January iS, 1894] 



NATURE 



2»1 



magnetic metah, the vortex atoms pair into molecules and mole- 

 cular aggregates in such way as to a large extent cancel each 

 other's magnttic fields; why in iron at ordinary temperatures 

 the molecular aggregates form so striking an exception to the 

 general rale is for some reason peculiar to tlie substance, which, 

 cjniidering the complex character of molecular aggregation in 

 solid'^, need not excite surprise. 



We have now to consider the cause of the pairing together of 

 atoms into molecules. It cannot be on account of the magnetic, 

 i.e. hydroiynamical, forces they exert on one another, for two 

 electric carrents would ihen come together so as always to rein- 

 force each other's magnetic action, and all substances would be 

 strongly magnetic. The ionic electric charge, which the 

 phenomena of electrolysis sho.v to exist on the atom, supplies 

 the attracting agency. Furthermore, the law of attraction be- 

 tween these charges is thatoftheiaversesquareofthedistance, and 

 between the atomic currents is that of the inverse fourth power ; 

 so ihaf, as in the equilibrium state of the molecule these forces 

 are of the same order of intensity and counteract each other, the 

 first force must have much the longer range, and the energy of 

 chemical combination must therelore be very largely electro- 

 static, due to the attraction of the ions, as von Helmholtz has 

 clearly made out from the phenomena of electrolysis and electro- 

 lytic polarisation. 



Bat in this discussion of the phenomena of chemical combina- 

 naiion of atoms we have been anticipating somewhat. All our 

 conclusion^, hitherto, relate to the eether, and are therefore 

 about electromotive forces. We have not yet made out why 

 two sets of molecular aggregates, such as constitute milerial 

 bodies, should attract or repel each other when they are charged, 

 or when electric currents circulate in them ; we have, in other 

 words, no.v to explain the electrostatic and electrodynamic 

 forces which act between conductors. 



Consider two charged conductors in the field ; for simplicity, 

 let their conducting quality be perfect as regards the very slow 

 displacements of them which are contemplated in this argument. 

 The charges will then always reside on their surfaces, and the 

 state of the electric field will, at each instant, be one of equi- 

 librium. The magnitude of the charge on either conductor 

 cannot alter by any action short of a rupture in the elastic quality 

 in the jether : but the result of movement of the conductors is to 

 cause a rearrangement of the charge on each conductor, and of 

 the electric displacement (/, g, h) in the field. Now the elec- 

 tric energy W of the system is altered by the movement of the 

 conductors, and no viscous forces are in action ; therefore the 

 energy that is lost to the electric field mast have been somehow 

 spent in doing mechanical work on the conductors ; the loss of 

 potential energy of the electric field reappears as a gain of 

 potential energy of the conductors. We have to consider how 

 this transfoimatioii is brought about. The movement of the 

 coaductors involve^, while it lasts, a very intense flow of ideal 

 electric displacement along their surfaces, ani also a change of 

 actual displacement of ordinary intensity throughout the dielec- 

 tric. The intense surface flow is in close proximity with the 

 electric tljws round the vortex atoms which lie at the surface ; 

 their interaction produces a very intense elastic disturbance in 

 the medium, close at the surface of the conductor, which is 

 distribute 1 by radiation through the dielectric as fast as it is 

 proluced, the elastic co.idition of the dielectric, on account of 

 its extreme rapidity of propagation of disturbances compared 

 with its finite extent, being always extremely nearly one of 

 equilibrium. It is, I believe, the reaction on the conductor of 

 these wavelets whicharecontiuually shooting out from its surface, 

 carrying energy into the dielectric, that constitutes the mechani- 

 cal forcive acting on it. Bat v/e can go further than this ; the 

 locality of this transformation of energy, so far at any rate as 

 regards the material forcive, is the surface of the conducor ; and 

 the gain of mechanical energy by the conductor is therefore 

 correcdy located as an absorption of energy at its surface ; there- 

 fore the forcive acting on the conductor is correctly de- 

 termined as a surface traction, and not a bodily forcive 

 throughout its volume. One mode of representing the dis- 

 tribu'.ion of this surface traction, which, as we know, gives the 

 correct am )unt of work for every possible kind of virtual dis- 

 placement of the surface, is to consider it in the ordinary electro- 

 static manner as a normal traction due to the action of the 

 electric fo'ce im the electric density at the surface ; we conclude 

 that this distril)ution of traction is the actual one. To recapitulate : 

 if the dielectric did not transmit disturbance sd rapidly, the 

 result of the commotion at the surface produced by the motion 



of the conductor would be to continually start wavelets which 

 would travel into the dielectric, carrying energy with them. 

 But the very great velocity of propagation effectually prevents 

 the elastic quality of the medium from getting hold ; no sensi- 

 ble wave is produced and no flow of energy occurs into the dielec- 

 tric. The distribution of pressure in the medium which would 

 be the accompaniment of the wave motion still persists, though 

 it now does no work ; it is this pressure of the medium against 

 the conductor that is the cause of the mechanical forcive. 



The matter is precisely illustrated by the fundamental apercit 

 of Sir George Stokes with regard to the communication of 

 vibrations to the air or other gas. The rapid vibrations of a 

 tuning fork are communicated as sound waves, but much less 

 completely to a mobile medium like hydrogen than to air. 

 The slow vibrations of a pendulum are not communicated as 

 sound waves at all ; the vibrating body cannot get a hold on 

 the elasticity of the medium, which retreats before it, preserving 

 the equilibrium condition appropriate to the configuration at 

 the instant ; there is a pressure between them, but this is instan- 

 taneously equali ed throughout the medium as it is produced, 

 without leading to any flow of vibrational energy. 



Now let us formally consider the dynamical system consisting 

 of the dielectric media alone, and having a boundary just inside 

 the surface of each conductor ; and let us contemplate motions 

 of the conductors so slow that the medium is always indefinitely 

 near the state of internal equilibrium or steady motion, that is 

 conditioned at each instant by the position and motion of the 

 boundaries. The kinetic energy T of the medium is the electro- 

 dynamic energy of the currents, as given by Neumann's formula ; 

 and the potential energy W is the energy of the electrostatic 

 distributioa corresponding to the conformation at the instant ; 

 in addition to these energies we shall have to lake into account 

 surface tractions exerted by the enclosed conductors on the 

 medium, at its boundaries aforesaid. The form of the general 

 dynamical variational equation that is suitable to this problem 

 is, for currents in incomplete circuits, and therefore acyclic 

 motions, 



S j {T- W) d/ + \dl i 57U dS = o, 



where Szv di represents the work done by the tractions acting on 

 the element dS of the boundary, in the virtual displacement 

 contemplated. If there are electromotive sources in cer- 

 tain circuits of the system, which are considered to introduce 

 energy into it from outside itself, the right-hand side of this 

 equation must also contain an expression for the work done by 

 them in the virtual displacement contemplated of the electric 

 coordinates. No.v this variational equation can be expressed in 

 terms of any generalised coordinates whatever, that are suffi- 

 cient to determine the configuration in accordance with what we 

 know of its properties. If we suppose such a mode of ex- 

 pression adopted, then, on coiducting the variation in the usual 

 manner and equating the coifticients of each arbitrary variation 

 of a coordinate, we obtain the formulae 



cl^dT 

 dt ^ 

 ■p _ d rt'T 



~d}'d^' 



In these equations <J> is a component of the mechanical forcive 

 exerted on our dielectiic system by the conductors, as specifietl 

 by the rule that the wo:k done by it in a displacement of the 

 system represented by 5;^, a variation of a single coordinate, is 

 ^dp : the corresponding component of the forcive exerted by 

 the dielectric system on the conductor is of course — *. Also 

 E is the electromotive force which acts from outside the system 

 in a circuit in which the electric displacement is e.; so that the 

 current in it is i ; the electromotive force induced in this circuit 

 by the dielectric system is - E. 



These equations involve the whole of the phenomena of 

 ordinary electrodynamic action^, whether ponderomotive or 

 electromotive, whether the conductors are fixed or in motion 

 through the medium : in fact, in the latter respect no distinction 

 appears between the cases. They will be completed presently 

 by taking account of the dissipation which occurs in ordinary 

 conductors. 



These equations also involve the expressions for the electro- 

 static ponderomotive forces, the genesis of which we have 

 already attempted to trace in detail. The generalised c im- 

 ponent, corresponding to the coordinate (p, of the electrostatic 



* = 



dT d\N 



N. [ 23 i- , VOL. 49] 



