Kinetic Theory of Gases. 605 



equations (3) may be neglected, and we may write 



JJJ . . . dx h dx c ... = n jjj dx b dy b dz b = ft N -\ 



for we now have \\\ dx a dy a dz a = £l, the volume of the con- 

 taining vessel. Expression (7) now becomes 



N 



q/(i«, i\iv) dxdy dz dudvdw (8) 



The first factor is the mean density of molecules in the 

 containing vessel, and the connexion between this result and 

 that usually obtained for a gas, whether homogeneous or not, 

 will be obvious. 



§ 13. Let us now revert to the general result of § 11. We 

 found in expression (7) the probability that there should be a 

 molecule (which we may now agree to call A) of which the 

 coordinates should lie between x and x + dx, &c. Let us now 

 find the probability that, in addition to this, there shall be a 

 second molecule having its six coordinates lying between x' 

 and x' + dx', &c. The probability that in addition to the 

 original condition satisfied by A, this second condition shall 

 be satisfied by B, can be deduced from expression (7) by 

 limiting the integration in the numerator to values of Xb lying 

 between x and x' + dx', values of yb lying between y' and 

 y' -rdy', and so on for z b) lib, t'b, Wb. The probability of which 

 we are in search will be N/(i/, v', w r ) du' dv' dw' times this 

 corrected probability; for this is the number of molecules 

 which can take the part of molecule B. The probability in 

 question is therefore found to be 



N 2 /(w, v, w)f(u r , */, w f ) dx dy dz du dv dw dx' dy' dz 1 du' dv' dw' 



JJJ ...dx a dx b dx c dx d ...' 



§ 14. In the special case in which R = this reduces, by 

 the method of § 12, to 



( 7S )/( w ? v 5 io)f(u', v', w') dx dy dz da dv dw dx' dy' dz du' ' dv' dw'. (10) 



§ 15. The probability given by expression (10) is exactly 

 that which would be found in a homogeneous gas, with the 

 help of the " molekular-ungeordnet " assumption. The whole 

 supposed point of this assumption is, however, to exclude a 



