702 Prof. P. Lowell on the Limits of the 



This being true for any attracting particle is true for their 

 aggregate. 



5. Sinlilarly in the case of the centrifugal force. Since 

 the part of it in the direction of x 



4tt 2 



IS " rp2 "'i 



47T 2 



that in y, rf^y, «-&<! that in 2 = 0, 



we have for the whole 



4?r 2 

 Xdx + Ydy + Zdz = - r . vz (xdx + ydy), 



2-7T 2 



and its integral = -rf^-(# 2 -|-t/ 2 )- 



Lastly, if p be a function of the forces, then it, too, becomes 

 immediately integrable. In this case dpi* an exact differential 

 and equilibrium is assured. A particular case of this is 

 where p — constant. 



6. Now the extremes possible for the law of density dis- 

 tribution in a rotating body are when its derivative is and 

 when it is co . 



In the first case /£> = constant and the body is homogeneous. 



7. To find the oblateness in this case we may take two 

 canals, the one from the centre to the pole, the other from 

 the centre to the equator. For equilibrium the pressures 

 due to the two must be equal. In this case it has been proved 

 that the figure of equilibrium is a spheroid by Maclaurin, 

 and extended by d'Alembert. 



The attraction of a homogeneous spheroid for a particle at 

 its pole is 



. 7 /l \/l — e 2 .sm- 1 e\ /K , 



=M? 7 — r ■ ■ • (5) 



where 6 = the semi minor axis 



^=the eccentricity of the generating ellipse 



v 



and a = the semi major axis. 



