We have next to combine the vibrations of two forks, one of which oscillates with twice the rapidity of the other; in other words, to determine the figure corresponding to the combination of a note and its octave. To prepare ourselves for the mechanics of the problem, we must resort once more to our pendulum; for it also can be caused to oscillate in one direction twice as rapidly as in another. By a complicated mechanical arrangement this might be done in a very perfect manner, but at present simplicity is preferable to completeness. The wire of our pendulum is therefore permitted to descend from its point of suspension, A, Fig. 175, midway between two horizontal glass rods, _a b_, _a′ b′_, supported firmly at their ends, and about an inch asunder. The rods cross the wire at a height of 7 feet above the bob of the pendulum. The whole length of the pendulum being 28 feet, the glass rods intercept one-fourth of this length. On drawing the pendulum aside in the direction of the rods, _a b_, _a′ b′_, and letting it go, it oscillates freely between them. I bring it to rest and draw it aside in a direction perpendicular to the last; a length of 7 feet only can now oscillate, and by the laws of oscillation a pendulum 7 feet long vibrates with twice the rapidity of a pendulum 28 feet long. I wish to show you the figure described by the combination of these two rates of vibration. Attached to the copper ball, _p_, is a camel’s-hair pencil, intended to rub lightly upon a glass plate placed on black paper and over which is strewed white sand. Allowing the pendulum to oscillate as a whole, the sand is rubbed away along a straight line which represents the amplitude of the vibration. Let _a b_, Fig. 176, represent this line, which, as before, we will assume to be described in one second. When the pendulum is at the limit, _b_, of its swing, let a rectangular impulse be imparted to it sufficient to carry it to _c_ in one-fourth of a second. If this were the only impulse acting on the pendulum, the bob would reach _c_ and return to _b_ in half a second. But under the actual circumstances it is also urged toward _d_, which point, through the vibration of the whole pendulum, it ought also to reach in half a second. Both vibrations, therefore, require that the bob shall reach _d_ at the same moment; and to do this it will have to describe the curve _b c′ d_. Again, in the time required by the long pendulum to pass from _d_ to _a_, the short pendulum will pass _to and fro_ over the half of its excursion; both vibrations must therefore reach _a_ at the same moment, and to accomplish this the pendulum describes the lower curve between _d_ and _a_. It is manifest that these two curves will repeat themselves at the opposite sides of _a b_, the combination of both vibrations producing finally a figure of 8, which you now see fairly drawn upon the sand before you.