Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

ALGEBRAICAL EXPRESSION FOR AN UNDULATION. 243
function of the time and of the distance of that point of rest
from some fixed point. Let x be the original distance of any
point in the line (a) from some fixed point: to find an expression for its disturbance at the time t, in a function of v
and t, consistent with the conditions of an undulation. By
the original description of an undulation (1), putting v for
the velocity of the wave's transmission, it is easily seen that
whatever be the state of disturbance of a particle whose original ordinate is x at the time t, the same state of disturbance must hold at the time t + t' for a particle whose original ordinate is w + vt'. Or if p express the form of the
function,
( (x, t) must = ( (v + vt, t + t'),
whatever be the value of t'. It will be found on trial that
()(vt - ) satisfies this condition, ( being any function whatever. For putting t + t' for t, and - + vt' for x, it becomes
(v. t + t'- t + vt') = (vt - i),
the same as before. But it may be found analytically thus.
Expanding the second side we have
d. >(x^t) 'd..cp(x,t)
(it) = (i,t) + d vt' + dt t' + &c.,
d. p (x, t) d. (, t)
or V --- + - -= 0O
dv dt
the general solution of which gives / (x, t) = ((vt - Zv)*.
7. This expression however is too general to be of much
use to us, and we will choose a particular form that will be
more convenient. Suppose we fix on this condition to determine the form of the function: the vibration of each particle shall follow the same law as the vibration of a cycloidal
pendulum. The distance of a cycloidal pendulum from its
place of rest is expressed by
* This is the expression found, by investigation from mechanical principles,
for the disturbance of the particles of air when sound passes along a tube of
uniform bore, or for the disturbance of an elastic string (as that of a musical
instrument) fixed at both ends.
16-2