21 
[ch. ii 
17 - 19 ] Statical Systems 
instants 
roblems 
3 occur, 
spoken 
atisfied, 
noment, 
;he con- 
process 
e speci- 
, the 
aineters, 
liter the 
uantities 
1 change 
tions (7) 
ntinually 
rtions of 
, * ” 
5 . 
sentation 
therefore 
im. The 
5 (7) will 
motion of 
nensional 
at a con- 
is exactly 
:ons. shall 
nk of the 
presented 
orizontal; 
let us for brevity call these “ level points.” On joining up a succession of 
level points, such as P,, P 2 , P ;! 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 l} S l} T 3 
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 /1 increases, 
ide. (Paris, 
Fig. 2. 
Fig. B.