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.

330 UNDULATORY THEORY OF OPTICS.
acting on it will tend to move it back in the same line in
which the displacement is produced. These lines we suppose
to be parallel to some lines determined by the form of the
crystal.
107. Now in general the displacement* of a particle
or a series of particles will not produce a force whose direction coincides with the line of displacement. For suppose
the disturbance in the direction of x to be X; that in the
direction of y to be Y: and suppose the corresponding forces
to be a X and b2Y. The tangent of the angle made by
b2 Y
the resultant force with the axis of o is 2-: but the tana2X
gent of the angle made by the direction of displacement
Y
with the axis of x is-: and these are different if a2 and
X
b2 are different. In the same manner if we supposed a displacement Z in the direction of z, and if it produced a force
c'Z, the tangents of the angles, made by the projection of
the resultant's direction on the planes of xo and yz with
a2X b2 Y
the axis of z, would be- and --: while those made
c Z c'Z
by the projection of the line of displacement would be -
and -.
108. Now suppose that, in fig. 26, MN is the front of
a wave: or by the definition of (20) and the assumptions
* We have spoken here of displacements as if the forces concerned in the
transmission of a wave were thus put in play by absolute displacements. It is
however plain from (103) that the forces on A really put in play are produced
by relative displacements: but it is evident that these forces are the same as
those that would be put in play by the absolute displacement
d2u h2
(2U -, - u') or d 2.-.
âdx2 2
In like nlanner, when the direction of displacement is any whatever, the quantity
>f (2u -,- u') in its proper direction may be resolved into the direction of the
co-ordinates, and the forces really acting on A will be the forces corresponding
to these spaces considered as absolute displacements.