Pressure in Fluids. 117 



Suppose that we have a cylindrical tube of indefinite length 

 filled with air; and let x be the ordinate of the position of rest 

 of a lamina made by planes perpendicular to the axis of the tube ; 

 y the same ordinate at the time t, — the whole motion being sup- 

 posed parallel to the axis of the tube, to which the ordinates are 

 measured parallel. 



Let p, v, p denote respectively the density, velocity, and pres- 

 sure at the time / at the point whose ordinate at that time is y. 



If D be the density of equilibrium, we shall have the following 

 equation of motion, viz. 



dP + D dx' K) 



an equation which holds without reference to any hypothesis as 

 to the nature of the law of pressure, resting on the simple as- 

 sumption that the fluid is a continuous substance. 



It is obvious that in any particular case of motion each of the 

 quantities p, p, v must be capable of being expressed in terms of 

 x and t, i. e. that we may assume 



p=f 2 {xt), 



From the last two, eliminating x, we may obtain 



t=f a (P> v )'> 

 and eliminating t, we may obtain 



®=fb(p> v )> 

 Substituting these values in the equivalent of p, we shall obtain 

 jo=funct. (p, v). 



We may hence perceive the gigantic character of the assump» 

 tion made in the received theory by taking, under all circum- 

 stances, p = d 2 p. 



All that we know of the pressure a priori is that p is a func- 

 tion of p and v, or, as we may put it, that 



p = ¥ a (p)+F b (p,v); 



where F A is some function of p and v which vanishes when v = 0. 

 Now, even if Mariotte's law established (which it does not*) 



* Mariotte's law has only been proved to hold when the air is in equili- 

 brium. In the last three cases of my former paper it is clear that the air 

 is not in equilibrium at the time when we have v = throughout the entire 

 mass, at which time it is shown in each case that Mariotte's law does not 

 hold. 



