OCCURRING- IN SPHERICAL HARMONIC ANALYSIS. 
393 
If now we expand the reciprocals of AP and BP in harmonics we arrive at the 
well-known theorems 
f|P,Q/ZS=0 
I 
4-77-R 3 
p ,Q4S=|^P,(co S «) 
(23) 
The last result, when properly considered, can be made to yield the integral of any 
two harmonics. For it can be easily seen from the foregoing work that if p { , p 2 are 
any two vectors drawn from the origin, then we have this very remarkable formula 
for Q; 
2i! Q,=pf +1 /V +1 2^ 
On account of the operator being an invariant we may suppose the axes to be any 
rectangular axes whatever. 
By what has just been shown we may throw this into the form 
d d d d d d 
d,)\ dxc, ' dy 1 dy 2 ' dz 1 dz„ 
; 1 
P 1 P 2 
where v l stands for 
2 i\ Q t = Pl i+1 P J +1 
1 d\x 1 
v dzj p 1 p 2 
d d 
drj 2 ’ dy j 
The general term of the value of 2 i\ Q, just found is 
i.e., 
or 
P\ +l p-2 +l 
(-1P2 
i ! v l 
—cr 
dr 
-cr d i+,T 1 
i — <x ! 
i + crl 
v i 
+cr 
d h 
dz 2 PiP. 
Pl i+1 p-2 i+1 (~ 
-1R2U 
d° 
dr° 
d}~ a 
d i+<r 1 
i — cr! ■i + cr! 
dvi 
drj 2 
dz. 
dz. 2 p xP , 
Pl i+1 pj +1 2i 
I (Jv 
* o2(r 1 
d° 
d l 
~ a d‘ 
i-a- l 
i — <r! i-\-o\ dr] 1 dz 1 dz 3 p x p 2 
It is obvious a term similar to this can be obtained with only ^ put for and for 
by merely changing the sign of cr in the first of the last three expressions. Adding 
the two terms and recollecting the definitions of the harmonics we may write the result 
2 il il il 
i — crl i + c! 
2 3ff 1 { (i, cr) (i, cr)'+p, <r]p\ crj} 
(24) 
Hence Laplace’s Coefficient as given in § 1. 
Reverting now to equation (23), let us take two fixed poles, and let Q„ Q/ be 
expanded according to Laplace’s Formula; we shall then be at once led to the well- 
known surface integrals of two tesseral harmonics. 
MDCCCLXXIX. 3 E 
