| z = | x² | + Bxy + | y² | + αx³ + βx²y + γxy² + δy³ + ... (8); |
| 2ρ | 2ρ′ |
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.[8] 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, 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). The error thus arising may be compensated by a rotation of the object-glass about one of the diameters y = ± x. The term in x²y corresponds to a deviation from parallelism in the same direction on both sides of the central line (fig. 14); and that in y³ 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
| 1 | + | 1 | = | 2cosφ | , |
| v1 | u | a |
where v1 = 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).