464 Proceedings of Royal Society of Edinburgh. [july 18 , 
mode of proving the theorem will he readily understood by seeing 
it applied to this illustrative example. Taking the permutation 
8, 4, 9, 3, 10, 7, 6, 5, 2, 1 , 
and interchanging 3 and 5 we have the permutation 
8, 4, 9, 5, 10, 7, 6, 3, 2, 1 , 
and thence by cyclical changes the permutation 
8, 4, 9, 5, 1, 2, 3, 6, 7 , 10, 
the number of alterations of sign thus being 
1 + (5 + 4 + 3 + 2 + 1) 
i.e. 1 + J(5x6), 
— an even number. 
Annexed to the theorem is the following corollary, which is not 
essentially sufficient from Hindenburg’s proposition regarding the 
sequence of signs, — 
If the permutations of 1, 2, 3, . . . , n - 1 be arranged after the 
manner in which numbers are arranged in ascending order of 
magnitude , and also in like manner the permutations of 1, 2, 3, 
. . . . , n — 1, n, then those permutations of the latter arranged set 
which begin with r, say , hare in order the same signs as the permu- 
tations of the former arranged set, or different signs, according as r 
is odd or even. (hi. 11.) 
For example, arranging the permutations of 1, 2, 3, each with its 
proper sign in front, we have 
+ 1, 2, 3 
-1, 3, 2 
-2, 1, 3 
+ 2, 3, 1 
+ 3, 1, 2 
-3, 2, 1; 
then arranging those permutations of 1, 2, 3, 4 which begin with 
3 say, each with its proper sign, we have 
+ 3, 1, 2, 4 
- 3, 1, 4, 2 
- 3, 2, 1, 4 
+ 3, 2, 4, 1 
+ 3, 4, 1, 2 
- 3, 4, 2, 1 ; 
(B) 
