4 Mr. S. H. Burbury on the Law of Probability 



given value depends on x n as well as on x x . But n being 



df(x ) 

 very great, these correlations, e.g. ' 7 , &c, are inappre- 



ciable, and in the ordinary theory are generally treated as 

 non-existent. 



7. The definition above given of correlation is purely alge- 

 braic. Generally, if x and y be two o£ the variables which 

 determine the state of a material system the parts of which 

 mutually act on each other, the time differential coefficients, 



dor dii 



t; and ~, are correlated by virtue of that mutual action. 



d x dzi 



For — and ~ are functions of the same variables, and 

 dt dt 



therefore correlated. And if the system be defined by n 

 generalized coordinates q x . . . q n , and their corresponding- 

 velocities qi...q n , the products q\'q 2 &c appear in the 

 expression for the kinetic energy, and therefore the mean 

 product q , q 2 is not generally zero, and whether or not the 

 means q x and q 2 are separately zero, q x and q 2 are correlated. 



8. Since correlations may, or may not, be negligible we 

 may suppose that Xi ... x n , or the things to which they relate, 

 have at every instant positions in space, and that the corre- 

 lation between any two of them, as x and x , is or is not 

 negligible, according to the distance which at the instant 

 separates them from each other. But this localization is not 

 essential to the general statement that correlations may or 

 may not be negligible. 



A General Problem stated. 



9. Let Si ... s be n quantities, which, until otherwise 

 stated, shall be each of zero dimensions, each of which varies 

 continuously between assigned limits. I assume them to be 

 in general correlated with each other, but that such correla- 

 tions may as regards any s, as s , be negligible for some or 

 for most of the others. The chance that they shall respec- 

 tively lie 



Si between r v and ?\ + dr h 

 s-2 9i r 2 5 ? r 2 + dr 2 , 



between r and r -f dr 



