THE 



LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE 



[SIXTH SERIES.] 



JANUAR Y 1909. 



I. On the Law of Probability for a System of Correlated 

 Variables. By S. H. Burbury, F.R.S* 



1. TN the works of Karl Pearson, Yule, and other writers 

 JL on the theory of evolution, the term correlation 

 has acquired a special meaning. For the purposes of this 

 paper I define it as follows : — Let x and y be two quantities, 

 each of which varies continuously between its own limits, 

 and let them be independent variables, in the general sense, 

 that if either be given, the other is not thereby determined, 

 but may vary continuously through a finite range of values. 

 That is, the system of x and y has two degrees of freedom. 



The chance that x shall lie between x f and rf + dx, that is the 

 number of cases in which, out of a very great number of cases, 

 it so lies, is a function of x\ saj fi(x f )dx. The correspond- 

 ing chance for y lying between // and >/ + dy shall be f 2 (y')dy. 

 It may be that, notwithstanding the system having two 

 degrees of f reedom, /i(V) is a function of y as well as of 

 x\ and / 2 (j/) is a function of x as well as of y\ so that 



— ^(V^O, and -j-f 2 (y)^0. If this is the case, x and y 



are correlated. The chance that simultaneously x shall lie 

 between X\ and x x + dx, and y between y l and y r -\-dy is of 

 the general form (^{x^^dxdy. If 



^/iO)=0, and^/ 2 (y) = 0, <f>(.vy) =f 1 { : v)f i (y), 



and x and y are not correlated. If they are correlated, 



* Communicated by the Author. 

 Phil Mag. S. 6. Vol. 17. No. 97. Jan. 1909. B 



