IN PHYSICAL ASTRONOMY. 
63 
pies are given in the Theor. Anal. vol. i. p. 362. Similar reductions are 
applicable to the terms in r multiplied by the first power of the 
eccentricities. 
In Laplace’s notation 
— lb 
— 2°I > 
2 
II 
A A (2) 
^i,2 — toi , 
2 
A. , = ii <3> 
-I* (0) 
— 2 u 3 > 
2 
A A W 
^3,1 — & 3 > 
2 
A -A & 
to$ t 2 to- y 
2 
2 
-lb (0) 
— 2 U 5 > 
2 
A _ A 0) 
"5,0 ®5 > 
2 
7 _ , (2) 
to 5,2 to 5 y 
2 
»m = L <S) 
2 
h —lb 
°$,0 2°3,1 
dS, (0) 
12 
“ 2 d a 9 
— b 3il -b 3f0 
a. 
I A — _ 
2 ^3,2 — 
d6, 
2 
d a, 
(l) 
a L 
"3,2 
a , 
I A — _ 
l 2 "3,3 — 
d 6 i (2) 
dT ’ 
— &3,3 — h “ 4 6 S,4 = - 
a, 
dt7 (3> 
2 
d a 
f a 
3 < — 
" ^5,0 — 
«, <0) 
2 
d a 
3 ^~^5,1 ^5,0 “"4 ^5,2^ 
d.i s W 
2 
d a 
/ a j 
1 h 
1 h \ — 
d.s s <2) 
2 
3 b S ,3 — 5^5,2— 4^5,4 j 
d . 6_ (3) 
u t 
l 5 ’ 2 
2 "5,1 
2 "5,3 J 
d a 
d a 
The numerical values of these quantities are given for the principal planets 
in the third volume of the Mecanique Celeste. 
The following numerical examples will serve to explain the expressions given 
above, and to show their accuracy, the results agreeing exactly with those 
given in the Mecanique Celeste. 
= l + j } - ' {* + f - h $*>•■ } <»< + «- «) 
— r, cos (nt — n t t + e — e ; ) See p. 55. 
* The coefficient of cos {n t + e — m) p. 52 line 1 J , and p. 55 line 3, should be 
