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

202 HISTORY OF THE THEORY OF DETERMINANTS
The other notation is his own, and is worthy of careful note.
It differs from Sylvester's in making use of row-numbers and
column-numbers not of the retained elements but of the elements
omitted, the said numbers being enclosed in brackets for the
purpose of recalling this difference. "Thus," he says (p. 352),
"the complete compound determinant of the first class may be
written
that of the second class
1/ 2\ A 30 /2 3\ \
1. 2.1 3..' 2 3 ''
and generally that of the ith class
(1 1 22... 2 / (t. i M
\\l 19 2...i 21 22.-.. 2 l / t2 \...
wherei y (~ -1 (.'. (i- +1),,
where M= 1.2...- -
and where, it should have been added, rs stands for the sth integer
in the rth combination of i integers taken from 1, 2,..., s.
These preliminaries having been attended to, a discussion of
the properties follows. The first five pages (pp. 353-358) and
two later pages (pp. 366-368) are mainly concerned with compound determinants of the first class (that is to say, with the
adjugate determinant), and they do not break fresh ground.
The same, however, cannot be said with reference to the next
two pages (pp. 358-360), which concern those of the second class.
The result first reached is that the complete determinant of this
class, namely,
(1 2) /1 3\ /2 3) }.
1 2) 1 3.... * * 2:3).... * * or A2 say,
f1 2..
-~1 2.. ),
where v= ~(n- l)(n -2). Although there is a semblance of
reasoning, no real proof is given. Passing then to any,first