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.

COLOURED FRINGES ABOUT SHADOWS. 305
performed as above: but a general explanation may be given
thus. Let P and Q be points similarly situated on the
two sides: the small waves diverging from their neighbourhood would, as in (46), produce bands by their interferences;
and the breadth of these, by (56), would be inversely as
the distance of P and Q. Consequently the nearer P and
Q are taken to the solid angle, the broader will the bands
be, and their form will therefore resemble the hyperbola.
In this we have omitted the effects of interference of other
portions of the light nearer to and further from the angle,
but as the omitted parts would at different points produce
effects nearly similar, it is probable that the general form
of the curves will be similar to hyperbolas.
78. Another instance is, if thle light fall on a very
narrow slit, coloured bands of much greater breadth are
thrown on the screen. The second case of (25) sufficiently
explains their origin. If the slit be triangular, it was observed by Newton that the bands are rectangular hyperbolas, the asymptotes being parallel and perpendicular to
the axis of the triangle. This appears from the same investigation, as the intervals between the bands are inversely
as b. the breadth of the aperture, or inversely as the distance from the geometrical shadow of the triangle's vertex.
79. In the following instance we may find the intensity at one point without much trouble. Suppose the aperture in (73) to be a round hole: to find the intensity at
that point of the screen which is the projection of its center.
Divide the circle into rings, the inner and outer radii of
one being r and r + r, or its surface being 27rrSr. The
distance of every point of this ring from the point of the
screen is
a+b
b + - r2 nearly,
2ab
and hence the whole displacement at the central point of
the screen is.rr sin 27 (vt ab -
X 2ab
20