OF SOUND. 



69 



the whole elasticity, this weight will be to the weight of 

 A as /i to NA ; and the force impelling A being A.NA ; DL, 

 this force will be to the weight of A as h to DL, or as 



;, ^^ to 1. Let there be two pendulums, of which the 



'DKq 

 lengths are h and AB, then with the same force, they will 

 vibrate in times which are as ^ h and ^/^ AB, and if the force 



in AB become h. ^^. , the time being inversely in the sub- 



DKq 

 duplicate ratio of the force, the vibrations will be as v^ A to 



^ AB. v/ f^^ ^ or as A to DK ; and in the time of 



[h.ABj 

 this semivibration in AB, tlie undulation will be transmitted 



through DA, therefore in a semivibration of h, it will be 

 transmitted through a space greater in the ratio of h to DK, 

 which will be to h as DA to DK, or as half the circum- 

 ference of a circle to its diameter ; and while a heavy body 

 falls through half /i, the undulation will describe /i (25g), 

 its velocity will therefore be equal to the final velocity of 

 the body falling through half A (23l). Accordfng to this 

 theorem the mean velocity of sound should be gifi feet 

 in a second, h being 2788O feet, but it is found to be nearly 

 1 130, which is one fifth greater than the computed velo- 

 city. The most probable reason that has been assigned for 

 this difference is the partial increase of elasticity occasioned 

 by the heat and rold produced by condensation and ex- 

 pansion. 



401. Theorem. The Tieight of the ba- 

 rometer will not affect the velocity of sound; 

 but, if the density vary, the pressure remain- 

 ing the same, the velocity will vary in its 

 subduplicate ratio. 



For the velocity varies in the subduplicate ratio of the 

 height of a homogeneous atmosphere, and that height re- 

 mains the same while the density is only varied by means 

 of pressure. 



Scholium. The velocity of the transmission of an im- 

 pulse through an clastic medium of any kind may be more 

 generally determined without the consideration cf any par- 

 ticular law for the variation of the density; and it may be 

 directly demonstrated, that the velocity, with which any 

 impulse is transmitted by an elastic substance, is equal to 

 that which is acquired by a heavy body in falling through 

 half the height of the modulus of its elasticity. The density 

 of the different parts of the medium, throughout the finite 

 space, which is affected by the impulse at anyone time, may 

 be represented by the ordinatcs of a curve ; that which cor- 

 responds to the natural density being equal to the height of 

 the modulus of the elasticity. Tlie force acting on any 

 small portion will be expressed by the difference of the 



ordinates at its extremities, that is, by the weight of a pojr- 

 tion of the modulus equal in height to that difference ; this 

 force is to the weight, which is to be moved, as the fluxion 

 of the ordinate to that of the absciss; and the velocity with 

 which the density increases will be as the difference of the 

 forces at the extremities of the portions, or as the second 

 fluxion of the ordinate of the curve ; and the increment of 

 the ordinate expressing (he density will be to the whole, as 

 half of its second fluxion to its first fluxion ; while therefore 

 the density varies so as to be represented by the mean of two 

 ordinates at a small distance on each side of the first ordi- 

 nate, tlie increment of the ordinate being represented by the 

 mean versed sine of the arcs, or half the second fluxion, of 

 the mean ordinate, the decrement of the space occupied by 

 the particles will be as much less as the fluxion of the ab- 

 sciss is less than the ordinate, and the whole velocity being 

 as much greater than the difference of the velocities, as the 

 force is greater than its fluxion, or as the first fluxion of the 

 ordinate is greater than its second fluxion, it follows that, in 

 the same time, the particles will actually describe a space 

 equal to half of the first' fluxion of the ordinate, diminished in 

 the ratio of the fluxion of the absciss to the ordinate; but if 

 the forcewere altered in theratioof the fluxion of the ordinate ' 

 to that of the absciss, so as to become equal to that of gra- 

 \ity, the space described would become equal to half the 

 fluxion of the absciss, diminished in the ratio of the fluxion' 

 of the absciss to the ordinate ; and if the time were increas- 

 ed in the ratio of the fluxion of the absciss to the ordinate, 

 the space described would be increased in the duplicate ra- 

 tio, and would become equal to half tlie ordinate ; and if a 

 point move each way through the curve so as to describe 

 an arc while the variation of density causes the ordinate to- 

 be diminished by a space equal to the mean versed sine, it 

 wouJd describe a space equal to the ordinate or the height 

 of the modulus, while half that space wouldbe described by 

 the action of gravity ^ consequently the velocity of the- 

 points would be initially equal to that of a heavy body fall- 

 ing through half the height of the modulus. And that it 

 would always remain equal to this velocity, so that the 

 density of the medium might always be expressed by the 

 mean ordinate, may be shown exactly in the same manner 

 as has already been done with respect to the motions of 

 waves and of vibrating chords. The variation of the velo- 

 city, and the change of place of the particles may be easily 

 deduced from the successive fonms of the curve representing 

 the density ; and the whole eft'ect may also be considered as 

 arising from the progressive motion of the. same curves 

 which express the cotemporary affections of the different 

 parts of the medium, and which will also show the succcs- 

 shestates of any one portion of it- at different times,. 



