Q6 Prof. A. Gray on the Calculation 



and we are able to calculate the integrals of even order 

 required for (2) by successive differentiation of A ( = l — xjr). 

 To find the integrals of odd order let us assume that 



A f %#) rf ,= |%,. 



(9) 



where A is a constant. Differentiating, we obtain from this 

 equation and (8) the relation 



and therefore also 



A{(l-^ZV. (*) -(K+8)^,h. 1 (« } 



= (2* + l)!a 2 Z' 2i . +2 (<£), (10) 



where //,= cos 0. 



The assumption made in (9) will be justified if the relation 

 expressed in (^10) holds for a constant value of A. Now if 

 Z; denote a zonal harmonic of any order i i we have by the 

 fundamental relations of zonal harmonics 



(aZi-z,., =-J(i-v)z/, 



Z 1 .- /t Z i _,= -i(l-^)ZV 1 . 



From these equations we find by elimination of Z*_i ? 



z i= 



and by elimination of Z*, 



Z i= J (/.ZZ-Z',..,) ;....'. (11) 



z^lczy-^zv.) (no 



Differentiating (11) and (11') with respect to fi and elimi- 

 nating Z/', we get the relation 



(l»/* 2 )ZVi-^(^ + l)ZU=-(t-l)Z/j 

 which, with 22^2 written for i, agrees with (10) if we put 



This verifies the assumption. 



