Mr dearie, 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 S, and the coefficient of mutual 
induction by M, the potential energy of T in the magnetic field of 
S is — M, when the currents in S and T are each of unit strength. 
Thus, if IF 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 0 in the plane of the paper as origin, let 
OA = a, OB = b. Let the angle AOB be 2y and let OB bisect it. 
Let OT 0 be the projection on the plane of the paper of OT ', the 
axis of the circle, and let T 0 OD = </>, and let the angle between 
OT and a straight line parallel to the wires be 0. Thus 6 is the 
co-latitude, and <j) the longitude, on a sphere, whose polar axis is 
parallel to the long wires. Let K be any point on OT and K 0 its 
projection. Then if OK = r, the coordinates of K are r, 6, </>. 
If we denote OK 0 by p, then the coordinates of K 0 in the plane of 
the paper are p, </>. 
We must now expand the magnetic potential due to S in a 
series involving p and </>. If the current in S, as seen from 0, 
