uccessive
convolutions envelop one another without intersection.

[Illustration: Fig. 19.]

The utility of the curve depends upon the fact that the elements of
arc represent, in amplitude and phase, the component vibrations due to
the corresponding portions of the primary wave-front. For by (30)
d[sigma] = dv, and by (2) dv is proportional to ds. Moreover by (2)
and (31) the retardation of phase of the elementary vibration from PQ
(fig. 17) is 2[pi][delta]/[lambda], or [phi]. Hence, in accordance
with the rule for compounding vector quantities, the resultant
vibration at B, due to any finite part of the primary wave, is
represented in amplitude and phase by the chord joining the
extremities of the corresponding arc ([sigma]2 - [sigma]1).

In applying the curve in special cases of diffraction to exhibit the
effect at any point P (fig. 18) the centre of the curve O is to be
considered to correspond to that point C of the primary wave-front
which lies nearest to P. The operative part, or parts, of the curve
are of course those which represent the unobstructed portions of the
primary wave.

Let us reconsider, following Cornu, the diffraction of a screen
unlimited on one side, and on the other terminated by a straight edge.
On the illuminated side, at a distance from the shadow, the vibration
is represented by JJ'. The co-ordinates oi J, J' being (1/2, 1/2),
(-1/2, -1/2), I^2 is 2; and the phase is 1/8 period in arrear of that
of the element at O. As the point under contemplation is supposed to
approach the shadow, the vibration is represented by the chord drawn
from J to a point on the other half of the curve, which travels
inwards from J' towards O. The amplitude is thus subject to
fluctuations, which increase as the shadow is approached. At the point
O the intensity is one-quarter of that of the entire wave, and after
this point is passed, that is, when we have entered the geometrical
shadow, the intensity falls off gradually to zero, _without
fluctuations_. The whole progress of the phenomenon is thus exhibited
to the eye in a very instructive manner.

We will next suppose that the light is transmitted by a slit, and
inquire what is the effect of varying the width of the slit upon the
illumination at the projection of its centre. Under these
circumstances the arc to be considered is bisected at O, and its
length is proportional to the width