14 
PROFESSOR CAYLEY’S SUPPLEMENTARY MEMOIR ON CAUSTICS. 
16. In the equation U = 0 of the curve, writing x—m— 0, the equation becomes 
that is, 
4(y-l) 3 +4m 2 (/-l) 2 =0, 
%*-!)*(/— i+*» a )=o, 
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=+\/l—rri l , that is, at the 
intersections of the line x—m= 0, with the circle x*-\-y*= 1. 
17. The maximum or minimum values of y correspond to the values 0— 0=^p 
0=^-5 0=x °f 0 5 and we have for 
4 4 
«=!■ 
s* 
II 
Mm 
<s 
bO! 
y— \/2— m, 
4 
*=-K/2, 
y= n/ 2 +m, 
0 = — , 
4 
x— ^x/2, 
y=—\/2—m. 
e=4 
x= i\/2, 
y=-*/2+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, m=\/ 2, m= f, 
with the intermediate values m>0<l, &c. ...and w>§. 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=\/ 2+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 > f 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. 
