SECT. XVII. HARMONIC SOUNDS. 141 



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, 

 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 conse- 

 quence 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 com- 

 municated to the moveable 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, running back again along the whole length, are again re- 

 flected at the other ends. And thus they will continue to run 

 backwards 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 or 

 pulse continually doubled back on itself by reflection at the fixed 

 extremities. 



Harmonics generally co-exist with the fundamental sound in 

 the same vibrating body. If one of the lowest strings of the 

 pianoforte be struck, an attentive ear will not only hear the 

 fundamental note, but will detect all the others sounding along 

 with it, though with less and less intensity as their pitch becomes 

 higher. According to the law of co-existing undulations, the 

 whole string and each of its aliquot parts are in different and 

 independent states of vibration at the same time ; and as all the 

 resulting notes are heard simultaneously, not only the air, but 



