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XXXIV. On a Method for deriving Mutual- and Self-In- 

 ductance Series. By S. Butterworth, M.Sc, Lecturer in 

 Physics, School of Technology, Manchester*. 



1. l^TUMEROUS series formulae have been obtained for 

 ll the mutual induction between coaxial circles f . 

 In general, these formulae have been completely determined 

 (i. e. the range of convergency and general term are known) 

 only in the case of circles far apart. 



It is proposed in the present paper to develop a method 

 which will yield the general term for formulae which hold 

 when the circles are close together. 



2. Let the radii of the circles be A and a, and let b be 

 the distance of their centres. Then the mutual induction 

 between them is given by the Elliptic Integral formula J 



M = 47r\/Aa{ (|-/Ak-|e l, . . (1) 



in which 



4Aa 



{A + a) 2 +b 2 



and K and E are complete Elliptic Integrals of the first and 

 second kind respectively with k as modulus. 



By differentiating (1) twice with respect to k, remem- 

 bering that 



dE_E-K dK_ E _K 



dk~ k ' dh ~~ k(l-7c 2 ) k ' 



, dM , d 2 W A v . ... 

 to obtain expressions for -jr and -,, y , and eliminating 



E and K between these expressions and (1), we find that 



P(l_P)^_/ : (i + p)^_3M = 0. . . (2) 



By the substitutions M = k*y, k 2 = x, (2) transforms into 



$($ + 2)y = ^ + §) 2 y, (3) 



in which 3 =#-7— . 

 doc 



* Communicated by the Author. 



t For a collection of such formulae, see Eosa, Bull. Bureau of 

 Standards, viii. p. 1 (1912). 

 t Maxwell, ' Electricity and Magnetism,' ii., art. 701. 



