462 Proceedings of Royal Society of Edinburgh. [sess. 
With the same convention we have quite generally, as is well 
known, 
C _ n 
1+2+. . .+(»-i),a ^i+2+. . .+(«-i)i«( 7 i-i)-a’ 
and therefore the theorem of § 14 
Y =V 
n,o n,lw{n-\)- & 
(25) There is still another problem which has for its solution 
exactly the same double series of numbers as we have tabulated in 
§16, viz., the problem of finding the number of positive integral 
divisors of ab 2 c 3 d 4 . . . loliere a,b,c,d, . . . are integers ptrime to one 
another. In fact, the related table in § 24 may be viewed either 
as a table of combinations, as was intended, or as a table of the 
said divisors. Of course, for the latter purpose, the insertion of the 
row of l’s referred to at the close of § 24 is a necessity, 1 being a 
divisor in every case — the co-factor, in fact, of the divisor of 
highest degree. 
(26) To bring out still more clearly the connection between the 
three problems, let it be observed that the table of combinations 
in § 24 is also by implication a table of the permutations of 1 2 3 
. . . arranged in order of the number of inverted-pairs which they 
contain. This fact is not so readily perceived as the identity of 
the combinations of § 24 with the divisors of § 25, but a little care 
suffices to establish it. 
Taking the third column of combinations or factors, viz. : — 
1 , 
a,b,c. 
ab,bb : ac,bc : cc. 
abb : abcfbc : accfcc : ccc. 
abbc : abccfbcc : acccfccc. 
abbcc : abcccfbccc. 
abbccc. 
and viewing each a, each 5, each c as representative of an inverted- 
pair of which 2, 3, 4 are respectively the first element, we translate 
the column into 
