47 0 Proceedings of Royal Society of Edinburgh. [sess. 
and y,x the pair of elements in the lower line, so that the resulting 
substitution is 
Then in beginning with a in the latter and telling off in linked 
fashion the items of the substitution, viz., a-into-/?, /3- into-y, 
we necessarily find that, instead of going as before from y to 8, 
and so on through the whole of the remaining elements, we go 
from y to a more advanced element in the upper line than S, viz., 
to i/f, and consequently reach more rapidly the element with which 
we began, viz., a, thus obtaining the shorter circular substitution 
which also is necessarily circular. Further, as the elements 
S,e, . . . ., 0 are p in number, the number of elements in the latter 
circular substitution is p + 1 . 
The case where the two interchanged elements are consecutive 
in the cycle should be noted in passing. One of the two com- 
ponent circular substitutions will then be monomial, the result of 
the change being to put the second element into its standard 
place. 
(38) From this it follows that — 
If in a given permutation of n elements tivo elements be inter- 
changed which are in the same cycle of substitution the resulting 
permutation will have one cycle more than the given permutation. 
Consequently, by continuing to make such interchanges, the 
resulting cycles may be made all monomial, and therefore be n in 
number. This means that each element would then be in its 
standard place, and that the number of such interchanges would be 
the excess of n over the original number of cycles. We thus learn 
that — 
and leaving the substitution 
'S,c, , 
-xA € > ••••.. 
