fj'20 PHILOSOPHICAL TRANSACTIONS. [ANNO 1800. 



compendious method of considering the problem. But when a chord vibrates 

 freely, it never remains long in motion, without a very evident departure from the 

 plane of the vibration ; and, whether from the original obliquity of the impulse, 

 or from an interference with the reflected vibrations of the air, or from the inequa- 

 bility of its own weight or flexibility, or from the immediate resistance of the par- 

 ticles of air in contact with it, it is thrown into a very evident rotatory motion, 

 more or less simple and uniform according to circumstances. Some specimens of 

 the figures of the orbits or chords are exhibited in plate 10, fig. 44. At the 

 middle of the chord, its orbit has always 2 equal halves, but seldom at any 

 other point. The curves of fig. 46, are described by combining together various 

 circular motions, supposed to be performed in aliquot parts of the primitive orbit : 

 and some of them approach nearly to the figures actually observed. When the 

 chord is of unequal thickness, or when it is loosely tended and forcibly inflected, 

 the apsides and double points of the orbits have a very evident rotatory motion. 

 The compound rotations seem to demonstrate to the eye the existence of secondary 

 vibrations, and to account for the acute harmonic sounds which generally attend 

 the fundamental sound. There is one fact respecting these secondary notes, which 

 seems entirely to have escaped observation. If a chord be inflected at ±, or -£-, or 

 any other aliquot part of its length, and then suddenly left at liberty, the harmo- 

 nic note which would be produced by dividing the chord at that point is entirely 

 lost, and is not to be distinguished during any part of the continuance of the 

 sound. This demonstrates, that the secondary notes do not depend on any inter- 

 ference of the vibrations of the air with each other, nor on any sympathetic agi- 

 tation of auditory fibres, nor on any effect of reflected sound on the chord, but 

 merely on its initial figure and motion. If it were supposed that the chord, when 

 inflected into right lines, resolved itself necessarily into a number of secondary 

 vibrations, according to some curves which, when properly combined, would ap- 

 proximate to the figure given, the supposition would indeed in some respects cor- 

 respond with the phenomenon related ; as the co-efficients of all the curves sup- 

 posed to end at the angle of inflection would vanish. But whether we trace the 

 constituent curves of such a figure through the various stages of their vibrations, 

 or whether we follow the more compendious method of Euler to the same purpose, 

 the figures resulting from this series of vibrations are in fact so simple, that it 

 seems inconceivable how the ear should deduce the complicated idea of a number 

 of heterogeneous vibrations, from a motion of the particles of air which must be 

 extremely regular, and almost uniform ; a uniformity which, when proper precau- 

 tions are taken, is not contradicted by examining the motion of the chord with the 

 assistance of a powerful magnifier. This difficulty occurred very strongly to Euler; 

 and La Grange even suspects some fallacy in the experiment, and that a musical ear 

 judges from previous association. But, besides that these sounds are discoverable 

 to an ear destitute of such associations, and, when the sound is produced by 2 

 strings in imperfect unison, may be verified by counting the number of their beats, 



