(vide fig. 6) --12--
3) The catadioptrical devices (v. fig.2)
To avoid the difficulties in cylindrical deforming of the secondary and to gain full corrected focal lengths, a third optical member in form of a lens is introduced. There are two possibilities, the coma- free mirror-system being the first of it.
To eliminate the overcorrected astigmatism (according to the formulae
7 and 8) we introduce between secondary and uncorrected second focus
a plano-convex-cylindrical lens (axis of the cylinder in the sagittal
diameter of the beam of rays, cylindrical surface against the second-
ary) of such power that the path of the meridional rays will be
shortened by the astigmatical difference k , while the convergence
of the sagittal rays will suffer no change. To eliminate at the same
step the astigmatical difference of focal lengths too (according to
the formulae 11 -17), it is necessary to install the lens in the
plane X, where the diameters of meridional and sagittal rays of the
cone between secondary and focus are equal. This actual plane lies
in a distance a1m from the meridional focus, which is given by
18) (vide fig. 6)
In this formula all values are known by the former calculations.
With aid of a1m and k we compute the necessary power and the
cylindrical r.o.c. of the lens
19)
20)
The focal length f3 results in each case so long (about 35 f1) that we haven't to fear any injurious chromatical or spherical aberration. But exactly in this fact the main difficulty of this catadioptrical device consists. The location of the lens resulting near by the second- ary, the accuracy of its cylindrical surface must be optimal and just this is the 'black sheep'. Cylindrical surfaces are scarcely to be finished to the same exactness as spherical surfaces and the longer the r.o.c. the more difficult the task. Deviations from the exact value of the r.o.c. are of less importance; they will solely cause a change in the power of the lens, which will be balanced by compara- tively small changes of the distance a1m. The condition of focal- lengths 18) then being violated, the resulting faults are negligible, even, in comparatively large instruments.