OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
395 
^2(l+m+n) — i. 
2\! 2 fi\ 2v' 
\ + l\ X—ll /x + ml jjb—ni\ v + n\ v—nl 
' d l d l d l d l \(d m d m d m d m \f d “ d u d n d n \ 
d&dVi d^d V J\d^d Vl d^ drj.Jxd^ d v , dl? 2 d.rjJ 
X 
^fx+v—m—71 (Jy+k—n—l ^A+/x— l— m 
dz x dz 2 dz s 
where the values of l, m, n range between 0 and X, p, v. 
The general operator just found we shall denote by the symbol tt (l, m, n). 
§ 10. If we expand each of the reciprocals of AP, BP, CP in harmonics we shall 
arrive at the general term of 2t+2 dimensions in It in a different form, viz. : it will be 
In order therefore that the general term as found by the work of the last article 
may correspond with this we must have 
l±-\-v~p 
v+X=q 
X -b /r == v 
The quantities X, p, v, as depending upon p, q, r, are therefore perfectly determinate, 
and the equation expressing the identity of the general terms may by Theorem i. be 
written 
v. 
47rlt 3 
2i + I! A! ji\ v\ 
ZttTr(l, m, n){Q p rfQ q r^Q r r{) 
(25) 
On examining the equations determining X, p, v, we see that in order to their being 
positive, p-\-q-rr must be an even number, and no one of the three quantities p, q, r, 
must be greater than the sum of the other two. 
If these conditions are not satisfied, then JJQ jO Q !? Q^S= 0 for integral values of p, q } r, 
wherever the poles of the harmonics may be. 
!|V !' !■■/< 
§ 11. Let us now suppose that the three points A, B, C are in the axis of z. Then 
the harmonic becomes P^, which we will suppose expressed by the series D. In like 
manner V q and P, may be similarly expressed. It follows that, in selecting from the 
3 E 2 
