432 Proceedings of Royal Society of Edinburgh. [sess. 
The other rule of signs dealt with is Cauchy’s, in which permuta- 
tion-cycles are counted instead of inversions. The existence of 
such cycles is the first point to be established, that is to say, it has 
to be shown that any 'permutation of 1 2 3 . . . n may be obtained 
from any other by the performance of one or more cyclical permuta- 
tions. Let 3271654 be the permutation sought,* and 2647513 the 
permutation from which it is to be derived. Placing the former under 
the latter, thus 
2647513 
3 2 7 1 6 5 4, 
we see that 2 has to be changed into 3, then seeking 3 in the upper 
line we see that it has to be changed into 4, similarly that 4 has to 
be changed into 7, 7 into 1, 1 into 5, 5 into 6, and 6 into 2, the 
element with which we started. Now the proof turns upon the 
simple fact that the elements in the two lines being exactly the 
same, by following a string of changes like this we are bound sooner 
or later to reach in the second line the element we started within 
the first. It may be that as here one cycle 
suffices for the second transformation ; but if not, as in the case of 
the two permutations 
2647513 
4 1 5 7 2 3 6, 
where the short cycle 245 is obtained, we turn to the remaining 
elements, and knowing that those in the first line are of necessity 
the same as those in the second, we see that the application of the 
same process to them must, for the same reason as before, lead to a 
cycle. The possibility of arriving at any permutation by means of 
cyclical permutations alone is thus made manifest. The next point 
to be established is that a cyclical permutation of r elements can be 
accomplished by r- 1 interchanges of pairs of elements. Little more 
than the statement of this is necessary. For if the elements of the 
* This is a paraphrase of Jacobi’s demonstration, which is not so simple as 
it might have been. The notation of substitutions, which Jacobi did not follow 
Cauchy in using, is here a great help toward clearness. 
