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.

262 UNDULATORY TIEORY OP OPTICS.
Case 1. Suppose X much greater than b. (This will
generally be the case with sound, as for all audible sounds
X varies from a few inches to several feet.)
Here - O will be a small arc, and will not differ
X
much from its sine: putting the arc for the sine, the
maximum of vibration becomes
cX 2 7rb cos 0 2cb, or -,5
7rrcosO r
which is the same for all values of 0.
Case 2. Suppose X much smaller than b. (This will
generally be the case with light.)
For the part nearly opposite to the entering wave,
cos 0 is very small, and
cX 27rb cos 2cb.sin -
7r cos 0 \X r
In other parts it is to be observed that the disturbance is
o when
27rb cos 0
- r or =o 2r, or = 2, or = 7r, &c.,
that is, when
X 2X 3X
cos O= 6 -, or =5 or = - &c.
2b 2b 2b
Hence there is a succession of points in which there is
absolute darkness. Of the intermediate parts, the brightest
will be found (nearly) by making
27b cos 0
sin = 1
cX
then the maximum of vibration is.Consequently
-r cos
the intensity of light at the brightest part of one of the bright
portions is to that of the part nearly opposite to the entering