264 The Configurations of Rotating Compressible Masses [ch. ix 
out of a series of stars of varying degrees of compressibility, the wholly 
incompressible mass is the first to become unstable. The physical meaning 
of this result can be seen from the general considerations advanced in § 197. 
A GENERAL THEOREM ON ROTATING MASSES. 
240. This part of our discussion may terminate with a general theorem 
which forms a simple extension of one originally given by Poincar6*. 
Let the motion of a mass which is rotating approximately as a rigid body 
with angular velocity co be referred to axes rotating with angular velocity <a, 
and let u, v, w be the components of velocity, which we assume to be small, 
of any element of the mass relative to these rotating axes. 
The equations of motion of any small element of the mass are three 
equations of the form (cf. equations (224T)), 
du 
dV , 1 dp 
T7 — ~ b ® # f 
dt dx p ox 
.(2401). 
Differentiating these three equations with respect to x, y, z and adding 
corresponding sides we obtain 
9 /1 9 p\ 9/1 dp 
d (du dv dw\ _ , 9 /1 dp\ d /1 dp\ 
dt \dx + dy + dz) 7r '^ p W dx \p dx) + dy \p dy) 
+ 
(l 
\p dz) 
IJ (lu + mv + nw) dS = J[J [2a> 2 — ^iryp] dxdy dz — dS, 
d_ 
dx \p dxj ' dy \p dy) ' dz 
Let us multiply by the element of volume dxdydz, and integrate 
throughout the whole of the rotating mass. On transforming the first and 
last integrals by Green’s Theorem, we obtain 
d_ 
dtjj - jjj ~ ■ - - jj p 
where the surface integrals extend over the whole surface of the mass, l, m, n 
are the direction-cosines of the outward normal to this surface at any point, 
and d/dv denotes differentiation along this normal. 
If A is the whole volume of the mass, the integral on the left measures 
the rate of increase of A, and the equation may be written in the form 
~ = (2a,--4 tt 7 p) A-jfi^dS (240-2), 
where p is the mean density of the whole mass. Since p vanishes at the 
boundary of the mass and must be positive at all interior points, dp/dv is 
necessarily negative, so that the last term is necessarily positivé. 
For the mass to be in a state of steady rotation, the left-hand member of 
the equation must vanish, so that we must have 
to 2 < 27T7p (240 3). 
This is Poincaré’s original theorem; whatever the arrangement of the 
mass, a rotation of speed greater than that given by equation (240'3) is 
inconsistent with a steady rotation. 
* Leçons sur les Hypothèses Cosmogoniques, p. 22.