169 -m] Points of Bifurcation 189 
particular points in this plane determined by equations such as (169‘7) will 
represent the configurations of equilibrium in this physical state. 
From its meaning the function W must be a single valued function of 
0 U 0 2 , and fx, so that the surfaces W = cons, in the three-dimensional space 
are necessarily non-intersecting surfaces. The condition that a configuration 
shall be one of equilibrium, as expressed by equations (169'6), is exactly 
identical with the condition that the tangent to the surface W - cons, shall 
be perpendicular to the axis of ¡i. If we make the axis of [x vertical, the 
configurations of equilibrium are represented by the points at which the 
tangents to the surfaces W =cons. are horizontal. We may speak of these as 
“level points.” Each level point represents a configuration of equilibrium 
corresponding to one value of the parameter /x. On joining up a succession 
of level points we get a line such that points on it represent configurations 
of equilibrium for different values of /x. Such a succession of configurations 
forms a “linear-series.” 
We have illustrated the meaning of our terms by using a simple system 
with only two co-ordinates 0 ly 0 2 , but the method is quite general. If the 
system has n co-ordinates we represent its configurations, as fx changes, in an 
imaginary space of (n + 1) dimensions, and soon reach the same result. 
As the value of any parameter fx changes, the configuration of the system 
will change, and so long as the system remains in equilibrium, its various 
configurations will lie on a linear series. 
Points of Bif urcation. 
171. As we pass along a linear series, the regular succession of points 
representing configurations of equilibrium may be broken in various ways. 
One obvious way is by a change in the direction of curvature of the IF-surfaces, 
resulting in the formation of a kink, such as is shewn occurring at the point 
R S T