238 PROFESSOR K. PEARSON AND MR. L N. G. FILON 



"We require to find 



,, the standard deviation of errors in o- lt 



(Tj) )! )> ~2) 



^V J2 > J) )! 5! '12) 



2,,, ., ,, pi = r^a-i/a-.i, 



which is the coefficient of regression of a:,,* 

 R, 1<ra , the coefficient of correlation between errors in cr l and or.,, 



<ri>'n> ' '' " ""l " *"> 2 ' 



^ cr 2 rjj )) 5* ) ^2 )> '12) 



S/,,, the standard deviation of errors in h } , 



Sft,) j) ;) ,) "2) 



H/,,;,.,, the coefficient of correlation between errors in h, and h 2 . 



It follows that R Vl , R,,^, R,, i<Ta , R,,^, R^., 12 , R,, a ,. 2 , are all zero, since, by (ii.) of p. 233, 

 G rt will vanish when the distributions of Xi and x., are symmetrical. These correla- 

 tions would not, however, vanish for the skew frequency distributions, which are of 

 most frequent occurrence in problems of heredity and fertility in man, &c. 



The first -stage in the investigation is to write down the second differentials of the 

 logarithm of z for all quantities occurring in it. 



We find 



7 <> 1 A 9/1 9 \ *T 9 \ [ J 



rtcrf erj [ 07 (1 r,- 2 ) cr^cr., (1 'J 2 ) J 



? ~ ~ T 1 ~ ~~ -f" "~ [ " a (' 



d*qogz) _ rw lt 



2i' t& ' ' > / 1 t *,2 \ T 



1*1 iX'jiicg ( J- ~r * j9 / I 



, ,, " , v , -r, rr, \ > 



[(1 - B) ! 0-10-2 (1 - nrJ 



l 1** 1 + 3?j 8 /M_ i ^_\ i 6? 'ia 



of (1 ??,) 



* See the memoir on " Heredity, Regression, and Panmixia," p. 268. 



