770 Dr. J. R. Wilton on the Solution of 



In addition we have O = at infinity, and 



at the singular points o£ the transformation. 

 We have immediately 



O =i { /&-'■ W(X„ T a ) +/ (6)-4*rJ(X» Y 2 ) 



^=i{/W+/W}-^J(.^y)+^{FW-F(r)}, 



where F and /are to be determined by the conditions stated 

 above. 



As a very simple illustration take the case of an elliptic 

 cylinder uniformly magnetized. We have, in this case, 



And it immediately follows that 



O =^{(E + 47raA)cosi7 + (E + 47r6B)sini7}, 



X2i = O — 4-7T sinh (-(&A cos^ + aB sin 77). 



In this result E and F are constants to be determined from 

 the values of ~d£l /3f and 'dkli/'d'n at the singular points. 

 We have 



—^ = - e~t | (E + 4wwA) cos 77 + (F + 4tt6B) sin 77I 

 — 47T cosh £(& A cos rj + aB sin 77) , 



^i = e -S J - (E + AttclA) sin 97 + (F + 4tt&B) cos 77 

 + 47r sinh £(6 A sin 77 — aB cos 77) . 



And these must both vanish when f = — \, sin 77 = 0. Thus 

 we have 



E = — 4-7T A (a 4- b e~ x cosh X) = — 4-7raA(a + 2b) I {a +• b), 

 -F=-4;7rB(b + ae^shihX)=-4:TrbB(2a + b)/(a + b). 

 And, finally, 



X2 = —A7r{ab/(a + b)}e~^(A cos 77 + B sin 77) , 

 Hi = — 4-7T sinh %(bA cos 77 + aB sin 77) + I2 C . 



