274 
ME. A. CAYLEY’S iVIEMOIE UPOX CAUSTICS. 
Write for shortness, 
(d— /3)(^— a) — (a— ci)(}}— (3) = VQGy 
(a—ci)(^—a)-j-(d—l3)(f! — 13)= OQGS, 
then V QGN is equal to twice the area of the triangle QGN, and if instead of being 
the coordinates of a point Q on the incident ray were cuiTent coordinates, the equation 
V QGN = 0 would be the equation of the line through the points G and N, ? . e. of the 
normal at the point of incidence ; and in like manner the equation □ QGN = 0 would 
be the equation of the line through G pei'pendicular to the line through the points G 
and N, i. e. of the tangent at the point of incidence. 
We have 
NG'=(a-a)^+(^-f3)^ 
and therefore identically, 
V(^^'+nQGN'. 
Suppose for a moment that <p is the angle of incidence and (p' the angle of reflexion 
or refraction ; and let be the index of refraction (in the case of reflexion ,«/= — 1), then 
writing 
and 
we have 
{b—^)(x—a) — {a—a){y—(3)=VqG'S 
(a — a)(x — a)-\-{h — /3)(?/ — /3)= □§'GN, 
VQGN . , VoGN . 
“NG.Gfy ’ 
and substituting these values in the equation 
sin^ sin^ <p' — 0, 
we obtain 
qG" VQGN'-^a^ QG' VgGN'=0, 
an equation which is rational of the second order in s, y, the coordinates of a point q on 
the refracted ray ; this equation must therefore contain, as a factor, the equation of the 
refracted ray ; the other factor gives the equation of a line equally inclined to, but on 
the opposite side of the normal ; this line (which of course has no physical existence) 
may be termed the false refracted ray. The caustic is geometrically the envelope of 
the j)air of rays, and for flnding the equation of the caustic it is obidously convenient to 
take the equation of the two rays conjointly in the form under which such equation has 
just been found, without attempting to break the equation up into its linear factors. 
It is however interesting to see how the resolution of the equation may be effected ; 
for this purpose multiply the equation by NG^, then reducing by means of a prerious 
formula, the equation becomes 
(V^GN'+D^GN') VQGN'-g,^(VQGN'+ °QGN') 
