Mr Searle, On the Coefficient of Mutual Induction, etc. 401 



When linear electric currents are in question, the magnetic 

 shells, by which they can be replaced, may be of any forms provided 

 they are bounded by the circuits. In this case the series for W is 

 convergent, if the least distance from the origin of any part of the 

 wire belonging to S exceed the greatest distance of any part of 

 the wire belonging to T. 



§ 6. We are now able to attack the problem of finding the 

 coefficient of mutual induction between the circle and the circuit 

 with two parallel sides of infinite length. Denoting the circle 

 by T and the other circuit by 8, and the coefficient of mutual 

 induction by M, the potential energy of T in the magnetic field of 

 $ is_— M, when the currents in 8 and T are each of unit strength. 

 Thus, if W denote the potential energy in this case 



M = -W (7). 



§ 7. Let the diagram (Fig. 2) represent lines drawn in a 

 plane which cuts at right angles the two parallel wires forming 



the infinite sides of S. Let A, B be the sections of the wires 

 Taking any point in the plane of the paper as origin, let 

 OA = a, OB = b. Let the angle AOB be 2,<y and let OD bisect it. 

 Let OT be the projection on the plane of the paper of OT, the 

 axis of the circle, and let T OD = (p, and let the angle between 

 OT and a straight line parallel to the wires be 6. Thus is the 

 co-latitude, and (p the longitude, on a sphere, whose polar axis is 

 parallel to the long wires. Let K be any point on OT and K its 

 projection. Then if OK — r, the coordinates of K are r, 6, </>. 

 If we denote OK by p, then the coordinates of 7i" in the plane of 

 the paper are p, (p. 



We must now expand the magnetic potential due to S in a 

 series involving p and cp. If the current in S, as seen from 0, 



