Integrated Variations.
In considering the various modes in which one effect may depend upon another, we must set in a distinct class those which arise from the accumulated effects of a constantly acting cause. When water runs out of a cistern, the velocity of motion depends, according to Torricelli’s theorem, on the height of the surface of the water above the vent; but the amount of water which leaves the cistern in a given time depends upon the aggregate result of that velocity, and is only to be ascertained by the mathematical process of integration. When one gravitating body falls towards another, the force of gravity varies according to the inverse square of the distance; to obtain the velocity produced we must integrate or sum the effects of that law; and to obtain the space passed over by the body in a given time, we must integrate again.
In periodic variations the same distinction must be drawn. The heating power of the sun’s rays at any place on the earth varies every day with the height attained, and is greatest about noon; but the temperature of the air will not be greatest at the same time. This temperature is an integrated effect of the sun’s heating power, and as long as the sun is able to give more heat to the air than the air loses in other ways, the temperature continues to rise, so that the maximum is deferred until about 3 P.M. Similarly the hottest day of the year falls, on an average, about one month later than the summer solstice, and all the seasons lag about a month behind the motions of the sun. In the case of the tides, too, the effect of the moon’s attractive power is never greatest when the power is greatest; the effect always lags more or less behind the cause. Yet the intervals between successive tides are equal, in the absence of disturbance, to the intervals between the passages of the moon across the meridian. Thus the principle of forced vibrations holds true.
In periodic phenomena, however, curious results sometimes follow from the integration of effects. If we strike a pendulum, and then repeat the stroke time after time at the same part of the vibration, all the strokes concur in adding to the momentum, and we can thus increase the extent and violence of the vibrations to any degree. We can stop the pendulum again by strokes applied when it is moving in the opposite direction, and the effects being added together will soon bring it to rest. Now if we alter the intervals of the strokes so that each two successive strokes act in opposite manners they will neutralise each other, and the energy expended will be turned into heat or sound at the point of percussion. Similar effects occur in all cases of rhythmical motion. If a musical note is sounded in a room containing a piano, the string corresponding to it will be thrown into vibration, because every successive stroke of the air-waves upon the string finds it in like position as regards the vibration, and thus adds to its energy of motion. But the other strings being incapable of vibrating with the same rapidity are struck at various points of their vibrations, and one stroke will soon be opposed by one contrary in effect. All phenomena of resonance arise from this coincidence in time of undulation. The air in a pipe closed at one end, and about 12 inches in length, is capable of vibrating 512 times in a second. If, then, the note C is sounded in front of the open end of the pipe, every successive vibration of the air is treasured up as it were in the motion of the air. In a pipe of different length the pulses of air would strike each other, and the mechanical energy being transmuted into heat would become no longer perceptible as sound.
Accumulated vibrations sometimes become so intense as to lead to unexpected results. A glass vessel if touched with a violin bow at a suitable point may be fractured with the violence of vibration. A suspension