128 Professor Hamiiton’s Third Supplement 
ISTIC FUNCTION, to the study of which may be reduced all the problems of mathema- 
tical optics. But the problem of connecting by general equations the direction and 
velocity of a ray with the direction and with the law of normal velocity of a wave, 
has been elegantly resolved by M. Caucuy, in the 50th Livraison of the Evercices 
de Mathématiques : and the formule which have been there deduced by considering 
the normal velocity as a homogeneous function of the first dimension of its three 
cosines of direction, may easily be shown to agree with the equations (D"). 
Theory of Fresnet, New Formule, founded on that theory, for the Velocities 
and Polarisations of a Plane WVave, or WVave-Element. New method of deducing 
the Equation of Fresnew’s Curved JV ave propagated from a Point in a Uniform 
Medium with. Three Unequal Elasticities. Lines of Single Ray-Velocity, and 
of Single Normal-Velocity, discovered by Fresne.. 
27. Let us now consider more particularly the undulatory theory of Fresnet. 
In that theory, the small displacements of the vibrating etherial points are confined 
to the surface of the wave, the ether being supposed to be sensibly incompressible, 
and so to resist and prevent any sensible normal vibration: and the tangential forces, 
which regulate the tangential or transversal vibrations, result in general from the 
elasticity of the ether, combined with this normal resistance. It is also supposed that 
the etherial medium has in general three principal unequal elasticities, corresponding 
to displacements in the directions of three rectangular aves of elasticity ; in such a 
manner that if we take these for the axes of co-ordinates, any small component dis- 
placements dx, ay, dz parallel to these three axes will produce elastic forces —a%éz, 
—b°ey, —c*éz parallel to the same axes, and equal to the displacements taken with 
contrary signs and multiplied by certain constant positive factors a’, b*, c’: and any 
small resultant displacement, 8/, in any other direction, having 62, ey, dz for its com- 
ponents or projections, will produce a corresponding elastic force — Hél, of which 
the components are —a*dx, —6°8y, —c’dz, and which has not in general the same 
direction as the displacement &, nor a direction exactly opposite to that. Light, 
polarised in any plane P, is supposed to correspond to vibrations perpendicular to 
that plane, and propagated without change of direction ; and in order that a vibration 
should thus preserve its: direction unchanged, while the plane wave or wave-element 
to which it belongs is propagated through the uniform medium with a normal velocity 
w, it is necessary and sufficient that the elastic force —H3/, when combined with a 
normal resistance arising from the incompressibility of the ether, should produce a 
tangential force —w°é/, in the direction opposite to the displacement 8/, and equal to 
this displacement taken with a.contrary sign, and multiplied by the square of the nor- 
