the edge of the 
 zinc strips and the distance it projected. The unfavourable 
 relation of the resistances was, therefore, not the cause of the 
 previous absence of sparking, for this relation had not been 
 changed ; but the wire being in the interior of tha conducting
 
 492 DYNAMICAL THEORY OF INDUCTION. 
 
 mass, was at first deprived of the influence coming from 
 without. Moreover, it is only necessary for us to surround 
 the projecting part of the wire with a little tinfoil in metallic 
 communication with the zinc strips in order to immediately 
 stop the sparking again. By this means we have brought the 
 copper wire back again into the interior of the conductor. If 
 we bend another wire into a fairly large arc round the pro- 
 jecting portion of the gutta-percha wire, the sparks will be 
 likewise weakened ; the second wire takes off from the first a 
 certain amount of the effect due to the outer medium. Indeed, 
 it may be said that the edge of the zinc strip itself takes away 
 in a similar manner the induction from the middle of the strip. 
 For if we now remove one of the strips, and leave the insulated 
 wire simply resting on the other one, we certainly obtain 
 sparks continuously in the wire ; but they are extremely weak 
 if the wire lies along the middle of the strip, and much 
 stronger when near its edge. Just as in the case of dis- 
 tribution under electrostatic influence the electricity would 
 prefer to collect on the sharp edge of the strip, so also here 
 the current tends to move along the edge. Here, as there, it 
 may be said that the outermost parts screen the interior from 
 outside influence." 
 
 The following experiments are somewhat neater and equally 
 convincing. Hertz inserted into the conductor transmitting 
 the waves a very thick copper wire, 1-5 metre long, whose 
 ends carried two circular metallic discs of 15cm. diameter. 
 " The wire passed through the centres of the discs ; the planes 
 of the discs were at right angles to the wire ; each of them 
 had on its rim 24 holes, at equal distances apart. A spark- 
 gap was inserted in the wire. When the waves traversed the 
 wire they gave rise to sparks as much as 6mm. long. A thin 
 copper wire was then stretched across between two corre- 
 sponding holes of the discs. When this was done the length of 
 the sparks sank to 8-2mm. There was no further alteration if 
 a thick copper wire was put in the place of the thin one, or 
 if, instead of the single thin wire, twenty-four of them were 
 taken, provided they were placed near each other through the 
 same two holes. But it was otherwise if the wires were 
 distributed over the rim of the discs. If a second wire was 
 inserted opposite the first one, the spark-length fell to l'2nim. 

 
 DYNAMICAL THEORY OF INDUCTION. 493 
 
 When two more wires were added midway between the first 
 two, the length of the spark sank to O5mm. ; the insertion of 
 four more wires still in the mean positions left sparks of 
 scarcely Olmm. long; and after inserting all the twenty-four 
 wires at equal distances apart, not a trace of sparking was 
 perceptible in the interior. The resistance of the inner wire 
 was nevertheless much smaller than that of all the outside 
 wires taken together. We have also a still further proof that 
 the effect does not depend upon this resistance. If we place 
 by the side of the partial tube of wires, and in parallel circuit 
 with them, a conductor in all respects similar to that in the 
 interior of the tube, we have in the former brilliant sparks, 
 but none whatever in the latter. The former is unprotected, 
 the latter is screened by the tube of wires. We have in this 
 an electro-dynamic analogue of the electrostatic experiment 
 known as the electric birdcage." 
 
 FIG. 163. 
 
 Hertz again altered the experiment, in the manner depicted 
 in Fig. 163. 
 
 "The two discs were placed so near together that they 
 formed, with the wires inserted between them, a cage (A) just 
 large enough for the reception of the spark-micrometer. One 
 of the discs, a, remained metallically connected with the central 
 wire ; the other, /?, was insulated from the wire by means of a 
 circular hole through its centre, at which it was connected to 
 a conducting-tube, y, which, insulated from the central wire, 
 surrounded it completely for a length of 1-5 metre. The free 
 end of the tube, 8, was then connected with the central wire- 
 The wire, together with its spark-gap, is once more situated 
 in a metallically protected space ; and it was only to be 
 expected, from the previous experiments, that not the slightest 
 electrical disturbance would be detected in the wire in which- 
 ever direction waves were sent through the apparatus. So 
 far, then, this arrangement showed nothing new, but it had 
 the advantage over the previous one that we could replace the-
 
 494 DYNAMICAL THEORY OF INDUCTION. 
 
 protecting metallic tube, y, by tubes of smaller and smaller 
 thickness of wall, in order to investigate what thickness is 
 still sufficient to screen off the outside influence. Very 
 thin brass tubes tubes of tinfoil and Dutch metal proved 
 to be perfect screens. Glass tubes were taken whicli had been 
 silvered by a chemical method, and it was then perfectly easy 
 "to insert tubes of such thinness that, in spite of their pro- 
 tecting power, brilliant sparks occurred in the central wire. 
 But sparks were only observed when the silver film was no 
 longer quite opaque to light, and was certainly thinner than 
 Y^g-mm. In imagination, although not in reality, we can 
 conceive the film drawn closer and closer round the wire, and 
 finally coinciding with its surface ; we should be quite certain 
 that nothing would be radically altered thereby. However 
 actively, then, the real waves play round the wire, its interior 
 remains completely at rest ; and the effect of the waves hardly 
 penetrates any more deeply into the interior of the wire than 
 does the light which is reflected from its surface. For the 
 real seat of these waves, therefore, we ought not to look in the 
 wire, but rather to assume that they take place in its neigh- 
 bourhood; and, instead of asserting that our waves are 
 propagated in the wire, we should be more accurate in saying 
 that they glide along on the wire. 
 
 " Instead of placing the apparatus just described in the cir- 
 cuit in which we produced waves indirectly, we can insert it 
 in one branch of the primary conductor itself. In such 
 experiments results similar to the previous ones are obtained. 
 Our primary oscillation, therefore, takes place without any 
 participation of the conductor in which it is excited, except 
 at its bounding surface ; and we ought not to look for its 
 existence in the interior of the conductor. 
 
 " To what has been said above about waves in wires we wish 
 to add just one remark concerning the method of carrying 
 out the experiments. If our waves have their seat in the 
 neighbourhood of the wire, the wave progressing along, a 
 single isolated wire will not be propagated through the air 
 alone ; but, since its effect extends to a great distance, it will 
 partly be transmitted by the walls, the ground, &c., and will 
 thus give rise to a complicated phenomenon. But if we place 
 opposite each pole of our primary conductor, in exactly the
 
 DYNAMICAL THEORY OF INDUCTION. 495 
 
 same way, two auxiliary plates, and attach a wire to each of 
 them, carrying the wires straight and parallel to each other 
 to equal distances, the effect of the waves makes itself felt 
 only in the region of space between the two wires. The wave 
 progresses solely in the space between the wires. We can 
 thus take precautions to propagate the effect through the air 
 alone or through another insulator, and the experiments will 
 be more convenient and free from error by this arrangement. 
 For the rest, the lengths of the waves are nearly the same in 
 this case as in isolated wires, so that with the latter the effect 
 of the disturbing causes is apparently not considerable. 
 
 " We can conclude from the above results that rapid electric 
 oscillations are quite unable to penetrate metallic sheets ot 
 any thickness, and that it is, therefore, impossible by any 
 means to excite sparks by the aid of such oscillations in the 
 interior of closed metallic screens. If, then, we see sparks 
 produced by such oscillations in the interior of metallic con- 
 ductors which are nearly, but not quite, closed, we shall be 
 obliged to conclude that the electric disturbance has forced 
 itself in through the existing openings. This view is also 
 correct, but it contradicts the usual theory in some cases so 
 completely that one is only induced by special experiments 
 to give up the old theory in favour of the new one. We 
 shall choose a prominent case of this kind, and, by assuring 
 ourselves of the truth of our theory in this case, we shall 
 demonstrate its probability in all other cases. We again take 
 the arrangement which we have described in the previous 
 section and drawn in Fig. 163 ; only we now leave the protect- 
 ing tube insulated from the central wire at 8. Let us now 
 send a series of waves through the apparatus in the direction 
 from A towards 8. We thus obtain brilliant sparks at A; 
 they are of similar intensity to those obtained when the wire 
 was inserted without any screen. The sparks do not become 
 materially smaller, if, without making any other alteration, 
 we lengthen the tube y considerably, even to 4 metres. 
 According to the usual theory it would be said that the wave 
 arriving at A penetrates easily the thin, good-conducting 
 metal disc a, then it leaps across the spark-gap at A, and 
 travels on in the central wire. According to our view, .on 
 the contrary, we must explain the phenomenon in the follow-
 
 496 DYNAMICAL THEORY OF INDUCTION. 
 
 ing manner. The wave arriving at A is quite unable to 
 penetrate the metallic disc ; it therefore glides along the disc 
 over the outside of the apparatus and travels as far as the 
 point 8, 4 metres away. Here it divides : one part, which 
 does not concern us at present, travels on immediately along 
 the straight wire, another bends into the interior of the tube 
 and then runs back in the space between the tube and the 
 central wire to the spark-gap at A, where it now gives rise- 
 to the sparking. That this view, although more complicated, 
 is still the correct one, is proved by the following experiments. 
 Firstly, every trace of sparking at A disappears as soon as- 
 we close the opening at 8, even if it be only by a stopper of 
 tinfoil. Our waves have only a wave-length of 3 metres ; 
 before their effect has reached the point 8 the effect at A has 
 passed through zero and changed sign. What influence, then, 
 could the closing of the distant end 8 have upon the spark at 
 A, if the latter really happened immediately after the passage 
 of the wave through the metallic wall ? Secondly, the sparks 
 disappear if we make the central wire terminate inside the 
 tube y, or at the opening 8 itself ; but they reappear when we 
 allow the end of the wire to project even 20cm. to 30cm. only 
 beyond the opening. "What influence couid this insignificant 
 lengthening of the wire have upon the sparks in A, unless the 
 projecting end were just the means by which a part of the 
 wave breaks off and penetrates through the opening 8 back 
 into the interior? Thirdly, we insert in the central wire 
 between A and 8 a second spark-gap B, which we also com- 
 pletely cover with a gauze cage like that at A. If we make 
 the distance of the terminals at B so great that sparks can 
 no longer pass across, it is also no longer possible to obtain 
 visible sparks at A. But if we hinder in like manner the 
 passage of the spark at A, this has scarcely any influence on 
 the sparks in B. Therefore, the passage of the spark at B 
 determines that at A, but the passage of a spark at A does 
 not determine that at B. The direction of propagation in the 
 interior is therefore from B towards A, not from A to B. 
 
 "We can moreover give further proofs, which are more 
 convincing. We may prevent the wave returning from 5 to 
 A from dissipating its energy in sparks, by making the spark- 
 gap either vanishingly small or very great. In this case the=
 
 DYNAMICAL THEORY OF INDUCTION. 49? 
 
 wave will be reflected at A, and will now return again from 
 A towards 8. In doing so it must meet the direct waves 
 from 8 to A, and combine with them to form stationary waves, 
 thus giving rise to nodes and ventral segments. If we succeed 
 in proving their existence, there will be no longer any doubt 
 as to the truth of our theory. For this proof we must give 
 somewhat different dimensions to our apparatus in order to 
 be able to introduce electric resonators into its interior. Hertz 
 therefore led the central wire through the axis of a cylindrical 
 tube 5 metres long and 30 centimetres diameter. It was not 
 constructed of solid metal, but of 24 wires arranged parallel to 
 each other along the generating surface, and resting on seven 
 equidistant and circular rings of strong wire, as shown in 
 Fig. 164. He made the requisite resonator in the following 
 manner: A closely-wound spiral of 1cm. diameter was 
 formed from copper wire of 1mm. thickness ; about 125 
 turns of this spiral were taken, drawn out a little, and bent 
 into a circle of 12cm. diameter; between the free ends an 
 
 FIG. 164. 
 
 adjustable spark-gap was inserted. Previous experiments 
 had shown that this circle responded to waves 3 metres long 
 in the wire, and yet it was small enough in size to admit 
 of its insertion between the central wire and the surface of 
 the tube. If now both ends of the tube were open, and the 
 resonator was then held in the interior in such a way that its 
 plane included the central wire, and its spark-gap was not 
 directed exactly inwards or outwards, but was turned towards 
 one end or the other of the tube, brilliant sparks of ^mm. 
 to 1mm. length were observed. On now closing both ends 
 of the tube by four wires arranged crosswise and connected 
 with the central conductor, not the slightest sparking remained 
 in the interior, a proof that the network of the tube is a suffi- 
 ciently good screen for our experiments. The end of the tube 
 on the side (3, that, namely, which was furthest away from the 
 origin of the waves, was now removed. In the immediate 
 
 K K
 
 498 DYNAMICAL THEORY OF INDUCTION. 
 
 neighbourhood of the closed end that is, at the point a, which 
 corresponds to the spark-gap A of our previous experiments 
 there were now no sparks observable in the resonator. But on 
 moving away from this position towards ft sparks appeared, 
 became very brilliant at a distance of 1-5 metre from a, then 
 decreased again in intensity, then almost entirely vanished at 
 3 metres distance from a, and increased again until the end 
 of the tube was reached. We thus find our theory borne out 
 by fact. That we obtain a node at the closed end is clear, for 
 at the metallic contact between the central wire and the 
 surface of the tube the electric force between the two must 
 necessarily vanish. It is different when we cut the central 
 conductor at this point just near the end, and insert a gap of 
 several centimetres length. In this case the wave will be 
 reflected in a phase opposite to that of the previous case, and 
 we should expect a ventral segment at a. As a matter of 
 fact we find brilliant sparks in the resonator in this case ; 
 they rapidly decrease in strength if we move from a towards 
 /3, almost entirely vanish at a distance of 1*5 metre, and 
 become brilliant again at a distance of 3 metres ; moreover 
 they give a second well-marked node at 4'5 metres distance 
 that is, 0-5 metre from the open end. The nodes and loops 
 which we have described are situated at fixed distances from 
 the closed end, and alter only with this distance ; they are, 
 however, quite independent of the occurrences outside the 
 tube, for example, of the nodes and loops formed there. The 
 phenomena occur in exactly the same way if we allow the 
 wave to travel through the. apparatus in the direction from 
 the open to the closed end ; their interest is, however, smaller, 
 since the mode of transmission of the wave deviates from that 
 usually conceived less in this case than in the one which has 
 just been under our consideration. If both ends of the tube 
 are left open with the central wire undivided, and stationary 
 waves with nodes and loops are now set up in the whole 
 system, there is always found for every node outside the tube 
 a corresponding node in the interior, which proves that the 
 propagation takes place inside and outside with, at any rate 
 approximately, the same velocity. 
 
 "On looking over the experiments which we have described, 
 and the interpretation put upon them, as well as the explana-
 
 DYNAMICAL TSEORY OF iNDVCTlOJt. 499 
 
 tions of the physicists referred to in the introduction, a 
 difference will be noticed between the views here put forward 
 and the usual theory. According to the latter, conductors are 
 represented as those bodies which alone take part in the pro- 
 pagation of electric disturbances ; non-conductors are the bodies 
 which oppose this propagation. According to the modern view, 
 on the contrary, all transmission of electrical disturbances 
 is brought about by non-conductors : conductors oppose a 
 great resistance to any rapid changes in this transmission. 
 One might almost be inclined to maintain that conductors 
 and non-conductors should, on this theory, have their names 
 interchanged. However, such a paradox only arises because 
 one does not specify the kind of conduction or non- conduction 
 considered. Undoubtedly metals are non-conductors of elec- 
 tric force, and just for this reason they compel it, under certain 
 circumstances, to remain concentrated instead of becoming 
 dissipated, and thus they become conductors of the apparent 
 source of these forces, electricity, to which the usual termi- 
 nology has reference." 
 
 15. Experimental Determination of Electromagnetic 
 Wave Velocity. Space does not permit us to make further 
 mention of the important and valuable work which has been 
 carried out in recent years in confirming and extending this 
 work of Hertz. We refer the reader specially, however, on 
 this subject, to Dr. Lodge's monograph on this subject.* 
 
 We shall conclude this chapter by presenting an abstract of 
 an interesting research by Messrs. Trowbridge and Duane on 
 the " Velocity of Electric Waves "f because it furnishes a 
 proof, having a high degree of accuracy, that the velocity of 
 an electromagnetic wave is identical with the velocity of light. 
 
 Broadly speaking, the method employed consisted in 
 establishing stationary waves in a conducting circuit and 
 determining the period of oscillation by photographing the 
 oscillating spark in a spark gap, and at the same time 
 measuring the wave-length of the stationary waves induced in 
 
 * " The Work of Hertz and some of his Successors." By Dr. 0. J. Lodge, 
 published by " The Electrician " Printing and Publishing Co., London, 
 t Phil, Mag., August, 1895 5 also The Electrician, Vol. XXXV., p. 712. 
 
 EK2
 
 500 DYNAMICAL T11EOHY OF INDUCTION. 
 
 a secondary circuit turned to resonance with the primary. In 
 the following paragraphs the description of their experiments 
 is taken from the Paper by Messrs. Trowbridge and Duane. 
 They say : 
 
 "The first point in the course of the investigation worth 
 detailed description is the production of electric waves along 
 parallel wires in such a manner that they are actually visible 
 to the eye. The arrangement of the apparatus to accomplish 
 this was as follows : 
 
 "A primary condenser, AB (Fig. 165), was held with its 
 plates in vertical planes by means of suitable wooden supports 
 (not represented in the figure), and was joined in a circuit, 
 B C, consisting of two wires about 75cm. long, placed 4cm. 
 apart. In reality this circuit B C should be represented as 
 perpendicular to the plane of the paper (which is taken as the 
 horizontal plane passing through the centre of the apparatus). 
 
 FIG. 165. 
 
 The plates of the condenser A B were sheets of tinfoil 101 X 
 40cm., glued to hard rubber sheets, and the dielectric between 
 them consisted of other similar sheets of hard rubber sufficient 
 in number and thickness to make the distance between the 
 condenser plates 4 -2cm. Outside the primary condenser plates, 
 and separated from them by hard rubber plates (total thickness 
 O-Gcm.), were two secondary plates, E and F, each 40cm. square. 
 To these plates was attached the secondary circuit EGJHF, 
 the form of which is represented in the figure. This latter 
 circuit consisted of copper wire, diameter O g 13cm., and its 
 total length from E to F was 4,200cm. A spark-gap with 
 spherical terminals 2- 5cm. in diameter was placed at C in the 
 primary circuit, and another spark-gap with pointed terminals 
 was sometimes inserted at J in the secondary circuit, although 
 this latter spark-gap had no effect upon the phenomena to be
 
 D YNA MICAL Til EOE Y OF IND UCTIO Y. 501 
 
 described. The primary condenser was charged by means of 
 a large Euhmkorff coil excited by five storage cells with a 
 total voltage of 10 volts. The current from these cells was 
 made and broken by an automatic interrupter. Every time the 
 primary condenser was charged a spark passed at C, causing 
 an oscillatory discharge. A convenient method of forming a 
 mental picture of the oscillation excited in the secondary circuit 
 is the conception of Faraday tubes elaborated by J. J. Thomson 
 hi his 'Recent Researches in Electricity and Magnetism.' 
 The oscillations of the primary acted inductively upon the 
 secondary and sent out groups of Faraday tubes which travelled 
 along the secondary circuit, with their ends on the wires, and 
 lying chiefly in the space between them. At the end J they 
 reversed their direction and travelled back along the circuit. 
 The period of oscillation of the primary circuit was altered, 
 until by trial it was found that groups of returning tubes met 
 groups of advancing tubes between the points G and H. As 
 the two sets of moving tubes were oppositely directed they 
 annulled each other and produced a node. Thus a system of 
 stationary waves is set up with a node at J, another node 
 at G H, and a ventral segment at K L. The method of dis- 
 covering when the circuits were in tune and of investigating 
 the shape of the waves will be described later. The point 
 to be noticed here is that the vibrations were sufficiently 
 powerful to cause a luminous discharge on the surface of 
 the wire where the accumulation of tubes was a maximum, 
 i.e., at K L, while at the nodal points J and G H the wire 
 was entirely dark. Still further, the wave formation could be 
 made apparent to the sense of hearing as well as that of sight ; 
 for, placing the ear within a few centimetres of the wire and 
 walking beside it, a distinct crackling sound could be heard at 
 the points K and L, whereas no such sound could be heard at 
 G, J and H. By placing bits of glass tubing on the wire the 
 sound was much intensified at the points K and L, and the 
 phenomena made more striking. It might be supposed that 
 by decreasing the capacity of the primary condenser, and 
 therefore the period of its oscillation, the secondary circuit 
 could be broken up into a new set of shorter stationary waves, 
 with nodes at J and at points somewhere near K, L, G and H, 
 and ventral segments between them. This was tried with
 
 602 DYNAMICAL THEORY OF INDUCTION. 
 
 perfect success, except that it was not possible to cause the 
 light at K and L to actually disappear. There was decidedly 
 less light at these points, however, than on either side of them. 
 The light, of course, is simply that which always appears 
 around wires carrying very high-potential currents, the in- 
 teresting point being that it appears in some places on the 
 circuit and not in others. The experiment showing how the 
 circuit breaks up in several different ways would form a most 
 beautiful lecture experiment. 
 
 " As a means of ascertaining when the circuits were in 
 resonance, and of investigating the form of the wave in 
 the secondary circuit, a bolometer similar to that designed 
 by Paalzow and Kubens* was used. 
 
 "The bolometer as an instrument for measuring electric 
 waves is so well known, that it is not necessary to state here 
 more than its fundamental principles. It consists essentially 
 of a well-balanced Wheatstone bridge, to one of the arms of 
 which are metallically connected two small conductors. These 
 conductors are brought near the circuit to be tested, and the 
 oscillating charges induced in them and sent through the arm 
 of the Wheatstone bridge develop enough heat to throw the 
 bridge out of balance. By moving the conductors along the 
 circuit different deflections are produced according to the 
 magnitude of the charges on the wire in their neighbourhood, 
 and thus an excellent estimate of the wave formation can 
 be obtained. In the present case the conductors that were 
 brought near the secondary circuit consisted of two pieces 
 of wire insulated with rubber, bent into circles of about 2cm. 
 radius, and fastened to a bit of pine-wood by means of a heavy 
 coating of paraffin. The two wires of the secondary circuit 
 passed through holes in this bit of wood in such a manner as 
 to pass through the centres of the two circles. In the early 
 part of the investigation the bolometer and galvanoscope were 
 placed at a sufficient distance from the oscillating circuits 
 to prevent any direct action of one on the other, and the 
 leads running from the circular conductors to the bolometer 
 consisted of long fine wires. Later, when longer circuits and 
 longer waves were experimented with, great inconvenience 
 
 * "Anwendung des Bolometrischen Princips auf Electrische itessungen,'* 
 Wied, 4nn., XXXVII., p. 529.
 
 DYNAMICAL THEORY OF INDUCTION. 503 
 
 was experienced from the long leads, since their relative 
 position had considerable effect upon the galvanoscope deflec- 
 tions. In order to obviate this difficulty, short leads of 
 heavily insulated wire were used, and the bolometer was 
 placed on wheels and moved along from place to place. A 
 bolometric study of the circuit just described showed the 
 character of the oscillation to be that mentioned namely, 
 nodes at the points J and G H, and a ventral segment at 
 KL. A careful run was made from one end of the circuit 
 to the other, which furnished data from which a very regular 
 curve was drawn. 
 
 " The insertion of a small spark-gap (1mm. to 3mm.) at 
 the point in the secondary circuit marked J (Fig. 165, p. 500) 
 had no appreciable effect upon the position of the nodal point 
 G H, or of the point of maximum accumulation K L. The 
 form of the wave was slightly altered for a metre on each 
 side of J, and the bolometer showed a slight accumulation 
 in the immediate neighbourhood of the spark-gap. This was 
 probably due to the charging of the spark terminals to a 
 sufficiently high potential to break through the dielectric. 
 The fact that the insertion of a spark-gap into a secondary 
 circuit in the manner described has no effect upon the length 
 of the waves set up in that circuit was tested for a number 
 of different cases (in none of which, however, was the length 
 of the waves greater than in the present case), and found to 
 be true in each one of them. 
 
 " In order to determine the time of vibration, we used a 
 concave rotating mirror, and the images of the oscillating 
 sparks were thrown on a sensitive plate. If the mirror 
 rotated about a horizontal axis the photographs showed 
 bright horizontal lines, perpendicular to which at their 
 extremities extended two series of dots. The distance 
 between successive dots was the distance on the plate 
 through which the image of the spark-gap moved during 
 the time of a complete oscillation. Hence, by determining 
 the speed of the mirror, and measuring the distances from 
 the mirror to the plate, the time of oscillation could be 
 calculated. To measure the sparks we used a sharp pointer, 
 moved at the end of a micrometer screw under a magnifying 
 glass of low power. The instrument was originally intended
 
 504 DYNAMICAL THEORY OF INDUCTION. 
 
 for microscopic measurements, and was very accurately con- 
 structed. The rotating mirror was driven by an electric 
 motor by means of a current from a storage battery of 
 extremely constant voltage. To give great steadiness a heavy 
 flywheel was attached to the axis of the mirror. The speed 
 of the mirror was determined to within about one part in 
 500 by means of an electric chronograph. This apparatus, 
 requiring great technical skill, was made for us by the 
 mechanician of the laboratory. The mirror consisted of a 
 thick piece of glass with a concave surface accurately ground 
 for this research and silvered by ourselves. 
 
 " There are many advantages in photographing the secon- 
 dary spark rather than the primary. In the first place, to 
 properly photograph a spark it is necessary to use pointed 
 terminals ; but experiment has shown that the waves excited 
 in a secondary circuit depend to a large extent upon the 
 character of the primary spark, and that the most active 
 sparks are those between metallic spheres with polished sur- 
 faces. It is true that waves can be produced by sparks 
 between points, but the oscillations are not so powerful or 
 well marked. In the second place, from the results obtained 
 by Bjerknes, one would expect the oscillations in the secondary 
 circuit to be much less damped than those in the primary. 
 This expectation has been fully realised. Photographs show 
 from ten to twelve times as many oscillations in the secondary 
 as in the primary. The longest secondary spark we counted 
 indicated 60 complete oscillations. In the third, and by no 
 means the least important case, the question how close the 
 resonance is does not affect the accuracy of the results. 
 By photographing the sparks in the secondary the period 
 of oscillation is determined, not of a circuit that is altered 
 until by trial it is found to have as nearly as possible the 
 same period of vibration as the circuit on which the length 
 of the wave is measured, but that of the circuit along which 
 the wave itself is actually travelling ; and hence the con- 
 clusions in regard to the effect of damping reached by 
 Bjerknes in his admirable Paper on ' Electric Resonance '* 
 do not affect the accuracy of the results. 
 
 * " Ueber electrische Resonanz," Wied. Ann., LV., p. 121 (1895).
 
 DYNAMICAL TIIEOEY OF INDUCTION. 
 
 505 
 
 " The great difficulty to be overcome is the production of 
 secondary oscillations that will produce sparks sufficiently 
 bright to photograph. It is comparatively an easy task to 
 photograph the primary spark, but in order to photograph 
 the secondary the dimensions of the circuit must be chosen 
 with great care. 
 
 " With a view to increasing the light of the spark, together 
 with the length of the waves, it seemed desirable to lengthen 
 the period of oscillation by enlarging the condensers rather 
 than by increasing the self-induction of the primary circuit. 
 A castor-oil condenser, therefore, was designed and con- 
 structed on the following plan : Eight plates (25cm. by 
 20cm.) were cut out of sheet zinc, and were held in vertical 
 planes side by side 2cm. apart by a suitable hard-rubber 
 
 D" 
 n c 
 
 
 B 
 
 A E G K iv 
 
 b 
 
 ' 
 
 
 
 
 
 
 
 J 
 
 
 
 1 
 
 
 h 1 
 
 ' F H 
 FIG. 166. 
 
 frame. The plates were entirely immersed in castor-oil con- 
 tained in a glass jar. They were connected together in the 
 manner shown in Fig. 166. The plates marked a, c and e 
 were fastened to the conductor A B, and formed one armature 
 of the condenser. Those marked d, f and h were joined to 
 CD, and formed the other armature. The two ends of the 
 secondary circuit E, G, J, H, F were fastened to the plates 
 h and <j. The plane of the secondary circuit was 50cm., and 
 that of the primary 8em. above the upper edge of the con- 
 denser plate. The total length of the secondary circuit from 
 one condenser plate through E, G, J, H, F to the other plate 
 was 6,888cm. The circuit consisted of copper wire (diameter 
 0-215cm.) supported at each end by suitable wooden frames, 
 and also once in the middle by hard-rubber hooks, fastened 
 by long pieces of twine to a wooden crossbar above. The
 
 506 DYNAMICAL THEORY OF INDUCTION. 
 
 distances from F to E and from K to L were 30cm., and a 
 spark-gap with pointed tin terminals was inserted at J. The 
 primary circuit consisted of copper wire (diameter 0-34cm.). 
 The distances between the two parts A B and C D were 45cm. 
 The portion BD contained a spark-gap with platinum-faced 
 spherical terminals, and was made so as to slide back and 
 forth, to and from the condenser. The motion of this movable 
 piece varied the self-induction, and therefore the period of 
 oscillation of the primary circuit. By this means the circuits 
 were brought into resonance. With certain arrangements of 
 the condensers the resonance was very sharp, and the position 
 of the movable portion could be determined to within 0-25em. 
 In the arrangement which was finally adopted the resonance 
 was not so sharp. Even in this case the distance of the 
 sliding part from plate a could not have been in error by more 
 than 2cm. The length 6ocm. was finally chosen for its value. 
 
 " The automatic current interrupter that worked so beauti- 
 fully in connection with the Hertz vibrator would not operate 
 well when used to excite the circuits just described. After 
 trying many devices, we finally adopted an ordinary reed 
 interrupter with a comparatively large hammer -and -anvil 
 arrangement, which gave little trouble. 
 
 " At first it was found impossible to produce anything but 
 a complex vibration in the secondary circuit when the spark- 
 gap was open. Some slight evidence of resonance was 
 obtained, but nothing of a decided character. When, how- 
 ever, the spark-gap was closed, very good resonance ensued, 
 and a wave the length of which could be measured to within 
 0-4 per cent, was excited. Some photographs were taken 
 of the spark in the secondary circuit, and they showed 
 immediately the character of the complex wave formation. 
 The secondary circuit could and did oscillate in three different 
 ways, and the ratios of the periods were those of the notes in 
 an open organ pipe, namely 1;2;3. Usually the lowest or 
 fundamental oscillation together with one of the overtones 
 was present ; but several sparks were noticed that furnished 
 unmistakable evidence of the simultaneous existence of all 
 three. We have observed in a circuit 10,000cm. long the 
 same peculiarities of oscillation, excited by a primary circuit 
 that, judging from its dimensions, could not have been in
 
 DYNAMICAL THEORY OF INDUCTION. 
 
 507 
 
 resonance with the secondary. It was evident that the oscil- 
 lation having a node between the points marked E and F 
 (Fig. 166) is that whose period is one third of the fundamental. 
 
 " A number of measurements of this period have been made, 
 and from these values the velocity of the waves has been 
 calculated. The results appear in the table below. As an 
 average of five measurements of the wave length, none of 
 which differed from the mean by more than 20cm., the value 
 5,888cm. was chosen. The distance from the mirror to the 
 photographic plate in each case except the last was 800- 1cm. 
 Each of the first five values in the second column of the table 
 is an average of 30 measurements of distances ranging in the 
 neighbourhood of 1cm. 
 
 " The last line in the table contains the results of measure- 
 ments on photographs of the primary spark instead of the 
 secondary. In this case the distance from the mirror to the 
 photographic plate was 311 -5cm. 
 
 " These results were published as a preliminary record in the 
 American Journal of Science for April, 1895. Since then the 
 authors succeeded in producing much better waves and much 
 more regular sparks, and discovered a phenomenon which 
 renders a measurement on a photograph over a space where 
 the dots are obliterated a questionable proceeding. The new 
 data have given a value for the velocity more in accord with 
 theory. 
 
 Number of revolu- 
 tions of mirror 
 per second. 
 
 Distance between two 
 successive points on plate. 
 Centimetres. 
 
 Velocity of waves. 
 Centimetres. 
 
 71-2 
 70-85 
 70-7 
 71-3 
 70-8 
 
 0-05608 
 0-05600 
 0-05532 
 005637 
 0-05611 
 
 2-819 x 10 10 
 2-810 x 101 
 2-835 x ID" 
 2-808 x IQio 
 2-808 x IQio 
 
 69-2 
 
 Average 
 0-05340 
 
 2-816 x IQio 
 2-988 x IQio 
 
 " Since the waves in the secondary were not well formed 
 when the spark-gap was inserted, it seemed desirable to try to 
 find an arrangement that would produce simultaneously a 
 good wave and a photographable spark. A number of con-
 
 508 DYNAMICAL THEORY OF INDUCTION. 
 
 erent 
 
 no tn 
 
 densers with plates of different sizes and shapes and diff< 
 substances for the dielectric were tried, and the apparatus to 
 be described was finally adopted. The difficulties to be over- 
 come were these. Too strong a reaction between the primary 
 and secondary condensers could not be employed, because the 
 increase in the damping of the primary due to the large 
 amount of energy drawn off by the secondary made good 
 resonance impossible. The amount of energy in the primary 
 at full charge must be much greater than that in the secondary. 
 On the other hand, the capacity of the primary condenser 
 must not be too great ; for the self-induction of the primary 
 circuit would have to be proportionately small, and this, too, 
 means an increase in the damping. The secondary con- 
 denser, too, must have a capacity of less than a certain 
 magnitude in order that the node may fall on the circuit and 
 not in the condenser plate. These points seem to indicate 
 that, small condensers are preferable to large ones ; but a 
 
 decrease in the size of the plates means a decrease in the light 
 of the secondary sparks, and the sparks are at best barely 
 photographable. Practically, therefore, our choice was much 
 limited, and the particular arrangement to give the best 
 results had to be selected by experiment after a long series of 
 trials. The arrangement and dimensions of the apparatus 
 finally adopted were as follows : 
 
 "Two metallic plates, a and b (Fig. 167), 30 x 30cm., placed 
 in vertical planes, formed the primary condenser. The dielec- 
 tric between them consisted of the best French plate glass 
 obtainable (K = 8 + probably) and was 2cm. thick. Outside 
 the plates a and b, and separated from them by a hard-rubber 
 dielectric (K = 2 + about) l-8cm. thick, were the secondary 
 plates, 26 x 26cm. The primary and secondary circuits were
 
 DYNAMICAL THEORY OF INDUCTION. 509 
 
 joined to the condenser plates as indicated in the figure. The 
 primary circuit lay in the horizontal plane passing through 
 the centres of condenser plates, and consisted of copper wires 
 0-34cm. in diameter. In order to control the period of 
 oscillation of the primary circuit, the portion B D containing a 
 spark-gap with spherical terminals was made, as before, so as 
 to slide along parallel to itself. The distance between the 
 straight portions A B and C D was 40cm., and the lengths of 
 A B and C D finally chosen for best resonance were 85cm. 
 Most of the secondary circuit lay in a horizontal plane 16cm. 
 above that of the primary. The lengths GE and H F, 
 however, were bent down and fastened to the middle points G 
 and H of the secondary plates. The circuit consisted of 
 copper wire (diameter O21ocm.), and its total length from G 
 through J to H was 5,860cm. At J was a spark-gap with 
 pointed terminals. With this apparatus we succeeded in 
 producing a very regular wave formation, as indicated by the 
 bolometer, even when there was a spark-gap at J. So many 
 curves have been plotted and published to illustrate the 
 characteristics of electrical waves that it does not seem worth 
 while to add to the number here. It will be sufficient to state 
 that the ratio of the maximum and minimum deflections in 
 the bolometer was about 15 : 1, and that there was a node at 
 J and another about 40cm. to the right of E and F. 
 
 "Upon photographing the secondary spark some curious 
 phenomena were observed. In the first place, the dots usually 
 appeared in pairs. There would be two black dots followed 
 by a space where two or three dots either appeared faintly or 
 were absent altogether ; after that two black dots would 
 reappear, followed again by a faint space, and so on for six or 
 seven repetitions. All this, of course, occurred in a single spark. 
 
 " The explanation that first presents itself is that the two 
 black dots are the result of the first two oscillations in the 
 primary circuit, which, owing to the damping, are much more 
 powerful than the others. If this were the true reason, the 
 first of the pair of dots always ought to be blacker than the 
 second, and every third dot ought to be the first of a pair. 
 This is not the case, however. On the other hand, the 
 phenomena cannot be explained as the result of a complex 
 vibration, for the bolometer readings, taken only a few
 
 510 DYNAMICAL THEORY OF INDUCTION. 
 
 minutes before the photographic plates were exposed, and 
 with exactly the same arrangement of apparatus, indicated 
 extremely regular waves. A clue to the mystery was furnished 
 by several sparks in which the dots made by one spark 
 terminal had the characteristics just described, whereas those 
 made by the other were quite regular. Following out this 
 hint, we found that the particular substances used for the 
 secondary spark terminals had a large effect upon the charac- 
 teristics of the photographs. We tried spark terminals made 
 of a number of different metals tin, aluminium, magnesium, 
 fuse-metal, &c. and finally adopted cadmium as productive 
 of the best sparks. In the case of cadmium the characteristics 
 described are much less marked, and we have succeeded even 
 in producing a few sparks in which no difference in blackness 
 could be detected between one dot and the next. The photo- 
 graphs from cadmium terminals, too, are far more distinct 
 and far more easily measured than those from terminals of 
 any other metal that we tried. 
 
 "An interesting question arose here as to whether the 
 distance between two successive dots would depend upon the 
 period of oscillation of the primary circuit if the secondary 
 were unaltered. To test this point the circuits were brought 
 into resonance, and a photograph taken. The self-induction 
 of the primary circuit was increased by about 20 per cent, of 
 its value, and a second photograph taken. In the first case 
 the distances between successive dots were all within 2 or 3 
 per cent, of the average obtained by measuring over several 
 dots and dividing by the number of intervening spaces ; 
 whereas in the second case the measurements of some of 
 the single spaces were from 8 to 12 per cent, greater than 
 before, the average from long measurements being the same. 
 This indicates that the vibrations of the secondary circuit are 
 not necessarily perfectly regular, and at a distance apart fixed 
 by the character of the circuit, but are to be looked upon 
 as a series of pulses started travelling along the circuit and 
 keeping at a distance from each other that is determined by 
 the exciter. Owing to the fact that the damping of the 
 primary is much greater than that of the secondary, the 
 seventh and eighth pulses started are too weak to obliterate 
 the first and second, which have travelled the length of the
 
 DYNAMICAL THEORY OF INDUCTION. 511 
 
 circuit and back. We should expect from this that the 
 bolometer throws, which measure the average length of the 
 wave, would not indicate a shifting of the node when the 
 circuits are thrown slightly out of resonance, but that the 
 minimum throws would be greater than when the circuits are 
 exactly in resonance. This, as is well known, is what happens. 
 " The improved sparks which the new arrangement of 
 apparatus and the use of cadmium as material for the 
 spark terminals have enabled us to produce, have brought 
 to light another interesting fact, namely, that even when 
 the best resonance is obtained and the most regular wave 
 formation is excited, the distances between the first three or 
 four dots are slightly greater than the distances between three 
 or four dots taken farther down the spark. The explanation 
 we offer for this is the following, and it applies as a criticism 
 to all cases in which waves are excited in a circuit by a 
 neighbouring circuit possessing a much larger damping factor: 
 The fact that the secondary waves last longer than the 
 primary oscillations means that the last times that the 
 waves travel over the circuit they do so under different end 
 conditions from the first few times. The capacity of the 
 secondary plates is slightly less after the primary spark has 
 stopped than it was before, and therefore the length of the 
 wires equivalent to the secondary plates is slightly less, and 
 it takes a shorter time for the waves to travel along the 
 circuit and back. Hence the observed decrease in the distance 
 between the spark points and a certain mixing up of the dots, 
 which occurs after the sixth or seventh oscillation (see Fig. 166). 
 The sixth dot in the figure, apparently following its predecessor 
 after about half an interval, is not a usual characteristic. In 
 the vast majority of sparks the first few dots are far more 
 powerful than those that follow them, and only occasionally 
 do sparks occur that indicate more than five or six good 
 complete oscillations. Hence these first few ossillations have 
 the preponderating influence in fixing the length of the waves 
 as indicated by the bolometer. In examining the sparks, 
 therefore, we measured from the first oscillation as far down 
 the spark as we could without passing over a space where dots 
 were obliterated ; and hence in every case we knew the 
 number of dots between the points from which measurements
 
 512 DYNAMICAL THEORY Of INDUCTION. 
 
 were taken, and did not have to assume that good oscillations 
 had occurred without affecting the plate. 
 
 " The following table, containing the results of our measure- 
 ments with the improved apparatus, explains itself. The 
 distance from the mirror to the photographic plate was 302cm. 
 in each case : 
 
 Number of 
 revolutions of 
 mirror per sec. 
 
 Distance be- 
 tween successive 
 points on plate. 
 Centimetre. 
 
 Time of 
 oscillation. 
 Seconds. 
 
 Length 
 of wave. 
 Centim. 
 
 Velocity of 
 wave. 
 Centim. per sec. 
 
 70-8 
 73-7 
 75-2 
 69-5 
 689 
 69-0 
 71-2 
 
 0-05028 
 0-05247 
 0-05536 
 0-05002 
 0-04900 
 
 o-oa974 
 
 0-05075 
 
 1-871 x 10 7 
 
 1-876 x 10 7 
 1-940 x 10 7 
 1-897 x 10 7 
 1-874 x 10 7 
 1-899 x 10 7 
 1-878 x 10 7 
 
 5,670 
 5,670 
 5,670 
 5,690 
 5.690 
 5,690 
 5,660 
 
 3030xl0 10 
 3-022 x 10 10 
 2-923xl0 10 
 3-OOOxlO 10 
 3 036 x 10 )0 
 2'996xl0 10 
 3014xl0 10 
 
 Average Value of Velocity S'COSxlO 10 
 
 " With the exception of three preliminary trials, which gave 
 values differing from the mean by 10 per cent, or by 12 per 
 cent., these are the only determinations we have made. In 
 some cases the waves in the circuit were just as good with the 
 spark-gap as without it. In others there was a decided wave 
 formation when sparks occurred, but the node was not quite 
 so well marked. For this reason, and since it did not appear 
 to make any difference in their length, the waves usually 
 were measured without the spark-gap. As the sparks were 
 quite regular, the difference in the bolometer readings must 
 have been due to Faraday tubes that were reflected from the 
 spark-gap without forming a spark and reversing themselves. 
 The variation in the number of revolutions of the mirror per 
 second is due to the fact that different cells were used to drive 
 the motor on different occasions." 
 
 As an example of the data taken to ascertain the position of 
 the node the authors give the following table. The top line 
 contains the distances of the bolometer terminals from a pair 
 of arbitrary fixed points on the circuit : 
 
 " Distances from fixed points 20cm. 40cm. 60cm. 
 
 T4-3 ... 4-0 ... 4-3 
 Bolometer deflections -U'5 ... 41 ... 4'4 
 
 V 
 
 4-5 
 
 Average deflections 4'43 
 
 4-0 
 
 4-03 
 
 4-2 
 4-3'
 
 DYNAMICAL THEORY OF INDUCTION. 513 
 
 The authors conclude with the following remarks : 
 "From these deflections the position of the node was 
 estimated. It appears from the best results that we have 
 obtained that the velocity of short electric waves travelling 
 along two parallel wires differs from the velocity of light by 
 less than O2 per cent, of its value. It has been shown 
 theoretically, that the velocity of such waves travelling along 
 a single wire should be the velocity of light approximately. 
 Our results, therefore, in a certain sense confirm the theory to 
 an accuracy within their probable error. Theoretically, too, 
 the velocity should be approximately equal to the ratio between 
 the two systems of electrical units. The average of the best 
 measurements of this ratio is 3-001, which is nearer the 
 average velocity obtained by us than it is to the velocity of 
 light." 
 
 T.7'
 
 CHAPTER VI. 
 
 THE INDUCTION COIL AND TEANSFORMEE. 
 
 1. General Description of the Action of the Transformer 
 or Induction Coil. In the previous chapters we have prepared 
 the way, by a general study of the phenomena of the induction 
 of electric currents, to enter upon a particular examination 
 of the structure and action of the induction coil and trans- 
 former. The most logical method of procedure would be to 
 trace first the historical development of these appliances from 
 the initial scientific principles and facts accumulated by the 
 early investigators. It will, however, be more advantageous 
 to the student to defer this historical survey to a later portion 
 of this treatise, and to direct attention at present to the actual 
 electrical and magnetic operations which go on in the induction 
 coil and transformer. 
 
 The induction coil and transformer, or converter as it is 
 sometimes called, consists essentially of two conducting 
 circuits which are both linked with a third or magnetic 
 circuit, the three circuits being called respectively the 
 primary circuit, the magnetic circuit, and the secondary 
 circuit. The magnetic circuit may consist wholly of material 
 having a magnetic permeability equal to that of air. A core 
 of this kind may be obtained by winding the primary and 
 secondary circuits on a ring of wood or on a paper tube, but 
 whatever may be the exact material used, a transformer 
 having a core made of a material, the magnetic permeability 
 of which is equal to that of air, is generally called an air 
 core transformer. Not very much interest attaches to the 
 actions of an air-core transformer, for the reason that all 
 practically-used transformers possess magnetic circuits con-
 
 THE INDUCTION COIL AND TRANSFORMER. 515 
 
 sisting either partly or wholly of iron. If the magnetic 
 circuit consists wholly of iron, the transformer is called a 
 closed-circuit transformer; and if it consists partly of iron 
 and partly of air, or other material of unit permeability, it is 
 called an open-circuit transformer. The ordinary induction 
 coil is of this last type. It has a core formed of a bundle of 
 iron wires, and the magnetic circuit lies partly through this 
 core and partly through the air outside it. The two conduct- 
 ing circuits consist generally of copper wires or bands insu- 
 lated in various ways and wound on the core in sections or in 
 overlying coils. In the chapter devoted to the practical con- 
 struction of the transformer, the various methods of carrying 
 this into effect will be described ; meanwhile it will suffice to 
 state that the two circuits, which are called respectively the 
 primary and the secondary circuits, are well-insulated con- 
 ducting circuits, the several turns of which are insulated from 
 each other, the two circuits as a whole being also carefully 
 insulated. The number of convolutions of each circuit may 
 be, and generally is, very different. These are briefly spoken 
 of as the primary turns and secondary turns. The iron core 
 is constructed of laminated iron or iron wire, and the thick- 
 ness or diameter of this is most usually about -013 or -014 of 
 an inch. The object of this lamination is to prevent the pro- 
 duction of local electric currents, called eddy currents, in the 
 iron, which would represent an energy loss ; but, as previously 
 explained, this lamination does not, of course, prevent the 
 hysteresis loss caused by the reversal of the magnetisation of 
 the core. 
 
 The general action of the transformer consists in the pro- 
 duction of a current, called a secondary current, by means 
 of the variation in the magnetic induction in a magnetic 
 circuit linked with it, and this induction is produced by 
 means of another current called a primary current, the 
 variation of the primary current producing a change of 
 magnetic induction in a core, or magnetic circuit, which in 
 turn creates an electromotive force in the secondary circuit 
 linked with it. 
 
 Assuming that periodic currents are employed, it is 
 evident, also, that the relative number of primary and 
 secondary turns will be an important factor in determining 
 
 LL2
 
 516 THE INDUCTION COIL AND TRANSFORMER. 
 
 the ratio between the mean-square value of the potential 
 difference across the primary terminals and that across the 
 secondary terminals of the transformer, and that it is in our 
 power to increase or diminish this ratio. It is of course 
 obvious, also, from first principles, that there can be no creation 
 of energy, but only a transformation, and we can only alter 
 the potential difference of the terminals of the two circuits at 
 the expense of a change of corresponding current strength. 
 
 The most fundamental and valuable quality of an induction 
 coil or transformer is, then, that it enables us to increase or 
 reduce electrical potential difference or current strength in a 
 definite ratio, and it is this transformation of energy which 
 gives the apparatus its name. Transformers may therefore 
 be classified according to the nature of the change in the 
 character of an electric energy supply they are intended to 
 produce. 
 
 Transformers may be constructed to act as (1) constant- 
 potential transformers, or (2) constant-current transformers, 
 and these may furthermore be divided into step-up transformers 
 or step-down transformers, according as they are designed to 
 increase or diminish in a certain ratio a potential difference 
 or a current. Thus a transformer may be designed to work 
 off a circuit of constant potential difference and to reduce 
 that pressure in a certain ratio, called the transformation 
 ratio. If it lowers the pressure it would be called a step- 
 down constant-pressure transformer. In the same way a 
 transformer may be employed to change a current strength 
 in a certain ratio, or to convert from constant pressure to 
 constant current. 
 
 The ordinary induction coil is a step-up transformer as 
 generally used. 
 
 It is unnecessary to make any special classification depend- 
 ing on the character of the change of current employed in 
 varying the induction, but it will be obvious to the reader 
 that a closed iron-circuit transformer can only be used with 
 alternating currents, and that for use with interrupted currents, 
 as in the case of the ordinary induction coil, an open iron 
 circuit or air core transformer must be employed. 
 
 Hence we have the following classification of transformers, 
 understanding by this term any electrical arrangement con-
 
 THE INDUCTION COIL AND TRANSFORMER. 517 
 
 sisting of two conducting circuits, both linked with a magnetic 
 circuit, and in which the variation of current in one circuit 
 gives rise to a production of electromotive force in the other : 
 1. Transformers may be 
 
 (a) Iron core transformers, with core wholly or partly of 
 
 iron; 
 
 (b) Air core transformers, with core wholly of non- 
 
 magnetic substance. 
 Iron core transformers may be 
 
 (c) Closed iron circuit transformers, with core wholly of 
 
 iron; 
 
 (d) Open iron circuit transformers, with core partly of 
 
 iron. 
 
 Transformers may be used 
 
 (e) To transform a potential difference in a constant ratio, 
 
 called constant-pressure transformers ; 
 (/) To transform a current strength in a constant ratio, 
 
 called constant-current transformers. 
 Transformers may be employed 
 
 (g) To raise pressure or current, called then step-up 
 
 transformers ; 
 (li) To lower pressure or current, called then step-down 
 
 transformers. 
 
 The above is not a complete or exhaustive classification, 
 but is sufficient to mark out broadly the various forms of 
 the apparatus. 
 
 In whatever form it is used, the primary current must give 
 rise to a variation of the magnetic induction in the core, and 
 this in turn gives rise to an electromotive force in the 
 secondary circuit. 
 
 The transformer or induction coil can evidently be operated 
 either by intermittent, continuous, or by alternating currents. 
 Whichever mode is adopted, the instrument is, of course, 
 essentially and merely an energy-translating device. The 
 current passing through the primary circuit magnetises the 
 core. The intermittance or reversal of the primary current 
 causes a variation or reversal of the magnetisation of the 
 core. The variation or reversal of the magnetic induction 
 in the core creates an electromotive force in the secondary
 
 sis fna INDUCTION COIL A~$D 
 
 circuit which is linked with it, and this sets up a secondary 
 current in the secondary circuit if it is closed. The energy 
 supplied to the primary circuit partly reappears in the secondary 
 circuit, and the difference is represented by energy losses, called 
 the copper losses, caused by the resistance of the conducting 
 circuits, and partly by energy losses in the core, called the 
 iron losses, and due to the hysteresis and eddy-currents set up 
 in it. 
 
 The complete examination of the transformer involves, 
 therefore, a knowledge of the manner in which the variation 
 of the primary current, the magnetic induction in the core, 
 the secondary current, and the secondary terminal potential 
 difference is taking place when a certain assigned and varying 
 primary terminal potential difference is created. It involves, 
 also, a knowledge of the magnitude of the energy losses above 
 described, and of the efficiency of transformation or the ratio 
 between the power given to the external secondary circuit and 
 the power given to the primary circuit. In addition to this, 
 the relation of the values of the primary and secondary 
 currents, and the primary and secondary terminal potential 
 differences, for various states of the transformer from no load 
 to full load, has to be ascertained. The following is, then, 
 a summary of the operations going on in the transformer or 
 induction coil which must be known before we can consider 
 the action as fully understood ; and the problem of transformer 
 construction is to predetermine these variables from certain 
 data, so as to foretell the result of construction. Given a 
 potential difference created between the primary terminals 
 of the transformer following any assigned law of variation, 
 we shah 1 have the following effects taking place as a conse- 
 quence, and the practical problem is to determine or predeter- 
 mine their mode and magnitude. 
 
 (1.) We have a primary current in the primary circuit 
 following a certain mode of variation in strength, and causing 
 a definite copper loss in the primary circuit ; 
 
 (2.) A magnetic induction in the core following a definite 
 mode of variation, and having a certain magnitude at every 
 instant, this magnetic induction causing, by its variation, 
 iron core losses due to hysteresis and eddy currents in the 
 iron core :
 
 THE INDUCTION COIL AND TRANSFORMER. 519 
 
 (3.) A secondary current and secondary terminal potential 
 difference following some definite law of variation and accom- 
 panied by a copper loss in the internal secondary circuit due 
 to its resistance and a contribution of power to the external 
 secondary circuit. 
 
 The transformer problem in all its completeness would be 
 solved if we could in all case predetermine the above effects 
 from the known primary potential difference. This, however, 
 is not capable of being effected in a perfect manner, for reasons 
 presently to be stated. 
 
 In early discussions of the transformer problem it was 
 customary to make arbitrary assumptions as to the mode of 
 the variation of the currents, the induction and potential 
 differences. These artificial assumptions did not, however, 
 assist real knowledge. The only useful method is to 
 endeavour, in the first place, to ascertain what does go on 
 inside the transformer, and then, on the basis of this analysis, 
 to construct as far as possible a true working theory of the 
 transformer. We have accordingly abandoned all discussion 
 of imaginary transformers with air cores and currents and 
 inductions, which are simple sine functions of the time and 
 base, for such theory as we are able to build up on an actual 
 knowledge of what does take place in the transformer. The 
 only scientific method of treating the problems involved is, 
 we repeat, first to endeavour to ascertain what are the actions 
 really taking place, and to make them the basis for further 
 reasoning. 
 
 2. The Delineation of Periodic Curves of Current and Elec- 
 tromotive Force. The method which has proved most fertile 
 in enabling us to understand the operations taking place in 
 the transformer is that which consists in graphically repre- 
 senting the form and relative position of the curves of periodic 
 current and electromotive force in the two circuits and 
 deducing that of the magnetisation of the core. When the 
 primary electromotive force is an alternating one, derived 
 from a single alternating-current dynamo which is accessible, 
 the method of obtaining a graph of the various periodic 
 quantities required is some modification of the arrangement
 
 520 THE INDUCTION COIL AND TllANSFOBMER. 
 
 first suggested by Joubert,* in which a circuit is closed for 
 a very short but assigned period during the phase, and puts 
 some electrical instrument intermittently into connection 
 with the circuit, so that it reads, not the mean-square value of 
 the current or potential difference, but the instantaneous value 
 at the assigned instant. The modern method of delineating 
 transformer curves is as follows : 
 
 Let the ordinates of the periodic curve in the Fig. 168 
 represent the varying potential difference between two con- 
 ductors connected to an alternator, and let a condenser be 
 connected across the circuit in series with a switch. If the 
 switch is permanently closed, the condenser has a flow of 
 current into and out of it, and the potential difference of its 
 
 FIG. 168. 
 
 terminals varies periodically. If, however, the switch is closed 
 intermittently at intervals which are equal to the periodic time 
 of the alternator, then the condenser has a series of short con- 
 tacts made with it, and its terminal potential difference is 
 equal to the instantaneous value of the periodic potential 
 difference of the circuit corresponding to the instant when the 
 contact is broken. The difference of the potentials of terminals 
 of the condenser has then to be determined. Several methods 
 may be adopted. The condenser plates may be connected to 
 the terminals of an electrostatic voltmeter, and the potential 
 difference of the condenser plates thus determined. The con- 
 denser may be discharged through a galvanometer, and the 
 
 * Comptcs Rendus of the Academy of Science, France, Vol. XCL, July, 
 1380, p. 161.
 
 THE INDUCTION COIL AND TRANSFORMER. 521 
 
 condenser plate discharge determined by the quantity of the 
 charge found in the condenser. We can thus charge the 
 condenser by a series of contacts made with the circuit, for a 
 very short time, always at the same position during the com- 
 plete cycle or period of the varying potential difference. The 
 condenser acquires, after a short time, a potential difference 
 between its terminals which is exactly equal to that of the 
 instantaneous value of the potential difference of the circuit at 
 the instant, when the contact is made. If, instead of connect- 
 ing the condenser across the circuit, it is connected to the 
 extremities of a suitable non-inductive resistance inserted in 
 any alternating - current circuit, we can obtain from its 
 terminal potential difference the instantaneous values of the 
 periodic current. 
 
 We have next to consider the various practical details of 
 the process. If the alternator is accessible, or if a single 
 alternator is providing the current, then the intermittent 
 contact may be made by an apparatus fixed on the shaft of 
 the alternator. If the alternator is not accessible, and if the 
 primary potential difference is derived from a battery of 
 alternators running in parallel, then it is necessary to operate 
 an intermittent contact by means of a synchronous alterna- 
 ting-current motor, driven from that part of the circuit which is 
 accessible. As this last method is capable of so much more 
 general application than that of the contact-maker driven off 
 the shaft of the alternator, we shall describe in some detail the 
 arrangement of a suitable motor and associated apparatus. 
 In addition to tbe early experiments by Joubert above 
 mentioned, the delineation of periodic curves of current and 
 electromotive force by means of an intermittent contact on 
 the shaft of the alternator was suggested and carried out by 
 Dr. Louis Duncan, and experiments by this method were effec- 
 tively conducted by Messrs. Duncan, Hutchinson and Wilkes, 
 and also by Prof. Eyan hi the United States.* It has also 
 been largely employed by Dr. J. Hopkinson, M. Blondel, by 
 
 * See a Paper by Messrs. Duncan, Hutchinson and Wilkes in the 
 Electrical World of New York for March, 1888, referred to in The 
 Electrician, Vol. XXL, p. Ill, June 1, 1888. Also see Prof. Harris J. Kyan 
 on Transformers in the Transactions of the American Institute of Elec- 
 trical Engineers, Vol. VIL, January, 1890.
 
 522 THE INDUCTION COIL AND TKANSFORMER. 
 
 the author, and by many others for alternating current 
 researches. The general arrangement of the apparatus 
 required for the complete study of the periodic quantities in 
 a transformer, under any circumstances, is as follows : The 
 principal instrument required is a small alternating-current 
 synchronous motor. A suitable form has been devised and 
 used by the author in investigations of this character,* and 
 a view of it is shown in Fig. 169 on next page. 
 
 The general details of the machine are as follows: The 
 motor, as constructed by the author, consists of two sets of 
 field magnets, MM, which are secured to two cast-iron discs. 
 Between these field magnets revolves a small armature, A, the 
 iron core of which is formed of a strip of very thin transformer 
 iron, wound up into a ring, the armature coils being wound 
 upon this ring. The armature coils are joined up in series 
 with one another, so as to give a series of contrary polarities 
 round the iron ring. The diameter of this armature is about 
 Gin. The field magnets have eight poles, and the armature 
 eight coils. The field-magnet cores are bobbins about 2in. 
 long and l|in. in diameter, and when joined up in series 
 in the proper manner the field magnets take a current of about 
 4 amperes to give them the proper amount of saturation. 
 The armature is carried upon a hard wood boss fixed to a 
 steel shaft. This steel shaft is carried through small ball 
 bearings like bicycle bearings, the shaft being borne upon 
 seven or eight balls carried in gun-metal cells. In order 
 to prevent any side shake of the armature, there are at the 
 opposite ends of the base cast iron pillars with a gun-metal 
 screw at each end, against which the rounded end of the 
 shaft bears. The shaft can thus be adjusted with great nicety, 
 and runs with great freedom from friction. The ends of the 
 armature coils are brought to two small insulated collars, 
 fixed on the shaft, against which press two light brass brushes, 
 marked B B, kept gently against the collars by means of an 
 expanding steel wire, W. On the armature shaft is an ebonite 
 disc, which carries a transverse steel slip let into it. Two 
 insulated springs, S S, are carried upon a rocking arm, H ; 
 the rocking arm can be traversed over through half a 
 
 * See The Electrician, Vol. XXXIV., p. 460, February 15, 1895, " On the 
 Delineation of Alternating Current Curves."
 
 THE INDUCTION COIL AND TRANSFORMER. 523 
 
 circumference, and is centred upon the gun-metal end screw, 
 which prevents side shake in the shaft. A pointer and 
 
 graduated scale, G, enables the exact angular position of the 
 contact springs, S S, to be determined.
 
 524 THE INDUCTION COIL AND TRANSFORMER. 
 
 One of the springs, S, is carried on a small adjusting screw, 
 so that one spring can be given a little lead over the other, and 
 in this manner the duration of the contact made when the 
 steel transverse piece passes underneath and electrically con- 
 nects the springs S S is determined. By means of a set screw 
 the springs can be lifted off from the ebonite disc, and their 
 pressure also adjusted. This little synchronising motor with 
 its attached contact-breaker forms the apparatus for determin- 
 ing the form of the current and electromotive-force curves. 
 The motor is started in step with the alternating current 
 flowing through the armature coils by passing round the end 
 of the steel shaft which projects at the end opposite to the 
 contact-breaker a tape or thin leather strap sprinkled with 
 rosin. To start the motor the following arrangements are 
 made : The field magnets are excited by current obtained 
 from a small secondary battery, or from any other constant 
 source of continuous current. The armature circuit requires 
 about 2 amperes to make it run properly. Let us assume that 
 the potential difference curve is to be taken from two 100 volt 
 alternating-current mains which come into a building. The 
 armature of the motor is joined across these mains in series 
 with two or three incandescent lamps placed in parallel. The 
 field magnets being excited in the proper direction by a con- 
 tinuous current from a few secondary cells, the operator passes 
 the strap or tape half round the shaft, and by pulling on one 
 side of the tape the motor can gradually be set in rotation 
 with an increasing speed. If the frequency of the alternating 
 current is, say, 100 c\j , then the 8-pole motor has to be brought 
 up to run at something approaching to 1,500 revolutions per 
 minute before it will drop into step ; but at a certain speed 
 the incandescent lamps in series with the armature begin 
 to blink, and by a little skill in adjusting the speed by 
 suitable pulls on the tape the motor will drop into step 
 and continue to run in synchronism with the circuits. If 
 the springs are then put down gently upon the revolving 
 contact piece, a contact is made from one spring to the other 
 at an assigned position during the phase of electromotive 
 force, depending on the position of the rocking arm. If the 
 maximum electromotive force to be read does not exceed 160 
 volts, then by far the most convenient instrument to employ
 
 THE INDUCTION COIL AND TRANSFORMER. 525 
 
 for reading the electromotive force, and the one which has been 
 constantly employed in these tests, is Lord Kelvin's vertical 
 or horizontal pattern multicellular voltmeter. As these volt- 
 meters only begin to read at about 60 or 80 volts, it is 
 necessary to add a constant electromotive force in series with 
 them, and this is done by employing a set of small secondary 
 cells. About fifty cells in one tray form a convenient arrange- 
 ment, provided they have contacts at every cell, so as to take 
 off any required electromotive force. The battery is joined up 
 in series with the electrostatic voltmeter, and the terminals of 
 the voltmeter are short-circuited by a condenser having a 
 capacity of about half a microfarad. This arrangement of volt- 
 meter and battery is then connected across the two points 
 between which the potential is to be determined through the 
 two springs S S. The motor being started, the needle of the 
 voltmeter takes a certain deflection, which is due to the electro- 
 motive force of the cells, plus the value of the difference of 
 potential between the mains at an instant depending upon the 
 position of the rocking handle. By blocking up the voltmeter 
 in this way, and using more or less cells as required, so as to add 
 a known amount to the electromotive force to be measured, the 
 electrostatic voltmeter can be employed to measure potential 
 differences over the whole range varying from zero to 160 
 volts in either direction. These observations are taken at 
 equal short intervals as the rocking arm H is swept over 
 through a quarter of a circle. It is possible to thus measure 
 the instantaneous values of the alternating potential difference 
 between the two points at equi- distant instants throughout 
 the phase. It has been found by experiment that this small 
 alternating-current motor, when working on the circuits of 
 any alternator of a size such as would be used in a generating 
 station, does not sensibly affect the form of the curve of electro- 
 motive force. The motor is only used as a means of making 
 the contact with a voltmeter at an assigned instant during the 
 phase. The current which passes through its armature is not 
 in any way measured or taken account of ; the motor simply 
 acts as a synchronising arrangement, which connects the 
 contact-breaker electrically to the distant alternator. 
 
 The synchronising motor can, therefore, be set to run in 
 step with any alternating-current circuit, and to make a
 
 526 THE INDUCTION COIL AND TRANSFORMER. 
 
 contact or close a circuit for a short instant during every 
 period at an assigned instant in the phase, which depends 
 on the position of the rocking arm carrying the contact- 
 making springs. 
 
 This apparatus may be employed to determine any of the 
 curves of current or potential of a transformer, as follows : 
 Let us suppose the transformer is one intended to be operated 
 with a primary terminal potential difference of 2,000 volts, 
 and that the secondary terminal potential difference is 100 
 volts. Across the primary terminals of the transformer a non- 
 inductive resistance is connected, which is divided into two 
 sections in the ratio of 1 to 19, and in series with the primary 
 circuit of the transformer is placed another non-inductive 
 resistance having such a magnitude that, when traversed by 
 the primary current of the transformer, it will create a fall 
 of potential of about 100 volts. The synchronising motor is 
 then suitably arranged to be operated from the same circuit 
 which supplies the primary current for the transformer, and 
 the armature circuit of the motor may be fed through a step- 
 down transformer, which reduces this circuit pressure to a 
 convenient magnitude. 
 
 The motor contacts are then arranged to close the circuit of a 
 voltmeter, which is placed across one or other of the resistances. 
 It is found necessary to connect a condenser across the termi- 
 nals of the voltmeter to increase its capacity, or else the 
 leakage of the voltmeter in the intervals between the moments 
 when the contact is made causes irregularity and uncertain 
 deflections of the instrument. The process of getting the 
 complete set of curves of current and electromotive force of a 
 transformer is then as follows : The curve of primary potential 
 difference is obtained by connecting the voltmeter through the 
 motor contacts across the smaller section of the divided resis- 
 tance which bridges over the primary terminals. The motor 
 being started, the voltmeter will read a potential difference, 
 which is one-twentieth of the whole primary potential diffe- 
 rence, and if the rocking arm of the motor is moved over step- 
 by-step the indications of the voltmeter will successively give 
 the values of this fraction of the primary potential differ- 
 ence corresponding to the different intervals of the whole 
 period.
 
 THE INDUCTION COIL AND TRANSFORMER. 527 
 
 In the same way the curve of primary current can be 
 obtained by connecting the voltmeter circuit across the termi- 
 nals of the resistance inserted in series with the primary 
 circuit of the transformer. The curves of secondary potential 
 difference and secondary current, if necessary, can be obtained 
 by connecting the contact-maker and voltmeter across the 
 secondary terminals of the transformer when closed by a 
 known non-inductive resistance. 
 
 The curves of current and potential thus obtained can be 
 set down in a chart, the horizontal abscissae in which repre- 
 
 4000 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 .. 
 
 /~ 
 
 
 
 ^0 
 
 )0 
 
 \ 
 
 
 
 
 
 
 
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 20( 
 
 \ 
 
 )Q 
 
 \ 
 
 
 
 
 
 
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 1 
 
 * 9 
 
 
 100 
 
 
 
 A 
 
 
 
 
 
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 ^ 
 
 ( 
 
 V- 
 
 \ 
 
 
 
 ^ 
 
 
 
 
 7 
 
 S 
 
 
 
 
 
 % 
 
 >. 
 
 3 
 
 \ \ 
 
 
 
 
 
 
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 1 
 
 
 
 
 
 100 
 
 : 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
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 w 
 
 
 
 
 
 20( 
 
 )0 
 
 % 
 
 
 ^ 
 
 
 
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 / 
 
 
 
 4.QOO 
 
 
 
 
 30 
 
 DO 
 
 
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 30 60 90 120 150 18 
 
 210 2^ 
 
 970 300 33 
 
 o se 
 
 
 FIG. 170. Primary Current, Primary Terminal, Potential Difference, 
 ind Induction Curves of Ganz 10 H.P. Transformer taken off a Kapp 
 Alternator by the Alternating-current Curve Tracer. 
 
 sent fractions of the complete periodic time, and the vertical 
 ordinates represent the instantaneous values of the potential 
 differences or currents. 
 
 In Fig. 170 is shown a set of curves taken from a Ganz 
 transformer connected to a Kapp alternator. The dotted curve 
 marked volt curve is the curve of primary electromotive force,
 
 628 THE INDUCTION COIL AND TRANSFORMER. 
 
 or difference of potential at the primary terminals of the 
 transformer. The dotted curve marked current curve is the 
 curve of primary current, and the dots show the actual posi- 
 tion of the observations. The figures on the horizontal line 
 indicate degrees of phase. The scale on the left-hand side is 
 the scale of volts, and that on the right-hand side the scale 
 of current. From the curves of current and potential differ- 
 ence we can obtain the curve of magnetic induction in the 
 core as follows : Let b be the induction density in the 
 iron core that is, the number of lines or unit tubes of induc- 
 tion per square centimetre of cross-section of the core. This 
 induction density will not in general be the same in all 
 parts of the core or the same at full load as at no load. If, 
 in the first place, we consiJer the case of the transformer 
 when the secondary circuit is open, we have a definite relation 
 between the current and the primary circuit, the magnetic 
 induction in the core, and the primary terminal potential 
 difference or electromotive force at any instant. Let i be the 
 instantaneous value of the current in the primary circuit, e 
 the instantaneous value of the primary potential difference, 
 and b the induction density in the core. If N x is the number 
 of turns of the primary circuit, B the resistance of the primary 
 circuit, and S the area of cross-section of the core, then from 
 the ordinary current equation for inductive circuits we have 
 the relation 
 
 1 dJ 
 or b = 
 
 In most cases of closed iron circuit transformers, the second 
 term or integral on the right-hand side of the last equation is 
 a very small quantity compared with the first term, and may 
 be neglected. Hence when/id* is small we can obtain the 
 value of b by integrating the primary E.M.F. curve, or by 
 finding the value offedt between proper limits. Take, for
 
 THE INDUCTION COIL AND TEANSFOEMEE. 529 
 
 instance, the case of the Ganz transformer, the curves of 
 which are given in Fig. 170. The resistance of the primary 
 circuit E is 2-5 ohms and the maximum value of the primary 
 current i is nearly 0-34 amperes. Hence the value of Rz 
 never exceeds 0-85 of a volt. The value of e, the primary 
 electromotive force, varies from to nearly 3,000 volts, 
 and hence at any instant, except very near the moment 
 when e is zero, the value of R i is quite negligible compared 
 with that of e. 
 
 In order to obtain the induction curve we have to integrate 
 the curve of primary electromotive force, and to do this 
 properly the following procedure must be followed. The 
 whole area included by the primary E.M.F. curve must be 
 obtained, and the integration of the curve must be started 
 from that point on the horizontal axis which corresponds 
 with the bisection of the area of the E.M.F. curve. Starting 
 from this point the area of the E.M.F. curve is obtained by 
 successive increments, and corresponding to the limit of each 
 increment an ordinate is set up whose length on some scale 
 is proportional to the whole area of the E.M.F. curve measured 
 from the abcissa corresponding to the semi-area of the curve 
 to the limit considered. This ordinate will then be an 
 ordinate of the curve of induction. In making this integra- 
 tion the area of the E.M.F. curve below the time axis must 
 be reckoned as negative. To obtain the absolute value of the 
 induction at any point, the area of the E.M.F. curve must be 
 reckoned out in volt-seconds and then divided by the value of 
 Nj S, S being measured in square centimetres. The result 
 must be multiplied by 10 8 to reduce to G.G.S. measure and 
 give the induction density in C.G.S. units. 
 
 In this manner the firm line curve which is marked induc- 
 tion curve in Fig. 170 was obtained. If the curve of secondary 
 terminal potential difference has been obtained, we can, by 
 a similar integration of this curve, obtain another induction 
 curve which is generally practically identical with that obtained 
 from the primary E.M.F. curve if the transformer secondary 
 circuit is unloaded, but which does not agree with it if the 
 secondary circuit is closed and a secondary current is being 
 produced therein. Into the causes of this we shall enter 
 later. The full set of transformer curves for the currents, 
 
 MM
 
 530 THE INDUCTION COIL AND TRANSFORMER. 
 
 potential differences and induction constitutes what may be 
 called the indicator diagram of the transformer, and shows us 
 all that is going on inside. The quick description of these 
 curves becomes, therefore, an important matter. Many investi- 
 gators have devised methods for expediting this process. One 
 effective method was described by M. A. Blondel* in 1891. 
 M. Blondel employs a rotating contact-maker with two brushes, 
 and the contacts are so arranged that a condenser is periodically 
 charged at a certain moment during the complete phase of 
 the potential and then immediately afterwards is discharged 
 through a galvanometer. The two brushes are fixed to an 
 arm movable about an axis co-axial with that of the revolving 
 motor or alternator, and this brush holder is revolved by 
 clockwork at a regular rate. Hence the galvanometer indicates 
 a current which is varied as the brush holder rotates. If the 
 brush holder is held at rest, the galvanometer has a series of 
 rapid charges from the condenser sent through it, and takes 
 a steady deflection. If the brush holder rotates, this deflection 
 varies from moment to moment, but at any instant is pro- 
 portional to the instantaneous potential at which the condenser 
 is being charged. If a mirror d'Arsonval galvanometer is 
 employed, and the image of an illuminated opening thrown 
 on a photographic scale which is moved transversely to the 
 motion of the spot of light, a photographic trace of the alter- 
 nating-current curve can be obtained. By using a pair of 
 contact-makers and two galvanometers the current and E.M.F. 
 curves can be delineated at the same time. Such a photo- 
 graphic record of the current and E.M.F. curve for an 
 alternating-current arc lamp worked off a Meritens alternator 
 is shown in Fig. 171. 
 
 A very similar arrangement has been described by Messrs. 
 Barr, Burnie and Eodgers.f These investigators employ a 
 revolving contact -maker of a particular kind. It is thus 
 described by them : The shaft of the alternator or motor is 
 fitted with a contact-making disc, and the contact brush is 
 moved slowly and continuously through its successive angular 
 positions. Contact is thus made each time at a slightly 
 
 * See La Lumiere Electrique, September 12, 1891, and September 16, 
 1893 ; also see The Electrician, Vol. XXVII., p. 603. 
 t See The Electrician, September 27, 1895.
 
 THE INDUCTION COIL AND TRANSFORMER. 531 
 
 different position of the armature. Thus the potential 
 difference at each contact differs slightly from that at the 
 preceding contact. This potential difference is used to charge 
 a condenser across the terminals of which is connected either 
 a reflecting electrometer or a high-resistance galvanometer. 
 
 The deflection of the instrument so used follows the value 
 of the potential difference of the wave form to be determined, 
 and accurately follows it, for the mean rate of variation of the 
 potential differences between the terminals of the condenser is 
 exceedingly small in comparison with the rate of change of the 
 electromotive force to be investigated. 
 
 FIG. 171. Photographic Trace of Current Curve I and Electromagnetic 
 Force Curve E of an Alternating Current Arc Lamp. 
 
 Fig. 172 shows the arrangement of the contact disc and 
 accessories, a galvanometer being used, but for which an 
 electrometer might be substituted. In this diagram, for the 
 sake of clearness, the vulcanite foundation work is omitted and 
 the brass only shown. ~D l is the contact disc, with knife edge 
 a'ad contact brush, which is rigidly fixed to the shaft of the 
 dynamo or motor. The rings D. 2 and D 3 , and the rods and 
 brushes Bj and B 4 , are mounted on a vulcanite sleeve loose 
 
 M M 2
 
 632 THE INDUCTION COIL AND TRANSFORMER. 
 
 upon the shaft, and are revolved slowly. The brush Bj is 
 joined to the ring D 2 , with which the brush B 2 is in permanent 
 connection, so that when the brush B! makes contact with 
 the knife edges the condenser G is charged to the potential 
 difference between the terminals T^ T 2 . The condenser is 
 throughout the whole revolution of the disc D x discharging 
 through the galvanometer G by way of ~B lt B 4 , B 5 , and the 
 resistance K. 
 
 As the brushes Bj and B 4 are moved slowly round, a suc- 
 cession of charges passes through the galvanometer, the value 
 of each of which is proportional to the potential of the con- 
 denser that is, to the potential difference of the points T lt T 2 . 
 This contact apparatus, in fact, performs the operation of 
 
 FIG. 172. 
 
 charging a condenser at a definite instant during the period 
 at the terminals T x and T a , which are the terminals of the 
 alternating-current circuit under investigation, and then 
 immediately afterwards discharges this condenser through a 
 galvanometer. The galvanometer, therefore, gives a steady 
 deflection which is proportional to the instantaneous potential 
 difference between the points T! and T a at the instant 
 corresponding to the moment when the contact with the 
 condenser is broken. 
 
 The rapidity with which the curves of instantaneous 
 potential can be determined depends to a large extent upon 
 the perfection of the insulation of the condenser and voltmeter.
 
 TEE INDUCTION COIL AND TKANSFORMEB. 533 
 
 If these leak to any sensible degree they lose charge in the 
 intervals between the contacts, and the resulting permanent 
 deflection is too small, and the time required for the volt- 
 meter to take its full steady deflection when the place of 
 contact is changed is greatly increased. Hence it is necessary 
 to examine this question of leakage carefully before placing 
 implicit reliance on the voltmeter and condenser actually 
 used. 
 
 The value of the instantaneous potential may also be 
 determined by balancing it against some point on a slide wire 
 down which a known fall of potential is created by a battery. 
 The arrangement known as a potentiometer consists of a uni- 
 form fine wire stretched over a scale down which a uniform 
 fall of potential is created by a cell or two of a secondary battery 
 attached to its extremities. If a sliding contact moves over 
 this wire, we can insert between one end of the potentio- 
 meter wire and this slider any source of electromotive force, 
 and, by moving the slider, balance the fall of potential down 
 any length of the slide wire against this other potential 
 difference. If the revolving contact-maker, connected in series 
 with a condenser, is placed across a proper section of a 
 divided resistance, which resistance is across the terminals of 
 the transformer, the contact-maker will close the circuit of the 
 condenser at equal periodic intervals and give it a potential 
 which depends upon the position of the contact of the contact- 
 maker. The potential of this condenser can then be measured 
 on the slide wire, and, knowing the value of the two sections 
 of the divided resistance, we are able to determine the value 
 of the instantaneous potential difference between the terminals 
 of the transformer. 
 
 A revolving contact-maker for determining alternating- 
 current and potential curves has also been devised by Prof. 
 Hicks.* In this instrument the same principle is adopted as 
 in the one just described. A revolving contact-maker connects 
 a condenser intermittently, but at definite instants in the period, 
 to a source of alternating potential, and then in between these 
 contacts discharges the condenser through a galvanometer. 
 The shifting of the brush contacts varies the galvanometer 
 deflection, but so that it is always proportional to the instan- 
 * See The Electrician, Vol. XXXIV., 1895, p. 698.
 
 534 THE INDUCTION COIL AND TRANSFORMER. 
 
 taneous value of the potential charging the condenser. Many 
 other forms of apparatus have been described, but the 
 principles are practically the same as those above referred to- 
 In all cases a condenser is charged through a contact-maker, 
 and the potential of the condenser determined by either a 
 galvanometer or voltmeter. A few practical suggestions 
 in connection with the construction of such contact-makers 
 may be useful. In the first place, if a condenser and 
 electrostatic voltmeter are used, care must be taken to see 
 that both are very highly insulated. The use of the 
 condenser is to act as a reservoir and supply the electrical 
 leakage of the voltmeter. If a condenser of large capacity 
 is employed, then the contact must be suitably prolonged, or 
 else the condenser will not be charged completely during 
 the contact. The contact springs sometimes give trouble by 
 making imperfect contact with the disc, and too much pressure 
 must not be applied to the brushes, or else they create a trail 
 of metallic deposit on the insulating disc. The author has 
 found a material called stabiiit a very suitable insulating 
 material for the construction of the insulating disc of the 
 contact-maker, and the contact piece may be a transverse slip 
 of steel or hard brass let into it. The contact springs are best 
 made of steel, tempered and well cleaned at the contact 
 surfaces. The contact-maker is best constructed by attaching 
 a circular disc of stabiiit or ebonite to the shaft of the motor 
 or alternator and turning it up very accurately on the shaft. 
 The metal contact slip is then let into the disc and a pair of 
 insulated springs are carried on a rocking arm which moves 
 round an axis co-axial with that of the motor or alternator. 
 As the disc revolves the contact slip passes under the springs, 
 and connects them together for an instant. These springs are 
 connected to the circuit of the voltmeter and condenser, so 
 that when the contact is made between the springs the con- 
 denser and voltmeter in parallel with it are connected to the 
 alternating circuit under test for a short instant. The instant 
 during the period when the contact is made can be varied by 
 rocking over the arm carrying the springs. 
 
 Prof. Kyan has suggested and employed a jet of salt water 
 as a means of making an electric contact. Through this jet a 
 steel needle passes at an assigned instant during the revolution.
 
 THE INDUCTION COIL AND TRANSFORMER. 535 
 
 With care and proper construction the steel spring contact- 
 maker works very well, and is much more convenient to use 
 than a contact in which a liquid jet is employed. 
 
 M. Blondel has described several forms of instrument, which 
 he calls oscillographs, for the direct representation by optical 
 means of the form of alternating-current curves, enabling us 
 to project on to a screen a luminous line having the form of 
 the alternating current curve. For a description of these we 
 must refer the reader to his Paper in the Comptes Rendus, 
 Vol. CXVL, No. 10, March -6, 1893, p. 502, and to The 
 Electrician, Vol. XXX., March 17, 1893, p. 571. 
 
 3. Discussion of Transformer Diagrams. The methods, 
 some of which have been described in the previous section 
 
 FIG. 173. Curve of Electromotive Force of Thomson-Houston Alternator 
 on Open Circuit.* 
 
 enable us, as it were, to look inside the transformer and observe 
 the nature and order of the electrical operations taking place 
 in it. We shall proceed to discuss some of the experimental 
 results which have been thus obtained. In the first place, it 
 must be noted that the curve of primary potential difference, 
 or as it is generally called, the curve of primary E.M.F., 
 ~*In Figs. 173, 174, 175, the dots represent the actual position of 
 observations.
 
 536 THE INDUCTION COIL AND TRANSFORMER. 
 
 depends upon the construction of the alternator producing the 
 electromotive force, and also upon the nature of the circuit, 
 whether inductive or non-inductive, which that alternator is 
 supplying. Any assumption that the curve of primary 
 potential difference is always a simple sine curve is very far 
 from true. The form of the curve of primary electromotive 
 force is not even a fixed and independent attribute of the 
 alternator. The form of the E.M.F. curve of the alternator 
 may be quite different when taken on open circuit to that 
 which it is when taken at the terminals of the alternator 
 when this last is loaded with an inductive or non-inductive load 
 of transformers. In Fig. 173 is shown the curve of electro- 
 
 FIG. 174. Curve of Electromotive Force of -Thomson-Houston Alternator 
 working on an Inductive Circuit. 
 
 motive force of a Thomson- Houston alternator at no load or 
 on open circuit, and in Fig. 174 the E.M.F. curve of the 
 same machine when actuating a load of transformers, the 
 secondary circuits of which are lightly loaded. It will be 
 seen that the second curve is quite different to the first, 
 and that neither of them is even approximately a simple 
 periodic curve. In Fig. 175 is shown the E.M.F. curve of 
 a Mordey alternator at full load on a water resistance. It is, 
 then, clear that no assumption must be made as to the con-
 
 THE INDUCTION COIL AND TRANSFORMER. 537 
 
 stancy of the form of the curve of electromotive force of any 
 alternator, but that the form of the curve of primary terminal 
 potential difference of the transformer under test must always 
 be determined. 
 
 FIG. 175. Curve of Electromotive Force of Mordey Alternator on 
 Water-Resistance Load. 
 
 2,000 
 
 60 120 180 240 
 
 Degrees of Phase. 
 
 FIG. 176. Primary E.M.F. and Primary Current Curves of Mordey 
 Transformer, on Open Secondary Circuit, supplied off Mordey Alternator, 
 with no other load. 
 
 We have, in the next place, to consider the case of the 
 transformer when the secondary circuit is open or unloaded, 
 and to inquire what under those circumstances is the form and
 
 538 THE INDUCTION COIL AND TRANSFORMER. 
 
 relative position of the primary terminal potential difference 
 curve and the primary-current curve. A number of examples of 
 such curves are given in the diagrams on pages 537 to 541. 
 In Figs. 176 to 183 are shown the primary-current curves 
 and primary E.M.F. curves for transformers on open secondary 
 circuit made by the Brush Electrical Engineering Company 
 
 2,000 
 
 120 180 240 
 
 Degrees of Phase. 
 
 FIG. 177. Primary E.M.F. and Primary Current Curves of Mordey 
 Transformer, 011 Open Secondary Circuit, supplied off Mordey Alternator, 
 furnishing Current also to other transformers lightly loaded. 
 
 2,000 
 1,000 
 
 l : 00t. 
 2,00 
 
 ,'* 
 
 Degrees of Phase. 
 
 FIG. 178. Primary E.M.F. and Primary Current Curves of Thomson- 
 Houston Transformer, on Open Secondary Circuit, supplied off Mordey 
 Alternator, with no other load. 
 
 and the Thomson-Houston Company, the electromotive force 
 being supplied by Mordey or Thomson-Houston alternators 
 in various states of load. The Mordey-Brush transformers
 
 THE INDUCTION COIL AND TRANSFORMER. 539 
 
 are 50 kilowatt size and the Thomson- Houston are 30 kilowatt 
 size. 
 
 It will be seen that the primary current under these 
 conditions always lags behind the curve of primary E.M.F. 
 or primary terminal potential difference. The primary current, 
 vrhen the secondary circuit of the transformer is open, is called 
 
 2,000 
 
 l.OOC 
 
 
 
 l.OOC 
 
 2,000 
 
 6 
 4 
 "2 
 
 
 2-* 
 4 
 6 
 
 Deyreet of Phase. 
 
 FIG. 179. Primary E.M.F. and Primary Current Curves of Thomson- 
 Houston Transformer, on Open Secondary Circuit, supplied off Mordey 
 Alternator, furnishing Current also to other Transformers lightly loaded.' 
 
 2,000 
 1,000 
 
 1,000 
 2.COO 
 
 o eu 120 180 
 
 Degrees of Phase. 
 
 FIG. 180. Primary E.M.F. and Primary Current Curves of Mordey 
 Transformer, on Open Secondary Circuit, supplied off Thomson-Houston 
 Alternator, with no other load. 
 
 the magnetising current of the transformer. Even if the 
 curve of primary potential difference is nearly a true sine 
 curve, the curve of primary current is not of a similar 
 character, but is always more irregular. The form of the 
 primary current curve depends not merely upon the form of
 
 540 THE INDUCTION COIL AND TRANSFORMER. 
 
 the primary E.M.F. curve, but upon the nature of the iron 
 used in the iron core and upon the structure of the trans- 
 former generally, so that the primary -current curves of two 
 transformers by different makers will have different forms of 
 
 3,000 
 
 2,000 
 
 1,000 
 
 
 1,000 
 2,000 
 
 8,000 
 
 300 
 
 360 
 
 60 120 ISO 240 
 
 Degrees of Phase. 
 
 FIG. 181. Primary E.M.F. and Primary Current Curves of Mordey 
 Transformer, on Open Secondary Circuit, supplied off Thomson-Houston 
 Alternator, furnishing Current also to other Transformers lightly loaded. 
 
 2,000 
 1,000 
 
 I 
 
 1,000 
 2,000 
 
 60 
 
 240 
 
 120 180 
 
 Degrees of Phase 
 
 Fia. 182. Primary E.M.F. and Primary Current Curves of Thomson- 
 Houston Transformer on Open Secondary Circuit supplied off Thomson- 
 Houston Alternator with no other load. 
 
 magnetising current curve, even if worked off the same 
 alternator. This is well shown in the curves in Figs. 176 and 
 178, in which a Brush and Thomson-Houston transformer are
 
 THE INDUCTION COIL AND TRANSFORMER. 541 
 
 worked off the same Mordey alternator. The curve of primary 
 E.M.F. is the same in each case, but the curve of primary 
 current is of a quite different form. This is brought about by 
 differences in the reluctance of the iron circuit producing 
 small differences in the form of the curve of magnetic 
 induction in the core. 
 
 By comparing Figs. 176 and 181 it will be seen that the form 
 of the curve of primary current is also dependent upon the 
 form of the curve of primary E.M.F. , and for the same 
 transformer the curves of current may be considerably altered 
 by supplying it off a different alternator, or off the same 
 
 60 
 
 860 
 
 120 180 240 
 
 Degrees of Phase. 
 
 FIG. 183. Primary E.M.F. and Primary Current Curves of Thomson- 
 Houston Transformer, on Open Secondary Circuit, supplied off Thomson. 
 Houston Alternator, furnishing Current also to other Transformers lightly 
 loaded. 
 
 alternator in different states of load. In some alternators, 
 such as the Mordey alternator, the armature reaction is very 
 small and the form of the curve of electromotive force given 
 by the machine is not very different whether the machine 
 is worked on open circuit or on fuU load, on water resistance 
 or on an inductive load. In the case of a machine with large 
 armature reaction, the form of the curve of electromotive 
 force will, under these various conditions, be greatly altered, 
 and hence the form of the primary-current wave of trans-
 
 542 THE INDUCTION COIL AND TRANSFORMER. 
 
 formers on open secondary circuit connected to it will be 
 -quite different also. 
 
 We pass on next to consider the form and position of the 
 curve of secondary terminal potential difference or secondary 
 E.M.F. when the transformer secondary circuit is open. 
 
 3000 
 2000 
 
 1000 - 
 2000 - 
 
 TIG. 184. The Primary E.M.F. Curve (firm line) and Secondary E.M.F. 
 'Curve (dotted line) of a Thomson-Houston Transformer taken off a 
 Thomson-Houston Alternator. The Secondary Curve is drawn to a scale 
 which makes its Maximum Ordinate equal to that of the Primary Curve, 
 and the Curves are seen to be identical in form. 
 
 FIG. 185. The same Primary and Secondary E.M.F. Curves, delineated 
 in Fig. 184, are here drawn with the Secondary Curve (dotted) reversed 
 and superposed on the Primary Curve to show its exact coincidence with 
 the Primary Curve. 
 
 It is found that this curve of secondary electromotive force 
 is, under these conditions, an exact copy on a reduced scale of 
 the curve of primary E.M.F., and that it is in exact opposition 
 to it in phase. In Fig. 184 are shown the curves of primary 
 and secondary terminal potential difference of a Thomson-
 
 THE INDUCTION COIL AND TRANSFORMER. 543 
 
 Houston transformer at no load. The primary terminal 
 potential difference curve or primary E.M.F. curve is repre- 
 sented in Fig. 184 by a firm line, and the secondary E.M.F. 
 curve by a dotted line. The secondary curve has been drawn 
 to such a scale that the ordinates of the secondary curve are 
 equal to those of the primary curve. In Fig. 185 the curve 
 of secondary E.M.F., represented by a dotted line, has been 
 reversed and drawn over the primary to show the exact coinci- 
 dence of the two curves. 
 
 FIG. 186. Primary E.M.F. Curve (I), Primary Current Curve (II) and 
 Secondary E.M.F. Curve (III) of 10-light Westinghouse Transformer on 
 Open Secondary Circuit 
 
 This constitutes one of the most valuable properties of the 
 transformer, viz., that it copies varying or periodic potential 
 difference exactly to a reduced or increased scale. Hence, if 
 we have a pair of terminals between which there is a periodi- 
 cally-varying potential difference having a \/mean-square 
 value of, say, 2,000 volts, and we attach the primary circuit 
 of a suitably- wound transformer to these terminals, we can 
 produce a periodically-varying potential difference of lower
 
 644 THE INDUCTION COIL AND TRANSFORMER. 
 
 or higher value, and the curve of which is an exact copy to 
 a reduced or increased scale of the original. We shall see 
 later on that useful applications can be made of this fact. 
 
 If the secondary circuit of the transformer is closed by a 
 non-inductive resistance, such as incandescence lamps, then the 
 Curve of secondary terminal potential difference or secondary 
 electromotive force undergoes a displacement and is brought 
 forward or lags behind the curve of primary electromotive 
 force. The reason for this is to be found in the magnetic 
 
 7 
 
 0-3 = 
 
 FiG. 187. Primary E.M.F. Curve (Ij, Primary Current Curve (II) and 
 Secondary E.M.F. Curve (II!) of 10-light Westinghouse Transformer, loaded 
 to one- tenth of full load. 
 
 leakage across the magnetic circuit which then takes place, 
 and which will be discussed in a later section. The act of 
 closing the secondary circuit of the transformer and produc- 
 ing a secondary current also effects a displacement in the 
 position of the primary- current curve. As the transformer 
 is loaded up the primary- current curve is displaced backwards, 
 so that the lag in phase between the primary current and 
 primary electromotive force is decreased. At full load the
 
 THE INDUCTION COIL AND TRANSFORMER. 545 
 
 primary current and secondary electromotive force are nearly 
 in opposition of phase. This is seen to be the case by 
 examining the series of curves in Figs. 186 to 189, which 
 were taken by Prof. Eyan from a small Westinghouse trans- 
 former, in which magnetic leakage is not by any means 
 absent. 
 
 We have next to consider the position of the curve of 
 magnetic induction. The curve of induction is obtained, 
 as already described, by integrating one or other of the 
 
 r 
 I sco 
 
 too 
 
 \ 
 
 1 
 
 0-4 
 
 FIG. 188. Primary E.M.F. Curve (I), Primary Current Curve (II) and 
 Secondary E.M.F. Curve (III) of 10-light Westinghouse Transformer loaded 
 to half load. 
 
 curves of electromotive force. The process of obtaining a 
 second curve, by taking as ordinates the area up to successive 
 absciss of a first curve and plotting these areas as new 
 ordinates to the limiting abscissae, is a process which always 
 has the effect of smoothing out irregularities in the original 
 curve, so that if the first curve is one not far removed in form 
 from a simple sine curve the second or integration curve will 
 be more nearly still a simple sine curve. 
 
 NN
 
 546 THE INDUCTION COIL AND TRANSFORMER. 
 
 An analytical proof of this is as follows : If the ordinate y 
 of a periodic curve is represented by a Fourier series, as it 
 can always be if periodic and single valued, then y may 
 be expressed by the series 
 
 y = A sin p t + B cos p t + C sin 2 pt + D cos 2 p t + &c., 
 -where A, B, C, &c., are constants. Hence, 
 
 yd = l 
 P 
 
 - 
 
 sin 2- cos 2ni + &c.. 
 
 
 / 
 
 D 
 
 'Bsmpt- A cospt + 
 
 C 
 
 - cos 2/j 
 
 1800 
 
 160 
 
 A 
 
 \ 
 
 r 
 
 FIG. 189. Primary E.M.F. Curve (\), Primary Current Curve (\\) and 
 Secondary E.M.F. Curve (\\\) of 10-light \Vestinghouse Transformer at 
 full load. 
 
 It will be seen that the result of the integration has been to 
 effect a change of phase of all the components and to weaken 
 the higher harmonics by diminishing the coefficients which 
 denote their amplitudes. Hence the process of forming a new 
 periodic curve by taking as ordinates the area of a first
 
 THE INDUCTION COIL AND TKANtFJBMER. 547 
 
 periodic curve up to successive abscissae always has the effect 
 of wiping out irregularities of form of the primary curve, and 
 yielding a curve more nearly a simple sine curve. It follows, 
 therefore, that the curve of magnetic induction is always less 
 irregular than the curve of primary or secondary electromotive 
 force from which it is derived. From what has been already 
 said, it will be seen that the curve of magnetic induction in 
 the core has its maximum value at the moment when the 
 electromotive force curve from which it is derived has its 
 zero value. We may, then, sum up the general facts about 
 transformer indicator diagrams by saying that when a trans- 
 former is at work we have 
 
 1st. A varying potential difference between the primary 
 terminals Avhich follows a certain wave form depending 
 (a) On the nature of the alternator; 
 (6) On the state of the load of that alternator, whether 
 
 full or light, inductive or non-inductive ; 
 (c) On the nature and construction of the transformer 
 
 connected to the alternator. 
 
 No assumptions must be made as to the form of this curve, 
 "but in every case its true form at the terminals of the trans- 
 former under test must be determined. The curve of primary 
 electromotive force has widely different forms in the cases met 
 -with in practice. 
 
 2nd. If the transformer has its secondary circuit open, we 
 have a primary current flowing into its primary circuit which 
 is called the magnetising current, and which lags in phase 
 behind the curve of primary electromotive force. As the trans- 
 former secondary circuit is loaded up this curve of primary 
 current is brought more into step with the primary electro- 
 motive force under the conditions that the load on the 
 secondary circuit is a non-inductive load. 
 
 The curve of primary current is an irregular periodic curve 
 the form of which is affected by the form of the curve of 
 primary electromotive force and by the nature of the trans- 
 former, and may have very different forms as these two 
 operating causes are changed. 
 
 3rd. We have a curve of secondary terminal potential 
 difference which is in exact opposition to the curve of 
 primary terminal potential difference when the transformer
 
 548 THE INDUCTION COIL AND TRANSFORMER. 
 
 is on open secondary circuit, and which is an exact copy 
 of the curve of primary potential difference to a reduced scale. 
 This curve of secondary potential difference may be shifted 
 forward in phase as the transformer secondary is loaded up, 
 so as to come more nearly into opposition with the curve of 
 primary current. 
 
 4th. We have a curve of magnetic induction, and this 
 induction is not the same in different parts of the core, or the 
 same on open secondary circuit as at full load. The form of 
 the curve is always more nearly a simple periodic curve than 
 is the form of the curves of primary and secondary terminal 
 potential difference. 
 
 Each of these curves heing a periodic single-valued curve, 
 can be expressed by a Fourier series and analysed into con- 
 stituent harmonics. Thus the ordinate ^ of the curve of 
 primary potential difference corresponding to any instant t 
 reckoned from the beginning of the phase can be expressed by 
 the series 
 
 i = E! sinp t + F! cos^> t + E 3 sin 3 p t + F 3 cos 3p t 
 
 + E 5 sin 5p t + F 5 cos 5p t + &cv 
 
 The constant or first term of the Fourier series is zero, 
 because the curve is always symmetrical above and below the 
 axis of time. Moreover, only the odd harmonic constituents 
 are present, viz., the harmonics whose wave lengths are one- 
 third, one-fifth, &c., of the fundamental wave length, and if by 
 any form of harmonograph we mechanically resolve any of these 
 transformer curves, we find that they can be quite adequately 
 represented by the first three odd terms of the Fourier series 
 that is to say, we can build up any transformer curve by 
 adding together the ordinates of three simple periodic curves 
 the wave lengths of which are in the ratio of 1;3; 5, the 
 amplitudes and relative positions being suitably chosen. The 
 reason for the absence of the even harmonic constituents viz., 
 those whose wave lengths are |, that of the fundamental is 
 to be found in the peculiar symmetry of these transformer 
 curves. On looking at any transformer curve it will be seen 
 that it is of such a character that, if the portion below the time 
 AXIS be considered to be reversed, we should get a repetition 
 of the same form. Thus, a curve of electromotive force
 
 THE INDUCTION COIL AND TRANSFORMER. 549 
 
 in Fig. 190, when so treated, becomes rectified into that 
 in Fig. 191. 
 
 On considering, then, the form of any curve, it will be seen 
 that the harmonic constituents must be such that if we move 
 forward 180deg. along the time axis the value of the ordinate 
 becomes negative but remains the same in magnitude. 
 
 If e represents the ordinate of any curve at any point 
 corresponding to an instant t, and if p as usual is 2?r n, where 
 
 FIG. 190. A Transformer Curve showing the typical symmetry of all 
 transformer curves. 
 
 Fm. 191. The same Transformer Curve shown in Fig. 190, but with the 
 second half of the wave rectified to show the typical symmetry ol 
 curve. 
 
 w is the frequency, then we can represent the value of e by 
 the series 
 
 + E 3 sin 3pt + ~F s cos 
 The harmonic constituents must be such that if we put 
 (pt + v) for pt the value of e becomes e. 
 
 It is easily seen that, since sin (pt + ir)=-sin pt and 
 sin {8(ji* + 7r)} = -sin8j? t, &c., whereas sin {2 (pt + ir)} 
 = sin 2 pt, the essential condition is that only the odd
 
 550 THE INDUCTION COIL AND TRANSFORMER. 
 
 harmonics must be present. Hence the expansion of the 
 ordinate of the real transformer curve can only contain the 
 1st, 3rd, 5th, &c., terms. As a matter of experience it is 
 found that any transformer curve met with in practice can very 
 
 FIG. 192. The Harmonic Analysis of the Curve of E.M.F. of a Thomson- 
 Houston Alternator at no Load. The thick Curve C is the Curve of 
 E.M.F., and the Curves marked HI, H 3 , H 5 , are the Harmonic Constituents- 
 
 with Wave Lengths in the ratio of 1, 3, 5. 
 
 Fio. 193. The Harmonic Analysis of the Curve C of E.M.F. of a. 
 Thomson-Houston Alternator partly loaded up on water resistance. The 
 Harmonic Constituents of the Curve are represented by the Curves marked 
 HL H 3 , H B . 
 
 nearly be represented by the first three odd terms of the series, 
 and hence any observed transformer carve can be very quickly 
 analysed into its constituents by the arithmetical process 
 explained on page 92. 
 
 By the use of mechanical harmonographs or analysers thia 
 can, of course, be very easily done, and an illustration ia
 
 THE INDUCTION COIL AND TEANSFOEMER. 551 
 
 given in Figs. 192, 193 and 194 of the E.M.F. curve of a 
 Thomson-Houston alternator so analysed. The curves given 
 in Figs. 192, 193 and 194 were analysed for the author by 
 Mr. G. U. Yule, with his mechanical harmonograph. It 
 will be noticed that when the curve C is symmetrical, the 
 harmonic constituents start from the same point, and have no 
 lag relatively to one another. 
 
 FIG. 194. The Harmonic Analysis of the Curve C of E.M.F. of a 
 Thomson-Houston Alternator, partly loaded up on Inductive Resistance. 
 The Harmonic Constituents of the Curve are represented by the Curves 
 marked Hj, H 3 , H 5 , with wave lengths in the ratio of 1, 3, 5. 
 
 4. Derivation of Curves of Power and Hysteresis. 
 From the curves of current, electromotive force, and induction 
 obtained as above described we can construct two other curves 
 which give us the variation of the total power supplied to the 
 transformer, and the total loss in the iron core per cycle. 
 These curves are obtained as follows : Let us assume that a 
 set of transformer curves has been taken when the trans- 
 former is on open secondary circuit. Taking the curves of 
 primary current and primary terminal potential difference, 
 we multiply together (as explained on page 153, 23, of 
 Chapter III.), the corresponding ordinates of the two curves 
 for abscissae taken at equidistant points on the time axis, 
 and set up a new ordinate representing the value of the 
 product ei, where e is the primary potential difference and 
 i the primary current at the same instant. This product set 
 off as an ordinate defines another curve called the power curve, 
 and the true mean ordinate of this power curve gives us the
 
 552 THE INDUCTION COIL AND TRANSFORMER. 
 
 mean power taken up in the transformer at no load. To 
 obtain the true mean ordinate of the power curve we have to 
 integrate the whole area included between the power curve 
 and the time axis, and to reckon those areas which lie above 
 the time axis as positive and those which lie below as negative. 
 The total area of the positive and negative parts algebraically 
 added, and divided by the length of the axis representing one 
 complete period, gives us the true mean ordinate of the power 
 curve. Hence, we can, from the transformer diagram taken on 
 open secondary circuit, determine the mean power taken up in 
 the transformer. The amount dissipated in heat in the copper 
 of the primary circuit is generally an exceedingly small 
 fraction of the total loss, and hence the mean power obtained 
 as above is practically the value of the power taken up in the 
 iron core. 
 
 The analytical expression of this fact is as follows : Taking 
 the fundamental equation for the transformer on open 
 secondary circuit, viz., 
 
 %-BA+SN r ll 
 
 at 
 we multiply the equation all through by i\ and obtain 
 
 a t?, 
 
 d t 
 
 or e 1I t = 
 
 If this last equation is integrated between the limits to 
 
 m 
 
 _ , where T is the complete periodic time, and each integral 
 
 Q 
 
 multiplied by , we obtain an expression for the mean power 
 given to the transformer during one half-period. Thus, 
 
 The first term on the left hand side represents the true 
 mean power given to the transformer in one half-period. The 
 second term represents the power dissipated as heat in the 
 primary circuit in one half-period, and the third term repre- 
 sents the power dissipated on eddy currents and hysteresis in
 
 THE INDUCTION COIL AND TRANSFORMER. 553 
 
 the core in the same time. If a horizontal line is taken, and 
 from an origin distances are set off right and left to represent 
 the varying values of the primary current i during the period, 
 and vertical ordinates corresponding to these abscissas taken 
 to represent the values of the induction density b in the core 
 at the same instant, then a curve will be defined which will be 
 a cyclic curve, and will give us the total core loss per cycle 
 when the numerical value of its area is multiplied by the 
 factor NjS. If the, iron core is well laminated, eddy current 
 loss will be practically absent ; and the value of this area, 
 therefore, will give us the true hysteresis loss in the iron. 
 
 4000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 f 
 
 7 
 
 1000 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 / 
 
 
 L 
 
 
 
 
 
 
 / 
 
 
 / 
 
 
 
 
 
 
 / 
 
 
 y 
 
 ' 
 
 
 
 
 I 
 
 t 
 
 -^ 
 
 X 
 
 
 
 
 
 
 3000 
 4000 
 
 
 
 
 
 
 
 
 
 
 4 
 
 3 
 
 2 
 
 i 
 
 \ 
 
 i 
 
 > 
 
 3 4 
 
 Magnetuiny Force. 
 FIG. 195. Hysteresis Curve of a Ganz Transformer. 
 
 Hence, such a curve is called the hysteresis curve of the core. 
 In Fig. 195 is shown the hysteresis curve of the Ganz trans- 
 former, so obtained from the current and induction curves of 
 the same transformer as given in Fig. 170. 
 In order to obtain the correct numerical value of the 
 hysteresis loss per cycle it must be noted that if all the quan- 
 tities S, i and b are measured in C.G.S. measure, the value oi 
 the integral S^J idb will give us, when taken round one 
 complete cycle, the value of the core loss in ergs during one
 
 554 THE INDUCTION COIL AND TRANSFORMER. 
 
 complete period. And this value has to be divided by 10 7 to 
 reduce it to joules. Since the integral SN 1 lidb can be 
 written l(N i) d(Sb), we see that the core loss per cycle in 
 
 joules can at once be obtained by taking the area of a loop 
 curve, the horizontal ordinates of which represent the periodic 
 values of the primary ampere-turns, and the vertical ordinates 
 the corresponding total core induction during one complete 
 period, taken in a unit equal to 10 8 C.G.S. units of magnetic 
 induction. 
 
 If the frequency is n, then n times the above integral gives 
 the loss in the core per second ; and this should have the same 
 numerical value as the mean ordinate of the power curve 
 which measures the same quantity. The practical rule, 
 therefore, for obtaining the core loss in the transformer due to 
 the hysteresis and eddy current loss which may be present 
 is as follows : Draw two axes at right angles ; on the hori- 
 zontal axis set off right and left from the origin distances 
 which represent the primary ampere-turns for the different 
 instants during one complete period. At these points set up ordi- 
 nates which represent the total induction in the core measured 
 in units each equal to 10 8 C.G.S. units of magnetic induction, 
 and complete the looped curve defined by these ordinates. 
 The area of this curve, measured in terms of the area of 
 a rectangle one side of which is the length taken to represent 
 one ampere-turn and the other side is the length taken to 
 represent 10 8 C.G.S. units of induction, will give the value 
 of the core loss per cycle in joules, and multiplication of 
 this value by the frequency n will give the mean loss of 
 power in the core in watts. The number so obtained will 
 agree closely with the value of the mean ordinate of the 
 power curve in those cases in which the copper loss in the 
 primary circuit when the transformer is not loaded can be 
 neglected. 
 
 The form of this hysteresis loop will depend upon the manner 
 in which the magnetic induction in the core varies with the 
 magnetising force, and, as we shall see presently, the area 
 of this hysteresis loop depends, amongst other things, upon 
 the form of the curve of primary impressed electromotive 
 force.
 
 THE INDUCTION COIL AND TBANSFOEMEK. 555- 
 
 5. The Efficiency of Transformers. If the secondary 
 circuit of the transformer is closed through a resistance, and 
 the transformer is therefore loaded up, the power given to the 
 primary circuit in part reappears in a transformed form in the 
 external secondary circuit. As by far the most frequently 
 presented case in practice is that in \vhich the resistance 
 which closes the secondary circuit consists of incandescence 
 lamps or other practically non-inductive resistances, we shall, 
 therefore, in the first instance assume that the external 
 secondary circuit is an inductionless resistance. Under these 
 conditions the secondary current is, to a close approximation, 
 in step or synchronism with the secondary electromotive force, 
 and the mean power given to the external secondary circuit is 
 measured by the product of the mean-square value of the 
 secondary current strength and the mean-square value of the 
 potential difference of the secondary terminals. If we denote 
 by P 2 the power thus given up to the external secondary 
 circuit, and similarly by P t the power given up to the primary 
 circuit, the ratio of P 2 to P 1 is called the efficiency of the trans- 
 former. This efficiency is generally expressed as a percentage, 
 and will be denoted by the symbol e. Hence 
 
 .-100%. 
 
 The difference between P l and P 2 is represented by the 
 power lost in the core and dissipated in the copper circuits of 
 the transformer. If the symbol Cj stands for the mean-square 
 value ( Vmean 2 ) of the primary current, and C 2 for that of the 
 secondary currents at any time, and if R x and E. 2 are the resis- 
 tances of these circuits when warm and at that time, then the 
 power wasted in the primary and secondary circuits respec- 
 tively is (V E 2 and C 2 2 R 2 , and if H is the core loss, viz., the 
 hysteresis and eddy-current loss, then 
 
 P 1 -P 2 =C 1 2 B+C 2 2 E 2 + H, 
 
 on the assumption that there are no eddy-current losses or 
 energy dissipations in the copper circuits, or in the iron case 
 or framework of the transformer. 
 
 One of the most important measurements, therefore, which 
 it is necessary to make in connection with transformers is the
 
 556 THE INDUCTION COIL AND TRANSFORMER. 
 
 measurement of the power given to the primary circuit. We 
 shall defer to a later chapter on transformer testing a full 
 discussion of the various methods which can be employed for 
 determining either the magnitude of the quantity P x or of the 
 difference P : - P 2 in the case of a transformer at any load. It 
 may suffice to state at present that one way in which this 
 measurement can be made is by means of a properly con- 
 structed wattmeter, which measures directly the power Pj 
 given to the primary circuit. The objection to this method is 
 that any error made in evaluating P x appears to the same 
 extent and percentage in the ratio of P 2 to P 1? and therefore in 
 the efficiency. Hence other methods have been devised for 
 measuring directly the difference P a - P 2 . However the value 
 of the efficiency may be determined, the results are best set 
 down in the form of an efficiency curve as follows : Each 
 transformer is constructed to give safely a certain output of 
 power to the secondary external circuit, which is called its 
 full load, and is stated generally in .watts or kilowatts. The 
 load on the secondary circuit in any other cases can be 
 expressed as a fraction of the full load. To draw an efficiency 
 curve for any transformer, a horizontal line is taken, on 
 which are marked off the decimal fractions of the full load, 
 
 and at these points are set up ordinates which represent the 
 
 p 
 
 percentage efficiencies at these loads, viz., the value of 100?, 
 
 PI 
 
 where P x is the power given to the primary circuit and P 2 is 
 the power given to the external secondary circuit. The ex- 
 tremities of these ordinates delineate the efficiency curve. 
 
 In Figs. 196 and 197 are shown the efficiency curves of 
 various transformers. It will be seen that the chief difference 
 is that the more modern transformer has a higher efficiency 
 at the low loads. A good transformer of any moderate size 
 should have at least 80 per cent, efficiency at one-tenth load. 
 And larger transformers of 15 and 20-kilowatt size and 
 upwards will reach to 90 per cent, efficiency or more at one- 
 tenth of full load. 
 
 Another method of delineating the efficiency is to plot the 
 difference P x - P 2 in terms of P 2 ; in other words, to plot a 
 curve the abscissae of which represent the secondary output 
 P 2 , and the. ordinates of which represent the total loss of
 
 THE INDUCTION COIL AND TRANSFORMER. 557 
 
 1 -2 -3 -4 '5 '6 -7 -8 '9 I'D 
 
 Fractions <tf Full Secondary Load. 
 FIG. 196. Efficiency Curves of Transformers. 
 
 2 -3 -4 '5 '6 '7 '8 '9 I'D 
 
 Fractions of Full Secondary Load. 
 FIG. 197. Efficiency Curves of Transformers. 
 
 In the above diagrams, horizontal distances represent the decimal 
 fractions of full secondary load, and vertical ordinates the percentage 
 efficiency corresponding thereto. The numbers against the curves refer 
 to the following transformers : 
 No. 9. 5 horse-power Ferrauti Transformer 
 12. 5 
 
 20. 15 
 23. 15 
 
 27. 20 
 
 31. 6 kilowatt 
 
 35. 
 39. 
 45. 
 16. 
 
 4-5 
 4 
 3 
 6-5 
 
 Mordey Transformer 
 
 Thomson-Houston Transformer 
 Kapp Transformer 
 
 1885 type. 
 1885 type 
 rewound. 
 1892 type. 
 1892 type 
 rewound. 
 1892 type. 
 1892 type. 
 1892 type. 
 1892 type. 
 
 " Hedgehog" (Swinburne) Transformer 1892 type. 
 Westinghouse Transformer ... 1892 type.
 
 .558 
 
 THE INDUCTION COIL AND TRANSFORMER. 
 
 power in the transformer, viz., 
 P, - P. 2 . In Fig. 198 is shown 
 such a curve drawn for a 
 6,500 - watt Westinghouse 
 transformer. The ordinates 
 of the upper curve give the 
 value of P! P 2 corresponding 
 to the secondary output P 2 . 
 
 One important question 
 which arises in this connec- 
 tion is whether the true iron 
 core loss by hysteresis remains 
 constant at all loads of the 
 transformer. This was at one 
 time denied. It has, how- 
 ever, been shown by careful 
 experiments that the hys- 
 teresis loss in the iron core 
 is sensibly constant at all 
 loads.* 
 
 The proof of this was 
 obtained by careful measure- 
 ments made of the total 
 energy loss P - P 2 for various 
 transformers. This value was 
 plotted down, as in Fig. 198, 
 in terms of the secondary out- 
 put P 2 . On the same diagram 
 was drawn a curve represent- 
 ing the total copper loss or 
 C 2 R loss for the primary 
 
 * The reader may be referred to 
 a Paper by the author in the Pro- 
 ceedings of the Institution of Elec- 
 trical Engineers, Vol. XXI., 1892, 
 entitled " Experimental Researches 
 on Alternate - Current Transfor- 
 mers," for full information on the 
 experimental methods by which this 
 question has been settled. See 
 also The Electrician, Vol. XXX., 
 pp.97, 120,162,446. 
 
 I 1 
 
 g 
 
 1 
 
 $ 
 
 i 
 
 o 
 
 
 D 
 
 t 
 
 S 
 
 n 
 
 8 
 
 
 ! 
 
 1 j 
 
 
 
 
 k 
 
 S 1 
 
 
 O 
 
 * 
 
 
 
 H i 
 
 
 \J 
 
 il 
 
 
 1 
 
 o 5 
 
 iM 
 
 
 51 
 
 H fe 
 
 
 
 S t fe 
 
 o 1 
 
 
 J 
 
 :3 
 
 
 K el 
 
 | 
 
 
 
 " 
 
 ? 
 
 
 <r 
 
 LI 
 
 
 
 Ft- 
 
 
 
 ig 
 
 
 
 | 
 
 ~ 
 
 
 i 
 
 s 
 
 
 S A8 S3SSOT 
 
 rrgrs^
 
 THE INDUCTION COIL AND TRANSFOEMEB. 559 
 
 and secondary circuits taken together. These two curves 
 are found to be sensibly parallel to each other through- 
 oui; their whole range, and hence the true iron core loss or 
 hysteresis loss is a constant quantity at all loads. This is 
 a necessary consequence of the fact that in constant potential 
 transformers as designed for ordinary electric lighting work 
 the induction in the core is constant at all loads, and this in 
 turn is a consequence of the fact that the resultant mag- 
 netising force in the core is constant for all loads. Generally 
 speaking, we may state that for all fairly well designed closed 
 iron circuit constant-potential transformers the iron core loss is 
 constant for all loads. This enables us to determine the 
 efficiency curve for any transformer of this description by three 
 measurements. If we measure the total power loss in the 
 transformer at no load we have the quantity which is constant 
 ac all loads. Call this loss in watts w. If, then, we measure 
 the resistances of the primary and secondary circuits and 
 correct these values so as to obtain the true resistances R x 
 and R 2 of the copper circuits at the final temperature reached 
 by the transformer when working, we can calculate the copper 
 losses (V R! and C 2 2 R 2 ior various values of the output of the 
 transformer. We can determine the value of the primary 
 current Cj corresponding to any value of the secondary current 
 O 2 to a sufficient approximation for this purpose by taking it as 
 
 equal to C 2 2?, where N z and N 2 are the number of turns of 
 
 Nj 
 
 the primary and secondary circuits respectively. Hence, the 
 total copper loss in the transformer is very approximately 
 equal to 
 
 and the total power P x given to the primary circuit conse- 
 quently corresponding to any secondary output C 2 V 2 , where 
 V 2 is the secondary terminal potential difference, is given 
 by the equation 
 
 and P 2 =C 2 V 2 . 
 
 Hence the ratio of P a to P lf or the efficiency at various
 
 560 THE INDUCTION COIL AND TRANSFORMER. 
 
 CTJrHcd-OvOCOCMr-IOLOlO 
 
 vOOiCCOCOKJiOOtOCDiO 
 
 Ov]03i-IOi-l[>a> 
 
 ooooooooooo
 
 THE INDUCTION COIL AND TRANSFORMER. 561 
 
 loads, can be calculated. This is a convenient and fairly 
 accurate method to adopt in the case where we are testing 
 large transformers. It is sometimes very difficult or impos- 
 sible then to obtain the necessary non-inductive load in the 
 form of incandescence lamps for very large loads such as 40 
 or 50 kilowatts, and in that case the above procedure may be 
 followed. The table on page 560 gives the results of a large 
 number of transformer efficiency measurements made by the 
 author in 1892, employing many forms of transformers then, 
 in use. 
 
 The table shows particularly what a great advance was- 
 made in transformer manufacture in the course of the seven 
 years between 1885 and 1892. 
 
 In the case of larger transformers the efficiency curves can 
 be made still more square-shouldered, and efficiencies of 
 over 90 per cent, obtained at one-tenth load. 
 
 Since the core loss at no load is an important factor in 
 determining the efficiency of the transformer, it is obvious 
 that no transformer can have a high efficiency at light loads 
 unless the core loss is small. The iron core loss or no-load 
 loss in the case of transformers of 30 kilowatt size and 
 upwards can now be made to be less than 1 per cent, of 
 the full secondary output. That is to say, it is possible to 
 make the iron core loss of a 50-kilowatt transformer not 
 more than 400 watts. In the case of smaller transformers, 
 from 1 to 15 kilowatts, the core loss will in general be from 3 
 to 1-3 per cent, of the full secondary output. Thus, a 1-kilo- 
 watt transformer, or one capable of giving out 1,000 watts in its 
 external secondary circuit, is a fairly good one if it has a core 
 loss of not more than 30 watts, or 3 per cent, of its full load ; 
 a 6-kilowatt transformer if it has a core loss of not more than 
 120 watts, or 2 per cent. ; and a 15-kilowatt transformer is 
 good if its core loss does not exceed 225 watts, or 1-5 per 
 cent. These figures will be a guide to the reader to know what 
 the core loss may be expected to be found in various cases. 
 
 We shall consider presently the causes which affect the 
 magnitude of the core losses. 
 
 6. Current Diagram of a Transformer. Let a horizontal 
 line be taken on which are set off distances proportional to
 
 
 
 1
 
 THE INDUCTION COIL AND TEANSFOEMEE. 563 
 
 the power output on the external secondary circuit, that is, to 
 the secondary load of a transformer, and let ordinates at these 
 points be drawn to any scale representing the magnitudes of 
 the primary and secondary currents, the scale of the primary 
 -current ordinates being taken so that if one unit of length 
 represents one ampere of primary current, and the scale of the 
 
 secondary current so that one unit of length represents 
 
 times the corresponding secondary current, then we shall 
 delineate the lines called the current curves. For any closed 
 circuit transformer of constant potential type these current 
 .are nearly two lines running nearly parallel to each other as 
 shown in Fig. 199. If C : stands for the mean-square value 
 of the primary current, and C 2 for that of the secondary 
 current, and if NI. and N 2 are the numbers of the primary and 
 secondary turns respectively, then experiment shows that a 
 good type of closed magnetic circuit constant potential trans- 
 
 former G! - ~ C 2 is a nearly constant quantity, and that this 
 
 i 
 
 difference is practically the same as the mean-square value of 
 the primary current when the transformer is not loaded. Let 
 this last be called q. Then 
 
 or <?! Wj = GI N! - C 2 N a . 
 
 In other words, the difference of the primary and secondary 
 ampere-turns at all loads is a constant quantity, and is equal 
 to the ampere-turns at no load. This is merely the expression 
 of the fact that the magnetomotive force acting on this 
 magnetic circuit is a constant quantity, and that therefore the 
 induction is constant as well. This is, however, not the case 
 lor open-circuit transformers. In Fig. 200 is shown the 
 current curves for a Swinburne "Hedgehog" transformer, and 
 it will be seen that the difference of the primary and secondary 
 ampere-turns is not constant, but increases as the load dimi- 
 nishes. This is a consequence of the fact that in the open 
 circuit transformer the difference of phase between the primary 
 and secondary currents is considerable at light loads, but 
 ibecom.es less as the transformer is loaded up, and that there- 
 
 oo 2
 
 664 THE INDUCTION COIL AND TRANSFORMER. 
 
 Table A. Test of a Westinghouse Transformer. 
 
 Power, 6,500 watts. Secondary volts, 100. 
 Frequency used, 82'5 periods per second. 
 Average final temperature of transformer, 96F. 
 Volts on primary circuit (V x ) = 2,400 (kept constant). 
 Primary circuit resistance = 5'95 ohms at 96F. 
 Secondary circuit resistance = 0'0108 ohm at 96F. 
 
 Secondary Circuit. 
 
 Primary Circuit. 
 
 II., 
 
 -*^ o >. 
 
 S? = 
 
 
 
 Power 
 
 
 
 Power 
 
 53^5'" 
 z ' 
 
 !*tS 
 
 Volts. 
 
 Amperes. 
 
 ;akenout 
 in watts, 
 
 Volts. 
 
 Amperes. 
 
 given in 
 watts 
 
 &2 
 a a || 
 
 5 II a 
 H .= 
 
 
 
 W 2 . 
 
 
 
 = Wi. 
 
 -S*^ 
 S& 
 
 
 101-0 
 
 
 
 
 
 2,400 
 
 0-050 
 
 95 
 
 95 
 
 
 
 100-9 
 
 1-00 
 
 101 
 
 
 0100 
 
 205 
 
 104 
 
 49-3 
 
 100-8 
 
 1-98 
 
 2CO 
 
 " ( 
 
 0-140 
 
 306 
 
 106 
 
 65-4 
 
 100-8 
 
 2-94 
 
 296 
 
 ' ( 
 
 0-180 
 
 401 
 
 105 
 
 73-7 
 
 100-7 
 
 3-87 
 
 390 
 
 | 
 " 
 
 0-218 
 
 493 
 
 103 
 
 79-1 
 
 100-7 
 
 4-79 
 
 482 
 
 
 0-250 
 
 597 
 
 115 
 
 80-7 
 
 100-7 
 
 8-00 
 
 806 
 
 w 
 
 0-382 
 
 920 
 
 114 
 
 87-6 
 
 100-4 
 
 10-15 
 
 1,019 
 
 " t 
 
 0-472 
 
 1,139 
 
 120 
 
 89-5 
 
 100-3 
 
 13-07 
 
 1,311 
 
 
 0-580 
 
 1,440 
 
 129 
 
 911 
 
 100-1 
 
 18-00 
 
 1,802 
 
 " 
 
 0-800 
 
 1,930 
 
 128 
 
 93-4 
 
 100-1 
 
 19-90 
 
 1,992 
 
 
 0-880 
 
 2,118 
 
 126 
 
 94-1 
 
 100-0 
 
 21-93 
 
 2,193 
 
 
 0-960 
 
 2,330 
 
 127 
 
 94-1 
 
 100-0 
 
 24-74 
 
 2,474 
 
 
 1-080 
 
 2,609 
 
 135 
 
 94-8 
 
 100-0 
 
 29-66 
 
 2,966 
 
 
 1-285 
 
 3,096 
 
 130 
 
 95-8 
 
 99-8 
 
 37-20 
 
 3,713 
 
 
 1-610 
 
 3,870 
 
 157 
 
 96-0 
 
 99-5 
 
 42-00 
 
 4,179 
 
 
 1-810 
 
 4,324 
 
 145 
 
 96-7 
 
 99-3 
 
 46-65 
 
 4,633 
 
 
 2-002 
 
 4,792 
 
 150 
 
 96-9 
 
 99-2 
 
 50-40 
 
 5,000 
 
 
 2-160 
 
 5,174 
 
 174 
 
 96-7 
 
 99-0 
 
 52-16 
 
 5,164 
 
 
 2-240 
 
 5,422 
 
 258 
 
 95-3 
 
 98-9 
 
 55-60 
 
 5,499 
 
 Jj 
 
 2-383 
 
 5,702 
 
 203 
 
 96-1 
 
 98-8 
 
 57-68 
 
 5,700 
 
 
 2-478 
 
 5,885 
 
 185 
 
 96-9 
 
 98-9 
 
 59-32 
 
 5,867 
 
 
 2-550 
 
 6,041 
 
 174 
 
 97-3 
 
 98-7 
 
 61-32 
 
 6,053 
 
 " 
 
 2-633 
 
 6,271 
 
 218 
 
 96-7 
 
 98-8 
 
 6216 
 
 6,142 
 
 || 
 
 2-672 
 
 6,344 
 
 202 
 
 6-8 
 
 8-7 
 
 63-00 
 
 6,218 
 
 M 
 
 " 
 
 2-700 
 
 6,426 
 
 208 
 
 96-8 
 
 987 
 
 64-CO 
 
 6,317 
 
 
 2-750 
 
 6,522 
 
 205 
 
 96-9 
 
 S8-6 
 
 64-74 
 
 6,384 
 
 " 
 
 2-775 
 
 6,598 
 
 214 
 
 96-9 
 
 fore there must be an increase in mean-square or maximum 
 value of the primary current, so that its greater value at light 
 loads is a compensation for the greater difference of phase 
 between the primary and the secondary current. This is on 
 the assumption that the secondary circuit is a practically non- 
 inductive circuit. If the carefully-drawn current diagram ot
 
 THE INDUCTION COIL AND TRANSFORMER. 565 
 
 Table B. Test of a Sicinburne "Hedgehog " Transformer. 
 Power, 3,000 watts. Secondary volts, 100. 
 Frequency used, 81'1 periods per second. 
 Average final temperature of transformer, 145F. 
 Volts on primary circuit (7^ = 2,400 (kept constant). 
 Primary circuit resistance = 24 '00 ohms at 145F. 
 Secondary circuit resistance = 0'051 ohm at 145F. 
 
 Secondary Circuit. 
 
 Primary Circuit. 
 
 11 
 
 
 
 
 Sis* 
 
 w 
 
 
 
 Power 
 
 
 
 Power 
 
 
 lift?! 
 
 Volts. 
 
 Amperes. 
 
 taken <>ut 
 
 Volts. 
 
 Amperes 
 
 given in 
 
 *<f 
 
 1 " 1 
 
 
 
 in watts, 
 W 2 . 
 
 
 
 watts Wj. 
 
 f " 
 
 W C 
 
 101-8 
 
 
 
 
 
 2,400 
 
 0-756 
 
 121 
 
 121 
 
 
 
 101-7 
 
 100 
 
 102 
 
 
 0-761 
 
 213 
 
 111 
 
 47-9 
 
 101-5 
 
 2-97 
 
 301 
 
 
 0-786 
 
 414 
 
 113 
 
 727 
 
 101-3 
 
 484 
 
 490 
 
 
 0-811 
 
 608 
 
 118 
 
 80-6 
 
 101-3 
 
 6-00 
 
 f07 
 
 
 0-829 
 
 730 
 
 123 
 
 83-1 
 
 101-2 
 
 8-00 
 
 810 
 
 
 0-862 
 
 943 
 
 133 
 
 85-9 
 
 101-0 
 
 10-20 
 
 I,0i0 
 
 
 0-915 
 
 1,161 
 
 131 
 
 886 
 
 100-9 
 
 12-00 
 
 1,211 
 
 
 0960 
 
 1,361 
 
 150 
 
 890 
 
 100-6 
 
 14-00 
 
 1,408 
 
 
 1-013 
 
 1,551 
 
 143 
 
 90-8 
 
 100-3 
 
 15-87 
 
 1,5-2 
 
 
 1-066 
 
 1,750 
 
 158 
 
 91-1 
 
 100-0 
 
 17-89 
 
 1,789 
 
 
 1-133 
 
 1,951 
 
 162 
 
 91-7 
 
 100-0 
 
 19-80 
 
 1,980 
 
 
 1-197 
 
 2,129 
 
 149 
 
 93-0 
 
 99-9 
 
 21-62 
 
 2,180 
 
 
 1,260 
 
 2.344 
 
 164 
 
 93-0 
 
 99-7 
 
 23-66 
 
 2,359 
 
 
 1-321 
 
 2,525 
 
 166 
 
 93-3 
 
 99-5 
 
 25-46 
 
 2,534 
 
 
 1-397 
 
 2,732 
 
 198 
 
 92-8 
 
 99-3 
 
 26-46 
 
 2,628 
 
 
 1-430 
 
 2,823 
 
 195 
 
 93-2 
 
 99-1 
 
 27-42 
 
 2,718 
 
 
 1-465 
 
 2.914 
 
 196 
 
 93-4 
 
 99-0 
 
 28-28 
 
 2,8CO 
 
 w 
 
 1-500 
 
 2,988 
 
 188 
 
 93-7 
 
 98-9 
 
 29-26 
 
 2,896 
 
 w 
 
 1-532 
 
 3,103 
 
 207 
 
 93-3 
 
 99-0 
 
 30-20 
 
 2,9,0 
 
 " 
 
 1-566 
 
 3,185 
 
 195 
 
 94-0 
 
 any closed-circuit transformer of constant-potential type are 
 examined, it will be seen that the difference between the ordi- 
 nates of the primary and secondary current curves or lines 
 increases slightly very near the origin, and as we shall see pre- 
 sently this is an indication of the fact that the primary current 
 is for all loads, except very small ones, practically in exact 
 opposition, as regards phase, to the secondary current. As an 
 example of the measurements of a complete test of a closed 
 circuit and open circuit transformer, we give on page 564 
 and above, in Tables A and B, the figures obtained for a 
 Westinghouse and Swinburne transformer respectively.
 
 566 THE INDUCTION COIL AND TEANSFORMEE. 
 
 7. The Power Factor of Transformers. If under any 
 conditions of load on the secondary circuit we measure the 
 true power P a being taken up by the primary circuit, and also 
 the mean-square (v'mean 2 ) value A of the primary current 
 and the mean-square value V of the primary terminal potential 
 difference, the ratio of P! to the product A V is called the 
 power factor of the transformer at that load. The product 
 A V is often called the apparent power or apparent watts given 
 to the transformer, and the value of Pj is the true power 
 or true watts given to the transformer. Hence the power 
 factor (F) is defined thus : 
 
 Power fac-\ _ True power in watts given to the transformer 
 tor (F) / Apparent power in watts given to the transformer 
 
 The above relation may be symbolically expressed by 
 writing 
 
 F = A or P 1 = (FA)V. 
 &. v 
 
 This last mode of writing it exposes the appropriateness of 
 the term "power factor" ; since we see that it is a factor by 
 which the value of the total mean-square value of the current 
 must be multiplied to obtain that current which, when multi- 
 plied by the mean-square value of the potential difference 
 V, will give the true mean power being taken up in the 
 circuit. Thus, if the power factor is denoted by F, this 
 signifies that the portion F A of the current A is effective in 
 conveying power, and the remainder (1 - F) A is ineffective, 
 or, as it is sometimes called, is the wattless component of the 
 current. Hence, we may, in imagination, divide the apparent 
 power A V into two portions: a part F AV, which is a measure 
 of the true power given to the circuit ; and a part (1 - F) AV, 
 which is the wattless or powerless portion. 
 
 The true power P t is obtained from the correct wattmeter 
 reading, and the apparent power A V is the value of the product 
 of the readings of an electrostatic voltmeter used to measure 
 the mean-square value of the potential difference and that of 
 an alternating-current ammeter used to measure the mean- 
 square value of the current. A very important constant with 
 respect to any transformer is its power factor at no load or on 
 open secondary circuit, and the Table on page 567 gives the
 
 INDUCTION COIL AND TRANSFORMER. 567 
 
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 Sj 
 
 1 
 
 I I 
 2 
 I 3 
 
 H I 
 
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 r 
 
 P4 
 
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 l- 
 
 cooocoocxicnuo 
 
 OSrHOUJOOOC- 
 
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 w il
 
 668 THE INDUCTION COIL AND TRANSFORMER. 
 
 values of the no-load power factor for various types of trans- 
 formers taken on certain alternators. 
 
 Generally speaking, it is found that the power factor of 
 most closed iron circuit transformers has a value lying between 
 0-5 and 0-8, but that for induction coils on open-circuit 
 transformers, such as the "Hedgehog" transformer, the power 
 factor is about one-tenth of the value for closed iron circuit 
 transformers. 
 
 It must not be supposed, however, that the power factor of 
 an induction coil or transformer has a constant and fixed 
 value for any particular transformer. The value of the power 
 factor is affected to a very considerable degree by the form of 
 the curve of primary terminal potential difference, and may 
 vary within wide limits according as the curve is varied in 
 form. 
 
 Power Factors of Trcmsformen taken off different Alternators at 
 the same Primary Voltage. 
 
 Transformer. 
 
 Size in 
 Kilow'tts 
 
 Power Factor. 
 
 Magnetising Currents 
 in Amperes. 
 
 On 
 
 Thomson- 
 Houston 
 Alternator 
 
 On 
 Mordey 
 Alternator 
 
 On 
 
 Thomson- 
 Houston 
 Alternator 
 
 On 
 
 Mordey 
 Alternator 
 
 Mordey -Brush . . . 
 Thomson- Hou ston 
 Mordey-Brush . . . 
 
 50 
 30 
 18 
 
 0-609 
 0-490 
 0685 
 
 0-704 
 0-536 
 0-751 
 
 0-668 
 0-562 
 0-326 
 
 0-623 
 0-569 
 0-332 
 
 In the Table above are given the power factors at no 
 load of three transformers taken off a Mordey alternator 
 having an E.M.F. curve similar to that shown in Fig. 175, 
 and a Thomson-Houston alternator having an E.M.F. curve 
 of the kind shown in Fig. 174, and it will be seen that the 
 power factors and magnetising currents of the transformers 
 are quite different in the two cases. 
 
 We must not, therefore, regard the power factor as an 
 absolute constant for the transformer, but as a function to 
 some degree of the form of the primary E.M.F. curve, although 
 at the same time dependent essentially upon the nature of 
 the magnetic circuit of the transformer.
 
 TEE INDUCTION COIL AND TRANSFORMER. 669 
 
 If the primary E.M.F. curve and current curve were both 
 -simple periodic or sine curves, then the power factor would be 
 simply the cosine of the angle of lag of the current behind the 
 electromotive force. If the transformer has its secondary 
 circuit loaded up, the power factor approximates to unity as 
 this loading takes place. 
 
 In the case of most closed-circuit transformers a very little 
 loading-up of the secondary circuit causes the power factor to 
 become unity, but in the case of an open-circuit transformer, 
 whilst the loading-up of the transformer increases the power 
 factor, it never actually reaches unity. 
 
 Table C. Test of a Westingliouse Transformer. 
 
 Primary volts, 2,400 (kept constant). 
 Secondary volts at no load = 101'G. 
 
 1 i 
 
 r 
 i 
 
 Secondary 
 Current, I 2 . 
 
 * 
 
 Copper Losses in Watts. 
 
 Total Secondary 
 Drop in volts 
 
 Primary Watts. 
 
 Power-Factor, 
 
 F= w; 
 
 Primary, 
 
 Ij2x5-95. 
 
 Secondary, 
 I 2 *x 0-0108. 
 
 1 
 
 True Power 
 = W. 
 
 Apparent 
 Power =Wj. 
 
 0-050 
 
 
 
 
 
 
 
 
 
 
 
 
 
 95 
 
 120 
 
 0-79 
 
 o-ioo 
 
 1-00 
 
 0-042 
 
 
 
 
 
 
 
 01 
 
 205 
 
 240 
 
 0-85 
 
 0-140 
 
 1-98 
 
 0-083 
 
 O'l 
 
 
 
 
 
 0-2 
 
 306 
 
 336 
 
 0-91 
 
 0-180 
 
 2-94 
 
 0-122 
 
 0-2 
 
 01 
 
 
 
 0-2 
 
 401 
 
 432 
 
 0-93 
 
 0-218 
 
 3-87 
 
 0-161 
 
 0-3 
 
 0-2 
 
 
 
 0-3 
 
 493 
 
 523 
 
 0-94 
 
 0-250 
 
 4-79 
 
 0-199 
 
 0-4 
 
 0-2 
 
 1 
 
 0-3 
 
 597 
 
 600 
 
 0-99 
 
 0-382 
 
 8-00 
 
 0-333 
 
 0-9 
 
 0-7 
 
 2 
 
 0-3 
 
 920 
 
 917 
 
 1-00 
 
 0-472 
 
 10-15 
 
 0-423 
 
 1-3 
 
 1-1 
 
 2 
 
 0-6 
 
 1,139 
 
 1,133 
 
 1-00 
 
 0-580 
 
 13-07 
 
 0-545 
 
 2-0 
 
 1-8 
 
 4 
 
 0-7 
 
 1,440 
 
 1,392 
 
 1-03 
 
 0-800 
 
 18-00 
 
 0-750 
 
 3-8 
 
 3-5 
 
 7 
 
 0-9 
 
 1,930 
 
 1.920 
 
 1-00 
 
 0-880 
 
 19-90 
 
 0-830 
 
 4-6 
 
 4-3 
 
 9 
 
 0-9 
 
 2,118 
 
 2,112 
 
 1-00 
 
 0-960 
 
 21-93 
 
 0-914 
 
 5-5 
 
 5-2 
 
 11 
 
 1-0 
 
 2,330 
 
 2,304 
 
 1-01 
 
 1-080 
 
 24-74 
 
 1-031 
 
 6-9 
 
 6-6 
 
 14 
 
 1-0 
 
 2,609 
 
 2,592 
 
 1-01 
 
 1-285 
 
 29-66 
 
 1-238 
 
 98 
 
 9-5 
 
 19 
 
 1-0 
 
 3,096 
 
 3,085 
 
 1-00 
 
 1-610 
 
 37-20 
 
 1-550 
 
 15-4 
 
 14-9 
 
 30 
 
 1-2 
 
 3,870 
 
 3,864 
 
 1-00 
 
 1-810 
 
 42-00 
 
 1-750 
 
 19-5 
 
 19-0 
 
 39 
 
 1-5 
 
 4,324 
 
 4,344 
 
 0-99 
 
 2-002 
 
 46-65 
 
 1-945 
 
 23-5 
 
 23-5 
 
 47 
 
 1-7 
 
 4,792 
 
 4,805 
 
 1-00 
 
 2-160 
 
 50-40 
 
 2-100 
 
 23-7 
 
 27-5 
 
 55 
 
 1-8 
 
 5,174 
 
 5,184 
 
 1-00 
 
 2-240 
 
 52-16 
 
 2-171 
 
 29-8 
 
 29-5 
 
 59 
 
 2-0 
 
 5,422 
 
 5,376 
 
 1-01 
 
 2-383 
 
 55-60 
 
 2-320 
 
 33-9 
 
 33-5 
 
 67 
 
 2-1 
 
 5,702 
 
 5,719 
 
 1-00 
 
 2-478 
 
 57-68 
 
 2-404 
 
 36-5 
 
 36-0 
 
 73 
 
 2-2 
 
 5,885 
 
 5,947 
 
 0-99 
 
 .2-550 
 2-633 
 
 59-32 
 61-32 
 
 2-474 
 2-560 
 
 38-7 
 41-2 
 
 38-1 
 40-6 
 
 77 
 82 
 
 2-1 
 2-3 
 
 6,041 
 6,271 
 
 6,120 
 6,319 
 
 0-99 
 099 
 
 2-672 
 
 62-16 ! 2-594 
 
 42-5 
 
 41-8 
 
 84 
 
 2-2 
 
 6,344 
 
 6,413 
 
 0-99 
 
 2-700 
 2-750 
 2-775 
 
 63-00 
 64-00 
 
 64-74 
 
 2-623 
 2-665 
 2-700 
 
 43-3 
 45-0 
 45-8 
 
 42-9 
 44-2 
 45-3 
 
 86 
 89 
 91 
 
 2-3 
 2-3 
 2-4 
 
 6,426 
 6,522 
 6,598 
 
 6,480 
 6,600 
 6,660 
 
 0'99 
 0-99 
 1-00
 
 570 THE INDUCTION COIL AND TRANSFORMER.. 
 
 This may best be illustrated by giving the figures of test of 
 two transformers, one of the closed-circuit type (Westinghouse) 
 and one of the open-circuit type (Swinburne) (see Tables C 
 and D). It will be seen that a very small loading of the 
 closed-circuit type suffices to bring the power factor up to 
 unity. 
 
 Hence it follows that for closed- circuit transformers the 
 apparent power given to the transformer is equal to the real 
 power at and beyond about one-tenth of full load. In Fig. 201 
 are shown three curves illustrating the gradual rise of the 
 power factor towards unity in the case of three types of 
 transformer. In the case of an open-circuit transformer at 
 no stage of the load is the true power taken up by the 
 transformer identical in value with the apparent power given 
 to the transformer. 
 
 Table D. Test of a Swinburne " Hedgehog " Transformer. 
 Primary volts, 2,400 (kept constant). 
 Secondary volts at no load = 102 '00. 
 
 1 
 
 
 Copper Losses in Watts. ^ Primary Watts. 
 
 . 
 
 L 
 
 I| 
 
 m 
 
 >?' 
 
 r 
 
 23 
 
 I! 
 
 4 
 
 rr^ 
 lc 
 
 CO"" 
 
 i- 
 
 If 
 
 ?*,K 
 
 i 
 
 >O 
 
 
 |x 
 
 | X 
 
 J> 
 
 3s 
 
 u 
 
 Ii 
 
 i * 
 
 1 
 
 
 pn 
 
 ^ 
 
 
 H Q 
 
 H 
 
 l 
 
 
 
 0-756 
 
 
 
 
 
 13-7 
 
 
 
 14 
 
 
 
 121 
 
 1,816 
 
 0-07 
 
 0-761 
 
 1-02 
 
 0-042 
 
 14-0 
 
 o-i 
 
 14 
 
 o-i 
 
 228 
 
 1,829 0-13 
 
 0-786 
 
 2-98 
 
 0-124 
 
 14-9 
 
 0-5 
 
 15 
 
 0-2 
 
 420 
 
 1,886 
 
 0-22 
 
 0-811 
 
 4-84 
 
 0-202 
 
 15-8 
 
 1-2 
 
 17 
 
 0-4 
 
 621 1,948 
 
 0-32 
 
 0-829 
 
 6-00 
 
 0-250 
 
 16-5 
 
 18 
 
 18 
 
 0-6 
 
 730 : 1,988 
 
 0-37 
 
 0-862 
 
 8-00 
 
 0-333 
 
 17-8 
 
 33 
 
 21 
 
 0-8 
 
 943 i 2,067 
 
 0-46 
 
 0-911 
 
 10-00 
 
 0-417 
 
 19-9 
 
 5-1 
 
 25 
 
 1-0 
 
 1,152 i 2,188 
 
 0-53 
 
 0-960 
 
 12-00 
 
 0-500 
 
 22-6 
 
 73 
 
 30 
 
 1-2 
 
 1,353 2,301 
 
 0-59 
 
 1-013 
 
 14-00 
 
 0-584 
 
 24-7 
 
 10 -C 
 
 35 
 
 1-3 
 
 1,538 2,432 
 
 0-53 
 
 1-066 
 
 15-88 
 
 0-663 
 
 27-2 
 
 12-9 
 
 40 
 
 1.5 
 
 1,746 2,559 
 
 0-68 
 
 1-133 
 
 17-89 
 
 0-746 
 
 30-9 
 
 163 
 
 47 
 
 1-7 
 
 1,932 ; 2,720 
 
 0-71 
 
 1-197 
 
 19-80 
 
 0-825 
 
 34-5 
 
 20-0 
 
 55 
 
 1-9 
 
 2,129 2,873 
 
 0-74 
 
 1-260 
 
 21-63 
 
 0-902 
 
 38-2 
 
 24-0 
 
 62 
 
 2-1 
 
 2,320 3,022 
 
 0-77 
 
 1-321 
 
 23-58 
 
 0-983 
 
 42-0 
 
 28-3 
 
 70 
 
 2-4 
 
 2,510 
 
 3,172 
 
 0-79 
 
 1-395 
 
 25-46 
 
 1-C60 
 
 46-6 
 
 33-1 
 
 80 
 
 2-7 
 
 2,706 
 
 3,350 
 
 0-81 
 
 1-426 
 
 26-38 
 
 1-098 
 
 48-9 
 
 35-4 
 
 84 
 
 2-9 
 
 2,829 
 
 3,422 
 
 0-83 
 
 1-460 
 
 27-28 
 
 1-136 
 
 51-2 
 
 380 
 
 89 
 
 3-0 
 
 2,890 
 
 3,501 
 
 0-83 
 
 1-498 
 
 28-21 
 
 1-175 
 
 54-0 
 
 40-8 
 
 95 
 
 3-0 
 
 2,993 i 3,596 
 
 0-83. 
 
 1-529 
 
 29-19 
 
 1-215 
 
 56-1 
 
 43-5 
 
 100 
 
 3-2 
 
 3,042 3,666 
 
 0-83 
 
 1-567 | 30-33 
 
 1-262 
 
 59-0 
 
 46-9 
 
 106 
 
 3-3 
 
 3,163 
 
 3,761 
 
 0-84
 
 THE INDUCTION COIL AND TRANSFORMER- 571 
 
 In Fig. 201 the three curves show the progress of increase 
 of the power-factor as the load on the secondary circuit is 
 progressively increased. The upper curve represents the ; 
 growth of power factor (F) for a 6,500-watt Westinghouse 
 transformer. Beginning at 0-8, it rises up to unity at about 
 one-tenth of full load. Hence at and after this load the 
 apparent watts are the same as the true watts, and the 
 real power taken up in the transformer is quite accurately 
 given by the product of the primary terminal pressure and the 
 primary current, mean-square ( ^/mean' 2 ) values being under- 
 
 FIG. 201. Relation of Power Factor to Secondary Output. 
 
 stood. For an open magnetic circuit transformer like the 
 "Hedgehog" the case is quite different. The power factor 
 begins at a value of 0-08 or 0-06, and it never rises up 
 above 0-8. Hence at no stage of the load is the real power 
 taken up by the transformer equal to the " apparent watts." 
 A transformer like the 4,000-watt Kapp appears to occupy an 
 intermediate position, and although it has a medium power 
 factor to start with, its power factor rises up to unity at 
 about half-load. The importance of this fact in alternate- 
 current station working is very great. It shows us, if we 
 have a station wholly supplied with transformers of the type 
 of Mordey, Westinghouse, Thomson-Houston, Ferranti, &c., 
 that the apparent power supplied to the transformers is equal 
 to the real power at any hour when all the transformers are 
 more than one- tenth loaded.
 
 572 THE INDUCTION COIL AND TRANSFORMER. 
 
 The reciprocal of the power factor of a transformer on open 
 secondary circuit is a measure of the reluctance of the mag- 
 netic circuit of the transformer. In the case of a transformer 
 with an air-iron magnetic circuit (open-circuit type) the reluc- 
 tance of the iron circuit is large and the reciprocal of the power 
 factor large also, and may be a number approximating to 
 16 or 17. In the case of a closed iron circuit transformer like 
 the Mordey transformer, with very short magnetic circuit and 
 very small reluctance, the reciprocal of the power factor is 
 very small, and will be a number approximating to 1-2 to 1-4. 
 The introduction of any bad magnetic joint into the iron 
 circuit, or the employment of iron of small permeability, im- 
 mediately decreases the magnitude of the power factor of that 
 transformer. Any joint or break in the magnetic circuit 
 accordingly increases the value of the reciprocal of the power 
 factor, and although this alone will not affect the total core 
 loss in the transformer, it is an indication of the increased 
 reluctance of the magnetic circuit. The advantage of a large 
 power factor is that it involves a small value of the magnetising 
 current of the transformer. In the case of an alternating 
 current station large magnetising current involves additional 
 waste of power in the passage of this current through the 
 distributing mains. This point will be discussed at greater 
 length in connection with the subject of alternating current 
 distribution. 
 
 8. Magnetic Leakage and Secondary Drop. If a trans- 
 former has the mean-square value of the potential difference 
 of its primary terminals kept perfectly constant, whilst at 
 the same time secondary currents of various magnitudes are 
 taken from its secondary coil by altering the resistance of the 
 external secondary circuit, we find that the mean-square value 
 of the potential difference between the secondary terminals of 
 the transformer changes with every change in the secondary 
 load. 
 
 The secondary terminal potential difference (S.P.D.) becomes 
 less as the secondary current and load increases. The diffe- 
 rence between the secondary terminal potential difference at 
 no load and at any load is called the secondary drop of the 
 transformer due to that load.
 
 THE INDUCTION COIL AND TRANSFORMER. 573 
 
 We may represent the variation of secondary drop with 
 secondary load hy a diagram as follows : Let a horizontal line 
 be taken on which are set off distances representing the 
 fractions of the full secondary load, and let vertical ordinates 
 set up at these points represent the value of the secondary 
 potential differences at these loads. For convenience sake 
 we may make these ordinates represent the magnitude of the 
 secondary terminal potential difference diminished by a certain 
 constant amount which is less than the least difference found 
 with full load. For instance, suppose the secondary terminal 
 potential difference at no load is 100 volts and at full load is 
 97 volts, we may make the vertical ordinates represent the 
 terminal potential difference minus 90 volts. The curve 
 
 fir 
 
 94 O 650 1300 1950 2600 3250 3900 4550 5200 5850 650Q- 
 
 Output in Secondary Watts. 
 
 a b is the horizontal line throngh a; Carve ac is the Curve of Drop due to 
 secondary res'stance ; Curve a d, that due to primary resistance ; and Curve ae w 
 the Curve of total Drop. 
 
 FIG. 202. Secondary Drop Curves of 6,500- watt "Westinghouse Transformer. 
 
 defined, as in Fig. 202, by the extremities of these ordinates 
 is called the secondary terminal volt curve, and it shows 
 in a graphical manner the gradually diminishing secondary 
 terminal potential difference as the transformer is loaded 
 up. This " secondary drop " arises from two causes. The 
 first is the loss of potential due to resistance, and the 
 second is the loss of secondary potential due to magnetic 
 leakage. Let the resistance of the secondary coil of the 
 transformer be represented by E 2 , and the secondary current 
 (mean-square value) be represented by (I 2 ). Then E 2 (^) 
 is the loss of voltage due to secondary resistance. If the 
 primary terminal potential difference is kept constant, then,
 
 574 THE INDUCTION COIL AND TBANSFOBMEB. 
 
 over and above the loss of secondary voltage due to the 
 resistance of the internal secondary circuit, there is a portion 
 of the secondary drop which is due to loss of voltage by the 
 resistance of the primary circuit, and, in addition to this, 
 the loss above mentioned, which is due to magnetic leakage. 
 Furthermore, the secondary drop is to a considerable extent 
 dependent, as will be explained presently, upon the form of 
 the curve of primary terminal potential difference. Hence 
 the difference between the potential difference of the secondary 
 terminals of the transformer at no load and full load, primary 
 potential difference being constant, is dependent on four 
 things, viz., upon 
 
 (1) The resistance of the primary circuit ; 
 
 (2) The resistance of the secondary circuit ; 
 
 (3) The magnetic leakage of the transformer as affected by, 
 
 (a) Its construction. 
 
 (6) The form of the curve of primary terminal poten- 
 tial difference. 
 
 The effect called the magnetic leakage in a transformer 
 may be generally described as follows : The primary current 
 creates in the iron core a certain total induction, or in usual 
 language creates a certain number of lines of induction in the 
 core which are linked with the primary circuit. The mag- 
 netising effect of the secondary current is at any instant 
 opposed to that of the primary, and hence creates an induction 
 in the core in an opposite direction. The resultant, or actual 
 induction in the core at any place is due to the difference of 
 the opposed magnetising forces acting on the core. When 
 the transformer has its secondary circuit open the magnetic 
 induction in the core is that due to the primary current only, 
 which is then generally called the magnetising current. When 
 the secondary circuit is closed and a secondary current produced, 
 the rise of induction in that part of the core enveloped by the 
 secondary circuit is delayed, and its maximum value is 
 reduced. The simplest way in which the effect of increasing 
 the secondary current of the transformer can be regarded is as 
 follows : Let us denote by the letter Z : the maximum value 
 of the total magnetic induction in the core which would be 
 produced by the primary current if it acted alone, and by Z 2 
 the same due to the secondary current, these values being 
 the inductions just within that part of the core enveloped by
 
 THE INDUCTION COIL AND TEANSFOEMEE. 575 
 
 the primary and secondary coils respectively. The whole of 
 the induction Z 1 which is linked with the primary coil turns 
 is not, however, linked with the secondary. Let a fraction, say 
 ft Z 1} of this primary induction escape linkage with the secon- 
 dary coil, and a similar fraction, say ft Z 2 , of the secondary 
 induction will escape linkage with the primary coil. Then the 
 total induction linked with the primary coil is Z x - Z 2 (1 - ft), 
 because the induction caused by the primary current is opposed 
 in direction to the induction caused by the secondary current, 
 and the inductions, like the two currents, are opposite in phase 
 and reach their maxima nearly coincidently. Also, for the 
 same reasons, the total induction linked with the secondary 
 circuit is 
 
 ft is called the coefficient of leakage. 
 
 The value of the total induction linked with the primary 
 circuit is therefore the product of the number of primary 
 turns Nj. and the resultant induction Z 1 -Z.,(l-ft), and, 
 similarly, the value of ths total induction linked with the 
 secondary circuit is given by the product of the number of 
 secondary turns N 2 and the resultant induction Z x (I- ft)- Z 2 . 
 Hence we have the relation, 
 
 The total linkage of primary circuits 
 
 and induction traversing it I _ N : {Z l -Z. 2 (l -ft)} _ rp 
 The total linkage of secondary | N 2 {Z a (1 - ft) - Z 2 f 
 circuit and induction traversing it J 
 
 It will be shown presently that this fraction T represents 
 the ratio of the mean-square value of the primary terminal 
 potential difference to that of the secondary terminal potential 
 difference. 
 
 This ratio, which is denoted by T, is called the transforma- 
 tion ratio of the transformer. Since the difference between 
 Z 2 and Z 2 remains nearly constant as Z 1 and Z 2 increase, it is 
 easily seen that the transformation ratio increases as Z l and Z 2 
 increase, subject to the condition that Z a Z 2 is nearly constant 
 at all loads. 
 
 Hence, if the mean-square value of the primary electro- 
 motive force is kept constant, that of the secondary potential
 
 576 THE INDUCTION COIL AND TRANSFORMER. 
 
 difference decreases as the currents, and therefore the 
 inductions, in the core increase, and this effect is called the 
 "secondary drop." 
 
 The predetermination of the magnetic leakage of a trans- 
 former is a matter of some difficulty, and can only be antici- 
 pated in certain limited cases. We can obtain the relation 
 between the leakage drop, the resistance drop, and the 
 total drop if we assume an approximately simple periodic 
 variation of the electromotive forces, currents and inductions, 
 as follows : 
 
 Let E : be the true resistance of the primary circuit and 
 B 2 that of the secondary circuit of the transformer, and let S 
 be the cross-section of the magnetic circuit or core. Let N : 
 be the number of primary turns, N 2 the number of secondary 
 turns, and a stand for the ratio of Nj to N a . Let b : be at any 
 instant the induction density in that part of the core enveloped 
 by the primary coil, and b. 2 that part enveloped by the secon- 
 dary coil; the difference between these inductions may be 
 called the density of the leakage of induction, and be denoted 
 by b. Hence 
 
 &-V-** 
 
 In other words, if S is the cross-section of the core, then 
 S b = S &! - S b z and S b represents that part of the induction 
 linked with the primary coil which is not linked with the 
 secondary coil. If <? x is the primary terminal potential diffe- 
 rence at any instant, and e 2 that of the secondary terminal at 
 the same instant, and % and i 2 the currents at the same 
 moment, then, by fundamental equations, we have 
 
 , ..... (146) 
 and = R 2 ? 2 + f 2 + SN 2 i^. . . . (147) 
 
 Let us write ~ = a and b = b 1 -b a ; 
 
 N 2 
 
 we have by elimination from the fundamental equations the 
 result 
 
 ^ + e a + E 2 / 2 -E 1 ^=SN 2 ^. . . (148) 
 a a at
 
 THE INDUCTION COIL AND TRANSFOEMER. 577 
 
 If e l varies in a simple periodic manner so that e l = E x sin p t, 
 then, since e z is always opposite in phase and similar in form 
 to v lt we must have 
 
 e z = - E 2 sin p t. 
 
 Moreover, when the transformer is fully loaded, the currents 
 *! and i 2 are in step with the electromotive forces e l and e t , 
 but * 2 differs ISOdeg. in phase from i^. 
 
 Hence i 1 = 
 
 We can also write b = - B sin^ t, because the leakage b is deter- 
 mined by, and is in step very nearly with, the secondary 
 current. Hence, by substitution of the above values in the 
 equation (148) we arrive at the equation 
 
 ?i - E 2 - E 2 1 2 - E^A sin p t = - S N 2 ;> B cos pt. 
 a a/ 
 
 The quantity S N 2 j9 B cos p t is the instantaneous value of the- 
 potential difference of the secondary circuit lost by leakage 
 that is to say, it is the measure of the amount by which the 
 secondary terminal potential difference would be increased if 
 there were no leakage. Hence the left-hand side of the above 
 equation represents the same thing. The factors which mul- 
 tiply the sin p t and cos p t respectively in the above equation 
 give, therefore, the maximum value of the "leakage," and there- 
 fore, when divided by \/2, represent the mean-square value. 
 
 Hence the quantity i -Eg-K^-R or, which 
 
 \ a. 
 
 comes to the same thing, the quantity 
 
 represents the mean-square ( ^/mean 2 ) value of the loss of 
 potential difference of the secondary circuit due to magnetic 
 leakage when the potential difference is measured in volts. 
 
 Tjl Tjl T T 
 
 The quantities -^i -4. -4r -j= represent the magnitude 
 of the currents and potentials as read in alternating-current
 
 578 THE INDUCTION COIL AND TRANSFORMER. 
 
 ammeters and voltmeters. We may denote these mean-square 
 values by the symbols (Ej), (E 2 ), (Ij), (I 2 ), and the values 
 which these mean-square potential differences and currents 
 have at full and at no .secondary load by the symbols (EJ,, 
 (E 2 ) f , (Ei),,, (E 2 ) , &c. 
 
 The total loss of secondary terminal voltage between full 
 and no secondary load will be given by the difference between 
 the values of the expressions 
 
 and {1 (EJ. - (E 2 ) - R 2 (I 2 ) - 1 
 
 The value of (I 2 ) is, of course, zero. 
 
 If the primary terminal potential difference is the same at 
 no load as at full load, we have for the secondary drop due to 
 magnetic leakage the expression 
 
 (E 2 ) - (E,),- (B, (I 2 V + 1 B, (I,),- - B, (U) - (149) 
 
 a a 
 
 The secondary terminal volts at full load being denoted by 
 (E 2 )/, and that at no load by (E 2 ) , we see that the quantity 
 (E 2 ) (E.,);- represents the total secondary drop due to all 
 causes. The quantity 
 
 therefore represents that part of the drop due to the resistance 
 of the primary and secondary circuits. 
 
 Hence we have the following rule for determining the drop 
 due to magnetic leakage : Add together the product of the 
 
 secondary resistance and secondary current and - multiplied 
 
 a 
 
 into the product of primary resistance and primary current, 
 after deducting from the last value the primary current at no 
 load. Subtract this sum from the total observed drop, and 
 the remainder is the secondary potential difference due to 
 
 Testing in this way a number of transformers, the author 
 found that where the primary and secondary circuits were
 
 THE INDUCTION COIL AND TRANSFORMER. 579 
 
 intermixed, the magnetic-leakage drop was small, but that 
 where the primary and secondary circuits were separated, and 
 in each consisting of one coil only, the magnetic leakage drop 
 was large. 
 
 In the diagram in Fig. 202 are shown three curves, by which 
 the three sources of secondary drop have been distinguished, 
 the lines ac, ad showing the curves of drop due respec- 
 tively to the primary and secondary resistance, and the curve 
 e showing the total drop. 
 
 In designing a transformer, it is not permissible to purchase 
 small core loss at the expense of large secondary drop. In a 
 proper specification for a transformer a limitation should be 
 put upon the amount of secondary drop allowed, and it is 
 usual to express it as a percentage of the normal potential 
 difference or voltage of the secondary circuit when the trans- 
 former is unloaded. 
 
 It is advantageous to so arrange the winding of the 
 secondary circuit that if the drop is, say, 2 per cent., and 
 the secondary-circuit voltage is 100, that the transformer shall 
 give 101 volts terminal pressure at no load, and 99 volts at full 
 load. In this way the full drop is divided and is not felt so 
 much in working on 100-volt lamps as if the transformer were 
 wound to give 100 volts at no load and 98 at full load. 
 
 9. Effect of the Form of the Curve of Primary Electro- 
 motive Force in the Transformer Efficiency and Currents. It 
 
 has generally been assumed by many of those who have written 
 on the subject of the alternate-current transformer that the 
 efficiency, power factor and secondary drop were characteristics 
 of the transformer only. It has already, in previous sections, 
 been suggested that the form of the curve of primary potential 
 difference or primary electromotive force had a considerable 
 effect in modifying the value of these quantities, and it will 
 now be necessary to examine the matter a little more in detail. 
 We will consider in the first place the effect of the form of 
 primary terminal potential difference upon the form of the 
 current curves and magnitude of the mean-square value of the 
 currents. 
 
 The widely -different forms which the primary current of a 
 transformer on open secondary circuit may have is shown in 
 
 pp 2
 
 580 THE INDUCTION COIL AND TRANSFORMER. 
 
 the diagrams in Figs. 203 and 204. Fig. 203 shows the 
 primary-current curve of a small transformer taken off a Ganz 
 alternator having a peaked curve of electromotive force. The 
 curves in Fig. 204 show the primary electromotive force and 
 primary- current curves of the same transformer taken from a 
 
 200 
 160 
 
 <: 
 
 
 
 60 
 
 
 
 OS 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 H 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / JA 
 
 3 * 
 
 *>x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 l/r 
 
 
 
 X, 
 
 
 
 
 
 
 
 
 
 
 
 
 xf 
 
 <J 
 
 iS" 
 
 
 \ 
 
 
 
 
 "^- 
 
 -~. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 ^ 
 
 
 
 a 
 
 
 
 
 
 
 
 
 
 
 ^~< 
 
 
 
 
 
 3 
 
 
 
 
 
 5 
 
 ^-. 
 
 >^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \j 
 
 40 50 60 70 80 90 100 110 120 130 140 
 
 Degrees of Phase. 
 
 FlO. 203. Primary Current Curve II and Primary Electromotive Forca 
 Curve I of a Transformer taken off Ganz Alternator. 
 
 ^4 
 
 10 20 30 40 50 60 70 SO 90 100 
 
 Degrees of Phase. 
 
 FIG. 204. Primary Current Curve II and Primary Electromotive Force 
 Curve I of the same Transformer taken off Wechsler Alternator. 
 
 Wechsler alternator. These and the following curves are 
 from an interesting Paper by Dr. G. Koessler.* 
 
 * "Das Verhalten von Transformatoren unter den Einflusse von 
 Wechselstromen Verschiedenen Periodischen Verlaufs." A Paper read at 
 the third annual meeting of the Verband Deutscher Electrotechniker, 
 Munich, July 6, 1895. See also The Electrician, Vol. XXXVI., 1895, p. ISO.
 
 THE INDUCTION COIL AND TRANSFORMER 581 
 
 Not only do the forms of the current curves differ when 
 taken with different-shaped electromotive force curves, but if 
 the primary electromotive force is kept at the same 'mean- 
 square value, and if the transformer is gradually loaded up, the 
 mean-square values of the primary current corresponding to 
 given secondary currents will differ if the curves of primary elec- 
 tromotive force have different forms. This is shown in Fig. 205, 
 where the ordinates represent the mean-square values of the 
 secondary and primary currents of one and the same trans- 
 former, taken off a Ganz and Wechsler machine respectively. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J 
 
 /, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 '/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / n 
 
 / 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 s , 
 
 
 . 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 'S 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fT 
 
 x 
 
 / 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 / 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 '/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 x 
 
 ' 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ; 
 
 x 
 
 / 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 5 
 
 l 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x 
 
 x 
 
 ' 
 
 \ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x 
 
 
 X 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^* r 
 
 ,<- 
 
 ? 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L^ 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0-8 1-2 1-6 2-0 2'4 2'8 8"2 
 Secondary Current Amperes. 
 
 3'6 4'0 44 
 
 FIG. 205. Current Diagram of a certain Transformer. Curve I, Primary 
 Current taken with Ganz Alternator. Curve II, Primary Current taken 
 with Wechsler Alternator. Curve III, twice value of Secondary Current. 
 Transformation ratio of Transformer = 2 :1. 
 
 It is thus seen that the peaked electromotive force curve gives 
 a primary current with smaller mean-square value than a 
 rounded curve. 
 
 The most important fact, however, is that the iron core loss 
 in the transformer, and therefore its efficiency, is sensibly 
 affected by the form of the curve and primary electromotive 
 force. In Fig. 206 are shown two curves, the ordinates of 
 which represent the total power given to a transformer for 
 Certain values of the secondary output represented by the
 
 582 THE INDUCTION COIL AND TRANSFORMER. 
 
 abscissae, the transformer being tested in the two cases on 
 the Ganz and Wechsler alternators. 
 
 There is between the two power lines a nearly constant 
 difference of ordinate, showing that the cause is to be sought 
 in the difference between the iron core losses in the two cases. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^. 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <; 
 
 '^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 > 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 .^ 
 
 z- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 -& 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 & 
 
 ** 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 100 200 30 4UO 500 
 
 Secondary output in Watts. 
 
 FIG. 206. Energy Diagram of Transformer. Curve I, Ganz Alternator. 
 Curve II, Wechsler Alte nator. 
 
 100 200 
 
 Secondary output in Watts. 
 
 FlG. 207. Efficiency Curve of Transformer. Curve I taken on Ganz 
 Alternator. Curve II taken on Wechsler Alternator. 
 
 It follows that both the efficiency curves and power-factor 
 curves of the transformer plotted in terms of the secondary 
 output will differ if the form of the primary electromotive 
 force curve is varied. 
 
 In Figs. 207 and 208 are shown the forms of the efficiency 
 and power-factor curves of the same transformer when taken 
 off the Ganz machine with peaked electromotive force curve 
 and the Wechsler machine with a rounded curve. 
 
 We find also that the secondary drop is considerably affected 
 by the form of the primary electromotive force curve. In
 
 THE INDUCTION COIL AND TRANSFORMER. 583 
 
 Fig. 209 are shown the secondary drop curves of the same 
 transformer taken on the above-mentioned alternators, the 
 primary electromotive force having in each case the same 
 constant mean-square value. 
 
 It is clear, therefore, that a peaked electromotive force curve 
 of the type given by the Ganz alternator causes a less iron 
 core loss but a greater magnetic leakage than does a curve of 
 a more rounded form similar to that of the Wechsler machine. 
 Numerous tests and experiments made by the author with the 
 Mordey and Thomson-Houston alternators had established this 
 fact prior to the appearance of the Paper by Dr. G. Roessler, 
 
 120 160 200 240 280 320 
 Secondary output in Watts. 
 
 400 460 480 
 
 FIG. 208. Power Factor Curves of a Transformer. Curve I taken on 
 Ganz Alternator. Curve II taken on Wechsler Alternator. 
 
 200 300 
 
 Secondary output in Wattt. 
 
 FIG. 209. Curve' of Secondary Drop of Transformer. Curve I taken on 
 Ganz Alternator. Curve II taken on Wechsler Alternator. 
 
 from which the above transformer diagrams, taken off the Ganz 
 and Wechsler alternators, are copied. It is generally true that 
 a sharp-peaked electromotive force curve gives a less hysteresis 
 loss in the iron than does a rounded or sine electromotive 
 force curve having the same mean -square value. 
 
 10. The Form Factor and Amplitude Factor of a Periodic 
 
 Curve. The above differences are closely connected with the 
 magnitude of the form factor of the curve of primary electro- 
 motive force. This quantity is denned as the ratio of the
 
 684 TEE INDUCTION COIL AND TRANSFORMEE. 
 
 square root of the mean of the squares of the equispaced 
 ordinates of a curve to the true mean value of the equispaced 
 ordinates. If we denote the first function, viz., the mean- 
 square value, by the letters R.M.S. (root mean square), and 
 the second function by the letters T.M. (true mean), then the 
 form factor of any single-valued periodic curve is defined as 
 follows : 
 
 The form factor = The R.M.S. value of equispaced ordinates = 
 The T.M. value of equispaced ordinates 
 
 Take, for instance, in the case of a simple sine curve, the 
 R.M.S. value of the equispaced ordinates is equal to the value 
 of the maximum ordinate divided by ^/2. The T.M. value 
 
 of the equispaced ordinates is equal to the value of the maxi- 
 
 o 
 mum ordinate multiplied by -. 
 
 7T 
 
 Since 
 
 -^=0-707 and f= 0-637, 
 
 the ratio of the R.M.S. value to the T.M. value for a simple 
 
 sine curve is 
 
 0-707 
 0-637 
 
 1-1. 
 
 For several other simple forms of curve the form factor, 
 R.M.S. value, and T.M. value are as below, the maximum 
 ordinate in each case being taken as unity : 
 
 
 T.M. value of 
 
 R.M.S. value 
 
 
 Curve. 
 
 ordinate as 
 
 of ordinate as 
 
 Form factor 
 
 
 fraction of max. 
 
 fraction of max. 
 
 /. 
 
 
 ordinate. 
 
 ordinate. 
 
 
 Sine 
 
 0-637 
 
 0-707 
 
 1-1 
 
 
 0-7854 
 
 0-835 
 
 1-063 
 
 Triangle 
 
 0-5 
 
 0-58 
 
 1-16 
 
 Rectangle 
 
 1-0 
 
 1-0 
 
 1-0 
 
 
 0-785 
 
 0'816 
 
 1-039 
 
 Parabola with axis 
 
 
 
 
 vertical 
 
 0-666 
 
 0-730 
 
 1-096 
 
 Two semi - parabolas 
 
 
 
 
 meeting at a cusp. . 
 
 0-33 
 
 0-447 
 
 1-35 
 
 The form factor of any curve can easily be obtained geome- 
 trically as follows : On one side of a straight line (see Fig. 210) 
 plot a wave diagram of the curve, and on the other side of the
 
 THE INDUCTION COIL AND TRANSFOKMEB. 585 
 
 line plot a polar diagram of the same curve, with its pole on 
 the line of reference. Then, by the proposition on page 193, 
 the radius of the semicircle, so drawn that its area is equal to 
 the area of the polar curve, is the E.M.S. value of the 
 ordinates, and the height of the rectangle described on the 
 base line, so that its area is equal to that of the wave curve, 
 gives the T.M. value of the ordinates. Hence the ratio of the 
 radius of the semi- circle to the height of the rectangle is the 
 form factor of the curve, which is represented by the wave or 
 , polar diagram. 
 
 \ 
 \ 
 
 u 
 
 f 
 
 
 1! 
 
 
 
 
 9 
 
 
 
 6 
 
 
 
 3 
 
 " 
 
 / X 
 
 
 \ 
 
 NS 
 
 , 
 
 
 
 ^ 
 
 S 
 
 / 
 
 ^ 
 
 ' 
 
 
 
 
 
 
 FIG. 210. 
 
 This form factor is an important quantity in the design of 
 alternators, and by suitably proportioning the width of arma- 
 ture coils and field poles the form factor can be varied within 
 wide limits. 
 
 It is evident that for the same R.M.S. value the form 
 factor will be greater if the curve is a sharp-peaked curve 
 than if it is a rounded curve like a semicircle or sine curve. 
 
 If some of the ordinates of any curve are increased so as 
 to form a peak, this operation, geometrically considered, 
 increases the R.M.S. value faster than it increases the T.M. 
 value, and so increases the form factor of the curve.
 
 586 THE INDUCTION COIL AND TEANSFORMER. 
 
 The amplitude factor of a periodic curve is defined as the 
 ratio between the root mean-square (B.M.S.) value of the 
 ordinates and the value of the maximum ordinate, or 
 
 The amplitude) _E.M.S. value of equispaced ordinates _ f 
 factor j Value of maximum ordinate 
 
 For the same E.M.S. value the amplitude factor is less for a 
 sharp-peaked curve than for a rounded or flat curve. 
 
 These two factors the form factor/ and the "amplitude 
 factory are important quantities in the case of periodic curves. 
 
 11. General Analytical Theory of the Transformer and 
 Induction Coil. It remains, then, to indicate the manner in 
 which the various periodic and fixed quantities concerned in 
 the action of the transformer are connected and how they can 
 be determined. 
 
 For convenience we may collect together the symbols 
 employed to represent the various quantities with which 
 we are concerned. 
 
 e 1 = The value of the primary terminal potential dif- 
 ference or primary E.M.F. at any instant. 
 E x = The maximum value of the same. 
 me 1 = The true mean (T.M.) value of e l during the 
 
 period. 
 
 ,Jme^ = Th& root-mean-square (R.M.S.) value of e l during 
 the period. 
 
 the same quantities for the secondary terminal 
 .- potential difference. 
 
 Ej = The resistance of the primary circuit. 
 E 2 = The resistance of the secondary circuit. 
 Nj = The number of turns on the primary coil. 
 N 2 = The number of turns on the secondary coil. 
 6 1= =The density of magnetic induction in the core 
 
 inside primary coil. 
 
 Bj = The maximum value of induction density b r 
 Z l = The total induction produced in the core due to 
 
 primary coil. 
 6 2 , B 2 , Z 2 are the same quantities for the secondary circuit.
 
 THE INDUCTION COIL AND TRANSFORMER. 58T- 
 
 **! = Primary current at same instant that the primary 
 
 terminal potential difference is e v 
 Ij = Maximum value of i r 
 ij = True mean value of i l during the period. 
 
 Root-mean-square value of ^ during the period. 
 m i 2 , J^ijz are the same quantities for the secondary 
 
 circuit. 
 
 /4 = Magnetic force due to the primary current i v 
 Hj = Maximum value of 7i r 
 Ji. 2 = Magnetic force due to the secondary current. 
 H 2 = Maximum value of A 2 . 
 X = Total power loss in watts in the iron core. 
 Y = Hysteresis loss in watts in the iron core per cubic 
 
 centimetre. 
 U = Eddy-current loss in watts in the core per cubic 
 
 centimetre. 
 
 V = Total volume of the iron core. 
 S = Cross-sectional area of iron core. 
 I = Mean length of magnetic circuit. 
 f= The form factor = E.M.S -r- T.M. value. 
 /7 = The amplitude factor = E. M.S. -5- maximum value. 
 n = The frequency. 
 p = 2;r n = the angular velocity. 
 T = periodic time = n~ l . 
 
 Then the fundamental equations are as follows : 
 When the secondary circuit is open and the transformer, 
 therefore, at no load, we have 
 
 l ..... (151) 
 
 The above equation holds good also when the transformer 
 is loaded up, provided we then interpret b l to mean the 
 resultant induction density in the iron core as affected by the 
 current in the secondary coil. 
 
 In all good modern closed iron circuit transformers the 
 value of K a ^ is so small at all times during the period, when 
 compared with e v that we may without sensible error write 
 
 Hence e l dt=SK 1 db l . . r . . (152)
 
 588 THE INDUCTION COIL AND TEANSFOEMEE. 
 
 If we integrate this last equation throughout one-quarter 
 period we have already seen that 
 
 SN -D H. i. ... . T 
 
 n 
 1 B 1 = I e L dt = 
 
 J n 
 
 4 
 
 but T = *. 
 
 n 
 
 Hence 4S N 1 B 1 w = .^ (153) 
 
 But if/ is the form factor of the curve of primary potential 
 difference, then 
 
 Therefore, from equations (153) and (154), we have 
 
 >/W7*? = 4/N 1 SB lf . . . (155) 
 
 which gives us the E.M.S. value of the primary potential 
 difference in terms of the maximum value of the induction 
 density in the core within the primary coil. 
 
 If the secondary circuit of the transformer is closed, and a 
 secondary current is being taken from the transformer, then 
 the currents, inductions and potentials are determined by the 
 two equations, 
 
 and = K 2 t 2 + c 2 + N 2 S ' (157) 
 
 If the secondary circuit is open, and hence R 2 i 2 equal to zero, 
 
 and Rj ^ practically negligible in comparison with N 2 S i-^, we 
 may write (156) and (157) 
 
 XT 
 
 and %= -j 
 
 and, therefore, as already shown, 
 
 Bp .... (158) 
 and m.<> 2 = 4wN 2 SB 2 ..... (159)
 
 THE INDUCTION COIL AND TRANSFORMER. 589 
 
 In the previous section we have shown that the total 
 induction S B 1 linked with the N x primary turns may be 
 expressed as (Z x - Z 2 + Z 2 y3), where Z l is the maximum value 
 of the total induction due to the primary current alone, and 
 Z 2 that due to the secondary current alone. 
 
 Hence, as before, writing Z l -Z 2 + Z 2 (3 for S B x , and 
 Z l -Z 2 -Z 1 (3 for S B 2 , we have, by substitution in equations 
 (158) and (159), the results 
 
 and m . e 2 = 4 n N 2 ( 
 
 It is an experimental fact that the curve of secondary 
 potential is an exact copy of the curve of primary potential at 
 no load, and very nearly also at any load ; hence the form 
 factors of the curves of primary and secondary potentials are 
 the same. Writing /for this form factor we have 
 
 and 
 
 Hence the transformation ratio of the transformer T is given 
 by the equation 
 
 T _ ^'^7_N 2 (Z 1 -Z 2 -Z 1 /?) , 
 
 - ' 
 
 Accordingly we see that the transformation ratio of the 
 transformer is never exactly equal to the ratio of the turns 
 unless the leakage coefficient (3 is zero, and that the trans- 
 formation ratio diminishes as Z x and Z 2 increase with load, 
 because their difference Zj Z 2 always remains approximately 
 constant at all loads, and is the mean core induction. 
 
 The leakage coefficient /? is a function of the form factor/, 
 such that /? is greater as / is greater. Thus, peaked primary 
 potential curves give greater secondary drop than rounded 
 potential curves, even if they have the same E.M.S. value. 
 
 From equation (155) we see that the maximum value of the 
 core induction B, either within the primary or secondary coil, 
 is smaller in proportion as the form factor / is greater, if the 
 B.M.S. value of the primary potential remains constant.
 
 90 THE INDUCTION COIL AND TfiANSFOKMER. 
 
 Hence the maximum value B of this core induction is less 
 for pointed or peaked primary electromotive force curves than 
 for rounded or flat curves, the R.M.S. value of the primary 
 terminal potential difference being constant. 
 
 This has been experimentally proved by Dr. Roessler in his 
 researches on the influence of the form of the potential and 
 current curves on transformer action. 
 
 In Fig. 211 are shown three curves. The curve marked I 
 is the magnetisation curve of a small transformer measured 
 with the ballistic galvanometer in the ordinary way. The curve 
 marked II is the curve of induction as obtained with alter- 
 nating currents, using a Wechsler alternator, and the curve 
 marked III that obtained in the same way, but by the use of 
 
 16,000 
 14,000 
 12,000 
 10,000 
 | 8,000 
 6,000 
 4,000 
 2,000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L*. ' 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 r ^~ 
 
 .^^ 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 /^ 
 
 r*> 
 
 , ' 
 
 
 
 
 ' 
 
 ftf 
 
 
 
 
 
 
 
 
 
 / 
 
 r /- 
 
 
 
 
 ^ 
 
 ** 
 
 
 
 
 
 
 
 
 
 
 i f 
 
 / 
 
 /v 
 
 
 ^ 
 
 P 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 f 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 1 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ! 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f ( 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 (/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 8 12 16 20 24 28 32 
 
 Magnetisitiy Force. 
 
 FIG. 211. 
 
 a Ganz alternator. The values of the maximum induction 
 B for the alternating currents are obtained from the primary 
 electromotive force curves, as already described. 
 
 It is seen that the curves of induction as obtained by the 
 alternating-current machines lie below that obtained by the 
 continuous currents and ballistic galvanometer. 
 
 In other words, for a given induction, the magnetising force 
 required is greater with alternating than with continuous 
 currents. There are two reasons for this : first, the existence 
 of eddy currents in the core, which, acting like smaller closed
 
 THE INDUCTION COIL AND TRANSFORMER, 591 
 
 secondary currents, increase the primary current, and, second, 
 the existence of magnetic leakage when alternating currents 
 are used. The curves show, however, that when the peaked 
 form of primary electromotive force curve given by the Ganz 
 machine is used, the induction corresponding to a given mag- 
 netising force is less than when the Wechsler machine with 
 rounded electromotive force curve is employed, the sameR.M.S. 
 values of the primary electromotive force being employed. 
 
 The root-mean-square values of the primary and secondary 
 currents, viz., ^/m . if and Jm . i^, are connected with the 
 maximum values of these variables by the equations 
 
 and 
 
 where g is the quantity already called the amplitude factor. 
 
 It has been shown that for peaked or pointed curves the 
 amplitude factor is smaller than for flat or rounded curves. 
 Hence for the same root mean-square value of the primary 
 and secondary currents the maximum values of these quantities 
 are greater for peaked current curves than for rounded or flat 
 curves. Hence the inductions created by these currents 
 respectively are greater that is, Z x and Z 9 will be greater for 
 peaked current and potential curves than for flat or rounded 
 curves. 
 
 Accordingly, whilst the respective primary and secondary 
 inductions Z l and Z are greater for electromotive force curves 
 with large form factors, their difference, Z t - Z 2 , which is the 
 resultant core induction B, is less. Hence we see that the 
 secondary drop, or increase of transformation ratio produced 
 by loading up the transformer must be greater when the 
 primary electromotive force curve is peaked than when it is 
 rounded, the same mean-square value of this last being 
 preserved constant. 
 
 We have, then, to discuss the form of the curves of primary 
 current under variations of form factor of the electromotive 
 force curve. 
 
 If /<! is the instantaneous value of the magnetising force due 
 to the primary current i v then 
 
 10
 
 592 THE INDUCTION COIL AND TRANSFORMER. 
 and on open secondary circuit we have 
 
 Hence ^-B^N.S,.., 
 
 ^4^8^.*... . . (161) 
 
 This last equation is true for all forms of primary electro- 
 motive force curves. 
 
 If the permeability of the iron was constant, the value of 
 
 1 would be constant and equal to a, but in practice it is not 
 dh i db, 
 
 found to be constant. We see, however, that ^- is the slope 
 
 of the geometrical tangent to the hysteresis curve at the 
 instant considered that is, it is the trigonometrical tangent 
 of the angle which the geometrical tangent to the hysteresis 
 curve makes with the positive direction of the axis of time, 
 and this is not found to be a constant quantity as we travel 
 
 round the hysteresis curve. The value of - 1, however, is not 
 
 dh^ 
 
 greatly affected by the form of the curve of e 1 . Hence, for 
 curves of primary electromotive force which have a peaked 
 form, and therefore a large maximum value, the value of 
 
 ?i, or the slope of the current curve, will be greater than for 
 d t 
 
 flatter curves of electromotive force. This is seen to be the 
 case by reference to Figs. 203 and 204, which show the no- 
 load primary-current curves and primary electromotive force 
 curves of the same transformer tested by Dr. Koessler on the 
 Ganz and the Wechsler alternator. 
 
 The exact predetermination of the form of the primary 
 current at no load from the curve of primary electromotive 
 force is, at any rate as yet, an impossible matter. It would 
 be an easy thing to predetermine if the hysteresis curve 
 always had the same form, but as this last is affected to 
 a considerable extent by variations in the quality of the 
 iron and of the reluctance of the magnetic circuit, it is
 
 THE INDUCTION COIL AND TRANSFORMER. 593 
 
 not of much use to make assumptions which are not justified 
 in practice. 
 
 A knowledge of the power factor of transformers of any 
 particular type will always enable us to make an approximation 
 to the value of the magnetising current if the total power 
 taken up in the core is known and the mean-square value of 
 the primary electromotive force. For if X is the total power 
 taken up in the core at no load, and ,Jm^ t J^T* are the 
 E.M.S. values of the primary electromotive force and current, 
 
 v- 
 
 and F is the power factor, then F = - j f ro m 
 
 which ,J m . ^2 can be obtained. 
 
 It is seen, however, that the primary current curve at no 
 load is always a more irregular curve than the curve of 
 primary electromotive force, and that for the same B.M.S. 
 value of the primary electromotive force the B.M.S. value of 
 the primary current at no load (the magnetising current) is 
 less for pointed or peaked potential curves than for flat 
 curves. 
 
 12. Iron Core Loss in Transformers and Induction Coils. _ 
 It has already been explained that two distinct causes of 
 energy dissipation exist in the iron cores of transformers and 
 induction coils viz., the magnetic hysteresis loss and the 
 eddy-current loss. The former of these is not affected or 
 diminished by any amount of lamination of the core, but 
 the latter can be reduced to a very small percentage of the 
 total loss by constructing the core of iron plates of thickness 
 not greater than 0-014 inch, the plates being separated from 
 each other by very thin paper, or a layer of paint or varnish. 
 In the chapter in the Second Volume of this Treatise devoted 
 to the Construction of the Transformer, the various practical 
 details connected with the core construction, and the pre- 
 determination of the core loss for plates or wires of given size 
 are considered. 
 
 Supposing, however, that the core is properly laminated, 
 and in planes parallel to the lines of induction in the core, 
 there will still be a certain dissipation of energy, by reason of 
 eddy electric currents set up in the iron as the induction 
 changes its direction. If we consider a small circuit described 
 
 QQ
 
 594 THE INDUCTION COIL AND TBANSFOBMEB. 
 
 anywhere in the iron plate, in a plane perpendicular to the 
 lines of induction, then, if e is at any instant the electromotive 
 force set up in this circuit by reason of the variation of the 
 induction through it, the mean rate at which energy is being 
 dissipated in this circuit must be equal to some constant, 
 multiplied by the value of the mean of the square of e. But 
 we have seen that if B is the maximum value of the induction 
 in the core, then the E.M.S. value of the electromotive force 
 of induction induced in the primary circuit is equal to the 
 value of the expression 4/N x .n S B, where/ is the form factor 
 of the curve of electromotive force. 
 
 Hence the mean-square value (w.e z ) of the electromotive 
 force of induction must be numerically proportional to/ 2 2 B 2 ; 
 also the same holds good for the eddy-current electromotive 
 force and rate of energy dissipation, and the eddy current loss 
 per unit of volume in the core measured in watts must be 
 proportional to the product of some constant and the 
 quantity /- ;i 2 B 2 . In other words, the eddy-current loss will 
 be equal to / 2 w 2 B 2 watts per unit of volume of the core, 
 where / is the form factor of the curve of primary electro- 
 motive force, n the frequency, and B the maximum value of 
 the induction. 
 
 Mr. Steinmetz has shown that the hysteresis loss in iron 
 cores can be represented by an arbitrary formula, expressing 
 the fact that the hysteresis loss per unit of volume of the 
 core is proportional to the product of a constant, the frequency, 
 and the maximum value of the induction raised to a power 
 very near to 1-6. Hence, if H is the hysteresis loss in the 
 core per unit of volume, 
 
 where 17 is called the hysteretic constant of the iron. 
 
 This law, although only an empirical one, deduced entirely 
 from observation, yet appears to be sufficiently exact to guide 
 practice within the limits of the range of induction density 
 employed in transformers. 
 
 Hence the total loss T in a transformer core of volume V 
 is given by the expression 
 
 / 2 B 2 ). . . . (162)
 
 THE INDUCTION COIL AND TRANSFORMER. 595 
 
 The eddy-current loss varies as the square of the maximum 
 value of the induction, and the hysteresis loss as the l-6th 
 power of the same. 
 
 Since the E.M.S. value (J^) of the primary electro- 
 motive force has been shown to be related to B by the 
 equation 
 
 we can substitute for B, in equation (162), its value in terms 
 of Vmej^i and we arrive at the equation 
 
 This last equation shows us that the eddy-current loss is 
 not affected by the form factor of the curve of primary electro- 
 
 s s s" 
 
 Maximum Value of Core Induction. 
 
 FIG. 212. Curve I taken with Ganz Machine. Curve II taken with 
 Wechsler Machine. 
 
 motive force, but that the hysteresis loss is affected by it, 
 because the form factor / appears in the hysteresis term of 
 the expression for T but not in the eddy current term. 
 
 Hence variation in the form factor of the curve of primary 
 electromotive force will alter the total core loss in the trans- 
 former, and make it less in proportion as the form factor is 
 greater. This has been pointed out both by the author and 
 by Dr. G. Eoessler, and is amply confirmed by experiment. 
 
 Hence the total core loss in a transformer is not an 
 absolute and fixed quantity, but depends upon the form of 
 the wave of primary electromotive force to a not inconsiderable 
 decree. 
 
 QQ2
 
 596 THE INDUCTION COIL AND TRANSFORMER. 
 
 This dependence of the core loss upon the form factor of 
 the curve of electromotive force is shown by the results of 
 Dr. Roessler's experiments embodied in the diagrams in 
 Figs. 206 and 212. From Fig. 212 it will be seen that at 
 a given induction the core loss is greater when the trans- 
 former is worked off the Ganz alternator than off the 
 "Wechsler alternator ; and from Fig. 206 it is shown that for 
 the same R.M.S. value of the primary potential difference 
 the core loss is greater with the Wechsler than with the 
 Ganz alternator. 
 
 Broadly speaking, pointed or peaked potential curves give 
 rise to greater core loss than rounded or flat primary potential 
 curves. 
 
 In the Second Volume of this Treatise, we return to the 
 discussion of these matters, and enter into more details as 
 to the practical considerations to which they lead in the 
 Construction of the Induction Coil and Transformer. 
 
 END OF VOLUME L
 
 APPENDIX. 
 
 NOTE A. (See page 19.) 
 
 The Magnetic Force at any Point in the Plane of a Circular 
 'Current. The magnetic force at any point in the plane of a 
 circular conductor conveying a current may be found by an 
 elegant geometrical method due to Mr. A. Kussell (see Tlw 
 Electrician, Vol. XXXI., p. 284). 
 
 FIG. 
 
 Let P be any point in the plane of a circular current, and 
 let F be the magnetic force at P due to a current of strength 
 i in the wire. Let d s be an element of length of the circle, 
 and let OR be the radius of the circle. Take any point R 
 on the circumference of the circle (see Fig. 1), draw the 
 diameter through P, and at R draw a tangent to the circle. 
 Then draw the radius R, and through P draw P N perpen- 
 dicular to the tangent. Join PR. Let OP = a, OR = R, 
 P R = r, P N =p, and the angle R P N = <.
 
 698 APPENDIX. 
 
 Then, by Ampere's law, the magnetic force F at P due to the 
 element d s of the circuit at R is equal to ^-^ ^ . Hence 
 
 But 2 } ds = 
 
 -r = V R 2 - a 2 sin 2 <f> - a, cos , 
 
 j , V B' - a' ritf^+a COB 
 
 2 - a 2 sin 2 4 d d> + 
 
 
 The second integral, taken between the assigned limits, is 
 
 _ 
 
 i/B a -a a sin 2 <d& is called an 
 . 
 
 elliptic integral of the second order, and represents the length 
 of the circumference of an ellipse which has for its centre,, 
 P for one of its foci and 2 R for its major axis. 
 
 Hence, if we describe an ellipse on the diameter of the 
 circle, with as its centre and P as its focus, the magnetic 
 force F at the point P due to a current i in the circle is equal 
 to the value of ^ 2 x * ne l en gth f foe circumference of thia 
 ellipse. 
 
 In practice we can easily describe this ellipse by means of 
 two pins and a thread, and then measure the length of its- 
 circumference with a measuring-wheel, such as is used for 
 .measuring distances upon a map, or by laying a thread round 
 the ellipse and measuring its length. 
 
 In this way a practical measurement can be made of the 
 magnetic force due to a circular current at any point in its 
 plane. 
 
 Up to the limit of =0'8R, the length I of the circum- 
 ference of the ellipse can be calculated approximately from the 
 formula
 
 APPENDIX. 599 
 
 NOTE B. (See page 35.) 
 
 The Total Induction in a Circular Solenoid. If we consider 
 a circular-sectioned ring to be closely wound over with turns 
 of wire, we obtain a circular solenoid. 
 
 Let a be the mean radius of the circular cross-section of 
 the solenoid, and let R be the radius of the circular axis of 
 the solenoid. The whole solenoid may be considered to be 
 resolved into elementary solenoids. Let d S be the cross- 
 section of one of these, and let x be the radius of the circular 
 axis of the circular elementary solenoid. 
 
 Let N be the total number of turns of wire on the ring. 
 
 Then, for the elementary solenoid, there are ^ turns per 
 unit of length. 
 
 The magnetic force in the interior of the elementary solenoid 
 is H, and 4 B -NI_2NI 
 
 Zwx~~ x ' 
 where I is the current in the solenoid. 
 
 Hence, if the medium is non-magnetic, the surface integral 
 of magnetic force is the measure of the induction. Hence the 
 magnetic induction in the interior of the elementary solenoid is 
 
 2 N I 
 
 d S. Hence the whole induction Q through the whole 
 
 solenoid is obtained by integrating this last expression over 
 the area of the solenoid, viz., n-a 2 . Therefore between proper 
 limits 
 
 Now the integral / taken over the circular cross-section can 
 J x 
 
 be shown to be equal to 2:r {E - 
 
 and hence Q = ^ N I {E - 
 
 This, then, is the expression which should be employed for 
 
 the total induction over the circular cross-section of the ring, 
 
 if the mean radius of cross-section a is not very small 
 
 compared with the radius of the circular axis E. 
 
 If is a small quantity, then, since
 
 00 APPENDIX. 
 
 we have E- v'R 2 a 2 = ^-p nearly when ^ is small, and hence 
 
 NI , 
 
 or 4- times the current turns per unit of length of solenoid 
 multiplied by the area of cross-section of the solenoid ; and 
 for a very large thin circular solenoid the magnetic force in 
 the centre is 4?r times the current turns per unit of length. 
 
 NOTE C. (See page 52. ) 
 
 The Calibration of the Ballistic Galvanometer. A galvano- 
 meter consists generally of two parts a magnet and a coil 
 of wire. The magnet may be fixed and the coil suspended 
 and movable ; or the magnet suspended and movable and 
 the coil fixed. If the arrangements are such that the movable 
 system when disturbed is brought to rest without vibration, 
 or with very few vibrations, the galvanometer is called a 
 damped or aperiodic galvanometer. If, however, the resistance 
 to motion is so small that the movable part, when disturbed 
 or made to oscillate, continues for a long time to oscillate with 
 very slowly decreasing amplitude of vibration, the galvano- 
 meter is called a ballistic galvanometer. 
 
 If a body is suspended so as to vibrate about a vertical axis, 
 and if when given an angular displacement about that axis 
 it returns when released to its original position, and oscillates 
 about it, the body will in general execute vibrations of 
 decreasing amplitude. If I is the moment of inertia of the 
 body about that axis, and if /* is the torque or couple which 
 tends to restore the body to its original position if displaced 
 round the axis by a unit angle, then, if at any instant the 
 angular velocity of the oscillating body is u, the equation of 
 motion, neglecting for the moment resistance to motion, is 
 
 where 6 is the angle of displacement at the instant when the 
 angular velocity is u>. 
 
 i n 
 
 Since o> = , we have as the equation of motion, 
 dt
 
 . APPENDIX. 6 
 
 and the time t of one complete vibration of the body will be 
 
 /* 
 
 This gives us the periodic time of oscillation of the body, 
 assuming that the resistance to motion due to friction is nothing. 
 
 If there is any small resistance to the motion, such as air 
 resistance or other causes of energy dissipation, the amplitude 
 of the swings of tbe vibrating body will gradually decrease. 
 If the experiment is tried of setting the needle or coil of a 
 ballistic galvanometer in motion, and observing the amplitudes 
 #i a'ai x s , &c., of successive swings to the right and left, it 
 will be found that x lt .c. 2 , u? s , &c., form a descending geometrical 
 progression, and that the values of log a\, log x. 2 , log #,, &c., 
 form a descending arithmetic progression. 
 
 If the difference of the logarithms of the amplitudes of two 
 successive swings, one to the right and one to the left, is 
 taken, this difference, denoted by A, is called the loyarithwic 
 decrement of the galvanometer, and it will be found that this 
 difference is approximately constant for any two successive 
 swings right and left during the progress of the decay of the 
 excursions. That is, log x 1 - log x = A, also log a? a - log x a = A, 
 &c. Hence, if the logarithm of the amplitude falls off or 
 decreases by an amount A in tbe course of the passage of the 
 needle or coil of the galvanometer from the extremity of one 
 swing to the right to the extremity of one swing to the left, it 
 may fairly be assumed that, if the coil or needle is started from 
 rest by a sudden blow, the excursion x it would make if there 
 were no resistance is related to the excursion x l it does actually 
 make with the actual resistance existing by the relation 
 
 If the base of the logarithms is 10, or the logarithms are 
 ordinary ones, then by the exponential theorem we have
 
 C02 APPENDIX, 
 
 where M is the modulus 2-308, or the multiplier for converting 
 ordinary logarithms into Napierian logarithms. M is the 
 logarithm of 10 to the base e, the base of Napierian logarithms. 
 
 If - is small, we may neglect and higher powers in com- 
 
 parison with -, and write 
 2 
 
 - A 
 
 10 2 =1+M^, nearly. 
 
 Hence | = 1 + M * nearly, 
 
 or XQ =Xil+ M ~ , nearly. 
 
 In other words, we can find what the excursion x d would be 
 if there were no air resistance by multiplying the actual first 
 
 excursion ^ by a factor ( 1 + M -\ called the correcting factor. 
 
 Hence, if the movable system of a ballistic galvanometer 
 receives a sudden blow or impulse when at rest, and if it then- 
 makes an excursion the angular magnitude of which is x lt and 
 if such excursion is slightly resisted by air friction, we can 
 eliminate the results of this and correct for the frictional resist- 
 
 ance by multiplying a^ by the correcting factor f 1 + M- j. 
 
 If the logarithmic decrement is measured directly in. 
 Napierian logarithms, then the correcting factor is simply 
 
 K) 
 
 where A. equals the Napierian logarithmic decrement, or the 
 logarithm of the ratio of one swing to the next. 
 
 Eeturning to the equation of motion of the needle or coil, 
 since we have at any moment 
 
 let us consider a small magnetic needle of magnetic length I 
 hanging in the centre of a coil so wound that when a current 
 flows through the coil there is a uniform magnetic field due 
 to the current in all the region within which the needle lies. 
 Let the normal position of the needle when at rest be such
 
 APPENDIX. 603 
 
 that the magnetic axis of the needle is at right angles to 
 the direction of the field due to the current in the coil. 
 
 Then let M be the magnetic moment of the needle, H the 
 strength of the controlling field, the direction of H being at 
 right angles to the field due to the current in the coil. 
 
 Let the needle be set oscillating by a sudden impulsive 
 couple acting upon it when at rest. Let 6 be the angular 
 displacement of the magnetic axis of the needle at any 
 instant t. 
 
 The restoring couple acting on this needle at that instant 
 is M H sin 6, and, by the equation of motion, 
 
 - 
 dt 
 
 Hence, Irfo> = MHsin Od6\ 
 
 d t 
 
 dO 
 but -=o, 
 
 Hence 
 
 Integrate the above equation from the instant when the 
 needle leaves its position of rest with a finite angular velocity 
 Ji until it reaches a displacement 6 and has a zero angular 
 velocity. We have 
 
 if 
 
 or 
 
 therefore fl = 2^ sin . ... (1) 
 
 If the impulse which starts the needle from rest with a 
 finite velocity is that due to the flow of a quantity of electricity 
 through the coil surrounding the needle, which flow is all over 
 before the needle has had time to move sensibly from rest, we 
 can obtain a relation between the quantity of electricity so 
 sent through and the excursion of the needle. 
 
 For let i be the current in the coil at any instant, and let 
 G be a constant depending upon the form of the galvanometer
 
 04 A PPEXDIX. 
 
 coils, such that G i is the magnetic field due to the current. 
 Then, since M is the moment of the needle, M G i is the whole 
 couple acting on needle, and the impulse of this couple is 
 M G i d t in a time d t. If the whole impulse is over before 
 the needle has had time to move, then the whole impulse of 
 the couple must be equal to the total gain of angular momen- 
 tum I d w taking place in the time d t. 
 
 Hence M G i d t = I d ; 
 
 but if dq is the quantity of electricity which has flowed 
 through the galvanometer coils in the time d t, we have 
 
 im& 
 
 dt 
 and 
 
 But since the whole impulse is over before the needle has time 
 to leave its position of rest, we have, by integrating from w = 
 to w = fi, the equation 
 
 where S2 is the angular velocity with which the needle leaves 
 its position of rest. But by equation (1), 
 
 therefore Q - jl sin |, 
 
 f\ 
 
 or Q = /,- sin - . 
 
 Hence we see that, when a quantity Q of electricity is dis- 
 charged suddenly through the coils of a ballistic galvanometer, 
 the whole discharge being over in a very short time compared 
 with the period of free vibration of the needle, the quantity 
 of electricity is proportional to a constant, k, called the ballistic 
 constant, multiplied by the sine of half the angle of excursion 
 of the needle. 
 
 If the galvanometer has a logarithmic decrement A, which 
 is moderately small, say not more than 10 per cent., then to a 
 clo.se approximation
 
 APPENDIX. 605 
 
 To determine k for any galvanometer, the easiest way is to 
 charge a condenser having a capacity of c microfarads to a 
 potential of v volts by placing it in contact with a battery of 
 known voltage, and then discharging this quantity c v micro- 
 coulombs through the ballistic galvanometer. If this quantity 
 c v gives a ' throw " 8, we have 
 
 2 
 Hence k is determined. 
 
 Such a calibrated ballistic galvanometer can immediately 
 be employed to measure a magnetic field. For if a loop of 
 wire having N turns is placed in a field of induction so that 
 the induction is linked with the circuit, and if the loop is 
 connected with the ballistic galvanometer, then on suddenly 
 withdrawing the loop from the field we shall get a " throw " 
 of the galvanometer which can be interpreted to mean so 
 much quantity in microcoulombs passing through the galvano- 
 meter. Then, by the principles explained on page 31, the 
 total change in induction linked with the circuit is equal to 
 the product of the resistance of the circuit and the quantity 
 set flowing through it, or 
 
 Induction x linkages = resistance x quantity. 
 
 If induction is measured in microwebers one weber being 
 10 s C.G.S. lines or units of induction, and one microweber 
 therefore 100 C.G.S. units we have the rule 
 
 Microwebers x linkages - microcoulombs x ohms ; 
 
 and if one microweber of induction passing through the loop 
 linked once with this circuit, is suppressed, it will cause one 
 microcoulomb of electric quantity to flow through the circuit 
 if its resistance is one ohm. 
 
 In employing the ballistic galvanometer to measure mag- 
 netic induction, it must be observed that the condition of 
 application of the above principles is that the whole change of 
 induction must be completed in a very short time compared 
 with the periodic time of the needle of the galvanometer. 
 
 To suggest, as is sometimes done, that the induction in the 
 field magnets of a dynamo can be measured by surrounding
 
 06 APPENDIX. 
 
 them with a loop of wire connected to a ballistic galvanometer, 
 and then short-circuiting or breaking the field circuit, is to 
 presuppose a most unlikely event, viz., that the time during 
 which such change of induction takes place is small compared 
 with the periodic time of an ordinary ballistic galvanometer. 
 
 If the galvanometer is one with a movable coil of the 
 d'Arsonval type, then it is better to standardize the galvano- 
 meter by means of a coil of known length, turns and 
 resistance, placed in the axis of a long cylindrical coil, the 
 turns per unit of length of which are krfown. The reason 
 for this is that if the galvanometer circuit is always closed the 
 movement of the coil in the strong magnetic field induces 
 currents in the galvanometer which resist its motion. Hence 
 a powerful damping action comes into play from this cause 
 alone. To ascertain what induction change caused a given 
 galvanometer throw we proceed thus : The standardizing 
 secondary coil should be joined up in series with the galvano- 
 meter coil and with the secondary or exploring coil, and it 
 should be placed in the axis of a long coil, of which the 
 field can be calculated. If, then, under the influence of 
 an unknown induction linked or unlinked with the exploring 
 coil we obtain a galvanometer throw 0, we can find out 
 what was the induction causing this throw by interrupting 
 or reversing a known current in the long standard field coil, 
 and thus linking or unlinking a known induction with the 
 standard secondary coil. If the current through the standard 
 iield coil is altered until it gives a similar throw 6 on being 
 interrupted or reversed, we know at once that the value of 
 the unknown induction linked or unlinked with the secondary 
 or exploring coil which gives an equal throw, must be the 
 same as that calculable induction linked with the standardizing 
 secondary coil.
 
 INDEX. 
 
 Action at a Distance, 10 
 
 lotions taking place in Transformers, 518 
 
 Admittance of Condenser, 186 
 
 Alternating Current Curve Tracers, 522 
 
 Alternating Current Curve Tracing, 530 
 
 Alternating Current Flow in Conducting 
 
 | Circuits, 492 
 
 Alternating Current Flow, Theory of, 293 
 
 Alternating Currents, Distribution of, in 
 
 \ Conductors, 306 
 
 Alternating Currents, Propagation of, 
 
 through Conductors, 292 
 Alternative Path, Experiment of the, 401 
 Ampere's Discoveries on Electromagnetism, 
 
 Ampere's Fundamental Law, 15 
 Amplitude Factor of a Periodic Curve, 583 
 Analysis of Compound Periodic Curve, 90 
 Analytical Theory of Transformer, 586 
 Apparent Power, 157 
 Apparent Power given to Circuit, 205 
 Ballistic Galvanometer, 51, Appendix Note 
 
 C., 600 
 
 Ballistic Galvanometer, Use of, 51 
 Bernstein's Researches on Induction, 251 
 Blaserna's Researches on Induction, 246 
 .Branch Circuits, Impedance of, 163 
 Cardew Voltmeter, 155 
 Circuit, Inductive, 110 
 Circuit, Magnetic, 33 
 Circuit Non-inductive, 110 
 Circular Current, Magnetic Force at the 
 
 Centre of, 18 
 
 Circular Solenoid, Magnetic Force in In- 
 terior of, 23 
 
 Classification of Transformers, 517 
 Clock Diagram for Inducing and Induced 
 
 Circuits, 177 
 Clock Diagrams, 141 
 Closed Circuit Transformer, 515 
 Closed Magnetic Circuit, 38 
 Coefficient of Mutual Inductance, 121 
 
 Coefficient of Self-induction, 119 
 
 Coil Induction, Theory of, 232 
 
 Complex Periodic Functions, 202 
 
 Compound Periodic Curve, 84 
 
 Compound Periodic Curve, Harmonic 
 Analysis of, 91 
 
 Condenser Equation, 183 
 
 Condenser, Flow of Current into, 182 
 
 Condenser, Time-Constant of, 184 
 
 Conducting Power for Lines of Force, 40 
 
 Conduction Current, 335 
 
 Confirmation of Maxwell's Theory by 
 Hertz, 477 
 
 Correcting Factor of Wattmeter, 168 
 
 Current and ElectroinotiveForce Curves, 81 
 
 Current Diagram of Transformer, 561 
 
 Current Equation, 126 
 
 Current Flow in Circuits having Capacity, 
 Inductance and Resistance, Initial Con- 
 ditions of, 199 
 
 Current Flow in Inductive Circuits, Initial 
 Conditions of, 194 
 
 Current Growth in Inductive Circuits, 123 
 
 Current Sheet, 339 
 
 Curve of Hysteresis, 60 
 
 Curve of Induction, Determination of, 528 
 
 Curves, Logarithmic, 130 
 
 Curves of Current and E.M.F. of Various 
 Transformers on Open Secondary Circuit, 
 538, 539, 540 
 
 Curves of Magnetisation, 51, 55, 58 
 
 Curves of Power and Hysteresis of Trans- 
 former, 551 
 
 Curves of Primary and Secondary Alternat- 
 ing Electromotive Force of Transformer, 
 542 
 
 Cycle, Magnetic, 60 
 
 Delineation of Periodic Curves of Current 
 and Electromotive Force, 519 
 
 Derived Curves, 103 
 
 Description of Simple Periodic Curve, 96 
 
 Description of Transformer Diagrams, 535
 
 COS 
 
 INDEX. 
 
 Determination of Values, "v," 359 
 Dielectric Constants of Various Substances, 
 
 351-353 
 
 Discovery of Electromagnetic Induction, 2 
 Discovery of Induced Currents, 2 
 Displacement Current, 334 
 Displacement Currents and Displacement 
 
 Waves, 338 
 
 Direction of Magnetic Force. 14 
 Dove's Experiments on Induction, 262 
 Effect of Closing Secondary Circuit of 
 
 Induction Coil, 271 
 Effect of Form of Curve of Electromotive 
 
 Force on the Efficiency of Transformers, 
 
 579 
 Efficiency Curves of a Transformer taken 
 
 on Various Alternators, 582 
 Efficiency Curves of Transformers, 557 
 Efficiency of Transformers, 555 
 Electrical Oscillations, 372 
 Electrical Researches of Faraday, 1 
 Electric Currents produced by Magnetism, 5 
 Electric Displacement, 334 
 Electric Elasticity, 334, 333 
 Electric Surgings, 412 
 Electric Waves in Wires, 461 
 Electrodynamic Induction, 3 
 Electromagnetic Energy, 120 
 Electromagnetic Gyroscope, 324 
 Electromagnetic Induction, 14 
 Electromagnetic Induction. Discovery of, 2 
 Electromagnetic Medium, 333 
 Electromagnetic Momentum, 118 
 Electromagnetic Radiation, 478 
 Electromagnetic Repulsion, 307 
 Electromagnetic Repulsion, Theory of, 315 
 Electromagnetic Rotations. 320 
 Electromagnetic Theory, 332 
 Electromagnetic Waves in Air, 469 
 Electromotive Force Curves, 81 
 Electromotive Force Diagram for Inductive 
 
 Circuit, 144 
 
 Electromotive Force of Induction, 72 
 Electromotive Force of Rotating Coil, 76 
 Electromotive Intensity, 337 
 Electro-Optic Phenomena, 478 
 Electrostatic Induction, 1 
 Electrotouic State, 7, 118 
 Elihu Thomson's Experiments of Electro- 
 magnetic Repulsion, 313 
 Energy Dissipation by Hysteresis, 64 
 Equation for Charge of a Condenser, 183 
 Equation for Periodic Current in Inductive 
 
 Circuit, 135 
 Equation for Rise of Current in Inductive 
 
 Circuit, 127 
 
 Equipotential Surface, 46 
 Ether, 333 
 
 Experimental Determination of Electro- 
 magnetic Wave Velocity, 499 
 
 Experimental Determination of Inst; 
 taneous Value of a Periodic Current. 62C 
 
 Experimental Determination of the Form 
 of Periodic Curves of Transformer. 527 
 
 Experimental Measurement of Periodic 
 Currents and Electromotive Forces, 154 
 
 Experimental Proof of Existence of Oscilla- 
 tory Discharge, 381 
 
 Experimental Researches on Alternating 
 Current Transformers, 558 
 
 Faraday's Copper Disc Experiment, 5 
 
 Faraday's Discoveries, 1 
 
 Faradav, Discovery of Self -Induction by r 
 111 " 
 
 Faraday's Electrical Researches, 1 
 
 Faraday 'sExperiment with the Iron Ring, 3 
 
 Faraday's Law of Induction, 31 
 
 Faraday's Ring Coil, see Frontispiece 
 
 Faraday's Theories, 6 
 
 Flow of Simple Periodic Currents into 
 Condenser, 182 
 
 Flux of Magnetic Force, 26 
 
 Force, Magnetic, 43 
 
 Form Factor of a Periodic Curve, 583 
 
 Form Factor of Various Curves, 584 
 
 Fourier's Theorem, 85 
 
 Function of the Condenser in an Induction 
 Coil, 390 
 
 Galvanometer, Ballistic, 51, Appendix Note 
 C., 600 
 
 General Analytical Theory of the Trans- 
 former. 586 
 
 General D.scription of the Action of 
 Transformer, 514 
 
 Geometrical Illustrations, 138 
 
 Graphic Representation of Periodic Cur- 
 rents, 139 
 
 Harmonic Analysis of Transformer Dia- 
 grams. 550 
 
 Harmonic Curve, Description of, 87 
 
 Hedgehog Transformer, Current Diagram 
 of, 562 
 
 Helrnholz's Researches on Induction, 252 
 
 Henry, Discovery of Electro-magnetism 
 by, 11 
 
 Henry, Joseph, 11 
 
 Henry Joseph, Researches of, 207 
 
 Henry's Discovery of Induced Currents of 
 Higher Orders, 219 
 
 Henry's Electromagnets, 11 
 
 Henry's Experiments on Self and Mutual 
 Induction, 207 
 
 Henry's Experiments with Electromagnets. 
 12 
 
 Henry's Investigations, 11 
 
 Henry, The, 122
 
 INDEX 
 
 Hertz's Confirmation of Maxwell's Theory, 
 
 449 
 
 Hertz's Electrical Papers, 461 
 Hertz's Experiments of Propagation of 
 
 Alternating Currents in Conductors, 492 
 Hertz's Researches on the Propagation of 
 
 Electromagnetic Induction, 418 
 Hertz's Resonator, 426 
 High Frequency, Effect of, in changing 
 
 Distribution of Current in Conductor, 
 
 294 
 
 Historical Introduction, 1 
 Hopkinson's Researches on Hysteresis, 65 
 Hughes' Experiments on Transmission of 
 
 Alternating Currents, 281 
 Hughes' Induction Balance, 274 
 Hughes' Induction Bridge, Theory of, 285 
 Hughes' Sonometer, 276 
 Hysteresis Curves, 60 
 Hysteresis Curve of Ganz Transformer, 553 
 Hysteresis, Dissipation of Energy by, 64 
 Hysteresis, Magnetic, 59 
 Hysteresis, Magnetic Variation of, with 
 
 Temperature, 68 
 
 Hysteresis, Value of, for complete Cycle, 62 
 Hysteresis, Values of, for various kinds of 
 
 Iron, 63 
 
 Impedance, 136 
 Impedance Diagram, 146 
 Impedance of Branch Circuits, 163 
 Impedance to Sudden Rushes of Current, 
 
 416 
 Impressed and Effective Electromotive 
 
 Forces, 143 
 
 Impulsive Discharges, 399 
 Impulsive Impedance, 417 
 Index of Refraction of Various Substances, 
 
 350 
 
 Induced Currents, 239 
 Induced Currents, Discovery of, 2 
 Induced Currents, Duration of, 246, 247 
 Induced Currents, Duration of, Helmholz's 
 
 Researches on, 254 
 Induced Currents, Felici's Researches on, 
 
 243 
 
 Induced Currents of Higher Orders, 219 
 Induced Currents, Investigations on, 226 
 Induced Currents, Leuz's Researches on, 
 
 243 
 
 nduced Currents, Magnetic Effect of, 231 
 Currents, Various Effects of, 230 
 ....jce, 108, 119 
 
 stance and Inductive Circuits. 107 
 
 iductance and Inertia, Analogies of, 109 
 
 iductance, Annulment of, by Capacity, 
 
 189 
 
 Inductance Bridge, 116 
 Inductance, Discovery of, 111 
 
 Inductance, Mode of exhibiting, 113 
 Inductance of Various Circuits, 123 
 Induction at a Distance, 2l4 
 Induction Balance, 274, 279 
 Induction Coil, Theory of, 232 
 Induction, Electromotive Force of, 72 
 Induction, Faraday's Law of, 31 
 Induction in closed Magnetic Circuits, 36 
 Induction in open Magnetic Circuits, 37 
 Induction, Magnetic, 24 
 Induction Mutual, 211, 
 Induction of Electric Currents, 3 
 Induction Phenomena in open Circuits. 
 
 424 
 
 Induction Phenomenon in Dieletrics, 449 
 Induction, Prevention of, 259 
 Induction, Willoughby Smith's Researches 
 
 on, 261 
 
 Inductive Circuit, 110 
 Inductive Circuits, Division of. Current 
 
 between, 159 
 
 Inductive Effects by Leyden Jar Dis- 
 charges, 224 
 Inductive Effects by Transient Electric 
 
 Currents, 222 
 
 Initial Conditions at Starting Current, 194 
 Instantaneous Value of Periodic Current, 
 
 135 
 Integral Calculus, Important Theorem in, 
 
 90 
 
 Intensity of Magnetisation, 42 
 Interference of Electric Waves, 471 
 Interference Phenomenon of Electric 
 
 Waves, 464 
 
 Iron Ring, Permeability of, 53 
 Joubert's Instantaneous Contact Maker,520 
 Kelvin, Lord, on Flow of Alternating 
 
 Currents in Conductors, 303 
 Kunz's Researches on Magnetic Hysteresis, 
 
 68 
 
 Lemma, Trigonometrical, 161 
 Lightning Protectors, 403 
 Light, Velocity of, 358 
 Line Integral of Magnetic Force, 25 
 Line of Magnetic Induction, 29, 43 
 Lin*'-; of Force cutting Conductor, 8 
 Lines of Magnetic Force, 7 
 Linkage of Circuits and Lines of Magnetic 
 
 Induction, 47 
 
 Linkages of Magnetic and Conducting Cir- 
 cuits, 49 
 
 Lodge, Dr., on Lightning Conductors, 401 
 Logarithmic Curves, 130 
 Magnetic Axis, 13 
 Magnetic Circuit, 33 
 Magnetic Circuit, Closed, 38 
 Magnetic Circuit, Open, 38 
 Magnetic Cycle, 60 
 
 BB
 
 610 
 
 INDEX. 
 
 Magnetic Force, 43 
 
 Magnetic Force and Magnetic Fields, 13 
 
 Magnetic Force at the Centre of a Circular 
 
 Current, 18 
 Magnetic Force in Interior of Circular 
 
 Solenoid, 23 
 Magnetic Force in the Interior of Straight 
 
 i Solenoid, 20 
 Magnetic Force near a Straight Conductor, 
 
 Magnetic Hysteresis, 59 
 Magnetic Induction, 24 
 Magnetic Induction, Line of, 43 
 Magnetic Induction, Tube of, 43, 44, 45 
 Magnetic Induction, Unit of, 30 
 Magnetic Leakage, 328, 572 
 Magnetic Leakage, Calculation of, 576 
 Magnetic Leakage, Causes of, 574 
 Magnetic Leakage, Rule for, 578 
 Magnetic Permeability. 27, 39 
 Magnetic Permeability, Determination of. 
 
 52 
 
 Magnetic Potential, 24 
 Magnetic Reluctance, 40 
 Magnetic Researches, 56 
 Magnetic Resistance, 33 
 Magnetic Resistance, Definition of, 34 
 Magnetic Resistance, Specific, 34 
 Magnetic Saturation, 57 
 Magnetic Screening, 255 
 Magnetic Screening, Faraday's Experi- 
 ments, 257 
 Magnetic Screening, Henry's Experiments 
 
 on, 258 
 
 Magnetic Screening, Theory of, 270 
 Magnetisation. Curves of, 51 
 Magnetisation, Intensity of, 42 
 Magneto-electric Induction; 6 
 Magnetomotive Force, 24 
 Mathematical Sketch of Fourier's Theorem. 
 
 89 
 
 Maxwell's Mode of shewing Inductance, 116 
 Maxwell's Theory of the Electromagnetic 
 
 Medium, 344 
 Maxwell's Theory of Molecular Vortices. 
 
 339 
 
 Mean - Square and Maximum Value of 
 , Periodic Current, Relation of, 137 
 Mean-Square Value of Ordinate to Curve. 
 . 103 
 Mechanical and Electrical Quantities, 
 
 Analogies of, 126 
 Slechanical Illustrations of the Properties 
 
 of the Electromagnetic Medium, 342 
 Molecular Vortices, 339 
 Mordey's Alternator, Curve of E.M.F. of. 
 
 537 
 Multiple-valued Function, 82 
 
 Mutual and Self Induction, 207 
 
 Mutual Induction between Telephone Cir- 
 cuits, 216 
 
 Mutual Induction of two Circuits, 173 
 
 Mutuvl Induction of two Circuits, Theory 
 of, 232 
 
 Non-inductive Circuit, 110 
 
 Open Circuit Transformer, 515 
 
 Open Magnetic Circuit, 38 
 
 Oscillations of Leyden Jar, 375 
 
 Oscillatory and Non-oscillatory Discharge, 
 379 
 
 Oscillatory Discharge Experimentally In- 
 vestigated, 381 
 
 Periodic Currents, 79 
 
 Periodic Currents, Graphic Representation 
 of, 139 
 
 Permeability, Curves of. 55 
 
 Permeability Magnetic, 27, 39 
 
 Permeability, Magnetic, at Different 
 Temperatures, 56 
 
 Permeability, Magnetic, corresponding to 
 various Magnetic Forces, 54 
 
 Polar Diagram for Periodic Currents, 190 
 
 Polar Diagram, Important Use of, 193 
 
 Power Curves, 150 
 
 Power Curves for Induction and Non- 
 inductive Circuits, 152 
 
 Power Factor, 206 
 
 Power Factor of Transformers, 566 
 
 Power Factor, Relation of, to Secondary 
 Output, 571 
 
 Power Factors, Influence of the Form of 
 Curve of Electromotive Force on, 568 
 
 Power Factors of Various Types of Trans- 
 formers, 567 
 
 Power of Periodic Current, Mean Value of, 
 147 
 
 Poyutiiig's Theorem, 483 
 
 Propagation of Alternating Currents 
 through Conductors, 292 
 
 Propagation of Currents along Conductors, 
 488 
 
 Propagation of Electromagnetic Energy, 
 481 
 
 Ratio of Electromagnetic to Electrostatic 
 Units, 355 
 
 Rayleigh, Lord, Induction Bridge devised 
 by, 298 
 
 Rayleigh, Lord, Investigations on Induc- 
 tion by, 289 
 
 Reactance, 146 
 
 Reaction of Closed Secondary Circuit on 
 the Primary. 271 
 
 Relation of Power Factor to Secondary 
 Output, 571 
 
 Reluctance, Magnetic, 40 
 
 Repulsion, Electromagnetic, 307
 
 INDEX. 
 
 611 
 
 Resistance, Magnetic, 33 
 
 Resonance Phenomenon, 428 
 
 Rise of Current in Circuit, 132 
 
 Roessler's Experiments on Transformers. 
 
 580 
 
 Rotating Coil, Electromotive Force of, 76 
 
 Rowland's Experiments on a Rotating 
 Electrified Disc, 357 
 
 Screening, Magnetic, 266 
 
 Secondary Drop, 572 
 
 Secondary Drop Curves of Transformer, 
 573 
 
 Self Induction, 207 
 
 Shunted Condenser in Series with Induc- 
 tive Resistance, 187 
 
 Shunted Condenser, Theory of, 396 
 
 Side Flashing, 407 
 
 Simple Periodic Currents and Electro- 
 motive Forces, 93 
 
 Simple Periodic Currents, Theory of, 79 
 
 Simple Periodic Curve, 83 
 
 ine Curve, 88 
 
 Sine Curve, Mean-Square Ordinate of, 101 
 
 Sine Curve, Value of Mean Ordinate of, 98 
 
 Single-varied Function, 82 
 
 Slow and Quick Cycle Hysteresis, 65 
 
 Solenoid, Magnetic Force in Interior of, 20 
 
 Sonometer, 276 
 
 Specific Inductive Capacity, 350 
 
 Specific Magnetic Resistance, 3^1 
 
 Straight Conductor, Magnetic Forces near 
 a, 15 
 
 Strength of Magnetic Field, 15 
 
 Strength of Magnetic Pole, 14 
 
 Surface Integral of Induction, 28 
 
 Symmetry of Current and Induction, 329 
 
 Symmetry of Transformer Diagrams, 549 
 
 Table of Efficiencies of Various Trans- 
 formers, 560 
 
 Tallies of Specific Inductive Capacity, 351 
 
 Telegraphing to Moving Trains, 217 
 
 Telephonic Induction, 216 
 
 Temperature Effect on Hysteresis, 68 
 
 Test of Swinbourne Hedgehog Trans- 
 former, 565 
 
 Te^t of a Westinghouse Transformer, 564 
 
 Theory of Air- Core Induction Coil, 174 
 
 Theory of Experiments on Alternative 
 Path, 413 
 
 Theory of Hertz's Experiments, 435 
 
 Theory of Induction Balance, 285 
 
 Theory of the Transformer, 586 
 
 Thomson-Houston Alternator, Curve of 
 
 K.M.F. of, 536 
 Time-Constant of Circuit, 131 
 
 Time-Con.jtant of Condenser, 184 
 
 Time of Quickest Discharge of Condenser, 
 
 386 
 
 Transformation Ratio of Transformer, 575 
 Transformer, Analytical Theory of, 586 
 Transformer Curves of Current and Elec- 
 tromotive Force, 538, 539, 540 
 Transformer, Current Diagram of, 561 
 Transformer Diagrams, 543. 
 Transformer Diagrams, Description of, 535 
 Transformer, General Action of, 514 
 Transformer, General Analytical Theory 
 
 of, 586 
 
 Transformers, Classification of, 517 
 Transformers, Efficiency Curves of, 557 
 Transformers, Efficiency of, 555 
 Transformers, Power Factor of, 566 
 Transmission of Alternating Currents 
 
 through Conductors, 281 
 Trigonometrical Lemma, 161 
 Trowbridge's Experiments on Electro- 
 magnetic Waves, 500 
 
 Trowbridge's Experiments on the Propaga- 
 tion of Electrical Oscillations, 431 
 True Power given to Inductive Circuit, 157 
 Tube of Magnetic Induction, 43 
 Unit of Inductance, 122 
 Unit of Magnetic Induction, 30 
 Value of Mean Ordinate Sine Curve, 98 
 Value of Mean-Square Ordinate of Sine 
 
 Curve, 101 
 
 Variable and Steady Flow, 79 
 Variation of Hysteresis with Temperature, 
 
 70 
 
 Vector Potential, Explanation of, 360 
 Vector Quantities, 27 
 Velocity of Light, 358 
 Velocity of Propagation of Electromagnetic 
 
 Disturbances, 354 
 Velocity of Propagation of Electromagnetic 
 
 Waves experimentally determined, 511 
 Velocity of Propagation of Vector Poten- 
 tial, 368 
 
 Wattmeter, Correcting Factor of, 168 
 Wattmeter Measurement of Periodic Power, 
 
 166 
 
 Wattmeter, Theory of, 157 
 AVave Diagram for Periodic Currents, 192 
 Waves in Rectilinear Wires, 458 
 Westinghouse Transformer, Current Dia- 
 gram of, 562 
 Westinghouse Transformer, Diagrams of, 
 
 544, 545 
 
 Willoughby Smith's Experiments on 
 Magnetic Screening, 262
 
 LONDON : 
 PRINTED BY CEOKGE TUCKER, SALI-SBl'RY COURT, FLEET STREKT. 

 
 IK 2792 
 F 62 a 
 " 1896 
 
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 Physic* 
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