Prof. Challis on the received principles of Hijdrodynamics. 311 



the reasoning would hold good if we supposed that A = a a , B = 0, 

 C = 0, D = 0, and a = l, or that p = a 2 p. 



This argument proves that it is allowable to make for fluid in 

 motion the hypothesis that the pressure varies as the density 

 always and at all points. The selection of this hypothesis is 

 suggested by the circumstance that, as is known from experiment, 

 it expresses the relation between the pressure and the density 

 for fluid of given temperature at rest. The complete verification 

 of the hypothesis would depend upon making a sufficient number 

 of satisfactory comparisons of consequences mathematically de- 

 duced from it with experiment. 



The foregoing argument receives confirmation by applying a 

 similar one to an incompressible fluid. In this case, the density 

 being constant, there are only the equations which give the 

 values of p, u, v, w, the number of which does not exceed the 

 number of the variables x, y, z, t. Consequently, as is other- 

 wise known, there is not necessarily a relation between p, u, v, w 

 which is independent of coordinates and the time. 



In my communication in the August Number I gave Mr. 

 Moon, credit for the originality of the process of reasoning by 

 which he obtained the equation p =f(p, v), which is the same as 

 the above equation ¥(p, p 3 u 3 v, w) = restricted to motion in 

 one dimension ; and I pointed out that this equation was inclu- 

 sive of every relation between p } p, and v by which the fluid 

 might be defined. Also I argued that Mr. Moon's reasoning is 

 necessary for proving the legitimacy of assuming that p = a p 

 for fluid in motion. It will be seen from the additional argu- 

 ments adduced in this letter that I still maintain these views, 

 notwithstanding that I gather from Mr. Moon's communication 

 in the September Number that he dissents from them. The 

 reasons he there gives for his dissent do not appear to me to 

 invalidate in any respect my arguments. To show this it may 

 suffice to advert to two points on which he seems mainly to rely 

 for his objections to the acknowledged principles of hydrody- 

 namics. 



(1) Mr. Moon founds an argument on the possibility of the 

 immediate juxtaposition of two densities one of which is double 



the other : in other terms, he admits that -~- may have an in- 



pdx 



finite value. But the a priori demonstration of the law of the 



equality of pressure of a perfect fluid in all directions from a 



given point, on which law the whole of analytical hydrodynamics 



depends, excludes infinite values both of the effective accelerative 



force and of -^-. See * Principles of Mathematics' &c, pp. 106 

 pdx r 



& 173. There are, it is true, physical conditions under which 



