84 Mr. Gwyther on a 



These displacements must now be resolved along a set 

 of fixed axes, most conveniently that set of which the axis 

 of X passes through P, thus 



4 = ^iCOsQ + >7isin0 



1^2 = ^isin0cos0 - ^icos0cos(^ + <^isin^ 



<^2 = sisin0sin^ - 7/iCos6sin(/) - <CiCOS0. 



We shall then have to affect each element of these expres- 

 sions by c'^sinddddfj) and integrate over the whole sphere. 

 I have retained p in the expressions up to this point to 

 show that we do not get an infinite expression when p is 

 small, but an expression proceeding in powers of p/X. I 

 shall now, however, put p = 0, and consider the displacement 

 at the origin only. 



Hence, remembering that 2^^ =t, we get 



= - ^sin/^(T + 2 QJ J r,U_isin20^04 

 «r2 / / rj^s'mddddf = ^sin/^(T + 2^) x 



\ I I TiV .iSmdcosdcos(f)ddd(p - I I yU -isinds'mfpdddf) [ 

 c" / / ^^^mddddcp = ^sin^/;(T + 24) x 



] / / r,U_isinOcos0sin(/)d^0^(^ '^11 yU_isinOcos(/)^0^/i/; - 



The values of these integrals depend on the value of ;/• 

 For the case n = 2. 



fi 



r,U_isin-a^/9^(^ = "^"^(Ao + P,o)sinocos/3. 



/ / nU_isin0cos0cos(/)^6^(/> - / / ^U_isin0sin 



iirKpdOdcp 



= + — (A2 + B2)cosacos/3 



/ I rXJ -i^inOcosdsirKpdddcp + I I ^U _ismdcos(l)ded(j) 



