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



In the same way it may be shewn that the values of a, /3, 7 satisfy the other given equations. 

 The function \^ may be considered at present as perfectly indeterminate. 



The method here given is taken from Prof. W. Thomson's memoir on Magnetism 

 (Phil. Trans. 1851, p. 283). 



As we cannot perform the required integrations when a, b, c are discontinuous functions of 

 a?, y, s, the following method, which is perfectly general though more complicated, may indicate 

 more clearly the truth of the proposition. 



Let A, B, C be determined from the equations 



^A d'A d^ 

 daf dy^ dx^ 



d'B dTB d'B 



H H h 6 = 0, 



daf dy" dz^ 



d'C d^C d'C 

 da^ dy" dz" 



by the methods of Theorems I. and II., so that A, B, C are never infinite, and vanish when ai, y, 

 or z is infinite. ^ 



Also let 



then 



id'A (PA drA 



I + - H 



\ dx' dy'^ dz 



If we find similar equations in y and z, and differentiate the first by x, the second by y, and 

 the third by z, remembering the equation between o, h, c, we shall have 



\dx' dy' dz'l \dx dy dz J 



and since A, B, C are always finite and vanish at an infinite distance, the only solution of this 

 equation is 



dA dB dC 



dx dy dz 



and we have finally 



d^_dy 

 dz dy 



with two similar equations, shewing that a, (3, y have been rightly determined. 



