4 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



the hypergeometrical series. His types I to V are continuous functions for these se- 

 ries. An attempt at extending this idea to variation surfaces vvas no doubt the next 

 step taken. If the object was to find continuous functions giving the multinomial- 

 and solid hypergeometrical series, it may easily be understood, that the attempt met 

 with great difficulties, though, at least in case of the multinomial, the difficulties 

 are surmountable. Continuous functions for the multinomial series have, in fact, 

 been deduced by the author in a work on the extension of the theorem of Ber- 

 noulli to two dimensions. 1 As the moments of the solid hypergeometrical series are 

 found without difficulty it was, however, possible to apply the series to correlation 

 surfaces without having recourse to a continuous function for it. This has been 

 done in recent years by L. Isserlis 2 . Now this application was made many years 

 after Pearson's work on skew correlation appeared, but even if it had been made 

 before, Pearson could have had no use for it. Really, the case would have been 

 the same had there been continuous correlation functions available, if they had been 

 built up on schemes giving rise to multinomial or hypergeometrical series. Pearsoists 

 was a work on non-linear regression, but as I have endeavoured to prove 3 in another 

 place correlation surfaces to which the multinomial or hypergeometrical series may 

 be applied must necessarily have linear regression. 



Hence it may be seen that though binomial and hypergeometrical series give 

 fairly general frequency curves their extension to frequency surfaces can cover only 

 very special cases. 



But, Pearson's theory of variation is not restricted to the above schemes. 

 He has devised a far more general method, based on a certain differential equation of 

 singular generality. For one variate the integration was easily performed, but for 

 two variates this has hitherto, so far as I know, proved impossible. At least there 

 has appeared up to date no theory resting on a similar basis. 



However, very general mathematical expressions for correlation functions are 

 now available. In recent years Charlier has developed a theory of skew variation 

 a part of which has been simultaneously advanced by Edgeworth. B}' Charlier 

 the theory has been shown to allow of a ready extension to correlation between any 

 number of variates. 4 



The starting point of Charlier is entirely different from that of Pearson, as 

 the former starts from the hypothesis of elementary errors. Charlier's theory 

 is called by himself the genetic theory of frequency. But the developments of Char- 

 lier have, indeed, properties that make them more or less independent of their hypo- 



1 S. D. Wicksell: Sonie theorems in the theory of probability, with special reference to their impor- 

 tance in the theory of homograde correlation. Svenska Aktuarieföreningens tidskrift, haft. 4 — 5, 191(5. Medde- 

 landen från Lunds Observatorium N:r 74. 



2 Ii. Isseklis: The application of solid hypergeometrical series to frequency distributions in space. Phil. 

 Mag., Sept. 1914. 



3 S. D. Wicksell: The application of solid hypergeometrical series to frequency distributions in space. 

 Phil. Mag., May 1917. 



4 For the work of Charlier see: Meddelanden från Lunds Astonomiska Observatorium, Ser. I, n:ris 25. 

 26, 27, 34. 43, 49, 50, 51, 52, 57, 58, 61, 66, 71, and Ser. II, n:ris 4, 8, 9. 14. Lund 1905 — 1915. 



