308 



Mr. U. Moon on Poisson^s Solution^ ^c,j [Dec. 13, 



aiLitrary function of f, conversely we musl have / equal to an arbitrary 

 function of F. Hence, in exactly the same way in which it is proved that 

 F must have the above form, we may equally prove that /, F,y* must seve- 

 rally have the same form. It clearly results, therefore, from the above 

 assumed solution (B), that the equations (A) are insoluble, except on the 

 assumption that the velocity may be expressed in terms of the density 

 alone ; ^. e. that we may assume 



But substituting this value of v in equations (A), the latter become 



which is^the most general solution of which the equations of motion are 

 susceptible ; and which, making allowance for the difference of the ordi- 

 nates employed in the two cases, y referring to the particle when in motion, 

 and X to its position of rest, is identical with that of Poisson. 



The failure of mathematicians to derive from the equations of motion of 

 the received theory a solution containing two arbitrary functions has 

 hitherto, I apprehend, been universally attributed to difficulties of integra- 

 tion. So far is this view from being well founded, however, that in a 

 postscript to my Paper it is shown that, assuming the pressure to follow 

 any law whatever, a solution of the equations of motion can be obtained 

 containing two arbitrary functions ; a result, however, which requires that 

 the expression for the pressure shall satisfy certain conditions, which con- 

 ditions are violated when the pressure is assumed to vary with the den- 

 sity alone. 



Whatever be the law of pressure, it must always be capable of being ex- 

 pressed in terms of x and t. Moreover, the velocity and density are in 

 like manner severally capable of being expressed in terms of x and t ; 

 whence it follows that we may always express x in terms of p and v, and 

 equally that we can express t in terms of p and v ; so that, whatever be 

 the law of pressure, we may assume it to be a function of p and v. 



Hence, assuming 



''=/(p)- 



whence we get 



and eventually, D being the value of p when v=0. 



