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There are three devices to change the above mentioned fundamental system into a Schiefspiegler l) the anastigmatic, 2) the coma-free, 3) the catadioptrical device

1) The anastigmatic device ( former called 'The Neo-Brachyt') ( All symbols vide fig.2) The inclination φ₁ of the incoming parallel beam of rays is determined by the condition for freedom of silhouetting. It must be the 'first ax-distance' 1) 2)

If this excentrical system is to be free from astigmatism the incli- nation φ₂ of the secondary must become 3) ^Note

When r1 equals r2 the formula 3) is simplified to 3a)

With φ₂ the 'second ax-distance' Δ′( important for the construction of the Schiefspiegler ) is determined 4)

The inclination φ₂ of the secondary eliminates the astigmatism of the oblique beam of rays coming from the primary, but isn't sufficient to abolish thoroughly its coma too. It remains therefore a residual coma 5)

Note: Calculation with the slide rule will do. r₂ is to be introduced with negative sign. The value of β results in the arcus of the angle. To get the angle β in seconds of arc multiply the result with the factor 206265. The accuracy of the result is very high and reaches that of a logarithmic-trigonometrical computation ( 7 places) within +- 1 to 2 %. For the above mentioned fundamental system the residual coma results of about + 4,8″ ( undercorrected ). This value changes only slowly in moderate changing the dimensions A, b of our fundamental system, but diminishes and enlarges rapidly by diminishing or enlarging the aperture ratio of the primary.

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Note: Arjan te Marvelde finds that equation 3, above, appears to be in error. The positions of f₁ and f₂ should be inverted, thus:
Arjan derives this result from equation 3 of page 20, the condition of astigmatism, by setting the term for the corrector to zero and solving for sin φ₂, dropping the imaginary factor that results from taking the square root of a negative number.

Guntram Lampert has found the equation on page 46 of Kutter's "Der Schiefspiegler". It matches, Arjan's corrected form. We can conclude that the orignal version of equation 3 in "The Schiefspiegler" is simply a typographic error.