SKCT. XVII. VIBRATION OP MUSICAL STRINGS. 135 



then vibrate twice as fast as the whole, but in opposite 

 directious ; the ventral or bulging segments will be alter- 

 nately above and below the natural position of the string, 

 and the resulting note will be the octave above C. When 

 a point at a third of the length of the string is kept at 

 rest, the vibrations will be three times as fast as those 

 of the whole string, and will give the twelfth above C. 

 When the point of rest is one fourth of the whole, the 

 oscillations will be four times as fast as those of the fun- 

 damental note, and will give the double octave ; and so 

 on. These acute sounds are called the harmonics of 

 the fundamental note. It is clear from what has been 

 stated, that the string thus vibrating could not give these 

 harmonics spontaneously unless it divided itself at its 

 aliquot parts into two, three, four, or more segments in 

 opposite states of vibration separated by points actually 

 at rest. In proof of this, pieces of paper placed on the 

 string at the half, third, fouith, or other aliquot points 

 according to the corresponding harmonic sound, will re- 

 main on it during its vibration, but will instantly fly off 

 from any of the intermediate points. The po.ints of 

 rest called the nodal points of the string, are a mere 

 consequence of the law of interferences. For if a rope 

 fastened at one end be moved to and fro at the other 

 extremity so as to transmit a succession of equal waves 

 along it, they will be successively reflected when they 

 arrive at the other end of the rope by the fixed point, 

 and in returning they will occasionally interfere with 

 the advancing waves ; and as these opposite undulations 

 will at certain points destroy one another, the point of 

 the rope in which this happens will remain at rest. 

 Thus a series of nodes and ventral segments will be 

 produced, whose number will depend upon the tension 

 and the frequency of the alternate motions communi- 

 cated to the movable end. So when a string fixed at 

 both ends is put in motion by a sudden blow at any^oint 

 of it, the primitive impulse divides itself into two pulses 

 running opposite ways, which are each totally reflected 

 at the extremities, and running back again along the 

 whole length are again reflected at the other ends. And 

 thus they will continue to run backward and forward, 

 crossing one another at each traverse, and occasionally 



