ON FARADAY'S LINES OF FORCE. 



199 



In the same way it may be shewn that the values of a, ft, y satisfy 

 the other given equations. The function \j/ 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 discon- 

 tinuous functions of x, y, z, 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 



d'A d*A d*A 



d'B d'B d*B 



d'C d'C d'C 



T-j + - r - r + -T T +c = 0, 



aar dtf dz 



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

 and vanish when x, y, or z is infinite. 



Also let 



_ dB dC dty 



~ dz dy dx ' 



n_dC dA d\ji 

 P = dx~~dz + dy' 



dA dB cfyr 



<y ^^ I * 



dy dx dz ' 



then 



+a. 



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 

 a, b, c, we shall have 



cP <P\ IdA dB d( 



d/3 dy _ d IdA dB dC\ (d*A d'A d*A\ 

 dz dy~ dx\dx dy dz) \dx* dy 3 dz' / 



d_ fdA_ dB d( 



