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C. Runge and K. W. Meissner, Handbuch der Astrophysik (Julius Springer, Berlin, 1933), Vol. 1, pp. 235–257.
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When the slit is not parallel to the grating rulings, it (except the midpoint) deviates a very small amount from the Rowland cylinder. Since the terms involving (cos2α/r) - (cosα/R) are no longer zero for all points (except the center of the slit) we must take into account this fact in our treatment. The variation of 1/r, Δ(1/r), is fortunately very small and Δ(1/r)= φΔz· tanα0/ (R2 cos2α0)~O(z0/R3). Therefore, only one term, -ωφΔz· tanα0/R2 comes into (10) as the correction [note that we retain the terms down to O(1/R3)].
For the 21-ft off-plane Eagle mounting vacuum spectrograph at the University of Chicago, W = 12.7 cm, z0= 12.7 cm. We have the ratio z0/R=0.02 and (W2/16)/z02=0.0625. Therefore, this assumption is not far from a practical case.
We are now considering a point in the image where the mthorder line due to light of wavelength (λ-Δλ) would have its central maximum intensity. In other words, λΔ is not a change in the wavelength but means displacement, which is measured in units of wavelength, of an image point from the position where the mth-order line of wavelength λ would have its central maximum intensity. It has the same sign as Δβ.
But if we assume uniform illumination over the grating and a perfect grating, then, for simplicity, all the δn's may be treated as equal.
In the case of the in-plane mounting, R asymptotically approaches 0.75W optm/σ as W≫Wopt (see Mack, Stehn, and Edlén, reference 8).
.The word "slit" used here is equivalent to the illuminated part of the slit.
W. R. Hamilton, "Geometrical Optics," Mathematical Papers (Cambridge University Press, London, 1931), Vol. 1, p. 17; J. L. Synge, "Geometrical optics (an introduction to Hamilton's method)," Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge University Press, London, 1937), No. 37, p. 17; J. L. Synge, J. Opt. Soc. Am. 27, 75 (1937).
Private communication with Dr. D. L. MacAdam. This comment resulted from examination of the manuscript of this paper by Dr. Herzberger and himself.
Strictly speaking g is not independent of l and such dependence becomes appreciable for grazing incidence cases. In that case (45) can give the correct answer only under certain restrictions, i.e., w, l ≤ ½W opt.
See (83) in his paper (reference 13). Equation (83) is misprinted and the exponent 2 should be put on the sines within the round bracket.
B. Edlén, Nova Acta Regiae Soc. Sci. Upsaliensis 9, No. 6 (1933).