1534 
are the loci of constant slope of the solutions of (109), while 
the dotted trajectories are the solutions themselves for various 
arbitrary starting points (or boundary conditions) of *X, and R. 
Examination of Figure 9 shows that all the trajectories, 
regardless o* their initial starting point, tend at larger values 
of R to approach very close to a certain central trajectory 
which has been denoted as the “equilibrium trajectory." It seems 
reasonable therefore, in view of the lack of any well knom, a 
priori boundary condition, to select this equilibrium trajectory 
ae the solution of equation (109) which is appropriate to our 
particular problem. 
Thus, differentiating equation (110) we have along 
any of the loci of constant slope: 
Rien! © fteidx =o 
oa AR 
(111) 
But for the solution we seek, ax /dR in (111) is given by 
equation (109). Therefore 
113 $x) _ Ftx)]1- sa 2uG 
oT RE. 
(112) 
is the equation of the equilibrium trajectory, and we could 
obtain % as a function of R by graphical solution of (112) at 
a@ succession of points. 
For practical purposes, it is more convenient to solve 
equation (112) once in order to obtain an appropriate starting 
