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.

PHYSICAL EXPLANATION OF DOUBLE REFRACTION. 329
going in the same direction, provided that one of these waves
consist of vibrations in one direction, and the other of vibrations in another direction, as if for instance in fig. 26
the directions of both waves were perpendicular to the paper,
but if one set of vibrations were in the direction up and down,
and the other ii the direction right and left. For the force
with which the particles act on each other depends on the
distance of the particles in the direction of the waves' motion,
and on their distance in the direction of the particles' vibration: and in the case supposed, the latter element is
different for the two waves, though the former is the same.
105. If the displacement of a particle, considered as in
any direction, be resolved into three displacements in the
directions of x, y, z, the variations of force il those directions produced by the alteration of a single particle (and
consequently the force produced by the whole system) are
the same as if the displacements in those directions had been
made independently. From this it easily follows that the
sum of any number of displacements causes forces equal to
the sum of the forces corresponding to the separate displacements: and then, by the reasoning in (10) and (11), any
number of undulations, produced by vibrations in different
directions, may co-exist without disturbing each other.
PIOP. 22. To explain the separation of common light
into two pencils by doubly refracting crystals: and to account for the polarization of the two rays in planes at right
angles to each other.
106. We shall assume for the state of the particles of
ether within a crystal, an arrangement similar to that described in (104), or at least possessing this property, that
there are three directions* at right angles to each other, in
which if a particle be disturbed, the resultant of the forces
* M. Fresnel has demonstrated that this must be the case when the small.displacement of a particle in the direction of any one of the co-ordinates produces
forces in the direction of all, represented by multiples of that displacement. This
is apparently the most general supposition that can be made. Memoires de 1'
Institut, 1824.