404 Mr Searle, On the Coefficient of Mutual Induction , etc. 
The simplest case is that in which the origin lies in the plane 
of the two straight wires and is midway between them. We then 
have b = a and 7 = J tt , so that 
2 
u 2m—i 
and then, if c < a, 
M = 87 rc 
(- O' 
> V-zm—i — 0 , 
1 C . n .,1.1c 3 . „ . 0 , 
s - sin t) cos 0 4- ~ — 7 — o sir V cos 30 
2 a r 2.4ft 3 r 
1.1.3 c 5 . r . 
4 - , — 7 ; - sm J 6 cos o 0 4 - . . . 
z 4 . 0 a 5 r 
By § 5, these series are convergent provided that the radius of the 
circle is less than the shortest distance from its centre to either of 
the infinite wires. 
If we introduce the additional limitation that = 0, so that 
now a diameter of the circle lies on a line cutting both long wires 
at right angles, the last expression for M reduces to 
M — 87 r cosec 0 {a — fa 2 — c 2 sin 2 0}, (c < a) 
as may be seen by expanding the square root (a 2 — c 2 sin 2 0)K 
APPENDIX [Added 14 April 1902]. 
The following method, of obtaining the expression 
W = g 0 Y 0 + g 1 Y 1 +g i Y 2 +... 
for the mutual potential energy of the systems S and T , seems 
preferable to that given in §§ 3, 4 of the foregoing paper. 
I will consider first the case in which every part of S is further 
from 0 than any part of T, so that s > t. 
Let the potential due to T be expressed at points on its axis 
OT (Fig. 3) by the series 
V 0 
#0 & 02 
r " t " r 2 ' ^ ' * * * * 
(r > t) 
Then the potential due to T at a point H, which lies, at a 
distance R from the origin, on a radius making an angle yjr with 
0T } is 
Y_ 0 ° 1 $1-^1 (V) I 9^2 (^) 1 
R + R 2 + R 3 
(R > t) 
where X = cos yfr. 
