Pressure of Gaseous Vibrations. 369 



or, if we prefer it, 



Po( Po/"0>„) + l\L* da . . . (25). 

 a I 4a 2 ) J 



The total energy of the length considered is 



pQ \ u 2 dx ; 

 and the result may be thus stated 



momentum = -J p0 '' \ p0 ^ + — j- x total energy (26). 



This may be compared with (10). If we suppose the long 

 cylinder of length I to be occupied by a train of progressive 

 waves moving towards the piston, the integrated pressure 

 upon the piston during a time t, equal to l/a, should be equal 

 to twice the momentum of the whole initial motion. The 

 two formulae are thus in accordance, and it is unnecessary to 

 discuss (26) at length. It may suffice to call attention to 

 Boyle's law, where f // (p ) = 0, and to the law of pressure 

 (15) under which progressive waves have no momentum. 

 It would seem that pressure and momentum are here asso- 

 ciated with the tendency of waves to alter their form as they 

 proceed on their course. 



The above reasoning is perhaps as simple as could be ex- 

 pected ; but an argument to be given later, relating to the 

 kinetic theory of gases, led me to recognize, what is indeed 

 tolerably obvious when once remarked, that there is here a 

 close relation with the virial theorem of Clausius. If <#, y, z 

 be the coordinates ; v m v y , v z the component velocities of a 

 material particle of mass m, then 



d 2 Zm.r ? ' 



with two similar equations, X being the impressed force in 

 the direction of x operative upon m. If the motion be what 

 is called stationary, and if we understand the symbols to re- 

 present always the mean values with respect to time, the last 

 term disappears, and 



iZmvl=-iSXx .... (27). 



The mean kinetic energy of the system relative to any 

 direction is equal to the virial relative to the same direction. 



Let us apply (27) to our problem of the one-dimensional 

 motion of a gas within a cylinder provided with closed ends. 



