CURRENTS IN CONDUCTING SHKLLS OP SMALL THICKNESS. 317 



(ityjdz being always of the same sign) at any point in the surface the resultant of F 

 and G coincides in direction with that of U and V, and, therefore, E/U = G/V. Also 

 IF + mG + "H = at every point, and, therefore, i/> 0. 



35. Again, if <f> be a function of z only, and if x, derived from it by the methods 

 above explained, do not contain z, the equations of condition reduce to two, namely : 



and these must necessarily be satisfied at every point, because the resultant of U 

 and V is the intersection of the tangent plane with a plane parallel to that of xy. 

 And this line is also the resultant of F dyjdx and G d^dy. We can then 

 determine <r/h in terms of K. As an example, let us take for our sheet the ellipsoid 



and make 



< = Az 



(see Professor LAMB'S paper, ' Phil. Trans.,' A., 1887, above referred to). 

 We have then 



V-A=, 



w = o, 



where ra is the perpendicular from the centre on the tangent plane. Also, at any 

 internal point, 



yv 

 f~1 ^_ A "_ 



and at any external point, 



dq dq_ 



y dP _ A*^" 



1 = A, ^ 3 , G = Aa -; -7- 



cr OQ a" aoQ 



<&* da* 



where 



dX 

 ? = 



and 



r d\ 



q ~ Jo x/{( s + X) (6 s + X) (c + 



