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

DETERMINANTS IN GENERAL (CAYLEY, 1843) 15
minants are constantly made use of. At the outset an important
notation is introduced which afterwards came to be generally
adopted. The passage in regard to it is:"Consider the series of termsx1 x2... An
A1 A2.... A,,
K1 K2... K,,
the number of quantities A,..., K being equal to qe(q < n). Suppose
q + 1 vertical rows selected, and the quantities contained in them
formed into a determinant, this may be done in
n(n - 1)... (q + 2)
1.2... (n-q- q1)
different ways. The system of determinants so obtained will be
represented by the notation
X1 X2.... X,,
A1 A2.... A,,,.,......
K1 KS.... K;
and the system of equations, obtained by equating each of these
determinants to zero, by the notation
X1 X2....
(3) 1 2 0."
K1 K2.. Ki
A theorem is next enunciated in regard to the expression of
any one of the determinants in terms of n - q of them.
"The n( — 2 1) (q +-) equations represented by this formula
1.2... (n-q-1)
reduce themselves to n -q independent equations. Imagine these
expressed by
(1)= 0, (2)=,...., (n - q)o0,
any one of the determinants is reducible to the form
e0(1) + 2(2) +...+ O_. q(1( -q)
where 1, 02,..., O,_q are coefficients independent of x, x2,..., x,."
No proof is given.