SHOETER ARTICLES AND DISCUSSION 



FORMULAE FOR THE RESULTS OF INBREEDING 



In connection with Pearl's recent valuable analyses of the 

 results of inbreeding (1, 2, 3), a comparison of these results with 

 those from self-fertilization is of interest. In my note on the 

 latter (4), I gave a formula for the rate at which organisms 

 become homozygotic through continued self-fertilization. This 

 occurs more slowly in the various types of inbreeding, but Pearl 

 gives no general formula for it. For purposes of comparison I 

 have worked out from Pearl's data the general formula for the 

 rate at which organisms become homozygotic through continued 

 brother b}' sister mat inn'; as such formula? appear to be of perma- 

 nent value, it is here given. 1 What the formula gives is, pre- 

 cisely, (1) the proportion of individuals that will be homozygotic 

 for any given character after any number of unbroken genera- 

 tions of such inbreeding, (2) the average proportion of the char- 

 acters of a given individual that will be homozygotic after any 

 number of unbroken generations of such inbreeding. The nu- 

 merical value so obtained may conveniently be called the co- 

 efficient of homozygosis. 



The formula turns out to be a combination of the successive 

 powers of 2, with the successive terms of the Fibonacci series, 

 which appears in so curious a way in various natural phenomena. 

 In this series every term is the sum of the two preceding terms, 

 the series beginning : 0, 1, 1, 2, 3, 5, 8, 13, etc. 

 Let x = the coefficient of homozygosis. 



n = the number of inbred generations (the number of 



times successive brother by sister mating lias 



occurred). 



fii ftt f*t etc., = the successive terms of the Fibonacci series 

 (thus A=0, f 2 = l, etc.). 

 Then the formula for the coefficient of homozygosis is : 

 _2»-i + /, .2»- 2 4-/ 2 -2»- 3 . . . etc. 

 • " 2" 

 (The terms in the numerator are continued until the exponent 

 of 2 becomes 0.) 



iln conversation, Dr. Pearl urged the publication of the present note, 

 otherwise I should not at this time have dealt with a matter which he has 

 under an .Isms 



698 



