400 Scientific Proceedings, Royal Dublin Society. 



Being compelled to select similar semi-factors from each parent, the 

 progeny must bear characters similar to those borne by the parents, and 

 must breed true. 



If, however, the parents are not similar, but differ in one or more pairs 

 of alternative characters, the progeny will receive mixed factors from their 

 parents, and wiU. not breed true as regards the differentiating characters, 

 though they will breed true as regards the others. 



If the parents differ in one pair of alternative characters, and we repre- 

 sent tlie dominant byX and the recessive by x, theprogen}' of their progeny, 

 i.e., their second crosses, split into two groups, one bearing X and the other.?, 

 and the number of individuals in the group bearing X is to the number in 

 the group bearing x in the ratio 3:1. 



If the parents differ in a second pair of alternative characters, say, Y 

 and y, the group bearing X on the one hand and that bearing x on the other 

 split each into two fm-ther groups, one bearing !F and the other y; and the 

 numbers bearing Y are to those bearing ^ as 3 : 1. Thus there are four 

 groups altogether, one bearing the characters XY, another Xy, anotlier xY, 

 and another xy, and the numbers of individuals in these gi'oups are in the 

 ratio 9:3:3:1. 



For every additional pair of alternative characters in which the parents 

 difEer, the number of groups into which their second crosses divide is doubled 

 — for one pair there are two groups, for two pairs four groups, for three pairs 

 eight groups, and so on — and the proportional numbers in each group expand 

 in accordance with a well-known mathematical formula. 

 This can be shown diagrammatically : 



X X 



For 1 pair : 3 1 



/\ /\ 



/ \ / \ 



/ \ / \ 



/ \ / \ 



Y y Y y 



For 2 pairs: 9 3 3 1 



/\ /\ /\ /\ 



/\ /\ /\ /\ 



Z % Z % Z % Z % 



Tor 3 pairs :27 9 9 3 "9 3 3 1 



/\ /\ /\ /\ /\ /\ /\ /\ 



-4 a A a A a A a A a A a A a A a 

 For 4 pairs: 81 27 27 9 27 9 9 3 27 9 9 3 9 3 3 1 



and so on. 



If we consider by way of example the case for three pairs of characters, 



we see that there are eight groups of second crosses ; and if we follow the 



