the caustic is an epicycloid in which
the radius of the fixed circle is twice that of the rolling circle
(fig. 2). The geometrical method is also applicable when it is
required to determine the caustic after any number of reflections at a
spherical surface of rays, which are either parallel or diverge from a
point on the circumference. In both cases the curves are epicycloids;
in the first case the radii of the rolling and the fixed circles are
a(2n - 1)/4n and a/2n, and in the second, an/(2n + 1) and a/(2n + 1),
where a is the radius of the mirror and n the number of reflections.

[Illustration: FIG. 1. c = a]

[Illustration: FIG. 2. c = [oo]]

[Illustration: FIG. 3. c = (1/3)a]

The Cartesian equation to the caustic produced by reflection at a
circle of rays diverging from any point was obtained by Joseph Louis
Lagrange; it may be expressed in the form

{(4c^2 - a^2)(x^2 + y^2) - 2a^2 cx - a^2 c^2 }^3 =
= 27a^4 c^2 y^2 (x^2 + y^2 - c^2)^2,

where a is the radius of the reflecting circle, and c the distance of
the luminous point from the centre of the circle. The polar form is
{(u + p) cos 1/2[theta]}^2/3 + {(u - p) sin 1/2[theta]}^2/3 = (2k)^2/3,
where p and k are the reciprocals of c and a, and u the reciprocal of
the radius vector of any point on the caustic. When c = a or = [oo]
the curve reduces to the cardioid or the two cusped epicycloid
previously discussed. Other forms are shown in figs. 3, 4, 5, 6. These
curves were traced by the Rev. Hammet Holditch (_Quart. Jour. Math._
vol. i.).

[Illustration: FIG. 4. c = 1/2a]

[Illustration: FIG. 5. c > a]

_Secondary caustics_ are orthotomic curves having the reflected or
refracted rays as normals, and consequently the proper caustic curve,
being the envelope of the normals, is their evolute. It is usually the
case that the secondary caustic is easier to determine than the
caustic, and hence, when determined, it affords a ready means for
deducing the primary caustic. It may be shown by geometrical
considerations that the secondary caustic is a curve similar to the
first positive pedal of the reflecting curve, of twice the linear
dimensions, with respect to the luminous point. For a circle, when the
rays emanate from any point, the secondary caustic is a limacon, and
hence the primary caustic is the evolute of this curve.

[Illustration: FIG. 6. a > c > 1/2a]

Ca