428 Proceedings of Royal Society of Edinburgh. [sess. 
(«i ~ a o )( a 2 ~ a o)( a s ~ %) • • • • {flu- a o) 
(a 2 - Oj)(a 8 - aj) .... (a n -af) 
(a 3 -a 2 ) .... (a n -a 2 ) 
(a n - a n _i) 
for the quantities a 0 > a v a 2 , ... . a n , while in the order here written, 
the definition stands as follows (pp. 285-286): — 
“Vocemus eas indicum 0, 1, . . . , n permutationes, pro 
quibus P valorem eundem servat, positivas ; eas pro quibus 
P valorem oppositum induit, negativas ; sive priores dicamus 
pertinere ad classem positivam permutationum , posteriores ad 
cla-ssem negativam. ” 
This implies of course that the original permutation 0,1, 2, ... . , n 
is to be considered positive; and, such being the case, there 
seems to be a certain appropriateness in applying the term negative 
to a permutation whose corresponding difference-product is of 
the opposite sign from the difference-product corresponding to 
0, 1 , 2 , .... ,n. 
The propositions which lead from the definition to Cramer’s rule 
may he enunciated as follows : — 
(a) One permutation performed upon another gives rise to a 
third, and the combined effect produced by performing the 
second and first in succession is the same as the effect 
of performing the third. 
(b) Two given permutations belong to the same class or to 
opposite classes according as the permutation by means 
of wdiich the one is obtained from the other belongs to 
the positive or negative class. 
(c) If the same permutation he performed on a number of per- 
mutations which all belong to one class, the resulting 
permutations will still all belong to one class, viz., the 
same or the opposite according as the operating permuta- 
tion is positive or negative. 
(d) The order of compounding a set of permutations is, as a 
rule, not immaterial. 
