RELATIVES ON THE SUPPOSITION OF MEN DELIAN INHERITANCE. 
413 
- -^[P'Q 2 (J 2 - IK) + PQ' 2 (J ' 2 - I'K')] = - 4^0' 2 (IP - KR ) 2 + p 2 (I'P' - K'R') 2 ]. 
Also collecting the terms in I and J, we find 
A[(P' + Jf + JQ) + (P + Q)(I'P' + J'Q')] 2 , 
which yields on eliminating J, 
^0'(IP - KR) +p(I' p ' - K'R')] 2 , 
while the result of collecting and transforming the terms in f is 
iRP # [P P '/h - PE'/is -- V'Wsx + RH'/ssl- 
Hence, if the frequency of the type DD' is unchanged 
^;pp'( IP - KR)(I'P' - K'R') + [PP'/jx- PR'/ 13 - p ' r / 3 i + RR'/bs] = pp Xi • • • (XIX, a) 
Now the corresponding equations for the types DR', RD', R'D' may be obtained 
simply by substituting K for I, R for P, and vice versa , as required ; and each such 
change merely reverses the sign of the left-hand side, substituting q or q' for p or p' 
as a factor. 
Combining the four equations 
A(ip - KR)(I'P' - K'R') = £[PP'/ U - PR '/ 13 - RP'/„ + RR '/„] ■ . (XX) 
so that the set of four equations 
A (ip - KR)(I'P' - K'R') =pp'f u = -m'A 3= -?p'/ 3 i = ^'/ 3 3 ■ • • (XXI) 
gives the whole of the conditions of equilibrium. 
13. Substituting now in (XVII), which we may rewrite,' 
we have 
i = * + 2[p ’(i\ -f)f u - nf - k)a 3 i 
K = k + 2[P '(*■' -/)/ 31 - R '(/ - /<0/ 33 ] 
IP - KR = zP - &R + X4(I p - KR)(I'P' - K'R') \p\i' -/) + q\j' - k')] = *P - &R + A(IP - KR), 
where 
or 
A(1 - A) = ^2(i'P' - k'K)[p'(i' - j ') + q(f - V)] 
-Ssft sine e/pjpjp’ 
AO -A)-,/ 
(XXII) 
It would seem that there is an ambiguity in the value of A, so that the same 
amount of assortative mating would suffice to maintain two different degrees of 
association : we have, however, not yet ascertained the value of V. Since this also 
depends upon A, the form of the quadratic is changed-, and it will be seen that the 
ambiguity disappears. 
