PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
499 
dv 
i + 3 
- 2(i + 2)v^ 
(„3 _ l)i + 3 
+ 
- 1 ) 
i + 2 
2^ + 2.^+^! 2‘+ 1(2^'+ 3) .'i + 1 r 
-1) 
i 3 
+ 
(v3 - lV+2 
/ \ i + 4 
A 
dvj 
But 
Q. = (-y 
2^ + \i + 3! 2‘ + 2(2t + ‘d).i + 2!' 
L(F2 - 1) 
1 + 4 
+ 
{v- - 1 ) 
i + 3 
V. 
Q' = - ’)“(£)'«■> 
2 * i ! 
therefore Q/ ^ ' = (“+ i, 
Q/ 
.2^ + Ki + llv 
( ) 0,2 _ 
(z." - lh(‘+2) 
(Oi + 3 f \i + l ^ t -1- X ■ 
Wi \ / (-..a „ 1 \i(j + 3) 
((,2 _ i)4(i + 3) [2^ + 4 + (2^ 4- 3) — I)] 
Q.‘"‘ = 
(39). 
These are all the functions which can be needed for the expression as far as of 
or of Q/ when s is less or equal to i. If s is equal to i, we shall liave terms 
IB~ qi + ^ or HyS' q\ + 4 , and these are the furthest. 
But it is well known that there is another expression for these functions of the 
second kind. 
The diffei'ential equation is 
(.= - \fj-, + 20,-’- - i)i - i{i + 1) - 1 ) - r 
dv 
©/ 
- 
- l)(.^ + 1 ),| + 2.^1 - i{i + 1 ).^ - 
= 0 , 
where ©f may be interpreted as meaning also Q/t 
Let us assume that 
[ Ydp 
J u 
is a solution, where may be interpreted as meaning also Q’, P*. 
Then since is a solution of the differential equation, we have 
- l)^[2V * + yAj 4. 2,(,= _ I)yv 
- / 3 [(.^ - !)(.= + 1 )( 2 V « + IS-Y) + 2 .=f'v' 
dv 
= 0. 
3 s 2 
