o the law of the secondary waves is thus answered by
Stokes. "Let [xi] = 0, [eta] = 0, [zeta] = f(bt-x) be the
displacements corresponding to the incident light; let O1 be any point
in the plane P (of the wave-front), dS an element of that plane
adjacent to O1, and consider the disturbance due to that portion only
of the incident disturbance which passes continually across dS. Let O
be any point in the medium situated at a distance from the point O1
which is large in comparison with the length of a wave; let O1O = r,
and let this line make an angle [theta] with the direction of
propagation of the incident light, or the axis of x, and [phi] with
the direction of vibration, or axis of z. Then the displacement at O
will take place in a direction perpendicular to O1O, and lying in the
plane ZO1O; and, if [zeta]' be the displacement at O, reckoned
positive in the direction nearest to that in which the incident
vibrations are reckoned positive,

dS
[zeta]' = ------ ( 1 + cos[theta]) sin[phi] f'(bt - r).
4[pi]r

In particular, if

2[pi]
f(bt - x) = c sin -------- (bt - x) (5),
[lambda]

we shall have

cdS 2[pi]
[zeta]' = ---------- (1 + cos[theta]) sin[phi]cos -------- (bt - r) (6)."
2[lambda]r [lambda]

It is then verified that, after integration with respect to dS, (6)
gives the same disturbance as if the primary wave had been supposed to
pass on unbroken.

The occurrence of sin [phi] as a factor in (6) shows that the relative
intensities of the primary light and of that diffracted in the
direction [theta] depend upon the condition of the former as regards
polarization. If the direction of primary vibration be perpendicular
to the plane of diffraction (containing both primary and secondary
rays), sin [phi] = 1; but, if the primary vibration be in the plane of
diffraction, sin [phi] = cos [theta]. This result was employed by
Stokes as a criterion of the direction of vibration; and his
experiments, conducted with gratings, led him to the conclusion that
the vibrations of polarized light are executed in a direction
_perpendicular_ to the plane of polarization.

The factor (1 + cos [theta]) shows in what manner the secondary
disturbance depends upon the direction i