the inclination of the secondary ray to the
direction of vibration and to the wave-front.

O x Q
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r| /
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P|/

FIG. 1.

The latter question can only be treated in connexion with the
dynamical theory (see below, S 11); but under all ordinary
circumstances the result is independent of the precise answer that may
be given. All that it is necessary to assume is that the effects of
the successive zones gradually diminish, whether from the increasing
obliquity of the secondary ray or because (on account of the
limitation of the region of integration) the zones become at last more
and more incomplete. The component vibrations at P due to the
successive zones are thus nearly equal in amplitude and opposite in
phase (the phase of each corresponding to that of the infinitesimal
circle midway between the boundaries), and the series which we have to
sum is one in which the terms are alternately opposite in sign and,
while at first nearly constant in numerical magnitude, gradually
diminish to zero. In such a series each term may be regarded as very
nearly indeed destroyed by the halves of its immediate neighbours, and
thus the sum of the whole series is represented by half the first
term, which stands over uncompensated. The question is thus reduced to
that of finding the effect of the first zone, or central circle, of
which the area is [pi][lambda]r.

We have seen that the problem before us is independent of the law of
the secondary wave as regards obliquity; but the result of the
integration necessarily involves the law of the intensity and phase of
a secondary wave as a function of r, the distance from the origin. And
we may in fact, as was done by A. Smith (_Camb. Math. Journ._, 1843,
3, p. 46), determine the law of the secondary wave, by comparing the
result of the integration with that obtained by supposing the primary
wave to pass on to P without resolution.

Now as to the phase of the secondary wave, it might appear natural to
suppose that it starts from any point Q with the phase of the