1879.] Mr. W. D. Niven. On certain Definite Integrals. 3 



The basis of the method is expressed by the evaluation of the 

 integral 



taken over the surface of a sphere of radius R, whose centre is the 

 origin of co-ordinates. This is easily shown to be 



l+5J(^+/3'+f)+ . . + f A 2 n (y + /3*+f) i + .. 



Hence it follows that, if V be any function which can be expressed 

 symbolically in the form 



x d_, d_ d 



e dx^dy dTV , 



where the differential coefficients of V belong to the origin, and 

 (x, y, z) is any point within the sphere, then 



rr y ,s=4^i^(i! + i!+i!)Vo a). 



jj o2i + l\dx chj ch) K J 



If we next consider a circle in the plane of x, y, whose centre is the 

 origin, and whose radius is R, we find, in like manner, 



taken round the perimeter, equal to 



The theorems (1) and (2) are applied in a variety of cases, of which 

 we will quote two results : — 



Let V n (ytt) be the zonal harmonic of the 7&th degree, having its pole 

 in the axis of z, and let m and n be integers, then 



(!) (l-^ 2 ) /rt P 2 »g?S = (-l)« , . , 4ttR- 



2m + 2n + ll m—n\n\n\ 



provided m be not less than n. In all other cases the integral vanishes, 

 the cases where the harmonic is of odd degree included. 



(2) j"((l-^)«-*P i „rZS = 



(a) »>«, (-1)- , 2 ro !2»!2»!2»-2>»! 2 ^ 



2~" l+ ,l n-\-m \ n — ml ml ml nl nl 



/ 1N . 2ml 2ml 2nl m—nl 

 (p) n<m, ( — l)' 1 2~~R 2 . 



