46 The Law of Distribution [CH. in 



Physical Interpretation. 



53. In 11 we discussed the conditions under which it is possible 

 logically to define the "density at a point" in a gas, in such a way that the 

 density is a definite continuous function of the position of the point at which 

 it is estimated. It was there found to be necessary for us to be able to divide 

 the gas into elements of volume, " which may, without appreciable error, be 

 supposed to be infinitely great in comparison with the distances between 

 neighbouring molecules, and at the same time infinitely small compared with 

 the scale of variation of the gas." We shall now see that when these con- 

 ditions are satisfied the quantity K admits of a physical interpretation which 

 is independent of the way in which the volume of the gas is divided up 

 into cells. 



In assuming the quantities a lt a 2 ...a n to be infinite, we have already 

 supposed the first of the above mentioned conditions to be satisfied by the 

 cells. If we assume the second also to be satisfied, we can speak of the 

 density of the gas at a point, and this density may, without appreciable error, 

 be taken to be constant throughout any single cell. In general let v denote 

 the molecular density, let v s denote the mean molecular density in the sth 

 cell, and v that in the whole gas. Then, obviously 



a s na s N 



and therefore ITv ................................... (89)< 



Also a s , the number of molecules in the sth cell, may be taken equal to 

 v s multiplied by the volume of the cell, so that 



a s = 1 1 \vdxdydz (90), 



where the integral extends throughout the sth cell. 



The value of K a is given by equation (64), in which (a s + ) may be 

 replaced by a s and therefore by expression (90). If we further use equation 

 (89), we find that (64) can be replaced by 



_.\ 



(91), 



in which the integration now extends throughout the whole volume of the 

 gas. The equation may also be put in the form 



* (92), 



expressing that K is the mean value of (V/VQ) log (v/v ) throughout the gas. 

 The results of 50 and 52 are, however, only concerned with the minimum 

 value of K, and comparisons between two different values of K. It is there- 



