deriving Mutual- and Self-Inductance Series. 279 



Since M, x, x 3 vanish with k, B = B 3 = 0. Also when k is 

 small 



M/47r-v/A^=7rP/16=7r^t/16= -mf/16 



by virtue o£ (1). 



Under the same conditions the right hand members of (6) 

 and (6 c) become 



Atft/32 and A 3 ^/32 respectively. 

 Hence A=— A 3 =27r. 



Using these values in (6) and (6 c) we find 



_2 / 02 Q2 K2 22 K2 H2 \ 



and 



_2 / 02 Q2 K2 92 *2 72 \ 



M -4^K l 4^ i+ TO'' , - 2 4: b .8.# (+ '-) < B > 



, . , „ k 2 4 A a 



m which yu/ — 



J-P (A-a) 2 + 6 2 * 



(A) is valid for all distances of the circles, but converges 

 most rapidly when the circles are far apart. It could readily 

 have been obtained by substituting the usual series for K 

 and E in (1). 



(B) converges only if P<^. When the circles are of 

 equal radius /u,= 2a/b, and the series is then identical with 

 one obtained by Havelock *. 



•» l5 x 2 , x± vanish when k is unity, and when either of the 

 independent variables is small (6 a), (6 b) } and (6 d) assume 

 the form A„ + B n log xn. Under the same conditions with 



M/4tt v /A^= log % - 2 = log 



h & ^/'x l 



= log — — — 2 = log 



Therefore 



A 1= log 4-2 = A 2 , A 4 = log . -2, 

 B 1 = B 2 = B 4 = — J. 



* Havelock, Phil. Mag. xv. p. 332 (1908). 



