of a homogeneous fluid mass that revolves upon an axis. 101 
by the action of all the forces that urge its particles. Let 
x, y, z, be three rectangular co-ordinates of a point K in the 
surface ; and put X, Y, Z, for the accelerating forces that act 
upon a particle at K respectively in the directions of the co- 
ordinates, and tending to diminish these lines. Suppose that 
K varies its position a little in the fluid’s surface ; then the 
condition that the resultant of the forces parallel to the co- 
ordinates, shall be perpendicular to that surface, is expressed 
by this equation, viz. 
Xdx-\-Ydy-\-Zdz = o. 
In order that the equilibrium be possible, the expression 
just set down must be a complete differential ; which subjects 
the forces X, Y, Z, to the criterion of integrability. This 
condition being fulfilled, the equation of the fluid’s surface 
will be, 
f(Xdx + Ydy + Zdz) =C, 
C being an arbitrary constant quantity. If, for the sake of 
brevity, we represent the preceding integral by <p, we shall 
have, 
<pz= c 
__ d (p Y __ ^ ® 2 d V 
dx’ dy’ dz 
Again, let 
y> = -/X 2 - Y’-f-Z*; 
t hen / is the resultant, of the forces X, Y, Z ; and it acts on 
a particle placed at K, in a direction perpendicular to the 
fluid’s surface, and tending inward. It is the gravity at that 
point. 
Suppose now that a stratum of fluid is laid upon the sur- 
face HKI, the thickness at K being equal to the indefinitely 
small line K S. The new pressure at K will be proportional 
