Ohlateness of a Rotating Planet. 707 



infinitely dense matter of infinitely small volume constituting 

 a finite mass. The curve expressing this distribution, instead 

 of being a straight line parallel to the axis of abscissae, as for 

 homogeneity, is asymptotic to the two axes of coordinate. 



Here again the forces are all functions of the distance, and 

 p is constant till it becomes infinite at the centre, a mere point, 

 being throughout a function of the distance. Throughout 

 the volume, therefore, the conditions for equilibrium hold. 



15. To find the resulting oblateness in this case we shall 

 proceed as follows : — 



Suppose as before two canals from surface to centre, the 

 one polar, the other equatorial. That equilibrium should 

 exist between the two it is necessary and sufficient that the 

 pressures produced by the fluid in these two canals should be 

 equal. This will give us the polar and equatorial diameters 

 and consequently the oblateness, whether the figure of equi- 

 librium be a spheroid or not. 



The pressure produced by a portion dp of the polar caual 

 at the pointy is 



p . dp, 



where P is the accelerative force, the section of the canal 

 being supposed unity. 



16. Let u denote the whole pressure at the point jo. Then, 

 since p is reckoned from the centre, an increase of distance 

 dp decreases the pressure by P. dp. Whence 



du = — P . dp* 



Since all the matter is supposed condensed at the centre, 



p= 



M 

 p- 



du = 



M , 

 - ., dp, 

 V 



w = 



*+c. 



and 



whence 



The pressure is nothing at the surface, therefore 



or 



C = 



M 



V 



2 Z 2 



