116 Physical Properties [CH. vi 



Hence in any element dv which is known not to be within a distance 



- of the boundary, or to be included in any one of the spheres surrounding 



z 



each molecule, the probability of finding the centre of a molecule is 



Ndv 



.(276). 



v T (Jr l)fira* 



If, however, the element is selected at random we must consider what is 

 the probability that the conditions postulated as to its not lying inside a 



sphere, or within a distance ^ of the boundary, shall be satisfied. 



Z 



The particular element of volume which is of importance for the calcula- 

 tion of the pressure is one of which the distance from the boundary is just 



greater than ^ . The second condition, therefore, is satisfied as a matter of 



course. To calculate the probability of the other condition being satisfied, 

 namely that the element dv shall not lie inside any one of the N 1 spheres 

 of radius a-, we notice that if it does lie in any one of these spheres, then the 



centre of the sphere, being at a distance not less than - from the boundary, 



z 



must be at least as far away from the boundary as the element dv. In other 

 words, if the sphere in question is divided into two hemispheres by a plane 

 parallel to the boundary, the element dv can only lie in that hemisphere 

 which is the nearer of the two to the boundary. 



Hence the probability that dv, selected at random, shall lie inside any 

 particular sphere is 



so that the probability that it shall not lie in any of the N 1 spheres in 

 question is, as far as the first order of small quantities 



V 



The product of this expression and expression (276) is 



i (^- 1 K 



, r 7 1 - - - 47TCT 3 



Ndv v 







This, then, is the probability that a molecule shall be found in the small 



element dv of which the distance from the boundary is = . As far as the 



z 



first order of small quantities the expression is the same as 



Ndv 1 



