## Ogx

g V l in which g is the distance from the Cassegrain focal plane, again defined as positive. Because E was originally defined by

T7I 1 yPr

in which H is the Lagrange-Helmholtz Invariant, the power of a thin lens only changes the metric of the equations by a scaling factor of the focal length. Since this affects all linear quantities equally, this has no effect on the relations derived below.

We can now consider the conditions under which all four aberrations of Eqs. (4.64) can be corrected. We note first of all that, while from (4.52) the expressions (S/)l and (S//)l for a thin lens are relatively complex, those for (S///)l and (S/y)l are very simple:

in our normalized system; and

where Kl is the power of the lens and n' the refractive index of the glass. (It should be emphasized that these are the "central" aberrations arising if the stop is placed at the thin lens). Furthermore, from Table 3.5

in which we have again

H2 = f '2 from the normalization, giving m2 /m2 + 1

We now apply (4.64), (4.65), (4.66) and (4.67) to the case of a classical Cassegrain telescope with a parabolic primary. Since Z and £ are zero, the correction of all 4 aberrations requires

(S///)cor = —f'(— Ef' ' and, substituting for E from (4.34)

From (4.65), if the corrector is a single lens

g for the condition for correction of astigmatism in a classical Cassegrain. For the correction of field curvature, (4.66), (4.67) and (4.68) give m-2 ( m2 + 1"

Combining (4.70) and (4.71) gives the condition for correcting all four aberrations in a classical Cassegrain with such a single thin lens:

This assumes, of course, that it is possible for the lens to contribute (S/)cor = 0 and (S//)cor = 2f' from (4.68). In fact, it is normally not possible, as was recognised in 1949 by Wynne [4.23] - we shall return to this point later. Consider first the condition (4.72). If we take g = f'/35, as assumed above for an aspheric plate corrector, and the values f' = +1, m2 = —4 and L = +0.225 from Table 3.2, then (4.72) gives

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