﻿Mr. R. Moon on the Theory of Sound. 197 



the velocity a{\ -he), which is represented by the equations (16) 

 and (17). The disturbance so propagated will retain the same 

 invariable form; and (17) shows that it will be either a conden- 

 sation or rarefaction, according as v is positive or negative. 



We have hitherto been considering what takes place to the 

 right of the original disturbance when the motion is represented 

 by (12), x being supposed positive when measured to the right. 



For points to the left of the original disturbance we shall have 



x — a(\-{-e)t <0, 

 and therefore 



F{a?-a(l + «)*}=0, f{x-a(\ + e)t} = \ 



for all values of t. 



Hence the motion to the left of the original disturbance will 

 be represented by the equations 



v= l^Y\x + a(l-e)t\~^[l-f{x-{-a(l-e)t\-] > . 



|= 1 V{x + *<L-e)t} + l[l-e+(l+e)f\x + a(l-e)t\ ]; 



and adopting a mode of treatment similar to that employed in 

 the former case, we shall find that when the equation (12) is ap- 

 plicable to the motion, there will be propagated with the velocity 

 a{\ — e) to the left of the original disturbance a disturbance re- 

 presented by the equations 



(l + e)v=^{x+a(l-e)(}-^^[l-f{x + a(l-e)t\] > 



— as=(l + e)v; 



from which last it results that the disturbance so propagated will 

 consist of a rarefaction when v is positive, and vice versa ; which 

 is the opposite of what obtains with regard to the disturbance 

 propagated to the right. 



If in (11) we take the lower sign, we get the alternative equa- 

 tion of motion, 



°-%-*~&-*& • • • <«> 



The results which occur when this equation is applicable to the 

 motion will obviously be found by changing the sign of e in the 

 results obtained on the assumption that (12) is the differential 

 equation applicable to the motion. 



In this case, therefore, we shall have a disturbance propagated 



