OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
387 
The value of the integral under discussion is accordingly 
k(k + 2) ,..(& + 2n —2) 
’ (:L-*)(3-») . . . (2W+1-A) 
and the above reasoning shows that must be less than 1 : it may, however, have any 
negative value. 
§ 4. To evaluate the integral now indicated, m and n being integers, we must 
modify our previous solution, for otherwise we should be led to infinite values of the 
differential coefficients at the origin. 
Let us consider the integral 
If V is a function which is symmetrical round the axis of z we may write this 
2-7T 
p 2m Yds 
where els is an element of arc of any section of the sphere of reference containing the 
axis of z, and p 3 -j-z 3 =BA 
Now the expression (5) gives the form of solution in this case. If we put 
Y=-j (z+jp cos i/r) 8 *# 
we have, in fact, 
{[(i— 4 fV^S=- 
4ttR 2 
2 2m+Zn m + n ! m + n ! 
X 
rP rp \ m+n 
o \dp + dz) C0S V') 2 *# 
Expanding the quantities under the integral sign, and performing the differentia¬ 
tions and integrations, we obtain 
_2t 
22»»+2» m _| -n\m\n \* 
where 
fc = 1 --12-12- &C ' 
— \ — 1) . . . ( — m — i + 1) 
1.2 . . . n 
3 D 2 
