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



Now the product of these represents the work done on account of this distribution of 

 media, the distribution of sources being determined, and taking in the terms in y and « 

 we get the expression Q for the total work done by that part of the whole effect at any 

 point which is due to the distribution of conducting media, and not directly to the presence 

 of the sources. 



This quantity Q is rendered a minimum by one and only one value of p, namely, that 

 which satisfies the original equation. 



Theoeem V. 



If a, b, c be three functions of ir, y, z satisfying the equation 



da db dc 



dx dy dz 



it is always possible to find three functions a, fi, 7 which shall satisfy the equations 



d/3 dy 



Let A =fcdy, where the integration is to be performed upon c considered as a function 

 of y, treating x and % as constants. Let B =fadz, C=jbdx, A' = jbdx, B> =fcdx, C = fady, 

 integrated in the same way. 



Then 



a=A-A' + f, 

 dx 



dy 



will satisfy the given equations ; for 



d(i dy rda rdc rdb f^^j 



dx dy J dy J dz J dy J dy 



and 



rda rdb rd<i , 



= / — dx + \ — dx + / — dx; 

 J dx J dv J dz 



dy 



_, d(B dy f^^ -, r^<^ J f^^ 

 d% dy J dx ' J dy J dz 



- a. 



8—2 



