I’ROFESSOR (4. H. DARWTN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
515 
It will be legitimate to develop pda in powers of sec'0 up to any given power, 
provided (V/)- involves as a factor such a power of cos~^ that the whole function to 
he integrated does not become infinite at the poles where ^ 
I shall at present limit the developments to the square of yd. 
We know that P'^ is of the same form as pq hut with the additional factor 
- sin- eY 
cos- e 
Sujipose then that 
Ilo + ySip 
+ ^'11.1 
and let 
7 
'fhen w'e put 
F, 
and 
F. 
eo.s- 6 
y = 1 — cos 2(f). 
P' — 1 -A si^^~ ^ j/ 
cos- 6 ^ ’ 
Now’ suppose that a function independent of 6, denotes one of the four 
or cr (Cq- or (Sq~ 
(^ — /3 cos 2^)' 
dlien in the cases involving ^J-functions and P-functions res])ectively, we have in 
alternative f()rm— 
*■' "• {% ’ f p (r,')-<o = Jl (I - /3)! j I A- -1^,’ Mhl,. 
If it he supposed that the development in pow’ers of sec- B is justifialde 
= cos 6 
1 + 
7 (/3 + /3-)' 
:— I Iq cos B -|— ^ 
COS” 6 
n„(7- 1) 
cos 6 
■ 3 
cos'- 6 - cos‘ 9 
[Ho + , 
+ cos B 
(■' IS 6 COS'" 9 
1^1(7 - 1 ) 
+ 
COS 9 
+ II.1 cos B 
And F.^ has a similar form, save that -y + I re])laces y — I, and y — 1, re})laces 
It is clear tliat unless IIq is divisible by cos'" B and by cos B, \F^dB and ^F/IB ^s■lll 
have infinite elements at the poles, and the development is not legitimate. 
Since P" = ~ (p~ — 1)', it follow’S that the power of cos B hy wdiich P^' is 
divisible increases as .s- increases. 
2 IT 2 
