x^2. FIG. 10.--y^2. FIG. 11.--x^3. FIG. 12.--xy^2.

/ / /
\ | | / | \ | | | |
/ / /

FIG. 13.--xy. FIG. 14.--x^2y. FIG. 15.--y^3.]

The effect of a gradual increase in the interval (fig. 9) as we pass
across the grating has been investigated by M. A. Cornu (_C.R._, 1875,
80, p. 655), who thus explains an anomaly observed by E. E. N.
Mascart. The latter found that certain gratings exercised a converging
power upon the spectra formed upon one side, and a corresponding
diverging power upon the spectra on the other side. Let us suppose
that the light is incident perpendicularly, and that the grating
interval increases from the centre towards that edge which lies
nearest to the spectrum under observation, and decreases towards the
hinder edge. It is evident that the waves from _both_ halves of the
grating are accelerated in an increasing degree, as we pass from the
centre outwards, as compared with the phase they would possess were
the central value of the grating interval maintained throughout. The
irregularity of spacing has thus the effect of a convex lens, which
accelerates the marginal relatively to the central rays. On the other
side the effect is reversed. This kind of irregularity may clearly be
present in a degree surpassing the usual limits, without loss of
definition, when the telescope is focused so as to secure the best
effect.

It may be worth while to examine further the other variations from
correct ruling which correspond to the various terms expressing the
deviation of the wave-surface from a perfect plane. If x and y be
co-ordinates in the plane of the wave-surface, the axis of y being
parallel to the lines of the grating, and the origin corresponding to
the centre of the beam, we may take as an approximate equation to the
wave-surface

x^2 y^2
z = ------ + Bxy + ------- + [alpha]x^3 + [beta]x^2y + [gamma]xy^2 + [delta]y^3 + ... (8);
2[rho] 2[rho]'

and, as we have just seen, the term in x^2 corresponds to a linear
error in the spacing. In like manner, the term in y^2 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 i