Series to Frequency Distributions in Space. 391 



number, the mean of the number of black balls in the second 

 sets of trials will be 



, _r'(n?— S) 



n — ! 



(i) 



Thus the mean value of s' for a given value of s is a linear 

 function of s, by which the linearity of regression is proved. 



It will be of some interest to see how the moments of 

 the hypergeometrical series, as found by Dr. Isserlis, are 

 consistent with the general conditions for linear regression, 

 and especially how these conditions are even contained in 

 the identical relations between the muments as far as such 

 have been deduced. First, we must, however, find the 

 relations between the moments that are the necessary 

 conditions of linearity of regression. Denoting by f and rj 

 the deviations from the means s—rq and s'—r'q, we must 

 (as from the above regression formula it is obvious that when 

 S = r<7 we have s r s z=r'q) give to the equation of the regression 

 line of, for instance, s' on s, the form 



V t =&* (2) 



Denoting the coefficient of correlation by p and the standard 

 deviations of s and s' by a and o-', we necessarily have 



l>=P°' a (3) 



The truth of formula (3) is well known and may be 

 demonstrated in the following way. Multiplying (2) with 



and summing for all values of the variable f, we obtain 



%fy%z(Z + rq 7 V + r<<j) = h^^z{t; + rq, V + r'q). (4) 



According to the signification of tj, as a mean of n for 

 constant f, this may be written 



V 



* 2&*(f + ''?> V+ r'q) = h%X^z^+rq, v + r'q) 

 or, denoting by />■■ the moments about the mean of z(s, s'] 



pu = bps 



.'II • 



As p= /— - -- '? we have thus proved the truth of (3). 



VP20P02 

 The equation of the regression line is hence 



v t = P^t (5) 



