1837.] Dr T. Muir on the Theory of Determinants. 517 
The third section breaks entirely fresh ground, its heading being 
Des Systemes de Quantites derivees et de 
leurs Determinans. 
Of the integers 1, 2, 3, . . n all the possible sets of p integers 
are supposed to he taken, and arranged in order on the principle 
that any one has precedence of any other if the product of the 
members of the former he less than the product of the members of 
the latter. The number n{n - 1) . . . . (n — p + 1) / 1. 2. 3 . . . . p 
of the said sets being denoted by P, the P th and last set would 
thus be 
n — p + 1, n —p 4- 2, . . . . ., n— 1, n. 
Now, any two of the sets being fixed upon, say the /P h and P h , the 
system of quantities (a Vn ) is returned to, and from it are deleted 
(1) all the “ ternies ” whose first index is not found in the set, 
and (2) all the “ termes ” whose second index is not found in the 
P h set. What is left after this action is clearly “ un systeme de 
quantites symetriques de l’ordre pf the determinant of which may 
be denoted by a For example, if fx = v= 1, all the a’s would be 
deleted whose first or second index was not included in the set 
1, 2, 3, . . p } and there would be left the system 
K-l a l’2 * * ’ * a VP 
S $2* 1 2 «•*«>« ^9 • p 
| &c. ...... 
g .... dp.p 
of which the determinant would be denoted by 
c py. 
a Vl • 
As any one of the P sets could be taken along with any other, pre» 
paratory to forming such a determinant, there would necessarily be 
in all P x P possible determinants. Arranged in a square as 
follows : — - 
( p ) 
C p) 
( V ) 
a vl 
a V2 .... 
a v p 
( p ) 
(p) 
( p ) 
a 2'l 
flf-22 .... 
a 2 F 
&c. 
...... 
( p ) 
(P) 
( P ) 
Op J 
2 .... 
a tv 
