To locate the tangential image of any point B situated on the incident ray, we drop perpendiculars from C onto the two sections of the ray, striking them at D and D0, respectively. Then the point of intersection of DD0 and EE0 is the point O. The line DD0 is found to be perpendicular to EE0. This point O replaces C when graphically locating the tangential image of B, so that the line from B to O crosses the refracted ray at the tangential image, while the line from B through the center of curvature C locates the sagittal image.

The proof of this is difficult. It is best to assume that the angle y in triangle BDO is equal to the angle y in triangle BPT; then the geometry of the two triangles leads to the regular Coddington equation for the tangential image.

11.1.5Astigmatism for the Three Cases of Zero Spherical Aberration

In Section 6.1.1 it was pointed out that a single spherical surface contributes no spherical aberration when the object is at (a) the surface itself, (b) the center of curvature of the surface, and (c) the aplanatic point. In Section 9.2.1 it was shown that the OSC also is zero for these three object points.

By means of the Coddington equations it is easy to show that at small obliquity the astigmatism contribution will be zero in cases (a) and (c), but when the object is at the center of curvature the astigmatism contribution is large and in the unexpected sense—that is, the convex front surface of a positive lens, for instance, contributes positive astigmatism when we would ordinarily have expected it to lead to an inward-curving field. This result is often of great significance, and it explains many anomalies, such as the flat tangential field of a Huygenian eyepiece.

11.1.6 Astigmatism at a Tilted Surface

If a lens surface is tilted through a given angle, the procedure outlined in Section 2.6 can be used to trace the principal ray, and the ordinary Coddington equations can be used to locate the astigmatic images along the principal ray. However, because of the asymmetry, the astigmatism at some angle, say 15 , above the axis will not be the same as the astigmatism at 15 below the axis, and to plot the fields it is now necessary to trace several principal rays with both positive and negative entering obliquity angles.

As an example, we will refer ahead to the design of a Protar lens (Section 14.4), and pick up the principal-ray data at several obliquities. We will next suppose that the rear lens surface has been tilted clockwise through an angle of 0.10 (6 arcmin), so that a ¼ 0.1. By comparing the field curves given in Figure 11.6