49G 
PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
But p'q = \ ] , p\ = - -h [i, 4} , p'g = {i, 4] {{, 6}, and 
3% + 0 + TP'i = 3w(5S + /”), 
3 ^ + = - 3-2 (- - 5), where S = {i + 1). 
Therefore the factors are 
])/ 
- (cos) = I — 3^/3- (5N“ + f) 
(36). 
— (sin) = I + {t- - 5) 
It is easy to verify that the other coefficients of iC" and are in fact reproduced. 
The notation adopted here and below for distinguishing the cosine and sine factors 
is perhaps rather clumsy, but I have not thought it worth while to take distinctive 
symbols for the factors in these cases, because they will not be of frecpient occurrence. 
When s = 1, 
{31 = [* ~ - -i\-/3Ul + cos 4^)] 1“®.^ + - i^cos2.i)|™G.^ 
= [ 1 T --i/ 3 M /3 + m ]“>+■• 
ffi' 
This must be equal to ^ j . 
Novr, Avith upper sign for cosine and lower for sine, 
p'a — ~ 3] [1 ± -i6^i ('>' + 1)] , p'o = fisiP 3] {i, 5] . 
Substituting for j^'g its values, we find with the upper sign 
L (cos) = 1 - 4^ - 1^- (p'g + i) = 1 - i/3 + -gi-yS' [^(^ + 1) - 10]. 
JJi 
And with the lower sign 
A (sin) = L + i ;8 - ifi' + i) = 1 + + -0-4/3- [{(i + 1) _ 10] 
(37). 
It follows that 
D] (cos) = 1 + [/ (^ + 1) - I 4] 
D] (sin) = 1 - - -oIf/3~p(t + 1) - 14] 
— (cos) 
= 1 -P + *^-p(^’+l)-8] 
[D] (cos)]- = 1 + - gi 2 -^'[i(/ + 1) — 16] 
TB (sin) 
Llfi 
— 1 + + 3^/5''[^' (^' + 1 ) “ 8] 
[D/ (sin)]- = I — -j- 1) — 16] 
(37). 
