Direction, of Sound through Air, 125 



wliicb ; integrated by Lagrange's method, gives 



«-=*{*+£<} • (8) 



Equations (7) and (8) together constitute a solution of (1) which, 

 making allowance for the difference of the independent variables 

 employed, is identical with that given by Poisson in the Journal 

 de VEcole Poly technique for 1807*. 



II. The foregoing result, it will be remembered, has been de- 

 rived on the assumption that (1) is derivable from a single one 

 of the equations (3) and its derivatives. I now propose to inves- 

 tigate the forms which I\ and F 2 must assume in order that (1) 

 may be derivable from a combination of equations (3) and their 

 derivatives. 



The differentiation of F 2 =0 gives us 



* When the motions are small the result is altogether different \ for in 

 that case (6) becomes 



0=F 1 K)+aF' 1 ( ? O v 

 which, the auxiliaries for its integration being 



=s dv ± adx x , = dx, = dt, 

 gives us 



Fi = <£{w, x, t}, 



where (o = v+aot x . Hence we have 



f 1 '(^)=^(»), F/a)=^'(o, 



Fi'(**)=±«tf>'(5), PiWfa+'Ws 

 and substituting these values, (5) becomes 



= <//<Y) + «</>'<»• 

 The auxiliaries for the integration of this last are 



= dx + adt, = da, 

 which give us 



3 ' ' tp=*p{(v±ax x ), (x±at)}, . . . , . . (8a) 



instead of, as in the case of the accurate equation treated of in the text, 



Hence, when the motions are small, we may assign to F 15 F 2 the alternative 

 values of <£ given by (8 a) ; or, which is the same thing, we may take for 

 our solution 



v+aa*=ifri(xjraf) 3 



v — ax x = ^ 2 (x — at), 

 which is identical with the ordinary solution given in this case. We cannot, 

 however, adopt the same course in the general case when the motions are 

 not small. 



