primary
wave, so that on arrival at P, it is retarded by the amount
corresponding to QP. But a little consideration will prove that in
that case the series of secondary waves could not reconstitute the
primary wave. For the aggregate effect of the secondary waves is the
half of that of the first Fresnel zone, and it is the central element
only of that zone for which the distance to be travelled is equal to
r. Let us conceive the zone in question to be divided into
infinitesimal rings of equal area. The effects due to each of these
rings are equal in amplitude and of phase ranging uniformly over half
a complete period. The phase of the resultant is midway between those
of the extreme elements, that is to say, a quarter of a period behind
that due to the element at the centre of the circle. It is accordingly
necessary to suppose that the secondary waves start with a phase
one-quarter of a period in advance of that of the primary wave at the
surface of resolution.

Further, it is evident that account must be taken of the variation of
phase in estimating the magnitude of the effect at P of the first
zone. The middle element alone contributes without deduction; the
effect of every other must be found by introduction of a resolving
factor, equal to cos [theta], if [theta] represent the difference of
phase between this element and the resultant. Accordingly, the
amplitude of the resultant will be less than if all its components had
the same phase, in the ratio

_ +1/2[pi]
/
| cos [theta]d[theta] : [pi],
_/-1/2[pi]

or 2 : [pi]. Now 2 area /[pi] = 2[lambda]r; so that, in order to
reconcile the amplitude of the primary wave (taken as unity) with the
half effect of the first zone, the amplitude, at distance r, of the
secondary wave emitted from the element of area dS must be taken to be

dS/[lambda]r (1).

By this expression, in conjunction with the quarter-period
acceleration of phase, the law of the secondary wave is determined.

That the amplitude of the secondary wave should vary as r^-1 was to be
expected from considerations respecting energy; but the occurrence of
the factor [lambda]^-1, and the acceleration of phase, have sometimes
been regarded as mysterious. It may be well therefore to remember that
precisely these laws apply to a secondary wave of sound, which can be
investigated upon the stric