Mathematical Contributions to the Theory of Evolution. 495 



Breadth of sknll :* ^ = 150-47, a l = 5'8488, t\ = '3-8871. 



Height of skull : m 2 = 133'78, a* = 4'6761, v z = 3'4954. 



Length of skull ; ra 3 = 180'58, <r 3 =: 5'8441, v 3 = 3'2363. 



Cephalic index, B/L : i a 83'41, 2 13 = 3'5794, Y 13 =. 4'2913. 



Cephalic index, H/L : i^ = 74 ; 23, 2 23 = 3'6305, V 23 = 4-8909. 



Cephalic index, H/B : ^ = 89-12, 2 21 = 4-1752, V 21 = 4'6849, 



The coefficients of correlation may at once be deduced : 



Breadth and length ; r 13 = (v^+v^ V 13 2 )/(2*? 2 t; 3 ) = 0'2849. 

 Height and length : r 23 = (t> 2 2 + v 3 2 V 23 2 )/(2v 2 v 3 ) = 0'0543. 

 Height and breadth ; r 21 = (y + v? V 21 2 )/(2i7 1 v 2 ) = 0'1243, . 



This is the first table, so far as I am aware, that has been published 

 of the variation and correlation of the three chief cephalic lengths.f 

 It shows us that there is not at all a close correlation between these 

 chief dimensions of the skull, and that a small compensating factor 

 for size is to be sought in the correlation oE height and length, i.e., 

 while a broad skull is probably a long skull and also a high skull, a 

 high skull will probably be a short skull, and a low skull a long skull. 



Without substituting the values of v 1} v 2 , t' 3 , ri 2 , ?'i 3 , r 23 in 

 (v), we can find />, or the correlation between breadth/length and 

 height/length indices from ; 



P = (V 13 2 -hV 23 2 -Y 12 2 )/(2Y 13 V 23 ). 



This follows at once from the general theorem given in my memoir 

 on " Regression, Panmixia, and Heredity," ' Phil. Trans./ vol. 187, 

 A, p. 279, or by substitution of the above values of r t2 , ri 3 , r& in (v), 

 we find : ; 



P == G'4857, 



If we calculate from (vi) the correlation between the same cephalic 

 indices on the hypothesis that their heights, breadths and lengths 

 are distributed at random, i.e., that our "imp "-has constructed a 

 number of arbitrary and spurious skulls from Professor Ranke's 

 measurements, we find : 



PQ 0-4008. 



It seems to me that a quite erroneous impression would be formed 

 of the organic correlation of the human skull, did we judge it by the 

 magnitude of the correlation coefficient (O4857) for the two chief 



* All the absolute measures given are in millimetres, and the coefficients of 

 variation are 'percentage variations, i.e., they must be divided by 100 before being 

 used in formulae (i), (ii), and (iii). 



f I hope later to treat correlation in man with reference to race, sex, and 

 organ, as I have treated variation. 



