OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
401 
If we denote by U the value of the expression on the right hand of the equation 
just found, we have further 
ff {p, a )(th /3)[n a+/3]cZS = 0 
(P> a )\jb PY r ’ a+/3]dS=—U 
&c. 
| |p yJ P 7 P,.P/i?S 
§ 14. We may remark in regard to the case which we have now to examine, that 
since P r P, may by the foregoing results be expanded in a series of the form 
A 0 P jw -|-A 1 P,._* +3 + . . . d-AJE\. — s+2m~\~ • • • “h A S I r+3 
where the value of A m determined by (26) is 
2 r — 2s + 4 to +1 r + ml r + m! 2m! 2r — 2s + 2m! 2s — 2m! 
2r+£m + l 2r + 2m! ml ml r—s + ml r—s + ml s—ml s—vil 
the above integral is reducible to the computation of a series of s terms of the form 
A w ||p y; P !? P,_, + o m dS 
The last integral is to be found by writing in (26) 
2 \= —p-\-q-\-r—s-\- 2m 
2/x=p — ( 2~\~ r ~ s ~\~ 2w 
2 v=p-\-q — r + s—2m 
We are thus led to a complete though practically laborious evaluation of jjP y ,P !? P,P,sdS. 
It will be observed that the result is zero unless y + q+ r+s be an even number. 
With the view of finding out how far the method of this paper is applicable in this 
case, and what difficulties stand in the way of its general application, we will now 
briefly apply it. It will be convenient to expand the operator in a somewhat different 
manner to that pursued in the case of the product of three harmonics. The operator 
was then expanded in a form which would render it useful for application in the case 
of the tesseral and sectorial system. If, however, the poles of the harmonic are all in 
the axis of z, a much simpler mode of expansion may be adopted. 
When the product of four harmonics is under consideration we have to discuss an 
operator of the form 
3 F 
MDCCCLXX1N. 
