} = r + a,

so that

x^2 = [lambda]ar/(a + r) nearly.

Since the distance to be travelled by the secondary waves is still r,
we see how the effect of the first zone, and therefore of the whole
series is proportional to a/(a + r). In like manner may be treated
other cases, such as that of a primary wave-front of unequal principal
curvatures.

The general explanation of the formation of shadows may also be
conveniently based upon Fresnel's zones. If the point under
consideration be so far away from the geometrical shadow that a large
number of the earlier zones are complete, then the illumination,
determined sensibly by the first zone, is the same as if there were no
obstruction at all. If, on the other hand, the point be well immersed
in the geometrical shadow, the earlier zones are altogether missing,
and, instead of a series of terms beginning with finite numerical
magnitude and gradually diminishing to zero, we have now to deal with
one of which the terms diminish to zero _at both ends_. The sum of
such a series is very approximately zero, each term being neutralized
by the halves of its immediate neighbours, which are of the opposite
sign. The question of light or darkness then depends upon whether the
series begins or ends abruptly. With few exceptions, abruptness can
occur only in the presence of the first term, viz. when the secondary
wave of least retardation is unobstructed, or when a _ray_ passes
through the point under consideration. According to the undulatory
theory the light cannot be regarded strictly as travelling along a
ray; but the existence of an unobstructed ray implies that the system
of Fresnel's zones can be commenced, and, if a large number of these
zones are fully developed and do not terminate abruptly, the
illumination is unaffected by the neighbourhood of obstacles.
Intermediate cases in which a few zones only are formed belong
especially to the province of diffraction.

An interesting exception to the general rule that full brightness
requires the existence of the first zone occurs when the obstacle
assumes the form of a small circular disk parallel to the plane of the
incident waves. In the earlier half of the 18th century R. Delisle
found that the centre of the circular shadow was occupied by a bright
point of light, but the observation passed into oblivion until S. D.
Poisson brought forward as an objectio