Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 263
Thus the equations (27) and (33) show that all the transformations of the one-parameter
group (26) are commutative with all translations.
8. It is evident either from the last-named property or directly from the form of
equations (27), that by varying t and this operating on a point (x1,... x,, z) with all
the transformations of the group (26), the point is changed successively into similar
surfaces and similarly placed. The point P0 is changed by the transformation whose
parameter is t, into the surface $. Operating on all the points P of 2 with the
transformation whose parameter is t2, these points P will be changed into congruent
surfaces that are similar and similarly placed to 2. These latter surfaces have an outer
envelope, a surface Il into which the surface l is changed by the second transformation.
The successive application or product of the two transformations is equivalent to the
transformation whose parameter is tl+ t2; the latter transformation carries the point P0
directly into the new surface Ci, and this surface must then be a similar and similarly
placed surface to 2.
The preceding geometrical operations and their results suggest the phenomena of
wave-motion in an elastic n + -dimensional medium. If such a space is filled with
such a medium in which motions originating at a point advance in different directions
with velocities depending only on the direction, then a center of disturbance P0 gives
rise to a series of waves similar and similarly placed with the common center of
similarity P0; accordingly the above geometric operations present a pure mathematical
interpretation of Huygens' principle for a non-isotropic elastic medium, and this principle
finds its equivalent in the fact that the oc' contact transformations (26) form a group.
9. The group (26) may be generalized and specialized.
1~. Much more general wave-motions nay be designed by using in a similar
manner the most general infinitesimal contact transformation defined by the characteristic
function
S1 (x x z, P... X, pl);
a simple geometric construction shows that the normal velocity of the wave is given
by the expression.2//1 +;p?2.
2~. The case applying to the optics of a double refracting crystal is given by
the particular form
1 1
= a 2 + a pi2, (=,...,n)........................,n)(34).
Observing that
f o =,2 -1 a.~,-.......................................... (35),
we have
- = - -a2Q-1...................................... (36);