16 
PROFESSOR CAYLEY’S SUPPLEMENTARY MEMOIR ON CAUSTICS. 
For ni= 1, the double tangent has the two coincident real intersections y—0, or it is 
in fact a triple tangent. 
For in < 1 > 0, the double tangent has with the curve two real intersections, viz. they 
are the points where the double tangent meets the circle 
And finally, for m— 0, the points in question unite themselves with the points of 
contact, the double tangent #=0 being in this case the common tangent at the two 
cusps #=0, y= + l. 
Added May 13, 1867.— A. C. 
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 cilrves ; 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. 
