479] 
FUNCTION IN THE LUNAR AND PLANETARY THEORIES. 
513 
Making these changes we have 
where 
n, ao'+/)+ ^ 2} i+/+ 2s V Tut—Hj-j')+0 7->~l(.j-j')+0 
n ( j + /) + s] V iu+f)+s rh \ u+f)+s ’ 
M -W-f)+e = , y n {Ki+iO + &} n (i (j +j') + g} 
^ ) ni^-^nKi+i' + s + ^ni^+^nHi+i' + s-^’ 
viz. this is (—) s into the product of two binomial coefficients, each belonging to the 
exponent % (j +j') + s- 
Particular Cases, j +j' — 0, 2, 4, 6, being those required in the Planetary Theory. 
Considering successively the cases j +j' = 0, 2, 4, 6, we have, first, 
ns 
D(j. -j) = S n ‘^ s *Vs(-y 
n^(s — 0) n|(s + 0)j 
Pr j+e 
which, developed as far as t? 6 , is 
(*) D U> ~j)= ? A ~ j 
- 1 v mB^ + B^) 
+ v *±(C-^ + 4,C-j + C-3-z) 
- 1 v s £ (ZH+ 3 + 9ZH+ 1 + 9ZH“ 1 + ZH" 3 ), 
where, and in what immediately follows, A, B, C, D are used to denote functions (not 
of (a, a'), but) of r, r'. 
Secondly, 
D(j, + 
( y n (s + 1) 
; ; uus-0)ui(s + d) + i 
which, developed to rf, is 
n(s + i) -j+1+e 
ni(s + d)n|(s-^) + l s + l 
(*) 
D(j> -j + 2) = 9? 2 | \ 
1.3 2 
"274 77 
. i (2C~ j+2 + 2CM), 
1.3.5 d 
+ 2.4.6 V 
. 1 (3ZH+ 3 + 9D~i +1 + 3ZH _1 )j. 
C. VII. 
65