A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 591 



Let a, /8, y be the components of magnetic intensity at any point due to 

 the first circuit, a, (?, y the same for the second circuit ; then the intrinsic 

 energy of the element of volume dV of the field is 



The part which depends on the product of the currents is 





Hence if we know the magnetic intensities / and /' due to the unit current 

 in each circuit, we may obtain M by integrating 



47T 



over all space, where 6 is the angle between the directions of / and /'. 



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 every- 

 where 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, 



To approximate more closely to the value of M, let a and a, be the radii of 

 the circles, and 6 the distance between their planes; then 



We obtain M by considering the following conditions: 

 1st. M must fulfil the differential equation 



=Q 

 da- db* a da 



This equation being true for any magnetic field symmetrical with respect to the 

 common axis of the circles, cannot of itself lead to the determination of M as 

 a function of a, a,, and 6. We therefore make use of other conditions. 



