Motion of Solid and Fluid Bodies. 183 
Hence the traces on the principal planes of the surface of wave-slowness consist 
of an ellipse and a curve of the fourth degree; which establishes an analogy be- 
tween the case of solid elastic bodies and light, where the corresponding traces 
are a circle and an ellipse. 
The surface of wave-slowness of symmetrical elastic bodies possesses singular 
points, of the nature of modes, in its principal planes, for the equation of the sur- 
face may be put under any of the forms 
b+A4 =0 
g++ yy, = 0 
d; + 2.4, = 0 
where the values of ¢,, @,, p; are 
¢, = (x—-1) [(p—1) (e—-1) — W] 
$, = (e@—1) [(p—1) (r-1) —@*] 
$s = (p—1) [(e—1) (R—1) — ¥] 
Now, if the equation of a surface be 
ux=o+2.¥=—0, 
it may be shown that, if the surface ¢ = 0 has any singular points in the plane 
z = 0, that these are also singular points in the surface w = 0; for the conditions 
for singular points are 
du du du 
Sa SUA dy — 0, GE is 0; 
and if z= 0, the part of w of the form (zy) will disappear from the three 
equations, 
ales LD agiy = 
dx 
which will then be of the form 
dp _ do_, do _ 
ree URL cane 
which are the conditions for singular points in the surface @¢ = 0. Hence, if any 
singular points of the surface @ = 0 exist in the plane z = 0, these will be also 
VOL. XXI. 2c 
