144 Mr. Moon on the Analytical Principles of Hydrodynamics. 



viz. the equation a 2 / u\ _ 



p= __ +x ^ + _j ; .... (2) 



he says : — " Putting on the left-hand side of this equation 



f(p, v) for p, it will be seen, since %( v+ -) is arbitrary in form 



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 arbi- 

 trary function of p." 



The step here taken is nothing other than eliminating p be- 

 tween (2) and the equation 



P=f(p>v), 



where f(p, v) is absolutely identical with the right-hand side of (2); 

 and no such result as funct. (p } v) = 0, or v=- funct. p, can be 

 derived from such elimination. For, when ^ in equation (2) is 

 spoken of as representing an arbitrary function, all that is meant 

 is that in every case of motion the pressure may be expressed in 

 terms of p and v in the manner indicated by (2), where ^ repre- 

 sents some function or other depending upon the particular cir- 

 cumstances of the motion. 



In any particular case of motion % will have a definite value ; 

 and to assume that in this particular case another relation exists 

 between p, p, v, viz. p =f{p, v) } where f(p, v) is not identical with 

 the right side of (2), is simply to beg the question at issue. 



To put the matter in a different light — we have to start with, 

 as Professor Challis admits, 



p =/, (x, t), p =/ 2 {x, t), v =f 3 {x, t) . 



How can it be possible, in general, to eliminate three quanti- 

 ties between these three equations so as to obtain the result 

 v = funct. p, or p = funct. p ? It is true that under particular cir- 

 cumstances we may do this, and under those circumstances no 

 doubt Mr. Earnshaw's equations and mine may coincide ; but the 

 ratio of the number of cases in which our equations differ to the 

 numbei; in which they agree must be simply infinite. 



I observe that Professor Challis refers to my solution of (1) 

 as a solution. I contend that it is the solution — the only one 

 which the equation admits of. 



I have no doubt that this could be shown directly, in the 

 manner I have elsewhere adopted to show that Poisson's integral 

 of the accurate equation of sound for motion in one direction 

 is the most general solution of that equation*; but it may be 

 proved much more compendiously as follows. 



The solution consists of the three equations 



* See Phil. Mag. for August last. 



