1887.] Dr T. Muir on the, Theory of Determinants, 
481 
+ a/3 y8 ^ 
— a@$y j 
— cry/SS ! 
-J- aydfi 
-f ad / 3 y 
— 0.57,8 
| 
J 
— / 3 ay 8 
+ fia$y 
+ /87a 8 
— fiy$a 
, — jSSay j 
-f fitiya :| 
y 
-f 70/68 I 
— 7 a8/8 
— 7,808 
-f 7/85 a 
+ 780,8 
— 78/50 
— 00/87 
4- 807/8 
+ S/607 
— 8,87a 
— dyafi 
+ §7 0 a J « 
A proof by tbe metliod of mathematical induction (so-called) is 
given that with these signs the sum of all the permutations of any 
group vanishes. 
Up to this point the essence of what has been furnished is a 
combined rule of term-formation and rule of signs. ( 11 . 5 -f in, 15.) 
In connection with it Bezout’s rule of the year 1764 may be 
recalled. 
The third problem is to determine the sign of any single per- 
mutation from consideration of the permutation itself. The solution 
is : — Under each letter of the given permutation put all the letters 
which precede it in the natural arrangement and which are not 
found to precede it in the given permutation ; and make the sum + 
or - according as the total number of such letters is even or odd. 
“ Exemp. Dafoe complexiones sint hse ; 
€yS 8 , Saey , eSya , S/2ey • 
Literse secundum I subjiciantur 
