contained in the square bracket of (3.126) and representing the limit case of £ in (3.41) when bs2 = 0, has very interesting properties. If we set f (—2) = +1, then (3.128) gives
Apart from the solution —2 = ±ro, given above from Eq. (3.128) for the limit case of a Mersenne-type, afocal telescope with a spherical secondary, Eq. (3.129) has two other roots —2 = —1.618034 and —2 = +0.618 034, the first corresponding to the Cassegrain (normal DK) case, the second to the Gregory case. The first value is the famous magic number, t, whose remarkable geometrical properties were first recognised by Pythagoras and led to the Fibonacci series in plant growth [3.19] [3.20]. The second root is, accordingly, simply the reciprocal of the first.
Another important limit case is given by —2 = ±1, whereby f (—2) = 0. From (3.126) the resulting form of the primary is parabolic: this must be the case since the spherical secondary has no optical power and has become a folding flat in the Cassegrain case (Fig. 2.15). The Gregory limit case with —2 = +1 was discussed in § 2.2.5 and corresponds to a secondary concentric with the primary image, a useless case in practice because of detector obstruction.
Figure 3.5 shows the complete function f (—2). The left-hand curve refers to the normal Cassegrain DK case. It has a very flat maximum at —2 = —3, but the function is virtually constant between +1.10 and +1.185 for the entire range of practical —2 values. From Eq. (3.126), this means the form of the primary is largely determined by the obstruction ratio Ra in all practical cases. Magnifications numerically less than unity give in both Cassegrain and Gregory cases a very steep rise to ro at —2 = 0; but practical systems in this range are uncommon. The unit values at —2 = —t and +1/t mean that Eq. (3.127) also applies in these cases: the asphericity of the primary depends strictly only on the obstruction ratio Ra. So the magic number also retains its powerful link with geometry in this optical case.
The arms of the function refer to the Cassegrain on the left and the Gregory on the right only in the normal case of a real final image. For virtual images they invert these roles, but such cases are rarely of practical significance.
We must now consider the field coma of the DK telescope. From Eq. (3.60) or Table 3.5, again neglecting the second term because of the condition
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