^52 



EEY. J. T. GIJLTCK ON DtVERGENT EVOLtlTIOif 



may expect that tbe three-quarter-breeds will also disappear 

 through fusion. 



In constructing my formula, it was found necessary to com- 

 mence by placing in the 1st generation of the half-breeds a more 

 or less arbitrary symbol; for the true symbol in each case is the 

 jBnal one reached in the wth generation when n is a very high 

 number. The chief interest therefore centres in what can be 

 accomplished through the use of this formula for the nth gene- 

 ration. It seems to me to furnish a method of reaching the 

 final proportion of pure breeding that will be produced by any 

 form of combination between Positive Segregation and Segregate 

 Fecundity, and to give results that would require thousands of 

 years of continuous experimenting to reach in any other way. 



Method of using Table III. (see p. 255). 



By supposing n to be an indefinitely high number, and by 

 giving different values to M, m, and we shall have the the means 

 of contrasting the number of the pure-breeds with that of the 

 half-breeds, when the process has been long continued under 

 different degrees of Positive Segregation and Segregate Pecundity. 



In the first place let us take a case in which there is no Segre- 

 gate Fecundity, that is M=m ; and for convenience in computa- 

 tion let us make M = l, w=l. In every case where m is not 



larger than M the fraction ^j^^^^ ^^^^ u^ity, and the 



sum of the geometrical progression of our formula will fall within 

 the limits of a number that can be easily computed by the well- 

 known formula 8 = ——, in which a is the first number of the 



j^rogression, which in this case is 1, and is the fraction we 

 are now considering. Supposing c—^-^, the fraction will be 



^^=t=?.---S = i^- become. S = jl,=^^=9. This 



number 9 is therefore equal to the sum of this progression and 

 can therefore be used as the value of the infinite progression in 

 the formula for the ^^th generation when ^ is a very high number. 

 Substituting these values we find that the ^^th generation of the 

 half-breeds equals the n\h generation of the pure forms, each 

 being equal to of A (M-Mc)'^-i. A(M-Mc)«-i is a 

 vanishing quantity, for M — Mc is less than 1. Every form is 

 therefore in time fused with other forms. But let us try higher 



