46 W1UKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



Pearson found, using moments up to the sixth order 



y x = 10,02 1 5 +0,1931 (X — M,) — 0,0498 (x — ?«,)" + 0,0017 1 (.v — m,) 3 , 



and using moments only to the fourth order 



y x = 10,3 6 63 +0,23 61 (a; — m,) — 0,0734 (x — »«,)-• 



When tried on the data the curves will be found to fit equally well, so we 

 shall compare only with the second, which depends 011 the same moments as our curve. 

 The result is 



Table VIII. 



X 



yx 1'earson) 



?A*(35) II 



Observation 



1 



4,458 



6,437 



5,300 



2 



5,724 



7,020 



5,833 



3 



6,842 



7,554 



7,790 



4 



7,813 



8,040 



8,050 



5 



6,638 



8,477 



9,473 



6 



9,315 



8,866 



8,436 



7 



9,8 16 



9,207 



8,596 



8 



10,229 



9,50 1 



10,207 



9 



10,466 



9,745 



10,701 



10 



10,555 



9,942 



11,027 



11 



10,498 



10,090 



10,953 



12 



10,203 



10,190 



9,100 



r: 



9,942 



10,242 



9,000 



11 



9,443 



10,245 



1 0,030 



15 



8,798 



10,201 



10,317 



The curves will be seen in plate 2. For a material amounting to 2010 

 individuals the fit is not verv satisfactory in the case of either curve. 



Pearson's example D, which is his last and concerns the correlation of the 

 number of branches to the whorl and the position of the whorl on the stem in 1448 

 Equisetum arvense, we shall not attempt to treat by our methods. Pearson has here 

 himself been obliged to have recourse to quartic regression, which requires that 

 moments up to the eight order shall be computed. Probably in order to apply the 

 J-function we should have to use equally high moments. Our formulae (34*) are, 

 however, not given in a form fit for the use of higher than the fourth moments. 

 Evidently there would be no difficulty in extending the formulae, but we presume 

 that the use of moments up to the eighth order is no gain for any theory. 



(14) Summary of results. In the preceding treatise we have found that, 

 assuming the ^4-function to give an adequate description of the correlation surface, 

 we may always express the regression of any characteristic of the arrays through a 



