Induction in Spheroids. 467 



When r — E=l very nearly, the left-hand side is large, 



.\ Q„(R) -775 is large compared with -^. 



Also 



= i 



2 li(K 2 -l) 



M _ PQ- + 4?r *) p ^Qn P«QQ . 



1 W ~A 2 K("K 2 -1) n ^ ' 1 , „ 7ri dP,' near1 ^ 





-£(l + 4ri)C ^g, x ;; dP|t , nearly.. 



Which is therefore evidently small in comparison with M 

 (M 1? M 2 . . . being of decreasing magnitudes) ; for 



M = (1 + 4^)^ 



= -§ (l+4rf)Co, 



and it cnn be proved, as is also a priori evident, that the C's 

 are of decreasing magnitudes. 

 Now 



C = ^2irmi{l+w\^^ cos/rV-/ VT^l] ^~ \ > 



taken between limits. 



In order to simplify this, we shall assume that the wire is 

 wrapped round the whole of one-half of the spheroid. This 

 amounts to taking jjl between the limits 1 and 0. 



In this case 



« = -^(«iCi— ac), 



and Co = -fcrnufl + £log^» - -\ 



where a x , q are axes of the outer confocal. 



If now X be the total depth of the layers of wire, since 



a x = a + X, 





-V 



= c (l+^) nearly, 



