PKOFESSOi; (4. IT. DAPWTN OX ELTJPROTDAL HAPMOXTG AXALY.^IS. 
.llC) 
Let us consider the case of = 2. 
Then 
IT, + /3n, + /3'n., = [P-^ - %,P - ^r^.P' + 
= (F)= - 2/3(5„PF + 
+ /3- [27„P5P« + (%P)^’+ (f/*P')= + 29 „ 7 .,PP'] 
(or tlie same witli accented q'n for the other case). 
From this it is clear tlurt IIq is divisible Im cos’*' 6 and tp by cos- 6, and the method 
of development is legitimate AA’hen .s = 2, but it is not so when .^ = 0 and .§ = 1. 
'fhe inAmstigation then se])arates into the general case, and the cases 5 = 0. 5 = 1. 
§ 14. Integration in the general ca.<e. 
We haA^e 
^ p. _ I3q,^_,p -^ - I3q.,^,-p -'- + /3'^7.,_,P'-^ + 
and 
= (P^Y - 2^(7 ,_,PF-^ + + 
Also 
PA'tv 
cos~0 
u I ' 
has the same form AA’ith accented ^’s. so that it Avill he 
\1 _ >-'/ J 
merely-necessary to accent the 7 ’s to obtain the second case. 
We haA^e then 
II, = {P)^ II, = _ 2(^,_,PP-^ + 7 .,,PP-^), 
II. = 2(7,_,P'P-^ + <i.,,,PP"^) + (v.-.P-^)^ + (7..2P'"7-’ 
+ 27,_,7„.,.,p--’P"m 
'I'hen since cos 010 = dp,. 
I\I0 = ^ 
• -1 
'+• (7 - i)(PA- 
-1 
1 -p- 
ijj. - 2 $ f Vy,_, P-P'-= + .,,.,,FP -*)iP 
+ )_, mwpr -t- /3' TTVTjT 'V 
/3. r M 7 - 1) - 2 PL^' - - + 'P + 2 3 ’P'- ■" •) J 
-2^1 - T—^- 
■. -1 
• + 1 
+ | 8 -f [ 27 .- . P P' 
.-1 
- 4 
+ '2q ,.. 4 P^P' ^ -f {q, _ 0 P" “)■ + {</., + 2 P' ~ "V + 2q_, _ -a],, + o P' - P' -] 
.... (60). 
And ^F:,<10 has the same form, hut Avith accented (j^., and Avith y 4- 1 replacing 
y — 1, and y — \ replacing f — y. 
