AND THE FIGURE OF A HOMOGENEOUS PLANET. 
145 
Maclaurin first demonstrated synthetically the equilibrium of an oblate 
elliptical spheroid when it revolves about the less axis with a certain angular 
velocity. In examining the equation of the surface of the fluid, D’Alembert 
discovered that it admitted of being solved more than one way, that is, he 
found that there are spheroids of different oblateness which will he in equili- 
brium with the same velocity of rotation ; and Laplace proved that there are 
two such spheroids and no more. Of this truth, first made known merely as 
a mathematical deduction from an algebraic equation, we have here attempted 
to give the physical explanation. 
Having now fully treated of the equilibrium of a homogeneous fluid, the 
order of discussion laid down would lead us to investigate that of one of vari- 
able density; but the length of this paper makes it advisable to reserve this 
part of our subject for another occasion. 
MDCCCXXXI. 
U 
