﻿560 Dr. S. R. Milner on the 



fraction of the whole, and therefore the error which \vill be 

 produced in the whole by treating these cases as though 

 their exponents E p — E P are small will be a second order 

 quantity which may be neglected. Secondly, E p — E p may 

 be large for certain positions of A p as the result of one or 

 more ions inside the sphere lying very close to the surface, 

 and thus over a certain region of the integration of (17) 

 being very close to A P . But in accordance with our initial 

 assumption (p. 557) a similar argument to that used above 

 applies here also. 



Treating E p — E p as small, we can write (17) as 



* Jo 



and in expanding the second exponential omit second order 

 terms. This gives 



P 1 (E)=G \ * f~^~Ml-E p +E p >ty p sin P v cW P 



«- o Jo 



= Ce~ Ep xItt, 

 since the integral of E p dcj) p sin 6 P d0 v over the surface of the 

 sphere is by definition equal to 4-7rE p . As the result of a 

 well-known electrical theorem, E p is equal to q v \r v multiplied 

 by the total charge inside A P 's sphere, i. <?., 



%= ^(qo + q, + fr-i) = ^ "i q,- ■ (18) 



'> P * p i—0 



We see that all the terms of (16) which before the inte- 

 gration involved the. distances r ip become changed into cor- 

 responding terms involving only the central distance r p of 

 the ion A p from A . The same argument now applies to the 

 integrations for A p _i, A p _ 2 , &c, down to A 1; so that when 

 they are carried out (16) takes the form 



P 2 (E) =k m e W r P *=° ° lirr^dr, 4mr„?dr m . 



For shortness, write 



a,=q/£qu h = K * (^)", ■ • • (P) 



so that 



m ap 4_ 3 



• P 2 (E) = k x e ~?i rv ~ 3 " 7 '" rfdri rjdr m . 



