Oblateness of a Rotating Planet. 709 



or reducing, pM - Mp _ 



~p l o ; 



differentiating twice we find 



S-* 



and our equation becomes 



?-?♦¥*- c«> 



lb'. Let q = -=^, as before, p denoting what the mean 



density of the mass M would be if that mass were spread 

 uniformly over the spheroid. 

 Then 



qir 2tt 2 _ 2tt* 



and 2tt- a 3 31 



T 4 b 



$ira'b *Trpa*bV MT" 



Oar equation thus becomes 



M = M 3M.jf 

 b a I ~T~' 



or 1 _ 1 3 i/ 



/; a 4 // 



which, since o=a(l— %), where ?/... is the oblateness of the 

 spheroid in this case, gives 



v*=h as) 



The simplicity of this result is very remarkable. 



19. As Hamy has shown, the figure of equilibrium in the 

 general case is not composed of concentric spheroids. In the 

 actual cases of the planets it is not so for another reason, 

 to wit : that in the Sun, Jupiter, and Saturn, different 

 latitudes rotate at different rates. As their equatorial spins 

 are the fastest, this tends to flatten them more in mid- 

 latitudes than would otherwise be the case. But the want of 

 a spheroidal figure does not affect the limits of the oblateness, 

 for in any case the spin must decrease to nothing at the poles. 



20. Evaluating now r} 2 for the several planets we find for 

 their oblateness at the lower limit, corresponding to central 



