466 Proceedings of Boy al Society of Edinburgh. [sess. 
INTERCHANGES. 
(32) Coming now to the subject of “ interchanges ” we have 
again to observe that the number of them is indefinite unless they 
are made in accordance with a more or less orderly plan. 
Interchanges may he divided into two kinds, effective and in- 
effective , an interchange being effective when it is practised upon 
two elements which are not in their standard places, and, as the 
result of it, one at least of them is brought into its standard 
place. An effective interchange may he singly or doubly effective , 
according as one or both elements are brought by it into their 
standard places. Thus, in the permutation 
3651427 
the interchange 6^2 would be doubly effective, 3^1 singly effec- 
tive, 4^2 and 2^7 ineffective. We may even hold that there 
are three degrees of inefficiency, the second degree being exempli- 
fied by 2 -*■5*7, which throws one of the elements out of its standard 
place, and leaves the other, which was not in its standard place at 
the outset, still in need of removal. 
If the interchanges be ail effective, the number, it will be found, 
is quite definite, and it is this number which is denoted by v. 
Thus, for the permutation 
2 4 15 3 
we have v = 4, 1 being brought into its standard place by the 
interchange 2-s*l, 2 by the interchange 4*=s>2, 3 by 4^3, and both 
4 and 5 by 4-^5. 
Of course we might vary this procedure by putting the elements 
in a different order into their standard places, but, as will be seen 
later, the number of necessary interchanges would not be altered. 
Thus, for the permutation 2 4 1 5 3, if we took the reverse order, 
4 interchanges would still be needed, viz., 5-v3, 4*^3, 3*^1, 
2 ^ 1 . 
(33) Where there are n elements in the permutations, the num- 
ber of needful interchanges cannot be greater than n- 1 in any 
case. For, supposing n - 2 effective interchanges have been neces- 
