RELATIVES ON THE SUPPOSITION OF MENDELIAN INHERITANCE. 
429 
other hand, all values down to zero are consistent with complete dominance, provided 
that the values of ^ are sufficiently small. 
We know practically nothing about the frequency distribution of these two ratios. 
The conditions under which Mendelian factors arise, disappear, or become modified 
are unknown. It has been suggested that they invariably arise as recessive mutations 
in a dominant population. In that case ^ would initially be very high, and could 
only be lowered if by further mutation, and later by selection, the recessive phase 
became more frequent. These factors would, however, have little individual weight 
if better balanced factors were present, until ^ had been lowered to about 10. In 
2 
face of these theories it cannot be taken for granted that the distribution of these 
ratios is a simple one. It is natural, though possibly not permissible, to think of 
their distributions as independent. We may profitably consider further the case in 
which the distribution is symmetrical, in which the factor of known a and d is 
equally likely to be more frequent in the dominant as in the recessive phase. 
For this case we combine the numerators and denominators of the two fractions 
3 P ?< /3 and W 2 
(p + q)' 2 d 2 - 2 (p 2 - <f)ad + ( p 2 + q 2 )d 2 (p + q) 2 a 2 + 2(p 2 - q 2 )ad + (p 1 + q 2 )d' 2 ’ 
and obtain the joint contribution 
2 pqd 
(p + ?) 2 a 2 + (p 2 + q 2 )d 2 ’ 
the curves for which are shown in fig. 2, representing the combined effect of two 
similar faetors, having their phases in inverse proportions. It will be seen that 
complete dominance does not preclude the possibility of low value for the dominance 
ratio : the latter might' fall below '02 if the greater part of the variance were con- 
tributed by factors having the ratio between p and q as high as 100 to 1. This ratio 
is exceedingly high ; for such a factor only one individual in 10,000 would be a 
recessive. We may compare the frequency of deaf mutism with which about one 
child in 4000 of normal parents is said to be afflicted. It would be surprising if more 
equal proportions were not more common, and if this were so, they would have by far 
the greater weight. 
The fact that the same intensity of selection affects the logarithm of - equally, 
whatever its value may be, suggests that this function may be distributed approxi- 
mately according to the law of errors. This is a natural extension of the assumption 
of symmetry, and is subject to the same reservations. For instance, a factor in 
which the dominant phase is the commonest would seem less likely to suffer severe 
selection than one in which the recessive phase outnumbers the other. But if 
symmetry be granted, our choice of a variable justifies the consideration of a normal 
distribution. 
