[145 
A MEMOIR UPON CAUSTICS. 
363 
145] 
curve lies 
e joining 
wo equal 
limiting 
> 1, the 
points of 
,t in the 
’ fig- 0- 
r at the 
and the 
igle cusp 
jhe form 
cusp on 
)s at its 
3 equally 
arly the 
on the 
the two 
asymptotes coincide with the axis of x\ one of the branches of the curve has wholly 
disappeared, and the form of the other is modified by the coincidence of the asymptotes 
Fig. 10. a = -—. Fig. 12. a = 1. 
V 2 
with the axis of x; it has in fact acquired a cusp at infinity on the axis of x (see 
fig. 12). When a < the curve consists of a single finite branch, with two cusps on 
the axis of x, and two cusps at the points of intersection with the circle through 
the radiant point; one of the last-mentioned cusps will be outside the reflecting circle 
as long as a>^; fig. 13 represents the case a = ^, for which this cusp is upon the 
reflecting circle. For a < i, the curve lies wholly within the reflecting circle, one of 
the cusps upon the axis of x being always within, and the other always without the 
circle through the radiant point, and as a approaches 0 the curve becomes smaller 
and smaller, and ultimately disappears in a point. The case a negative is obviously 
included in the preceding one. 
Several of the preceding results relating to the caustic by reflexion of a circle 
were obtained, and the curve is traced in a memoir by the Rev. Hamnet Holditch, 
Quarterly Mathematical Journal, t. I. [1857, pp. 93—111]. 
46—2