192 
Non-Spherical Masses—Dynamical Principles [ch. yii 
thick lines represent series of stable configurations, and thin lines series of 
unstable configurations, the series PQ being assumed to be stable in every case 
173. Suppose that in any physical problem /x changes very slowly, the 
direction of change being that represented by an upward movement in our 
diagrams. From what has already been said, it is clear that the following rule 
will trace out the sequence of stable states which will be followed by the 
system as /Lt varies. 
Start from a configuration in the diagram which is known to be stable 
and follow a path along linear series of equilibrium so as always to move 
upwards, and so as always to cross over from one series to another at a point 
of bifurcation. So long as we can do this we are following a sequence of con 
figurations which is always stable. When it becomes impossible to do this any 
longer, a value of /x has been reached beyond which no stable configurations 
exist, and if the physical conditions continue to change so that /x attains to a 
still higher value, the statical problem gives place to a dynamical one; it is 
no longer a question of tracing out a sequence of gradual secular changes, but 
of following up a comparatively rapid motion of a cataclysmic nature. 
At each point of bifurcation there is necessarily a certain amount of 
indefiniteness in the path which will actually be followed. For instance, in 
fig. 20 (i), the system on arriving at Q may proceed either along QT or along 
QR, both being equally consistent with the maintenance of stability, and so 
far as can be seen equally likely. 
This complication causes no difficulty in actual problems. It arises from 
the obvious circumstance that a general discussion of stability, although 
competent to determine when stability ceases, cannot in general determine 
what will happen after stability has ceased. A general discussion of stability 
will readily shew that a top spinning slowly on its point is in unstable equi 
librium, but it cannot determine in which precise direction the top will first 
fall to the ground. 
174. In his classical paper* in which the theory of linear series and points 
of bifurcation was first developed, Poincare used analytical methods to obtain 
results identical with those just given. 
If /x is the only parameter which can vary, the potential energy W of the 
system may be written in the form 
0«* ••• 0n, p) 
and the configurations of equilibrium are given by the equations 
••• 0», /0-0, etc. (174-1). 
* l.c. ante.