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XL VI. The Application of Solid Hij per geometrical Series ta 

 Frequency Distributions in Space. By L. Isserlis, B.A. * 



[Plate VII.] 



§1. r |"^HE connexion between the frequency distribution of 

 JL one variable character, and the hypergeometrical 

 series is well known. It has been discussed exhaustively 

 by Professor Karl Pearson in a series of memoirs in the 

 Phil. Trans, from 1895 on, where a series of frequency 

 curves are studied which may be described as parallels to 

 the hypergeometrical series. 



In a paper entitled " On certain properties of the hyper- 

 geometrical series, and on the fitting of such series to 

 observation polygons in the theory of chance," Phil. Mag. 

 1899, pp. 236-2i6, Pearson showed how to determine from 

 a given frequency distribution the constants of the corre- 

 sponding hypergeometrical series. The method of moments 

 was employed, and the fitting of the series was equivalent to 

 the statement of a problem of chance with a theoretical 

 distribution of events similar to the actual distribution. 



Although much progress has been made with the study 

 of the normal surface, no general theory of frequency 

 surfaces analogous to Pearson's Skew frequency curves 

 exists at present, and this paper is an attempt to make a 

 first step towards such a theory by solving the problem of 

 fitting a double hypergeometrical series to a frequency 

 distribution with two variable characters. 



§ 2. The corresponding chance problem may be stated as 

 follows : — 



A bag contains n balls of which pn are white and qn are 

 black ; r balls are drawn and 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 



{n) r+r , 



We may denote this by Nc(s, /), 

 where 



/h\_h(h-l)(h-2)...(h-k + l) 



w = ~ -p. — 



and 



(h) k = h(h-l)(h-2) . . . (A-* + l). 



Communicated by Prof. Karl Pearson, F.R.S. 



