Momentum and Pressure of Gaseous Vibrations. 365 



For the mean value of is at the piston we have only to 

 write pi for p in (4) and integrate with respect to t. And at 

 the piston U = 0. 



For the mean of the whole 'length I of the cylinder (parallel 

 to a?), we have to integrate with respect to x as well as with 

 respect to t. And in the integration with respect to x the 

 first term of (4) disappears, inasmuch as the mean density 

 remains the same as if there were no vibrations. Accordingly 



H'sMr- 



2 dx dt 



,r ,/// v M (n N, / C np-Pof dxdt f(pi-Po) 2 ; \ (6) 



the terms on the right being of the second order in the 

 quantities which express the vibration. 

 Again, 



JOi-Po) dt=${f( Pl ) -f(p ) }dt 



=Pof l M§ p ^ dt+p y'M ^ {j ^f^ dt '> 



so that by (5) 



. . . . °(6). 



The three integrals on the right in (6) are related in a way 

 which we may deduce from the theory of infinitely small 

 vibrations. If the velocity of propagation of such vibrations 

 be denoted by a, then f / (p o ) = a 2 . By the usual theory we 

 have 



TT-'i£ P-Po - 1 # / 7 \ 



d#> p ~ a 2 dt ' ' ' ' {i) ' 



If we suppose that the cylinder is closed at so = and at 

 x — l^ZL normal vibration is expressed by 



, S7rx sir at .„. 



0=cos-p. cos-j— ' . • . . (b), 



Phil. Mag. S. 6. Vol 10. No. 57. Sept. 1905. 2 C 



