[ 17 J 
III. Researches in Physical Astronomy. By John William Lubbock, Esq., 
V.P. and Treas. R.S. 
Read December 9, 1830. 
In last April I had the honour of presenting to the Society a paper containing 
expressions for the variations of the elliptic constants in the theory of the 
motions of the planets. The stability of the solar system is established by 
means of these expressions, if the planets move in a space absolutely devoid 
of any resistance *, for it results from their form that however far the ap- 
proximation be carried, the eccentricity, the major axis, and the tangent of 
the inclination of the orbit to a fixed plane, contain only periodic inequalities, 
each of the three other constants, namely, the longitude of the node, the longi- 
* When the body moves in a medium which resists according to any power of the velocity, the 
contrary obtains, the major axis and eccentricity acquiring a term which varies with the time, while 
the longitude of the perihelion and longitude of the epoch have only periodic inequalities. This 
results from the equations given in the former part of this paper, Phil. Trans. Part II. 1830, page 340. 
n — I ra+ 1 
A /-r 
o rn ( * ^ 
V 
f 1 +e cos v } 
^ (1— ecosu) 1 
Zca \ a ) 
1 1 — e cos u ) 
n dy 
n — 
1 
n 
-1 
de 
= ~ 
*«(f) 2 
J 
l+e cos v \ 
. 1 — e cos v 3 
2 A 
\~2~ ) cos d o 
n— 
1 
n - 
-l 
e die 
= — 
*•(*) 2 
5 
l+ecos u\ 
1— e cost/ ) 
Vl-e 8 • i 
— sin v d u 
n 
n— 1 
71—1 
— d -ct 
= 2< 
<i) 2 1 
l+ecosu| 2 
1 — e cos t/ ) 
ri-e 9 cost/-) , 
71+ 1 
Cl+e 
\l-e 
COS l/~| 
COS V J 
r 2 =i + 
( n 
+ l)ecosu 
+ (n + l)*e 9 cos 2 w 
n— 1 
cl+e 
\l-e 
COS V ] 
COS V \ 
i 2 
1 =! + 
( n 
— l)ecosu 
+ (n — l) 2 e 2 cos 2 o 
MDCCCXXXI. 
D 
