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 2cos[phi]
-- + - = ---------,
v1 u a

where v1 = AQ', u = AQ, a = OA, [phi] = 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 [phi], 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 [phi]' = [phi], there are a
series of spectra with various values of [phi]', 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 = [phi], AOP = [omega],

QP^2 = u^2 + 4a^2sin^21/2[omega] - 4au sin 1/2[omega] sin (1/2[omega] - [phi])
= (u + a sin[phi] sin[omega])^2 - a^2 sin^2[phi] sin^2[omega] + 4a sin^2 1/2[omega](a - u cos[phi]).

Now as far as [omega]^4

4 sin^2 1/2[omega] = sin^2[omega] + 1/4sin^4[omega],

and thus to the same order

QP^2 = (u + a sin [phi] sin [omega])^2
-a cos [phi](u - a cos [phi]) sin^2[omega] + 1/4 a(a - u cos[phi]) sin^4 [omega].

But if we now suppose that Q lies on the circle u = a cos [phi], the
middle term vanishes, and we get, correct as far as [omega]^4,

/ / a^2 sin^2[phi] sin^4[omega]\
QP = (u + a sin[phi] sin[omega]) / ( 1 + --------------------------- );
\/ \ 4u /
so that

QP - u = a sin [phi] sin [omega] + 1/8 a sin[phi] tan[phi] sin^4 [omega] (9),

in which it is to be noticed that the adjustment necessary to secure
the disappearance of sin^2[omega] is sufficient also to destroy the
term in sin^3[omega].

A similar expression can be found for Q'P - Q'A;