of the Induction- Coefficients of Coils. 

 Thus we have, neglecting constants as before, 



4!a 2 f ^ <fe 



B* 4 j 



67 



(8') 



Thus (8) and (8') give by the same process all the required 

 integrals. Taken together they give the theorem 



{m§ d ^ { -iy^h^, ■ ■ ■ 02) 



1 r t+2 y ' 1 1 a 2 d^- 1 ' v y 



where i is any positive integer. A similar theorem holds of 

 course, mutatis mutandis (that is, with a substituted for a, 

 f for #, and p for r) , for the harmonics in ft, and can be used 

 as indicated above for the calculation of the second integrals 

 in (2). 



The first seven derived functions of A are as follows : — 



Ao=l-f, 



^ 2 A _ 3a 2 a 

 ^^~ ^ ? 



^= 3.5^(4^-3^), 



3 2 .5-^(8a 4 -12aV + « 4 ), 



5 5 A _ 



^° = 3 2 . 5 ^ (56^ 4 - 14(U'V + 35a 4 ), 



r* 7 A rr 2 



° ° = -3 2 .5^- 5 (44^ 6 -1680^a 2 + 840^ 2 a 4 -35a 6 ). 



Substituting these values' in (8), (8'), or (12). and using the 

 results in (2), we obtain 



T=7r 2 nn f a 2 u 2 {K 1 k l Z 1 ($) + K 2 & 2 Z 2 (0) +...}, . (13) 

 F2 



