AND THE FIGURE OF A HOMOGENEOUS PLANET. 
119 
and they may not be at eveiy point the same functions of the co-ordinates in 
all the figures, of which it is susceptible ; but, notwithstanding the equilibrium 
may still be possible, because this indispensable condition may be fulfilled when 
figures of a certain class are induced on the mass. In such cases, the deter- 
mination of the equilibrium necessarily requires two distinct researches ; of 
which one is to find out what are the particular figures into which the mass 
must be moulded, so as to make the accelerating forces at every point the same 
functions of the co-ordinates. After these figures have been found, we can 
apply to them the equations expressing the conditions of equilibrium, and ac- 
complish the mathematical solution of the problem. But if it shall appear that 
no figure whatever capable of fulfilling both the conditions laid down above 
can be induced on the fluid, the equilibrium will be absolutely impossible. 
In the usual exposition of this theory, the equilibrium is made to depend on 
conditions that do not exactly coincide with those at which we have arrived. 
According to Clairaut and all other authors who have written on this sub- 
ject, it is necessary, and it is sufficient, for the equilibrium of a homogeneous 
fluid, first, that the expressions of the accelerating forces possess the criterion 
of integrability ; secondly, that the resultant of the forces in action at all the 
parts of the outer surface which are at liberty, be directed perpendicularly to- 
wards these surfaces. We may throw out of view what regards the criterion 
of integrability, about which there is no difference of opinion, and which in 
reality is always fulfilled by the forces that occur in physical researches. The 
perpendicularity of the forces to the outer surface is a property of the differen- 
tial equation of that surface, and will necessarily take place whenever it is pos- 
sible to form that equation. Nothing more is required for forming the equa- 
tion mentioned, than that the accelerating forces at every point of it be ex- 
pressed by the same functions of the co-ordinates of the point.* It follows 
* The forces are perpendicular to every surface in which the pressure is constant. The outer sur- 
faces are those at every point of which there is no pressure. In all the questions that have occurred, 
the forces at the outer surface of the fluid are the same functions of the co-ordinates of the point, what- 
ever geometrical figure the fluid is supposed to assume ; and on this account the equation of the outer 
surface can be formed without reference to any particular class of figures. But this is not sufficient ; 
for, according to the fundamental assumption laid down by Clairaut himself, the theory of equilibrium 
cannot be applied, unless the forces be the same functions of the co-ordinates of their point of action 
in every part of the mass. 
