520 
PROFESSOR O. il. DARWIN OX ELLIPSOIDAE HARMONIC ANALYSIS. 
Also ^F.ydB has a similar form with accented ^/’s, y + 1 for y — I, and y — ?> for 
2 - r- 
When we substitute for y its value 1 — cos 2f/), and write as before 
O < + 1 ) 
An T ’ 
the last term in \^L\dd becomes 
+ ;8. + 1) - ««-./,[L - -hUil 
h i' 'S • 
)]! 
. . ( 68 ). 
Also the last term in becomes 
+ ^ • , • r ~Ti 12 [1 + 1-/3 (N + 0)] — cos2(/)[l + f,/3(i + 3)]}. . (68). 
•s I - S . 
But it will appear later that we only need the parts of tliese terms which involve 
cos 2f/) developed as far as the first power (.)f ^8 ; lienee in both cases u e may write the 
latter term Inside ) } simply as — cos 2</). 
Our general formuhe for the </ coeliicients ajiply for all values of s down to = 3, 
inclusive, altliough the result for .v = 3 needs jiroper interjiretation. Hence the 
present result applies down to .s = 3, inclusive. 
I have just re-detined N, and 1 remind the reader that 
(, - 1)/(, + l)((: +-j) 
^ =- irzi - 
Then if in (68) we introtluce for the i/'s their values, we find that the coefficient of 
the term in ^ is 
-7. 
( < + •;>) { i + “ I) 
+ + {i — ■'<)(/ — 
1) = - - n. 
The coefficient of the term in j3~ is 
, f (t -f- 1) ( i — ,s + 2) (y — 6' + .1) (y — a + 1) I (6 — *■) (( — •8 — 1H < — — 2) y 
6 41 1 ,, ..V H 
— ) 
( S - 1)(.8 - 2 ) 
(s + I) (.8- + 2) 
{i - - ^+2) (y + 6+ l)(/+,S'+2)(/ - i>)(( 
- 6- - l ) 
.. -t - .5 + 2)(y - .s)y - .8 - I) 
0- -1) 
If this be reduced by a process similar to that employed in § Uf we find 
