Motion of Solid and Fluid Bodies. 191 
which will be found useful in discussing the conditions at the limits for this par- 
ticular case. Previously, however, to examining the limits, it is necessary to 
consider the laws of propagation ; and here I shall confine my attention to the 
particular integral for plane waves already used. In the present case equations 
(29) become 
P=n(32°+y7?+2’), Pp =Nn(2??+1), 
Q=n(37 + a’ +42), Q’ = Nn(2m?>+ 1), 
Ro=N(32°+ 2° +7’), rR’ = n(2n?+ 1), (53) 
F = 2nyz, F’ = 2nmn, 
G =2naz, Gc’ = 2nln, 
H = 2nxry, H’ = 2nlm 
Substituting these values in (28), we obtain 
ev cosa = N((2/° + 1) cosa + 2/mcosB + 2ncosy), 
ev-cosB = N((2m>+ 1) cosB + 2mncosy + 2/mcose«), (54) 
ev'cosy = N((2n*-+ 1) cosy + 2/ncosa + 2mncosp); 
from which we deduce, by multiplying by cosa, cosp, cosy, respectively, and 
adding 
ev’ = N(1 + 2(/cosa + mcosp + neosy)*). (55) 
It is evident, on inspection, that cosa = /, cos8 = m, cosy = n, will satisfy equa- 
tions (54), and consequently that one set of vibrations are normal to the wave- 
plane, and the other two in the wave-plane. The equation (55) will give for the 
two cases respectively, 
ev’ = 3N, 
ev? = N, 
the greater velocity belonging to the normal vibration. 
These results may also be deduced from the surface of wave-slowness (31), 
for, by substituting in this equation the values (53), and expanding, it becomes 
[Sn(o* +y* +2’) — 1) [Oty +2") —1P = 0, (56) 
which shows that it is composed of two concentric spheres, whose radii are 
1 1 ’ : 
Ue ax which, therefore, represent the wave-slownesses of the corresponding 
vibrations. 
VOL. XXI. 25D 
