Motion of Solid and Fluid Bodies. Te 
the directions of the axes of this surface are the three possible directions of mole- 
cular vibration for the given wave-plane ; so that, for a given direction of wave- 
plane, three waves are possible, the corresponding vibrations of the molecules of 
the solid being at right angles to each other. The truth of this construction 
appears from the relations, 
1 1 l 1 ] ] 
P= — t= — 2 = —— FF =, a =, Be, 
P, Pi, Pin Lip ihr US 
which are easily proved. Also, since we may suppose all the parts of the vibra- 
ting body contained in a plane parallel to dv + my + nz = 0, to be of the same 
density during the progress of the moving plane which contains them, we shall 
have the three velocities of wave-propagation inversely proportional to the three 
axes of the ellipsoid (30).* 
If wave normals were drawn from the common centre of the ellipsoids (29) in 
every possible direction, and the corresponding ellipsoids (30) constructed for 
each direction, and if on each normal three intercepts were measured, recipro- 
cally proportional to the axes of the ellipsoid, the locus of the extremities of these 
intercepts would be the surface of wave-velocity, or locus of feet of perpendiculars 
from centre on tangent planes of wave-surface; and the surface formed by radii 
vectores, which are the reciprocals of the wave velocities, will be the surface of 
wave-slowness, or the reciprocal polar of the wave-surface, and a knowledge of its 
properties will serve all the purposes of the wave-surface itself; it is analogous to 
the index-surface in the theory of light. The surfaces of wave-velocity and wave- 
slowness may be determined by the following considerations : 
The cubic equation whose roots are the squares of the reciprocals of the axes 
of the ellipsoid (30) is, 
(P’—s) (Q’—s) (2’—s) — ¥?(P’—s) — 6”(Q’—s) — n”(R’—s) + 2r’e'h’ = 0. 
If in this equation we substitute (a* + 7° + 2’) for s, and & 4 =| for (/, m,n), 
P 
we shall have the equation of the surface, whose radii vectores measure the wave 
* After having obtained this construction, I learned that M. Cauchy had given a similar one 
for a particular case; but he has not applied his results to their proper object, and has been mis- 
led in his attempt to account for the phenomena of light by his equations: his method, also, is 
quite different from that used in this Paper. 
