30 



Mr. S. M. Jacob. Inbreeding in a [Mar. 18, 



Disregarding for the present the absolute fertilities in each generation, 

 we see that the families of brethren or first siblings can be written in the 

 symbolic form 



[p 2 (2A) + 2pq(A + a) + q2(2a)Y, 

 where the ordinary algebraical operations are first carried out, and then 

 interpreted in the way noted. 



Similarly, it is easily verified that the second sibling groups of families 

 can be written symbolically in the form 



O 4 (4 A) + 4fq (3 A + a) + 6 f<f (2 A + 2 a) + Apf ( A + 3 a) + j* (4a)] 2 . 

 Let us assume that this result is generally true. We would then have the 

 ?ith sibling groups given by 



[p*" (2»A) + rdf'-iq {2» - 1) A+ a} + r C 2 f~\ 2 {(2»- 2) A + 2a} + ... 



+ rCtf- V {(2" -0 A + la} + . . . + f (2»a)] 2 . (7) 

 We will prove that if this result holds for the nth. siblings, it will be 

 true for the (% + l)th siblings, and thus, inductively, that it will be 

 universally true. For brevity write 2 n = w. 



First of all the types of families in the (% + l)th generation will be 

 obtained, and then their frequencies. 



The general type of family in the wth generation is given by 

 [(« - h) A + ha] [(w -l 2 )A + l&l 

 where h and l 2 have any values from to w. 

 This gives a group of type 



(w — h) O — l s ) (A A) + { h O — h) + h(w—h)} ( Aa) + hh (aa). 

 This family group written in operational form is — 



{2 (w-h)(w-l 2 ) + h (w-l 2 ) + l 2 (w-h)} A 



+ {k (w — 1 2 ) + l a (w — h) + 2 hh) a 

 = {2(w-h)iv-(w-! 2 )h + (iv-h)! 2 } A + w(h + l 2 )a 

 = w(2w — / 1 — l 2 )A + w(li + l 2 )a, 

 where U + l 2 can take any value from to 2w. 



Thus the (%+l)th sibling group will be formed of types of family produced 

 by two operators of the form 



(2 to— r) A + ra, 

 where r has any value from to 2 w. 



Thus the types of family groups in the (?i+l)th sibling generation are 

 given by 



{ r T[(2w-r)A + ra]y 



