398 Mr Searle, On the Coefficient of Mutual Induction, etc. 
On the Coefficient of Mutual Induction for a circle and a 
circuit with two parallel sides of infinite length. By G. F. C. 
Searle, M.A., Peterhouse, University Lecturer in Physics, and 
Demonstrator in Experimental Physics. 
\Bead 3 March 1902.] 
§ 1. Maxwell, in the chapter on Circular Currents in his 
Treatise on Electricity and Magnetism, has shewn how to express, 
in the form of a series, the coefficient of mutual induction for two 
circular circuits, whose axes meet in a point at any angle 6, the 
successive terms of the series containing the factors 
P o (cos0), P x (cos 6), ..., 
where P n (cos 6) is the Legendre coefficient or “ surface zonal har- 
monic” of the nth order. In the present commuoication, by using 
a process in principle identical with that employed by Maxwell, 
I obtain the coefficient of mutual induction when one of Maxwell’s 
circles is replaced by a circuit having two infinitely long parallel 
sides, the shortest distance between the sides being finite. 
A telephone circuit with its pair of parallel wires is a practical 
approximation to such a circuit. The series in which the result is 
expressed involves the two angular coordinates, which determine 
the direction of the axis of the circle relative to the two parallel 
sides, in the form of “ surface sectorial harmonics,” these functions 
playing the same part in the present problem as the Legendre 
coefficients play in Maxwell’s problem. 
§ 2. It may be useful to give a general explanation of the 
principle of the method before proceeding to the detailed calcula- 
tion for the problem in hand. 
If we take two systems, S, T , of matter, attracting according 
to the law of the inverse square, of which T is symmetrical round 
a straight line, we can apply the method to calculate the potential 
energy of T in the field of S provided that the least distance, s , 
from the origin of coordinates to any “particle” of S exceed the 
greatest distance, t, from the origin to any particle of T , and hence 
I shall not restrict the explanation to systems giving rise to 
magnetic force. 
§ 3. Let TOT', Fig. 1, be the straight line about which the 
system T is symmetrical, and let 0 be taken as the origin of co- 
