448 Mr. S. Butterworth on the Coefficients of 



5. Unequal circles. 



Let the radii of the circles he a and A and let x be the 

 distance of their planes. Then if x is large and the circles 

 are coaxial, the mutual induction is * 



-, 27r 2 a 2 AV 1 3 A 2 , 3.5 AV 3 . 5 . 7 A V , \ 

 in which 



(ii) 



Ki=l + y 2J 



K2=i+3 5+£' 



K i =l + 6j+6j + ^ 



K 



»=F(— «-l, — «, 2, jjj, 



where F(a, /3, 7, «) denotes the hypergeometric series 

 a/3 *(tt+l)£(ff+l ) 



Hence for non-coaxial circles the mutual induction is 



M 



VA 2 /- 

 r 3 V 



2ttVA 2 / t , 3 A 2 3. 5 A 4 3.5. 7 A 6 



2 ~ 2 ~? Kl± * + 2^ "7 K2n ~ 2T476 V 



=rK s P« + 



\ (12) 



reducing for coplanar circles to 



M 1= =- 



1+ ^^ Kl+ 271*76 7* 



+ 



3 2 .5 2 .7 2 A' 



2 . 4 2 . 6 2 . 8 



^K 3 +...). (13) 



Formulae (11), \0-2), and (13) converge if r>A+a, 

 i. e. in formula (13) if one circle is entirely outside the 

 other. 



When x is small it is convenient to choose the difference 

 in radii of the two circles as the unit of length. Then, 



* Havelock, loe. cit. 



