1894.] on Ellipsoids and Anchor-Rings. 161 



where flj is the potential due to the sheet on its positive side and 

 Hg that on its negative side, J,^^ denoting a Bessel's function of 

 order m. The co-ordinates used here are cylindrical and 



ds' = dr^^r''d<^'' + dz\ 



Now 47r^ = 0,-0,(^ = 0) 



^{A^-A^ e'^*/^ {qr) cos m(^, 



and (3) becomes 



(7 



47r 



= ipqr {A^ + ^ i) /,„ cos mj>, 



(A - A) j|: ('• |:) + ^ J|.} ^.. eos '»■#' 



= ipqr {A^ - 



or 



d'J 1 dJ ( 2 Wl^\ r /^ 



^•e.smce _+-._ + (^^ - -j J = 



Again, because ^— is continuous at the sheet 



whence substituting in (6) we find 



27^^•p 



^^-~'^^-~fyo-+27r*>-^« ^^^' 



and rij fig and thence <J> are completely determined. 



If -(4o = there is no external magnetic disturbance, and we 

 must then have by (7) 



q<T + '^irip = 0, 



qa 

 or *^=-2^' 



therefore 0^= A^J^^{qr)e '^'^ ^ cosm^, 



+ g(z--t) 



Og = — A^J^^^(qr) e ^ stt / ^^g ^-^^ 



Since any function of r, 6, ^ can be expanded in a series of 

 terms of this type, we see that if at the time ^ = 0, Oj = F(r, 6, z), 

 then at the time t 



fi. = ^jr,«.(. + |^*)}, 



VOL. VIII. PT. III. 12 



