1907-8.] 
Dr Muir on General Determinants. 
687 
( a i a 2 
a 3 l 
>J, v, y, 
( 
a 
h 
9 \ 
\ ^2 
ap 1 
h 
b 
f 
C 3 
g 
f 
c 
are made to stand for 
(«j £ + a 2 r] + a 3 £)a> and aa 2 ,+ by 2 + cz- + 2fyz + 2 gzx + 2 hxy 
+ Gi£ + btf] + b s £)y 
+ + c s9* 
respectively, and that the latter is also denoted by 
{a, b, c,f, g,h$x, y , z) 2 , 
and the binary cubics 
ax d + 3 bx 2 y + 3 cxy 2 4 - dy 3 , ax 3 4 - bx 2 y + cxy 2 4 - dy 3 
by 
(a, b, c, d\x, yf , (a, b, c, d\x, yf 
respectively. 
We may suggest for consideration in passing the following order of 
ideas, as leading up to Cayley’s contracted mode of writing a set of linear 
equations. First, a row of separate quantities e.g. (a, b, c, . . . ) ; second, 
the statement of the identity of two rows, e.g. (a,b,c, .... ) = (x, y, z, . . . .), 
or simply a, b, c, . . . = x, y, z, . . . ; third, the so-called product of two rows, 
e.g. (a, b, c, . . . $ x, y, z, . . . ) ] fourth, a square of separate quantities, i.e. 
a matrix ; fifth, the result of multiplying a matrix and a row being a row. 
It is unfortunate that, from the point of view of notation merely, this does 
not at once suggest, in the sixth place, the result of multiplying two matrices, 
where, as Cayley is careful to point out, the multiplication is row-by-column 
and not row-by-row. 
Brioschi, Fr. (1854). 
[La Teorica dei Determinanti, e le sue principali applicazioni ; 
del Dr Francesco Brioschi ; viii + 116 pp. ; Pavia. Translation into 
French, by Combescure; ix + 216 pp. ; Paris, 1856. Translation into 
German, by Schellbach; vii+102 pp. ; Berlin, 1856.] 
This, the second separately published text-book on determinants, is mainly 
on the same lines as the first, but is marked by greater attention to verbal 
and logical accuracy. It consists of an historical preface and eleven short 
chapters or sections, seven of the latter being devoted to determinants in 
general, and the remaining four to special forms. 
Sylvester’s umbral notation is given in the form 
j «] «2 ‘ * a n I 
I a 2 * • • a n | , 
