Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

304 UNDULATORY THEORY OF OPTICS.
simplest cases is, to find the intensity of light produced by
the shadow of a plate bounded by a straight line. If y
is parallel to the edge, and x for the edge = 0, then the
limits of the first integration are
from y =-o to y = +,
and those of the second
from e =O to = c.
This case has been fully considered by M. Fresnel, and he
has arrived at this conclusion. If a plane be drawn through
the bright point and the edge of the plate, and if the intersection of this with the screen be called the geometrical
shadow: then on the dark side of the geometrical shadow
the intensity of the light diminishes rapidly without increasing at all, and soon becomes insensible: but on the
bright side the light increases and diminishes by several
alternations before it acquires sensibly its full brightness,
forming a succession of several bands on the bright side
of the geometrical shadow. And as, for the same point,
the limits for which the tabular numbers are taken are
different for different values of X, and as the factor of the
whole varies with X, the intensity of the differently coloured
lights will be differently proportioned at different points,
and thus the bands will be coloured, nearly as in (52).
This phenomenon, known by the name of Grirnaldi's coloured fringes, had long been observed, and an imperfect
explanation was given by Newton. In Fresnel's Mémoire
sur la Difraction it was shewn, from accurate measures
with various values of a and b, to be a consequence of the
theory of undulations, (Memoires de 1' Institut, 1821).
77. Another instance is, if the form of the plate be
a square corner, then besides the bands on the outside of
tie geometrical shadow there are seen within it hyperbolical
curves as in fig. 21. The accurate investigation* may be
* In this and the preceding case, it is necessary to consider the effect produced by small waves diverging from distances sensibly different. In the investigation we suppose that the absolute effect of each of these is the same as
the effect of a wave diverging from a surface of equal extent at a smaller distance. This is manifestly incorrect: but it produces no sensible error in the
result, for the reason mentioned in (29).