138 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
brium, the first of these forces, namely, the resultant of the centrifugal force 
and the attraction of the mass a b c, is perpendicular to the surface a b c, and 
destroyed by the resistance of the fluid within that surface ; and from this it 
follows that the attraction of the stratum upon a m, must likewise be perpen- 
dicular to the same surface. For, if it acted obliquely to the surface a b c, it 
might be resolved into two partial forces, one perpendicular, and the other 
parallel, to the plane touching the surface ; and as there is no obstacle to op- 
pose the latter force, it would cause the molecule am to move, which is con- 
trary to the equilibrium of the whole mass ABC. It appears therefore that 
two distinct and independent conditions are required for the equilibrium of 
the fluid mass : for all the particles situated in any interior surface abc simi- 
lar to the outer surface, and similarly posited about the centre of gravity G, 
must be urged perpendicularly to the surface in which they are contained, not 
only by the resultant of the centrifugal force and the attraction of the interior 
mass, but likewise by the attraction of the exterior stratum of fluid. 
Conceive three planes intersecting at right angles in the centre of gravity 
of the mass, one of them being perpendicular to the axis of rotation P Q : let 
x, y , z represent the coordinates of the molecule a m, and r = J* 2 + y 2 + * 2 , 
its distance from G, x being parallel to P Q ; and put V (r) for the sum of the 
quotients of all the molecules of the whole mass ABC, divided by their re- 
spective distances from am : further, let V' (r) denote the same thing relatively 
to the interior mass a b c, that V (r) does relatively to the whole mass ABC: 
then V (r) — V' (r) will be the sum of the quotients of all the molecules of 
fluid contained in the stratum between the two surfaces, divided by the re- 
spective distances of the molecules from a m. According to the known pro- 
perties of this function, the partial attractions of the stratum upon am, in the 
directions of x, y, z, and tending to lengthen these lines, will be respectively 
equal to 
d . (V (r) — V' (r)) d. (V (r) - V' (r)) d. (V(r) - V' (r)) 
dx ’ dy dz 
Take any point {x dx, y + dy, z + dz) in the surface a b c, at the infinitely 
small distance d s from a m : then the resultant of the foregoing attracting 
forces in the direction of ds will be equal to 
