﻿Mr. R. Moon on the Theory of Sound. 195 



We have here, in effect, two equations of motion*, of which 

 the first, viz. 



-3+-a-*3 • • • • m 



has for its general integral 



y = ${x~a{e+ \fl+e*)t} + ^{x-a{e- </T+?)t} \ 

 or if ae be small in comparison with a, 



y=<t>{x-a(l+e)t}+>^{x!+a(l-e)t} ; 

 whence we have 



dy 



dt = - a ( l + e )${ x ~< l + e )t}+a{l-e)f , {x + a(\-e)t} j 



dy_ 



dx ~~ 



<fj{x-a(l + e)t} + ^'{x + a{l-e)t}. 



y. (13) 



J 



If ${x),j\x) be the respective values of % d jL w h en t=0, we 

 shall have 



V(x)=-a{l + e)cl>>(x)+a(l-e)f'(x) t 



f(x)= Q'W+Vfr); 



whence we get 



Substituting these values in (13), and neglecting terms involving 

 e 2 , we get 



,4 = -f ( 1 + ^)F^-«(l + ^}-«/^-«(l + ,)^-i 

 dt 2L+(l-e)V{x + a{l-e)t\+af{* + a(l-e)t\l' 



_y = i 



dx 2 



r i 

 - -Y{*-a(l+e)t\ +(l-e)f\x-a(l + e)t\ 



_+--F{x + a(l-e)t}+(l + e)f{x + a(l-e) t l, 



Suppose that when ^=0 the disturbance is confined to the length 

 / measured from the origin to the right, then 



F(a?)=0, if x>l or < 0, 



anvn^i^f.° fthe ^ i8 \° b ? ado P ted as representing the motion in 

 any particular case will hereafter be explained 



02 



