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 
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