Motion of Solid and Fluid Bodies. 187 
Yo Vwi Vc belonging to the wave whose vibrations are normal to its 
plane, and /;, /4, belonging to the waves in which the vibrations are in the 
plane of the wave. Similar conclusions will, of course, hold for waves parallel] 
to the other planes of symmetry. It may be observed that, in the case of 
these waves, the direction in which the motion is propagated through the 
solid (which corresponds to the ray in the theory of light) coincides with the 
perpendicular on wave plane; for it may be proved that the three axes of co- 
ordinates pierce the surface of wave-slowness in eighteen points, at each of which 
the tangent plane is perpendicular to the radius vector; for the equation of the 
tangent plane is 
du du OU 
oe ek a oe 7) 
u = O being the equation of the surface; but, if we make 4 =0, y= 0, it is 
: d 
easily seen that = 05 w= O, and consequently the tangent planes at the 
points where z pierces the surface are given by the equation 
du 
as (z’—z)=9, 
which denotes a plane parallel to (, 7). 
In general, when a wave-plane passes in any direction through the solid, the 
vibrations of the molecules for which such a direction of wave-plane is possible 
will not lie in the wave-plane ; but there are some cases besides the case of waves 
parallel to the principal planes, in which the vibrations are, one normal to the 
wave-plane, and the other two lying in the wave-plane. 
Let us consider the case of a wave-plane parallel only to one of the principal 
axes (z); the auxiliary ellipsoid becomes for this case 
Pa + ay + R22 + 2n,cy = 1, 
where 
Pp, = aP + nm’, H, = 2nlm, 
Q, = Bm’ + ni’, because 2 = 0, 
R, = M2? + Lm’. 
This equation proves that for every position of waves parallel to an axis, there is 
one vibration in the wave-plane parallel to the axis. 
