SECT, xvii.] VIBRATION OF MUSICAL STRINGS. 157 



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 fundamental 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 vibrat- 

 ing 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, fourth, or other 

 aliquot points according to the corresponding harmonic sound, 

 will remain on it during its vibration, but will instantly fly 

 off from any of the intermediate points. The points 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 oc- 

 casionally 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 communicated to 

 the movable end. So, when a string fixed at both ends is put 

 in motion by a sudden blow at any point of it, the primitive 

 impulse divides itself into two pulses running opposite ways, 

 which are each totally reflected at the extremities, and run- 

 ning back again along the whole length are again reflected at 

 the other ends. And thus they will continue to run back- 

 wards and forwards, crossing one another at each traverse, 

 and occasionally interfering, so as to produce nodes ; so that 

 the motion of a string fastened at both ends consists of a wave 



