1910.] ia Mendelian Population Mating at Random. 45 
to the particular constituent, protogenic, hybrid, or allogenic, each member of 
the pair possesses. It will be more convenient, however, to make use of the 
fact that the number of pairs of uncle and nepbew will be equal to the 
number: of pairs obtained by taking any one of the children from given 
parents with any one of the grandchildren, minus the number of pairs of 
parents and offspring among such pairs. This latter can be obtained more 
simply than by Table LV. For Table I enables us to pick out parent and 
offspring in pairs and arrange them according as each member of the pair is 
(AA), (Aa), or (aa). But we require the number of such pairs in the next 
generation. Allowing for the change in the fertility from 16s to 4¢, it is clear 
that the number in the second generation will be obtained from the number 
in the first by writing 4¢ for 16s and multiplying by 16s (p+gq)*. From 
Table I we can at once pick out the following terms for the first generation. 
Parent (AA), offspring (AA)...... p*. 168+ 2p%¢ . 8s, 
peice); . (CD) ane 2p*¢.8s+ pq’. 16s, 
eeied((ACAD), (OD) eee 0, 
a CA), 3 (CAAY. S.. 2p?q¢.8s+4pq?.4s, 
enn (CA) if (CAG) enone 2p°¢ .88+4p?¢?. 88+ 2pq3.8s, 
o (Aq), 4, CD) ane 4n?q? .48+ 29°. 8s, 
ee (aa), . (AA) ci. 0, 
a (aa); F (AG) 05.65. pq? .168+2nq°.8s, 
(aa), - (GQ) Fete 2pq?.8s+q*.16s. 
In obtaining these expressions, one parent only was considered. From the 
symmetry of the table, if both parents are to be taken into account, each 
term must be multiplied by two. Since this factor will occur throughout 
the work, it can be omitted. 
Collecting the above terms, and applying the modification necessary to 
transform the first generation into the second, we have, for this latter 
generation, the frequencies :— 
Parent (AA), offspring (AA)...... 64st p (p+q)’, 
san? (CAS): . GAG) ee 64st p*9 (p+¢)°, 
yx _ (AA), i (ACIS Becca 0, 
wo (Ae, s GAGA i is, 64st p79 (p+ @)’, 
a Ae) ie CAG) cis: 64st pq (p+ ¢q)°, 
eA), 5 (GO) eco 64st pg? (p+ 49), 
ae). 45 GAA) sees 0, 
(ae), i (CAG) sce 64st ng? (p+q)’, 
Pe Ae.) ” (GOP ene 64st g3 (p+q)°. 
From Table IV we have now to pick out and arrange the various pairs 
