ON THE KINETIC THEORY OF GASES. 221 



which they move. Can it be taken for granted that the 

 only effect of increasing density with which we need con- 

 cern ourselves is the shortening of the mean free path ? 

 And if so what is the shortening of the mean path ? 

 Tait goes so far as to say that when the density reaches 

 a certain point, an almost discontinuous diminution takes 

 place in the mean free path. For which statement the 

 reader may consult Tait (part iv., p. 265). 



34. It appears to me that one effect of increasing den- 

 sity will be that the velocities of spheres which are near 

 to one another will no longer be independent of one an- 

 other, as in the ordinary rare medium they are assumed to 

 be. Given that sphere A has positive velocity U in any 

 direction, there will be a presumption, becoming greater as 

 the density increases, that a neighbouring sphere B has 

 some positive velocity in that same direction. 



That statement can be proved as follows : Suppose a 

 sphere whose velocity is w in any direction, say that of x, 

 to undergo a collision. What is the value of the expecta- 

 tion of its resolved velocity in x after collision ? or (which is 

 the same thing when the spheres are equal) that of the 

 other colliding sphere after collision ? Its accurate form can 

 easily be found in the form of an integral for all values of 

 the pre-collision velocity of the other sphere. For the pre- 

 sent purpose I need only show that it is positive. Let V 

 be the common velocity, p the half relative velocity of the 

 two spheres. Let V = OC, and about C describe a sphere 

 with radius p. 



If w = OO, the pre-collision velocity of the other sphere 

 is OO', where OCO' is a % 



diameter of the p sphere. 

 After collision all directions 

 of the diameter OCO' are 

 equally probable. We see 

 that ( 1 ) if V>p J 2, the cosine 

 of the ano-le between the 



original direction of either J ^~ ~~ y 



sphere and its direction after collision is necessarily positive, 

 (2) if V <p J 2, the value of the expectation of it is positive, 



