EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
423 
Appendix. 
On the Energy and Stability of the System. 
M. Poincare has shown in his admirable memoir, referred to in the Summary, how the dynamical 
stability of a rotating fluid system in relative equilibrium depends on the energy. Certain factors in 
the expression for the energy, which he calls coefficients of stability, are there proved to afford the 
required criterion. 
It will now be shown how in this case these coefficients of stability are determinable, at least as far as 
spherical harmonic analysis permits. The results will also cast an interesting light on the methods by 
which the equations to the two masses have been obtained. 
The task before us is to determine the “ exhaustion of potential energy ” of the two masses in 
presence of one another as due to the deformation of each from the spherical figure by yielding to 
gravitation and to centrifugal force. 
The work will be rendered simpler by the introduction of a new notation. Let ns write, then, as the 
equations to two shapes, which are not necessarily together a figure of equilibrium :— 
1 = 1+2 — 
ft jfc=2 2/i 
T> 00 
? = 1 + 
A ic= 
k + 1 fA\s /cU*- 
nic 
m 
"A2H1MUU« f Ar IU_ P s*W t+ „ ] 
Tc-i 2k — 2 \c / \o) 1 li k B k J 
(i-) 
It will be observed that these equations have the same form as (72), but that the constants introduced 
are different from the h, l, m, e , which were determined, so that the figui’es might be figures of equilibrium. 
At present we do not assume that (i.) do represent figures of equilibrium. 
The energy lost may be divided into several parts :— 
e 1? the energy lost by the mass a yielding from its spherical figure to the gravitation of the mean 
sphere a. 
e 2 , the exhaustion of mutual energy of that layer of matter on the mass a which constitutes its 
departure from sphericity. 
e g , the loss of energy due to the deformation of the mass a in presence of the mean sphere A. 
JZ), B o, I? 3 , the similar quantities for the mass A. 
(Ue) 4 , the loss of mutual energy of the two layers in presence of one anothei\ 
e 5 , the loss of energy due to the deformation of the mass a in the presence of centrifugal force due 
to rotation tv. 
_Z? 5 , the similar loss for A. 
1st. e x is equal and opposite to the work required to raise each element of the layer on n through half 
its own height against the gravity due to the mean sphere a. This gravity is -wu. Co-latitude and 
longitude being denoted by 0, 0, let dsr = sin 0 clO d<fi, an element of solid angle. In effecting the inte¬ 
grations, the properties of spherical harmonic functions are used without comment, viz.:— 
4tt 
2F+ r 
k + 2 ! 
W{Wjc dvr = 0, 
Ik + 1 2 . k - 2! ’ 
'w/ c + j dvr = 0. 
j Wi Sh 
WJc C>"lVje + o 
dw — 
Then, taking only a typical term of the first of (i.), 
e i = 
dvr 
2 k + 1 o 
(2k - 2) 2 ' 
1c + 2! 
2 .k — 2! 
