THE EEY. S. EAENSffiYW ON THE MATHEMATICAL THEOET OF SOUND. 141 
begins at after twice passing through - ends at will have two stationary points 
in its type, \dz. those where Between these points the wave will be stationary 
though constantly changing type ; heyond them progressive. 
23. But instead of supposing the piston to generate such a wave as this, let us sup- 
pose it to begin from the velocity zero, and according to any proposed law (continuous 
of course) increase its velocity till it becomes infinite ; and let us consider the state of 
the medium at this moment. 
Denote by A and B the places of the piston where its velocity became respectively 
\/ ^ and infinite. Then whatever was the law of motion from A to B, and whether AB 
be great or small, provided it remains of finite length, the density at A will remain 
unchanged and equal to y? and the velocity of every particle as it passes by A will be 
equal to The mass of air also which will rush through the section of the tube at 
A ufiU be — ^ ; and this, be it observed, cannot be made either more or less by causing 
the piston to move in a different manner from A to B. It is also equally independent 
of the law of the piston’s motion before it reached A. Hence the mass of air that flows 
through the section at A is altogether independent of the law of the piston’s motion 
throughout its whole course. 
24. Now let us inquire what quantity of air rushes through any other section of the 
tube. In every part where there is motion the same relation between density and velo- 
u 
city obtains, \iz. "^^5 ^-nd consequently the quantity which rushes through any 
section is at the rate of 
per second. 
It is obHous this admits of a maximum value, which in the usual manner we find to be 
S§o 
- — 
£ 
at which value u=s / \jj and ^=y- 
25. Hence one part of the tube cannot supply air to another part faster than at this 
rate ; and consequently the greatest possible mass of air passes through the section at A : 
and it may be stated as a general property of motion through a tube, that a gas cannot 
S s/ 
be conveyed through a tube faster than at the rate of - — ^ cubic feet per second of gas 
of the density 
Hence the escaping powers of different gases through equal tubes are proportional to 
the velocities with which they respectively transmit sound. 
26. Since this result is independent of the law of velocity of the air, both before and 
after passing the section A, we are entitled to say that air cannot rush through a pipe 
S s/ 
of finite length, even into a vacuum, faster than at the rate of ^ cubic feet per 
MDCCCLX. 
u 
