278 
ME. A. CAYLEY’S MEMOIE EPOX CAESTICS. 
or reducing, the equation is 
af^ + j/' — 2 y tan 9 + tan^ :p — /f=0. 
Whence, considering tan <p as the variable parameter, the equation of the envelope is 
that is, 
or 
— k ^ k '^ = 0 , 
■f! t 
^ 2 — ^/2 — J - 
is the equation of the secondary caustic or conic ha\ing the caustic for its evolute. The 
radiant point, it is clear, is a focus of the conic. 
VII. 
Let the equation of the refracted ray be represented by 
XzrH-Y3/+Z=0, 
— ^ sin ip 
A . X . — 7 
V 
from which we obtain 
we have 
V 1 — /c^ sin'^ <p 
: 1 : — tan (p. 
X2 — Y2~Z2 
for the tangential equation of the caustic ; or if we represent the equation of the 
refracted ray by 
-f- T ^ ^ = 0, 
then we have 
X2— Y2 — F 
for the tangential equation of the caustic. 
Fig. 1. Fig. 2. 
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