612 Mr. J. H. Jeans on the 



so long as K 1? K 2 are both small. We now see that in 

 this case the whole value of the integral arises from the con- 

 tribution supplied by an infinitesimal range of values of K 

 near to and including the lower limit. It is obvious that the 

 proof can be extended so as to include all values of K whether 

 great or not. 



§ 24. Returning to the original generalized space, the 

 following propositions will now be seen to be true : — 



(i.) All except an infinitesimally small fraction of the 

 whole space has a value of K which < e, where 6 is a small 

 positive quantity, and this is true however small e may be. 



(ii.) K is positive at every point — either finite or vanish- 

 ingly small — and the mean value of K averaged throughout 

 the whole space is vanishingly small. 



(iii.) Of that part of the space for which K>K , where 

 K is some finite positive quantity, all except an infinitesimal 

 fraction of the whole has a value of K which < K + e, 

 where e is a small positive quantity, and this is true however 

 small e may be. 



§ 25. Let us now suppose that the edges of our cells are 

 so great compared with the average distance between two 

 molecules that we may legitimately regard the number of 

 molecules in each cell as infinitely great, but that at the same 

 time these edges are so small compared with the scale on 

 which the density of the gas varies, that we may regard the 

 density as constant over a large number of adjacent cells. 

 If the cells can be constructed so that these suppositions are 

 simultaneously possible, it will be possible to give a definition 

 of the density at a point in the gas which shall be logically 

 sound — we define the density at any point P as the number 

 (or mass) of molecules in the cell which contains P, divided 

 by the volume of this cell. 



Let us denote the density at a point by p and the mean- 

 density of the gas by p . By hypothesis, p is appreciably 



constant throughout a cell. An integral such as \\\ p dm dy dz 



taken throughout a cell may accordingly be replaced by pco. 

 The value of p/p in the srh cell may be replaced by a s /a Qy 

 when the system is of class A (§ 19). 



We now see that instead of defining K by equation (13) 

 we may suppose it defined by 



n p *\p J 



where the summation extends throughout all the cells, and 



