412 
R. A. FISHER ON THE CORRELATION BETWEEN 
five of which are independent. The unknowns are thus reduced to four, and we 
shall use f w, fiz, fz\, /%?,■, since any involving a 2 in the suffix can easily be eliminated. 
We have further 
I = i + 2 (P'i'fu + -Q'j'fiz + R'/c'/ ls ) 1 
J =j + 2 (P't'./a i + 2Q '// 22 + R7</ 28 ) [ (XV 1 1) 
K = h + 2 (PV/ 3I + 2Q '// 82 + R7c'/ 33 ) ) 
in which the summation is extended over all the factors except that one to which 
i,j, k refer. Since we are assuming the factors to be very numerous, after substitut- 
ing their values for the f ’ s we may without error extend the summation over all the 
factors. The variance defined as the mean square deviation may be evaluated in 
terms of the f ’ s 
V = 2(Pi 2 + 2Q./ 2 + R& 2 ) + 22{PP'(1 + fi i)m' + 8 other terms}, 
which reduces to 
2 (Pi 2 + 2Q j 2 + Rft 2 ) + 22{PF*.y u 4- 8 other terms}, 
so that 
V = 2 (Pil + 2Q,;J + RfcK) (XVIII) 
12. We can only advance beyond these purely formal relations to an actual 
evaluation of our unknowns by considering the equilibrium of the different phase 
combinations. There are forty-five possible matings of the nine types, but since we 
need only consider the equilibrium of the four homozygous conditions, we need only 
pick out the terms, ten in each case, which give rise to them. The method will be 
exactly the same as we used for a single factor. Thus the matings DD' x DD' have 
the frequency 
M(I+I') a 
PF.PF.(l+/ 11 )(l+/ 11 )e V , 
which for our purpose is equal to 
P 2 P' 2 [l + 2/ u + if (I + 1') 2 ]. 
Collecting now all the matiugs which yield DD', we have for equilibrium 
P 2 P' 2 [i + 2/ n + if(I + 1') 2 ] + 2P 2 P'Q'[l +/u +fn +.f(I + T)(I + J')] 
4 2PQP 2 '[l +/ n +/ 2l + ft(I + I')(J + 1')] + 2PQP'Q'[l +/ n +/ 22 + ^(1 4- I')(J + J')] 
4- 2PQP'Q'|^1 +/ 12 +/ 21 + ^(1 + J')(J + 1')] + P 2 Q' 2 [l + 2/ 12 + ^(1 + J') 2 j 
+ Q 2 P' 2 |^1 +. 2 f 2l + A(J + iqsj + 2PQQ' 2 [l +/ 12 +/ 22 + £(I + J'.)(J + J')] 
+ 2Q 2 P'Q'[l +/ 21 +/ 22 + ^(J + I')(J + J')] + Q 2 Q' 2 [l + 2/22 + ^(J + J') 2 ] 
= PP'(l.+/u) 
Now since 
(XIX) 
(P + Q) 2 (P' + Q') 2 - PP'(P + 2Q + R)(P' 4 - 2Q' + R') = (Q 2 - PR)P' + (Q' 2 - P'R')P + (Q 2 - PR)(Q' 2 - P'R| 
the terms involving only P and Q, reduce (XIII) to the second order of small 
quantities, 
