﻿Mr. R. Moon on the Theory of Sound. 193 



That equations (6) constitute a true solution of the problem 

 (and therefore the true solution unless the possibility be enter- 

 tained of there being other solutions possessed of the requisite 

 degree of generality) may be verified as follows. 



Eliminating <£ from the secoud of equations (6) by differen- 

 tiation in the usual manner, we get 



~dt + P *dt + D W + p»5/" * * () 



The first of equations (6) gives us 



dp _ a 2 dp f /dv -_ a dp\ 

 dx ~~ p 2 dx "■ \dx p 2 dx) 



= ±4+<^4£^l) (9, 



We have also the analytical condition 



^ ( d V\— d (dy\ . 

 dx\dt)~dt\dx/ ; 



dy , du D 

 or, since v=-~- and ~ == — > 

 at dx p 



dv D <//> 



cfe — p 2 eft 



Substituting this value of -7- in (9), it becomes 



Dividing this last equation by D and subtracting from (8) we get 



0= * + 1 ±, 



dt D dx 



which is identical with (1), the true equation of motion. 



The same mode of treatment, with the same result, may be 

 applied to the third of equations (6). 



I shall now proceed to draw certain conclusions from the fore- 

 going formulae in the case where the motions are small. 



The first of (6) gives us 



p=~j+x(»-±-) 



= + av + af v ± .- j + %( v ± -j 



— +uv-\-7r(v+-b 

 Phil. Mag. S. 4. Vol. 37. No. 248. Mar. 1869. 



