1887.] Dr T. Muir on the Theory of Determinants. 507 
nombre des substitutions circulates 4quivalentes a la substi- 
tution 
A 2 3 n\ 
\« P y O' 
Ce produit devra etre affecte du signe + , si n - g est un 
nombre pair, et du signe — dans le cas contraire.” (in. 18). 
Thus if the sign of the term 
a Q'l a §'2 ^3-3 a i'4 a 9’5 a 2'Q a b'7 a 4’S ^7*9 
in the determinant 
S( ± a vl ct 2 '2 a z’3 ' • • • ^ 9 * 9)5 
be wanted, we write the series of first suffixes 6 , 8 , . . . under the 
corresponding suffixes of the “ principal product,” that is to say, 
under the series 1, 2, 3 . . .9, obtaining the interchange 
/I 2345678 9\ 
\6 8 3 1 9 2 5 4 7/; 
this we separate into circular interchanges, finding them three in 
number, viz., 
/3\ /5 7 9\ /I 2 4 6 8\ 
\3/» *9 5 7/’l6 8 1 2 4/; 
and the determinant being of the 9 th order, we thence conclude that 
the desired sign is ( - ) 9 " 3 , i.e., + . In connection with this subject 
a modification of Cramer’s rule is given, no reference being made to 
“ derangements ” at all. Put into the fewest possible words it is — 
The sign of the term a a . x ap. 2 . . . . a$. n is the same as the sign of 
the difference-product of the first suffixes , that is, the sign of 
(ft ~ a ) (y “ a ) 
For example, the sign of 
( 111 . 19). 
a 6'l a 8'2 a 3'3 a l*4 a Xb a 2'6 a b'7 a 4‘8 a 7‘9 > 
above sought, is the sign of the difference-product of 
6, 8, 3, 1, 9, 2, 5, 4, 7 
i.e., the sign of 
(7 - 4) (7 - 5) (7 - 2) (7 - 9) (7 - 1) (7 - 3) (7 - 8) (7 - 6) 
x (4 - 5) (4 - 2) (4-6) 
x (5-2) (5-6) 
x (8-6) 
