test mechanical principles.

The recomposition of the secondary waves may also be treated
analytically. If the primary wave at O be cos kat, the effect of the
secondary wave proceeding from the element dS at Q is

dS dS
------------- cos k(at - [rho] + 1/4[lambda]) = ------------- sin k(at - [rho]).
[lambda][rho] [lambda][rho]

If dS = 2[pi]xdx, we have for the whole effect

_[oo]
2[pi] / sin k(at - [rho])x dx
- -------- | ---------------------,
[lambda] _/ 0 [rho]

or, since xdx = [rho]d[rho], k = 2[pi]/[lambda],

_[oo] _ _
/ | |[oo]
-k | sin k(at - [rho])d[rho] = | -cos k(at - [rho])| .
_/r |_ _|r

In order to obtain the effect of the primary wave, as retarded by
traversing the distance r, viz. cos k(at - r), it is necessary to
suppose that the integrated term vanishes at the upper limit. And it
is important to notice that without some further understanding the
integral is really ambiguous. According to the assumed law of the
secondary wave, the result must actually depend upon the precise
radius of the outer boundary of the region of integration, supposed to
be exactly circular. This case is, however, at most very special and
exceptional. We may usually suppose that a large number of the outer
rings are incomplete, so that the integrated term at the upper limit
may properly be taken to vanish. If a formal proof be desired, it may
be obtained by introducing into the integral a factor such as
e^-h[rho], in which h is ultimately made to diminish without limit.

When the primary wave is plane, the area of the first Fresnel zone is
[pi][lambda]r, and, since the secondary waves vary as r^-1, the
intensity is independent of r, as of course it should be. If, however,
the primary wave be spherical, and of radius a at the wave-front of
resolution, then we know that at a distance r further on the amplitude
of the primary wave will be diminished in the ratio a:(r + a). This
may be regarded as a consequence of the altered area of the first
Fresnel zone. For, if x be its radius, we have

/
{(r + 1/2[lambda])^2 - x^2} + \/ {a^2 - x^2