PEOFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
487 
In the cases EES, OES, OOC, EOS we put (/.,± 2 h= thus find the 
s 
coefficients for P’. 
The coefficients for (C, in EEC, EES, OOC, OOS are determined by 
Vi — 2/i 
Pi — 2rt + 2 
Pi + 2n 
Pi + 2n - 2 
{'t, s ‘Jn + 1} V-s - 
— ^ "4” Srt — I j- 
2m + 2 
Vs + 2h 
9s + 2n - 2 
The coefficients for C, S in OEC, OES, EOC, EOS are determined by 
'P s — 2n 
^ / 
P « — 2ii + : 
P s + 2n 
P's + 2-/1 - ; 
1 9s - 2i> 
{fi S - 2n + 2} 9s -2a+ 2 
= — {i, s -fi 2n!- 
V« + 2/1 - 2 
It follows that if we put q,, = I and = I 
p.-,. =(-TTr- 
Ps — 2/1 ) 
{i, s — 2)1 -h 1} {fi -s — 2n + 3} ... {i, s — 1} 
2'^s + ->,i = ^ S" 2'^ — t} (O •'»' + — '^1 • • • + If ^s + 2/1 ' 
1 
P'i-2n = (-)" 77 
Ps - 2« 
{i, s — 2/1 + 2} {i, s — 2?/, + 4} ... p; s} 
p's + 2n = {~Y {h ■'> + 2/i} {7, 6- + 2// — 2} . . . {?, 6' + 2} 5'^ + 
When .s = U, q.i/qo is equal to that wliicli would be given Ijy the general formula for 
when we put in it n =■ 1, ■§ = 0. Hence it follows that the qs for s — 0 have 
-V.1+2K 
9s +2/i-2 
double the values given by the general formula. 
If we change the sign of s, the two continued fractions in the equation for /3cr are 
simply interchanged. Hence /3 (t is unchanged when s changes sign. Also, since 
[i, t] is equal to [— i — 1, t], ^cr is unchanged when — i — 1 is written for ?. A 
consideration of the forms of the q'% and 2 ^’s shows that + 2 /.- is equal to 
i — s'. 
. ^ ^, q^ _ 2 t P'' ~ and therefore 
'p: 
2 -.!lPr^’ 
