58 Mr maxwell, ON FARADAY'S LINES OF FORCE. 



result deducible from Art. (30), since the volume of each cell is inversely as the square of 

 the velocity (Arts. J 2, 13), and therefore the number of cells in a given space is directly 

 as the square of the velocity. 



Theorem IV. 



Let a, /3, y, p be quantities finite through a certain space and vanishing in the space beyond, 

 and let k be given for all parts of space as a continuous or discontinuous function of xyz, 

 then the equation in p 



d If dp\ d !/_ dp\ d 1/ dp\ 

 Txlk"- d-v)-'jyk['^~dy)^lilcV-irzr^''P=''^ 



has one, and only one solution, in which p is always finite and vanishes at an infinite 

 distance. 



The proof of this theorem, by Prof. W. Thomson, may be found in the Cambridge and 

 Dublin Math. Journal, Jan. 1848. 



If a /3 7 be the electro-motive forces, p the electric tension, and k the coefficient of resist- 

 ance, then the above equation is identical with the equation of continuity 



da^ dbi dCi 

 dx dy dz ' ' 



and the theorem shews that when the electro-motive forces and the rate of production of 

 electricity at every part of space are given, the value of the electric tension is determinate. 



Since the mathematical laws of magnetism are identical with those of electricity, as far as 

 we now consider them, we may regard a^y as magnetizing forces, p as magnetic tension, and p 

 as real magnetic density, k being the coefficient of resistance to magnetic induction. 

 The proof of this theorem rests on the determination of the minimum value of * 



where V is got from the equation 



rfT d'V d'V 



and p has to be determined. 



The meaning of this integral in electrical language may be thus brought out. If the pre- 

 sence of the media in which k has various values did not affect the distribution of forces, then the 



"quantity" resolved in x would be simply — and the intensity ^ — . But the actual quan- 



1 / dp\ dp 



tity and intensity are -la - —I and a - — , and the parts due to the distribution of media 



alone are therefore 



1 dp\ dV dp dV 



k dxj doe dco dx ' 



