48G 
PROFESSOTi G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
When .S’ = 0, the equation becomes 
/3o- = 
ending when i is even (EEC) with 
and when i is odd (OEG) with 
4.D + / 3(7 - 4.22 + /So- - __ . 
- W {h {i i - 1 } 
- i/3- {i, i - 1} {i, i - 2f 
{i — 1)' + /Scr 
W^hen s = I the equation has two forms, which may, however, be written together. 
If the upper sign refers to cosines (OOC, EOC) and the lower to sines (OOS, EOS), 
the equations are :— 
/Icr i + 1) — 
2} K3} 
1^2 4} 51 
ending when i is even (EOC, EOS) with 
4.1.2 + yStr - 4.2.3 + /So- - ... 
— i — 1} {i,i— 2} 
(i - ly - 1 + 0c 
and when i odd (OOC, OOS) with 
- i} {i, i - 1} 
“ r- - I + 0a 
It might appear at first sight that a difficulty will arise in the interpretation of 
these results when i is small, for the numbers in the denominators of the fractions 
increase, and yet it is possible that the number at the end should be smaller than that 
at the beginning; thus ajjparently the fraction ends before it begins. But this 
difficulty does not really arise, because in such cases the numerator will always be 
found to vanish, and thus the whole fi’action disappears. For example, in the last case 
specified, if s = I, f = 2 the denominators, according to the formula, begin with 
8 + /3(t and end with 0 + 0(t ; but the fraction has for numerator {*2, 2} {2. 3). 
which vanishes. 
When 0cr has Ijeen determined we find the q’s by the formulae- - 
2qs-2n _ — {b .S — 2)1 + 2} \i, s — 2ii 4- 1} — 2?d {i, 6’ — 2n — 1 } 
qt- 2 n + 2 4:n{s — n) — 0a — 4:{n + 1) {s — n — 1) — 0a — ' 
2qst + _ _ 1 4 0~{i, s + 27t 4- 2} {/, y + 2?i + 1 } 
2$ + 2n — 2 4% (.s 4" ^i) 4* 00" — 4(ti + 1) (-S 4- M 4" 1) 4" 0^ — ... 
The terminations of the continued fractions are as specified above in the equation 
for 0a-. 
By forming continued products of ratios of successive q’s, we can find all the q’s as 
multiples of q^, and qs = 1. 
In the cases EEC, OEC, OOS. EOS, these are the required coefficients for 
