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.

246 UNDULATORY THEORY OF OPTICS.
(v2 being a constant, = mgH in the common notation, and X
being the disturbance from the state of rest), and the solution is
X = q (vt -.) + q (vtt + a),
where the form of the functions is to be determined by the
initial circumstances. Or if we suppose the wave of air to
move only in one direction, the expression for the disturbance
will be p(vt - x). And this may be divided into several
different expressions,
(vt - X) + X (v t - )) + v (v t - ) + &c.
where the form of each is to be determined by the initial
circumstances, or by the cause of the undulation. If there
was only a single original cause for the undulations, there
would be only a single term xr (v t - ) to be preserved. But
if there were several distinct original causes for the undulations, there would be a single term corresponding to each
of these to be preserved, and the whole disturbance would
be the sum of all these terms. And it is particularly to
be remarked, that the whole disturbance thus found as the
effect of all the original causes together, is precisely the sum
(with their proper signs) of a number of disturbances, each
of which would have been produced by one of the original
causes acting separately.
11. Now if we examine to what this property of the
solution of the differential equation (namely that it can be
broken up into several parts all similar to each other and
to the whole) is due, we find it is owing to the circumstance that the differential coefficients of u were raised only
to the first power in the equation, or (to express it in
other words) that the equation was linear. For the differential coefficient of the sum of a number of functions is the
same as the suni of the differential coefficients: but the
square of the differential coefficient of a sum of functions
is not the same as the sum of their squares, &c. If then
the differential coefficients (and the unknown quantity itself
if it enters in the equation) be all of the first dimension,
the substitution of a sum of functions is the same as the
sum of their substitutions separately, and therefore if each
of those functions satisfies the equation, their sum will