326 NEWTON S PRINCIPIA. 



** (3 -2e 2 )vr=~? . 



We have already seen that m is nearly g--^, whence we find 



1 

 &quot;232* 



If a curve be constructed of which the abscissa is e and the 

 ordinate the left hand side of the above equation, it will 

 be found to resemble from e = to e=\ s the line OEBC 

 where O C = 1. Take O G = fro, and draw G E F parallel 



to the axis of X, then we see 

 that the curve cuts this straight 

 line twice, and each of these 

 x points corresponds to a value 

 of e that will satisfy the above equation. It would appear 

 that for the same value of co there are two values of e, GE 

 and GF, which are consistent with equilibrium. But it 

 does not follow that if a mass of fluid be set in motion 

 with an angular velocity on that it can take either of these 

 forms. There is another condition to be satisfied. There 

 is a certain principle in mechanics, called the conservation 

 of areas, which teaches us that &quot; if any number of bodies 

 revolve round a centre, and are acted on only by their mu 

 tual attraction and by forces directed to the centre, the sum 

 of the products of the mass of each by the projection on 

 a given plane of the area which it describes round that 

 centre bears a constant ratio to the time.&quot; Hence the fluid 

 must take up such a form that this ratio shall be the same 

 as that in the fluid as originally set in motion. It requires 

 but little consideration to perceive that for two forms so 

 different that one is nearly a sphere and the other ex 

 cessively elliptical, it is impossible that the same angular ve 

 locity could sweep out the same areas in the same time. 



This form of equilibrium of the earth is stable, for if 

 the form were, by any chance, to become less spherical, 

 by the principle of the conservation of areas, the angular 



