88
same, all the angles of refraction must likewise be equal, and the line , must, like the radius of curvature, vary as ; the triangle will be every where similar to itself, so that the angle will always be of a given magnitude, and since touches the caustic in , this curve, having the angle between the radius vector and tangent constant, must be an equiangular spiral, with for a pole.
The constant angle may easily be calculated by means of tables, for if be a tangent, we have Then being found in terms of by the formula for (p. 82.) we have two sides and the included angle of the triangle to determine another angle.
115.We may now pass on to the caustics formed by rays refracted twice, such as those passing through a sphere or lens.
Let (Fig. 125.) be the centre of a sphere of glass into which rays enter parallel to the diameter .
We have seen (p. 66.) that the principal focus is at the distance of a radius and a half from the centre: the extreme ray is refracted into the direction making the sine of two-thirds of the radius, and emerges in a tangent to the sphere at . The form of the caustic is found to be such as having cusps at When the sphere is rarer than the surrounding medium, the form of caustic is such as represented in Fig. 126.[errata 1]
116.When parallel rays are refracted through a double-convex lens the form of the caustic is such as shown in Fig. 127 and 128, in the latter of which the caustic touches the lens, as on account of its great thickness the extreme rays do not pass through it.
A double-concave lens gives a caustic of a similar form but on the other side, (Fig. 129 and 130.).
As to other cases of the caustics given by the sphere and by lenses, it is not worth while to dwell upon them as the subject is of little or no practical utility.
It will, however, be as well to make one or two remarks connected with this subject.
117.Let (Fig. 131.) be a refracting surface, the caustic, the extreme rays touching the caustic inErrata
