• OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
399 
that if we suppose the differentiations performed by the whole of the operator, and if 
one of the resulting terms is 
*)a(q, P)b{r, y)c 
where the suffixes indicate that the angular coordinates of A, B, C are substituted 
in the harmonics to which they are respectively attached, then we have in all cases one 
of the quantities cl, /3, y equal to the sum of the other two. This we see from the 
expanded form of the operator just given, and it is a result which was to be expected, 
for supposing Q^, Q ? , Q,. severally expanded according to Laplace’s Formula we should 
have in the surface integral a series of terms of which the type involves the integral 
cos a (f) cos /3(f) cos y<f)d(f > 
and this integral vanishes unless the above conditions be complied with, the case where 
a-|-/3-f-y=0 being of course out of the reckoning. 
In accordance with the explanation just given we may now put 
(27) 
If then we substitute in tt (l, m, n) for l and on the values (3 ff - n and a — n, we see that 
n may range in numerical value between 0 and the least of the integers A.— (3, y — cl, v, 
and may be positive or negative. The operator tt, in fact, becomes 
on — M = a m + 72 = a j 
l-\-n=/3 ° l—n — (3 j- 
and /. /-fm=a + (3 J 
/ _ [ p2~ (a+ ^—' 
2A! 2 [jl'- 2v\ 
A + /3 + ?i! A— /3 — nl /x + cl — n\ y —« + a! v + n\ v — 
01: 
/ d* d?_ dy d?_ d a+ P \ dr+ v 
Wi W; dvi <A 2 d.%,j ^ dz y 
a rfv+K—fi y 
dz 2 d~ A 
We have therefore now to consider a series in which any term is the product of (1) 
an invariable power of 2, (2) an invariable operator, (3) a variable coefficient whose 
parameter is n. If we denote the first two of these factors by the symbol sr, then the 
sum of all such terms is ra-K, where 
K = 
2A! 2 yl 2v\ 
W 
fX V - cl'. ZZ + A - (3'. A + yU +«+/ 3 ; 
and W is the coefficient of x~ K y~ ,l z 2v in 
(—1 ) i {y—zY +v - a {z~-x) v+K -\x—yY +,x ' ¥lA ' & 
(--1)' (y—zy~ a (z—xy~ p (x—y) r+,t+l3 
that is, in 
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