298 
ME. A. CAYLEY’S ^^lEMOIE UPON CAUSTICS. 
in which z may be considered as the variable parameter ; hence the equation of the 
caustic may be obtained by equating to zero the discriminant of the above function of z; 
but the discriminant of a sextic function has not yet been calculated. The equation 
would be of the order 20, and it appears from the result previously obtained for parallel 
rays, that the equation must be of the order 12 at the least ; it is, I think, probable that 
there is not any reduction of degree in the general case. It is however practicable, as 
will presently be seen, to obtain the tangential equation of the caustic by refraction, 
and the curve is thus shown to be only of the class 6. 
XXVII. 
Suppose that rays proceeding from a point are refracted at a circle, and let it be 
required to find the equation of the secondary caustic : take the centre of the circle as 
origin, let c be the radius, |, n the coordinates of the radiant point, a, j3 the coordinates 
of a point upon the circle, jm. the index of refraction ; the secondary caustic will be the 
envelope of the circle, 
+ — - {(I— =0, 
where a, j3 are variable parameters connected by the equation a^+j8^— c®=0 ; the equa- 
tion of the circle may be written in the form 
— (f + + c") — 2(|!A"a- — |)a 
But in general the envelope of Aa-1-B(3+C=0, where a, (3 are connected by the equation 
— c^=0, is c^A^+B^) — c^=0, and hence in the present case the equation of the 
envelope is 
which may also be written 
If the axis of x be taken through the radiant point, then ^=0, and uniting also i=a. 
the equation becomes 
{ fJ!^%x^ -\-if —c^)—a^-\-c^Y=: {{x—ay-\-y-]-, 
or taking the square root of each side. 
{ljJ^{x‘^ -\-y‘^ ~c^)—a^ -yd^] =2cyj\/ {x-af-^-y ^ ; 
whence multiplying by 1 — ^ and adding on each side we have 
or 
which shows that the secondary caustic is the Oval of Descartes, or as it will be con- 
venient to call it, the Cartesian. 
