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.

INTENSITY OF LIGHT PASSING THROUGH A HOLE. 259
PROP. 8. A succession of waves, whose fronts are parallel
to the screen CD (fig. 5) in which is the comparatively
small opening AB, is moving towards the screen: to find
the magnitude of vibration on any point G of the semicircle CGD.
24. Let H be the center of AB and of the semicircle,
and let HG = r, CHG = 0, HA = b: divide AB into a great
number of small parts, and let the distance of one at Z
from H be z, and its breadth iz: then
ZG = /(r2 - 2r cos 0. + 2).
When a wave comes to AB, consider separately the parts
corresponding to the small divisions of AB. It seems reasonable to suppose that each of these small parts will cause
a diverging wave of equal intensity for all values of 0.
For if the medium were air, and a rush of particles took
place as in fig. 1, from EF, the only effect in the small
part under consideration would be to cause a condensation
of air: and this would cause a wave of equal intensity for
all values of 0. If the vibrations were like those of fig. 2,
and perpendicular to the plane of the paper, the same thing
appears evident: if parallel to the plane of the paper, it is
not so clear what the proportion of intensities would be.
The maximum of vibration when the wave reached G would,
if it followed the same laws as in air*, vary nearly as ZG
Now all the little waves which originate from the. different
* When a wave of air diverges symmetrically through any given solid angle,
if r be the original distance of any particle' from the center, r + u its distance
at the time t, r + h the original distance of a second particle; then the distance of the latter at the time t will be
r' + u + h ( +
nearly: and the particles which formerly occupied a volume proportional to r2h
now occupy a volume proportional to
(r+u)2h(l dur)
or nearly proportional to
(r2+2ru)h( l-d or to r2h.d\ -2r d
d2)1 orto r d(r+-+
Consequently,
17-2