171, 172] Points of Bifurcation 191 
point are all turned in the same direction, this direction being that of 
W -decreasing. 
Suppose, for instance, that in fig. 17 W increases as we pass upwards, and 
that the concavities for all sections of the TF-surface through P 1 are turned 
in the same direction as that shewn in the diagram. Then the configuration 
represented by the point P 1 will be one of stable equilibrium. 
On passing along a series such as PQS in fig. 17 or 19, one of the sections 
must clearly change the direction of its concavity as we pass through the 
point Q at which a kink is first formed on the TF-surfaces. Thus configurations 
which were initially stable give place to unstable configurations on passing 
through points such as Q. Thus we see that a principal series such as PQS 
loses its stability on passing through a point of bifurcation. 
If Pi,P 2 , P 3 represent stable configurations in fig. 17, the concavities of 
all the curvatures at these points must be turned downwards. The same is 
then true at the points R lt R 2 , R 3 and T lt T 2} T 3> so that the configurations 
represented by R li R a ,R 3 and T 1 } T 2 ,T 3 will also be stable. Thus stability, 
which leaves the principal series PQS at Q, may be thought of as passing to 
the branch series RQT. Thus we see that there is an exchange of stabilities 
at the point of bifurcation Q. 
In fig. 19, on the other hand, we find that if the configurations represented 
by P lt P 2 , P 3 are stable, then those represented by R u R 2 ,R 3 and T l ,T 2 , T 3 
will be unstable, in addition to those represented by S lf S 2> S 3 . In this case 
there is a disappearance of stability at the point of bifurcation Q. 
In fig. 18, it is clear that if P 1} P 2 , ... are stable, then U u U 2 , ... must be 
unstable; while conversely if U u U 2 , ... are stable, then P 15 P 2 ,... must be 
unstable. Thus in moving along a linear series a loss or gain of stability 
occurs on passing through a point such as Q at which y is a maximum. But 
in a physical problem, yu, will continually change in the same direction, and 
the physical phenomenon which will accompany the passing of p, through 
its value at Q will be a complete disappearance of two sets of equilibrium 
configurations. 
These results are shewn diagrammatically in the following figures, in which