of a homogeneous fluid mass which revolves upon an axis. 123 
urging it from the axis, will be « xV^H - a51( ^ the ei ^' eCL 
of the same force to lengthen b and c, will be equal to 
a 
x Vb * + c’ x 
and 
VVTS " X " " "T - * y/F 
and u c. Hence, the forces acting upon the attracted point, 
and tending to increase a, b, c, are respectively, 
d. V(r) 
da ’ 
d.V(r) , , 
-IT 1 + " ' 6 > 
d . V (r) 
. 
c x 
or to « 6 
db 
— j— CO • Cm 
Now, the resultant of these forces must be perpendicular to 
the level surface of which a , b, c are the co-ordinates ; which 
condition is expressed by this differential equation, viz. 
<LXS±da+ dc + u(bdb + cdc) = o : 
da 1 a 0 a c 
and the integral of this, viz. 
V(r) + -f(6 2 + r 2 ) = C, 
is the general equation of all the level surfaces. Let p de- 
note the cosine of the angle which the line r makes with the 
axis of rotation ; and the foregoing equation will become, 
V(r)+i.r*( i-^) = C. 
And if we put R for the radius of the outer surface of the 
fluid body, we shall have, for the equation of that surface, 
V(R) + H-xi(i-ri = C,...(A) 
which is one condition of the equilibrium of the fluid mass. 
The equation just found is an essential condition of the 
equilibrium, although it is not the only one. As it merely 
expresses that the resultant of all the accelerating forces is 
perpendicular to the fluid’s surface, it is not confined to a 
homogeneous body, nor to one entirely fluid, but is true in 
