(17).
[pi]v [pi]^3 v^5 [pi]^5 v^9

The corresponding values of C and S were originally derived by A. L.
Cauchy, without the use of Gilbert's integrals, by direct integration
by parts.

From the series for G and H just obtained it is easy to verify that

dH dG
-- = - [pi]vG, -- = [pi]vH - 1 (18).
dv dv

We now proceed to consider more particularly the distribution of light
upon a screen PBQ near the shadow of a straight edge A. At a point P
within the geometrical shadow of the obstacle, the half of the wave to
the right of C (fig. 18), the nearest point on the wave-front, is
wholly intercepted, and on the left the integration is to be taken
from s = CA to s = [oo]. If V be the value of v corresponding to CA,
viz.

/ / 2(a + b) \
V= / ( ---------- ).CA, (19),
\/ \ ab[lambda] /

we may write

_[oo] _[oo]
/ / \^2 / / \^2
I^2 = ( | cos 1/2[pi]v^2.dv ) + ( | sin 1/2[pi]v^2.dv ) (20),
\ _/v / \ _/v /

or, according to our previous notation,

I^2 = (1/2 - Cv)^2 + (1/2 - Sv)^2 = G^2 + H^2 (21).

Now in the integrals represented by G and H every element diminishes
as V increases from zero. Hence, as CA increases, viz. as the point P
is more and more deeply immersed in the shadow, the illumination
_continuously_ decreases, and that without limit. It has long been
known from observation that there are no bands on the interior side of
the shadow of the edge.

[Illustration: FIG. 18.]

The law of diminution when V is moderately large is easily expressed
with the aid of the series (16), (17) for G, H. We have ultimately G =
0, H = ([pi]V)^-1, so that

I^2 = 1/[pi]^2V^2,

or the illumination is inversely as the square of the distance from
the shadow of the edge.

For a point Q outside the shadow the integration extends over _more_
than half the primary wave. The intensity may be expressed by

I^2 = (1/2 + Cv)^2 + (1/2 + Sv)^2 (22);

and the maxima and minima occur when

dC dS
(1/2 + C_v) -- + (1/2 + S_v) -- = 0,
dV dV

whence

sin 1/2[pi]V^2 + cos 1/2[pi]V^2 = G (23).