Mathematical Contributions to the Theory of Evolution. 493 



(4) Proposition HI. To find the coefficient of correlation of two indices 

 in terms^ of the coefficients of correlation of the four absolute measurements 

 and their coefficients of variation. 



Let ajj/afc and x^x^ be the two indices. 



X l+.-_ 

 ra 2 m 4 



m 3 

 if we neglect terms of the cubic order. 



(5) Thus we have expressed p in terms of the four coefficients of 

 correlation and the four coefficients of variation of the absolute 

 measurements which form the indices. 



We may draw the following conclusions : 



(i.) The correlation between two indices will always vanish when 

 the four absolute measurements forming the indices are quite uncor- 

 related, 



(ii.) If two of the organs are perfectly correlated, let us say made 

 identical : for example, the third and fourth, so that r& =. 1, and v 3 = 

 1-4, we find 



p == / ~^ / - __ ___ ....... (v). 



v Vi -f v$ 



This is the coefficient of correlation between two indices with the 

 same denominator (#i/#3 and # 2 /# 3 ). 



The value of p in (v) does not vanish if the remaining organs be 

 quite uncorrelated, i.e., r 12 = r J3 = r m = 0. In this case 



This is the measure of the spurious correlation. For the special 



