200 Non-Spherical Masses—Dynamical Principles [ch. vn
but changes as the moment of inertia of the mass changes; if the motion
is referred to axes rotating with a uniform velocity, the rotation of the freely
rotating mass may lag behind that of the axes and the relative co-ordinates
x, y, z may increase without limit although the configuration remains stable.
It is therefore important to express the conditions of stability in a form
which does not involve the constancy of <w.
When the mass is rotating freely in space, G = 0 in equation (177*3) so
that M is constant. The elimination of &> from equations (176*7) and (176*8)
leads to
T=T S +
M 2
27’
where
JJ 2
Using the values of T R , U and 1 already obtained in equations (176*4) to
(176*6), we find that
2 IT S = [Swj (a?! 2 + y x 2 )] [2 w 2 (x£ + y 2 2 4- i 2 2 )]
- [tm 1 0æ iÿi - y x x i)] [Swia (x 2 y 2 - y 2 x 2 )]
= l^m x m 2 [(^¿2 + y 2 y x y + (x 2 x x + y x y 2 y
+ («iÿ 2 - xéù 2 + (« 2 ÿi - + ¿i № + y?) + ** + 2/i 2 )].
This expression, being a sum of squares, is necessarily positive. Thus, since
I is necessarily positive, T s is always positive. Since I does not involve
x, y, i, T 8 of course is quadratic in x, ÿ, z.
The equation of energy, T+W— cons., now assumes the form
r. + r+“ ! = c° n s (183-1).
This is of the same form as the former equation (181*2), T s replacing T R
and W + M 2 /2 1 replacing W — -|<u 2 /. By the argument already used in § 181,
it appears that configurations for which
M 2
F + ^ (183*2)
is an absolute minimum (M being kept constant) will be thoroughly stable,
while configurations for which this expression is not an absolute minimum
will be secularly unstable, and may or may not be ordinarily unstable.
184. The last two sections have shewn that as we pass along a linear
series of configurations of equilibrium of a rotating system, starting from a
part of the series which is known to be stable, the configurations will become
secularly unstable as soon as
W — |co 2 / (o> = constant) (184*1)
or W + |M 2 // (M = constant) (184*2)
ceases to be an absolute minimum, the former expression referring to the
case in which the mass is compelled by external forces to rotate at a constant
