Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

DIFFERENCE OF RETARDATION OF THE TWO RAYS. 361
154. In estimating then the displacement in the ether
produced by these two separate pencils after emergence frorn
the plate of crystal, if we represent that which is produced
by the ordinary ray by a multiple of
27r
sin — (vt - ),
we must represent that produced by the extraordinary ray
by a multiple of
2 7r 2 7 27rl\
sin A {vt - (x - I)} or sin.- (v t - () + - }.
For the extraordinary ray is less retarded than the ordinary
ray by the space I in air, and therefore the displacement
really caused by the extraordinary ray will correspond to
that which would have been produced at a space less advanced by 1* if they had been equally retarded.
PRoP. 32. A plate of Iceland spar &c. bounded by planes
perpendicular to the axis of the crystal (as in Prop. 31) is
placed between the polarizing and analyzing plates fig. 24:
to investigate the intensity of the light in various parts of
the image seen after reflection at B.
155. In fig. 35 conceive the direction of any ray to be
perpendicular to the paper: let the plane passing through
this ray and through the axis of the crystal {which in (111)
we have termed the principal plane for that ray} make with
the plane of first polarization the angle (: and let the plane
of polarization at the analyzing plate (which we shall call
the plane of analyzation) make with the plane of first polarization the angle a. Let the vibration in the rays as
first polarized be represented by
* The same expression applies when two plates cut in the same way from
crystals either of the same or of different kinds are applied together, I being
now the space by which in the combination the extraordinary ray is less retarded
than the ordinary ray. If in both plates the extraordinary ray is less retarded
than the ordinary ray, or in both more, the effect of the combination is that of
a thick plate: if in one it is less and in the other more retarded, the effect is
that of a thin plate.