ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
299 
It is proper to remark, that the Cartesian consists in general of two ovals, one of which 
is the orthogonal trajectory of the refracted rays, the other the orthogonal trajectory of 
the false refracted rays. In the case of reflexion, the secondary caustic is a Cartesian 
having a double point; this may be either a conjugate point, or a real double point 
arising from the union and intersection of the two ovals ; the same secondary caustic may 
arise also from refraction, as will be presently shown. 
XXVIII. 
Reverting to the original form of the equation of the secondary caustic, multiplying 
1 t (P" \ Cp / (P \ ^ P 
by ^ M — ^ j and adding on each side ^(1— } , the equation becomes 
or extracting the square root. 
Combining this with the former result, we see that the equation may be expressed 
indifferently in any one of the four forms. 
It follows, that if we write successively 
<^=a. 
c'=c. 
( 1 ) 
1 
a — — ? 
a 
C = 
A*- 
c 
a 
[a) 
t « 
a =-21 
H- 
f ^ 
C=-i 
H'' 
1 
“ft 
(/3) 
a=a 
a 
a 
c 
(y) 
1 
a=—^ 
a 
C—C, 
CjU. 
a 
(^) 
II 
1 « 
C =-5 
a 
c/x. 
( 0 . 
2 R 2 
