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.

INTERFERENCE OF UNDULATIONS. 247
satisfy the equation. But if they are raised to a higher
power, the substitution of the sum is not the same as the
sum of the substitutions, and therefore if each function
satisfies the equation, their sum will not.
12. If now we retrace the steps of the investigation
for air, it will be seen that the linearity of the differential
equation depends upon this physical fact, that upon altering
by a small quantity the relative position of particles, the
forces which they exert undergo variations very nearly proportional to that small quantity. And in any other case
where this holds, the equations will be linear; and the
wave-disturbance of any particle, produced by a number of
agitating causes, will be the sum of all the wave-disturb.
ances which these causes would singly have produced. We
can hardly conceive any law of constitution of a medium
in which undulations are propagated, where this does not
hold, and we shall therefore suppose it to be true for light.
13. Taking it then as a fact that the disturbance of
every particle produced by two co-existent undulations will
be the sum of the disturbances which they would produce
separately, we will consider the nature of the disturbance
produced by the superposition of two such undulations as
those treated of in (7) and (8), each of which is represented geometrically by fig. 1, if the vibrations are in the direction of the wave's transmission, and by fig. 2, if they are
perpendicular to that direction. For convenience of figure,
we will suppose them of the latter class: but ail that we
say will apply as well to the former. We will suppose
the length of a wave the same in both undulations. In
fig.3, let the Italic letters of (a) represent the state of an
undulation, at the time T, where the law of vibration is
a.sin-(vt - x + A),
and let (f3) represent the state of another undulation at the
same tirne where the law of vibration is
b sin - (v t - + B).
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