xvi PREFACE TO PART II. 



We have seen that, on a sphere of radius unity, 



also Y = m ( I p:) ^ H and Z = (n+l) H'". 



Hence p. Y: - X? = (w + in) H~ ~ l , 



i / 1 *\i T r m TT' 



also (1 -ft 8 )" i =mH n . 



From these formulae we find 



_i 

 and also 



p ,-,_,,, ; i / Y- t 1 (77"' +> Y " P 



J-i J -i J -i 



These definite integrals reduce to 



ri 



n(n+l) 



J-i 



Hence since Z"', =(ti+l) II'" , we have 



i ri 



JX:Y d , = f _ ( i - 



-i -i -i 



Putting n, for TO in the above equations we get 



and 

 Hence 



0* 17 - *:) (/* n- - ^;;) + (/* r; + jr : ^ y- +x + i- Y Y 



-(n + m) (n, + m) H'^H^ + 1 ( n - m) (n, - m) 





