and thus, if Q'A = v,
Q'AO = [phi]', where v = a cos [phi]', we get

QP + PQ' - QA -AQ' = a sin[omega] (sin[phi] - sin[phi]')
+ 1/8 a sin^4 [omega] (sin[phi] tan[phi] + sin[phi]' tan[phi]') (10).

If [phi]' = [phi], 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 [phi]' is not necessarily equal
to [phi]; 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 [-+]
m[lambda] (m integral), and therefore without influence, provided

[sigma] (sin[phi] - sin[phi]') = [+-] m[lambda] (11),

where [sigma] 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 [phi] = 0, and then by (11) the
value of [phi]' in the m^th spectrum is

[sigma] sin [phi]' = [+-] m[lambda] (12).

If [omega] now relate to the edge of the grating, on which there are
altogether n lines,

n[sigma] = 2a sin [omega],

and the value of the last term in (10) becomes

1/16 n[sigma] sin^3[omega] sin[phi]' tan[phi]',

or

1/16 mn[lambda] sin^3[omega] tan [phi]' (13).

This expresses the retardation of the extreme relatively to the
central ray, and is to be reckoned positive, whatever may be the signs
of [omega], and [phi]'. If the semi-angular aperture ([omega]) be
1/100, and tan [phi]' = 1, mn might be as great as four millions
before the error of phase would reach 1/4[lambda]. If it were desired to
use an angular aperture so large that the aberration according to (13)
would be injurious, Rowland points out that on his mach