for Regression, $c., in the case of Skew Correlation. 485 



say. In normal correlation o^x/l R^ is the standard deviation of 

 an X r array, corresponding to any given types of X 2 and X 8 . In 

 general correlation it may be regarded as the mean standard deviation 

 of the Xx-arrays from the plane 



or as the standard error made in estimating Xi from x z and #3 by 

 relation (12). 



The quantity R is of some interest, as it exactly takes the place of 

 r in the residual expressions (7). R! may, in fact, be regarded as a 

 coefficient of correlation between x\ and (#vc 3 ) ; it can only be unity 

 if the linear relation (9) or (12) hold good in every case. 



The quantities 6 12 , &i 3 , &c. (the others may be written down by 

 symmetry), may be termed the net regressions of Xi on #2, Xi on # 3 , 

 &c. If we write 2 for 1 and 1 for 2 in the value of 6 12 , we have 



621 being the the net regression of x z on x t . In normal correlation, 

 612 and 6 2 i are the regressions for any group of X^s or X 2 's associated 

 with a fixed type of X 3 's. Hence, in this case (normal correlation), 

 the coefficient of correlation for such a group is the geometrical mean 

 of the two regressions, or 



a quantity that may be called the net coefficient of correlation 

 between a? x and cc 2 .* The similar net coefficients between x\ and # 3 , 

 Xz and ofc, may be written down by interchanging the suffixes. 



In normal correlation yo 12 is quite strictly the coefficient of correla- 

 tion for any sub-group of X/s and X 2 's, whatever the associated type 

 of X 3 's. In generalised correlation this will not be so, and /> 12 can 

 only retain an average significance. 



The method does not appear to be capable of investigating changes 

 in the net coefficient as we pass from one type to another, but it may 

 be noted that whatever the form of the correlation, p l2 retains three 

 of the chief properties of the ordinary coefficients : (1) it can only be 



* My quantities, J 12 , b ls , &c., were termed by Professor Pearson (" Regression 

 &e.," ' Phil. Trans.,' A, vol. 187 (1896), p. 287), "Coefficients of double regression," 



and quantities like i^-^, #13--, &c., "coefficients of double correlation." My 

 <TI ff l 



quantities p he did not use. Having named the p's " net correlation," it seemed 

 most natural to rename the J's " net regressions," as the Vs and p's are correspond- 

 ing quantities. 



Some of my results given above were quoted by Professor Pearson in his paper 

 (loc. cit., notes on pp. 268 and 287). 



