190 
Non-Spherical Masses—Dynamical Principles [ch. vii 
Q in fig. 17. On any surface on which this formation has just occurred, there 
will be three adjacent level points such as Ri, S ly T lt 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. Poincard calls a point such as Q a “point 
of bifurcation.” 
The succession of level points can also be broken—or rather deviated—as 
shewn in fig. 18. In this case, as y increases, two linear series such as P x P 2 Q 
and Ux U 2 Q approach one another and finally coalesce in the point Q and 
then disappear. A point such as Q in this figure may conveniently be described 
as a “turning point.” 
A third possibility, shewn in fig. 19, is only a variant of that shewn in 
fig. 17, and again leads to two linear series crossing one another in a point 
of bifurcation Q. Other minor variations may occur, but the principle possi 
bilities are those shewn in figures 17, 18 and 19. 
Stability and Instability. 
172. Every point on a linear series is a configuration of equilibrium; a 
question which is of the utmost importance in cosmogonical problem is whether 
this equilibrium is stable or unstable. Confining our attention to one particular 
state of the system, and so to one of the planes y = cons, the condition that a 
particular configuration of equilibrium in this plane shall be stable is that 
the value of W at the point in question shall be a minimum. Hence the 
configuration represented at any point on a linear series will be stable if the 
concavities of the different vertical sections of the W-surface through this