nearly the same pitch be struck at the same time, we hear the resultant
sound alternately swell and die away, because the vibrations caused by the
two notes are sometimes going in the same direction, and after an interval
are going exactly in opposite directions. Diagrammatically we should
represent the vibrations by two waves, as below; the upper wave goes
through its period seven and a half times between A and D, the lower only
six times; and it is easily seen that at A and C the waves are
sympathetic, at B and D antipathetic. At A and C the compound vibration
would be doubled; at B and D reduced to insensibility. The point is so
important that perhaps a numerical illustration of it will not be
superfluous. The waves are now represented by rows of figures as below.
The first series recurs after every 6, the second after every 7.

[Illustration: FIG. 8.]

First Wave 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1
Second Wave 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3
-------------------------------------------------------------
Combined Effect 2 4 6 8 7 5 3 3 5 7 7 6 4 4 4 6 6 6 5 5 5 5 5 5 5 6 6 6 4 4 4
Great disturbance. Calm.
-----------------------------------------------------------------------------

First Wave 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2
Second Wave 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3
-------------------------------------------------------------

Combined Effect 6 7 7 5 3 3 5 7 8 6 4 2 4 6 8 7 5 3 3 5 7 7 6 4 4 4 6 6 6 5 5
Great disturbance.

[Sidenote: Illustration from ocean travel.]

Adding the two rows together, the oscillations at first reinforce one
another and we get numbers ranging from 2 to 8 instead of from 1 to 4; but
one wave gains on the other, until it is rising when the other is falling,
and the numbers add up to a steady series of 5's. It will be seen that
there are no less than seven consecutive 5's, and all the variation seems
to have disappeared. But presently the waves separate again, and the
period of great disturbance recurs; it will be seen that in the "combined
effect" the numbers repeat exactly after the 42nd term. Now those
unfamiliar with the subject may not be prepared for the addition of one
physical wave to another, as though they were numbers, but