On Systems of Rays. 117 
Let A, B, C; be three successive points, at finite intervals, on one common lumi- 
nous path. Let the rectangular co-ordinates of these three points be a’, 7/, 2 for 4; 
x, y,2for B; ands, y, z,for C. Let V (A, B) denote the integral /vds taken 
from the first point 4 to the second point B; let Y (B,C) denote the same integral, 
taken from the second point B to the third point C; and similarly, let Y(4, C) be 
the integral from 4 to C, which is evidently equal to the sum of the two former, 
V (A, C)=V (A, B)+V (B, C), CA) 
so that, if we put for abridgment 
Mes, BV, V(b, ©)= Vi; (B”) 
we shall have, by the continuity of the integral, 
Vi (A, C)E=V+V,. (C”) 
If we do not suppose that the intermediate point B is a point of sudden reflexion or 
refraction, the final direction of the part (4, B) will coimcide with the initial direc- 
tion of the part (B, C), and the final direction-cosines a B y of the one part will be 
equal to the initial direction-cosines of the other ; considering V therefore, as usual, 
as a function of wy z x’ y' z' xy, and V,as a function of x,y, 2,“ y z x, we have, by 
our fundamental formula (4), 
pipemediee el, 18K 280 ot OV ie Leo a) OP ey 
2 Bee Se a. ge a Sele 7 
Sean Gus ace? gy SG ay, ees Pope ee cp") 
that is, we have 
8V+8V,=0, (E") 
for any infinitesimal variations of the co-ordinates x y z, and therefore, to the accuracy 
of the first order, 
V (A, BY+V(B, C)=V (A, B)+V(B, C)=V (4,6) (FO 
B’ being any new intermediate point infinitely near to B, and the path (B’, C) being 
not in general a continuation of the path (4, B’). If therefore we regard the 
extreme points 4, C, as fixed, but consider the intermediate point B as variable and 
as not necessarily situated on the path (4, C’), the function V+V_, or 2 fvds, 
composed of the two partial and now not necessarily continuous integrals (B”), will 
acquire what may be called a stationary value, when the paths (4, B) (B, C) become 
continuous, that is, when the intermediate point B takes any position on the path 
(A, C) from one given extreme point to the other: since then the change of this 
function will be infinitely small of the second order, for any infinitely small alteration 
BB, of the first order, in the position of the point B. The stationary value thus 
determined, namely, / (A, C), might be called, by that customary latitude of expres- 
VOL. XVII. 2H 
