352 THE AMERICAN NATURALIST [Vol. L 



Before considering the means by which differential 

 viability may be reduced in crosses of multiple stocks, it 

 may not be out of place to explain two methods I devised 

 for getting a more correct estimate of the gametic ratio in 

 back-crosses involving only two pairs of linked factors. 



Let us say that the gametic ratio is r(AB):r(ab): 

 s(Ab) : s(aB). Assume that when A is present the viabil- 

 ity of the flies is reduced so that only A' per cent, of 

 those which would otherwise survive, now come to ma- 

 turity, and assume that factor B lessens the output to B' 

 per cent, of what it otherwise would be ; similarly a and b, 

 when present, lower the output to a' per cent, and b' per 

 cent., respectively. Then the relative number of AB indi- 

 viduals which survive will be rA'B' (per cent, marks are 

 omitted for brevity) ; the relative number of Ab will be 

 sA'b', etc. The actual, observed, numbers will be some 

 multiple (k) of these relative numbers; thus the number 

 of AB individuals actually found will be krA'B', the actual 

 number of Ab will be ksA'b', etc. It can now be shown 

 that the true gametic ratio (r : s), which it was desired to 

 find, may be derived by the formula 

 \ AB X ab 

 \Ab X Ab 



(using Ab, ab, etc., to denote the number of AB observed, 

 of ab observed, etc.), for, substituting the above values of 

 AB, ab, etc., in this formula, we obtain 



Ikr A'B' X kra'V ' WA'B'a'b' = \? _ r 

 ^ksA'b' X ksa'B' \kh r A'b' a' B' \? s ' 

 This formula should be used only when the smallest 

 class has not a very large probable error, for, by multiply- 

 ing the value of this class in the formula, we give the 

 entire result a probable error proportional to that of the 

 smallest class. Another objection to the formula is that it 

 assumes that each factor produces the same specific lower- 

 ing of viability, independently of whatever other factor it 

 comes into combination with ; this is not always true, since 

 factors often produce different effects when in different 

 combinations. 



