Mr Searle, On the Coefficient of Mutual Induction, etc. 403 
§ 8. Turning now to the circle, we have to find the magnetic 
potential at points on its axis, due to unit current, in the form 
Vo 
^ + £i + £ 2 .. 
r r 2 r 3 
We notice at once that g 0 — 0, since at a great distance the current 
acts as a small magnet ; the leading term consequently varies 
inversely as the square of the distance. Hence the term F 0 will 
not appear in the expression for M. 
I now suppose that the centre of the circle coincides with 0. 
If c be the radius of the circle, and if the current in it appear 
from a point T, on the “ axis ” of the circle, to circulate in the same 
direction as the hands of a watch, then the magnetic potential 
at a point on the axis, for unit current, is 
V 0 = -2tt (l-— ; 
\ V r 2 + c 
a 
a ,'lc 2 1 . 3 c 4 1 . 3 . 5 c 6 
‘ 7r .2 r 2 2.4 r 1 + 27 T. 6 r 6 + 
(r > c) ... (11). 
Comparing (11) with (1) we have 
^0 = ^2 = ^ = ... =0 
' 1 2 0 1.3 4 0 1 . 3.5 ( 
jr, = — 27 r . ^ c, ^ = 2^ 0 c 4 , g s = - 2^ 0 76 c< 
( 12 ). 
§ 9 . Remembering that M = — W, where W is here the 
mutual potential energy when a unit current flows in each coil, we 
have by (6), (10) and (12) 
M — 47 rc 
- c sin 6 (u x cos — sin 0) 
A 
13 1 “I 
• O ° 3 S i n3 0 ( u 3 C0S 30 ~ V 3 s ^ n 30) + • • • 
A . t o 
(2m)! 
= 4 t rc 2 (- l)™" 1 ^ sin 2 ”*" 1 6 
\ / 2 2m m! m! (2m — 1) 
cos (2m - 1) <p sin ( 2 m - 1) y J 
sin (2m — 1)0 cos (2m — 1) 7 
where m ranges from 1 to 00 , and c < a and c < b. 
/Q2m—i 
q2171 1 \ ~ 
U 2 ™ -1 
1 
CJ- 1 
VOL. XI. PT. V. 
29 
