406 Mr Searle, On the Coefficient of Mutual Induction, etc . 
If every part of S be nearer to the origin than any part of T, 
and if the potential due to S, at distant points on OT , be denoted 
by 
Y _ Ui _j_ Vi U? _j_ 
r r 2 r 3 
and the potential due to T, at points on OT near the origin, by 
U 0 = G 0 + (tj r 4- G. 2 r 2 + ... , 
then we can shew, in a similar manner, that the mutual potential 
energy of S and T is 
W = G 0 y 0 4- 0 1 y 1 + G. 2 y. 2 + . . . . 
Prof. T. J. I’A. Bromwich has kindly pointed out to me that 
the connexion between the equations (1) and (2) requires a word 
of comment. The point is that if S n be the sum of the series (2) 
to n terms, the quantities 
Lt cos \p=i ( Pt n = qo >S n ) and Lt. ft=00 (Lt cos ^ = i 
are not necessarily equal unless the series (2) be uniformly conver- 
gent. It is however easily shewn that this condition is satisfied 
provided r be definitely greater than t. 
If M be any particle of T , whose polar coordinates with 
respect to OT are R, cf>, then the potential at distant points on OT 
is expressed by a series, whose ?rth term is XMR n P n (cos <p)/r n+1 
or g n f n+1 , the summation including all the particles of T. Hence 
| g n | rj> t n 1j \M\. When T is a system of electric currents, flowing 
in circles whose common axis is OT, it can be shewn that 
| 9n ^ 4 t Tt^llil 
where t is the greatest distance from 0 to any wire of the system. 
If now we take two lengths k, l, such that k > t and l >k, then 
| g n P n (cos yfr)/l n+1 1 < g n /k n+1 1 for all values of cos yjr, the extreme 
values + 1 included. But, by the inequalities just mentioned, the 
series g 0 /k + g-^jk 2 + ... is absolutely convergent, and hence when 
r <(: l the series (2) is uniformly convergent for all values of cosy/r. 
Thus, provided r be definitely greater than t, we may put cos^=l 
either before or after the summation. 
