CONSTRUCTION OF 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 oforder. If wemakesix 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 A B eg 
their own kind. ‘These numbers constitute the a b G 
four terms of the third line. The first, second, a c b 
and fourth lines were constructed in the same way, c a b 
but for the construction of the tenth line in this b a c 
way I estimated that several years of constant b c a 
writing would be required. The remaining lines c b a 



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.—(1) 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. 
* Also see pp. 99-100 of this volume. 
