OF THE PERTURBATIONS OF THE PLANETS. 
225 
times and by different methods, at last succeeded in overcoming- the diffi- 
culties of this great problem. 
In this paper the utmost rigour of investigation has been strictly preserved. 
No admission or supposition has any where been made for the sake of simpli- 
fying calculation or of obtaining a result more readily. The procedure that 
has been followed likewise makes it easy to change the form of the differentials 
of the elements of the orbit, as occasion may require. Thus it is obvious from 
the formulas (C), and from other formulas, that the variations of all the ele- 
ments may be expressed by means of the three functions ^ ; or, 
R d R d R 
by means of the three ; or, by any two of the differentials of R 
relatively to a, e, s, w, and one of the two, relatively to i and N ; which re- 
mark is useful in the theory of the comets. 
There is this advantage in expressing the differentials of the elements by 
means of the function R, that inspection alone discovers the nature of the 
terms that enter into every formula. But it is not enough to know the form 
of the terms, we must likewise attend to their convergency. In the present 
state of the heavens there is no difficulty in this respect, because the eccentri- 
cities of the planetary orbits, and their inclinations to the ecliptic are found to 
be small, and it is upon the smallness of these quantities that the convergency 
of the series into which R is developed, mainly depends. In the present cir- 
cumstances of the planetary system, the formulas afford the utmost possible 
facility for computing the inequalities of the elliptic elements. After all, the 
inquiry is difficult enough when it is carried beyond a first approximation ; for 
in the second stage of the process every element that enters into a formula 
being itself a collection of sines or cosines, it is not easy to be assured of the 
nature of the quantities arising from the combination of so many complex 
expressions. 
If we extend our views and consider the stability of the system of the world, 
it is necessary to begin with establishing the convergency of the terms into 
which R is expanded. The mathematical form of these terms will always be 
the same ; but unless their total amount can be estimated with sufficient ex- 
actness by a limited number of them, the human understanding can come to 
no solid decision. Now this will depend upon the effect of the perturbations 
in changing the eccentricities and the inclinations of the orbits to the ecliptic. 
2 G 
MDCCCXXXII. 
