Kinetic Theory of Gases. \ 611 



<%, rt 2 , &c, being integral, will, however, be greater than the 

 unrestricted minimum for K, so that the infinitesimal value 

 of this former minimum will be positive. 



Hence it follows that, except at one point in our subsidiary 

 space and points which are in its immediate neighbourhood, 

 e-^K is vanishingly small. The density at this point is there- 

 fore infinite in comparison with that elsewhere. 



§ 23. There now three possibilities between which we have 

 to discriminate. If we draw a small region in our subsidiary 

 space, inclosing the minimum value of K, the density inte- 

 grated throughout this small region may be either vanishingly 

 small or finite or infinitely great in comparison with the 

 integral density taken throughout the remaining space. It 

 will be found that the last possibility is the true one. 



To prove this, we expand K in the neighbourhood of the 

 minimum value K = 0. We have (equation 17) 



K=i2^1og|* (22) 



ll s CIq Lin 



If we put a 5 =a + e s , where e s /a is small, we get 

 a a 



a 

 whence, since Se s = 0, 



° a a n 2\aJ 



= 2^©- W 



K 



We may now suppose the small region inclosing the 

 minimum value of K to be bounded by K = /e, where k is a 

 small positive quantity, and the equation of this boundary in 

 rectangular coordinates will, by equation (23), be 



w 



Znfc. 



Hence we see that jy...rff 1 df 2 ... taken inside the region 



K</e, is proportional to K n ~ l so long as k is small. By 

 differentiation, we find that the value of the integral from K 

 to K + ^K is proportional to K" -1 ^K so long as K is small. 

 Hence, except for a multiplying factor, the integral (16) 

 may be written in the form 



K=K a 



J^-nkk"- 1 ^ (24) 



