RECENT STATISTICAL THEORIES 677 



on the first, which we shall use in special cases.- It is also well known 

 that to obtain the distribution-function in some of the independent 

 variables from the distribution-function of all of them, it is necessary 

 to integrate the latter over the entire range of all the other variables: 

 in such a case as is symbolized by equation (1), the distribution in 

 pz would be obtained by integrating / with respect to the first five 

 variables over the entire range of each. 



The product dxdydz is an element of volume in ordinary or coordinate- 

 space; the product dpxdpydpz is an element of volume in momentum- 

 space, in which each particle is represented by a point having for its 

 coordinates in a Cartesian frame the values of its momenta; the 

 product dxdydzdpxdpydpz is an element of volume in phase-space. 

 The function / describes the distribution of the assemblage in this 

 phase-space of six dimensions. In some cases — for instance, that of 

 electrons in a metal not at an even temperature, and that of oscillators 

 — we shall have to think continually of this six-dimensional space. 

 In others — whenever we deal with photons, and whenever we consider 

 atoms or electrons in a region where neither temperature nor potential 

 varies from place to place — we shall be able to assume that the distri- 

 bution in the coordinate-space is uniform (that / is independent of x, 

 y, z) and to dismiss it from mind, and to derive the distribution in 

 the three-dimensional momentum-space quite separately as if there 

 were no other. Even in these simplest cases it would no doubt be 

 more consistent to operate always in the phase-space. Unhappily 

 the human mind is so constructed, that no matter how much it may 

 ratiocinate about space of six dimensions or six trillion, it always 

 visualizes in space of three. 



In an equation such as (1), the differential element or the product 

 of such elements which terminates the right-hand member must be 

 neither too large nor too small. If it is so large that / varies con- 

 siderably from one point in it to another, then its multiplier, which 

 is by definition the mean value of / in the said element, must be com- 

 puted by the methods of integral calculus. If on the other hand it is 

 so small that it contains only a few of the corpuscles, then the product 

 of/ into its size may be many times as great or many times as small 

 as the number which it does contain. This is easily perceived by 

 proceeding to the absurd limit of dividing the space into say ten 

 times as many elements as there are corpuscles, so that in at least 



2 Let Ml, Ui, ■• • represent the variables of the first set, vi, V2, ••• those of the 

 second; let f{ui, Un, •••) and F{v\, V2, •••) stand for the distribution-functions in 

 the two sets; then 



77/- \ II ^("1- "=' ■ ■ ■"* 



r\vi, V2, ■ • •) = f{vi,V2, ■ ■ ■, -TT— • 



d{Vi, Vi, ■ ■ ■) 



44 



