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

LESS COMMON SPECIAL FORMS (BORCHARDT, 1859) 459
Resuming consideration, but proceeding on a different tack, he
arrives at Sylvester's "rule," namely, that (0, 1, 2,..., n) is
"gleich der Summe aller nicht-cyclischen Producte, die aus je
jener ^n(n+ 1) Elemente (i lc) gebildet werden konnen." Unlike
Sylvester, however, he is careful to give a justification of it
based on four observed facts, namely, (1) that (0, 1, 2,..., n)
is unaltered by interchanging any two of the umbrae; (2) that
the coefficient of the term (01)(02)... (On) is 1; (3) that none
of the terms is free of the umbra 0; (4) that, as already
mentioned, the cofactor of (01) is (0+1, 2,..., %)* As the
proof, which extends to two pages (pp. 119-120), applies only
to the case of axisymmetry, it need not be given.
Lastly, the number of terms in the development of (0, 1, 2,..., n)
is investigated, the result obtained agreeing with Sylvester's.
We may note for ourselves in passing that the first three of the
basic facts of the proof are, like the last, most readily appreciated
by observing the determinant form, the case where n = 3, namely,
10+12+13 -12 -13
-21 20 +21 +23 - 23
-31 -32 30+31+32
being amply sufficient. Thus, increasing any column by all the
others, and thereafter increasing the corresponding row by all
the other rows, we obtain the first result, learning at the same
time that it only holds when axisymmetry exists; the second
is self-evident; and the third follows from the fact that the
aggregate of the terms which are free of 0, being got by
deleting 10, 20, 30, is expressible as a vanishing determinant.
(8) MISCELLANEOUS SPECIAL FORMS.
CAYLEY, A. (1845).
[On certain results relating to quaternions. Philos. Magazine,
xxvi. pp. 141-145; or Collected Math. Papers, i. pp. 123-126.]
Assuming that in each term of the development of a deter*As (01) occurs only in the element a1 - (11), its cofactor is the primary minor
obtained by deleting the first row and the first column, and this is seen to be
(0 + 1, 2...., n) by definition.