1910.] a Mendelian Population Mating at Random. 49 
(15) Somatic Correlation—Table VI gives, measuring from the “not A” 
row — 
gx (p+29) 
(p+qy 
and Geen et ZY — 
NCS 24 
Again the correlation depends upon the relative values of p and g. The 
limits between which 7 can lie for different values of » and g are 0 and 
8s—1 1 8s—1 1 
16s—1° 2° 16s—1/ 
for r is from 0 to 4, with the value + when p = 4g. 
If p = q, its value is Putting s infinite, the range 
vo] 
(C.) Correlation between First Cousins. 
(16) From Table IV we can pick out every pair of first cousins, and 
arrange them in groups according as each member of the pair exhibits the 
constituent (AA), (Aa) or (aa). But, again, it is more convenient to obtain 
these groups indirectly. For the number of pairs of cousins in any group will 
be equal to the number of pairs which can be taken from all children having 
the same two grandparents, minus the number of pairs of brothers among 
such children. Since the population is stable during succeeding generations, 
the latter can be obtained at once from the expressions given in (7) by writing 
4¢ for 16s in that table and multiplying by the factor 16s(p+q)*. We 
therefore have the following expressions for the number of pairs of brothers 
among all grandchildren :— 
First Second 
brother. brother. 
(AA) (CAFO ee 32st (p+q)* [(4¢—1) pt + (4¢—2) pg + (¢— 1) p77], 
CAME CAG)... 2. 64st (p+)! [4t p?q + 2t p?¢?], 
(AA) (COD: nee 64st (p+q)' t p?¢q?, 
(Aq) (Ce) eae 32st (p+q)* (4t—2) p'¢ + 4(38t—1) p?¢? + (4¢—2) pg? ], 
(Aq) (Ga) Soe 64st (p+q)! [2¢ p2¢? +4 pq? |, 
(aa) (CU) Seren: 32st (p+ q)* [(¢—1) p?q? + (4¢—2) pq? + (4¢—-1) g4]. 
The number of pairs of all possible grandchildren (ze. siblings, together 
with cousins) can be easily obtained from inspection of Table IV. From 
symmetry, only six combinations are now required. Three of these are 
given in detail, the final expressions only of the other three being written 
down. Throughout the detailed expressions & is written for the product s.t. 
VOL. LXXXIII.—B. 1D) 
