1904 - 5 .] 
Dr Muir on General Determinants. 
941 
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. 
For example, the value of 
a b 
c d 
e f 
g h 
will be found by taking the (D2) 3 arrangements, as below, 
ab ab ab ab ab ab ab ab 
cd dc cd dc cd dc cd dc 
ef ef fe fe ef ef fe fe 
gh gh gh gh hg hg Kg Tig , 
the signs of cd , ef , gh being supposed + , those of dc , fe , hg 
will be each - . Consequently the sum of the terms will be 
expressed by 
aceg • bdjli - adeg ■ bcfli — acfg • bdeli + adfg • bceh 
- aceh • bdfg + adeh • bcfg + acfh • bdeg - adfh • bceg 
It will be observed that the example here given is the quadratic 
function which Cayley would have denoted by 
/ V \ 
\ 0 0 0 0 I 
\ ] 1 1 1 / ’ 
and which, on the supposition that generally Y a ,^,y,s =V a + ) g + v +5 
and in particular that V 0 = a, V 1 = b , Y 2 — c , . . . . would represent 
ae-bd — bd + c 2 — bd + c 2 + c 2 — db 
i.e. ae - ibd + 3c 2 . 
In his applications of the theory of commutants to that of £ forms 5 
Sylvester first uses differential operators as umbrae, speaking, for 
example, of the commutant 
a_ a_ a_ 
dx 1 ’ dg 1 5 dz 1 5 
a „ 
dx 2 ’ dy 2 ’ dz 2 ’ 
