1894.] on Ellipsoids and Anchors-Rings. 



may be written 



167 



47r 



{a^f^-a,e:) 



(p^-^)l.+(p-/^r,$ 



Jip' - K') ip' - F) {pi' - ¥) (F - p:') {h' - v') {F - v') ^» ^^^ ^» ^''^ 



^ ~J{p? - If) (F - p:') {h? - v') (F - v"") 



{A^ + A,)E:E^{t.)ESv\ 

 or since p = a at the sheet 



drf 



4^(A^»-A^Jj(/>'^--^) 



+ {p'-p^')~}^E^{f.)EM\ ^^^^• 



= - iplfc^ {p? - v^) {A^ + A^) E:E^ {p) E,^ (v) J 

 Now (see Heine, Kugelfmictionen) 



^(i^) ^ _ {,, (,, + 1) ^^ - g (J^ + F)} ^„ (;.), 



and ^^^^= [n{n + l)v'-q(h' + F)]E^(v), 



where g is a constant which is known when the form of E^^ is 

 known. Hence 



= - (p:' - v') [n (n + l)p'-q (h' + ^0]• 

 Substituting this result in (21) we have 



£-(A^F„-A^EJ{n{n + l)a^-q{If + F)}] ^22). 



= ipbV(fi^-v^)(A.^ + A„)E:\ 



Since -y- is continuous in crossing the surface of the ellipsoid 

 dp 



therefore 



a,f:=a,e:- 



A,_ A^^ A^F„-A,E,, 



e:~f: e:f-f:e^ 



.(23), 



