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.

FORM OF WAVE IN BIAXAL CRYSTAL. 339
equation expressing the two surfaces (which are in fact one
continuous surface) is
(x2 + y2 + z') (a2'2 + b2y' + c'z2) - as(b2 + c2) 2
- b'(a' + c2) y2 - c (a2 + b') z' + a2b2C2 = o.
This cannot be resolved into factors, and therefore cannot
express a sphere and any other surface, as in (111) and
(113). Consequently neither of the rays is subject to the
law of ordinary refraction. This conclusion might also have
been drawn from the observation that neither of the velocities found in (121) is constant. The direction &c. of the
two rays when light is incident on a surface of the crystal
are found exactly as in (114), using the surface above mentioned instead of the sphere and spheroid, and finding the
two positions of the tangent plane passing through the line
projected in A'.
Before leaving this investigation we must remark that
this theory is imperfect in the same degree as the explanation of refraction. In every uniaxal crystal, we believe,
tile axis is the same for all the colours, but the ratio of
a to c is not the same for different colours. In biaxal
crystals generally the direction of the three axes is the
same for different colours, but the ratio of a, b, c, is not
the same, and consequently the position of the optic axes
(122) is not the same for different colours, though the optic
axes for all colours are in the same plane. And it has
been discovered by Sir John Herschel that the direction
of the three axes is in some instances different for different
colours, and then the optic axes for different colours are
not all in the same plane.
PROP. 25. Light polarized in the plane of incidence
falls on a refracting surface of glass &c.: to find the intensity of the reflected and the refracted ray.
126. The three next investigations which we offer to
the reader cannot be considered as wholly satisfactory. The
extreme difficulty of mathematical investigation into the state
of particles at the confines of two media prevents us from
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