OF SOUND. 



when vibrating separately-) in the same manner as if the 

 chord were prolonged without end by a repetition of si- 

 milar portions, of which the alternate ones arc in an in- 

 verted position. 



398. Theokem. The times, occupied by 

 the similar vibrations of elastic rods, are di- 

 rectly as the squares of tlieir lengths, and in- 

 versely as their depths. 



If the length vary, the force at a given depression will 

 vary inversely as its cube, and the weight will vary as the 

 length, consequently the relative force will be inversely as 

 the fourth power of the length ; and where the spaces are 

 given, the times are as the square roots of the forces. The 

 weight is also directly as the depth, and the force as its 

 cube ; the accelerating force is therefore as the square of 

 the depth, and the time inversely as the depth. 



SCHOHUM. It may be shown that the accelerating force, 

 which acts on any point of an elastic rod, is as the difference 

 of the curvature at the given point from the sum of the 

 curvatures at equal small distances on each side, that is, as 

 the second fluxion of the curvature, or ultimately, as the 

 fourth fluxion of the ordinate. In the harmonic curve, the 

 second fluxion of the curvature, aswell as the second fluxion 

 of the ordinate, is proportional to the ordinate itself; hence 

 it follows that a rod infinitely long being bent into a series 

 of harmonic curves, each of its points would reach the basis 

 at the same instant ; that a finite rod, loosely fixed at both 

 ends, might vibrate in a similar manner ; and that a ring is 

 also capable of similar vibrations, if it be divided into any 

 even number of vibrating portions. The time of such a 

 vibration may be thus determined. The extremities of the 

 rod, when loosely fixed, may be considered as simply sub- 

 jected to a transverse force, since the curvature ultimately 

 disappears : the sum of these transverse forces being equal 

 to the whole of the forces which urge the rod towards the 

 basis, and each of them being expressed by the area of one 

 half of the curve. Now the curvature of a bar fixed at one 

 end, and depressed a little by a weight at the other, increas- 

 es uniformly in advancing towards the fixid point, until the 



radius of curvature there becomes f32i) : and this 



120/ ^ 

 curvature may be represented by the ordinate of the har- 

 monic curve produced until it meets the tangent at the ori- 

 gin ; so that the radius of curvature at the vertex of the har- 



... , , ll>m 



monic curve will be greater than m the same ratio 



Via/ 



as the produced ordinate is longer than the original ordinate, 



that is, as the quadrant of a circle is greater than the tadius, 



and will therefore be equal to ; but the radius of 



•248/ 



curvature is also — (398), I being the length, and y 



^ ,■ i*afn . , 19ft' 



the ordinate, therefore y=,-r-; — , or, smce 2=:2a, r—'- 

 bbc'm blc^m 



The weight of the element of the rod j;' is to m as x' to the 



height of thi modulus h, and is therefore n-r-m.andtht 



h 



force urging it is to/i as the area corresponding to i', is to 



yl 

 half the area of the curve, that is, as yx* to — , or as x' to 



c 



I . . , cfx' , , /■J'c'mv , cfx" . 



— , and is equal to —-; but/=— — -= , and -i-— is to the 



. , iic'my m , Ibc'hy 



weight as ,' j to -r-x , or as ^ to l ; the time 



I'll* h 121' 



of vibration is therefore at much less than that of a pendu- 



bc^ ny\ 

 lum of which the length isy, as — v'l-.-j is greater 



than unity, and as much less than that of a pendulum of 



which the length is 12 A, as — — is greater than unity, and 



the time of a complete vibration is to the time of falling 

 through 64 as 2l^ to Ich, 



399. Definition. A sound, of which, 

 the number of vibrations in a second is any 

 integer power of 2, is denoted in music by 

 the letter c. 



Scholium. Hence we may form a table of the number 

 of vibrations of each note in a second. 



i 



221 



_____ ^^ 



8 

 C 



1 1^ ' ' 3i 04 138 



c 



= 4 

 c c 



iia 1024 3048 32768 



