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The Rev. 8S. Hauauron on the Equilibrium and 
Now, the equation which determines the axes of the ellipsoid lying in the 
plane (2, 7) is 
H,tan*d + (P, — Q,). tand —H, = 0. 
If we make tang = 7 we shall have the condition necessary to be fulfilled, in 
order that the other two vibrations may hie, one in the wave-plane, and the other 
normal to it; this condition is 
Im[(a — 3n)P? — (8 — 3n)m*] = 0. 
The factors 7 = 0, or m= 0, give the axes (2, 7), which have been already dis- 
cussed, and the third factor gives the equation 
A— 3N 
B—3n’ 
tan*d = 
which shows that there are two lines in the plane (7,7), making equal angles 
with the axis of x, such that if a wave-plane pass through the solid perpendicular 
to either of them, the three vibrations possible for that direction of wave-plane 
will be two transversal and one normal. 
Thus the existence of nine directions in the solid, for which the vibrations 
are in the wave-plane and normal to it, has been proved ; viz., the three axes of 
symmetry, and two directions in each plane of symmetry. 
To show the effect produced by the existence of nodes in the surface of wave- 
slowness, it will be necessary to consider a wave in its passage from one solid to 
another; the construction for the refracted wave is as follows: describe the sur- 
faces of wave-slowness (A,B) for both media, having a common centre in the 
plane which separates the two media; produce the normal to the incident wave 
to meet the corresponding sheet of its own surface (A), and from the point of 
intersection let fall a perpendicular on the separating plane, this perpendicular 
will in general pierce the surface (8) in three points; and the corresponding 
radii vectores will be the normals to the three refracted waves, the perpendicu- 
lars on the corresponding tangent planes being the directions of the refracted 
rays. Suppose the perpendicular should pierce the surface (B) m a node, the 
line joining the centre of (B) with the node will be the normal to refracted 
wave; but there will be an infinite number of rays, which will form the sides of 
a cone of the second degree, having its vertex at the centre of the surface of* 
