J\IE. A. CATLET’S MEMOIR UPON CAUSTICS. 
291 
XXI. 
The equation of the caustic by reflexion of a circle is 
Suppose flrst that 3^=0, we have 
{(4a^- 0, 
i. e. 
X- 
— a a 
'2a+f 
or the curve meets the axis of x in two points, each of which is a triple point of inter- 
section. 
Write next x^-\-y^=a^, this gives 
{(4a^—l)a^—2ax—a^y=0, 
and consequently 
x= — a(l~2a^) 
y = + 2a ^\/ 1 — a^, 
or the curve meets the cii’cle x^-^-'f — 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, 
-\-]ffx—2*l a^y\x^ )^ = 0 ; 
and rejecting the factor {x^-\-y‘^f^ this gives 
(4a^-l)V-f {(4a^-l)^-27a^}y^-6(4«^-l)^«.r=0; 
or reducing, 
(4a*— 1)V— (1 — — 6(4a^— ; 
and it follows that there are two asymptotes, the equations of which are 
(4«2_l)t 
y=- 
V \~a^[Sa^-\-\)y 4«2-1 
X- 
3a 
Represent for a moment the equation of one of the asymptotes by y=A{x—a), then 
the perpendicular from the origin or centre of the reflecting circle is Aa-^/v/lfi-A*, and 
A«= 
3a V 4a^ — 1 
, ,, (I-a^)(l + 8a^)^+(4a^-l)^ 27 
(l-a2)(l + 8a2)2 “(1 — a' 
(l-a®)(l + 8a2)2 
\/l+A* — - 
3 \/3a 
Vl-a^{l + 8a^) 
and the perpendicular is — = \/ 4a*— 1, which is less than a if only a*<l, i. e. in every 
V o 
case in which the asymptote is real. 
2 q2 
