—' " linn ---i—WF Ij'luMif 
145] 
A MEMOIR UPON CAUSTICS. 
337 
and refracting circle which give rise to identically the same caustic, [see post, xxvill]. 
The memoir is divided into sections, each of which is to a considerable extent in 
telligible by itself, and the subject of each section is for the most part explained 
by the introductory paragraph or paragraphs. 
I. 
Consider a ray of light reflected or refracted at a curve, and suppose that £, y 
are the coordinates of a point Q on the incident ray, a, ¡3 the coordinates of the 
point G of incidence upon the reflecting or refracting curve, a, b the coordinates of 
a point N upon the normal at the point of incidence, x, y the coordinates of a 
point q on the reflected or refracted ray. 
Write for shortness, 
(&-£)(£-«)-(a-a) (y — /3)= V QGN, 
(a-a) ( |-a) + (&-/3) (y -/3) = DQGN, 
then V QGN is equal to twice the area of the triangle QGN, and if f, y instead of 
being the coordinates of a point Q on the incident ray were current coordinates, the 
equation V QGN — 0 would be the equation of the line through the points G and N, 
i.e. of the normal at the point of incidence; and in like manner the equation 
□ Q6rW=0 would be the equation of the line through G perpendicular to the line 
through the points G and N, i.e. of the tangent at the point of incidence. 
We have 
NG 2 = (a-af + (b-l3)\ 
QG 2 =(Z-*T + (v-/3 )\ 
and therefore identically, 
NG 2 • Q(f = V QGN 2 + □ QGN. 
Suppose for a moment that 0 is the angle of incidence and cf>' the angle of reflexion 
or refraction; and let y, be the index of refraction (in the case of reflexion y = — 1), 
then writing 
(b — ,8) (x —a.) — (a —a) (y — /3)= V qGN, 
{a — a) (x — a) + (b - (3) (y — /3) = □ qGN, 
and 
qG 2 = (x - af + (y - /3f, 
we have 
43