AND THE FIGURE OF A HOMOGENEOUS PLANET. 
121 
every part of the mass, the same functions of the coordinates of the particle ; 
and unless this be verified, the theory of equilibrium cannot be applied. In a 
homogeneous planet in a fluid state, there are forces which prevail in the in- 
terior parts and vanish at the surface ; and, as Clairaut’s theory notices no 
forces except those in action at the surface, it leaves out some of the causes 
tending to change the figure of the fluid, and therefore it cannot lead to an 
exact determination of the equilibrium. 
II. Application of the foregoing Theory to the Question of the Figure of the 
Planets. 
7. Having now explained the general theory of the equilibrium of fluids at 
sufficient length, I proceed to apply it to the question of the figure of the 
planets, in which it is required to determine the equilibrium of a fluid entirely 
at liberty, and unconfined by any obstacle or support. The problem is one of 
considerable difficulty. It is necessary to distribute the investigation under 
distinct heads. It would otherwise be impossible to preserve perspicuity and 
precision of ideas in an inquiry essentially different in different hypotheses. 
The equilibrium of a homogeneous fluid must occupy our attention before that 
of one having its density variable. For although it may at first appear that 
the latter problem is the more general, and includes the former, yet it will be 
found that the equilibrium of a fluid of variable density, depends upon that of 
a homogeneous fluid, and is deducible from it. And even with regard to 
homogeneous fluids, distinctions must be made, because what is required for 
the equilibrium varies with the nature of the accelerating forces. In this 
respect we distinguish these two general cases, of which we shall treat in two 
separate problems ; First, when the accelerating forces depend only on the co- 
ordinates of their point of action, and are explicitly known when the coordi- 
nates are given ; Secondly, when the accelerating forces depend not only upon 
* 
the coordinates of the particle on which they act, but likewise upon the figure 
of the whole mass of fluid ; as happens for the most part when the particles 
attract or repel one another. 
MDCCCXXXI. 
R 
