deriving Mutual- and Self-Inductance Series. 281 



in which 



cf> " = log 4^-2 

 +i -+o -1~3 + 1 



2 - <Pl - o - K - 



A" A" - 1 X 1 



</>;i —<pn-l = 



1 



2?i+l 2?i — 3 



J 



(0) and (D) are valid for all distances of the circles, 

 but converge most rapidly when the circles are close 

 together. 



(E) converges only if k 2 >^. The first five terms of this 

 formula have been given by Rosa (loc. cit. p. 16). 



5. Self-induction of a Solenoid. 



Take the diameter of the solenoid as unit of length, let the 

 axial length of the solenoid be z 3 and the number of turns 

 be nz. Then, if M represents the mutual induction between 

 two equal coaxial circles at distance z, the mutual induction 

 between the solenoid and an equal circle in the plane of its end 



is n 1 M dz, and the self-induction required is 



2n*Cdz\*Mdz. 



Jo Jo 



Now from (3 d), since y 4 =M, #4=— z 2 , 

 so that by integration 



