12 The Law of Distribution of Velocities [CH. n 



quantity of it. In this case, we take a volume V and suppose M to be 

 the mass enclosed within it. The homogeneity of the matter now ensures 

 that as V is jnpreased indefinitely, the ratio J//F will approximate to 

 a definite limit p r and. as before, we define the density of the matter to 

 be the value of the limit p. 



The gas of the Kinetic Theory will, in general, be neither continuous 

 nor homogeneous. It will therefore be impossible to frame a general 

 definition upon the model of the two foregoing definitions, since to do this 

 we should have tp suppose the element of volume to become infinitely 

 great and infinitely small at the same time. But with reference to the 

 actual conditions of nature this objection is not serious. We can find an 

 element of volume which may, without appreciable error, be supposed to 

 be infinitely great in comparison with the distance between neighbouring 

 molecules, and at the same time infinitely small compared with the scale 

 of variation of density of the gas. For instance, the density of a gas may 

 generally be supposed homogeneous throughout a cube of edge equal to 

 one millimetre, while such a cube is large compared with the scale of 

 molecular structure, containing, as has already been mentioned, about 

 4 x 10* molecules in the case of a gas under normal conditions of pressure 

 and temperature. 



The ratio of the mass contained in an element of this kind, to the 

 volume of the element, will give the mass-dcngity of the gas. If we sub- 

 stitute "number of molecules whose centre is contained in" for "man 

 contained in," the definition gives the molecular-density of the gas. We 

 shall find it convenient to denote the mass-density by p and the molecular- 

 density by 9. If m is the mass of each molecule, we have 



It will be seen that this definition of density is not logically perfect, 

 but it will be admitted that it is adequate for practical use. The difficulty 

 of obtaining a logically perfect definition has been discussed by Burbury*. 

 A similar difficulty is of common occurrence in statistical work : consider, 

 for instance, the statement "the density of population in parts of London 

 is as high as 105 per acre." 



12. If il is a volume throughout which the density is sensibly constant, 

 the number of molecules of which the centres are contained within this 

 volume would, if the foregoing definitions were logically perfect, be fir. 

 As the definitions are not perfect, we must examine within what limits 

 the statement is true, that the number of molecules is fijr. It is certainly 

 not literally true, for neither fi nor * will in general be integers, while 



* & H. Burbuiy, Kimrtit Theory <jf Goes, p. 3. 



