248 Mr. 11. Moon on Objections recently made 



fluid whatsoever wc shall have always, or generally, 

 p=f{xt), p = a*f{xt). 



Eor proof of these latter propositions I must refer to the in 

 tegral (elsewhere derived*) of the equation of motion, 



1 dp 



U dt* + D dx 



(1) 



(2) 



viz. the trio of equations, 



where u = v+ -, where a is an arbitrary constant, and '$ :f typ ^ 2 



are arbitrary functionsf. Suppose that when / = 0, p and v are 

 represented by the equations p=f 2 (x), v=f 3 (x) ; then we shall 

 have, when £ = 0, from the first of (2), 



p —£-M f ^m}- - • • (3) 



Hence, although p = a' 2 f <2 (x) is an admissible assumption here, 

 since by means of <p we can satisfy the equation 



yet the number of different expressions for p which are equally 

 admissible is simply infinite — since, if in (3) we put/, (a?) forjp, 

 where/, denotes any function whatever, the equation so result- 

 ing may equally be satisfied by properly assuming <f>. The 

 above establishes that there is no fluid for which always, or ge- 

 nerally, p — a^p. But, dismissing for the present this argument, 

 if in general we have p~ funct. [p t v), I w r ould ask upon what 

 philosophical principle can we be called upon to accept, or even 

 be invited to collect evidence in favour of, the law p = d 2 p ) in 

 support of which not the vestige of an a priori argument can be ad- 

 duced, to the entire exclusion of a countless number of other laws 

 which a priori have precisely the same claim upon our attention V 



* See Phil. Mag. vol. xxxvi. p. 27. I regret that, in two subsequent 

 papers in the Philosophical Magazine, p is erroneously written for D in the 

 denominator of the fraction which occurs on the right-hand side of the 

 second equation of the solution. 



f That equations (2) satisfy (1) may be verified by differentiating 

 with respect to both x and t either of the last two of equations (2); 

 then eliminating \jr\ or yjr' 2i as the case may be, from the equations so ari- 

 sing, and, finally, by eliminating <p' from the last result by means of the 

 derivative with respect to x of the first of equations (2). 



