Mr maxwell, ON FARADAY'S LINES OF FORCE. 61 



The function yj/ is to be determined from the condition 



da d^ dy /d" £_ ^\\ . 



if the left-hand side of this equation be always zero, >^ must be zero also. 



Theorem VI. 



Let a, b, c be any three functions of a:, y, z, it is possible to find three functions a, /3, y and 

 a fourth V, so that 



da d3 dy 

 dio dy dz 



d)3 dy dV 



and " = :i -r + "T » 



dx dy dx 



dy da dV 

 dx dis dy ' 



Let 



_da _d0 dV 

 dy dx dz 



da db dc 



dx dy dz "' 



and let V be found from the equation 



then 



satisfy the condition 



d^V d'V d'V 



dV 

 '""'-di' 



b'.b-'I, 

 dy 



dV 



c = c - — - , 



da db' dc 

 dx dy dz 



and therefore we can find three functions J, B, C, and from these a, /3, y, so as to satisfy the 

 given equations. 



Theorem VII. 



The istegral throughout infinity 



Q = fffidiai + bi(ii + Cxyx)dxdydz, 



