OF SOUND. 



6S 



SECTION VI. OF SOUND. 



396. Theorem. When a uniform and 

 perfectly flexible chord, extended by a given 

 weight, is inflected into any form, difiering 

 little from a straight line, and then suflPered 

 to vibrate, it returns to its primitive state in 

 tlie time which would be occupied by a 

 heavy body in falling through a height which 

 is to the length of the chord as twice the 

 weight of the chord to the tension ; and the 

 intermediate positions of each point may be 

 found by delineating the initial figure, and 

 repeating it in an inverted position below the 

 absciss, then taking, in the absciss, each 

 way, a distance proportionate to the time, 

 and the half sum of the corresponding ordi- 

 nates will indicate the place of the point at 

 the expiration of that time. 



We may first suppose the initial figure of the chord to be 

 a harmonic curve ; then the force impelling each particle 

 will be proportional to its distance from the quiescent po- 

 sition, or the base of the curve. For the force acting on 

 any element i' ia to the whole force of tension p, as the ele- 

 ments' tothe radius of curvature r (299), therefore the force is 

 inversely as the radius of curvature, or directly as the cur- 

 vature, that is, in this case, as the second fluxion of the 

 ordinate (195) ; but the second fluxion of the ordinate of 

 the harmonic curve is proportional tothe ordinate itself; 

 for the fluxion of the sine is as the cosine, and its fluxion 

 again as the sine ; the force being therefore always as tlie 

 distance from a certain point, as in the cycloidal pendulum, 

 the vibrations will be isochronous, and the ordinates will be 

 proportionally diminished, .so that (he figure will be always 

 a harmonic curve. Now calling the length of the chord «, 

 and the greatest ordinate y, tlie ordinate of the figure of 

 sines being to the length as the diameter of a circle to its 



a 

 c>rcuniference,.or ::: — , the radius of curvature of the har- 

 c 



iBonic curve will be — , and the force'acting on the ele- 

 ccy 



ment z' will be ; but the weight of the chord being 



aa 



q, ihat of z' is il ,and the force is to the weight as 



ccyp 



to 



q, or as ff^ to 1: therefore the time of vibration will be 



to that of a pendulum of the length y as 1 to v' ^^1^1\ 

 and to that of a pendulum of the length atn a ratio as much 

 less as yfy is less than v'a, or as 1 toc.^/". But the 



time of the vibration of a pendulum of the length a is to the 

 time in which a body would fall through half a, as c to 

 1, consequently a single vibration of the chord will be per- 

 formed in the time of falling through ? 3, and a double vi- 



bration in the time of falling through 2a.?. Now the ele- 



P 

 ment z', moving according to the law of the cycloidal pen- 

 dulum, describes spaces which are the versed sines of arcs, 

 inc'reasing equably (259), and the difference of the sine at 

 any point from the half sum of the sines of two equidiffer- 

 ent arcs, is in a constant ratio to the versed sine of the dif- 

 ference, therefore, by taking the half sum of two equidistant 

 ordinates, we find the space remaining to be described, after 

 a time proportionate to the absciss. If tlie base be divided 

 into two equal parts, and a harmonic curve be described on 

 different sides of each part, the same demonstration is ap- 

 plicable to both parts, as if they were two separate chords : 

 since the middle point will always be retained at rest by 

 equal and opposite forces ; and nothing prevents us from 

 combining this compound vibration with the original one, 

 since, by adding together the ordinates, we increase or di- 

 minish the fluxions and increments, in proportion to the 

 spaces that are to be described, and the same construction 

 of two equidistant ordinates, will determine the motion of 

 each part. Such a compound figure may be made to pass 

 through any two points at pleasure, and it may easily be 

 conceived, that by subdividing the chord still further, and 

 multiplying the subordinate curves, we may accommodate 

 it to any greater numhcr of points, so as to approximate in- 

 finitely near to any given figure ; by which means the pro- 

 position is extended to all possible forms. 



Scholium. If the initial figure consist of several equal 

 portions crossing the axis, the chord will continue to vibrate 

 like the same number of separate chords ; and it is some- 

 times necessary to consider such subordinate vibrations as 

 compounded with a general one. It usually happens also 

 that the vibration deviates from its plane, and becomes a 

 rotation, which is often exceedingly complicated, and may 

 be considered as composed of various vibrations in different 

 planes. 



397. Theorem. The chord and its ten- 



VOL. II. 



K 



