358 
A MEMOIR UPON CAUSTICS. 
[145 
Represent for a moment the equation of one of the asymptotes by y = A (oc — a), 
then the perpendicular from the origin or centre of the reflecting circle is Aa a V1 + A 2 , 
and 
Aa — 
l+A 2 = 
Vl + A 2 = 
3a V4a 2 — 1 
VT^(l+8a 2 )’ 
(1 - a 2 ) (1 + 8a 2 ) 2 + (4a 2 - l) 3 
(1 - a 2 ) (1 + 8a 2 ) 2 
3 V.3a 
27a 2 
(1 — a 2 ) (1 + 8a 2 ) 2 ’ 
VI - a 2 (1 + 8a 2 ) 
and the perpendicular is ^gV4a 2 —1, which is less than a if only a 2 < 1, i.e. in every 
case in which the asymptote is real. 
The tangents parallel and perpendicular to the axis of x are most readily obtained 
from the equation of the reflected ray, viz. 
(— 2a cos 6 + 1)« + 
a cos 26 — cos 6 
sin 6 
y + a = 0 ; 
the coefficient of x (if the equation is first multiplied by sin 6) vanishes if sin 6=0, 
1 \/4a 2 — 1 
which gives the axis of x, or if cos 6 = —, which gives y = ± ^ , for the tangents 
AiCt Ld 
parallel to the axis of x. 
The coefficient of y vanishes if a cos 26 — cos 6 = 0; this gives 
cos 6 = 1 i V ^ a + 1 sin 6= (4a 2 — 1 + V8a 2 + 1), 
4a 8 a 2 v 
and the tangents perpendicular to the axis of x are thus given by 
_ — 2a 
"-TTv^rr 
these tangents are in fact double tangents of the caustic. In order that the point of 
contact may be real, it is necessary that sin 6, cos 6 should be real; this will be the 
case for both values of the ambiguous sign if a > or = 1, but only for the upper 
value if a < 1. 
It has just been shown that for the tangents parallel to the axis of x, we have 
V 4a 2 — 1 
y = ± 
2 a 
the values of y being real for a > \: it may be noticed that the value y = 
V 4a 2 — 1 
2a