“ If x and y be co-ordinates in the plane of the wave-surface, the axis of y being parallel to the lines of the grating, and the origin corresponding to the centre of the beam, we may take as an approximate equation to the wave-surface
and, as we have just seen, the term in x2 corresponds to a linear error in the spacing. In like manner, the term in y2 corresponds to a general curvature of the lines (fig. 10) , and does not influence the definition at the (primary) focus, although it may introduce astigmatism. ”

“ The absence of chromatic aberration gives a great advantage in the comparison of overlapping spectra, which Rowland has turned to excellent account in his determinations of the relative wave-lengths of lines in the solar spectrum (Phil. Mag., 1887) .
For absolute determinations of wave-lengths plane gratings are used. It is found (Bell, Phil. Mag., 1887) that the angular measurements present less difficulty than the comparison of the grating interval with the standard metre. There is also some uncertainty as to the actual temperature of the grating when in use. ”

“ This is the ordinary formula for a reflecting plane grating, and it shows that the spectra are formed in the usual directions. They are here focused (so far as the rays in the primary plane are concerned) upon the circle OQ′A, and the outstanding aberration is of the fourth order.
In order that a large part of the field of view may be in focus at once, it is desirable that the locus of the focused spectrum should be nearly perpendicular to the line of vision. ”

“ B0 we see that no lateral spectrum can surpass the central image in brightness; but this result depends upon the hypothesis that the ruling acts by opacity, which is generally very far from being the case in practice. In an engraved glass grating there is no opaque material present by which light could be absorbed, and the effect depends upon a difference of retardation in passing the alternate parts. It is possible to prepare gratings which give a lateral spectrum brighter than the central image, and the explanation is easy. ”

“ If the incident rays make an angle θ with the normal (fig. 6) , and the diffracted rays make an angle φ (upon the same side) , the relative retardation from each element of width (a + d) to the next is (a + d) (sinθ + sinφ) ; and this is the quantity which is to be equated to mλ. Thus
The “deviation” is (θ + φ) , and is therefore a minimum when θ = φ, i.e. when the grating is so situated that the angles of incidence and diffraction are equal. ”

“ The error of ruling giving rise to α is one in which the intervals increase or decrease in both directions from the centre outwards (fig. 11) , and it may often be compensated by a slight rotation in azimuth of the object-glass of the observing telescope. The term in γ corresponds to a variation of curvature in crossing the grating (fig. 12) .
When the plane zx is not a plane of symmetry, we have to consider the terms in xy, x2y, and y3. The first of these corresponds to a deviation from parallelism, causing the interval to alter gradually as we pass along the lines (fig. 13) . ”

“ If we now suppose half the grating cut away, so as to leave 1000 lines in half an inch, the dispersion will not be altered, while the brightness and resolving power are halved.
There is clearly no theoretical limit to the resolving power of gratings, even in spectra of given order. But it is possible that, as suggested by Rowland, [2] the structure of natural spectra may be too coarse to give opportunity for resolving powers much higher than those now in use. ”

“ The latter found that certain gratings exercised a converging power upon the spectra formed upon one side, and a corresponding diverging power upon the spectra on the other side. Let us suppose that the light is incident perpendicularly, and that the grating interval increases from the centre towards that edge which lies nearest to the spectrum under observation, and decreases towards the hinder edge. ”

“ From (5) we see that, when the light falls perpendicularly upon a grating (θ = 0) , there is no spectrum formed (the image corresponding to m = 0 not being counted as a spectrum) , if the grating interval σ or (a + d) is less than λ. Under these circumstances, if the material of the grating be completely transparent, the whole of the light must appear in the direct image, and the ruling is not perceptible. From the absence of spectra Fraunhofer argued that there must be a microscopic limit represented by λ ”

“ A much easier method, applicable to glass originals, is that of photographic reproduction by contact printing. In several papers dating from 1872, Lord Rayleigh (see Collected Papers, i. 157, 160, 199, 504; iv. 226) has shown that success may be attained by a variety of processes, including bichromated gelatin and the old bitumen process, and has investigated the effect of imperfect approximation during the exposure between the prepared plate and the original. ”