166 

 where 



Mr Mayall, On Garrent Sheets, especially [Feb. 26, 





{p^-h^){p^-M 



M' = \ 



(^-jj^f^^-v^ 



F' 







o\) 



a, b, c being the semi-axes of the ellipsoidal sheet. 

 Equation (3) is in this case 



dfM M dfju dv N dv 



MN d d ,^ „ , 



.(19). 



This equation may be easily solved if o- = kL where k is con- 

 stant. This is equivalent to supposing the current sheet to be a 

 homoeoidal shell, for the thickness of such a shell at any point is 

 proportional to the perpendicular from the centre on the tangent 

 plane at that point, i.e. inversely proportional to L, so that if the 

 sheet be made of a homogeneous conducting substance its specific 

 resistance at any point will vary as L. 



Let ^, = A,EME^XH)K{y)^'\ 



and assume O, = A^F^ (p) E^ (fi) E^ (v) e'P\ 



n,=A,EMEn(H')K{^)^^\ 



where E^ and F^ denote Lame's functions of the first and second 

 kind respectively, and 



dp 



^„(,)=(2. + l)^„(,)/^^^ 



.(20). 



Then 47r^ = (4,F - A,EJ ^„ (/.) ^„ (v) e^^' 



\E^ and F^ are here written as abbreviations for E^ (a), F^ (a)]. 

 Substitute this value of ^ in (19) and there results 

 /C..TT , ^Ad NL d d LM d\ „ , .„ , ^ 



MN 



= - i^' ip (A, + A,) e:E^ (/.) ^„ (.), 



which if we introduce two quantities 77, ^ defined by 

 diJb .. f dv 



H 



bJ{fM'-}f){k'-/M') 



— a^) Jo 



J(h' - v') (F - v') 



