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. 



[Read 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 9, the 

 successive terms of the series containing the factors 



P (cos6>), P.icosO), ..., 



where P n (cos 6) is the Legendre coefficient or " surface zonal har- 

 monic" of the nth order. In the present communication, 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 j)roblem. 



| 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 be taken as the origin of co- 



