Page 252

252     Wisconsin Academy of Sciences, Arts, and Letters.
towards the center. The molecule will, therefore move in this
line, and not in a circle; and if the plane of the circle E H I H'
be the bounding surface of the crystal, or the surface of erner-
gence of the light, I G will mark the azimuth of the molecular
movements of the emergent ray.
  But if the planes of E H I H' do not pass through the point of
intersection of the spirals it must cut each spiral in a different
point. The figure is drawn to represent this more general case,
the points of intersection with the spirals being severally L and K.
  By joining L K and drawing the radius G I perpendicular to it,
G I will bisect the angle G L K and M', at the intersection of G
I and L K will be the position of the molecule in the plane E E
L I K, which, if the tangential force P only were acting, would be
at L, and if the tangential force Q only were acting, would be at
K. The tangential forces acting at the moment on this molecule
will not be represented by I P' and I Q', but will be tangents at K
and L.
  Now, as D H, the distance between the planes AD B and E H
I, is a larger part of the length of an entire turn of the spiral M S
N K than of the spiral M F L N', the line G I will fall on the
right of G H, the position it would occupy if the two undulations
were equal in length. We may therefore say, as before, that if
the plane E E I were the surface of emergence of a ray from a
crystal, in which it had been subject to the action of the forces
supposed, its plane of polarization, G I, would be turned towards
the right from its original azimuth. The plane of polarization
turns, therefore, in the direction of the winding of the closest spi-
ral, or of the ray of shortest undulation; but it turns in the diree
tion of the gyration of the ray of longest undulation.
  This rotation of the plane, thus demonstrates that the two rays
advance with unequal velocities in the axis of quartz -a remark-
able fact which is not true of any crystal which produces plane
polarization only. It also enables us to determine the relative
velocities, or to ascertain the index of rotatory polarization. For
since nG I bisects the angle between the points K and L, which
mark the relative degrees of advancement of the two rays in their
respective rotations, if we take a thickness 0, which produces a