PROFESSOR CLERK MAXWELL OX THE ELECTROMAGNETIC FIELD. 507 
S' is a surface bounded by the current B, and l, m, n are the direction-cosines of the 
normal to the surface, the integration being extended over the surface. 
We may express this in the form 
M=jM-2^sin 0 sin 0 sin (pdS'ds, 
where d& is an element of the surface bounded by B, ds is an element of the circuit A, 
g is the distance between them, 0 and 0 are the angles between g and ds and between 
g and the normal to dS' respectively, and <p is the angle between the planes in which 
0 and 0 are measured. The integration is performed round the circuit A and over the 
surface bounded by B. 
This method is most convenient in the case of circuits lying in one plane, in which 
case sin 0=1, and sin<p=l. 
111. Third Method. M is that part of the intrinsic magnetic energy of the whole 
field which depends on the product of the currents in the two circuits, each current 
being unity. 
Let a, /3, y be the components of magnetic intensity at any point due to the first 
circuit, a!, (3 1 , y' the same for the second circuit; then the intrinsic energy of the 
element of volume dV of the field is 
£((«+«?+(|3+/ 3')’+(7+r') ! )'iV. 
The part which depends on the product of the currents is 
f(a«'+/3/3 '+yy’)dV. 
4 7T 
Hence if we know the magnetic intensities I and I' due to unit current in each circuit, 
we may obtain M by integrating 
J^lul T cos m 
4 7T r 
over all space, where 0 is the angle between the directions of I and I'. 
Application to a Coil. 
(112) To find the coefficient (M) of mutual induction between two circular linear 
conductors in parallel planes, the distance between the curves being everywhere the same, 
and small compared with the radius of either. 
If r be the distance between the curves, and a the radius of either, then when r is 
very small compared with a, we find by the second method, as a first approximation, 
M=Lra(log e ^— 2V 
To approximate more closely to the value of M, let a and a x be the radii of the circles, 
and b the distance between their planes ; then 
r 2 =(a — «,) 2 +§ 2 . 
3 z 
MDCCCLXV. 
