472 Proceedings of Royal Society of Edinburgh. [sess. 
complete the cycle is the omitted item a'-into-a, so that we have 
only one circular substitution, 
(41) If in any permutation of n elements there be k cycles of sub- 
stitution, n — k is the smallest number of interchanges necessary to 
transform it into the standard permutation. 
To transform the permutation into the standard permutation 
implies that the number of cycles is to he increased to n. Now, 
any two elements of the permutation must be either in the same or 
different cycles, and we have seen that the interchange of two, 
which are in different cycles, does not lead to an increase in the 
number of cycles. To attain our end with the fewest possible 
number of interchanges, we must, therefore, in every case choose 
two elements which are in the same cycle. When we do this, 
however, n - k such interchanges are known to he necessary and 
sufficient. The lowest possible number is thus n - k. 
(42) It follows, therefore, from § 39, that the lowest possible 
number of interchanges necessary to transform a given permutation 
into the standard permutation is secured by using only effective 
interchanges. This is important from a practical point of view, be- 
cause effective interchanges are easily recognised. To seek for 
pairs of elements which are in the same cycle of substitution would 
he much more troublesome. Besides, it implies that the cycles of 
substitution are known ; and, if this be the case, the consideration 
of interchanges is in practice unnecessary. 
As a matter of theory, however, it is curious to note the manner 
in which, when we follow the more troublesome process and in 
doing so use an ineffective interchange, we are compensated at a 
later stage by an additional doubly-effective interchange. Thus, 
taking the permutation 
my , . • 
\a,/3,y, . 
•, 9 • • • •> 
. ., X’ x f / ’ 0) ’P'>y> • • • •> x 
51762843 
with its circular substitutions 
pi 2 5\ 
\5 1 2/ 
8 6 4 7' 
3 8 6 4, 
