22 General Dynamical Principles [CH. n 



two linear series such as P 1 P 2 Q and U^.^Q coalesce in the point Q and then 

 disappear: It will be convenient to refer to a point such as Q in this figure 

 as a " turning point." 



Still a third possibility is shewn in fig. 3 ; this however is only a variant 

 of fig. 1, and again leads to two linear series crossing one another in a point 

 of bifurcation Q. Other minor variations may occur, but the principal possi- 

 bilities are those shewn in figures 1, 2 and 3. 



Stability and Instability 



20. Every point on a linear series is a configuration of equilibrium ; the 

 equilibrium may be stable or unstable. Confining our attention to any one 

 of the planes /JL = cons, the condition that a particular configuration of equi- 

 librium in this plane shall be stable is that the value of W at the point in 

 question shall be a minimum. Hence, for stability, the concavities of the 

 different vertical sections of the TF-surface through this point must all be 

 turned in the same direction, and this direction must be that of TF-decreasing. 



Suppose for instance that in fig. 1 W increases as we pass upwards, and 

 suppose 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 con- 

 figuration represented by the point P will be one of stable equilibrium. 



On passing along a series such as PQS in fig. 1 or 3, it is clear that one 

 of the sections must change the direction of its concavity as we pass through 

 the point Q at which a kink is first formed on the T7-surfaces. Thus con- 

 figurations which were initially stable give place to unstable configurations 

 on passing through Q. It appears that a principal series such as PQS loses 

 its stability on passing through a point of bifurcation. 



In fig. 1, it is clear that if P lt P 2 , P 3 represent stable configurations, then 

 the configurations represented by R l} R 2) R 3 and T ly T 2 , T s will also be stable. 

 Thus stability, which leaves the principal series PQS at Q, may be thought 

 of as passing to the branch series EQT. Thus there is an exchange of 

 stabilities at the point of bifurcation Q. 



In fig. 3, on the other hand, it appears that if the configurations repre- 

 sented by P lt P z , P 3 are stable then those represented by R l} R 2 , R 3 and 

 T,, T 2 , T 3 will be unstable, in addition to those represented by S lt S 2 , S s . In 

 this case there is a disappearance of stability at the point of bifurcation Q. 



In fig. 2, it is clear that if P lt P 2 , ... are stable, then U lt U 2 , ... must be 

 unstable; while conversely if U l , U 2 , ... are stable, then P lt P 2 , ... must be 

 unstable. Thus in moving along a linear series there is a loss of stability on 

 passing through a point such as Q at which ^ is a maximum. But in a 

 physical problem, //, will continually change in the same direction, and the 



