from which it follows that
For the classical Cassegrain given in Case 3 of Table 3.3 with m2 = -4, Eq. (3.93) gives for the form of the secondary bs2 = -2.778, a strong hyperbola. In the classical Cassegrain, the foci of this hyperbola obviously lie at the primary and secondary images I1 and I2 of Fig. 2.12. Since, for a given radius r2 and diameter 2y2, bs2 is a direct measure of the asphericity from Eq. (3.11), it is clear why the manufacture of Cassegrain secondaries, combined with the test problems resulting from its convex form (see RTO II, Chap. 1), remained unsolved for 180 years after the original proposal by Cassegrain.
Similarly, Eq. (3.93) gives for the classical Gregory of Case 7 in Table 3.3, with m2 = +4, an elliptical form for the secondary with bs2 = -0.360. However, the asphericity on the surface is not necessarily smaller than that on the Cassegrain secondary, since the radius r2 is steeper for the same value of m2 because the quantity P in Eq. (2.84) is less in the Gregory solution. The manufacture of the Gregory secondary is easier because its concave form is more convenient for testing - see RTO II, Chap. 1.
If m2 ^ Eq. (3.93) gives bs2 ^ -1, giving the Mersenne telescopes with confocal paraboloids (Table 3.6). For a given diameter and curvature, therefore, the manufacture of the Cassegrain secondary will ease with increase of |m2|. However, if increase of |m2| implies reduction in diameter (obstruction ratio Ra), as for a fixed final image position in the focal case according to Eq. (2.72), then this is not necessarily true, since the asphericity must be achieved over a smaller diameter - the aspheric function becomes "steeper".
The field coma S// for the classical Cassegrain or Gregory is given from Eq. (3.60) and Table 3.5. The last term fr1 (-/' z+LC)
is in all cases zero if the stop is placed at the primary, giving spr1 = 0. But it is anyway zero for the classical Cassegrain because S/ = 0 by virtue of Z = C = 0. This is, again, a statement of the stop-shift formulae of Eq. (3.22). Since any useful telescope must be corrected for spherical aberration, the field coma is always independent of the position of the stop.
For the classical telescope forms, £ = 0 from Eq. (3.91), so that the field coma is given from Eq. (3.60) by
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