172 The Rev. S. Hauveuton on the Equilibrium and 
velocities ; and if a radius vector be drawn from the centre, it will pierce this sur- 
face in three points, and the lengths of the three radii vectores measure the three 
velocities of wave-propagation possible for the given direction. This surface, 
however, being only the locus of feet of perpendiculars or tangent planes to 
wave-surface, is of very little use, but it enables us to determine the reciprocal 
polar of the wave-surface ; for, if the radii vectores be made the reciprocals of 
their lengths, the direction remaining the same, we shall have for the locus of 
their extremities the surface of wave-slowness. Its equation is, 
(p’p'—1) (a'p°—1) (R/p'—1) — F“p'(P’p’—1) —G"*p'(a'p—1) — "p'(R'p?—1) 
Beis iG) se 
or: + 2reHu.p = 0, 
(P—1) (@—1) (r—1)—F*(p—1) —6?(Q—1) — n°(R—1) + 2FGH = 0, (31) 
(P; Q, Rs F, G, H) being the quantities given in (29). 
This surface is of the sixth degree, and has three sheets, corresponding to the 
three velocities of wave-propagation, and determines not merely the laws of pro- 
pagation of plane waves in a solid, but also enables us to give a construction for 
the direction of waves refracted in passing from one solid into another. 
With a point in the surface of separation (supposed plane) as centre, construct 
the two surfaces (31) for both solids; produce the normal to the incident wave 
to meet its own surface, and from the point in which it pierces it, let fall a per- 
pendicular on the separating plane; this perpendicular will pierce the surface 
(31) of the second solid in three points; the lines drawn from these points to 
the centre are the normals of the three refracted waves, and their lengths are 
inversely proportional to the wave velocities. 
The directions of the internal reflected waves may also be easily found by means 
of this surface ; for we have only to produce the perpendicular backwards to 
meet the surface (31) of the first solid; and the three lines joining the centre 
with the three points of intersection will be the normals to the three plane waves 
reflected back into the first solid. These and all other constructions for the 
direction of reflected and refracted waves may be easily proved by the properties 
of the wave-surface and the reciprocal properties of the surface of wave-slowness, 
which, for such constructions as these, serves all the purposes of the wave-surface 
itself. The whole theory of wave-surfaces, in light, or in elastic solids, is only a 
development of Huygen’s construction for uniaxal crystals. 
