478 
PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
+ rj',[P^+“ — 2 crP^' + [i, s]{i\ s — l]P*-“] 
+ Sy8"^,_ 2 » [P-^ - 2 crp- 2 '* + {i, 6' - 27^} -\i, s - 2n - 1} p- 2 «- 2 ] 
+ + - 2 o-P^' + -" + {i, s + 2n} {i, s + 2n - 1 } P+-"-■] = 0 . 
The coefficients of the P’s must vanish separately. This gives from the coefficients 
of P'*’ ~ and P'' the following :— 
2\_4.n{s — n) — ^cr](/,_ 2 « + 
+ [i, s — 2n + 2 ] [/, .s- — 2n + + 2 = 0 , 
— 2 [477 (.S' + n) + /3a ] 5's + 2n + <2s + 2n - 2 
+ + 277 + 2 j {7, 6 * + 277 + l] ^/s + 2rt + 2 = 0 - 
These ecpiations may be written in the form 
— 2 )! 
(?•« — 2n + 2 
{t .s — 2n + 21 {{, s — 2n + 1} 
4/1 (s — '77) — /3a + }/3~ 
I 02 I 22.1 - 2n - 2 \ 5 
') 
2s + 2ii 
2s + 2n — 2 
2.-2,/ / 
1 
4?i(.s' + 7 ;) + l3a — s + 2)1 + 2} {'7, s + 2/7 + 2} f 
22s + 2h + 2 \ 
\ 2s + 2ii I 
(16). 
Wlience by continued application, the continued fractions 
22s ~2„ _ — {h s — 2/7 + 2} {i, s — 2n + 1} s — 277 } { 7 ’, s — 2/7 — 1} 
2 - 2 „ +2 4//(.s — 77 ) — yScr — 4(77 + l)(s — 77 — 1) — ... 
-iP{7,s - 2/7 - 2/- + 2HL7/- 2// - 2/' + P 
4(77 + 7 ')(.S- — 77 — /’) + i/ 3 - 
•2s — 2a — 2r — 2 
2s - 2 n - 2r 
22s + 2ii^ 
2s + 2,1 - i 
I 
s + 2/7 + 2} {i, .9 + 2 77 + 1 } 
4(77 + 1) (s + 77 + 1) + /3cr — ... 
(Ip- 
477 (s + 77 ) + I3a ~ 
— s + 277 + 27’) {-7, s 4- 2/7 + 2/’ — 1) 
4(77 + 7’)(s 4- 77 + 7’) — 1/3'^{i,s + 2/7 + 2r + 2} { 7 , ,s + 2/7 + 2/- -r 
^2»~ + 2a + 2)-+2 \ 
2s+2n+2r 
We must now consider what I may call the middle of the series, which corre¬ 
sponds with n = 0 . In this case eacli of the E’s contributes one term and tlie //,. term 
ii’ives another. The result is 
or 
— 2aqs + /3qs-2 + ^qs + 2{'i, ^ + 2] (' 7 , *■ + 1 } — 0 , 
fSo- = i/3’ ('^) + ifim s + 2} li s + 1} = 
Since 2qs_.2/qs and 2qs + ^/q^ are expressible as continued fractions, we have an 
et^uation for ^a, if the continued fmctiojis terminate. 
