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



function*. On the other hand, I have elsewhere f shown that, 

 adopting for the pressure the only expression which the facts of 

 the case authorize us to assume (that, namely, embodied in (4)), 

 a solution can be obtained possessing a degree of generality not 

 merely equal to all that had been considered requisite, but even 

 greater than had been conceived to be attainable. 



The solution to which I refer is contained in the three follow- 

 ing equations, viz. : — 





(6) 



p \ p. 



where a is an arbitrary constant ; %, <j>, and i]r are arbitrary 

 functions ; and the form off is defined by the equation 



/W ~JxV) W +2« 



For the details of the method by which this result has been 

 obtained I must refer to my former paper. Its general cha- 

 racter may be described as follows. 



Equation (4) enables us to put (1) under the form 



n_ d*y 1 (dp dp dp dv\ 

 W + V\fy"di + dv"di)'' 



or, since v= -—• and p==D . 



o=^ + -L.^. d<2y dp ^y 



dt 2 D do dx dt dp dx 



'*9 



(7 



Regarding (7) simply as a partial differential equation between 



x,t,y, -j-> J— requiring solution, the most obvious mode of treating 



it is to attempt its solution by Mongers method. I have shown 

 that, in order that (7) may admit of two integrals of the kind 

 given by Mongers method, two equations of condition must be 

 satisfied involving p, p, and v only; both of which are satisfied 

 by the relation between p, p, and v contained in the first of 

 equations (6). When that relation, holds, (7) will have for first 

 integrals the last two equations of the same group. 



* It would be easy to point out the cause of this failure. Upon this 

 branch of the subject, however, I do not propose to enter at present, 

 t See the Philosophical Magazine for August 1868. 



