

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 57 



Thus, for example, there would be out of the total population of possibilities 4" : 

 1 purely allogenic individual, n X 3 individuals with n 1 allogenic couplets ; 2n of 

 these would have one heterogenic couplet, and n would have one protogenic 

 couplet. 



Generally there would be 3* c H>tt0 individuals with n s allogenic couplets, and 

 these individuals would be distributed according to the terms of the binomial 

 (2v + iv)'. 



We are thus able to write down at once the number of any class of individual that 

 can appear in the segregating generation. For example, how often do individuals 

 like u H ~P~ivf-itfi appear, i.e., individuals with n p q allogenic, p heterogenic, and 

 <2 protogenic couplets ? 



To answer this problem all we have to do is to pick out the coefficient of ii u ~i 

 in the above multinomial, and the result is 



We are thus fully able to predict how many individuals of each kind ought to 

 occur when a population of perfect n-eouplet heterozygotes are crossed. 



Corollary (i.). Let us consider only the number of allogenic couplets in the 

 distribution of the segregating generation. If we were " pure Mendelians " we should 

 for the purpose of character classification make v = w, as the heterogenic couplet 

 would then give the dominant character. But without doing this we can assume 

 v and w to be non-it's. 



Hence the distribution of allogenic characters in the population follows the simple 

 binomial 



4" ( + f )" 



Thus we see that the distribution would be a skew binomial closely approximating' 

 to my skew curve of Type 111.,* and becoming indefinitely close to a normal 

 distribution of the form 



y y<fr*" t 



when the number of couplets, n, is indefinitely increased. 



For any Value of n the mean of the skew binomial as measured by the formula of 

 the memoir on " Skew Variation,"! 



= n + 1 - (1 + f) = in, 



i 



and the Standard deviation = \/T$n. 



Thus the mean number of allogenic Couplets in the members of the segregating 

 generation is J of the total number of couplets. 



* 'Phil. Trans./ A, vol. ISO, p. 373. 

 t Ibid., p. 346; 



VOL. com. A. i 



