368 
A MEMOIR UPON CAUSTICS. 
[145 
XXIX. 
The preceding formulae, which were first given by me in the Philosophical Magazine, 
December 1853, [124] include as particular cases a preceding theorem with respect 
to the caustic by refraction of parallel rays, and also two theorems of St Laurent, 
Gergonne, t. xviii., [1827, pp. 1—19] viz. if we suppose first that a=c, i.e. that the 
radiant point is in the circumference of the refracting circle, then the system (a) shows 
that the same caustic would be obtained by writing c, -, 1 (or what is the same 
thing — 1) in the place of c, c, g, and we have 
Theorem. The caustic by refraction for a circle when the radiant point is in the 
circumference is also the caustic by reflexion for the same radiant point, and for a 
reflecting 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 = cg, 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 
... c 2 
formula (8) shows that the same caustic is obtained by writing ~, c, 1 (or what is 
the same thing — 1) in the place of a, c, g 
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 reflexion for the same circle considered as 
a reflecting 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 g as varying from 
a small to a large value (positive or negative values of g give the same curve), we 
see that when g is small the curve consists of two ovals, one of them within and 
the other without the refracting circle (see fig. 14). As g increases the exterior oval 
continually increases, but undergoes modifications in its form ; the interior oval in the 
first instance diminishes until we arrive at a curve, in which the interior oval is reduced 
to a conjugate point (see fig. 15); then as g continues to increase the interior oval 
reappears (see fig. 16), and at last connects itself with the exterior oval, so as to 
form a curve with a double point (see fig. 17); and as g increases still further the