No. 594] THE MECHANISM OF CROSSING-OVER 353 



The two difficulties encountered above are largely 

 avoided by the second method, which involves making two 

 different kinds of crosses in preparation for the linkage de- 

 termination : i. e.y cross AB by ab, and what may be termed 

 the " contrary cross," Ab by aB. A back-cross of the Fj 

 from the first cross gives the gametic ratio r (AB) : r (ab) : 

 s (Ab) : s (aB) ; and the other cross results in gametes 

 showing the proportion s (AB) : s (ab) : r (Ab) : r (aB). 

 Suppose that w per cent, of AB individuals are viable, 

 x per cent, of ab, y per cent, of Ab, and z per cent, of aB. 

 Then in the first cross the observed ratio would be 

 rw (AB) :rx (ab) : sy (Ab) :sz (aB), and, in the second 

 cross, sw (AB) :sx (ab) :ry (Ab) :rz (aB). The numbers 

 actually observed in the crosses would be some multiple of 

 these ratios, but a different multiple in the two cases. 

 Thus we could designate the numbers actually observed in 

 the first cross as krw(AB) : krx(ab) : ksy(Ab) : ksz(aB), 

 and the numbers in the second cross as csw(AB) : csx(ab) : 

 cry(Ab):crz(aB). 



In this case the ratio r : s may be obtained by the follow- 

 ing formula : 



iABj X Ah 

 \AB 2 X Ah 



(using the symbol ABj to denote number of AB observed 

 in the first cross, Ab 2 to denote number of Ab observed in 

 the second cross, etc. ) . Now, the value of ABj has already 

 been given as krw, of Ab 2 as cry, etc. Substituting these 

 values in the above formula, we obtain 



\krw X cry _ j i^kcwy _. jr 2 _ r _ _ 



Besides this formula involving AB and Ab, there are 

 three similar formulas which will also give the gametic 

 ratio, namely: 



l ABi X aB 2 j abt X Ah . [ o&i X oft 



VAB 2 X aBi ; \ab2 X Ah ' ^ah X aB x ' 

 That formula should usually be chosen which contains the 

 largest number of individuals in its smallest class, for this 

 would usually have the least probable error. 



