204 Theory of a Non-Conservative Gas [CH. ix 



is taken through a collision which occurs at time t = 0, has to be equal to /. 

 Then we may take 



provided we put c = in the limit. For this value is such that U vanishes 

 except when t = 0, and we have 



Ic 



/: 



dt = I (494). 



In this case U, regarded as a function of the complex variable t', has two 

 infinities occurring at t = ic, ic, and each of residue + _ - . Hence there 



is only one infinity t = ic, occurring within the contour of integration in 

 fig. 16, and for this /3 = c. The right-hand member of equation (492) in this 

 case is therefore 



(495). 



Putting c=0 we notice that the expression becomes independent of p. This 

 is as it should be ; when the collision is instantaneous, the forces of restitution 

 in the molecule do not have time to come into play, and therefore do not 

 affect the result of the collision. 



Taking c small, but not quite zero, equation (493) can be supposed to 

 represent the value of U for an encounter which is not quite instantaneous. 

 Here U vanishes except through a small interval of time comparable with c 

 and occurring when t = 0. The effect on expression (495) of taking c small 

 instead of zero, is to decrease it in the ratio e~ 2pc , and the forces of restitution 

 now come into play. The quantity /3 which in this case is equal to c is, as 

 we have seen, comparable with the time through which an encounter lasts. 

 It will be seen that the interpretation of ft which has been obtained in this 

 special case, will hold in every case as regards order of magnitude, and hence we 

 conclude that the rate of dissipation of the energy of a gas may reasonably be 

 expected to be slow, provided that the product of the time occupied by a 

 collision and the frequency of vibration is small. 



The physical principle upon which this result rests would of course 

 have, been obvious enough without a mathematical discussion. What would 

 not have been obvious is the extreme rapidity with which the rate of 

 dissipation decreases as p is increased. 



242. Before we can determine the extent to which this principle applies 

 to a gas, we must form an estimate of the duration of a collision. Let us at 

 first suppose that the molecules are surrounded by spheres of molecular 

 action, satisfying the conditions mentioned in 82. The figures of 158, as 

 well as other experimental results to be mentioned later, shew that the 



