MR. A. CAYLEY’S MEMOIR UPON CAUSTICS. 
801 
caustic by refraction of parallel rays, and also two theorems of St. Laurent, Geryonne^ 
t. xviii., viz. if we suppose first that «=c, i. e. that the radiant point is in the circum- 
ference of the refracting circle, then the system (ce) shows that the same caustic would 
be obtained by writing <?, -, 1 (or what is the same thing —1) in the place of c, c, [Jj, 
and we have 
Theorem. The caustic by refraction for a circle when the radiant point is in the 
circumference is also the caustic by refiexion for the same radiant point, and fur a 
refiecting circle concentric with the refracting circle, but having its radius equal to the 
quotient of the radius of the refracting circle by the index of refraction. 
Next, if we write a=CfJb, then the refracted rays all of them pass through a point 
which is a double point of the secondary caustic, the entire curve being in this case the 
orthogonal trajectory, not of the refracted rays, but of the false refracted rays; the 
formula {1} shows that the same caustic is obtained by writing — , c, 1 (or what is the 
same thing —1) in the place of a, c, and we have 
Theorem. The caustic by refraction for a circle when the distance of the radiant 
point from the centre is to the radius of the circle in the ratio of the index of refraction 
to unity, is also the caustic by refiexion for the same circle considered as a refiecting 
circle, and for a radiant point the image of the former radiant point. 
XXX. 
The curve is most easily traced by means of the preceding construction ; thus if we 
take the radiant point outside the refracting circle, and consider /z, as varying from a 
small to a large value (positive or negative values of f/j give the same curve), we see 
that when is small the curve consists of two ovals, one of them within and the other 
Pig. 14. 
without the refracting circle (see fig. 14). As ui increases the exterior oval continually 
