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.

260 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS.
These transformations are obviously characterized geometrically by the property of
changing the surface-element (xi,..., xn, z, pl,..., pn) into the surface-element
(X/,... Xn,, z', p'., pn') in such a manner that the point of the second lies on the
normal to the first and at a constant distance t from its surface. They transform the
surface-elements of a point into those of a sphere, and change parallel surfaces into
such.
6. As Lie has pointed out for ordinary space the theory of wave-motion in an
isotropic elastic medium is intimately related to the one-parameter group of dilatations of
the space filled by the medium.
Consider a wave-motion originating at a center of disturbance P0 of an isotropic
n + 1-diinensional elastic medium; in an interval of tirne the motion will have advanced
to all points P of a sphere whose center is at P0 and whose radius is t, say, in
precisely the same manner as the dilatation (21) would change the surface-elements of
the point P0 into those of the last-named sphere. Every point P of this sphere can
now be regarded as the center of new elementary waves which in a second interval
of time, say tl, will have advanced to spheres of equal radii t, about the points P as
centers. These elementary waves have an outer envelope, which by Huygens' principle
is the identical wave that would have been developed from the original center P0 in
the total time elapsed. But in exactly the same manner the dilatation
pitl t,
= l + + I =V - i pf=, '... (i=l,..., n)......(22)
carries every point P of the sphere about P0 into a sphere of radius t, about P as
center, so that the sphere of center P0 will be changed by the dilatation (22) into
the sphere of center P0 and radius t1 +t0, that is into the sphere into which the point
P0 is changed by the successive application of the dilatations (21) and (22).
Thus the principle of Huygens finds its mathematical expression in the fact that
all dilatations form a one-parameter continuous group.
The importance of this particular group of contact transformations is further
exhibited by observing that reflections and refractions from one isotropic medium to
another are contact transformations which leave the infinitesimal dilatation invariant;
the reflections have the additional property of being commutative with the latter. To
establish these facts it is only necessary to make the ordinary illustrative constructions
in a space of n + 1 dimensions and apply the principle that all the surfaces of a
complex f that touch a surface b have in general an envelope cP, and hence the
passage from q to P is a contact transformation.
7. Let the characteristic function be an arbitrary function of pi,..., pn, say
nzi=nt(a, (pi, P,.......................................... (23);
the infinitesimal transformation defined by I is represented by the equations
Sxi = nT,Ït, Sx = 2pn, - nI, Spi = 0,...... (i= l,..., n)...............(24).