Mutual Induction of Eccentric Coils. 449 



when the circles are coaxial *, 



15 (1 + *)V X 31\ 1 n , 



S\/Aa 



in which X= log e 



VI -h^ 



The transformation of (14) to the formula for the non- 

 coaxial case requires the determination of a solution of 

 Laplace's equation, which will reduce to 



at all points on the axis of x. This solution will now be 

 found. 



6. In cylindrical coordinates {x, p, (/>) and with ^j- 2 =0, 

 Laplace's equation is 



¥ + v + f¥ =u (15) 



Transforming to spheroidal coordinates by putting 



af = fiv, /o = (l-/* 2 )*(l + z/% . . (15a> 

 (15) becomes 



a possible solution of which is 



'.0 



-2AJ> s O)P», . (17) 

 in which i= V — 1- 



When /u=l, that is, when /3 = 0, v = a', (17) reduces to 



Y='^^-Q n (iv)-%A S -P s (iv), . . (18) 



* Havelock, loc. cit. ; Rosa, Bull. Bureau of Standards, viii. p. 15 

 (1912). 



Phil. Mag. S. 6. Vol. 31. No. 185. May 1916. 2 H 



V= ^{P^)Pn(^)}-Pn(^)Q«(^) 



