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<ds, the integration being along the original path A0 ...
A; and similarly the optical distance between A0B0 and B is
represented by [int] ([mu] + [delta][mu])ds, the integration being
along B0 ... B. In virtue of (4) the difference of the optical
distances to A and B is
_ _
/ /
| [delta][mu]ds (along B0 ... B) - | [delta][mu]ds (along A0 ... A) (5).
_/ _/
The new wave-surface is formed in such a position that the optical
distance is constant; and therefore the _dispersion_, or the angle
through which the wave-surface is turned by the change of
refrangibility, is found simply by dividing (5) by the distance AB.
If, as in common flint-glass spectroscopes, there is only one
dispersing substance, [int] [delta][mu] ds = [delta][mu].s, where s is
simply the thickness traversed by the ray. If t2 and t1 be the
thicknesses traversed by the extreme rays, and a denote the width of
the emergent beam, the dispersion [theta] is given by
[theta] = [delta][mu](t2 - t1)/a,
or, if t1 be negligible,
[theta] = [delta][mu]t/a (6).
The condition of resolution of a double line whose components subtend
an angle [theta] is that [theta] must exceed [lambda]/a. Hence, in
order that a double line may be resolved whose components have indices
[mu] and [mu] + [delta][mu], it is necessary that t should exceed the
value given by the following equation:--
t = [lambda]/[delta][mu] (7).
8. _Diffraction Gratings._--Under the heading "Colours of Striated
Surfaces," Thomas Young (_Phil. Trans._, 1802) in his usual summary
fashion gave a general explanation of these colours, including the law
of sines, the striations being supposed to be straight, parallel and
equidistant. Later, in his article "Chromatics" in the supplement to the
5th edition of this encyclopaedia, he shows that the colours "lose the
mixed character of periodical colours, and resemble much more the
ordinary prismatic spectrum, with intervals completely dark interposed,"
and explains it by the consideration that a]]>
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