1884.] equations of the electromagnetic field. 131 



For this action depends on the value of 

 — [cos (ds, dcr) + £OS (r, ds) cos (r, da-) 



+ k {cos (ds, da) — cos (r, ds) cos (r, da)}], 



ds, da- being the elements of current and r the distance between 

 them, and if k is infinite this expression is infinite also, though by 

 integrating throughout a space whose boundaries are infinitely 

 distant from the point considered it leads us to 



F > = "///(" ^5T 5S? + ") I dx ' d,J ' d2 '' 



which on putting —j- = and — fik -r- = J gives 



dJ 1 



+ fi ill - dxdydz ', 



so that F 1 is finite. 



Since however we cannot adopt an infinite value for the action 

 of one element on another we are forced to conclude that to 

 reconcile Maxwell's theory with that of Helmholtz we must have 

 J=0, and this condition will be satisfied, for any value of k which 

 is not infinite, whenever <I> = ; thus the assumption <£> = is 

 sufficient for our purpose independently of the value of k provided 

 at least it be not infinite. 



In this case from (26) we have -^ (\7 2 ^) = 0, and comparing 



this with (30) we get -~ = 0, so that no free electricity is produced 



in the medium by the electromagnetic induction ; there will be no 

 normal wave, and the quantity k disappears from our equations. 



From a physical point of view this condition <3> = 0, at least in 

 a dielectric, seems the reasonable one. For <E>, according to 

 Helmholtz, is the potential of the electrification produced through- 

 out the dielectric by the given electromotive forces. So far as we 

 know it is not possible to generate electricity by induction in the 

 interior of a dielectric ; the distribution produced will, like that in 

 the similar magnetic problem, be solenoidal, and the density of the 

 electricity which gives rise to the potential <I> will therefore be 

 zero everywhere, and <E> will be zero. 



9—2 



