If we define as the “dispersion” in a particular part of the spectrum the ratio of the angular interval *d*θ to the corresponding increment of wave-length *d*λ, we may express it by a very simple formula. For the alteration of wave-length entails, at the two limits of a diffracted wave-front, a relative retardation equal to *mnd*λ. Hence, if *a* be the width of the diffracted beam, and *d*θ the angle through which the wave-front is turned,

*ad*θ = *mn d*λ,

or

dispersion = *mn*/*a* (7).

The resolving power and the width of the emergent beam fix the optical character of the instrument. The latter element must eventually be decreased until less than the diameter of the pupil of the eye. Hence a wide beam demands treatment with further apparatus (usually a telescope) of high magnifying power.

In the above discussion it has been supposed that the ruling is accurate, and we have seen that by increase of *m* a high resolving power is attainable with a moderate number of lines. But this procedure (apart from the question of illumination) is open to the objection that it makes excessive demands upon accuracy. According to the principle already laid down it can make but little difference in the principal direction corresponding to the first spectrum, provided each line lie within a quarter of an interval (*a* + *d*) from its theoretical position. But, to obtain an equally good result in the *m*^{th} spectrum, the error must be less than 1/*m* of the above amount.^{[1]}

There are certain errors of a systematic character which demand special consideration. The spacing is usually effected by means of a screw, to each revolution of which corresponds a large number (*e.g.* one hundred) of lines. In this way it may happen that although there is almost perfect periodicity with each revolution of the screw after (say) 100 lines, yet the 100 lines themselves are not equally spaced. The “ghosts” thus arising were first described by G. H. Quincke (*Pogg. Ann.*, 1872, 146, p. 1), and have been elaborately investigated by C. S. Peirce (*Ann. Journ. Math.*, 1879, 2, p. 330), both theoretically and experimentally. The general nature of the effects to be expected in such a case may be made clear by means of an illustration already employed for another purpose. Suppose two similar and accurately ruled transparent gratings to be superposed in such a manner that the lines are parallel. If the one set of lines exactly bisect the intervals between the others, the grating interval is practically halved, and the previously existing spectra of odd order vanish. But a very slight relative displacement will cause the apparition of the odd spectra. In this case there is approximate periodicity in the half interval, but complete periodicity only after the whole interval. The advantage of approximate bisection lies in the superior brilliancy of the surviving spectra; but in any case the compound grating may be considered to be perfect in the longer interval, and the definition is as good as if the bisection were accurate.

Fig. 9.— |
Fig. 10.— |
Fig. 11.— |
Fig. 12.— |

Fig. 13.— |

The effect of a gradual increase in the interval (fig. 9) as we pass across the grating has been investigated by M. A. Cornu (*C.R.*, 1875, 80, p. 655), who thus explains an anomaly observed by E. E. N. Mascart. 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. It is evident that the waves from *both* halves of the grating are accelerated in an increasing degree, as we pass from the centre outwards, as compared with the phase they would possess were the central value of the grating interval maintained throughout. The irregularity of spacing has thus the effect of a convex lens, which accelerates the marginal relatively to the central rays. On the other side the effect is reversed. This kind of irregularity may clearly be present in a degree surpassing the usual limits, without loss of definition, when the telescope is focused so as to secure the best effect.

It may be worth while to examine further the other variations from correct ruling which correspond to the various terms expressing the deviation of the wave-surface from a perfect plane. 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 *x*² corresponds to a linear error in the spacing. In like manner, the term in *y*² 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.^{[2]} If we suppose that everything is symmetrical on the two sides of the primary plane *y* = 0, the coefficients B, β, δ vanish. In spite of any inequality between ρ and ρ′, the definition will be good to this order of approximation, provided α and γ vanish. The former measures the *thickness* of the primary focal line, and the latter measures its *curvature*. 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*, *x*²*y*, and *y*³. The first of these corresponds to a deviation from parallelism, causing the interval to alter gradually as we pass *along* the lines (fig. 13). The error thus arising may be compensated by a rotation of the object-glass about one of the diameters *y* = ± *x*. The term in *x*²*y* corresponds to a deviation from parallelism in the same direction on both sides of the central line (fig. 14); and that in *y*³ would be caused by a curvature such that there is a point of inflection at the middle of each line (fig. 15).

All the errors, except that depending on α, and especially those depending on γ and δ, can be diminished, without loss of resolving power, by contracting the *vertical* aperture. A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating.

The explanation of the difference of focus upon the two sides as due to unequal spacing was verified by Cornu upon gratings purposely constructed with an increasing interval. He has also shown how to rule a plane surface with lines so disposed that the grating shall of itself give well-focused spectra.

Fig. 16. |

A similar idea appears to have guided H. A. Rowland to his brilliant invention of concave gratings, by which spectra can be photographed without any further optical appliance. In these instruments the lines are ruled upon a spherical surface of speculum metal, and mark the intersections of the surface by a system of parallel and equidistant planes, of which the middle member passes through the centre of the sphere. If we consider for the present only the primary plane of symmetry, the figure is reduced to two dimensions. Let AP (fig. 16) represent the surface of the grating, O being the centre of the circle. Then, if Q be any radiant point and Q′ its image (primary focus) in the spherical mirror AP, we have

where *v*_{1} = AQ′, *u* = AQ, *a* = OA, φ = angle of incidence QAO, equal to the angle of reflection Q′AO. If Q be on the circle described upon OA as diameter, so that *u* = *a* cos φ, then Q′ lies also upon the same circle; and in this case it follows from the symmetry that the unsymmetrical aberration (depending upon *a*) vanishes.

This disposition is adopted in Rowland′s instrument; only, in addition to the central image formed at the angle φ′ = φ, there are a series of spectra with various values of φ′, but all disposed upon the same circle. Rowland’s investigation is contained in the paper already referred to; but the following account of the theory is in the form adopted by R. T. Glazebrook (*Phil. Mag.*, 1883).

In order to find the difference of optical distances between the courses QAQ′, QPQ′, we have to express QP − QA, PQ′ − AQ′. To find the former, we have, if OAQ = φ, AOP = ω,

QP² = *u*² + 4*a*²sin²½ω − 4*au* sin ½ω sin (½ω − φ)

= (*u* + *a* sin φ sin ω)² − *a*² sin²φ sin²ω + 4*a* sin² ½ω(*a* − *u* cosφ).

- ↑ It must not be supposed that errors of this order of magnitude are unobjectionable in all cases. The position of the middle of the bright band representative of a mathematical line can be fixed with a spider-line micrometer within a small fraction of the width of the band, just as the accuracy of astronomical observations far transcends the separating power of the instrument.
- ↑ “In the same way we may conclude that in flat gratings any departure from a straight line has the effect of causing the dust in the slit and the spectrum to have different foci—a fact sometimes observed.” (Rowland, “On Concave Gratings for Optical Purposes,”
*Phil. Mag.*, September 1883).