Mutual Induction of Eccentric Coils. 451 



7. On applying the results of Section 6 to (14) and 

 inserting the values o£ P»(/a), P„(w), the mutual induction 

 between non-coaxial circles is found to be 



M = 4 V Aa[x-2 + l 6 i- a {(x-^ 2 - X2 } 



-i&M(*- §)*-*}+-]. (22) 



in which 



(U/tlVl+v 2 



*2=|-|P 2 WP2H 



= i{ (3 + 2/* s + 3/a 4 ) -2^(1 + 6/^-15^) 



+ 1^(3-30^ + 35/**)}, 



%2 =^(l- / .){(l- y a)-v 2 (l4-7 / ,)}, 



%4= 9 3 g( 1 -/^){3(l-^)(7 + 2 / ,+ 7^j 



+ z/ 4 (21 + 241/*-113/* 2 -533/* 3 )}, 



and from (15 a) 



(A 2 , —v 2 are the roots of 



t 2 -t{l-p*-X 2 )-X 2 = Q. .... (23) 



In (22) the difference in the radii of the two circles is 

 unity. If A~a = c, then replace in (22) 1/Aa by c 2 /Aa, 

 multiply by c, and make yu, 2 , — v 2 the roots of 



cH 2 -t(c 2 - P 2 -x 2 )-x 2 =--0. . . . (23 a) 



To test the formula, let c approach zero. The limiting 

 value of /jl is xj '^/ x 2 + p 2 = x/r, and that of cv is r. Using 

 these in (22), we obtain formula (7). 



2H2 



