146 Dr. T. J. I'a. Bromwich on 



The line-integral of the magnetic force is readily seen to be 



•?2 r<i2 ct-2 



No 



rn 2 c l -2 C* f* 2 



[B/fycfy + l [Cy] #-1 [B/3]^-) [Cy] dZ. 

 J n x JZ-l J *li "lr 



[Cy] /;2 -[Cy] ?h = r 2 |-(C y )^ 



r ?2 d 



9i 



and fl 



Thus the line-integral becomes 



m 



g(Cy)-^(B/5) 



Using Ampere's relation connecting the line -integral of 

 magnetic force with the total electric current through the rect- 

 angle, and Maxwell's hypothesis for displacement currents, it 

 is clear that the last integral must be equal to 



■^ BCdndi;. 



J, A * 2 



And so the first of equations (E) is proved ; the second and third 

 follow by symmetry. Similarly equations (M) are deduced from 

 Faraday's circuital relation applied to the same rectangle. 



It will be proved in §2 below that, under certain con- 

 ditions, the most general solutions of the equations (E) 

 and (M) can be derived by superposition of two special 

 solutions, which can be obtained by writing first a = 0, and 

 secondly X = 0. 



We proceed to find the first of these solutions, corre- 

 sponding to <* = ; then the first equation in (M) gives 



BT -s£ CZ =I' (1-1) 



where P is an arbitrary function. 



If we substitute from equations (1*1) in the second and 

 third equations in (E) we see that they become 



. • • (1*2) 



In order to proceed further with the solution we shall now 

 restrict the coordinates by assuming that A is a function 



