Series to Frequency Distributions in Space. 393 



Obviously, as the regression has already been shown to be 

 strictly linear, it should be found on deducing the moments 

 of higher order than the third that the hypergeometrical 

 series is subject to the general identities 



Pa,lP20 = Pa+l,0PlV 

 Pl t (SP02=Po,fi+LPll' 



Of course, these are not the only relations possible to find, 

 also relations between the pure marginal moments are at 

 hand. 



The application of solid hypergeometrical series to 

 correlation surfaces must, as we have shown, be confined to 

 cases of strictly linear regression. That this has not been 

 observed by Isserlis is evident, as otherwise he would have 

 mentioned it, or, at least, he would not have attempted to 

 apply the series to a case of decidedly curvilinear regression, 

 as in the example of the correlation of the ages of bachelors 

 and spinsters at the epoch of marriage. As regards the 

 example of the numbers of trumps in whist, the regression 

 is linear^ but there is an error in the computation of the 

 correlation coefficient, which is —0*3305, not —0*2559. 



Note I. — The chance problem that gives rise to the above- 

 mentioned multinomial series is the following : A bag contains 

 n balls. The balls are either white or black, besides being- 

 marked by either an even or an odd number. Of the balls 

 npi are black and even, np 2 are black and odd, np z are white 

 and even, and np A are white and odd ; r balls are drawn and 

 each ball is replaced after draiving. This is repeated N times. 

 If N is a large number, the theoretical frequency of sets 

 with s black balls and s' even balls is the coefficient of x s y s ' 

 in the development of 



(p x xy +p 2 x + p s y + pt)r. 



The moments of this series are deduced in my memoir in 

 the Meddelanden frcui Lands Astronomiska Observatonum 

 cited above. The regression is strictly linear. If the balls 

 are not replaced there arises a series in which the terms are 

 certain sums of the terms of a hypergeometrical series in 

 three dimensions. Hereby the regression will still be strictly 

 linear, as the even balls in samples of s=S black balls come 

 forth as if they had been drawn in S trials from a bag con- 

 taining all the black b;Jls and in ?• — S trials from a bag con- 

 taining all the white balls of the initial bag. The mean of 

 the number of even balls in samples of S black balls will 



