On Systems of Rays. 95 
Condition of Intersection of Two Near Final Ray-lines. Conical Locus of the 
Near Final Points, in a variable medium, which satisfy this condition. Investi- 
gations of Matus. Jllustration of the Condition of Intersection, by the. Theory 
of the Auxiliary Paraboloid, for Ray-lines from an Oblique. Plane. 
19. Returning now to the system of final ray-lines (C™) from an oblique plane 
(B"), let us consider the condition necessary in order that one of these near final 
ray-lines (C’) may intersect the given final ray-line or axis of z. This condition 
may be at once obtained by making x and y vanish in the equations (C'”), and then 
eliminating z; it may therefore be thus expressed, 
8 By. (8, B\s 
an. (2 an fe 57) oe 45 5 + q swt 
oa oa oa oa 
=yy.4 (E+ pZ)w+ (E+ aut, (Q*) 
or more concisely thus, on account of the equation of the oblique plane (B"), 
8B... Bs. Bo, 
dw, (Sax +5. ay +5. 8) 
=ty. (Baw + Say +% 82) 3 (R*) 
that is, 
da 88 = dy 8a; (S"*) 
it is therefore necessary and sufficient, for the intersection sought, that the near final 
point B’ should be on a certain conical locus of the second degree, determined by 
the equation (#"), between the co-ordinates dx, dy, dz. A conical locus of this kind, 
appears to have been first discovered by Matus. That excellent mathematician and 
observer had occasion, in his Zraité D’ Optique, to make some remarks on the gene- 
ral properties of a system of right-lines in space, represented by equations of the 
form 
a—xv_y-y _z—2 
i me 
> 
in which m, m, 0, are any given functions of the co-ordinates 2’, y/, z', of a point 
through which the line is supposed to pass, and by which it is supposed to be deter- 
mined ; and he remarked that the condition of intersection of a line thus determined, 
with the corresponding near line from a point infinitely near, was expressed by an 
equation of the second degree between the differentials of the co-ordinates 2’, y', 2’, 
which might be considered as the equation of a conical locus of the second degree for 
the infinitely near point. The theory of systems of rays which was given by Matus, 
differs much, in form and in extent, from that proposed in the present Supplement ; 
