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.

292 UNDULATORY THEORY OF OPTICS.
is a.ce2f: and so on. Also the wave entering at H is behind
that which entered at E by the sanie quantity V as before.
Hence the sum of the vibrations will be
a.cf { sin (v t - x) + e2 sin (v t- - V) + &c.}
X AX
sin -(vt - ) - e2 sin (vt - + V)
X X
-= a.cf.
2wr
1 -2e2cos - V+ e4
X
Treating this in the same manner as in (65), the intensity of light is found to be
a-(l _ e-)2
(1 - e)2 4e sin2
68. The proportional variations of this expression are
much smaller than those of the expression of (65); its
greatest value being a2, and its least a - 2). The
(i + e")"
absolute variations are however exactly the same: and in fact
the sum of the two expressions is always = a". This is
expressed by saying that one of the intensities is complementary to the other. This relation spares us the necessity
of examining every particular case of the value of D. If
for any particular value of D the expression of (65) is
maximum for any particular colour, that of (67) is minimum
for the same colour: and so on. Thus if for some value
of D the expression of (65) gave maximum intensity of
red light, less of yellow, the mean intensity of green, less
of blue, and nothing of violet (the mixture of which would
produce a ricli yellow): then the expression of (67) would
give the minimum intensity of red light, more of yellow, the
mean intensity of green, more of blue, and the maximum
of violet (the mixture of which would produce a greenish
blue diluted with much white). It is to be remarked that
in the case of transmitted light the colours can never be so
vivid as in reflected light, bcause none of the colours ever