342 Ef*'V. J. T. GULrCK ON 



have now to consider tlie degree of probability that these iden- 

 tical varieties will make the same combinations with each other 

 in the different districts. I shall not attempt to give a complete 

 answer ; but by carrying the computation through several steps, 

 I shall sufficiently exhibit the extreme improbability that, even 

 when identical varieties succeed in propagating in the different 

 districts, they will combine with each other in the same way and 

 in the same proportions. 



As in the case of the 10 varieties that have been under consi- 

 deration, 5 is more likely to be the number of varieties that suc- 

 ceed than any other number, 5 is most likely to be the number 

 of successful varieties in each district when the varieties liappen 

 to be the same in each district ; and we will therefore begin with 

 that number. If, now% -we suppose that there are 5 varieties in 

 each district, and that there is the same chance in the case of 

 each variety that it will breed with any one of the other varieties, 

 as there is that it will be segregated and breed by itself, we shall 

 find that in 120expei'iinents there will probably be 1 occasion in 

 which all the varieties of one of the districts will be segregated 

 from each other, and 10 occasions in which three of the varieties 

 will be segregated, and 20 occasions in Avhich two will be segre- 

 gated, and 45 occasions in which one wdll be segregated, and 44 

 occasions in which none will be segregated *. These probabilities 

 are expressed by the fractions yi^, JJ^j, j\?jj, '^-^, and y%. And 

 the probability that the same varieties will be intercrossed and 

 the same ones segregated in each district is r^o 5 while the pro- 

 bability that some one particular set of segregations and inter- 

 crossings that is designated in advance will occur in both districts 

 ig (j^y. For example, the probability that all the 5 varieties 

 in one district will be segregated is y^ ; and the probability that 

 all in both districts will at the same time be segregated is (y|o)'. 



But the two districts may correspond by the complete failure 

 of all varieties to propagate, in which case they will continue to 

 correspond. Again, there may be but one variety in each district 

 that succeeds in propagating, and that the same, in which case 

 there will be no chance for diversity of Amalgamation in the 

 different districts, at least not before a diversity of subordinate 

 segregations has first arisen. Again, if the same two varieties 



* These figures are found in the 5th line of the Permutational Triangle. See 

 Appendix. 



