QP − u = a sin φ sin ω + 1⁄8a sin φ tan φ sin4 ω (9),
in which it is to be noticed that the adjustment necessary to secure the disappearance of sin²ω is sufficient also to destroy the term in sin³ω.
A similar expression can be found for Q’P − Q′A; and thus, if Q′A = v, Q′AO = φ′, where v = a cos φ′, we get
QP + PQ′ − QA -AQ′ = a sin ω (sin φ − sin φ′)
+ 1⁄8a sin4 ω (sin φ tan φ + sin φ′ tan φ′) (10).
If φ′ = φ, the term of the first order vanishes, and the reduction of the difference of path via P and via A to a term of the fourth order proves not only that Q and Q′ are conjugate foci, but also that the foci are exempt from the most important term in the aberration. In the present application φ′ is not necessarily equal to φ; but if P correspond to a line upon the grating, the difference of retardations for consecutive positions of P, so far as expressed by the term of the first order, will be equal to ± mλ (m integral), and therefore without influence, provided
σ (sin φ − sinφ′) = ± mλ (11),
where σ denotes the constant interval between the planes containing the lines. 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. For this purpose Rowland places the eye-piece at O, so that φ = 0, and then by (11) the value of φ′ in the mth spectrum is
σ sin φ’ = ± mλ (12).
If ω now relate to the edge of the grating, on which there are altogether n lines,