9i6 THE HEREDITY OF THE BLOOD GROUPS 



and, finally, since p+q+r = i , 



(i-"^^0+B) + (i- i/O+A) + i/0 = i . 



Bernstein now takes for each population the observed values of O, A, B, and AB 

 and substitutes them in the two formulae : 



(A+AB).(B+AB)=AB , ' (i) 



(i-^ 0+B) + (i-i/0+A)+i/0 = i. (2) 



He finds that in formula (i), which, he asserts, tests the two-factor hypothesis of von 

 Dungern and Hirschfeld, the left side of the equation is always greater than the right. 

 On the other hand, he finds that in formula (2) the calculated value of p+q+r comes 

 out very nearly equal to i . His conclusion is that the two-factor hypothesis is wrong, 

 and that the three-multiple-allelomorph hypothesis is correct. He points out that 

 critical cases for the testing of his theory should occur in the marriages of group AB 

 persons. On his hypothesis such unions (with any of the groups) can never produce 

 group O children as the recessive gene R is of necessity absent in group AB. In ad- 

 dition it is evident that marriages of group AB with group O can produce neither 

 group O nor group AB children as the individual genes A and B here can only unite 

 with the recessive genes R to produce group A and group B. On the other hand, on 

 the two-factor hypothesis, unions of group AB may result in children of any group. 



Table VII allows us to make immediate comparison of the number of exceptions to 

 the two hypotheses. There are 27 children out of 5,187 in which Group A or B appears 

 though it was absent in both parents, giving against any Mendelian interpretation 

 which assumes the dominance of A and B, .5% exceptions. With these we may com- 

 pare those specific exceptions which apply only to the Bernstein theory. There are 51 

 exceptions in AB marriages to this theory out of a total of 771 children from such 

 marriages, giving 6.6% of exceptions. There are then 13 times as many exceptions 

 proportionately against the Bernstein formula as against the dominance of A and B 

 on any hypothesis as to genes. 



The genetical evidence requires that judgment be reserved. What can be done 

 with the mathematical computations of Bernstein? Does the failure of the observa- 

 tions to fit the formula (A-f-AB) • (B+AB) = AB mean that the two-factor hypothesis 

 is untenable? Here the question arises: Under what conditions are mathematical 

 formulae binding in the study of the biological phenomena? 



When the facts underlying a given phenomenon are thoroughly understood, a 

 mathematical formula may be employed to express concisely the interrelationship of 

 the factors and may then actually lead to the discovery of new laws. This situation 

 does not exist in the blood group investigations. There are other possible, indeed 

 plausible, explanations of the failure of calculation to match actual percentages. For 

 example, Hirschfeld' has pointed out the remarkable fact that of the fifty-five re- 

 corded cases of children arising from O mothers and AB fathers there is not a single 

 AB child, while there are on record nineteen cases of AB children from AB mothers 



'Hirschfeld, L.: Zlsclir. f. ImmiinitdtsforscJmng. u. ex per. Therap., 43, 4S5. 19:35; Ergebn. v. 

 Weichliardl, 8, 367. 1926. 



