239-241] Rate of Dissipation of Energy 203 



&, /8 2 ... it appears that when p is very great the term with suffix 1 will 

 be the preponderating term in the sum on the right-hand of equation (489). 

 We may accordingly write the equation in the form 



(490), 



ap 



and, since the modulus of a product is equal to the product of the moduli of 

 its factors, it follows at once that 



(491). 



Since, from equation (475) ap 2 =6, it follows that expression (484), which 

 gives the mean increase of the energy of vibration, becomes 



9-TT 2 



' (492), 



and the right-hand member depends on p only through the factor e~ 2pftl . It 

 therefore appears that, subject to the assumptions we have made, the energy 

 of the vibrations set up in the molecules decreases exponentially with p. 

 This immediately shews what an enormous range of values is possible for the 

 rate at which the translational motion becomes dissipated through these 

 vibrations. For instance, the two values of the expression given by equation 

 (492) which correspond to p^ = 200 and p^ = 100 stand in the ratio e" 200 , or 

 about 10" 88 . It will, therefore, be seen that there is no difficulty in account- 

 ing for the slowness of the process of dissipation in a gas, provided we can 

 attribute a sufficiently high value to pfii. 



241. The values of p are known with great accuracy, being the fre- 

 quencies of the light vibrations emitted by the molecule. 



To determine the values of /3 X we should require to have full knowledge 

 of the forces which come into play at an encounter of molecules. This is, of 

 course, beyond our reach, but without it we can arrive at an estimate of 

 the order of magnitude of ft^ which, although not exact, is sufficient for our 

 purpose. 



Let us begin by considering a collision between two elastic spheres. In 

 this case the forces may be supposed to act - instantaneously. We do not 

 require to know the variations in the magnitude of the force-function (476) 

 which occur throughout the collision: all that is significant is its value 

 integrated throughout the collision. We shall therefore get accurate results 

 by assigning to this force-function any form which is such that the collision 

 is instantaneous, and that its value integrated through the collision has 



a certain given value. Suppose, for instance, that I Udt, where the integral 



