RELATIVES ON THE SUPPOSITION OF MENDELIAN INHERITANCE. 
411 
must be" twice as frequent , as the first, and if I, J, and K are the means of the 
distributions of the three phases, 
4Q 2 ^ = 4PRe^ r . 
Since ^ and ^ are small quantities, we shall neglect their squares, and obtain 
the equation 
pu 02 o 9 J2 ~ IK 
PR - Q- = QV— y 
(XIII) 
If, as before, the two types of gamete are in the ratio p : q, the frequencies of the 
three phases are expressed by the equations 
It is evident that 
p 2l22 J 2 -IK' 
P=P + j^ 2 g> y — 
n , 2 J 2 - IK 
Q =pq-p-q*p 
p o , o 2 J 2 - IK 
R = g 2 + V y 
PI + 2QJ + RK = 0 
(XIV) 
(XV) 
and this enables us, whenever necessary, to eliminate J, and to treat only I and K 
as unknowns. These can only be found when the system of association between 
different factors has been ascertained. It will be observed that the changes produced 
in P, Q, and R are small quantities of the second order : in transforming the quantity 
2 9 J 2 - IK 
we may write — (p 2 I + g ,2 K) for 2pqJ, leading to the form 
^(p 2 i-g 2 K) 2 , 
which will be found more useful than the other. 
11. The nine possible combinations of two factors will not now occur in the 
simple proportions PP', 2PQ', etc., as is the case when there is no association ; 
but whatever the nature of the association may be, we shall represent it by intro- 
ducing new quantities, which by analogy we may expect to be small of the second 
order, defined so that the frequency of the type 
that of 
and that of 
DD' is PP'(I +f n ), 
DH' is 2PQ'(1 +/ 12 ), 
DR' is PR'(1 +/ 13 ), 
and so on. 
Formally, we have introduced nine such new unknowns for each pair of factors, 
but since, for instance, the sum of the above three quantities must be P, we have 
the six equations 
P'/n + 2 Q '/ i2 + R'/ 13 = 0 P/ n + 2Q/ 21 + R/ 31 = 0 ^ 
P /21 + 2Q '/ 22 + R7 23 = 9 P /12 + + R / 3 2 = 9 > . . (XVI) 
P/31 + 2Q/ 32 + r/ 33 = 0 py ^ + 2 Q / 2 3 + p/ 33 “ ® ' 
