of a homogeneous fluid mass that revolves upon an axis. 119 
Proposition III. 
If a homogeneous fluid mass fulfil the two above-men- 
tioned conditions, it is in equilibrio. 
Let the equation of the outer surface of the fluid mass, be 
<P = C ; 
<p representing a function of three rectangular co-ordinates, 
x, y, z. Then the accelerating forces parallel to the axes of 
x,y,z, will be respectively equal to '•> an< ^ the con- 
dition that the resultant of these forces is perpendicular to 
the surface of the fluid will be expressed by the differential 
equation, 
gdx + ¥/y + £dz = o. 
We shall suppose that the whole fluid mass is divided into 
thin strata by level surfaces, which are determined by making 
the constant quantity C vary by insensible degrees in the 
equation of the outer surface. Farther, let the thickness of 
the uppermost stratum be denoted by the line k, drawn per- 
pendicular to the outer surface from a point of which the 
co-ordinates are x,y,z ; and let x — S x, y — Sy , z — S z, be 
the co-ordinates of the other extremity of k in the under sur- 
face of the stratum. The equation of this surface will be 
found by substituting x — $ x, y — Sy, z — Sz, in place of 
x,y, z, in the function <p, and by changing C into C — SC ; 
it will therefore be, 
— — — ijz = C — fC: 
T ax ay J at 
and by subtracting this from the equation of the outer sur- 
face, we get 
