METHOD OF VARIATIONS. 69 



alter the intervals of the strokes so that each two suc- 

 ve strokes act in opposite manners they will exactly 

 neutralize each other, and the energy expended will be 

 turned into heat or sound at the point of percussion. 

 Exactly similar effects occur in all cases of rhythmical 

 motion. If the musical note C is sounded in a room con- 

 taining 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 periods of 

 their vibrations, and one stroke will sooner or later be 

 opposed by one contrary in effect. All phenomena of 

 resonance arise from this coincidence in time of undu- 

 lation. The air in a pipe closed at one end, and about 

 12 inches in length, is capable of vibrating 512 times in 

 >nd. 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 would be 

 transmuted into heat and become no longer perceptible 

 as sound. 



These accumulated vibrations may 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 excess of vibration. In the same way 

 a suspension bridge may readily be broken down if a com- 

 pany of soldiers walk across it in steps the intervals of 

 which happen to agree with the intervals of vibration of 

 the bridge itself. But if they break the step or march 

 with very different time, they may have no perceptible 

 effect upon the bridge. In fact if the impulses com- 

 municated to any vibrating body are exactly synchronous 



