592 A DYNAMICAL T11EOKY OF THE ELECTROMAGNETIC FIELD. 



2ndly. The value of M must remain the same when a and a, are exchanged. 

 3rdly. The first two terms of M must be the same as those given above. 

 M may thus be expanded in the following series : 



(113) We may apply this result to find the coefficient of self-induction 

 (L) of a circular coil of wire whose section is small compared with the radius 

 of the circle. 



Let the section of the coil be a rectangle, the breadth in the plane of 

 the circle being c, and the depth perpendicular to the plane of the circle being 6. 



Let the mean radius of the coil be a, and the number of windings n ; 

 then we find, by integrating, 



X = r I M (xy x'y) dx dy dx' dy", 



where M(xy x'y'} means the value of M for the two windings whose coordinates 

 are xy and x'y' respectively; and the integration is performed first with respect 

 to x and y over the rectangular section, and then with respect to x' and y 

 over the same space. 



L = 4mi'a {log. + -5j - \ ( - 7) cot 26 - ^cos 20 - Jcot'fllogcos 6 - Jtan'fllogsin 



T I u \ *J o O o 



Here a= mean radius of the coiL 

 r= diagonal of the rectangular section = \/6* + c'. 

 ,, 0= angle between r and the plane of the circle. 

 ,, n= number of windings. 



The logarithms are Napierian, and the angles are in circular measure. 





