s symmetrical on the
two sides of the primary plane y = 0, the coefficients B, [beta],
[delta] vanish. In spite of any inequality between [rho] and [rho]',
the definition will be good to this order of approximation, provided
[alpha] and [gamma] vanish. The former measures the _thickness_ of the
primary focal line, and the latter measures its _curvature_. The error
of ruling giving rise to [alpha] 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
[gamma] 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^2y, 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^2y 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 [alpha], and especially those
depending on [gamma] and [delta], 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.

[Illustration: 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