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

DETERMINANTS IN GENERAL (BELLAVITIS, 1857) 95
the multipliers referred to can actually be found by solving a
set of simultaneous linear equations.
The multiplication-theorem for determinants A, A2 of the third
order he seeks to establish (~ 31) by partitioning the productdeterminant into twenty-seven determinants, and showing that
the sum of the six which do not vanish is AlA2.
Chio's theorem of 1853 is introduced (~ 38) by noting that the
resultant of +by+c 0 = 2)
a~.x+b4.y+cr = O (~'=1,2,3)
may be viewed as the resultant of
aI b2y + I C2% = 0
lab3 y + I^ C3 = 3
and that therefore
II 2a, I~ I a, C2 | | -.
|al b2 iala C! must be a multiple of a, b2 C3.
That it is so he proves by diminishing the 2nd and 3rd columns
of I a b2 C3 by b/acc times the 1st column and cl/al times the
1st column respectively. Further, he points out (~~ 39, 40) a
practical application, namely, in evaluating a determinant whose
elements are given in figures.
The adjugate determinant (unfortunately renamed associato)
is dealt with (~ 55-58) in connection with the solution of a set
of simultaneous linear equations, the special cases being considered where the determinant of the set is 1 and 0. In the
former special case he notes the theorem, The adjugate of the
product of two unit determinants is identical in all its elements
with the product of the adjugates of tthe said determinants;
and in the latter the theorem all but reached by Jacobi in 1835
and 1841, In a zero determinant the cofactors of the elements of
a row are proportional to the cofactors of the elements of any
other row.
Cauchy's " clefs algebriques " (chiavi algebriche) are expounded
at some length (~~ 81-88).
In the last three paragraphs he draws attention to the existence
of expressions which may be viewed as "determinanti simbolici,"
his first kind being those in which symbols of differentiation take
the place of elements; e.g. the negative of the expression