444 
Proceedings of Royal Society of Edinburgh. [sess. 
decompose this into circular substitutions 
7\ 4 8 2 6\ 
75 7 9\ 
V6 1 4 8 2/, 
\3/, 
\9 5 7/: 
and the number of the latter being 3, and the total number of 
elements in the permutation being 9, we take for our sign-factor 
(-i) 8 - 3 . 
Cauchy’s rule may thus be formulated : — If k be the number of 
circular substitutions necessary to transform the given permutation 
into the standard permutation , and n be the number of elements 
in either permutation , the sign of the given permutation is ( — ) n ~ K . 
(4) The fourth rule appeared in 1831, its author being J. E. 
Drinkwater. In his paper “ On Simple Elimination ” in the 
Pliilos. Mag ., x. pp. 24-28, he says : — 
“ Any permutation may he derived from the first by con- 
sidering a requisite number of figures to move from left to 
right by a certain number of single steps or descents of a 
single place. If the whole number of such single steps 
necessary to derive any permutation from the first be even, 
that permutation has a positive sign prefixed^to it : the others 
are negative. For instance, 4 2 13.... n may he derived 
from 1 2 3 4 .... n by first causing the 3 to descend 
below the 4, requiring one single step : then the 2 below the 
new place of the 4, another single step : lastly, the 1 below 
the new place of the 2, requiring two more steps, making in 
all 4. Therefore this permutation requires the positive sign.” 
This amounts to saying, that, If /x be the number of moves necessary 
to transform the given permutation into the standard permutation , 
the sign of the given permutation is ( - f. 
To a certain extent Rothe, to whom the rule of interchanges 
has been attributed, may he considered a co-discoverer with Drink- 
water, one of Rothe’s corollaries being (p. 272) : — 
“ Ensteht eine Permutation dergestalt aus einer andern, 
dass ein einziges Element aus seiner Stelle genommen, und in 
eine andere Stelle gesetzt wird, so hahen beyde Permutationen 
einerley Zeichen, wenn der Unterschied der Stellen gerade, 
