200 ON FARADAY'S LINES OP FORCE. 



and since A, B, C are always finite and vanish at an infinite distance, the 

 only solution of this equation is 



dA dB dC 



and we have finally 



dz cty 



with two similar equations, shewing that a, ft, y have been rightly determined. 

 The function ^ is to be determined from the condition 

 da dft dy = (d* d?_ f*\ .. 



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



THEOREM VI. 



Let a, b, c be any three functions of x, y, z, it is possible to find three 

 functions a, /J, y and a fourth V, so that 



dx dy dz ~ 



d0 dy dV 

 and a = ^ _ ^ + 



dz ay dx 



j _dy da dV 

 ~ dx dz dy ' 



C:= _ + 

 dy dx dz 



Let 



da db dc 



and let V be found from the equation 



d'V 



