RELATIVES ON THE SUPPOSITION OF MENDELIAN INHERITANCE. 403 
in rather different proportions to those in which they exist in the whole population. 
The latter will be represented now by P, 2Q, R, while P, 2Q, R stand for the pro- 
portions in some particular array under consideration. 
Consider a population which is the same in every respect as the one we are 
dealing with save that all its members have one particular factor in the heterozygous 
phase, and let us modify it by choosing of each array a proportion P which are to 
become dominants and to increase by a — d, and a proportion R which become recessive 
and diminish by a + d : the mean is thereby moved to the extent m-d. 
Of those which after this modification find themselves in the array with 
deviation x, the dominants formerly had a deviation x — a + m, the heterozygates 
x — d+m, and the recessives x + a + m, and since the variance of the original popula- 
tion was cr 2 — a 2 , the frequencies of these three types are in the ratio 
(x-a + on ) 2 ( x-d+m ) 2 (x + a+m ) 2 
Pe 2(<r2 -“ 2) : 2Qe V (ff2 ““ 2) : ’ 
or, when is great compared to a, so that may be neglected, 
P = p[l 
Q = Q|"l + -JJL - m)J 
R = e[i -^(a+m)] 
(III) 
giving the proportions in which - the phases occur in the array of deviation x. 
3. Hence the members of this array mating at random will have offspring 
distributed in the three phases in the proportion 
P 2 ^l + - jn)J + PQ^2 + | 2 (a - m + d - m ) J + Q 2 [l +^(d - m) J, 
PQ |^2 + ^(a-m + d - m)J + 2Q 2 |^1 + d - m)J + PR^2 - ^(2m)J + QR^2 + '^{d - m - a - m)J, 
Q 2 [ 1+ ^2^ + _ ^( a + m )]> 
and therefore the deviation of the mean of the offspring is 
2ri(PR-Q 2 ) + ^PQ(a - c?) 2 -j- 2PR(a 2 - d 2 ) + QR)a + d) 2 + (PR - Q 2 )d(d - m) J . 
Omitting the terms in (PR — Q 2 ), which for random mating is zero, the regression 
due to a single factor is 
|^PQ(a - d) 2 -f- 2PR(a 2 - d 2 ) + QR(a + e£) 2 J (IV) 
4. To interpret this expression, consider what is involved in taking a, d, —a as 
representing the three phases of a factor. Genetically the heterozygote is inter- 
mediate between the dominant and the recessive, somatically it differs from their 
