PROFESSOR G. H. DARWIN OX ELLIPSOIDAL HARMONIC ANALYSIS. 
467 
It will be observed that // is independent of (3, and that it has the same form as in 
spheroidal harmonic analysis when /3 vanishes. Since /x^ is less than 1 and greater 
than , a- and y are real. ' 
I /3 
In all the earlier portion of this paper I always write p- — 1 and not 1 — /x", so 
as to maintain perfect svmmeti'v with respect to v and y. 
W e now have 
1 .1 7 I + /I 
I - f - , 
A.;' = Z- (^ ^ 
A3^- 
1 - / 9 / ’ 
2k-/3 cos- cf, 
Bp - - i ) , = A'V; , 
B/ = 1), Cp = ZV; 
1 - /3 
-p 2 2k~^sm^4> 
=-1-/3 ’ 
f\ o _ 7^1 ^cos 20 
1-/3 
Let us denote the differential operators involved in our equations, thus 
Then 
1 ). = (1 - 
Dj = - ,/-l.(l - 
1 + yd \L o T U 
1 + yd \b o - ^. <l 
i X V3L3C3 d 
ku^ du^ 
Dy = (1 — ^cos2^)- 
, d 
■ct 
d 
kl\ 
vylu^ (1 — 13)- 
AoB.,Co 
dcj) 
d 
/■D., 
?6/h<2 (1 — ^y- ’ 
A0B3C3-4 - = v/-l . - 
^ ^ (1 — /Sf 
(2). 
Hence oui- differential equations are 
DpLh 
1 - /3 
- 
l-/3~ 
(^■+ 1)-'+ A 
Uj, a similar equation with suffix 2, and 
':/■ I , ^ 1 - /3 cos 20 , /c-' 
'0 + !)■ 1 -/3 +3 
u„. 
/3 ' /S 
Let us replace k~ by another constant such tiiat 
(i(. + 1),.* + |)(i - i3) = {{i + i)[^=(i - ft - 1] + »• - y3<^. 
so that 
K- 
k^ 
i{i + ]) — s- A (3a 
1 - ft 
3 o 2 
