PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
533 
Hence we have* 
2r ! 
i + ?■ ' 
(;’ !)- r + s ! i — r\ 
cos''' 0 
Now suppose 
Then clearly 
Therefore 
= V y,,—2 COS''' 6 , 
(P,)'- = S ao^ cos'-'' 6. 
v, '—• 
' o-*" r — 1 ! (?• !P r + 1 ! * i — r !" t — 1 ! 
^^■ir = {-y : 
2?'! i + r! 
93r Ip i—r\ 
u cos-''+- 6 + y8/cos-'' 0 
L '■=' Ci- c 
y=- 7 ,,_o - 
r r=l j -Jtt 
= - y2r-2 [ (cos'^’'^+/ 3 ^ COS“''“' 6 ) Add, 
r=l J — jTT 
M ’•=' 
G 
;y r=i cos-''+2^+/3AcO3'''0 
.=ia2rl -::- du 
G ,= 
0 J —^TT 
(82). 
(82). 
(83). 
The evaluation of these integrals depends on two integrals only, namely, j 
and jcos'"^. Add, and these will be considered in the next section. 
cos^" 0 
dd 
f cos^'‘0 f 
- ^ dO and 1 cos"" 6Add. 
I will denote these integrals D and E respectively, and I propose to find their 
values in series proceeding by powers of /c". 
The usual notation is adopted where n(a’) is such a function that it is ecpial to 
.Tn(a; — 1); accordingly when a: is a positive integer n(a-) — a:!. 
Since /c'" is less than unitv 
00 1 3. 2'—1 
■2 • 2 
r ! 
and since 
[ COS'” d sin-'' ddd = it - 
J -iir 
1 ^ 2r-l X A 2'(-l 
2 . .. 2 . 2 • 2 • • • 
?i-hr I 
rrn<i^'^0 13 2n-i or, /A 3 2;—ly: 
D = J dd = 7r 2 AA-: - A ^ (2 • -2 • • • —) K 
0 
(>i-h l)(n 4-2)... (?i-^-r)r I 
^ Mr. Hobson kindly gave me this proof when I had shown him the series which I believed to 
hold true. 
