HOMOGENEOUS FLUID AT LIBERTY. 505 



rotation : and / being the distance between the assumed particle and the molecule 

 dm, we shall have 



/ = Jipc - a^f + (3/ - yr + (^ - z')\ 



The attraction of spheres, according to the law supposed in this problem, being 



the same as if all the matter were collected in the centre, we may adopt for the unit 



of mass a sphere of the fluid having its radius equal to the unit of distance ; and for 



the unit of force, the attraction of the sphere upon a point in its surface ; then the 



direct attraction of the molecule d m upon the particle at the distance f, will be -j^ ; 



and the partial attractions urging the particle inward in the directions of ^, y, ;s, will 

 be respectively equal to 



dm X — x' dm y — 1/ dm z — s^ 



'P ' ~7^' 'T ' ~f~' 7^ ' ~7~' 



Now if we observe that 



X — a^ df y —y' df z — z' df 



~~J ~" 'dx' f ~ T^' f ~ rf7' 



it will readily appear that the sums of the attractive forces, with which all the mole- 

 cules of the mass urge the particle inward in the respective directions of <r, y, z, may 

 be thus commodiously expressed : 



, /%dm - pdm , pdm 



"■St J-Jt JJt 



dx ' dy ' dz * 



the integral extending to all the molecules of the mass. 



The attraction of the sphere at its surface being represented by unit, the velocity 

 communicated by that force in the infinitely short time d t, will hel X dt; and if 

 the time of one entire revolution about the axis of rotation be denoted by T, the 

 velocity generated by the centrifugal force at the distance of unit from the axis in the 



time dt, will be -^^ X dt\ wherefore the centrifugal force acting at the distance of 



unit from the axis of rotation, and estimated in parts of the unit of force, will be 

 equal to 



rpa — S. 



At the distance of ^y^ + z^ from the axis, the centrifugal force will therefore be 

 g X s/y^ + «^ ; and the resolved parts of it which urge the particle in the prolon- 

 gations of y and z, will be equal to s x y and z X z. 



Now if X, Y, Z represent the total forces tending inward and urging the assumed 

 particle in the directions of the coordinates, we shall have 



