144 Physical Properties [CH. vi 



Averaged over all instants of time from t = to t = T, this equation 

 becomes 



1 f t=T 1 rd ~\t=r 



1 f t=T 



- 



Tj t =o 



-- f~ T ^(xX + yY+zZ)dt ...... (333). 



T J t = 



In the steady motion of a gas, the quantities 



+ f + z 2 ) and 2 (xX + yY + zZ\ 



are approximately constant throughout the motion. Hence as we increase r 

 indefinitely in equation (333), the first and last terms will remain approxi- 

 mately constant, while the middle term tends to vanish. Taking T sufficiently 

 large, equation (333) reduces to 



^md i =-^(xX^yY+zZ} .................. (334), 



in which both sides, which are in any case constant except for the slight 

 departures from the steady state which occur in the motion of the gas, are 

 averaged over a time sufficient for them to be regarded as sensibly constant. 

 The mean value of ^^(xX + yY + zZ) has been termed by Clausius the 

 virial of the system. We have therefore shewn that when a gas moves, 

 undisturbed from its steady state, the kinetic energy of its motion is equal 

 to its virial. 



161. The virial depends solely on the forces acting upon the molecules, 

 and not upon the motion of the molecules. In the case of a gas these forces 

 consist of the pressure exerted upon the gas by the walls of the containing 

 vessel, and the forces exerted by the molecules upon one another. 



If dS is an element of the surface of the containing vessel, and I, m, n 

 the direction cosines of its outward normal, the pressure of the element dS 

 exerts upon the gas a , force of which the components are IpdS, mpdS, 

 npdS, so that the value of that part of ^xX which is contributed by the 



pressure will be 1 1 IpxdS. The present treatment compels us to assume 

 that the pressure is the same at all points of the containing vessel. If we 

 make this assumption the quantity just obtained may be written p \\lxdS. 

 Hence the contribution of the pressure to the virial is, in all, 



^p I \(lx + my -f nz) dS 



by Green's Theorem, 



=l^a 



where O is the volume of ^the vessel. 



