183-185] Secular Instability 201
rate ay, while the latter refers to the case in which the mass is rotating freely
in space.
The theory of linear series and stability developed in §§ 170—173 will be
exactly applicable to the problem of the secular stability of a rotating mass,
provided W is replaced in the argument of those sections by the appropriate
one of expressions (184T) or (184-2). Secular stability is lost at a “turning
point ” or “ point of bifurcation.” At a turning point stability is lost entirely;
at a point of bifurcation it may be lost or may be transferred to the branch
series through the point according as the branch series turns downwards or
upwards in the appropriate diagram. And, finally, to determine the positions
of “ points of bifurcation ” and of “ turning points ” we need only express
the appropriate one of the two above expressions as a sum of squares, and
the “ points of bifurcation ” and “ turning points ” occur whenever one of the
coefficients vanishes.
For instance, to determine at what stage a mass rotating freely in space
becomes secularly unstable, we calculate W + \ M 2 /I for any configuration
which is arrived at by a small displacement from a configuration of relative
equilibrium, M being kept constant throughout the displacement, and express
the excess over the equilibrium value as a sum of squares in the form
8 (W + £M 2 /J) = £ (b,0* + b 2 0J + ... + b n . x 6> 2 n _i) (184-3).
If the configuration under consideration is secularly stable, W + \ M 2 //
must be an absolute minimum, so that b 1} b 2 , ... b n ^ Y must all be positive.
As the various parameters which determine the physical state of the system
change, the values of b 1} b 2 , ... change. Secular stability persists so long as
b 1} b 2 , ... all remain positive, but is lost as soon as one of them becomes
negative.
Examples of Secular Instability.
185. Some interesting examples of secular instability have been worked
out by Lamb*.
As a first instance consider a spherical bowl, which is made to rotate with
* Proc. Roy. Soc. A, lxxx. (1907), p. 170.
J
3