.(2). 



1894.] on Ellipsoids and Anchor-Rings. 159 



Hence l^^HO) - ^iOB) ■. 

 and similarly —{FA)-^(Ha) = -'^. '^ , V (1). 



^((?B)-|(i-^) = 



Suppose now that one of the orthogonal surfaces, e.g. a = a^, 

 is formed of a conducting substance whose specific superficial 

 resistance at any point is cr. 



Let ^ be the current function at that point, F, G, H the 

 components of the vector potential due to the current sheet, 

 FoGqHo those due to the external disturbing system, 11 and flo 

 the corresponding scalar potentials, and -v/r the function known 

 as the potential of free electricity when the motion is steady. 

 Then the equations of electromotive force are 



d^ d ,^ ^ . d-yjr I 



""Gdc^-dt^^-^^'^-Bdbl 

 d^ d ,rj. jy. dyjr 



-''Bdb=~dt^^^^'^-Gdc^ 



Again, equation (1) gives 



hence on eliminating G + Gq and H + Ho by means of (2), we 

 have 



db\~B~db)'^Tc\G Tc)'"!: dtda^^^^^^'^'-^'^^' 



in which a must be put equal to a^ after the differentiation with 

 respect to a has been performed. 



Since 12 is known in terms of ^ this equation is sufficient 

 to determine either of them, but it is more convenient in 

 applying the equation to particular cases to use the conditions 



that n is discontinuous at the surface a = a,, while -7— is con- 



da 



tinuous ; thus 



47r^ = Hi - nA 



and ^1^^, K«=ao), 



da da ) 



where fij is the potential due to the sheet on its positive side, 

 and Oa that on its negative side. 



