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

DETERMINANTS IN GENERAL (SYLVESTER, 1852) 69
tion, and, as was natural, did not wish that share to be lost sight
of. It made clear that the two workers had during the year been
deeply engrossed in what Sylvester then called the 'calculus of
forms,' that they had been in close communication with one
another, and that Sylvester's discovery that the function
ace + 2bcd c- cac2 - b2e - c
could be expressed as a commutant, namely,
/(O O
1 1
2 2
by considering 00=a, 01=10=b, 02=11=20=c, 12=21=d,
22 =e had led Cayley to the conception of intermutants.
The famous paper which we have now reached, and which was
doubtless completed very shortly after Cayley's, contains the
results-numerous and suggestive-of Sylvester's labours. The
only section, however, which directly concerns the theory of
determinants is the third, bearing the heading " On Commutants."
It opens with a page regarding the simplest species, " the wellknown common determinant," and then proceeds:"If, instead of two lines of umbrae, three or more be taken, the same
principle of solution will continue to be applicable. Thus, if there be
a matrix of any even number r of lines each of n umbrse
al b.... 11
a b2....
ar 1),.
a, b,..... I,,
the first may be supposed to remain stationary, and the remaining r - 1
lines each be taken in 1-.2... n different orders: every order in each
line will be accompanied by its appropriate sign + or -; and each
different grouping in each line will give rise to a particular grouping
of the letters read off in columns. The value of the commutant
expressed by the above matrix will therefore consist of the sum of
(1.2... n) - terms, each term being the product of n quantities
respectively symbolised by a group of r letters and affected with the
sign + or - according as the number of negative signs in the total
of the arrangements of the lines (from the columnar reading off of
which each such term is derived) is even or odd.