AND THE FIGURE OF A HOMOGENEOUS PLANET. 135 
A 3 A 2 
and consequently they cannot be equal to /, unless ^r 3 and be both less than 
unit : and supposing that h is the least of the three axes, the two values of/ 
will not be equal, unless B' = C', and h! = h!', in which case both the formulas 
coincide in one, viz. 
/= 
In conclusion, it follows that the figure of the fluid in equilibrium is an oblate 
elliptical spheroid of revolution, of which the equation is 
* 2 + (; y 2 + * 2 ) h2 > 
the mass turning about h the less axis, and the relation between the centrifugal 
force and -p the ratio of the axes, being determined by the equation 
/= B'-£a\ 
The complete solution of the problem is now brought to the discussion of this 
last equation ; and as this is a question purely mathematical, but slightly con- 
nected with the physical conditions of the equilibrium, which we have under- 
taken to investigate, we shall refer to the Mecanique Celeste of Laplace and 
to the Theorie Analytique du Systeme du Monde of M. de Pontecoulant, in 
which works this point is amply treated. 
The foregoing solution, being perfectly general, proves that the equilibrium 
is possible only when the elliptical spheroid is oblate at the poles. When the 
spheroid is oblong, and the axis of rotation h greater than the other axis h\ the 
expression that must be equal to the centrifugal force is negative ; and as that 
force is essentially positive, the equilibrium becomes impossible. 
It will not be necessary to retrace the steps of the foregoing analytical pro- 
cess of reasoning, in order to show synthetically that the equilibrium will be 
secured if the conditions deduced be fulfilled. For, as soon as such a figure is 
found as will make the forces that actually urge every particle of the mass the 
same functions of the coordinates of their point of action, this problem comes 
under the hypothesis of the first one, and may be demonstrated in the very 
same manner. 
