1894.] on Ellipsoids and Anchor-Rings. 165 



Thus ^=^ (^1 - A,) J.n&^'e^i' ^^^ m6 



47r cos 



sm 

 - ipJ^&vte^iz me 



A COS 



= A, 



o- 1 - — + 2' ) + ^TripaJJKJ \ —j-^ 



Oi J J a, '^"m 



Again, the whole effect inside the cylinder is given by 



sm 



^ + rL, = {A„ + A,)J,n{qr)e'P'e±'i^ mO 



cos 



A, 



<^{^'^+q')j'm(qr) 



dr 



where tan -^^ = 



sm 

 = ^0 cos xJm (qr) e* '^*~^* e^i^ mO, 



4!7rpaJJKJ — ^^ 





shewing as in the case of the sphere a field diminished in the 

 ratio cospj; : 1 with a lag in phase equal to 27r/% of a complete 

 period. 



For free currents of the above type we may put Ao = in (18), 

 and we then have 



7?^ \ c gLt* 



-^ + ^'^ j + 4^7ripaJ,„'KJ I ^^, , 



. -"l^ + g 



ip = 



4,7raJJKJl fj 



which gives a real value for the modulus of decay. 



(5) Let the current sheet be now an ellipsoid, taking as ortho- 

 gonal co-ordinates of any point the semi-major-axes (p/jiv) of the 

 quadrics confocal with the given ellipsoid which pass through that 

 point, so that 



ds' = Udp" + M'dfi' + N'dv\ 



