226 Prof. K. Pearson. On the Ancestral Gametic [Apr. 2, 



I write this for brevity 



p 2 (AA) + 2pq(Aa) + q 2 (aa), (ii) 



and this constitution remains permanent in all successive matings. Hence 

 the standard deviations of the gametic constitutions remain the same 

 generation after generation, and the correlation coefficient is in every case 

 equal to the slope of the regression line. I shall determine the slope of this 

 line which will give the correlation and show that the regression is truly 

 linear in each case. 



(3) I consider first the effect of individuals of each special type mating 

 with the general population (ii). 



(a) Type (AA) : the array of offspring is (p + q) [p (AA) + q (Aa)], 

 (6) Type (aa): „ „ (p + q) [p (Act) + q (aa)], 



(c)Type(Aa): „ „ %(p + q)[p(AA) + (p + q)(Aa) + q(aa)]. 



Thus, in seeking what any differentiated group 

 h (AA) + 1 3 (Aa) + 1 2 (aa) 



produces when mated with the general popvdation, i.e. when mated at 

 random, all we have to do is to replace (AA), (Aa) and (aa) by the above 

 three expressions respectively. 



In this manner I obtained the array of offspring clue to any parent, any 

 grandparent and any great grandparent. These at once allowed me to 

 reach the general law of distribution, and, assuming this, one multiplication 

 by the general population (ii) demonstrated by induction the validity of the 

 results reached. These are as follows : — 



I term ?tth parent any individual n generations back in the direct ancestry : 

 thus a 1st parent is the father or mother ; a 2nd parent, a grandparent ; 

 a 3rd parent, a great grandparent, and so on. 



(i) If the nth parent be an (AA), then the array of offspring due to random 

 matings is 



p 2 (p + ? )2(»-D (J 2 -)»-i {(2"-^ + q)p (AA) + [(2»- l)p + q]q (Aa) 



+ (2"- l -l)q 2 (aa)}. 



(ii) If the nth parent be an (Aa), then the array of offspring is 



in) (P + 2) 2( " _1> (I ;) re-1 {[(2 K - VP + Sip ( AA) + [p* + 2 (2»- l)pq + q 2 ] (Aa) 



+ [p + (2»-l)q]q(aa)}. 



(iii) If the nth parent be an (aa), then the array of offspring is 



q 2 (p + q) 2 {n ~ ^ (k) n ~ 1 { (2" _ 1 - 1 )f (A A) + [( 2" - 1 ) q +p] p (Aa) 



+ (2"- 1 q+p)q(aa)}. 



