6 1 8 PHILOSOPHICAL TRANSACTIONS. [ANNO 1800. 



41, 42 ; the demonstration is deducible from the properties of the circle. It is 

 remarkable, that the law by which the motion of the particles is governed, is 

 capable of some singular alterations by a combination of vibrations. By adding to 

 a given sound other similar sounds, related to it in frequency as the series of odd 

 numbers, and in strength inversely in the same ratios, the right lines indicating a 

 uniform motion may be converted very nearly into figures of sines, and the figures 

 of sines into right lines, as in pi. 9, figs. 39, 40. 



12. Of the Frequency of Vibrations contituting a given Note. — The number of 

 vibrations performed by a given sound in a second, has been variously ascertained; 

 first, by Sauveur, by a very ingenious inference from the beats of 2 sounds; and 

 since, by the same observer and several others, by calculation from the weight and 

 tension of a chord. It was thought worth while, as a confirmation, to make an 

 experiment suggested, but coarsely conducted, by Mersennus, on a chord 200 inches 

 in length, stretched so loosely as to have its single vibrations visible; and, by 

 holding a quill nearly in contact with the chord, they were made audible, and were 

 found, in one experiment, to recur 8.3 times in a second. Ey lightly pressing the 

 chord at -f of its length from the end, and at other shorter aliquot distances, the 

 fundamental note was found to be £ of a tone higher than the respective octave of 

 a tuning-fork marked c : hence the fork was a comma and a half above the pitch 

 assumed by Sauveur, of an imaginary c, consisting of I vibration in a second. 



13. Of the Vibrations of Chords. — By a singular oversight in the demonstration 

 of Dr. Brook Taylor, adopted as it has been by a number of later authors, it is 

 asserted, that if a chord be once inflected into any other form than that of the 

 harmonic curve, it will, since those parts which are without this figure are impelled 

 towards it by an excess of force, and those within it by a deficiency, in a very short 

 time arrive at or very near the form of this precise curve. It would be easy to 

 prove, if this reasoning were allowed, that the form of the curve can be no other 

 than that of the axis, since the tending force is continually impelling the chord to- 

 wards this line. The case is very similar to that of the Newtonian proposition 

 respecting sound. It may be proved, that every impulse is communicated along a 

 tended chord with a uniform velocity ; and this velocity is the same which is in- 

 ferred from Dr. Taylor's theorem ; just as that of sound, determined by other me- 

 thods, coincides with the Newtonian result. But, though several late mathema- 

 ticians have given admirable solutions of all possible cases of the problem, yet it 

 has still been supposed, that the distinctions were too minute to be actually ob- 

 served ; especially, as it might have been added, since the inflexibility of a wire 

 would dispose it, according to the doctrine of elastic rods, to assume the form of 

 the harmonic curve. The theorem of Euler and De la Grange, in the case where 

 the chord is supposed to be at first at rest, is in effect this : continue the figure 

 each way, alternately on different sides of the axis, and in contrary positions ; then, 

 from any point of the curve, take an absciss each way, in the same proportion to 

 the length of the chord as any given portion of time bears to the time of 1 semi- 



