YOL. XC.] PHILOSOPHICAL TRANSACTIONS. 6lQ 



vibration, and the half sum of the ordinates will be the distance of that point of 

 the chord from the axis, at the expiration of the time given. If the initial figure 

 of the chord be composed of 2 right lines, as generally happens in musical instru- 

 ments and experiments, its successive forms will be such as are represented in 

 plate 10, figs. 47, 48 : and this result is fully confirmed by experiment. Take 

 one of the lowest strings of a square piano-forte, round which a fine silvered wire 

 is wound in a spiral form ; contract the light of a window, so that, when the eye 

 is placed in a proper position, the image of the light may appear small, bright, and 

 well defined, on each of the convolutions of the wire. Let the chord be now made 

 to vibrate, and the luminous point will delineate its path, like a burning coal 

 whirled round, and will present to the eye a line of light, which, by the assistance 

 of a microscope, may be very accurately observed. According to the different 

 ways by which the wire is put in motion, the form of this path is no less diversified 

 and amusing, than the multifarious forms of the quiescent lines of vibrating plates, 

 discovered by Professor Chladni ; and is indeed in one respect even more interest- 

 ing, as it appears to be more within the reach of mathematical calculation to deter- 

 mine it; though hitherto, excepting some slight observations of Busse and Chladni, 

 principally on the motion of rods, nothing has been attempted on the subject. 

 For the present purpose, the motion of the chord may be simplified, by tying a 

 long fine thread to any part of it, and fixing this thread in a direction perpendicu- 

 lar to that of the chord, without drawing it so tight as to increase the tension : by 

 these means, the vibrations are confined nearly to one plane, which scarcely ever 

 happens when the chord vibrates at liberty. If the chord be now inflected in the 

 middle, it will be found, by comparison with an object which marked its quiescent 

 position, to make equal excursions on each side of the axis ; and the figure which 

 it apparently occupies will be terminated by 2 lines, the more luminous as they are 

 nearer the ends, plate 10, fig. 49. But if the chord be inflected near one of its 

 extremities, fig. 50, it will proceed but a very small distance on the opposite side 

 of the axis, and will there form a very bright line, indicating its longer continuance 

 in that place ; yet it will return on the former side nearly to the point from which 

 it was let go, but will be there very faintly visible, on account of its short delay. 

 In the middle of the chord, the excursions on each side the axis are always equal ; 

 and beyond the middle, the same circumstances take place as in the half where it 

 was inflected, but on the opposite side of the axis ; and this appearance continues 

 unaltered in its proportions, as long as the chord vibrates at all : fully confirming 

 the non-existence of the harmonic curve, and the accuracy of the construction of 

 Euler and La Grange. At the same time, as M. Bernoulli has justly observed, 

 since every figure may be infinitely approximated, by considering its ordinates as 

 composed of the ordinates of an infinite number of trochoids of different magni- 

 tudes, it may be demonstrated, that all these constituent curves would revert to 

 their initial state, in the same time that a similar chord bent into a trochoidal curve 

 would perform a single vibration ; and this is in some respects a convenient and 



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