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.

250 UNDULATORY THEORY OF OPTICS.
motion whatever. To understand this clearly, we must
consider what is meant by the expression
a. sin {2 (v t - ) + B
or, (in this case of destruction of the motion)
a sin {- (v t - V) + A A 7r}.
X J
This is the same as
a sin { (vt- - +A.
Now this is exactly the same expression as
a. sin - (v t- ) + }
putting iv =F instead of x. That is, the expression for the
disturbance in this second undulation, if B = A = 1800, is the
same as that in the first, provided instead of x we take
q: -. That is, one of the undulations may be represented
by the same construction as the other, provided we suppose
it in advance or in arrear of the other by half the length
of a wave. The undulations (f3) and (S), or (y) and (e)
in figures i and 2 have this relation to one another. And
it will very easily be seen in fig. 2, that if we compound
(() with (/3) by a process similar to that which we used in
fig. 3, (13), the elevations of particles in (Î) will correspond
to equal depressions in (/3), and vice versa, and consequently by their combination the particles will all be
brought to their original position. The same will be true
after the time - when (3) has been changed to (y), and at
4
the same time (c) has been changed to (e); and at every
other time: and therefore there will be continued rest.
Thus we arrive at the extraordinary conclusion that one
undulation may be absolutely destroyed by another with