14 Proceedings of the Royal Irish Academy. 



rise to a force tc + Vn], where '■ is the surface-density. By the aid 

 of the boundary conditions one form of tliis is 



■ c'-(l -c--(F<i'v)=) + c-V 

 i - {bavy c 



where v {Tv = 1) is the normal to the boundary; and as we are dealing 

 with surface-distributions, we must take one-half of this in computing the 

 total force. It may be noticed that tliis force is entirely normal, so that 

 for a sphere such forces have no tendency to cause rotation. The current 

 < = (q -^ fu can be obtained from the boundary-conditions. However, 

 if harmonicoids of the first type only are present, (q is " irrotational," and 

 can be calculated from c by the diflerential equation.^ 



V{p-' VpV,,) = 0. 

 For example : if t = f„(p), a solid harmonic of degree n, then 



v=/;.(p) = 



may be written 



71 (n + 1)] 



~7 



so that the current is 



-pVpV d^ 

 n (n + 1) dt 



Consider the case of the isolated sphere. We have 

 c = e„(l + 2c-'Spa - 2ac^SpS . . .) 

 where £ = 47ra'e„ and the cuiTent 



'o = c^ic-'VpVpa . . .). 

 Hence 



e'(l - c-H Vapf) + c-V 



= c„'[(l + 2c^Spi - 2a(r>SpSy' + c^{<f + 2c-'<T,S>rT + c-'VpYpSy] 

 [1 + (SivYc-^ + (SavYcr* + . . .] 



= .„= [1 +' 2c-'(25/)«T + ff' -1- {S„vy) - iac-'SpS + ...]. 

 It is clear that only terms of odd degree will contribute to the final result, 

 so that on multiplying by 2n-i', and integiating, we find 

 21^ 



\p-^Vpvy.'l^^\f„(p)-o, 

 Mp). 



( - of + «C"''(T + ...!• 



It will Ije noticed that this is the same a.s if tlie charge on the sphere wa.s 

 uniform and fixed. The energy wasted and conserved is the same (to this 



