58 PROFESSOK K. PEAESON ON A GENERALISED THEORY OF ALTERNATIVE 



Corollary (ii.). The distribution of heterogenic couplets in the segregating 

 generation is given by the symmetrical binomial 



The mean number is therefore \n, and the standard deviation = v/i^. 



This is a symmetrical binomial, and approaches extremely closely, even for a fairly 

 small value of n, to the normal curve. We see that if any character depends solely 

 upon heterogenic couplets, the distribution will be nearly normal, and the variability 

 slightly greater than one depending on the allogenic (or, of course, the protogenic) 

 couplets only. 



To sum up, then, so far as the distribution of characters depending upon allogenic 

 or heterogenic couplets goes, we may say that a generalised theory of the pure 

 gamete leads us to those normal and skew distributions of frequency with which 

 biometric studies of variability have made us already familiar. It would not be 

 possible to base a crucial experiment on the existence or non-existence of such 

 frequency distributions. The generalised pure gamete theory would, however, 

 account for their appearance, which, of course, a purely descriptive statistical theory 

 cannot do. On the other hand, distributions diverging much beyond the errors of 

 random sampling from binomials of the above types would tell pro tatno against the 

 pure' gamete theory in its above form. The presence of binomials of two types only, 

 (I + ?)" :lll( i (i + )"> ought to be capable of detection, even if it would not already 

 have been discovered, had it been the rule. 



(4.) PROPOSITION II. To determine the Distribution of the Offering of the 

 Segregating Generation, supposing them to Mate at Random and 'without 

 Differential Fertility. 



The solution of this problem may be reached as follows : 



Suppose P any male, and Q any female, say each of n 1 couplets, producing an 

 array of offspring, which we will denote by R ; now suppose an additional couplet, the 

 /i th , added to both male and female zygote. The male may be now : 



P + a,^,,, or P + a fl A H , or P + Aa,, or P + A H A U ; 

 and the female may be 



Q + a,a, or Q + a,,A,,, or Q + A,,a,,, or Q + A, A, ; ' 



that is, we get 4x4 new mating individuals, with 4x4x4 new offspring 

 possibilities. 



Now consider the first father P + a K a n ; the possibilities which arise from mating 

 him with the four mothers are the array K of offspring combined with any one of the 



