58 The Law of Distribution [CH. iv 



since we have supposed the molecule A to be at the point x, y, z, we may say 

 that 7(6, c, ...) is a function of x, y, z only. Equations (117) and (118) now 

 become 



= Nf(u, v, w}dudvdwdxdydz r , , ' ' 1 1 ^- ...(122). 



v I (a, o, c, a ...) 



-fS = jyy( u> v ^ w }f(u' } v ' } w'}dudvdwdxdydzdu'dv'dw'dx'dy'dz' ^rj- ~ L r^- N 

 v 1 (a, o, c, a ...) 



(123). 



From equation (122) we see that the density of molecules of class A 

 at x, y, z may no longer be taken to be 



vf(u, v, w) dudvdw 

 but must be taken to be 



Vif(u, v, w) dudvdw, 



where Vl = N /*V'"\ -..(124), 



I (a, b, c...) 



N 



a quantity which reduces to ^ , and therefore to v, when the molecules are 



aZ 



infinitely small. In general v 1 is a function of x, y, z but it is not a function 

 of u, v, w. We may conveniently refer to v^ as the " effective molecular 

 density " at the point x, y, z. When we require to specify the point x, y, z 

 at which v l is estimated, we shall replace Vi by v x>y>z . 



The " expectation " of the number of molecules in the whole vessel is 

 equal to the total number of molecules actually present in the vessel, so that 



1 1 1 v x,y,z dxdydz = 



Thus v is the mean value of v x , y ,z averaged throughout the vessel. We shall 

 see later the importance of the distinction between v x>yiZ and v. 



From equations (122) and (123) we obtain 



Vp Vq _ Vpq I (b, c,d ...)I(a > c,d...} 

 v v v I {c, d ...)I(a, b, c, d ...) " 



Hence given that the molecule A is in position at x, y, z the probability 

 that a second molecule B has a position at x', y', z' is not that which would 

 be given by the assumption of molecular chaos, but is equal to this value 

 multiplied by 



/(c, d ...} I(a, b, c, d ...) 



7(6, c,d. ..)/(, c,d...y 



a function which is symmetrical as regards the x, y, z coordinates of A and 

 B, and which is independent of the velocities of the molecules A and B. 



