A History of the Theories of Aether and Electricity (1910) — part 16 of 29

The correct theory of the energy of magnetic and electro- magnetic fields is due mainly to W. Thomson (Lord Kelvin). Thomson's researches on this subject commenced with one or two short investigations regarding the ponderomotive forces which act on temporary magnets. In 1847 he discussed t the case of a small iron sphere placed in a magnetic field, showing that it is acted on by a ponderomotive force represented by

• grad cR~, where c denotes a constant, and R denotes the magnetic force of the field ; such a sphere must evidently tend to move towards the places where E' is greatest. The same analysis may be applied to explain why diamagnetic bodies tend to move, as in Faraday's experiments, from the stronger to the weaker parts of the field.

• We suppose all transitions to be continuous, so as to avoid the necessity for writing surf ace -integrals separately.

tCamb. and Dub. Matb. Journal, ii (1847), p. 230; W. Thomson's Papers on Electrostatics and Magnetism, p. 499; cf. also Phil. Mag. xxxvii (1850), p. 241.

R 2

24:4 The Mathematical Electricians of the

Two years later Thomson presented to the Koyal Society a memoir* in which the results of Poisson'a theory of magnetism were derived from experimental data, without making use of the hypothesis of magnetic fluids ; and this was followed in 1850 by a second memoir,f in which Thomson drew attention to the fact previously noticed by Poisson,J that the magnetic intensity at a point within a magnetized body depends on the shape of the small cavity in which the exploring magnet is placed. Thomson distinguished two vectors ;§ one of these, by later writers generally denoted by B, represents the magnetic intensity at a point situated in a small crevice in the magnetized body, when the faces of the crevice are at right angles to the direction of magnetization ; the vector B is always circuital. The other vector, generally denoted by H, represents the magnetic intensity in a narrow tubular cavity tangential to the direction of magnetization ; it is an irrotational vector. The magnetic potential tends at any point to a limit which is independent of the shape of the cavity in which the point is situated ; and the space-gradient of this limit is identical with H. Thomson called B the " magnetic force according to the electro-magnetic definition," and H the " magnetic force accord- ing to the polar definition " ; but the names magnetic induction and magnetic force, proposed by Maxwell, have been generally used by later writers.

It may be remarked that the vector to which Faraday applied the term " magnetic force," and which he represented by lines of force, is not H, but B ; for the number of unit lines of force passing through any gap must depend only on the gap, and not on the particular diaphragm filling up the gap, across which the flux is estimated ; and this can be the case only if the vector which is represented by the lines of force is a circuital vector.

• Phil. Trans., 1851, p. 243 ; Thomson's Papers on Elect, and Mag., p. 345.

t Phil. Trans., 1851, p. 269 ; Papers on Elect, and Nay., p. 382.

I Of. p. 64.

§ Loc. cit., § 78 of the original paper, and § 517 of the reprint^

Middle of the Nineteenth Century. 245

Thomson introduced a number of new terms into magnetic science — as indeed he did into every science in which he was interested. The ratio of the measure of the induced magnetiza- tion I,-, in a temporary magnet, to the magnetizing force H, he named the susceptibility ; it is positive for paramagnetic and negative for diamagnetic bodies, and is connected with Poisson's constant kp* by the relation

3 if t\jp

= SFTv

where K denotes the susceptibility. By an easy extension of Poisson's analysis it is seen that the magnetic induction and magnetic force are connected by the equation

B = H + 47rl,

where I denotes the total intensity of magnetization : so if I0 denote the permanent magnetization, we have

B = H + 47rl» + 47rl,,, = )uH + 47rI0,

where //, denotes (1 + 4™) : //, was called by Thomson the permeability.

In 1851 Thomson extended his magnetic theory so as to include magnecrystallic phenomena. The mathematical founda- tions of the theory of magnecrystallic action had been laid by anticipation, long before the experimental discovery of the phenomenon, in a memoir read by Poisson to the Academy in February, 1824. Poisson, as will be remembered, had supposed temporary magnetism to be due to " magnetic fluids," movable within the infinitely small " magnetic elements " of which he assumed magnetizable matter to be constituted. He had not overlooked the possibility that in crystals these magnetic elements might be non-spherical (e.g. ellipsoidal), and symmetri- cally arranged ; and had remarked that a portion of such a crystal, when placed in a magnetic field, would act in a manner depending on its orientation. The relations connecting

• Cf. p. 65.

246 The Mathematical Electricians of the

the induced magnetization I with the magnetizing force H he had given in a form equivalent to

( Ix = aHx + b'ffy + c"ffz, Iy = a"Hx + bHy + c'HZ) Iz = a'Hr + b"Hy + cHz.

Thomson now* showed that the nine coefficients a, b' ', c" . . ., introduced by Poisson, are not independent of each other. For a sphere composed of the magnecrystalline substance, if placed in a uniform field of force, would be acted on by a couple : and the work done by this couple when the sphere, supposed of unit volume, performs a complete revolution round the axis of x may be easily shown to be 7rH(l - H^j IP) (- &" + c). But this work must be zero, since the system is restored to its primitive condition ; and hence ~b" and c must be equal. Similarly e" = a, and a" = bf. By change of axes three more coefficients may be removed, so that the equations may be brought to the form

777" T TT T TT

JC ~ Kl/Zx, Iy = K-lJily, 1Z = Ka/Zz,

where KI, KZ, K3 may be called the principal magnetic suscepti- bilities.

In the same year (1851) Thomson investigated the energy which, as was evident from Faraday's work on self-induction, must be stored in connexion with every electric current. He showed that, in his own words, f " the value of a current in a closed conductor, left without electromotive force, is the quantity of work that would be got by letting all the infinitely small currents into which it may be divided along the lines of motion of the electricity come together from an infinite distance, and make it up. Each of these ' infinitely small currents ' is of course in a circuit which is generally of finite length ; it is the section of each partial conductor and the strength of the current in it that must be infinitely small."

• Phil. Mag. (4) i (1851), p. 177: Papers on Electrostatics and Magnetism, p. 471.

t Papers on Electrostatics and Magnetism, p. 446.

Middle of the Nineteenth Century. 247

Discussing next the mutual energy due to the approach of a permanent magnet and a circuit carrying a current, he arrived at the remarkable conclusion that in this case there is no electrokinetic energy which depends on the mutual action ; the energy is simply the sum of that due to the permanent magnets and that due to the currents. If a permanent magnet is caused to approach a circuit carrying a current, the electromotive force acting in the circuit is thereby temporarily increased ; the amount of energy dissipated as Joulian heat, and the speed of the chemical reactions in the cells, are temporarily increased also. But the increase in the Joulian heat is exactly equal to the increase in the energy derived from consumption of chemicals, together with the mechanical work done on the magnet by the operator who moves it ; so that the balance of energy is perfect, and none needs to be added to or taken from the electrokinetic form. It will now be evident why it was that Helmholtz escaped in this case the errors into which he was led in other cases by his neglect of electrokinetic energy ; for in this case there was no electrokinetic energy to neglect.

Two years later, in 1853, Thomson* gave a new form to the expression for the energy of a system of permanent and temporary magnets.

We have seen that the energy of such a system is represented

by

where p0 denotes the density of Poisson's equivalent magnetiza- tion for the permanent magnets, and <f> denotes the magnetic potential, and where the integration may be extended over the whole of space. Substituting for pn its value - div I0,f the expression may be written in the form

• J

<£ div Io dx dydz ;

*Proc. Glasgow Phil. Soc. iii (1853), p. 281; Kelvin's Math, and Phys. Papers, i, p. 521. t Cf . p. fi4.

248 The Mathematical Electricians of the

or, integrating by parts,

(!«, . grad <£) dx dy dz, or - J (H . I0) dx dy dz.

Since B = yu,H + 47rI0, this expression may be written in the form

-— (H. offJJJ

but the former of these integrals is equivalent to

fff (B . grad <£) dx dydz, or - <£ div B dx dy dz,

which vanishes, since B is a circuital vector. The energy of the field, therefore, reduces to 1 BIT,

integrated over all space; which is equivalent to Thomson's form.*

In the same memoir Thomson returned to the question of the energy which is possessed by a circuit in virtue of an electric current circulating in it. As he remarked, the energy may be determined by calculating the amount of work which must be done in and on the circuit in order to double the circuit on itself while the current is sustained in it with constant strength; for Faraday's experiments show that a circuit doubled on itself has no stored energy. Thomson found that the amount of work required may be expressed in the form \Li*, where i denotes the current strength, and L, which is called the coefficient of self-induction^ depends only on the form of the circuit.

It may be noticed that in the doubling process the inherent

• The form actually given by Thomson was

— fff (E? — lA d-d Sir}}} :-.)

which reduces to the above when we neglect that part of I2 which is due to the permanent magnetism, over which we have no control.

Middle of the Nineteenth Century. 249

electrodynamic energy is being given up, and yet the operator is doing positive work. The explanation of this apparent paradox is that the energy derived from both these sources is being used to save the energy which would otherwise be furnished by the battery, and which is expended in Joulian heat.

Thomson next proceeded* to show that the energy which is stored in connexion with a circuit in which a current is flowing may be expressed as a volume-integral extended over the whole of space, similar to the integral by which he had already represented the energy of a system of permanent and temporary magnets. The theorem, as originally stated by its author, applied only to the case of a single circuit; but it may be established for a system formed by any number of circuits in the following way : —

If N8 denote the number of unit tubes of magnetic induction which are linked with the &h circuit, in which a current is is flowing, the electrokinetic energy of the system is JSJV,^; which

may be written |2/r, where /r denotes the total current flowing

through the gap formed by the rth unit tube of magnetic induc- tion. But if H denote the (vector) magnetic force, and H its numerical magnitude, it is known that (l/4?r) J Hds, integrated along a closed line of magnetic induction, measures the total current flowing through the gap formed by the line. The energy is therefore (l/8?r)S jffds, the summation being extended over all the unit tubes of magnetic induction, and the integra- tion being taken along them. But if dS denote the cross-section of one of these tubes, we have BdS = 1, where B denotes the numerical magnitude of the magnetic induction B : so the energy is (1 1 'Sir) SBdS / Hds ; and as the tubes fill all space, we may replace 'S.dSjds by ^dxdydz. Thus the energy takes the form (l/8?r) JJf BHdxdydz, where the integration is extended over the whole of space ; and since in the present case B = pH, the energy may also be represented by (Il8v)ffffjjrdxdydz.

• Nichols* Cyclopaedia, 2nd ed., 1860, article " Magnetism, dynamical relations of; " reprinted in Thomson's Papers on Elect, and Mag., p. 447, and his Math, and Phys. Papers, p. 532.

250 The Mathematical Electricians of the

But this is identical with the form which was obtained for a field due to permanent and temporary magnets. It thus appears that in all cases the stored energy of a system of electric currents and permanent and temporary magnets is

-' dxdydz,

where the integration is extended over all space.

It must, however, be remembered that this represents only what in thermodynamics is called the " available energy " ; and it must further be remembered that part even of this available energy may not be convertible into mechanical work within the limitations of the system : e.g., the electrokinetic energy of a current flowing in a single closed perfectly conducting circuit cannot be converted into any other form so long as the circuit is absolutely rigid. All that we can say is that the changes in this stored electrokinetic energy correspond to the work furnished by the system in any change.

The above form suggests that the energy may not be localized in the substance of the circuits and magnets, but may be distri- buted over the whole of space, an amount (pH2 /Sir) of energy being contained in each unit volume. This conception was afterwards adopted by Maxwell, in whose theory it is of fundamental importance.

While Thomson was investigating the energy stored in connexion with electric currents, the equations of flow of the currents were being generalized by Gustav Kirchhoff (b. 1824, d. 1887). In 1848 Kirchhoff* extended Ohm's theory of linear conduction to the case of conduction in three dimensions ; this could be done without much difficulty by making use of the analogy with the flow of heat, which had proved so useful to Ohm. In Kirchhoff s memoir a system is supposed to be formed of three-dimensional conductors, through which steady currents are flowing. At any point let V denote the " tension " or " electroscopic force " — a quantity the significance of which

*Ann.d. Phys. Ixxv (1848), p. 189: Kirchhoff's Ges. AbhandL, p. 33.

Middle of tke Nineteenth Century. 425l

in electrostatics was not yet correctly known. Then, within the substance of any homogeneous conductor, the function V must satisfy Laplace's equation V- V= 0 ; while at the air-surface of each conductor, the derivate of V taken along the normal must vanish. At the interface between two conductors formed of different materials, the function V has a discontinuity, which is measured by the value of Volta's contact force for the two conductors ; and, moreover, the condition that the current shall be continuous across such an interface requires that Jed VfoN shall be continuous, where k denotes the ohmic specific conductivity of the conductor, and 3/3^ denotes differentiation along the normal to the interface. The equations which have now been mentioned suffice to determine the flow of electricity in the system.

Kirchhoff also showed that the currents distribute them- selves in the conductors in such a way as to generate the least possible amount of Joulian heat ; as is easily seen, since the quantity of Joulian heat generated in unit time is

where k, as before, denotes the specific conductivity ; and this integral has a stationary value when V satisfies the equation

a /ar\ a

Kirchhoff next applied himself to establish harmony between electrostatical conceptions and the theory of Ohm. That theory had now been before the world for twenty years, and had been verified by numerous experimental researches ; in particular, a careful investigation was made at this time (1848) by Kudolph Kohlrausch (b. 1809, d. 1858), who showed* that the difference of the electric " tensions " at the extremities of a voltaic cell, measured electrostatically with the circuit open, was for different cells proportional to the electromotive force

*Ann. d. Phys. Ixxv (1848), p. 220.

252 The Mathematical Electricians of the

measured by the electrodynamic effects of the cell with the circuit closed ; and, further,* that when the circuit was closed, the difference of the tensions, measured electrostatically, at any two points of the outer circuit was proportional to the ohmic resistance existing between them. But in spite of all that had been done, it was still uncertain how " tension," or " electro- scopic force," or " electromotive force " should be interpreted in the language of theoretical electrostatics ; it will be remembered that Ohm himself, perpetuating a confusion which had originated with Volta, had identified electroscopic force with density of electric charge, and had assumed that the electricity in a conductor is at rest when it is distributed uniformly throughout the substance of the conductor.

The uncertainty was finally removed in 1849 by Kirchhoff,f who identified Ohm's electroscopic force with the electrostatic potential. That this identification is correct may be seen by comparing the different expressions which have been obtained for electric energy; Helmholtz's expression^ shows that the energy of a unit charge at any place is proportional to the value of the electrostatic potential at that place ; while Joule's result§ shows that the energy liberated by a unit charge in passing from one place in a circuit to another is proportional to the difference of the electric tensions at the two places. It follows that tension and potential are the same thing.

The work of Kirchhoff was followed by several other investigations which belong to the borderland between electro- statics and electrodynamics. One of the first of these was the study of the Leyden jar discharge.

Early in the century Wollaston, in the course of his experi- ments on the decomposition of water, had observed that when the decomposition is effected by a discharge of static electricity, the hydrogen and oxygen do not appear at separate electrodes ; but that at each electrode there is evolved a mixture of the

• Ann. d. Phys, Ixxviii (1849), p. 1.

f Ib. Ixxviii (1849), p. 506 ; Kirchhoff's Get. Abhandl, p. 49 ; Phil. Mag. (3), xxxvii (1850), p. 463.

I Cf. p. 242. § Cf. p. 239.

Middle of the Nineteenth Century. 253

gases, as if the current had passed through the water in both directions. After this F. Savary* had noticed that the discharge of a Ley den jar magnetizes needles in alternating layers, and had conjectured that " the electric motion during the discharge consists of a series of oscillations." A similar remark was made in connexion with a similar observation by Joseph Henry (ft. 1799, d. 1878), of Washington, in 1842.f " The phenomena," he wrote, " 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 restored." Helmholtz had repeated the same suggestion in his essay on the conservation of energy : and in 1853 W. Thomson J verified it, by investigating the mathematical theory of the discharge, as follows : —

Let C denote the capacity of the jar, i.e., the measure of the charge when there is unit difference of potential between "the coatings ; let R denote the ohmic resistance of the discharging circuit, and L its coefficient of self-induction. Then if at any instant t the charge of the condenser be Q, and the current in the wire be i, we have i = dQ/dt ; while Ohm's law, modified by taking self-induction into account, gives the equation

Eliminating i, we have

an equation which shows that when IFC < 4Z, the subsidence of Q to zero is effected by oscillations of period

27T

(1- * \LC 4Z

• Annales de Chiniie, xxxiv (1827), p. 5. tProc. Am. Phil. Soc. ii (1842), p. 193.

J Phil. Mag. (4) v (1853), p. 400 ; Kelvin's Math, and Phys. Papers i, p. 540.

254 The Mathematical Electricians of the

This simple result may be regarded as the beginning of the theory of electric oscillations.

Thomson was at this time much engaged in the problems of submarine telegraphy; and thus he was led to examine the vexed question of the " velocity of electricity " over long insulated wires and cables. Various workers had made experiments on this subject at different times, but with hopelessly discordant results. Their attempts had generally taken the form of measuring the interval of time between the appearance of sparks at two spark-gaps in the same circuit, between which a great length of wire intervened, but which were brought near each other in order that the discharges might be seen together. In one series of experiments, performed by Watson at Shooter's Hill in 1747-8,* the circuit was four miles in length, two miles through wire and two miles through the ground ; but the discharges appeared to be perfectly simultaneous; whence Watson concluded that the velocity of propagation of electric effects is too great to be measurable.

In 1834 Charles Wheatstone,f Professor of Experimental Philosophy in King's College, London, by examining in a revolving mirror sparks formed a,t the extremities of a circuit, found the velocity of electricity in a copper wire to be about one and a half times the velocity of light. In 1850 H. Fizeau and E. GounelleJ experimenting with the telegraph lines from Paris to Eouen and to Amiens, obtained a velocity about one- third that of light for the propagation of electricity in an iron wire, and nearly two- thirds that of light for the propagation in a copper wire.

The first step towards explaining these discrepancies was made by Faraday, who§ early in 1854 showed experimentally that a submarine cable, formed of copper wire covered with

• Phil. Trans, xlv (1748), pp. 49, 491. t Phil. Trans., 1834, p. 583. ; Comptes Rendus, xxx (1850), p. 437.

§ Proc. Roy. Inst., Jan. 20, 1854: Phil. Mag-., June, 1854: Exp. Res. iii, pp. 508, 521.

Middle of the Nineteenth Century. 255

gutta-percha, " may be assimilated exactly to an immense Leyden battery ; the glass of the jars represents the gutta- percha ; the internal coating is the surface of the copper wire," while the outer cgating corresponds to the sea-water. It follows that in all calculations relating to the propagation of electric disturbances along submarine cables, the electrostatic capacity of the cable must be taken into account.

The theory of signalling by cable originated in a corre- spondence between Stokes and Thomson in 1854. In the case of long submarine lines, the speed of signalling is so much limited by the electrostatic factor that electro-magnetic induc- tion has no sensible effect ; and it was accordingly neglected in the investigation. In view of other applications of the analysis; however, we shall suppose that the cable has a self-induction L per unit length, and that E denotes the ohmic resistance, and C the capacity per unit length, Fthe electric potential at a distance x from one terminal, and i the current at this place. Ohm's law, as modified for inductance, is expressed by the equation

9^ Tdi _>.

• -^- = L — + Ri ;

dx dt

moreover, since the rate of accumulation of charge in unit length at # is - di/dx, and since this increases the potential at the rate - (l/C^difix, we have

'dt dx

Eliminating i between these two equations, we have 1 82F

which is known as the equation of telegraphy*

Thomson, in one of his letterst to Stokes in 1854, obtained this equation in the form which applies to Atlantic .cables, i.e., with the term in L neglected. In this form it is

• "We have neglected leakage, which is beside our present purpose.

t Proc. Roy. Soc., May, 1855 : Kelvin's Math, and Phys. Papers, ii, p. 61.

256 The Mathematical Electricians of the

the same as Fourier's equation for the linear propagation of heat : so that the known solutions of Fourier's theory may he used in a new interpretation. If we substitute v - /,2»<V - i -j- \x

y — t/ >

we obtain

A, = ± (1 + -v/^l) (nCR)l J

and therefore a typical elementary solution of the equation is V = e-(nCR^x sin \2nt - (nCR)^x}.

The form of this solution shows that if a regular harmonic variation of potential is applied at one end of a cable, the phase is propagated with a velocity which is proportional to the square root of the frequency of the oscillations : since therefore the different harmonics are propagated with different velocities, it is evident that no definite " velocity of transmission " is to be expected for ordinary signals. If a potential is suddenly applied at one end of the cable, a certain time elapses before the current at the other end attains a definite percentage of its maximum value ; but it may easily be shown* that this retardation is proportional to the square of the length of the cable, so that the apparent velocity of propagation would be less, the greater the length of cable used.

The case of a telegraph line insulated in the air on poles is different from that of a cable ; for here the capacity is small, and it is necessary to take into account the inductance. If in the general equation of telegraphy we write

V = enx^~l + Pl, we obtain the equation

R (R* n* \i

2l f (±L* ~ CL) ;

as the capacity is small, we may replace the quantity under the radical by its second term : and thus we see that a typical elementary solution of the equation is

F= e i siu n{x - (CL)-1* t};

• This result, indeed, follows at once from the theory of dimensions.

Middle of the Nineteenth Century. 257

this shows that any harmonic disturbance, and therefore any disturbance whatever, is propagated along the wire with velocity (CL}~\ The difference between propagation in an aerial wire and propagation in an oceanic cable is, as Thomson remarked, similar to the difference between the propagation of an impulsive pressure through a long column of fluid in a tube when the tube is rigid (case of the aerial wire) and when it is elastic, so as to be capable of local distension (case of the cable, the distension corresponding to the effect of capacity) : in the former case, as is well known, the impulse is propagated with a definite velocity, namely, the velocity of sound in the fluid.

The work of Thomson on signalling along cables was followed in 1857 by a celebrated investigation* of Kirchhoff's, on the propagation of electric disturbance along an aerial wire of circular cross- section.

Kirchhoff assumed that the electric charge is practically all resident on the surface of the wire, and that the current is uniformly distributed over its cross-section; his idea of the current was the same as that of Fechner and Weber, namely, that it consists of equal streams of vitreous and resinous elec- tricity flowing in opposite directions. Denoting the electric potential by V, the charge per unit length of wire by e, the length of the wire by I, and the radius of its cross-section by a, he showed that Fis determined approximately by the equationf

V = 2e log (I/a).

• Ann. d. Phys. c (1857), pp. 193, 251 : Kirchhoff's Ges. Abhandl., p. iai ; Phil. Mag. xiii (1857), p. 393.

t His method of obtaining this equation was to calculate separately the effects of (1) the portion of the wire within a distance e on either side of the point con- sidered, where e denotes a length small compared with J, but large compared with o, and (2) the rest of the wire. He thus obtained the equation

where the integration is to be taken over all the length of the wire except the portion 2e : the equation given in the text was then derived by an, approximation ,. which, however, is open to some objection.

S

258 The Mathematical Electricians of the

The next factor to be considered is the mutual induction of the current-elements in different parts of the wire. Assuming with Weber that the electromotive force induced in an element ds due to another element ds' carrying a current i' is derivable from a vector-potential

,.3 •

Kirchhoff found for the vector-potential due to the entire wire the approximate value

w = 2i log (//a),

where i denotes the strength of the current ;* the vector- potential being directed parallel to the wire. Ohm's law then gives the equation

ldw

where k denotes the specific conductivity of the material of which the wire is composed; and finally the principle of conservation of electricity gives the equation

di _ _de .

dx~ ~di'

Denoting log (I/a) by y, and eliminating e, i, w from these four equations, we have

82F 1 d*V 1 8F

which is, as might have been expected, the equation of telegraphy. When the term in 3 V/dt is ignored, as we have seen is in certain cases permissible, the equation becomes

82F lF

• This expression was derived in a similar way to that for F, by an intermediate formula

2 c ci'ds'

w = 2i log -- h — cos 6 cos Q , & a J r

where 6 and Q' denote respectively the angles made with r by ds and ds'.

Middle of the Nineteenth Century. 259

which shows that the electric disturbance is propagated along the wire with the velocity c* KirchhofF s procedure has, in fact, involved the calculation of the capacity and self-induction of the wire, and is thus able to supply the definite values of the quantities which were left undetermined in the general equation of telegraphy.

The velocity c, whose importance was thus demonstrated, has already been noticed in connexion with Weber's law of force ; it is a factor of proportionality, which must be introduced when electrodynamic phenomena are described in terms of units which have been defined electrostatically ,f or conversely when units which have been defined electrodynamicallyj are used in the description of electrostatic phenomena. That the factor which is introduced on such occasions must be of the dimensions (length/time), may be easily seen : for the electrostatic re- pulsion between electric charges is a quantity of the same kind as the electrodynamic repulsion between two definite lengths of wire, carrying currents which may be specified by the amount of charge which travels past any point in unit time.

Shortly before the publication of Kirchhoff s memoir, the value of c had been determined by Weber and Kohlrausch§ ; their determination rested on a comparison of the measures of the charge of a Leyden jar, as obtained by a method depending on electrostatic attraction, and by a method depending on the

• In referring to the original memoirs of Weber and Kirchhoff, it must he remembered that the quantity which in the present work is denoted by e, and which represents the velocity of light in free aether, was by these writers denoted by c/V'2. Weber, in fact, denoted by c the relative velocity with which two charges must approach each other in order that the force between them, as calculated by his formula, should vanish.

It must also be remembered that those writers who accepted the hypothesis that currents consist of equal and opposite streams of vitreous and resinous electricity, were accustomed to write 2t to denote the current-strength.

f i.e., defining unit electric charge as that which exerts unit ponderomotive force on a conductor at unit distance which carries an equal charge ; and then defining unit current as that which conveys unit charge in unit time.

% i.e., defining unit current by means of the ponderomotive force which it exerts on an equal current, when the two currents flow in circuits of specified form at a specified distance apart.

§ Ann. d. Phys. xcix (1856), p. 10.

S 2

260 The Mathematical Electricians of the

magnetic effects of the current produced by discharging the jar. The resulting value was nearly

c = 3*1 x 1010 cm./sec.;

which was the same, within the limits of the errors of measure- ment, as the speed with which light travels in interplanetary space. The coincidence was noticed by Kirchhoff, who was thus the first to discover the important fact that the velocity with which an electric disturbance is propagated along a perfectly- conducting aerial wire is equal to the velocity of light.

In a second memoir published in the same year, Kirchhoff* extended the equations of propagation of electric disturbance to the case of three-dimensional conductors.

As in his earlier investigation, he divided the electromotive force at any point into two parts, of which one is the gradient of the electrostatic potential </>, and the other is the derivate with respect to the time (with sign reversed) of a vector- potential a ; so that if i denote the current and k the specific conductivity, Ohm's law is expressed by the equation

i = k (c2 grad <£ - a).

Kirchhoff calculated the value of a by aid of Weber's formula for the inductive action of one current element on another; the result is

where r denotes the vector from the point (x, y, z), at which a is measured, to any other point (x, y, z") of the conductor, at which the current is i' ; and the integration is extended over the whole volume of the conductor. The remaining general equations are the ordinary equation of the electrostatic potential

V2<£ + 4irp = 0

(where p denotes the density of electric charge), and the equation of conservation of electricity

| + div i = 0. ot

• Ann. d. Phys. cii (1857), p. 529 : Ges. AbhandL, p. 154.

Middle of the Nineteenth Century. 261