Direction, of Sound through Air. 123 



I. Supposing: (1) to be derivable from one only of equations 

 (3)*, as, for instance, the first, the form of Fj can readily be 

 determined. For, differentiating F^O with respect to t and x 

 separately, we get 



0rf lW +I»,«3r+l»*Msa ? ^ 



d 7 u 

 the elimination of -. — — between which gives 

 dx at 



(4) 



0=I»F,(/) -p.wf.w +*W$ - i>J» 2 5 ; 



and in order that this may coincide with (1), we must have 



F 1 WF 1 (0-F' 1 ( a jF 1 W=O. ... (5) 



The first of these gives us 



¥',(«,) a 



or 



= F 1 (« I )+^F 1 (,); .... (6) 



the auxiliary equations for the integration of which by Lagrange's 

 method are 



= dv+~ d.* xi = dx, 0~dt; 



~~*x 



whence we have 



v±ahg e ct x — const., 



x = const., 



t ~ const. ; 

 and therefore 



Y l {txvu x )=z^{(v±ahg 6 ^) } x > t\ 



= ${<», x,t\, 

 suppose, where co =v + a log 6 ot x . 



But this value of F 2 must satisfy (5); hence, since 



* This is, of course, not a necessary supposition. It. will obviously be 

 sufficient if (1) can be derived from any combination of the pair of equa- 

 tions constituting the solution and their derivatives. 



K2 



