516 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



the distance/ will he/dm ; and the cosines of the angles which/ makes with .r, i/, z 



being 



X — x^ y — j/ z — sf 



~T~' ~7~' ~7~' 



the partial attractions, directed inward^ and parallel to x, y, z, will be 



dm {x — y), dm {y — y'), dm {z — z') ; 



and, by integrating, the sums of the like attractions of all the molecules of the mass 

 are obtained, viz. 



xfdwj — fi^dm, yfdm—fy'dm, %f dm — f t! dm. 



Now, by the property of the centre of gravity, we have 



f x' dm = 0, fy' dm = 0, fz' dm = : 



wherefore, the attractions of the whole mass respectively parallel to x, y, z will be 



equal to 



x/dm, yfdm, zfdm, 



or simply to .r, 3/, a, because/ dm'v^, the unit of mass. 



Let g denote the centrifugal force at the distance 1 from the axis of rotation, and 

 estimated in parts of the unit of force ; then the action of this force urging the par- 

 ticle in the prolongation of y and z will be equal to zy and zz. 



Now, if X, Y, Z denote the whole accelerating forces acting parallel to x, y, z, we 



shall have 



X = ^, Y= {\-i)y, Z={\-z)z', 



which forces are therefore known functions of the point of action. Representing the 

 intensity of pressure by jo, we obtain 



— dp = X dx -{• (1 — g) . (y dy -\- zdz) ; 



and, by integrating, 



C-p = ^ + (l-.) .^, 



which equation determines the pressure at the interior points of the fluid. The equa- 

 tion of the figure of the mass in equilibrium is obtained by making p = 0, viz. 



Supposing, therefore, that g is less than 1, or that the centrifugal force at the 

 distance 1 from the axis of rotation is less than the attraction of the mass collected 

 in a point at the same distance, the fluid in equilibrium will have the form of an 

 oblate elliptical spheroid of revolution. 



As this problem is different from the first only in the law of attraction, it may be 

 alleged that the methods of solution should be similar. There would be no difficulty 

 in applying to it the same investigation employed in the first problem ; but in what- 

 ever manner we proceed, the distinction between the two cases will remain unchanged. 



