19-21] 



Statical Systems 



23 



physical phenomenon which will shew itself as //. passes through its value at 

 Q will be a complete disappearance of two sets of equilibrium configurations. 

 The results obtained may be shewn diagram matically in the following 

 figures, in which 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. 



Q 



u 



/ 



,'R 



IS 



Q 



P 



(iii) 



\ 



T'I 



21. Suppose that //. changes very slowly in any physical problem, and 

 for definiteness let the direction of change of JLC be that represented by an 

 upward movement in our diagrams. From what has already been said, it is 

 clear that we have the following rule for tracing out the sequence of stable 

 states which will be followed by the system as //. 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 do this we are following a sequence of 

 configurations which is always stable. When it becomes impossible to do 

 this any longer, a value of //, has been reached beyond which no stable con- 

 figurations exist, and when the physical conditions change so that /a 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. 4 (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. In actual fact there may even be 

 more indefiniteness than this ; our figures are two-dimensional diagrammatic 

 representations of (?i+ l)-dimensional spaces, and the line RQT in our figures 

 may very possibly represent a surface in the (n + l)-dimensional space. 



These apparent complications cause no difficulty in actual problems. 

 They arise from the obvious circumstance that a general discussion of stability, 



