Mathematical Contributions to the Theory of Evolution, 399 



Hence the more nearly 1— %e qi = unity the more nearly the offspring 

 has the full character of its selected parentage. I venture to call 

 this expression the stability of the stock. It is a measure of the stock 



breeding true. 



Lastly, to find the standard deviation of the array = <x () v/R/R 00 

 we have only to express R in terms of tbe minors of its first row, or, 



R = R + flRoi + />2^02 • • • • +/>2^02; 



R/R 00 = l-c(e 12a 2 + e 22 a 4 + e 32 « 6 + .... + e^l) 



1-0*3 (e 1? +^ + 7<= 35 



r). 



4 "* 22 

 In the limit when q = oo 



R/R 00 = 1-0-3 xi(l + i + T V + . • • • ) = l—0"2 = 0-8, 

 and \/R/Roo = 0-8944. 



The following table has been calculated from these formula? 



(xvii). 



Table of Pedigree Stock according to Gralton's Law. # 



amber 

 of 



snera- 

 ions. 



Eatio of 

 variability of 



offspring to 

 that of whole 



population. 



'9055 

 -8946 

 '8945 

 '89445 

 -8944 

 0-8944 



-8944 



Partial regression coefficients. 



0-6 



-5122 

 0-5015 

 -5002 

 -5000 

 -5000 



-5000 



0-2927 

 -2553 

 -2507 

 -2501 

 -2500 



-2500 



0-1459 

 0*1276 

 -1253 

 0-1250 



0-1250 



-0729 

 0-0638 

 0-0627 



0-0625 



-0365 

 -0319 



0-0312[ 



-0182 



0-015625 



Stability. 



s(4 



-6000 

 (0-5) 

 -8049 

 (0-75) 

 0-9027 

 (0-875) 

 0-9514 



(0 9375) 

 0-9717 



(0 '9687) 

 -9879 



(0 -9844) 



* To save possible labour, in case it should ever be needed to investigate the 

 partial regression coefficients for more generations, I place here the ratios of the 

 first six f's : 



tq-iq/eqq = 1'75; *q—2q\zqq = 3 4375 ; Sq-^cjeqq = 6-859,375. 

 *q-hql*qq = 13-714,843; e 2 - 6 qjfqq = 27'428,709. 



