Mathematical Contributions to the Theory of Evolution. 489 



was always less than the standard error when only x z was taken into 

 account, unless 



/>13 = 0. 



We may now prove the similar theorem that when we use three 

 variables, x zi 3 , *i, on which to base the estimate, the standard error 

 will be again decreased, unless 



Pli = 0. 



The condition that S(?r), in our present case, shall be less than 

 S(r 2 ) in the last, is, in fact, 



2 2 + r 13 2 + r M - n^-r^Vu 2 -r 13 V 24 2 -| 



Wl 

 J 

 13 2 2r 12 r 13 r 23 )(l--r 23 2 r 24 2 ?' 



This may be finally reduced to 



0, 

 that is /> 14 2 > 0. 



The treatment of the general case of n variables, so far as regards 

 obtaining the regressions, is obvious, and it is unnecessary to give it 

 at length. 



We can now see that the use of normal regression formulae is quite 

 legitimate in all cases, so long as the necessary limitations of inter- 

 pretation are recognised. Bravais' r always remains a coefficient of 

 correlation. These results 1 must plead as justification for my use of 

 normal formulas in two cases* where the correlation was markedly 

 non-normal. 



" Mathematical Contributions to the Theory of Evolution. On 

 a Form of Spurious Correlation which may arise when 

 Indices are used in the Measurement of Organs." By 

 KARL PEARSON, F.R.S., University College, London. Re- 

 ceived December 29, 1896, Read February 18, 1897. 



(1) If the ratio of two absolute measurements on the same or 

 different organs be taken it is convenient to term this ratio an index. 



If u =/!(#, y) and v =/ 2 (^, y) be two functions of the three variables 

 a/*, 2/, 0, and these variables be selected at random so that there exists 

 no correlation between #,?/, y,z, or z,x, there will still be found to 



* ' Economic Journal,' Dec., 1895, and Dec., 1896, " On the Correlation of Total 

 Pauperism with Proportion of Out-relief." 



