(}2 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



given allogenic constitution are not either symmetrical or skew binomials, but of a 

 much more complex character. The only exception is the array of offspring of pure 

 allogenic fathers,* which is given by 



4" X 4" X (%u + %)* 



This is a symmetrical binomial. This result is, of course, of special interest, for it 

 gives us the distribution of offspring if the hybrid offspring were at any time crossed 

 with the pure allogenic race, which was one of the original factors of the hybridisation. 

 The deviation from binomial distribution in the above arrays ought to be further 

 considered, for if this deviation should turn out to be very significant, it would form a 

 convenient test for any generalised theory of pure gametes. 



Corollary (ii.). If we sum the above expressions for the array of offspring of all 

 fathers of p allogenic couplets for values of s from o to n, we have the total offspring 

 population 



= 4" X 4" . Sc,, ,, U< (2V + 



= 4" X 4" X (U + 2V + W)" 



4" x 4" X (u + 2v + ')", 



a result we have already found in Proposition II. as giving the distribution of the 



total offspring population. 



(G.) PROPOSITION IV. -To find the Mean Number of Allogenic Couplets in the 

 Offspring of all Fathers having in their Constitution s-allogenic Couplets. 



By the first corollary to the last proposition the distribution of such offspring is 

 given by 



4" X 4" . c, , i + Iv 



where 17 is written for \ (v -f w), a quantity which is unity so long as we consider not 

 the distribution, but the total number of the non-allogenic couplets. Now this is 

 clearly the sum of a number of symmetrical binomials in ^u + ^v, and may be put 



= 4" X 4" . S c,,, v - (%u + ly)"-'' (217)''. 



i = 



Now the means of each of these binomials can be found from the general theory of 



the binomial, t If we take our origin at n~+T allogenic couplets, with a frequency 

 zero, the mean of the first binomial, or 



* Or, of course, the array of sons from pure protogenic fathers, 

 t 'Phil. Trans.,' A, vol. 186, p. 373. 



