290 
ME. A. CAYLEY’S ^^IEMOIE TJPOX CAUSTICS. 
which give the coordinates of a point of the caustic in terms of the angle A which deter- 
mines the position of the point of incidence. The values in question satisfy, as they 
should do, the equation 
We have, in fact. 
^ 4 «3(cose-«)3 _ 
from which it is easy to derive the equation in question. 
XX. 
If we represent the equation of the reflected ray by 
XA‘-fY?/-f «=0, 
then we have 
and thence 
X= — 2a cos d-{-l 
„ «cos2fl— cos9 
sin fl ’ 
(X— 1)^— 4a^= —4a® sin® 0 
X®+Y®=^v^ (1 — 2a cos ^+a®) 
X-f-«^=l— 2acos ^+a®, 
and consequently 
(X®-fY®){(X-l)®-4a®}+4a®X+4a^=0, 
or, what is the same thing, 
{X(X-l)-2a®}®-fY®{(X-l)®-4a®} = 0, 
which may be considered as the tangential equation of the caustic by reflexion of a circle ; 
or if we consider X, Y as the coordinates of a point, then the equation may be considered 
as that of the polar of the caustic. The polar is therefore a curve of the foiudh order, 
having two double points deflned by the equations X(X— 1)— 2a®=0, Y=0, and a thu’d 
double point at infinity on the axis of Y, i. e. three double points in all ; the number of 
cusps is therefore 0, and there are consequently 4 double tangents and 6 inflections, and 
the curve is of the class 6. And as Y is given as an explicit function of X, there is of 
course no difficulty in tracing the curve. We thus see that the caustic by reflexion of a 
circle is a curve of the order 6, and has 4 double points and 6 cusps (the cii’cular points 
at infinity are each of them a cusp, so that the number of cusps at a finite distance is 4) : 
this coincides with the conclusions which will be presently obtained by considering the 
equation of the caustic. 
