ny phase-difference which may
arise at neighbouring striae is multiplied in proportion to the total
number of striae.

The theory was further developed by A. J. Fresnel (1815), who gave a
formula equivalent to (5) below. But it is to J. von Fraunhofer that we
owe most of our knowledge upon this subject. His recent discovery of the
"fixed lines" allowed a precision of observation previously impossible.
He constructed gratings up to 340 periods to the inch by straining fine
wire over screws. Subsequently he ruled gratings on a layer of gold-leaf
attached to glass, or on a layer of grease similarly supported, and
again by attacking the glass itself with a diamond point. The best
gratings were obtained by the last method, but a suitable diamond point
was hard to find, and to preserve. Observing through a telescope with
light perpendicularly incident, he showed that the position of any ray
was dependent only upon the grating interval, viz. the distance from the
centre of one wire or line to the centre of the next, and not otherwise
upon the thickness of the wire and the magnitude of the interspace. In
different gratings the lengths of the spectra and their distances from
the axis were inversely proportional to the grating interval, while with
a given grating the distances of the various spectra from the axis were
as 1, 2, 3, &c. To Fraunhofer we owe the first accurate measurements of
wave-lengths, and the method of separating the overlapping spectra by a
prism dispersing in the perpendicular direction. He described also the
complicated patterns seen when a point of light is viewed through two
superposed gratings, whose lines cross one another perpendicularly or
obliquely. The above observations relate to transmitted light, but
Fraunhofer extended his inquiry to the light _reflected_. To eliminate
the light returned from the hinder surface of an engraved grating, he
covered it with a black varnish. It then appeared that under certain
angles of incidence parts of the resulting spectra were _completely
polarized_. These remarkable researches of Fraunhofer, carried out in
the years 1817-1823, are republished in his _Collected Writings_
(Munich, 1888).

The principle underlying the action of gratings is identical with that
discussed in S 2, and exemplified in J. L. Soret's "zone plates." The
alternate Fresnel's zones are blocked out or otherwise modified; in
this way the original compensation is upset and a revival of li