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XI. On certain Definite Integrals occurring in Spherical Harmonic Analysis and on 
the Expansion, in Series, of the Potentials of the Ellipsoid and the Ellipse. 
By W. D. Niven, M.A., Fellow of Trinity College, Cambridge. 
Communicated by J. W. L. Glaisher, M.A., E.P.S. 
Received April 3,—Read May 1, 1879. 
§ 1. The object of this paper is to explain a method of dealing with a class of results, 
some of which are of frequent occurrence and some of considerable importance, in the 
solution of Laplace’s equation in series. 
The fundamental theorem on which the method depends is expressed by the inte¬ 
gration of 
jje“ + ^ + n7S 
over a sphere whose centre is the origin of coordinates and whose radius is It. This 
sphere we shall call the sphere of reference. 
By change of axes the above integral takes the form 
SttRP e zV ( a’+/3’+ v ' ] dx 
J — E 
and its value is 
27rR 
gTt \7(a 2 +/3 2 +y~) ^-EV(a 2 + |3 2 -f-y 2 ) 
V / (a 2 + /3 2 + 7 3 ) 
or, m series, 
4ur{i+f( a =+/3"-+ y =)+ ... .. 
•} 
(1) 
( 2 ) 
It follows that if Y be any function whose value may be expressed for all points 
within the sphere by a convergent series, according to Taylor’s theorem, or by the 
symbolical form 
d , d , c? TT . 
0 x ~ +y— +z—V 
c dx dy dz y 0 
then the integral jfVdS taken over the surface of the sphere is 
» R 2 ' [cV , <P , d?\\ T 
ilrn p2rn{dx + T v + te) v » 
3 c 2 
( 3 ) 
