66 



OF SOUND. 



sion remaining the same, the time of vibra- 

 tion is as the length ; and if the tension be 

 changed, the frequency will be as its square 

 root : the time also varies as the square root 

 of the weight of the chord. 



It has been shown, that the time varies in the subdupli- 

 cate ratio of the force, that is, of the tension directly, and 

 of the weight inversely ; and since the weight varies as the 

 length, the equivalent space will vary as the squate of the 

 length, and the time of describing it simply as the length. 



Scholium. The properties of vibrating chords have 

 been demonstrated in a more direct and general manner by 

 means of a branch of the fiuxionary calculus which has been 

 called the method of variations, and which is employed in 

 comparing the changes of the properties of a curve existing 

 at once in its different parts, with the variations which it 

 undergoes in successive portions of time from an alteration 

 of its form. An example'of this mode of calculation has 

 already been given in the investigation of the motions of 

 waves (395), and it may be applied with equal simplicity to 

 the vibrations of chords, and to the propagation of sound, 

 notwithstanding tlie intricacy and prolixity with which it 

 has been always hitherto treated. It may be shown that 

 every small change of form is propagated along an extended 

 chord with a velocity equal to that of a heavy body falling 

 through a height equal to half the length of a portion of the 

 chord, of which the weight is equivalent to a force produc- 

 ing the tension, and which may be called the modulus of 

 the tension ; ' and that the change is continually reflected 

 when it arrives at the extremities of the chord ; and from 

 this proposition all the properties of vibrating chords may 

 be immediately deduced. 



For the force, acting on any small portion of the chord, 

 being to the tension as its length to the radius of curvature, 

 and its weight being to the tension as its length is to the 

 modulus of tension, the force is to the weight as the length 

 of the modulus to the radius. By this force the whole por- 

 tion is initially impelled, since the change of curvature in 

 its immediate neighbourhood is inconsiderable with respect 

 to the whole : and it will describe a space equal to its versed 

 sine, which is to the arc as the arc to the diameter, in the 

 time in which a body falling by the force of gravity would 

 describe a space as much less, as the modulus of tension is 

 greater than the radius, that is, a space which is to the arc 

 as the arc to twice the modulus ; and if the time be in- 

 creased in the ratio of the arc to the modulus, the space 

 described by the falling body will be increased in the du- 

 plicate ratio, and will become equal to half the modulus : 



If tlierefore a point move in the original curve with such a 

 velocity as to describe the arc, while its versed sine is d^ 

 scribed by the motion of the chord, it would describe the 

 length of the modulus while a heavy body would descend 

 through half that length, and its velocity will therefore be 

 equal to that which is acquired by a body falling through 

 half the length : and supposing a point to move each way 

 with such a velocity, the successive places of the given 

 point of the chord will be initially in a straight line be- 

 tween these moving points. The place of the given point 

 will also remain in a straight line between the two moving 

 points as long as the motion continues. For the figure of 

 the curve being initially changed in a small degree accord- 

 ing to this law, each of the points of the chord will be 

 found in a situation which is determined by it, and its mo- 

 tion will be continued in consequence of the inertia of the 

 chord, and will receive an additional velocity from the ef- 

 fect of the new curvature. The space described in the first 

 instant being equal to the mean of the versed sines of the 

 arcs included by the two moveable points, the velocity, as 

 well as the second fluxion of the versed sine, may be repre- 

 sented by twice that mean : the increment of this velocity 

 in the next succeeding position of the curve will be repie- 

 sented by the new mean of the versed sines, which is al- 

 ways half of the mean of the second fluxions of the ordi- 

 nates on each side ; for the extremities of the new ele- 

 mentary arcs being determined by the bisections of two 

 equal chords removed to the distance of the arc on each 

 side, the versed sine of each is half of the excess of the in- 

 crement on one side above the increment adjoining to the 

 corresponding one on the other side, and the sum of the 

 versed sines is therefore half the sum of the differences of 

 the increments from the contiguous increments on the 

 same side, consequently the fluxion, or rather the variation 

 of the velocity, which is represented by twice the mean 

 versed sine, is equal to the half sum of the second fluxions 

 of the original curve at the parts in which the moveable 

 points are found, and the second fluxion or variation of the 

 space, which is as the variation of the velocity, is equal to 

 the mean of the second fluxions of the ordinates ; there- 

 fore the space described is always equal to the diminu- 

 tion of the mean of the ordinates. And the same mode of 

 reasoning may be extended through the whole curve. If 

 the initial figure be such that two of its contiguous portions, 

 lying on opposite sides of the absciss, are similar to each 

 other, and placed in an inverted position, it is obvious that 

 the point in which they cross the axis must remain at rest, 

 consequently its place may be supplied by a fixed point, 

 and either portion of the cun-e will continue its motion. 



