1910.] ia Mendelian Population Mating at Random. 43 
Hence le Wel = Yo—- 93 = 7, 
from which it follows that the regression is linear and the correlation is 
8s—1 
16s—V 
(9) Somatic Correlation.—In this case, from Table III, we have, measuring 
from the “ not A” row, 
as (16s—1)p?+4(16s—1) p?g +(68s—5) pq? +(16s—2) 9g 
- (16s—1)(p+9)? (p+ 29) 
12sp? + 16 spqg 
(16s—1)(p+q)’ 
y Lp pa 29) 
(p+) 
= AES oa ee) ae 
oe Cee Ira, 
Y= 
The somatic correlation therefore depends upon the ratio of p to q, @.e. upon 
the ratio of the number of individuals in the population possessing the 
dominant character to the number possessing the recedent. For diiterent 
values of » and g the above value of 7 can range between cee 
1 16s—2 , .. 20s—3 .* 
9 16s _-1- If p = ¢ its value is Clos) 
and 
(10) In considering the value to be given to s in these formule in order 
that they can.be compared with statistical results, we must remember that 
the latter are obtained from families of varying size, from which we take 
selections of fraternal couples. We must therefore look upon our observa- 
tions as forming a random sample of all the possible pairs of brethren 
obtained from an indefinitely large number of matings. That is, we must 
make s indefinitely large. Then we get the gametic correlation to be 4, and 
the range for the somatic correlation from }4 to $, with the value ;3, when 
p=¢. 
* This particular value of the somatic fraternal correlation was reached by Prof. 
Pearson in his memoir in the ‘ Phil. Trans.,’ A, 1904, vol. 203. In that memoir he dealt 
with somatic characters only, and he assumed the population to be generated from a 
series of hybridisations and not from a combination of individuals exhibiting the 
protogenic, allogenic and hybrid constituents in certain proportions. But he was more 
general in his supposition that the character depended upon m Mendelian couplets, and 
not on a single one. 
