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.

LIGHT PASSES THROUGH RECTANGULAR APERTURE. 301
7r. 7r 82
fjcos- and fsin2 2
for different values of s, and there is no difficulty in applying them to the nuinerical expression of the first integral in
our problem. But as the integral thus found is given in
numbers, and not in an expression involving y and which
can be expressed in terms of x, it is seldom possible to
perform the second integration.
75. In certain cases however thib may be effected. For
example, suppose the aperture to be a parallelogram whose
sides are 2e and 2f in the direction of i and y, the line
AO passing through its center. Let
a+b aq 2\ 52
Xab Y a+b) 2
a-q / Xab dy / Xab. Y 7r a+ b; aq 2 / \ab rs2.* S y c- - -a+ > V/ -. Lcos --;
y \À ab * a+b) (a b) Js 2
and this is to be taken from
y= -f to y= +f,
and therefore the value of the integral from s - h to s + h is
dU d3U h3
2- h2 —+ - -+&c.
ds ds3 12.3
d2S ha
or 2Sh +2 - — + &c.
ds' 2.3
which can be easily calculated: and by taking h small enough, one or two terms
will he sufficient. In this way the values of the integral can be computed for
successive limits, and the sum will be the integral up to any given value of s.
A table of, cos -and J sin — will be given at the end of this Tract.
2 2