366 Lord Rayleigh on the Momentum and 



where s is any integer, giving 



^ki^ dt=a ^(j^^fi = ^^ (9)t 



the integrations with respect to x in (9) being taken from 

 to I, that is over the length o£ the cylinder. 



The same conclusions (9) follow in the general case where 

 4> is expressed by a sum of terms derived from (8) by attri- 

 buting an integral value to s. The latter part expresses the 

 equality of the mean potential and kinetic energies. 



Introducing the relations (9) into (6), so as to express the 

 mean pressure upon the piston in terms of the mean kinetic 

 energy, we get as the final formula 



Among special cases let us first take that of Boyle's law, 

 where p = a 2 p, so that 



f{p,) = a\ /"(.Po)=0. 



We have at once 



J^-i^^oJJ 11 ^ • • • (11) 



The expression on the right represents double the volume- 

 density of the kinetic energy, or the volume-density of the 

 whole energy, and we recover the result of the former 

 investigation. 



According to the adiabatic law 



PlPo=(p/Po) r (12); 



so that 



fW=M, ,r W =M(lri) . . (13) . 



Po po" 



Hence from (10) 



Jto-pJtf^Ky+i^JpJ^f . . (14). 



The mean pressure upon the piston is now i(y-f-l)of the 

 volume-density of the total energy. We fall back on Boyle's 

 law by taking 7= 1. 



It appears therefore that the result is altered when Boyle's 

 law is departed from. Still more striking is the alteration 

 when we take the case treated in ' Theory of Sound ' § 250 



