462 
A SUPPLEMENTARY MEMOIR ON CAUSTICS. 
[359 
16. In the equation JJ = 0 of the curve, writing x — m = 0, the equation becomes 
4 (y 2 — l) 3 + 4 in 2 (y 2 — l) 2 = 0, 
that is 
4 (2/ 2 — i) 2 (2/ 2 — 1 + rri 2 ) =0, 
and the line (x — m) = 0 is thus a double tangent to the curve, touching it at the 
points x = m, y— +1, and besides meeting it at the points x = m, y = ± Vl — m 2 , that 
is, at the intersections of the line x — m = 0, with the circle x 2 + y 2 = 1. 
17. The maximum or minimum values of y correspond to the values 0 = lir, 
0 = f 7r, 0 — \7r, 0 = of 0 ; and we have for 
0 = \ir, x = V2, 
# = f 7T, x = — \^2, 
0 = f 7T, x = — \^2, 
0 = \iT, x = £V2, 
2/ = V2 — m, 
y = V2 + m, 
y = — V 2 — m, 
y = — V2 + m. 
18. It is now easy to trace the secondary caustic; we may without loss of 
generality assume that m is positive, and the values to be considered are 
m = 0, m= 1, 2, m = |, 
with the intermediate values m>0<l, &c. ... and m>f. I have for convenience 
delineated in the figure only a portion of each curve, viz. the figure is terminated at 
the negative value x = — \^2, which corresponds to the maximum value y = V2 + m; 
as x increases negatively, the value of the ordinate y diminishes continuously from 
this maximum value, becoming = 0 for the value x = — 2 —m, and the curve at this 
point cutting the axis of x at right angles; this is a sufficient explanation of the 
form of the curves beyond the limits of the figure. Moreover the curve is symmetrical 
in regard to the axis of x, and I have within the limits of the figure delineated 
only one of the two halves of the curve. 
19. For m > | the cusps are both imaginary, the nodes both real, but one of 
them is an isolated point or acnode (shown in the figure by a small cross). The 
curve has an interior loop, as shown in the figure, and there is also the acnode lying 
within the loop. 
For m — f, there is still an interior loop, but the acnode has united itself to the 
loop, the point of union, although presenting no visible singularity, being really a 
triple point equivalent to a node and two cusps. And in all the cases which follow 
there are two real cusps.