40 The Law of Distribution [CH. m 



Now Sa g = N, so that S (a g + ^) = -ZV+ |-w, and hence it will be found that 

 the foregoing equation may be transformed into 



_ 



a = log ,i - _ log 27r^- S (a. + 4) log ...... (63). 



It will be convenient to write 



........................ (64), 



-* ' S = 1 * 



so that 6 a , the fraction of the generalised space which represents systems of 

 class A, is given by 



ri?n 



O a =- -^e-* ........................ (65). 



(2irN) * 



41. It is now necessary to represent the various possible classes of 

 systems in a new generalised space. Let us imagine a space of n dimensions, 

 in which the position of a point is specified b}^ n rectangular coordinates 

 X 1} X 2 ...X }1 . Then we may suppose a system of class A represented in this 

 space by the point 



X, = a lf .X 2 = a 2 , ........ , X B = a ........................ (66). 



Not every point in this generalised space will represent a possible class of 

 system, for we are supposing a^, a 2 ... to be integers, and their sum to be N. 

 The different possible classes will be represented by those points for which 

 Xi, X 2 ... X n have integral values such that 



x, + x a + ... + x n = N .............................. (67). 



Just as in two-dimensional space a linear equation between the coordinates 

 represents a line (of one dimension), or as in three-dimensional space a linear 

 relation between the coordinates represents a plane (of two dimensions), so in 

 the present w-dimensional space, the linear relation (67) represents a region 

 of (n 1) dimensions. All values of the coordinates which satisfy (67) will 

 represent points in this (n l)-dimensional region, and therefore all possible 

 classes of systems may be represented in this region. The representation is 

 still limited to points for which the x's have integral values, and these points 

 are perhaps most easily thought of as the intersections of systems of parallel 

 straight lines. 



If we now place a mass a (cf. equation 65) at the point representing 

 class A, and a similar mass for every other class, then the various possible 

 classes will be represented both in nature and magnitude in this new 

 generalised space. 



42. Consider the function K, a continuous function of position in this 

 new generalised space, defined by 



1 s=n v 



^ = ^2 & + i)log^? ........................ (68). 



