64 Mb maxwell, ON FARADAY'S LINES OF FORCE. 



Let us now consider the conditions of the conduction of the electric currents within the 

 medium during changes in the electro-tonic state. The method which we shall adopt is an 

 application of that given by Helmholtz in his memoir on the Conservation of Force*. 



Let there be some external source of electric currents which would generate in the con- 

 ducting mass currents whose quantity is measured by a^ 63 Cj and their intensity by aj jSa 72- 



Then the amount of work due to this cause in the time di is 



dtjjj{a./i2 + 62/82 + c./yi)daidydz 

 in the form of resistance overcome, and 



dt d 



47r dt 



Jjjiazaa + hfio + c^y^dxdydz 



in the form of work done mechanically by the electro-magnetic action of these currents. If 

 there be no external cause producing currents, then the quantity representing the whole work 

 done by the external cause must vanish, and we have 



dt jjf(«2a2+ ^A + c.,y2)da!dydx+ — — Jfjictiaa + b.,(i^ + c.i'y^)dxdydz, 



where the integrals are taken through any arbitrary space. We must therefore have 



a-ift^ + 60/82 + CjYjj = — — (a,ao + h^fi^ + c^yo) 



for every point of space; and it must be remembered that the variation of Q is supposed due to 

 variations of oq fio 7o' ^"^ ^^^ "^ '^^ ^2 '^2- ^^ must therefore treat a.^ b^ c^ as constants, and 

 the equation becomes 



\ A>Tr dt I V 4,7r dt J \' 'iir dt J 



In order that this equation may be independent of the values of a^ h.^ c^, each of these co- 

 efficients must = ; and therefore we have the following expressions for the electro-motive 

 forces due to the action of magnets and currents at a distance in terms of the electro-tonic 

 functions, 



°^ 4.7r dt ' 47r dt ' '^' 47r dt ' 



It appears from experiment that the expression --" refers to the change of electro-tonic state 



of a given particle of the conductor, whether due to change in the electro-tonic functions 

 themselves or to the motion of the particle. 



If Op be expressed as a function of x, y, x, and t, and if x, y, % be the co-ordinates of a moving 

 article, then the electro-motive force measured in the direction of x is 



1 /dao dx da^ dy daa dz da^\ 

 47r V dx dt dy dt dz dt dt J 



• Translated in Taylor's New Scientific Memoirs, Part 1 1. 



