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.

DOUBLE REFRACTION OF BIAXAL CRYSTALS. 337
force of displacement into two, one parallel to the direction
of displacement, the other perpendicular to it (not necessarily perpendicular to the front of the wave), and shall
neglect the latter. If the direction of displacement make
angles X, Y, Z, with the axes of x, y, z, this resolved
force, as in (lii), is
displacement x (a2 cos' X + b cos2 Y + c' cos2Z).
Construct a surface of which the latter factor is the radius,
which we shall call the surface of elasticity; it is easily
seen that the radius is the squared reciprocal of the radius
I 1 1
in the ellipsoid whose axes are -, -
120. Make a section by the plane front of the wave
through the center of this surface; the radius vector of the
section will be the square of the reciprocal of the radius vector
in the corresponding section of the ellipsoid, that is in an
ellipse; and this section of the surface of elasticity will
therefore be a curve symmetrical with respect to its greatest and least dianieters, which are at right angles.
121. The radius vector of this section in any direction
represents the resolved part, in that direction, of the force
produced by displacement in that direction, the neglected
part being perpendicular to that direction and not necessarily perpendicular to the front of the wave. If now we
examine the direction of displacement in which the neglected part is perpendicular to the front of the wave, it
is found that the greatest and least diameters above alluded to are the only ones which satisfy this condition.
Consequently the vibrations must be resolved into two, parallel respectively to these diameters; and these will produce
the two rays. Their velocities will be represented by the
square roots of the values of those semi-diameters.
122. In two positions of the front of the wave, and
no more, the section becomes a circle. Whatever then is
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