the received principles of Hydrodynamics. 161 



of the comparisons that admit of being made satisfactorily. One 

 contradictory experiment would be fatal to its truth. As far as 

 my acquaintance with the solutions of hydrodynamical problems 

 extends, I have seen no reason to doubt the exactness of that 

 equation for air of given temperature in motion, within a wide 

 range of density. 



The legitimacy of the hypothesis that p = a?p having been 

 proved by an argument for which Mr. Moon is himself respon- 

 sible, it must be under some misapprehension that he contends 

 by other arguments that that equation is not admissible. The 

 fact is, these arguments are vitiated by the circumstance that in- 

 ferences, drawn from an investigation which embraces perfect 

 fluids of all kinds, are treated as applicable to a particular fluid 

 without previously introducing any condition defining the fluid. 



To illustrate the foregoing views, I propose to discuss an in- 

 stance of the application of the equation p = a 2 p which is consi- 

 dered by Mr. Moon to lead to absurd conclusions, viz. that in 

 which " a vertical cylinder closed at its lower end has an air- 

 tight piston, which is capable of working freely in the upper part 

 of it, and is exactly supported by the air beneath ; and at a given 

 time a given weight is placed upon the piston." The question 

 is, to determine what motion of the piston is thereby produced. 

 From what has been already said, it will be assumed that the 

 equation p — funct. (p and v) justifies making the hypothesis 

 that p = a 2 p for air in motion; and the solution of the problem 

 will accordingly proceed as follows. 



Let the weight of the piston be M^, and the additional weight 

 mg ) and let the distance of the under surface from the bottom 

 of the cylinder be a when the weight is added, and z 1 after the 

 time t. Also let k % p Q be the whole pressure of the fluid upon 

 the piston before the weight is put on, and k~p- t that at the time 

 }. In general, if p be the pressure at any point of the fluid at 

 the height z above the bottom of the cylinder at any time /, we 

 shall have^= funct. (z x t), and therefore 



(dp\_cty 

 \dt)~ dt 



dp dp dz 

 + ~dz"dt 



The factor ~ 3 expressing the variation of the pressure from 



point to point at a given time, depends partly on the circum- 

 stance that the impulse immediately given by the bottom of the 

 piston to the contiguous fluid is propagated, so that for any other 

 position the consequent pressure is a function of z and t, and 

 partly on the variation of the pressure at the given time due to 

 the force of gravity. As the total variation of pressure referable 

 to these two causes will always be comparatively very small for 



