558 



THE AMEBIC AN NATURALIST [Vol. LV 



^iK on Ehle (1911) In dt'^oibtcl i cw m a\1u(]i th( i in^c of 

 variation m the length ol heads ol wheat m tlie i„ considerably ex- 

 ceeded tlie combined ranges of the two parents. Hayes (1912) has 

 found a similar case in the number of leaves m tobacco, and Emerson 

 and East (191.3) have s(>en the same phenomenon in the length of 

 intcrnode and total length of stalks in maize. It seems probable that 

 such transgressive variation may be the rule rather than the cxccp- 



to be exix3cted that a large number of plural determiners aflfectmff 

 such a character shall all act m tlie same direction or that the parent 

 having the highest development of the given character shall generally 



diMl . I imul r itl 1 1 . nt . 1 1 1 1 il t til numbei ot 

 plui .1 1 t nniiui^ ulii h 1 1 n 1. u^l t i y rlui, with le pe< t 



TVe might even derive a matlicmatical expression for the prob- 

 ability that the parents would stand at the extremes of total 

 {renetie variabilitv, by assnminor tliat the two parental types are 

 taken at random. This would be perhaps a fair assumption, since 

 the number of factors can not be determined by inspection. 

 There would then be, if we let n represent the number of factors 

 involved in the cross, n \/2 ways in which the event in question 

 can not happen and only one way in which it can happen ; hence 

 the probability that all the factors would be present in the larger 

 parent and absent in the lesser parent would be as 1 to nl/2. 

 In the specific case of East and Emerson's corn, cited by Castle 

 as having about 15 factorial differences, (71= 15), there would 

 be, therefore, 633,477,184,000 chances to one against all of these 

 size-modifying factors being present in the larger parent and 

 absent in the smaller parent. When we let n = 50 or 150, to 

 agree with the numbers indicated for the rabbit crosses, the 

 chances become practically infinitesimal and we must fairly 

 conclude that it has never happened, and never will happen, 

 that a cross involving so many independent size differences has 

 been made, or will be made, between individuals standing at the 

 opposite extremes of the total potential genetic variability. 



The remark made in my paper on duplicate genes, ret^arding 



drawn from changes in the F, stiimhii-d .ifviations. The 



