366 
A MEMOIR UPON CAUSTICS. 
[145 
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 / c 2 \ . a? ( c 2 Y c 2 
by —11 -) and adding on each side — 1 - J + ~ g {(# — a) 2 + y 2 }, the equation 
fX \ CL / fX \ CL J CL 
becomes 
{(• - if + A-6 ''(—«)■+»■+1 (i -1)}‘. 
or extracting the square root, 
\!- S) ■ 
Combining this with the former result, we see that the equation may be expressed 
indifferently in any one of the four forms, 
* - if + ' y ' = a ~ “ )S + y’ + K“ " f ) ' 
-3 + *-? 
/¿v p a 
c 2 \ 2 
X I 
a 
) + y' + {~ a+ ~a)^ [“-"A) + ti‘ + A~ + = 
It follows, that if we write successively 
a' — a , 
c = c , 
/ 
= /* 
/ c 2 
/ c 
c 
a = ~ , 
c — —, 
= — 
a 
a 
. a 
/ 0 
/ 
1 
a = —, > 
C =- , 
= — 
, a 
a 
a — a , 
c =- , 
c 
, _ c 1 
/ 
/ 
CfJi 
(.1 — , 
a 
c = c , 
/* 
a 
, a 
/ Cl 
a 
a = -5 , 
c = - . 
/*' 
— — 
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