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.

242 UNDULATORY THEORY OP OPTICS.
the distance from a to a', a' to a", &c. at T, or the distance from d to d', d' to d", &c. at T +-, &c.) is wholly
independent of the extent of vibration of each particle. For
if each particle vibrated only half as far as is now supposed,
still at T, a would be a point where the particles are most
condensed, and a' would be the next point where they are
most condensed, and a" the next, &c. The interval between
similar points of two waves (which we shall call the length
of a wave, and shall always denote by the letter X) would
be the same as at present: the only difference would be
that the particles about a, a', &c. would not be so closely
condensed, nor those about g, g', &c., so widely separated
as at present. Similarly the length of a wave in fig. 2
would be unaltered if the vibration of the particles were
altered in any ratio: the only difference would be that the
elevation of the high points and the depression of the low
points would be altered in that ratio.
Plto. 3. The length of a wave depends on the velocity of transmission, and on the time of vibration of each
particle.
5. In the cases both of fig. 1 and of fig. 2 (and in every
other conceivable case of a continued sequence of waves) we
see that every particle has returned to the same state at
T + Tr as at T, that is, that the vibration of every particle
is completed in the time r. But in this time the wave has
appeared to glide over a space equal to the interval between
corresponding points of two waves, or X. Hence we find,
Space described by the wave in the time of vibration
of a particle = X.
Velocity of wave =...
o w = time of vibration of a particle
PROP. 4. To express algebraically the transmission of
an undulation.
6. The quantity for which we shall seek an expression
is, the distance of any point from its point of rest, in a