Motion of Solid and Fluid Bodies. 189 
wave-slowness, and reciprocal to the tangent cone at the node. Also, there may 
be one ray, and an infinite number of waves; for if we consider that there are 
four tangent planes of the surface (B), which touch the surface along an ellipse, 
it is evident that there might be a cone of refracted wave normals, of the second 
degree, whose base is one of the ellipses of contact, while there would only be one 
ray, whose direction is the perpendicular on tangent plane: to find, in this case, 
the cone of incident waves which will be refracted into a single ray, we must pro- 
ject, by perpendiculars to the plane of separation, the ellipse of contact of surface 
(8) upon (a); then the cone whose base is the projection will be the cone of in- 
cident wave normals; while the cone whose base is the ellipse will be the cone of 
refracted wave normals, and the unique direction of the ray will be the perpendicu- 
lar to the plane of the ellipse. 
II. If the molecules of the body be arranged symmetrically round one axis, 
the differential equations of motion will still be the same as (47), with the follow- 
ing relations among the constants : 
Al="Bi= ONG L = ™. 
In these equations the axis of z is the axis of symmetry ; two of these relations 
are evident, a= 8, L = M; and the third may be thus proved. The function 
(¥,), in the coefficients (11), is in general a function of (p, @,); but since, in 
the present case, everything is similar for all planes passing through the axis of 
symmetry, F, will not be a function of @, therefore the expressions for A, B, N 
may be integrated once with respect to ¢, 
A= B= 2S costada, N = 2§\S\F,cos*acos*Bdw ; 
or, substituting for cosa, cosp, their values, and assuming 
QO = 2SSr,sin’p*dpdo, 
we shall have 
N= BS D.cos'pild, N= { 2-sin'geos'gde, 
where Q is independent of @; and, finally, integrating these two expressions, we 
shall obtain 
ONS 
These relations introduce corresponding simplifications into the surface of wave- 
