Mr Searle, Or the Coefficient of Mutual Induction , etc. 399 
ordinates for the two systems S and T . Then the semi-infinite 
straight line OT is called the 
“ axis” of the system T\ the “axis” 
thus extends in one direction only 
from 0. Now let the potential 
due to T be expressed at any point 
K on its axis by the series 
r.=?+$+§+-o->*) ox 
Fig. 1. 
where OK = r. Then by Legendre’s theorem the potential at any 
point R on a radius OR, which makes an angle yjr with the axis 
OT, is 
y = g.r.(co sf) + y,P,( cost) + (r>t) (2)j 
the series (1) and (2) being “absolutely” convergent if r>t 
where t is the greatest distance from 0 of any particle of the 
system T. 
But (2) is the potential at (r, yjr) due to a system, Q, of singular 
points of orders 0, 1, 2, and moments g 0 , g%... placed at 0, 
every particle of each singular point lying on the axis OT. Thus, 
at all points outside the sphere r = t, the potentials due to T and 
to Q are equal. 
Now, if W be the potential energy of T in the field of S, 
the potential energy of S in the field of T is also W. But, 
when s > t, so that every particle of S lies outside the sphere 
r = t, centred at the origin 0, the potential energy of S in the 
field of T is equal to the potential energy of S in the field of 
Q. This last is equal to the potential energy of Q in the field 
of S. 
Considering the system Q, let m, m', ... be the masses of the 
particles which form the whole system of singular points, and let 
h, h', ... be their distances from 0 in the direction OT. Then the 
potential due to Q, at any distant point on OT, is 
r m ^ /I h h? \ 
V 0 = 2 j = Xm - + 
i — h \r r- r 3 J 
'Zmh 
Hmh 2 
+ 
(3). 
But the values of V 0 given by (1) and (3) must be equal. Hence, 
equating coefficients of the powers of 1/r in the two series, we 
obtain 
2m = g 0 , Xmh = g x , Sm/i 2 = ^, &c (4). 
