1904 - 5 .] 
Dr Muir on General Determinants. 
943 
‘ determinant of a substitution ’ : and on the following page there 
occurs the passage — 
“ Quindi dal noto teorema della moltiplicazione delle 
determinant e facile dedurre che, se si chiama A la deter- 
minante del prodotto delle due sostituzioni ( h ) e (li) [whose 
determinants are D and D'], avremo 
A =D1)'; 
cioe la determinate del prodotto di due sostituzioni e eguale 
al prodotto delle loro determinanti .” 
Bruno, F. Faa di (1852, May). 
[Demonstration d’un theoreme de M. Sylvester relatif a la 
decomposition d’un produit de deux determinants. Journ. de 
Liouville (1), xvii. pp. 190-192.] 
The theorem is that which appeared in the Philosophical 
Magazine for August 1851, and which is there formulated in the 
umbral notation as follows : — 
| a, a 2 ... . a n 
lb 1 b 2 ... . b n 
i x { ? a J ■ ■ 
. . CL n | 
J \ /?! P 2 . . 
• • Pn ) 
a^ a 2 a n 1 j a 2 a n 
&i b 2 ... bp @0p + 2 • • ■ @6n * ' fio j 2 • • • fiop b p+1 bp^_2 • • • b n 
where 6 1 , 0 2 , . . . , 0 n are ‘ disjunctively ’ equal to 1 , 2 , ... n . 
Faa di Bruno prefers to write it in the form 
(±«? 
01 ™02 
^ . . . a*") • 2( + sf 1 . . . b* n ) 
0 n / V — 1 2 p n / 
, 0 p J*p + i „'!'p + 2 
P + 1 P + 2 
using two other sets of letters like 0 1} 0 2 , . . . 0 n . This change 
in notation being allowed for, the new proof is in general character 
exactly the same as the old ; it is, however, more concise and more 
clearly set forth. It starts with the fact that any term arising 
from the expansion of the typical product on the right-hand side 
may be written 
01 02 
a 1 
d>p \fj p+ 1 tin i//! jp2 
p p+\ n 01 0-2 
npp 702>+l 7 0P+2 7 0n 
°0p °0p + 1 %+ 2 * ‘ * °6n 
