17-19] 



Statical Systems 



21 



R 



let us for brevity call these "level points." On joining up a succession of 

 level points, such as P,, P 2 , P 3 in fig. 1, we obtain a "linear series."- 



Points of Bifurcation 



19. The regular succession of such points as we pass along a linear 

 series may be broken in various ways. One obvious way is by a change in 

 the direction of curvature of the TF-surfaces, resulting in the formation of a 

 kink, such as is shewn occurring at 

 the point Q in fig. 1. On any surface 

 on which this formation has just 

 occurred, there will be three ad- 

 jacent level points such as R, S lt T^ 

 in the figure. The original linear 

 series PQ will accordingly become 

 replaced by three linear series such 

 as QR, QS and QT as soon as we 

 pass above the point Q at which 

 the kink first forms. It is readily 

 seen that at Q two of the series 

 QR and QT must run continuously 

 into one another, and so in effect 

 form a single new series, while the 

 series QS may be regarded as a 

 continuation of PQ. We may accordingly suppose that there are two linear 

 series PQS and RQT crossing one another at the point Q. A point such as 

 Q is called by Poincare a " point of bifurcation." 



Another and more usual way in which the succession of level points can be 

 broken or rather deviated is shewn in fig. 2. In this case, as /x- increases, 



Fig. 1. 



Fig. 2. 



