OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
389 
Theorem ii. relates to the differentiation of the reciprocal of r 2<r+1 and may be 
stated thus : 
(- 1 )-’:, 
cX'-* 
1 2^cr! 
cl‘~ a ’ 
d\d\ . . . dhi_ a 7 
>2cr+l 
■ d? Q, 
r 7,—(T Cl 
2 o\ dJiylli^ . . . dhi_ a \ dp? 
( 21 ) 
Theorem iii. If f 
Id d d 
\dx dy dz 
be any homogeneous operator of the i th degree 
operating on a homogeneous expression <-[> (x, y, z) of the i th degree, then 
. (22) 
This theorem is obviously true, and though stated for only three quantities, x, y, z, 
is true for any number. 
Theorems i. and ii. are proved as follows: Let OP = ?% QP=p and QjPQ = 7r — 9, 
where Q is any point near to P. Then 
_1_1 
OQ r 
fAi+ 
Keeping r fixed let us perform the operation (7) on the two sides of this equation 
and then put p= 0, in which case OQ becomes r. It is to be observed that with a 
homogeneous operator like (7), it is immaterial what point is taken for origin of 
coordinates. If then we take P for origin when we are dealing with the right hand 
side of the above expansion, we see that the first i terms will disappear by repeated 
differentiations, and the terms beyond the i-\-l th disappear in consequence of the 
zero value of p. The theorem stated therefore follows, the substitution of r for p 
making no difference in a result from which x, y, z are finally made to disappear. 
Theorem ii. may be proved in the same way, if instead of the expansion for the 
reciprocal of OQ, as above, we take the expansion for the reciprocal OQ 2 ' r+1 , found by 
differentiating the above series cr times with regard to /x. 
DIFFERENT FORMS OF TESSERAL AND SECTORIAL HARMONICS. 
§ 7. In several of the following investigations frequent use will be made of different 
forms of the zonal and tesseral harmonics. With the view of classifying these various 
forms, and of bringing the various expressions for the tesseral and sectorial system 
into harmony with the corresponding expressions for the zonal, a proof is here given 
which will be a direct illustration of the foregoing theorems. 
The most important forms of the zonal harmonic P, are 
1 d l 
2V'W s - 1)i 
(A) 
