and the figure of a homogeneous planet. 
137 
sions of the two bodies. Wherefore the joint action of all the forces is to urge 
the two particles in similar directions with intensities proportional to the linear 
dimensions of the bodies. And as the same thing is true of all particles simi- 
larly situated in the two bodies, if there be an equilibrium in one case, there 
will be an equilibrium in the other ; for the forces which urge the particles of 
one body are in no respect different from the forces which urge the particles of 
the other, except in being all increased or all diminished in the same given 
proportion. 
This lemma being premised, let ABC repre- 
sent a mass of homogeneous fluid in equilibrium, 
by the attraction of its particles in the inverse pro- 
portion of the square of the distance, and a cen- 
trifugal force caused by revolving about the axis 
P Q. The axis P Q will pass through G, the cen- 
tre of gravity of the mass. For, abstracting from 
any motion or force common to all the particles, 
that centre may be considered at rest ; and, as 
the attractive forces of the particles balance one 
another at that point, the centrifugal force must likewise vanish at the same 
point. 
Let any radius G B, drawn from the centre of gravity to the surface of the 
fluid, be divided in a given proportion at b ; and supposing it to turn round 
G so as to be directed successively to all the points in the outer surface of the 
fluid, the radius G b , being always the same part of G B, will describe an in- 
terior surface similar to the outer one, and similarly posited about G. And be- 
cause the whole mass A B C is in equilibrium, it follows from the lemma that 
the interior mass a be, which is similar to the whole mass, and revolves with 
it about the common axis P Q, will be separately in equilibrium, supposing the 
exterior stratum of matter were taken away or annihilated. 
In the interior surface a be assume any molecule am: the forces that act 
upon a m are ; first, the resultant of the centrifugal force and the attraction of 
the mass abc ; secondly, the attraction of the stratum of fluid between the 
two surfaces. Because the interior body of fluid abc is separately in equili- 
MDCCCXXXI. 
T 
