1898-99.] Dr Muir on a Single Term of a Determinant. 467 
sary to put the first n - 2 elements into their standard places, the 
remaining two elements must at the worst be in each other’s 
place, and therefore the interchange of them will be doubly 
effective. 
(34) The fundamental proposition in regard to “ interchanges ” 
is as follows : — 
If X and fx be any two elements of any permutation of 
1, 2, 3, . . , , n, and d be the number of elements lying between them 
and intermediate to them in value , the interchange X^/x icill in- 
crease or diminish the number of inverted-pairs by 2d + 1 according 
as X is less or greater than /x. 
Let the given permutation be 
X y 
and consider first the case where X>y .. Then in comparing the 
numbers of inverted-pairs before and after the interchange, the 
elements preceding X and those following /x may clearly be 1-eft 
out of account ; that is to say, we have only got to consider the 
portion 
X /x. 
Now, if X be moved so as to follow immediately after /x, thus giving 
the permutation 

the number of inverted-pairs is thereby diminished by d + 1 : and 
if to effect the interchange spoken of in the enunciation /x be next 
moved to the former place of X, thus giving the desired permutation 
/* K 
the number of inverted-pairs is further diminished by d. Hence 
the total diminution of the number is 2d + 1, as was to be proved. 
The reasoning for the case where X</x is exactly similar.* 
* This proposition is a modification of one of Rothe’s. See p. 268 of his 
Memoir, or Muir’s “ History of Determinants,” p. 56. Rothe’s proof is very 
engthy. 
