122 Mr. Ivory on the figure requisite to maintain the equilibrium 
upon one another. We are therefore to conclude, that the 
two conditions of the proposition are sufficient to ensure the 
equilibrium of the fluid mass, and that both are necessary to 
produce the effect. 
5. Having established the physical properties of a homo- 
geneous fluid mass in equilibrio, the investigation of its figure, 
which is now brought within the power of analysis, is not 
attended with much difficulty. 
If a homogeneous body of fluid revolve about an axis, and 
be in equilibrio by the centrifugal force and the attraction of 
its particles, the axis of rotation will pass through the centre 
of gravity. This point is to be supposed at rest ; since it is 
not the effect of any external forces that we have to consider, 
but merely the mutual action of the particles upon one 
another. If one of the planes of the co-ordinates pass through 
the centre of gravity at right angles to the axis of rotation, 
the other two will intersect one another in the same axis. Let 
a, b, c, denote the co-ordinates of an attracted point ( which 
must be considered as some small particle of the fluid con- 
taining a given quantity of matter) placed any where in the 
mass, a being parallel to the axis of rotation ; and put 
r — V a + b z -j- c % . Suppose also that V ( r) denotes the sum 
of all the molecules of the body divided by their respective 
distances from the attracted point : then - -T,— , 
1 da do dc 
will be the accumulated attractive forces exerted upon the 
attracted point by the whole mass, in the directions of a, b, c, 
and tending to increase these lines. Again ; let u be the 
centrifugal force at the distance from the axis of rotation 
equal to unit ; then, the distance of the attracted point from 
the same axis being equal to vV-fi- , the centrifugal force 
