OF EQUILIBRIUM OF A ROTATING I^IASS OF LIQUID. 
;3I5 
Ps'Qs' = ■ 6 
Sin® 7 
K cos^ 7A^ r 1 + /c 
4. 
F- 
2 ( 1 - Fk'-) 
k*k'^ 
E 
, /I + - (1 + «QsiiF7\tan7] 
+ ( UUI j ^'1 • (^'U- 
2 '4. 
K'^K ^ 
In (30) 
/ = |[1 + (1 - -f An 
« 4 - 52 ^ 
In (31) 
</ = i[l + T (1 - 
^ 1-422 
In (32) 
= i[2 -f T (4 - ^)^], 
0 . 4 - 522 
Bifurcation of Jacobi's Ellfsoid. 
If a mass of liquid be rotating like a rigid body about an axis, x, with uniform 
angular velocity w, the determination of the figure of equilibrium may be treated as a 
statical jiroblem, if the mass be subjected to a potential ^or (^3 _{_ zf 
The energy lost in the concentration of a body from a condition of infinite disper¬ 
sion is equal to the potential of the body in its final configuration at the position of 
each molecule, multiplied by the mass of the molecule and summed throughout 
the body. In the proposed system, as rendered a statical one, it is necessary to add 
-|- z^) to the gravitation potential before making the summation. If A denotes 
the moment of inertia of the body about x, this latter portion of the sum is ^Aor, and 
is therefore the kinetic energy of the system. 
If dnii, dm^ denote any pair of molecules and distance between them, and 
E the energy lost, we have 
(o~ 
If the system had been considered as a dynamical one, the expression for the 
energy of the system, say U, would have resembled that for E, but the former of 
these terms would Iiave presented itself with a negative sign. 
It is clear that the variation of ^A(J, when the moment of momentum is kept 
constant, is equal and opposite to the variation of the same function when the 
angular velocity is kept constant. 
The condition for a figure of equilibrium is that U shall be stationary for constant 
moment of momentum, or E stationary for constant w, in both cases subject to the 
condition of constancy of volume. The variations in question lead to identical results, 
and I shall proceed from the variation of E. 
2 s 2 
