1898 - 99 .] Dr Muir on a Single Term of a Determinant. 465 
verted-pairs ” in which 1, 2, 3, ... are respectively the second 
elements, so that the total number of inverted-pairs is a + j3 + y + ... 
Now the number of “moves” necessary to transform the permuta- 
tion into the standard permutation is the number necessary to put 
1 into the first place, 2 into the second place, 3 into the third 
place, and so on. But the number necessary to put 1 into the first 
place must be a, because from consideration of the inverted-pairs 
we know that there are exactly a integers preceding 1. Again, 
the number of “ moves ” necessary to put 2 into the second place 
must be /3, because at the outset there were ft integers greater than 
2 preceding 2, and these could not be affected by the movement of 
1 into its standard place. Similarly the number of moves neces- 
sary to put 3 into the third place is seen to be y, and so on ; so 
that the total number of “moves” is a + /3 + y + . . . , that is to 
say, is the same as the number of “ inverted-pairs.” 
(30) If in the preceding proof we had arranged the inverted-pairs 
differently, viz., if we had begun with those in which the highest 
element, say 5, came first, then taken those in which 4 came 
first, and so on, the total number being thus partitioned into 
a' + /3' + y' + . . . we could have proved in the same manner that 
the number of moves necessary to put 5, 4, 3, . . . into their respec- 
tive places would have been a,(3',y', . . . 
Ih this way we see that the number of moves necessary to trans- 
form a given permutation into the standard permutation is the 
same whether ice put the elements in order , or in reverse order , into 
their standard places. 
(31) Further, the lowest possible number of moves is secured when 
the elements are put in order , or in reverse order , into their stan- 
dard places. 
No move can free us of more than one inverted-pair; and, there 
being 8 inverted-pairs, it follows that the least possible number of 
moves is 8. But 8, as we have just seen, is the number of moves 
made when we proceed in orderly fashion ; the theorem is thus 
established. 
