51H PROFPISSOTI G. H. DARWIN ON ELlilPSOlDAT. HARMONIC ANALYsrs. 
'rherefore* * 
1 Pip.'+ 2A- r /;]>i ^;pi + 2t'' 
, between limits i 1, 
r-i- 1 Pi]*.' + 2A- 
4Z:(.s “b k) j ^ ^ dfi = (1 — /x"') 
_ 1 1 — 
Again since by (11) 
P' + 2/. _ _ Oj 
r? 
P- 
<h 
= 0 . 
is 'Ik 
it follows tliat 
1 - p?- 
• + 1 pi pi + 2/. 
= AP'+ - + BP = -'■ -f CT' 
R Vi ■ 
J-i (1 - yxb- 
djx = 0, nnless k — 0 oi' 1. 
(l’0~ , pip.' + 2 
It remains to tind the integrals of . 
^ 1 — \r (1 — /r-)- (1 — /iO- 
\¥e liave seen in (II) (transformed to accord with onr present definition of PO 
that 
1 
— yU- HS (S + 1 ) 
xf - r 
+1. 
pi 1 {f '^} {b ^ ^} p, - 
4.9 ( .S — 1) 
Hence 
fr“A‘'^=17T7TT)A^“''^+i 
(DO- 
(•^ + 1).' 
'i(i + 1) 
.9 - - 1 
+ 1 
f(P)^r7p 
+ 
i, .9} \ i, s - Iffp, 
iPT*--dp. 
J (1 — /ji 
1 r RH 
>.< + 2 
4s (.s + 1) j 1 — /d 
f (' + 1) 
s3-l 
+ 1 
+ 
46-(.9 - 1) 
JPO- 
1 - /P 
{i, H {b* - 1) 1 RH—- 
4s (s - 11 J 1 - ^ 
a 
r p,p. 1-2 I 
lrr^'''' = 
4s (.9 + 1 ) J 1 — IJL 
/-p' + 2p 
4 + i 
i {i + 1) 
s'^ -T^ 
+ 1 
- 1 - 
rpipi + 2 
i, s} (b s - 1 I- fl'-’ - 4R-’ + - 
4s (s — 1) 
rP-’-4P-’ + - , 
1- ~d^. 
J 1-/2- 
The first of tliese involves integrals already determined ; on introdncing them on 
1 ?* “(“ -9 ; 
the rigid, and redncing we find the result to be — , -- . 
Tlie first and last terms of the second iidegral vanish, and the integral is clearlv 
i (f -f 1) 
s- - 1 
+ 1 
1 / -1- s; 
s ’ — s ! ' 
'I'he second and third terms of the tliird iidegral vanish, and the whole is clearlv 
1 2 + .. + 2: 
4.s-(s + l)(s + 2) i-s-2l 
* I owe thi.s method of liiiding the.se last two integrals to Mr, llORsox, 
