AND THE FIGURE OF A HOMOGENEOUS PLANET. 
141 
remainder of the solution is deduced entirely from these equations, it would 
be superfluous to repeat here what has already been fully explained. The same 
procedure as in the first investigation will prove, that the figure of the fluid in 
equilibrium is exclusively an oblate elliptical spheroid of revolution turning 
about the less axis h, and that the ratio jp of the two axes is derived from the 
centrifugal force by means of the equation 
h 2 
/= B' - p A - 
10. The level surfaces of the mass in equilibrium are properly the interior 
surfaces similar to the outer surface, and similarly posited about the common 
centre. Such surfaces agree with Clairaut’s definition ; for they are perpen- 
dicular to the resultant of the forces urging the particles contained in them, as 
appears from the differential equation (8), which is common to them all. But 
as every particle ’within the mass is acted upon by several forces, it may be- 
come a question whether there are not other interior surfaces besides those 
similar to the outer one, which possess the properties of being equably pressed, 
and of being perpendicular to the resultant of the forces in action. It is this 
point that we are now to investigate. 
Suppose that ABC represents an oblate elliptical 
spheroid of homogeneous fluid in equilibrium by re- 
volving about the axis P Q ; and let a b c be an oblate 
elliptical spheroid within ABC, the centres, the less 
axes, and the equators of the two figures coinciding : 
taking any particle (x, y, z) of the interior mass a b c, 
the attractions of the whole mass ABC urging the 
particle in the respective directions of the co-ordinates, 
may, as before, be represented by 
A' x, B ' y, B'z: 
and in like manner the attractions of the interior mass ab c upon the particle, 
may be denoted by 
A" x, B "y, B "z: 
