of the Induction- Coefficients of Coils. 65 



But this is of the form 



a> = 27r(A + A lt ?/ 2 + A 2 / + ...), ... (5) 



where 27rA is the value of (o for y=0, so that A =l— a/r. 

 Now (o must satisfy Laplace's equation, which, since there is 

 symmetry round the axis of the circle, is for the present case, 



|^ + |^ + L|^ =0 (6) 



^x 2 Br y dy 



Differentiating (5) and substituting in (6) we find 



^o + ^A 1 ^A 2 



^x 2 + ^x 2 y + ~bx 2 y +c " 



+ 2A x + 3 . 4A 2 / + 5 . 6A 3 y 4 + . . . 



+ 2 A 1 + 4A 2 # 2 + 6 A s y 4 + . . . = 0. 



The coefficients of the different powers of y in this series 

 equated separately to zero give 



A-_I^ A-_L^° a_ 1 d 6 A 



Al_ 2 2 ^ 2 ' 2 ~2 2 .4 2 d^' 3 ~ ^TS 2 gF"' * * • 



so that 



--ArJA,- ^ ^ + grp a? -■ ■ •> • • (7) 



Comparing this with (4) we have 



a2 Z/W = 5 2 A 0| 



o. 2 Z/W_B 4 Ao 



5 , 32 Z6 , W_B 6 A 0i 



7** B^ 



Thus we have, neglecting constants, 



2 i "2VYJ 7 



J r 4 dx ~ a 





i s A ( 



S 3 'J 



(S) 



-dor 

 Phil. Mag. S. 5. Vol. 33. No. 200. Jan. 1892. F 



