53G 
PROFESSOR C4. II. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
It remains to consider these two F series. 
^ _ (1 — ?! — e) (2 — H — e) ... (?• ——e) . 7 ! 
[1.3...(2r-l)P/c2r 
n-l 
^ (-r.w 
(?! — 1 + e) (?i-2 + e) ... (?! —?■+ e) /• ; 
_Ip .••! ...(2/'-l)P K-^ 
2-'' (// — 1 + e) ... (1 + e) e (1 —e)—e) 
When ?’<!? 
_T_ ^ 1 A _ 
(?i. —1 + e) (?i —2 + e) ...(?! —?’ +e) ('/! —1)(?? —2) ...(?! — ?’) \ ^ t 
When ?’>n, put r = n4-'''’, arid 
1 
1 
LI 
(?i— 1 + e) (ft — 2 + e) ... (I + e) e (1 — e) ... (s — e) ?! — 1 ! .s ! e 
Also when ?’=r!-|-.s' 
'-W + '^r+,7 
2-'-. r \ 
Thus 
,^[1.3 ...(2?-DP (2ft !p r(^v^ + l)(^>^ + S)...(2ft + 2^-l)P 2, 
■ 2^"(ft!p'^ 22^??.+ ])(/? +2)... («+.<;) ’ 
s = 0 
22'(?! + 1) . . . ( ft -l-.s).S' ! £ 
It follows that we mav write the first term of D as follows 
97) ! Jin 
(“)"23«(„ !p p + w+1, K') 4- 
2 ft ! 
22«ft; ?! -1 ! (- y [1.3 .. ■ (2r- D]2 , 
2ft ! 
, V '- 
^ 93//7),_1 
(?!.— 1 ) ...(?! —?’).?■ ! 
K' 
r/■ _)» "" ^ in y [( 3ft+ 1) (2ft + 2r-l)P 
^ ( ■’ 22" (ft :)2 ^ 2-^{n + D ... (?! + -s) . .s ! j_^ t 
" 1*1 '"1 
X7 + ^7-^X^_3+ilog 4 
1 1 
The first of these terms becomes infinite when e = (). 
Turnino- to the second F ^xe have 
... * [2/! +1 + 2e) (2ft + 3 + 2D ■.. (2» + 2g-1 + 26)]2 „ 
22'*(ft +1 + D(ft +2 + e) ... (?i+ s + e)..«! ^ 
_ ^[(2ft+1) (2ft+ 3) ... (2ft + 2g-DP r. 
'/! + .< -I n 
2-’(n +1) (ft + 2) ... (?i + $). .S’ 1 
■f+4eX ^7 -eX 
L /I + 1 I! + 
K". 
