For a complete history and bibliography, as well as the clearest (Runge's) general discussion of the theory of the concave grating, see Kayser, Handbuch der Spektroskopie," vol. 1, pp. 450–482 (1900).

That is, the discrepancy in path would be less than λ/4. The relationship between this criterion and the results of this investigation, is discussed in footnotes 8 and 12.

The practical extension into three dimensions is obviously the cylinder, for the photographic plate, cut from a plane, can not take a second curvature.

By extending the method of Kayser, l.c., pp. 456–458, to *c*≠0.

By carrying the method of Section IV, which considers the three dimensional case, to a higher degree of accuracy, it can be shown that the light scattered from the whole length of one ruling has a resultant amplitude vector δ_{n} whose magnitude is practically independent of *y* (so long as *y* > >*r*, the case in any ordinary grating) and has the same phase dependence on *y* as the light coming from the line *z*=0. This statement is true to a higher degree of approximation when *c, c*′=0 (small source near the slit, and central axis of photographic plate). Hence the simplification made here is justified.

The term independent of *y* is dominant here, and the linear term is next in magnitude. But the resolving power will be seen in the equations below to depend upon the phase differences between the path [equation] throughout the range of points *P* (i.e., of ordinates *y*). Since *r* is independent of *y*, (*r*+*r*′) can not be expected to contribute anything interesting in the present discussion. The essence of the design of the concave grating, as well as of the plane grating, is that the factors exp(2π*in*σ*m*(λ-Δλ/λσ) in the expression for *I* below, are in phase for all integers *n*, if Δλ=0; i.e., if *A*′ is at the *m*th central maximum for wave length λ. The virtue of restricting *A, A*′ to the Rowland circle is that under this restriction the second and third power terms cancel out. This shows why the fourth power term must be carried. The fifth power term is unimportant in the consideration of the resolving power (but see Section V for its effect upon the position of the central intensity maximum for a line). The introduction of the limit *H*→∞, below, is not justified physically, of course, so long as we have neglected higher powers. But in practical cases *H* is always less than 10, usually much less, and within this range our approximations seem reasonable.

Cf. C. M. Sparrow, Ap. J. 49, 65; 1919.

The dashed circles tangent to the curves of constant α at the origin are the familiar curves ∫_{e}^{iαη}*d*_{η} that one obtains by the usual analysis, neglecting the fourth power term. Now we can see the relationship between this work and the previous Δ(path) >λ/4 approximation (footnote 2). Treating the grating as though it were plane, but limited in width by the condition that the path discrepancy be less than λ/4, is identical (cf. equations (5) and (6) and Kayser. I.c., p. 458) with using segments of circles [equation] in the amplitude plane, but limiting *H* to (π/2)^{¼}=1.120. Paradoxically, the more accurate method (Fig. 2, supplemented by a series expansion for *w*(0, *H*)) shows that part of the *second* Huyghens zone up to *H*=1.177=(0.611π)^{¼} i.e., almost to Δ(path)=5λ/16, actually contributes *positively* to the amplitude at the central maximum.

We are indebted to Prof. R. E. Langer for aid in putting this derivation into rigorous form.

β is a function of Δλ through its dependence upon cos θ′; hence it varies with α. The dependence is so slight, however, that we are quite well justified in calling β a constant for moderate values of α.

It is often possible to resolve two lines with Δλ considerably smaller than Δ_{1}λ. The fraction is calculable in special cases from the "undulation condition," C. M. Sparrow, Ap. J. 44, 76; 1916. We use a new criterion for resolution rather than Sparrow's because it seems at least as useful practically, and we prefer to extend rather than abandon Rayleigh's classical one. Several workers, e.g., A. Schuster, Ap. J. 21, 197; 1905 and S. K. Allison, Phys. Rev. 38, 203; 1931, have used simpler criteria, which are safe only when the outer portion of the intensity pattern can be neglected, which is far from true for the concave grating (see Fig. 3). It is well known that when two lines are nearly superimposed their maxima may be shifted, and the problem is further complicated by photographic and psychological phenomena; but here we are neglecting all these complications.

In the Δ(path) >λ/4 approximation, *H*_{eff}=*H* up to *H*=(π/2)^{¼}=1.120, the end of the first Huyghens zone, beyond which point *H*_{eff} is constant. For such a simple approximation this gives very good results. The ratio, *R* (simple approximation): *R* (this paper) is always somewhat more than unity. For all *H* > 1.2 it is never more than 1.05, and for *H*→∞ it is 1.27. The worst discrepancy is in the neighborhood of *H*=1.5, where the ratio is 1, 6.

The flutings are much too far apart to be accounted for as secondary maxima of the sort shown in Fig. 3.

Kayser, l.c., p. 459.

i.e., the value of Δλ in equation (3) corresponding to the shift in the intensity maximum from the position given by equation (11).

We might be expected to express the resolving power or "purity" of the instrument in terms of Schuster's "purity factor" (Encyclopedia Brittanica, 9th ed., "Spectroscopy," and Ap. J. 21, 197; 1905). But the complicated case of the concave grating spectrograph does not seem to be amenable to such simple treatment, so we have been compelled to limit our considerations to cases where either the slit or the grating is predominant in limiting the instrument's resolving power. We are indebted to Drs. R. O. Rollefson and A. E. Whitford for profitable discussions of the case of finite slit width.

In this analysis we have treated the line on the photographic plate as though it were due to diffraction from a uniformly lighted plane grating of width 2*Y*_{coh}, neglecting: (1) the variation of δ_{n} within the region 2*Y*_{coh} due to the shape of the slit diffraction pattern on the grating; and (2) the higher order aberrations in the grating which would give rise to a 2*Y*_{coh}_{eff}> 2*Y*_{coh}. These approximations are justified by the result below, that the whole diffraction pattern from the slit broadens the geometric image from the slit by only ten percent.

It might be worth while to consider the possibility of improving the resolving power of the system for a narrow wave-length region by deliberately widening the slit. The relationships that allow this possibility can be seen best by reference to the example, Fig. 6. The ordinates, for each wave length, are proportional to *H*. In the right hand part of the picture, the ratio between the width of the grating ("plane grating") and the effective width curve drawn ("concave grating, asymptote") equals *H*/0.88. But in the left hand part, the effective width, as limited by the size of the slit, takes the place of the whole width in determining *H*. So, near the intersection of the "concave grating" curve and the "slit" curve, *H* is small again, and relatively large oscillations of *H*_{eff} about the asymptotic value occur. The resolving power in the neighborhood of the intersection of the theoretical curves for the slit and for the grating, can not very well be discussed in detail, on account of the complicated nature of the illumination function δ_{n}. But it might be improved by experimental comparison for different values of *s*.

At the shortest wave lengths, the increased penetrating power of the rays would be expected to cause an asymmetric broadening of the lines, proportional to sin θ′ and to the penetration of the rays, and limited (unless there were considerable reflection at the surface of the glass) by tan θ′ times the thickness of the photographic (Schumann) emulsion. Also, the photographic resolving power of the plate itself might limit the resolving power of the system. In our example neither of these influences is important.