Mutual Induction of Eccentric Coils. 



445 



from this formula we must replace the terms x n by r"P m and 

 the terms x n log e x by 



|^, ^(pjog e , + ^), 



for these clearly satisfy Laplace's Equation, and since 



BP„ 



~dn 



0, P n =l when = 0, 



they reduce to x n and x n \og e x respectively when = 0. 

 In applying this transformation the following explicit 



values of -^- 5 are required : 



dP» 



in which 



3n - P « lo «-Kl+l0+^ .-.. (5) 

 ,u, — cos#, 



*»= 2 {r^( p »- p »- 1 )- w^i) c p .- p ~> 



+ • 



-)" +1 ^Ti)( p »- p «)} 



and in particular 



+*=- | (1— /*)(! + 7 A*)i 



^ 4 = ^(l- / ,)(21 + 241^-113 / , 2 -o33/. 3 ), 



1 



(6) 



(7) 



^ 6= _^(l_^)(185-2957 A 6+3728/^ 2 + J8008 A 6 3 



-3247^ 4 -18107^ 5 )J 



On making the necessary substitutions in (4) the required 

 mutual induction is given by 



M ^ 9-L 3 



-r~ =\ — 2 4- -T-r 

 4tt lb 



15 



{P 2 (x-^). t2 } 



+raw*W>-K)-*} 



in which 



(128)' 



. . . • • 



X — log, 



(8). 



16 



<l + H>) 



