SYSTEMS OF MATING 45 



The corresponding correlations in the preceding generations are symbolized by F', 

 r' ss , and r' 00 . Double primes are used for the second preceding generations. Tracing 

 the connecting paths from inspection according to the basic principle of path analysis, 



F = r os = (1/4)(2F' +r' ss + r' 00 ) 

 r ss = r 00 = (1/4)(2F' + 2). 



F can be solved easily in terms of preceding F's by substitution, F = (1/4) 

 (1 + 2F' + F"); but it will be convenient in the more complicated cases considered 



Fig. 14. Brother-sister mating, gamete diagram. 



tfc ¥% oi 



F F 



later to replace the correlations by the panmictic indices by substituting r = 1 — P tj 

 and using P 00 for both P 00 and P ss . 



Thus 



This can be written: 



or merely 



P os = (1/2)P;, + (\/2)P' 00 . 

 P 00 = (\/2)P' 0S . 



P os = (\/2)P' os + (1/4)/*. 



(hlh ) = (h'/2h ) + (h"IVi ) 



h = (l/2)h' + (1/4)/*". 



The changes in heterozygosis can thus be written by inspection for any number of 

 generations. Starting from a cross giving h = 1, the well-known series — 1/1, 1/2, 2/4, 

 3/8, 5/16, 8/32, 13/64... — is obtained in which each numerator is the sum of the two 

 preceding numerators (Fibonacci series) if the denominator is doubled in each genera- 

 tion. The ratio of successive terms rapidly approaches constancy. What this ratio 

 is can readily be found by putting hjh' = h'jh" = x. 



x - (1/2) + (1/4*). 

 4x 2 - 2x - 1 = 0. 



x = (1/4)(1 ± V5). 



