940 Proceedings of Royal Society of Edinburgh. [sess. 
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 + 2 bed - ad 2 - b 2 e - c 3 
could be expressed as a commutant, viz., 
/0 0 \ 
1 1 
\2 2 J 
by considering 00 = a, 01 = 10 = /;, 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 well- 
known 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 umbrae 
«1 
b i •• 
.. 
a 2 
h ■■ 
.. i 2 
a r 
b r . . 
.. l r 
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) r ~ l 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 
