ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
285 
rco 
\x=±iy, s 
1 
!/=±p’ 
(1 + 8 /^") 
x=-\ 
^/‘-i 
— 3 'v/.S/x. 
I 
V "T 
each of the points of the first system being a quadruple point of intersection, each of the 
points of the second system a triple point of intersection, and each of the points of the 
third system a single point of intersection. 
The points of intersection with the axes of x, and the points of triple intersection with 
the circles x^-\-y^=l and x^-\-y^=^, are all of them cuspidal points; the two circular 
points at co are, I think, triple points, and the other two points of intersection with the 
line oo , cuspidal points, but I have not verified this : assuming that it is so, there will be 
a reduction 54 accounted for in the class of the curve, but the curve is, in fact, as will be 
shown in the sequel, of the class 6 ; there is consequently a reduction 72 to be accounted 
for by other singularities of the curve. 
XIV. 
It is ob\ious from the preceding formulae that the caustic stands to the circle radius 
- in a relation similar to that m which it stands to the circle radius I, ^. e. to the 
refracting cu’cle. In fact, the very same caustic would have been obtained if the circle 
radius ^ had been taken for the refracting circle, the index of refraction being ^ instead 
of !«,. This may be shown veiy simply by means of the iiTational form of the equation as 
follows. 
The equation of the caustic by refraction of the circle radius I, index of refraction |tx/, 
is, we have seen, 
( I — = ( I — (Jiff + I —(/T^y^f- 
Hence the equation of the caustic by refraction of the circle radius c\ index of refrac- 
tion lO.', is 
or, what is the same thing. 
-(/J^d 
which becomes identical with the equation of the first-mentioned caustic if (jtJ=d= — 
Hence taking c instead of I as the radius of the first circle, we find, — 
Theorem. The caustic by refraction for parallel rays of a circle radius c, index of 
