312 
ME. A. CAYLEY’S JLEMOIE UPON CALSTICS. 
In the case of reflexion, /x= — 1, and the equation becomes 
4(^+2/^ — 1)^— 27^7^= 0. 
Comparing this with the equation of the caustic, it is easy to see. 
Theorem. In the case of parallel rays and a reflecting circle, there is a secondary 
caustic which is a curve similar to and double the magnitude of the caustic, the position 
of the two curves differing by a right angle. 
XLI. 
The entire system of the orthogonal trajectories of the refracted rays might in like 
manner be determined by finding the envelope of the circle (where, as before, a. (3 are 
variable parameters connected by the equation 
a)"+(y— /3)^} —(a-j-my=0. 
1 The result, as far as I have worked it out, is as follows, \’iz. — 
(3 — 12 [nf + %n\^x + + 2 /^)] + [1 — + 2m^ — 
— ([1 — 2jm»^+ 2m^ — 2^^(a^+3/®)][9 + 18?n® + 36mjU/^^+ 1 +^^)] 
— 5 4 [m^ + — 3 ^*)] — [1 — ~ 2|W-^(^ 4-2/^) = 0 5 
which, it is easy to see, is an equation of the order 8 only. Added Sept. 12. — A. C.] 
