390 Dr. Wicksell : Application of Solid Hyper geometrical 



have proved to be of only little harm to the generality of 

 the frequency-curves Generated. Now, the writer of these 

 lines has reason to think that the limitations in case of the 

 application to correlation surfaces is of far greater im- 

 portance. The fact which is the ground for the opinion of 

 the writer is the following : All surfaces of correlation described 

 by aid of the multinomial or hyper geometrical series must 

 necessarily have linear regression. With regard to the multi- 

 nomial correlation function, which is the coefficient of afy*' 

 in the development of {p\0cy-\-p^ +P$y+P±f'> the fact has 

 already been demonstrated in my paper in Svenska Aktuarie- 

 foreningens Tidskrift, Nr. 1-5, 1916 (see also Meddelanden 

 fran Lands Astronomiska Observatorium, 1916). 



In order to prove our proposition also in case of the solid 

 rrypergeometrical series, we must first recall the formulation 

 of the corresponding chance problem. In Dr. Isserlis' own 

 words it is : a bag contains n balls of which np are white 

 and nq are black ; r balls are drawn «nd not replaced ; a 

 second draw of r balls is made. This is repeated N times. 

 If N is a large number, the theoretical frequency of s black 

 balls in the first draw and s' in the second is (in a somewhat 

 different notation) 



N.r!r'!(yji)l(/w) \(n-r-r')\ 



sis' I (r — s) ! (r'-s')l (qn — s-s') I Qpn—r — r' + s+s') In V 



Galling this function t(s, s'), the moments p>\j may be 

 deduced either, as does Dr. Isserlis, with the aid of a system 

 of differential equations or more directly by performing the 

 summations 



s,s' 



by which method recursion formulae for the moments are 

 easily derived. Especially it will readily be found that the 

 mean values of s and s' are 



p\ = rq ; p' 01 =r'q. 



Thus it is seen that the means of the number of "lucky" 

 events are equal to the product of the number of trials and 

 the probability of a "lucky" event at the beginning of the 

 drawings. Now, it is an easy task to show that the regression 

 is linear. Indeed, if from the N sets of drawings we pick 

 out all that have given a certain number, say 5 = 8 black 

 balls in the first r trials, they will all have that in common, 

 that the second set of r' drawings has been extracted from a 

 bag that contained n — r balls, of which only qn — S were 

 black. Hence, for these samples, if N be a large enough 



