362 
A MEMOIR UPON CAUSTICS. 
[145 
XXIV. 
The equation 
{(4a 2 - 1) (x 2 + y 2 ) — 2ax — a 2 } 3 — 27a 2 y 2 (x 2 + y 2 — a 2 ) 2 = 0 
becomes when a = co (i. e. in the case of parallel rays), 
(4# 2 + 4y 2 — l) 3 — 21 y 2 = 0, 
which may also be written 
64^6 + 48^ (4y 2 - 1) + 12a; 2 (4y 2 - l) 2 + (8y 2 + l) 2 (y 2 - 1) = 0. 
XXV. 
It is now easy to trace the curve. Beginning with the case a = oo, the curve lies 
wholly within the reflecting circle, which it touches at two points; the line joining 
the points of contact, being in fact the axis of y, divides the curve into two equal 
portions; the curve has in the present, as in every other case (except one limiting 
case), two cusps on the axis of x (see fig. 6). Next, if a be positive and > 1, the 
general form of the curve is the same as before, only the line joining the points of 
contact with the reflecting circle divides the curve into unequal portions, that in the 
Fig. 6. a = oo. Fig. 7. ci> 1. 
neighbourhood of the radiant point being the smaller of the two portions (see fig. 7). 
When a = 1, the two points of contact with the reflecting circle unite together at the 
radiant point; the curve throws off, as it were, the two coincident lines x = 1, and the 
order is reduced from 6 to 4. The curve has the form fig. 8, with only a single cusp 
on the axis of x. 
If a be further diminished, a <1> 
V2 
the curve takes the form 
shown by fig. 9, with two infinite branches, one of them having simply a cusp on 
the axis of x, the other having a cusp on the axis of x, and a pair of cusps at its 
intersection with the circle through the radiant point; there are two asymptotes equally 
inclined to the axis of x. 
In the case a = —=, the form of the curve is nearly the 
v2 
same as before, only the cusps upon the circle through the radiant point lie on the 
axis of y (see fig. 10). The case a<-^>| is shown, fig. 11. For a = \, the two