CONSTRUCTION OP THE PERMUTATIONAL TRIANGLE. 241 



3. Construction of the Permutational Triangle. 



In my paper on ' ' Divergent Evolution "* I referred to the permuta- 

 tional triangle, which I had constructed in order to determine the prob- 

 ability of extinction that would, under certain conditions, result from 

 complete segregate fecundity, when unaided by any form of positive 

 segregation. The first four lines of the table were obtained by direct 

 observation on the permutations of letters arranged to represent the 

 pairing of animals entirely lacking in instincts or qualities that secure 

 the pairing together of those of one kind. 



For example, let A, B, C represent three females of three varieties 

 of pigeons, and a, b, c three males of the same varieties, all occupying 

 one aviary. Now, supposing they are devoid of segregating instincts, 

 and that they all pair, what are the probabilities concerning the pair- 

 ing of the males with their own kind? These will be clearly shown by 

 arranging the letters representing one of the sexes in one fixed order, 

 placing the letters representing the other sex underneath in every 

 possible permutation of order. If we make six experiments the proba- 

 bility is that in two cases none, in three cases one, 

 in no case two, and in one case three, will pair with 

 their own kind. These numbers constitute the 

 four terms of the third line. The first, second, 

 and fourth lines were constructed in the same way, 

 but for the construction of the tenth line in this 

 way I estimated that several years of constant 

 writing would be required. The remaining lines 

 here given were, therefore, constructed according 

 to the following rules, which were discovered by studying the first 

 four lines. The discussion of different methods of constructing the 

 permutational triangle, and the interesting properties of the same 

 when constructed, must be deferred; but I may say here that I 

 believe it will be found an important instrument for estimating a large 

 class of probabilities. 



One method of constructing any line of the permutational triangle from 

 the preceding line. (i) Of any given line, any desired number, except 

 the first, may be obtained by multiplying the preceding number of the 

 preceding line by the factor of the given line and dividing the result by 

 the figure marking the degree of correspondence of the column of the 

 desired number. (2) The first number of any line is one less or one 

 more than the second number of the same line, according as the factor 

 of the line is an odd or an even number. 



A B 



a b 



* Also see pp. 99-100 of this volume. 



