:224 SEE— THE EXISTENCE OF PLANETS [April 23, 



of his predecessors. Though they did not deal with definite cases 

 ■of extreme oblateness, the great mathematicians of the eighteenth 

 century made it clear that very rapid rotation would be adequate 

 to produce disc-shaped figures of equilibrium. 



This subject is quite fully discussed by Laplace in the " Me- 

 •canique Celeste,"- where the following theorems are announced. 



Any homogeneous fluid mass of a density equal to the mean density of 

 the earth, cannot be in equilibrium with an elliptical figure, if the time of its 

 rotation be less than 0.10090 day. If this time be greater, there will always 

 "be two elliptical figures, and no more, which will satisfy the equilibrium. 

 If the density of the fluid mass be different from that of the earth, we shall 

 have the time of rotation, in which the equilibrium ceases to be possible, 

 with an elliptical figure, by multiplying 0.10090 day by the square root of the 

 ratio of the mean density of the earth to that of the fluid mass. Therefore 

 with a fluid mass whose density is a quarter part of that of the earth, which 

 is nearly the case with the sun, this time would be 0.20180 day, and if the 

 •earth were supposed to be fluid and homogeneous, with a density equal to a 

 ninety-eighth part of its present value, the figure it must take to satisfy its 

 present rotatory motion, would be the limit of all the elliptical figures, with 

 which the equilibrium could subsist. 



What Laplace here points out was in fact established by 

 Maclaurin in his '' Treatise on Fluxions," Edinburgh, 1742. For 

 if k- be the gravitational constant, the density of the mass and w 

 the angular velocity of rotation, then it was proved that for 



the two possible figures of equilibrium coalesce into one, but for 



^r^ > 0.22467, 

 2Trk-a 



there is no ellipsoid of revolution which is a figure of equilibrium. 

 For very small values of w^/2irk-a, there are two distinct ellipsoids 

 which are figures of equilibrium, one of them being nearly spherical 

 and the other very oblate, the limits, for 0^ = 0, being respectively 

 a sphere and an infinite plane.^ 



== Liv. III., Chap III., § 20. 



*Tisserand's " Mecanique Celeste," Tome II., chap. VI. 



