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.

254 UNDULATORY THEORY OF OPTICS.
the same form as that for a cycloidal pendulum. But the
two expressions
9 7rv 27 X frX
a sin{ t+ (A } andbsin At+ (B- 2t }
v v
cannot be combined into that form, unless - =, which
there is no reason to think true. The consideration therefore of waves of different lengths tnay be kept perfectly separate, as their ultimate effect will be the same as the sum
of all their separate effects, without any possibility of their
destroying or modifying one another.
20. The reader is requested to attend to the conventional signification of the following terms.
By a wave we mean all the particles included between two
which are in similar states of displacement and of motion.
For instance, in any one of the cases (3), (y), (s), (e), (~)
of fig. 1 or 2, the particles included between b and b' form
a wave: or those between f' and f" form a wave: &c. It
is easily seen that a wave includes particles in every possible
state of displacement and of motion consistent with undulatory vibration.
The length of a wave we have explained to be the distance between two particles similarly displaced and moving
similarly. The interval, in time, of two waves (that is, the
interval between the arrival of two successive waves at the
sanie point), it will be recollected, is the same as the time
of vibration of any particle, (5).
By the phase of a wave, we shall denote the situation of
a particle in a wave, considered as affecting its displacement
and motion. For instance, b and b' in fig. 1 or 2 are in
similar phases, because their displacements are equal, and
their motions are also equal. But in (/3) fig. 2, b and f are
not in the same phase: for though their displacements are
equal, their motions are in opposite directions. Similarly f
and h are not in the same phase, for their displacements are
different though their motions are equal. It is readily seen