26 Prof. Ghallis's further Discussion of the 



and value, that v is an arbitrary function of p, and, by conse- 

 quence, on substituting for v in f(p, v), that p is also an arbitrary 

 function of p. Hence, by Mr. Moon's own reasoning, there 

 may be an unlimited number of different relations between p and 

 p, each defining a different fluid ; and evidently among these the 

 ordinary relation p = cPp must be included. 



Again, since v = — and p= D I —-) , and it has just been 



shown that v is an arbitrary function of p, it follows that 



l=KI) •••/*) 



By differentiating equation (2) with respect to t, we obtain 



d? \ \dx)) dx v { * 



which equation has the same generality as that from which it 

 was derived. 



The equations (2) and (3) are given by Mr. Earnshaw at the 

 commencement of a paper " On the Mathematical Theory of 

 Sound/' contained in the Philosophical Transactions (vol. cl. 

 p. 133), which is chiefly devoted to the discussion of the parti- 

 cular case in which, ~ being always equal to — , we have 



df^^d^ 



dx> dt* ~ dx*' [ } 



this equation being that which (3) becomes on the hypothesis 

 of Boyle's law. Now the equation (1) is identical with the equa- 

 tion (4) on the same hypothesis. For since p = J)(-~ ) , and 

 p = a 2 p } it follows that 



dx dx ' dy 2 ' 



dx~* 



dt) 

 whence by substituting for —- in (1) the equation (4) results. 



At the end of his paper Mr. Earnshaw briefly indicates the 

 mode of treatment of the case in which 



&y - (dy dy\ d?y 



dt 2 ~ Jl \dx dt) dx* 



assuming, as before, that -^=Fr-^V 1 shall now show that 



