the Propagation of Motion through Elastic Mediums. 327 

 a-h.l.q=fVdx- ^ - 4-, and -^=4^. 



S "^ rff 2 ' dx 



(Lagrange, Mcc. Anal, part ii. sect. xii. Art. 8.) 



P is the accelerative force impressed on the fluid. In the case 

 in which P = 0, M. Poisson has obtained a particular in- 

 tegral of the first of these equations, from which it may be in- 

 ferred that V = a-fi .1 . q = fix — at — vt). (1) 



{Journal de V Ecole Poly technique^ cah. xiv.) 

 Also when P is constant, by Monge's method of integration 

 may be obtained, 



v = ahl.g ^Vt=.f(x— at — vt ->r ~\ (2) 



Each of the two systems of equations (1) and (2) refers to 

 propagation in a single direction ; and it will follow from the 

 general equation (A) that in each of the two cases the velocity 

 of propagation is exactly a. This result may also be obtained 

 from the equations themselves; for if in (1) x and t be made 



to var\\ so that o does not alter, it will be found that — — = 

 •" 5 ' dt 



a + V. But — obtained on this hypothesis is the velocity 



of propagation together with the velocity of the fluid. Hence 

 the velocity of propagation is a. If in (2) x and / be made to 

 vary so that g remains unaltered, and consequently so that 



d c 



dv = V d t, the resulting value of—— is again a + v. This 



subject I have considered more at length in a recent com- 

 munication to the Cambridge Philosophical Society ; at pi'e- 

 sent I wish to make another kind of application of the equa- 

 tion (A). 



Experience has shown that the velocity of propagation in 

 air of the same temperature is constantly the same, and is in- 

 dependent of the magnitude of the condensations and motions. 

 The precision with which slight modifications of the disturb- 

 ing causes are appreciated by the ear, is very probably depen- 

 dent on the absolute uniformity of propagation of every indi- 

 vidual portion of the aerial undulations, and the constancy of 

 their ti/pe during their progression. Assuming that they are 

 of this nature, the equation (A) will be applicable to aerial 

 propagation. Now, by the general consideration of fluid mo- 

 tion in space of one dimension, the two following equations are 

 obtained: 



(Lagrange, as above cited.) 

 Combininir 



