PROFESSOR G. 11. DARWIN OX ELLIPSOIDAL HARMONIC ANALYSIS, 
501 
It we write 
V is very large, 
o.v' ani tlie first of our equations (40) becomes, when 
7 • 1 ” ^ F dv 
■^1+1 = 77 ';.2 = 6''' - 77T 
1 1 
a 2 i +,1 
Therefore 0’ — {2,1 1 ) ay, and since the a, y in the case of the Pt Q' only differ 
from these in the accenting of the 7 ’s we have 
^ + .9! 
l — S: 
, /I ^ — S ' 
1 +^'i-=i_,+ 2! + --- 
n -L Q — 9 1 
1 + ^' 1 ’- + 
E'' = the same witli accented g’s. 
Effecting the multiplication of the series 
' 2 / “f" 5 ; 
i — si 
\ , ni i — si , i + s - 2\ 
1 + P ( <ls -2 ; _ ^ J_ 9 , + fb-2 
i — S + 
i + sl 
, i — sl i + s 2' 
+ 2 „ 9 ! + 5'^ + 2 s] 
l — s — 1 
, n:> I i — sl i + .s — 4 ! -i — sl 
+ ^M'/^-S--,sT4! + 7 
■i + -s! 
i - s - il 
^ + § + 4! { — s!'i + s — 2! 
O Q^+i .• I „i “1 (],S-2<Js-2~ 
^ + s! 
■« 7 s! 'i — s + 
> ! 
■i — .s! -i + .9 + 2 ! 
i — sl i s + 21 
+ ^^ + 2 'Zj +2 _ 5 _ + .s! Y—s + 2 ! 
■i — s! ^ + .s — 2 ! 
d“ <Js + 2<2 s-2 77 
= the same with accented q’s. 
If we substitute for the q’s their values, the coefficient of /3 inside [ ] in the 
expression for is 
(i 7 s)(^i 7 'S — 1) (1 — -5 7 l)(i — 9 7 2) (i — 9)(// — s 1) 
s — 1 
s — 1 
+ 
+ 
s 7 1 
(z: 7 s 7 l)(i 7 5 7 2)' 
s 7 1 
s — 2 
In the expression for £■* the first pair of these terms are multiplied by —^—, and 
the second pair by 
s 7 2 
