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

204 HISTORY OF THE THEORY OF DETERMINANTS

From these cases of compound determinants of the second
class, the author passes to those of the 1th class, but contents
himself with stating only two results. The first is that the
complete determinant of this class, denoted as above by
* i 2122 2 y1 U2 * or Ail say,
I1 12... 1t 21 22... 2i % M2 * i
l 2...
l 2... -5
wher,. (n-l)(n-')... O(n-;+1)
where v,- 1)(n-) a result which agrees
1.2... i-
with that obtained on putting n=0 in Sylvester's general
theorem, of March 1851; and the other is that any first minor
of A,, say the minor
/21 22... 31.2 * i ^tl * * *.Mi
1 12. 1 l 2. '21 1...P i 2... n.
He appends, however, the words "and so on," and tells us that
"other formulae may be written as required."
Two theorems of Sylvester's are next given, the one being the
general theorem just alluded to, and the other that contained in
the paper of 16th December, 1852. In the case of the former
he varies the notation, and probably by reason of the abovementioned serious misprint of an in for an r in the original he
misses Sylvester's meaning, and makes an incorrect statement.
In the case of the other no risk of this kind is incurred, because
he takes the unusual course of reproducing Sylvester's words
letter for letter to the extent of almost two pages (pp. 361-363).
The original two pages, however, being, as we have seen, not
without evidence of Sylvester's carelessness, this course also was
unsafe.
Lastly, he shows how the multiplication-theorem may be
deduced from Sylvester's first theorem of March 1851 regarding
compound determinants, by taking as an example of the latter
the identity