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

COMPOUND DETERMINANTS (SYLVESTER, 1879) 201

SYLVESTER, J. J. (1879).
[Sur les determinants composes. Crelle's Journ., lxxxviii. pp.
49-67; or Collected Math. Papers, iii. pp. 456-473.]
At the outset (pp. 49-53) Sylvester rehearses his previous work
on the subject, the appearance of it being slightly changed because
of a fresh notation: for example, the theorem regarding the evaluation of the mtl compound takes the form
Ect *... i,,a$.. aX, _ (aa2.. a)(- 1 -atl A2... M a
and the 'extended' mth compound is written
[b1b92. l. b 1a ]. X [ /1a3'' P ^,Ci2... aC(,]
lie also illustrates diagrammatically the degeneration of one of his
theorems (Hist., ii. pp. 197-200) into the ordinary multiplicationtheorem, the product of two rows or two columns
(a1, a2 a, 4a, a a,,13, a4)
being representable in the form
I... a1
1.. t2. 1. 3
1 a4
(11 C2 C 3 4
There then comes his new but improved generalization (pp. 53-55),
the subject of which is already familiar to us from the writings of
others, namely, the expression of a minor of the mtll compound of
I a,, I in terms of minors of I a,,. Reiss, who had tackled the
problem in 1867, is not referred to. In effect Sylvester says that
every minor is so expressible if it be constructed so as to have each
of its elements containing a rows and a columns chosen from a fixed
rows and a fixed columns of the generating determinant, /3 rows and,3 columns from b other fixed rows and columns of the generating
determinant, and so on, the number of lines in each element thus
being
ca+3+. +,
the number of lines in the generating determinant
a+b+... +z,