300 OUTLINES OF NATURAL 



But though it was thus demonstrated that the parts 

 of a homogeneous fluid, on which the figure of the 

 oblate spheroid just described was any how indu- 

 ced, would be in equilibria, yet it was not shewn 

 conversely, that, whenever an equilibrium takes 

 place in such a fluid mass, the figure of the mass 

 ftiust be the oblate spheroid in question. D'ALEM- 

 BERT indeed shewed, that there are more sphe- 

 roids than one in which the state of equilibrium 

 may be maintained ; and this result, though it 

 was not observed by MACLAURIN, might have 

 been inferred from his solution. LE GENDRE af- 

 terwards proved, that the solids of equilibrium. 

 must always be elliptic spheroids, and that in ge- 

 neral there are two spheroids which satisfy the 

 conditions. 



In the case of a homogeneous mass of the mean den- 

 sity of the Earth, revolving in the space of 23 h 56' 

 4", one of the spheroids is that which has been 

 mentioned ; the other, is one in which the equato- 

 rial diameter is to the polar, as 681 to 1. Mem. 

 Acad. des Sciences, 1784. LA PLACE has added 

 the limitation which follows. 



303. A fluid and homogeneous mass, of the 

 mean density of the Earth, cannot be in equili- 

 brium with an elliptic figure, if the time of its 

 rotation be less than 2 h 25 m 17 sec ; if the time 

 of revolution is greater than this, there will al- 

 ways 



