411) 



THE AMERICAN NATURALIST 



[Vol. L 



Formulae IV are not to be used in connection with ob- 

 served F 2 frequencies except when the latter approximate 

 closely the form demanded by formula I, that is, when the 

 first term of the observed frequencies equals approxi- 

 mately the sum of the second and third terms plus three 

 times the fourth term. 



In cases of repulsion, where the fourth term of the 

 zygotic series is always relatively small and, therefore, 

 where the first term should be only slightly greater than 

 the sum of the second and third terms, it may happen that 

 the sum of the first and fourth terms, E, is actually less 

 than the sum of the second and third terms, M. In such 

 cases, formula? II (and consequently formulae III and IV 

 also) can not be employed, for, if E is less than M the 

 quantity under the radical {E — M) is negative and has 

 no real root. In such cases, the gametic ratio must be 

 calculated by means of the coefficient of association. 



The method here suggested for calculating gametic 

 ratios from observed frequencies never gives quite the 

 same results as that obtained by the association-coefficient 

 method except when the observed series approaches 

 closely the form demanded by formula I. Naturally, 

 then, the more widely the observed frequencies depart 

 from this form the greater the difference between the 

 results given by the two methods. Since the coefficient of 

 association gives reliable results if the tables to be used 

 with it are based upon sufficiently small differences in the 

 gametic ratios employed in its preparation, it follows that 

 the methods proposed in this paper give only approxi- 

 mate results. It is also true, therefore, that the nearer 

 the observed frequencies approach the form of formula I, 

 the closer the approximation obtained bv formulae II (or 

 III and IV). 



The two methods have been applied to numerous cases 

 taken from published accounts of linkage studies and the 

 goodness of fit tested by the method suggested by Harris 

 (1912). The differences, o — c, between the observed fre- 

 quencies, o, and the calculated frequencies, c, of the sev- 



