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XXII. Researches in physical astronomy. By John William Lubbock, Esq. 
Fellow of the Royal Society. 
Read April 29, 1830. 
In the first volume of the Mecanique Celeste, Laplace has given expressions 
for the variations of the elliptic constants, which are true when the square and 
higher powers of the disturbing force are neglected ; and he has proved, upon 
the supposition that the planets move in the same direction, in orbits nearly 
circular and little inclined one to another, that the eccentricities and inclina- 
tions vary within small limits, thereby demonstrating within these conditions 
the stability of the planetary system. But these conditions are not necessary 
to the stability of a system of bodies, subject to the law of attraction, which 
obtains in our system. I have given in the following investigation the expres- 
sions for the variations of the elliptic constants, which are rigorously true 
whatever power of the disturbing force be retained ; and it is easy to con- 
clude from the form of their expressions, that however far the approximation 
be carried, the eccentricity, the major axis, and the tangent of the inclination 
of the orbit to a fixed plane, contain no term which varies with the time ; 
their variations are all periodic, and they oscillate therefore within certain 
limits. This theorem is no longer true if the planet moves in a resisting 
medium. 
I have also given some equations which obtain when an angle is taken 
for the independent variable, which in the elliptic movement is the eccentric 
anomaly, which are of remarkable simplicity, and which, as far as I know, 
have never been noticed, and the development of the disturbing function 
R to the quantities involving the squares and products of the eccentricities 
inclusive. 
2 u 
MDCCCXXX. 
