48-52] The Method of General Dynamics 45 



'only true provided K^ and K 2 are both small. This condition, however, 

 does not require that NK shall be small, or even finite. 

 The general identity 



N^- \ K ~~*e- NK K* dK=Y( n ^ 1 } (87), 



.' K=Q \ Z / 



holds for all values of N. When, in the limit, N is made infinite, the 

 expression on the right-hand will of course remain finite, and the whole 

 value of the integral on the left-hand arises from the contributions made 

 to this integral by the infinitesimal range of values of K which is in the 

 immediate neighbourhood of K= 0. 



It follows that expression (86) vanishes unless K l = 0, and that it is equal 

 to unity when 'K^ = and K 2 is not equal to zero. 



50. Hence it follows that all except an infinitesimal fraction of the 

 original generalised space represents systems for which K has its minimum 

 value, or for which its value is only infinitesimally different from this. 



51. Transforming expression (86) by the use of equation (87) we see 

 that the fraction of the generalised space which represents systems for which 

 K is less than K is 



/* 





e -NK R 2 

 A'=0 



and hence that the fraction which represents systems for which K is greater 

 than K<i is 



*-'r**&** dK 



=*- -i^-- (8). 



e -NK K 2 dK 



r=o 



As in the former case, the whole value of this fraction arises from a 

 contribution made to the integral in the numerator by a small range of 

 values for K in the immediate neighbourhood of the lower limit. 



When K 2 is not small, expression (88) will require correction. The 

 surfaces K = constant are no longer spheres as in 45, and the consequence 

 of this is that every integrand in the numerator of (88) requires to be 

 multiplied by a correcting factor. This factor, however, is always finite 

 and is independent of N. Hence it follows that the result stated in the last 

 paragraph is true independently of the smallness of K 2 . 



52. It therefore appears that of the systems for which K^K lt where 

 K! is some finite positive quantity, all except an infinitesimal fraction have 

 values of K which are either equal to K l 01 differ from K only by an 

 infinitesimal quantity. 



