On Systems of Rays. 119 
These two separating planes P’ P” contain each the ray-line or element of the 
path (A, B, C) at B ; and they divide the near points of less from those of greater 
action, or those of shorter from those of longer time, when the continuous integral 
V+V,=V (A, C) ts not greater than all, or less than all, the adjacent values 
of the sum = fvds. ‘The directions of these planes depend on the positions of the 
points 4, B, C; so that if we consider 4 and B as fixed, but suppose C to move 
along the prolongation (8, C’) of the path (4, B), the separating planes P’, P’, 
will in general revolve about the ray-line at B. They will even become imaginary, 
when by this motion of C' the quantity @ becomes > instead of < 0, so as to satisfy 
the condition of existence of a maximum or minimum of the function V+ V”,; and 
in this transition from the real to the imaginary state the two separating planes P’ P” 
will close up into one real transition-plane P, determined by either of the two follow- 
ing equations, 
eV oe ot B 
oS ho ge) (8x — 82) + “Gag oo bz), 
2 4 (O”) 
OeiGeapih =r (ee 2 8z) + Gor 
Ny 
8), 
while the corresponding position of the und C, which we may call by analogy a 
transition-point, will satisfy the ee 
a 
Q=0, that is, Geet 
gl OnVeona 
1 yy = Goat oe = 
We are now prepared to us te a remarkable connexion between the transition- 
planes and transition-points to which we have been thus conducted by the considera- 
tion of the maxima and the minima of the function V+ V,, and the condition of 
final and initial intersection of two near luminous paths. For these conditions of 
intersection may be obtained by supposing that not only the point B, having for co- 
ordinates x y z, is on a given path (4, C’), so as to satisfy the equations (D"), but 
that also an infinitely near pot B’, having for co-ordinates x +éx, y +éy, x +8z, is 
on another path of the same colour connecting the same extreme points 4 and C, so 
as to give the differential equations 
ov 3V.. . 8V SV, ; 
= 2 f — ete ee (Q”) 
and since these last equations may be reduced, by the relations (G@"), to the forms 
(0"7), we see that when the conditions of initial and final intersection of a given path 
(A, B, C) with a near path (4, B’,C) are satisfied, and when we consider the initial 
point A as fixed, the near intermediate point B’ must be in a transition-plane P of the 
form (O"), and the final point of intersection C must be a transition-point of the form 
(P"). Continuing therefore to regard the initial point A as the fixed origin of a system 
