144 COEFFICIENTS OF INDUCTION. 



we have 



i t>V 



r rW />dV 



= --V/Z^=27T^ 2 <//*= 



]} *r Ji ^r 



M= 



Taking for V the value given by the second of the equations (44), 

 in which we might replace u by #', a by a', and then, making r = u, 

 we get 



M 



The integrations should be made starting from /* = cosa. Re- 

 placing the integrals by their values (24), we get 



(46) M = 4 7rV sin 2 a sin 2 a' f ^ X'^a'JX'^a) + . . . 



If the origin is at the centre O' of the small circle, it is sufficient 



2 

 to make u = a! and sin a = i. Replacing sin 2 a by , and calling 



S and S' the surfaces ira 2 and ira' 2 of the two circles, 



(47) M=-^ 



(-i) 1.3.5 



We thus arrive at the expansion, the first terms of which have 

 been given in (761). 



776. Suppose now that the axes (Fig. 146) of the two circular 

 currents cut in C at the angle 6, and consider at the point P an 

 element d<r of the shell o- which corresponds to the surface current S. 



Calling \' the angle O'CP, A the angle OCP, and < the angle 

 of the two planes O'CO and PCO, we may again write 



sn 



