290 Prof. K. Pearson. Mathematical Contributions [Jan. 20, 



r i\rt x = ?>f 2 = r, r xt2 = r 1Jt] = r\ 

 r xy = p, r ''t x u — r, ir = <r x = a-,,, 



„l-r 2 -r 2 -r' 2 + 2rn-' 

 V = j-^ 



l-r 8 2;y-r(^ + r^) 



^ 1 _ r 2 _ r 2 _ r >2 + 2m .' l-r 2 - f 2 - r' 2 + 2rrr' ™ 



This is the full solution of the first problem. 



We see that in order to solve it, it is necessary : 



(i.) To find the correlation p of the homologous pairs as if they 

 were simple homotypes. 



(ii.) To find the correlation r between the growth periods of each 

 pair of homotypes. 



(iii.) To find the correlation r between the character and the period 

 of growth. 



(iv.) To find the correlation / between the character of one homo- 

 type and the period of growth of its fellow. 



Now these correlations can be found at once by the usual statistical 

 processes, if the data are forthcoming. 



(3.) I propose to illustrate this on material, which, although not 

 homotypic, is so analogous that it brings out all the important features. 



We will determine the correlation between the head-length of 

 brothers, such length being measured on school boys of all ages, from 

 4 to 19.* It will be clear that we have here all the difficulties of the 

 homotypic problem — resemblance due to common origin obscured by 

 differences in the period of growth of each individual. 



Table I gives the correlation of pairs of brothers without regard to 

 their differences of age. 



Table II gives the correlation between age and length of head in 

 the same individual. 



Tables IIIa and IIIb gives the correlation between the age of one 

 brother, and the length of head of the second. 



Table IV gives the correlation between the ages of pairs of brothers. 



These tables have been prepared by taking off from the brother- 

 brother data papers of my school measurement records all the avail- 

 able pairs of cases falling into each series. Thus in some cases the 

 ages of both brothers were given, but not the head measurement of 

 one or other ; in other cases the head measurements of both, but the 

 age of one or other would fail, or again the age of one and the head 

 measurement of the other might be all the information available. 

 Thus the total number of cases and the frequency distribution varies 

 slightly from one table to a second. 



* The measurements form part of the material obtained with the assistance of 

 a grant from the Royal Society Government Grant Committee. 



and we find 



