200 REPORT—1885. 
and that hence 
ein. Bip seee ue) 
Ss zl: cos (raydv+ ¢ (I Ge ww!')dv . (73) 
It is proved that é and wo’, w’’, w'”’ satisfy the equation 
leh 
(74) 
2 
= 6277? w 
and hence, by Poisson’s solution, 
t ara ‘ 
— {| F(at)do +  & {trea} peace) 
where f and F are the initial values of é and dé/dt respectively. 
If, then, the values of 6 and dé/dt, w and dw/dt be given initially 
everywhere, the last equation, with the similar one for w, enable us to find 
d and w at any moment throughout the space considered, and then the 
equation (73) give us &, n, and ¢. 
In solving the equations for 6, w, it is clear that if we first find the 
part of the solution due to the initial velocity, the part due to the initial 
displacement may be obtained by substituting in the solution for the 
initial velocity the initial displacement, and then differentiating with 
regard to the time; and this proposition is proved generally for a system 
in which the forces depend only on the configuration of the system, and 
which is executing small vibrations about an equilibrium position. 
The integrals are then modified by suitable transformations. 
For 2, we have £;= of where Wy = -z|||- dv. 
/ us 
Thus — 4rw is the potential of matter distributed throughout space 
with density 6, and finally it is shown that 
y= — Pall (up@ + Voy + woz) a (rat) : eG76) 
where wo, Vp, Wy are the initial values of the velocities at the point 2’, y’, 2’, 
at which dv is an element of volume, 7 the distance between 2’, y’, 2’ 
and @, y, z, the point at which W is to be found. From this ¢, can be found, 
and in a similar manner £,. The terms £,, 7, ¢, arise from a wave of 
dilatation which is in general set up by any arbitrary displacement, and 
which travels through the medium with velocity a. If the initial disturb- 
ance be such that 09 = dé, /dt = 0 everywhere, then this wave will not be 
formed. 
The terms £5, no, f arise from a wave of distortion which traverses 
the medium with velocity b. If a disturbance be produced at a point O, 
and last there for a time 7, then the motion at a point P, at a distance r 
from O, will not commence until after an interval t, where t=r/a, P will 
be disturbed by a wave of dilatation lasting for an interval 7; it will be 
disturbed by the wave of distortion after a time 7/b, and this disturbance 
will last for an interval r. 
