430 Mr. J. H. Jeans on the 



If we write 7 for the angle between the radii r and r', 

 so that 



cos 7 = cos cos 6 f + sin 6 sin f cos (0 — <//) , 



we may expand %i (R) in the form 



Xl (R) = lr„P„(cos 7 ), 



where r n is a function of r, r' and of the coefficients which 

 occur in %i(R). When R is so great that %i(R) becomes the 

 same as R"" 1 , we have 



r n =r n lr ,n+1 or r ' n /r n+1 , according as r< or >/. . (2) 



We can further expand the biaxal harmonic P w (cos 7) in 

 the usual form, and so obtain 



Jl = 731=0 



where 0™=2|n — m/Jrc + m or I, according as m is different 

 from or equal to zero. 



If we substitute this value for %i(R) in E 1? it appears that 

 every term will vanish upon integration, except products o£ 

 pairs of terms which are such that n and m are numerically 

 the same for each. Performing the remaining integrations, 

 we obtain 



E^SSKTPro*)^, (3) 



where 



F - - - 2S1K h ^'' v " p ' ) + r "p'^" : + »>'-)} t ^'- w 



The integral occurring in E 2 is that part of the potential 

 of the whole atom upon a unit charge at the point r, 6. <£, 

 which arises from the term Xi( r ) ^ n ^ ne potential-function. 

 This is obviously a function of r only; so that if we denote it 

 by Vj, we have 



2 ^r 



The remaining part of the potential-function may be treated 

 in a similar way. If we expand % 2 (R) * n the f° rm 



00 

 X 2 (R)=2M > „(cos7), 

 



we find for the potential of the whole displacement upon a 

 unit mass initially at r, 0, (j> the value E 3 + E 4 , where 



E 3 =22G,7P^y wiCP , 



