The theory of determinants in the historical order of development, by Sir Thomas Muir.

DETERMINANTS IN GENERAL (BRIOSCHI, 1854)

91

QP = HrKri + H,2Kr2 +.... + H,.K.,,,
where Krs stands for what Q becomes when its sth row is replaced by the rth row of P.*
In his treatment of the minors of the adjugate determinant
Brioschi (pp. 36-39) closely follows Spottiswoode; that is to say,
from a set of linear equations he derives one result, then from
the adjugate set another result, and finally draws a deduction
from a comparison of the two. His thus obtained extension
of Spottiswoode's theorem is open to the same criticism as
Spottiswoode's extension of Jacobi's.
The section (~ 7) on " determinanti di determinanti" is founded
on Cauchy, and contains known extensions of two or three
theorems above given in the notation of differentiation.
CANTOR [M. B.] (1855, March).
[Theoreme sur les determinants Crameriens. Nouv. Annales de
Math. (1), xiv. pp. 113-114.]
The theorem in question may be formulated thus-If the
permutations of 1, 2, 3,,... n be arranged in order of magnitude
as if they were integral numbers, the sign of the kth permutation
is independent of n. Reference is appropriately made to Reiss'
paper of 1825, but the theorem is virtually contained in Hinderburg's rule of the year 1784.
Another author who dealt with the 'rule of signs' in this
year was Mainardi; his paper is referred to along with a kindred
one by Zehfuss of the year 1858.
*Brioschi does not note the independent importance of his second set of
equations, which may be condensed into
P aQ 8Q. aQ
a - Hr] = -b + r2 +... +Hrn n,s
Cars C1l, ab2s
and which, when r, s= 1, i and n=3, is
al a2 3 ba b 1 b b 2 a3 2 _ b1 b2 P3 2 3 + 7l b 3 2 3 a
Cth. b = bb Y3y3 b -w a. Y,3
where 3 3h fis ro fPis13.O
Y71 72 73 C c2 C3 c'1 ~2 3 Cl c2 Cs3
This, however, may be viewed also as a case of Sylvester's theorem, namely,
where the first row of P is 1, 0, 0.