No. 615] THEORIES OF CROSSING OVER 



251 



x, we shall have, in these first chromosomes of the pair, 

 A and a in the following proportions : 



(l-x)A 



Similarly, in this same chromosome we shall find B 

 and b distributed in the following proportions : 



y& 

 (i-y)£ 



(Thus, if A and a interchange in two fifths of all cases, 

 then after interchange we shall, in the first chromosome, 

 find a in two fifths of the cases, A in three fifths; and 

 similarly for B.) 



What will then be the proportions of the various com- 

 binations of the two pairs of factors ? It will evidently be 



xa-f(l — x)A, multiplied by 

 jb + (l-y)B 



= xyab + x(l- y)aB + y(l- x)Ab 

 + (l-x)(l-y)AB 



The cross-overs are aB and Ab, the proportion of which 

 is evidently: 



x(l-y)+y(l-x)=x + y-2xy 



The same result will be reached if we consider the 

 second chromosome of each pair (that which originally 

 contained a and b) ; so that the same proportion holds for 

 both together. This, therefore, gives us our formula for 

 the cross-over ratio in terms of the exchange ratios of 

 the two pairs. It is essentially the same formula em- 

 ployed by Goldschmidt (1917), giving the same results, 

 but written in more perspicuous form. 



Let us therefore recapitulate in algebraic form the es- 

 sential points. 



