

MATHEMATICAL CONTRIBUTIONS TO THE T&EORY OF EVOLUTION. 245 



To find its value we adopt a method, which we give on this first occasion at 

 length, as it will be frequently used in the sequel. 

 Take logarithmic differentials 



Ap, A/-,, AO-, A<TO 



7T ! ' ^' '~<rT ~^' 



Square and divide by n after summing 



S (Ap,y _ SjA^ S (Ao-,) 2 S (Ao-,)- 28 (A/- u Ao-,) 2S(Ar li Ao-s) -'S (Ao-, Ao-.,) 

 w .s 7M ,2^ 7lr 2 1ta -i nr.cr nr 



Now, remembering the definitions of standard deviations and coefficients of 

 correlation, this may be written 



- - 



^iL 4. i^iL _ f*i*2i5D?4 a. rs!!sc: 



" 1 O'lO'a ''l2"l 



Now all the quantities on the right have already been found in equations 

 (xv.) to (xviii.). Hence, substituting, we have 



pi 



I fence 



Thus the probable error of a regression coefficient 



- '67449 * 



This is of fundamental importance for testing the significance of results obtained 

 by applying the theory of regression to problems in heredity, panmixia, &c. 



The probable percentage error in a regression coefficient = ~ , and 



V n M2 



hence is small if the correlation be close, and increases rapidly if the correlation be 

 small. This again illustrates the point to which reference has been, made in another 

 memoir,* namely, that when only a few individuals can be measured, the most 

 reliable results for the purposes of the quantitative theory of evolution are to be 

 found from the measurements of the most highly correlated organs. 



* PEARSON and LEE : " Con-elation in Civilised and Uncivilised Races," ' Roy. Soc. Proc.,' vol. 61, 

 p. 345. 



