On the true Theory of Pressure as applied to Elastic Fluids. 71 



IV. To indicate the manner in which the instan- 

 taneous pressure maybe represented mathematically. 



V. To show the bearing of the proposed correc- 

 tion on the received theory of sound. 



A B C D is a vertical cylinder closed at the base 

 A B, and having an air-tight piston C D capable 

 of moving freely in the upper part of it. 



Below the piston the tube is filled with air, which 

 at the time t is wholly free from impressed velocity, 

 but in which the density varies in the following 

 manner : viz., from A B up to an imaginary hori- 

 zontal plane E F, the density is uniform ; while 

 from E F the density gradually increases up to C D, 

 in such a manner that the effective force at every 

 point of the air between E F and C D is exactly the 

 same, and equal to/*. Above the piston a vacuum 

 exists. The piston is supposed to have weight, 

 but, for the sake of simplicity, the air under the 

 piston is supposed to be unaffected by gravity. 

 The weight of the piston is supposed to be such 

 that the effective force on each particle of the 

 piston is the same as that on each particle of the 

 mass of fluid E C, viz./". 



If the pressure exerted by the air which originally occupied the 

 space A F on that which originally occupied the space E C were to 

 continue during the time t x the same that it was at the time t, every 

 particle of the former mass of air (which we will designate as the air 

 in A F) would during the time t, be under the action of the same 



ft 2 

 length of path, n\z. j ~- 



A 



effective force/, and would therefore in that time describe the same 



and on this supposition no change would 



take place in the density of the air in E C during the time t x . But, 

 according to the received theory, the pressure of the air in A F on 

 that in E C will continue unchanged until the density of the part of 

 the air in A B which abuts on the common boundary of the two 

 masses of fluid has changed. Hence change in the density of the 

 air in A F must precede change in the density of E C. 



On the other hand, so long as the pressure of the air in E C on 

 the air in A F remains unchanged, the air in A F will remain at rest, 

 and will therefore undergo no change of density. But as, according 

 to the received theory, the pressure of the air in E C on the air in 

 A F depends on the density of the part of the air in E C which abuts 

 on the common boundary of the two masses of air, it follows that 

 change in the density of the air in E C must precede change in the 

 density of the air in A F. 



But we have before proved the exact contrary, viz. that change in 

 the density of the air in A F must precede change in the density of 



* This tfill be the case if i-/=/, or putting »=a 2 p, a 2 log 2 p=A+e; where 

 p ax 

 p denotes the density at the distance x measured vertically, and c is a const. 



