464 
A SUPPLEMENTARY MEMOIR ON CAUSTICS. 
[359 
In all the above forms the double tangent x — m touches the curve at the points 
y = ± 1, but the other two intersections of the double tangent with the curve are 
imaginary. 
For m = 1, the double tangent has the two coincident real intersections y = 0, or 
it is in fact a triple tangent. 
For m<l>0, the double tangent has with the curve two real intersections, viz. 
they are the points where the double tangent meets the circle x 2 + y 2 = l. 
And finally, for m = 0, the points in question unite themselves with the points of 
contact, the double tangent x = 0 being in this case the common tangent at the two 
cusps x = 0, y=± 1. 
Added May 13, 1867. 
20. As remarked in the original memoir, p. 312, the secondary caustic, in the 
last-mentioned case m =0, is a curve similar to and double the magnitude of the 
caustic itself (viz. the caustic for parallel rays reflected at a circle), the position of 
the two curves differing by a right angle. 
The secondary caustics corresponding to the different values of m form, it is clear, 
a system of parallel curves; and, by the remark just referred to, it appears that this 
system is similar to the system of curves parallel to the caustic for parallel rays 
reflected at a circle.