1894] on Ellipsoids and Anchor-Rings. 171 



Let us find an expression which can represent the potential 

 due to the sheet. Assume 



n^ = ta„(ae)P,,Bmnd,) 



n^ = lbAcr0)Q,.smn0] ^^^^' 



where the X denotes summation with respect to n, and a„, b„ 

 are constants ; fl, and fl^ will then satisfy Laplace's equation 

 and since P., and Q are finite and continuous and P does not 

 become infinite at any point outside the surface nor Q^^ within 

 it, Xlj and O^ may represent the potential due to a current sheet 



dCl 



provided a^^ and h^^ are so determined as to make -j- continuous 



dcr 



at the surface. This requires that 



d 



X , [{<^0) (anPn - KQn) sin nO] = 



(1(7 



for all values of when a = a„, 



I.e. 



{^ (ad) ^*»^" - ^«^«^ + ^^^^ ^'^»^"' - ^"^»'^} '^" ''^=^' 



where ;S^ = sinh a and Pn'Qn denote —r^ and -^^ respectively, or 



da da 



S [S {cinPn - hnQn) + 2 (aOy {anPn " b^Qn')] sin nO = 0...(27). 



In place of «,,,&« introduce now two new constants Xn and /*,, 

 defined by 



&n — ^?i^« + f^nPnlS) 



in which /S, P,j, Q,i etc. have their values at the surface, then 



and (27) becomes 



t{fin-'^'Xn{<Tdf]Qva.ne = Q (29), 



or S {i^n — 2X„ ((7 — cos 6)] sin ?i^ = 0, 



i.e. % {/*„ sin ?i^ — 2(7\„ sin tz^ + X„ sin ?i + 1 ^ 



+ ?i,i sin n — 1 6] = 0, 



