MUTUAL INDUCTION OF TWO CIRCULAR CURRENTS. 145 



The potential V of the shell a-' in the direction CP is a function 

 of the angle A'; the second of these equations (44) gives 



'\ n + l 



From this we get for the point P, where r=u, 



27TU sm^a ^ / . 



- j X n( a ) X n( A )- 



The coefficient of mutual induction is then 



M = - 2*u' sin' a' $ (^y Xj J X^A' 



The double integration should be made from to 2ir for the angle <, 

 and from to a for the angle A ; the angle A' satisfies further the 

 equation 



cos A' = cos A cos 6 + sin A sin cos </>. 



Integrating first in respect of <, we have, from a theorem of 

 Legendre, 



consequently 



M = - A' sin* 



Replacing finally the definite integral by its value given by 

 equation (24), and i - /x 2 by sin 2 a, we get 



(48) M-4-rV sin'a sin^a'^- X',,(a')X; i (a)X n (e). 



We obtain then the coefficient of induction of the two circular 

 currents whose axes cut under an angle 6, by the value for the 

 parallel currents, by multiplying respectively each of the terms by 

 the polynomial XnW f tne sam e order as that of which this term 

 already contains the differentials. 



VOL. II. L 



