AND THE FIGURE OF A HOMOGENEOUS PLANET. 
143 
ture of the interior spheroid. If it be more oblate than the exterior spheroid 
ABC, A" will be greater than A', and the attraction (A' — A") x tending from 
the equator, the pressure of the canal a G m will be outward and opposed to 
that of the canal AamM. On this supposition, therefore, the whole action 
of the matter exterior to the spheroid ab e will cause a pressure upon the mole- 
cule a m, equal to p — p 1 . By subtracting the equations (11) and (12) we get 
p-p'=C-a-A!'^- ( 13 ) 
and we have now to inquire whether a spheroid can be found that will satisfy 
this equation, on the supposition that p — p' is the same at all the points of 
the surface of the spheroid. 
The equation (13) evidently comprehends the level surfaces, which are similar 
and similarly situated to the upper surface ABC: for, on the supposition that 
the figures are similar, we have A' = A", B' = B", p' = C', and the equation 
(13) is identical to the equation (12) which, by giving different values to p, de- 
termines all the level surfaces. The equation (13) is similar in its form to the 
equation (12), A" and B" being the same functions of the excentricity of the 
spheroid a be, that A' and B' are, of the excentricity of the spheroid ABC; 
and the centrifugal force f enters alike into both equations. It is therefore 
evident that the solution of the latter, supposing p constant, and the solution 
of the former supposing p — p' constant are both contained in the equation, 
/= B'-pA': 
h 3 
and, as from this two values of ^ are in general obtained, one of these results 
determines the spheroid ABC and its level surfaces, and the other determines 
the interior spheroid ab c, the surface of which sustains the same pressure at 
every point by the action of the exterior fluid, and which is therefore sepa- 
rately in equilibrium. 
There is this difference between the level surfaces and the other surfaces of 
equable pressure, that the former spread through the whole mass and ultimately 
coincide with the upper surface, whereas the latter, on account of the dissi- 
milarity of figure, are confined to a part of the mass. Of the two spheroids 
