490 
PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
r> I o--' {l s — If ^ Q., {i, s + 1} {i, s + 2} 
/ 3 <^ = - -M --- 
{i + 1) - 
S- - 1 
^r/3' 36- — 2i(^ + I) — 
If this expression were inserted in we should obtain qs ±-2 correct to yS'. But 
9s 
since the next apj^roximation would only introduce /3^, it follows that qs±z would be 
correct to inclusive. Now enters in the functions with a factor /3, and there¬ 
fore this approximation would give results correct to inclusive. Since the similar 
operation could he applied with equal ease in all the cases in which the continued 
fractions assume special forms, it follows that this degree of accuracy is very easily 
attainable. However, the forms of the coefficients would be rather complicated, and 
it would render the subsequent algebra so tedious that I do not propose at present to 
carry the approximation beyond 
It now suffices to put cr = 0 in the denominators of all the continued fractions, 
whereby the coefficients are determined, except in the cases of 6 = 1, s = 3, where 
(r = ± 
:ie gener 
qs-i = 
^i{i +1). 
il case Ave have 
1 {h s} {'/, s — 1} 
9.s + i — 
- 3} 
1 \ 
® s - 1 
fi,s} {i, . - 1} - 2} lo. 
8 (.s -f 1) ’ 
1 
'P-i — 
128 (s - 1) - 2) 
5 qs+i — 
128 (s + l)(s + 2) ’ 
s - 2 
S -\r 2 
6- ^^-2’ 
q s+i, — 
t 
s — 4 
/ 
s + 4 
II 
1 
o 
9s+i = 
s + 
— {'0 6-1-1} 
ps-t — 
8 (.s - 1) ’ 
Ps^Z — 
8(.s+l) ’ 
{i, 6’} {i, 6' 2} 
\i, S -r 1} {O A -(- 3} 
ps-^ - 
1 
r—i 
1 
CO 
ps + i, — 
128 (A + l)(s + 2) ’ 
{0 6’ - 1} 
/ 
— {i, A 4- 2} 
P ■s -2 — 
8.(s - 1) ’ 
P s + z — 
8 (A 4- 1) ’ 
_ f 
{i, s - 1} {h s - 3} 
r-/ 
{i, A 4 2} {i, A + 4} 
P S-i - 
128 (a- - 1) (s - 2) ’ 
P s+i, — 
128 \^A -L 1) (a 4- 2) y 
When 5 = 0, we double the results given by the general formula and find 
9.-1 ~ \ ■> ’ Pi ~ T {h I I'J = T’i'8 {h I} {^5 3} . 
• (^-)- 
There are no q ,', p/ , and p/ = — ^ {f, 2}, {i, 21-Jf, 4} 
. . (28). 
