isplacements of the same
particle at the end of time t, measured in the directions of the three
axes respectively. Then the first of the equations of motion may be
put under the form

d^2[xi] /d^2[xi] d^2[xi] d^2[xi]\ d^2 /d^2[xi] d^2[eta] d^2[zeta]\
------ = b^2( ------ + -------- + ------- ) + (a^2 - b^2)---( ------- + -------- + --------- ),
dt^2 \ dx^2 dy^2 dz^2 / dx \ dx^2 dy^2 dz^2 /

where a2 and b2 denote the two arbitrary constants. Put for shortness

d^2[xi] d^2[eta] d^2[zeta]
------- + -------- + --------- = [delta] (1),
dx^2 dy^2 dz^2

and represent by [Delta]^2[chi] the quantity multiplied by b^2.
According to this notation, the three equations of motion are

d^2[xi] d[delta] \
------- = b^2[Delta]^2[xi] + (a^2 - b^2) -------- |
dt^2 dx |
|
d^2[eta] d[delta] |
-------- = b^2[Delta]^2[eta] + (a^2 - b^2) -------- > (2).
dt^2 dy |
|
d^2[zeta] d[delta] |
--------- = b^2[Delta]^2[zeta] + (a^2 - b^2) -------- |
dt^2 dz /

It is to be observed that S denotes the dilatation of volume of the
element situated at (x, y, z). In the limiting case in which the
medium is regarded as absolutely incompressible [delta] vanishes; but,
in order that equations (2) may preserve their generality, we must
suppose a at the same time to become infinite, and replace a^2[delta]
by a new function of the co-ordinates.

These equations simplify very much in their application to plane
waves. If the ray be parallel to OX, and the direction of vibration
parallel to OZ, we have [xi] = 0, [eta] = 0, while [zeta] is a
function of x and t only. Equation (1) and the first pair of equations
(2) are thus satisfied identically. The third equation gives

d^2[zeta] d^2[zeta]
--------- = --------- (3),
dt^2 dx^2

of which the solution is

[zeta] = f(bt - x) (4),

where f is an arbitrary function.

The question as t