MATHEMATIOAIj CONSIDEKATIONS 89 



therefore, frequently cause a general reduction in vigor. 

 It has never been held that all hereditary factors 

 are equally involved in this effect on vigor. Some 

 are considered to be wholly without effect. The fact 

 remains, however, that the increased growth and extra 

 vigor commonly resulting from hybridization as a mass 

 effect is intimately associated with Mendelian phe- 

 nomena, and its expression is roughly proportional to the 

 number of heterozygous factors present and disappears 

 when homozygosity is brought about. 



The reduction of the number of heterozygous allelo- 

 morphs in an inbred population is automatic and varies 

 with the closeness of inbreeding. In self-fertilization it 

 follows the well-known Mendelian formula by which any 

 heterozygous pair forms in the next generation 50 per 

 cent, homozygotes and 50 per cent, heterozygotes in re- 

 spect to that pair. Since the homozygous allelomorphs 

 must always remain constant and the number of hetero- 

 zygous factor combinations is halved each generation and 

 one-half added to the homozygous class, the reduction in 

 the number of heterozygous elements proceeds as a vari- 

 able approaching a limit by one-half the difference in 

 each generation. The curve illustrating this approach to 

 complete homozygosity is shown as No. 1 in Fig. 24. 



As lEast and Hayes ^^ have said: "Mendel, in his 

 original paper, showed that if equal fertility of all plants 

 in all generations is assumed, and, furthermore, if every 

 plant is always self -fertilized, then in the nth generation 

 the ratio of any particular allelomorphic pair {A, a) would 

 be 2"-lAA:2Aa:2"-laa. If we consider only homo- 

 zygotes and heterozygotes, the ratio is 2" -1 : 1. Of 

 course, the matter is not quite so simple when several 



