CHAPTER II. 



THE LAW OF DISTRIBUTION OF VELOCITIES. 



I. THE STATISTICAL METHOD. 



n 10/ THE mathematical difficulties of the subject commence when we 

 .^ attempt to discuss the law according to which the velocities of the mole- 

 s cules are grouped about their mean value.) We are of course at liberty 

 to consider an imaginary gas in which tne velocities are grouped at the 

 outset according to any law we please, but in general every collision which 

 occurs will tend to change this law. The problem before us is to investigate 

 whether there is any law which remains, on the whole, unchanged by 

 collisions; and if so whether the velocities of the molecules' of a gas, 

 starting from some arbitrarily chosen law, will tend after a sufficient time, 

 to obey some definite law which is independent of the particular law from 

 which the gas started. 



There are two totally distinct methods of attacking these problems, and 

 these are given in this chapter and the next, the relation between them 

 being discussed in Chapter IV. The present chapter contains the classical 

 method of which the development is due mainly to Clerk Maxwell and 

 Boltzmann (see 60 below). 



The definition of Density. 



11. There is no difficulty in defining the density of a continuous sub- 

 stance. If we take a small volume v, enclosing a given point P, and denote 

 by ra the mass of matter contained within this volume, then the assumption 

 of continuity ensures that as the volume v shrinks until it is of infinitesimal 

 size, while still enclosing the point P, then the ratio m/v will approach 

 a definite limit p, and we define the density at the point P as being the 

 value of the limit p. 



Again, when, as in the Kinetic Theory, the matter is composed of discrete 

 molecules, there is no difficulty in defining density if the matter is homo- 

 geneous and if also it can be supposed that there is an infinitely great 



