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.

266 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS.
The geometric construction shows that we have here the long known construction
by which the apsidal surface of a given surface is derived. Accordingly the above
contact transformation is possessed of the very important property of changing ellipsoids
into Fresnel wave surfaces.
The finite equations of the transformation (46) expressing z', xi', pi' as functions
of z, xi, pi may be obtained without difficulty by the method of ~ 7. If this transformation be combined with those of the one-parameter group (37) we shall have o02
contact transformations which change the points of space of any dimensions into the
wave surfaces of that space.
12. This suggests the interesting problem of finding all those contact transformations
which change every wave surface into a wave surface, that is, those contact transformations which leave the family of all wave surfaces invariant.
Analytically the problem may be approached either by determining the finite
transformations or the infinitesimal transformations which leave the partial differential
equation of the wave surface invariant. From either starting point the difficulties in
the way of integrations to be effected are well-nigh insurmountable. This ought not to
be surprising since all contact transformations of ordinary space changing plane into
plane have not been determined (though Lie has found all those that change surfaces
of constant curvature into surfaces of constant curvature irl ordinary space, and lately
the most general contact transformation leaving unaltered the family of developable
surfaces of n+ 1-dimensional space has been found).
An indirect method for finding contact transformations transforming wave surfaces
into such may be employed by using the results of a beautiful memoir of M. Maurice
Lévy, "Sur les équations les plus générales de la double refraction compatibles avec la
surface de l'onde de Fresnel," Conptes Rendus, t. 105, pp. 1044-1050.
Without making any assumption whatever relative to the nature of a luminous
vector Lévy proposes to find its most general form compatible with the Fresnel wave
surface. His problem narrows itself to determining the most general expressions of the
second derivatives, with regard to the time, of the three components of the luminous
vector as functions of the various second derivatives of these components with regard
to the coordinates of the point of the medium which produces the light, by means of
the condition of reproducing the equation of velocities and hence the wave surface.
The equations to be invariant in this method are more numerous, but simpler in
form than the partial differential equation of the surface of waves.
For reference Lévy's system of equations is appended here. Letting u, v, w be
the components of the luminous vector, t the time, x, y, z the coordinates of the
point of the medium which produces the light, a, b, c the reciprocals of the principal
indices of refraction, a, /3, 7 three arbitrary constants, and X,,/, z three other arbitrary