A MEMOIR UPON CAUSTICS. 
365 
[145 
Take 
coordi- 
y the 
writing 
145] 
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 order 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, £, y the coordinates of the radiant point, a, /3 the coordinates 
of a point upon the circle, yu the index of refraction; the secondary caustic will be 
the envelope of the circle, 
y 2 {(x - a) 2 + (y - /3) 2 } - {(I - a) 2 + (y- /3) 2 } = 0, 
where a, /3 are variable parameters connected by the equation a? + ¡3 2 — c 2 = 0 ; the 
equation of the circle may be written in the form 
y 2 (x 2 + y 2 + c 2 ) - d 2 + y 2 + c 2 ) - 2 (y?x - f) ct - 2 (y 2 y - v ) ¡3 = 0. 
But in general the envelope of Aa + B/3 + C =0, where a, ¡3 are connected by the 
equation a 2 +/3 2 — c 2 = 0, is c 2 (A 2 + B 2 ) — G 2 = 0, and hence in the present case the equation 
of the envelope is 
[y 2 (x 2 + y 2 + c 2 ) — d 2 + y 2 + c 2 )} 2 = 4c 2 {(y 2 x — |) 2 + (y?y — ??) 2 }, 
which may also be written 
[y? (x 2 + y 2 — C 2 ) — d 2 + y 2 — G 2 )) 2 = 4c 2 y? {(x — f) 2 + (y — y) 2 }. 
If the axis of x be taken through the radiant point, then y =0, and writing also 
% = a, the equation becomes 
{,y 2 (x 2 + y 2 — c 2 ) — a 2 + c 2 } 2 = 4c 2 //. 2 {{x — a) 2 + y 2 \; 
or taking the square root of each side, 
{y, 2 (cc 2 + y 2 — c 2 ) — a 2 + c 2 } = 2c/x V(x — a) 2 + y 2 ; 
whence multiplying by 1 - and adding on each side c 2 (y — -') +(x — ci) 2 + y 2 , we have 
y 2 j(^ - + 2/j = 0* -a) 2 + y 2 + c(y- -^) j , 
or 
y\J [x - ^ + y 2 = \/(x - a) 2 + y 2 + c , 
which shows that the secondary caustic is the Oval of Descartes, or as it will be con 
venient to call it, the Cartesian.