145] 
A MEMOIR UPON CAUSTICS. 
357 
and a third double point at infinity on the axis of F, 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 F 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 circular 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. 
XXI. 
The equation of the caustic by reflexion of a circle is 
{(4a 2 — 1) (x 2 + y 2 ) — 2ax — a 2 } 3 - 27a 2 y 2 (x 2 + y 2 — a 2 ) 2 = 0. 
Suppose first that y = 0, we have 
{(4a 2 — 1) x 2 — 2 ax — a 2 } 3 = 0, 
i.e. 
— a a 
X ~2Y+1’ X ~ 2a —\ ’ 
or the curve meets the axis of x in two points, each of which is a triple point of 
intersection. 
Write next x 2 + y 2 = a 2 , this gives 
{(4a 2 — 1) a 2 — 2 ax — a 2 } 3 = 0, 
and consequently 
x = — a (1 — 2a 2 ), 
y = ± 2a 2 Vl — a 2 , 
or the curve meets the circle a? +y 2 — a 2 = 0 in two points, each of which is a triple 
point of intersection. 
To find the nature of the infinite branches, we may write, retaining only the terms 
of the degrees six and five, 
(4a 2 — l) 3 (x 2 + y 2 ) 3 — 6 (4a 2 — l) 2 a {oc 2 + y 2 ) 2 x — 27 a 2 y 2 (x 2 + y 2 ) 2 — 0 ; 
and rejecting the factor (x 2 + y 2 ) 2 , this gives 
(4a 2 — l) 3 x 2 + {(4a 2 — l) 3 — 27a 2 } y 2 — 6 (4a 2 — l) 2 ax = 0 ; 
or reducing, 
(4a 2 — l) 3 x 2 — (1 — a 2 ) (8a 2 + l) 2 y 2 — 6 (4a 2 — l) 2 ax = 0 ; 
and it follows that there are two asymptotes, the equations of which are 
= (4a‘ - l) j ( _ 3a 1 
Vl-a 2 (8a 2 + l) 1 4a 2 -l)