134 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
0 = V(R)+4(/ + * 2 )-C, 
and making the result coincide with the geometrical equation of the figure, we 
shall obtain the expressions of the axes in terms of the centrifugal force. But 
it will be more simple to use the differential equation, 
rf.V (R) 
which expresses the perpendicularity of the forces to the outer surface. The 
QllclIltltlCSj 
rf.V(R) d. V(R) d. V(R) 
dx ’ dy ’ dz ’ 
ire the attractive forces of the ellipsoid, urging a particle of the surface in di- 
rections parallel to the axes ; and these forces, by the nature of the ellipsoid, 
ire proportional to the coordinates of the point on which they act, and may be 
represented by A' x, B ' y, C z, the coefficients A', B', C' being known quan- 
tities depending upon the ratios of the axes of the ellipsoid; wherefore, these 
values being substituted in the differential equation, we shall have, 
A' x d x + (B' — f) y d y + (C' — f) z d z =. 0 ; 
and by integrating, 
W — f c ’ — f 
x 2 + ■ y 2 + -- z 1 = constant. 
Now, if /?, h', h" represent the axes of the ellipsoid, h being that about which 
the fluid revolves, the equation of the surface of the figure will be, 
* 2 + ^ 2 +^ 2 = A 2 ; 
and with this equation the foregoing one must be made to coincide. On ac- 
count of the arbitrary constant, we have only to equate the coefficients of y 2 
and and the resulting formulas may be thus written, 
/=B'-~A' f=C-~A’. 
But, on examining the functions that A', B', C' stand for, it will readily appear 
that the expressions on the right side of the two formulas will not be positive, 
