206 
Proceedings of the Royal Society of Edinburgh. [Sess. 
calls it the “complete determinant of the i th class,” a name equivalent there- 
fore to the more modern “ (n — i) th compound.” 
One of the notations employed is essentially the same as Sylvester’s — 
that is to say, he uses 
J li 2 1 . . . 1 2 2 2 ... u 2 1» 2 n ... u n \ 
L 1 1 2i ... 1 2 2 2 ... u 2 l n 2 n . . . u n J 
for what at a later date would have been written 
li 2 1 . . . u-j 
\ l 2 l ... u Y 
1 2 2 2 
. . . u c 
| R ^ n • • • 
f n ^ n • • • 'LL r 
The other notation is his own, and is worthy of careful note. It differs 
from Sylvester’s in making use of the row-numbers and column-numbers 
not of the retained elements but of the elements omitted, the said numbers 
being enclosed in brackets for the purpose of recalling this difference. 
“ Thus,” he says (p. 352), “ the complete compound determinant of the first 
class may be written 
that of the second class 
Id) (!) 
{(! D (! 
(I)}' 
C 5 ) 
\2 3/ 
; ; 
and generally that of the i th class 
f /I i i-2 • • • 2 2 . . . 2 a /7q ^2 • • • 
1 2 ••• 1/ % 2 2 ... 2j V Ml H ... lx Jf i 
_ n(n -l) ... (n-i+l) „ 
where 
1 ■ 2 
and where, it should have been added, r stands for the s th integer in the 
r th combination of i integers taken from 1, 2, . . . , n. 
These preliminaries having been attended to, a discussion of the 
properties follows. The first five pages (pp. 353-358) and two later pages 
(pp. 366-368) are mainly concerned with compound determinants of the 
first class (that is to say, with the adjugate determinant), and they do not 
break fresh ground. The same, however, cannot be said with reference to 
the next two pages (pp. 358-360), which concern those of the second class. 
The result first reached is that the complete determinant of this class, namely, 
or A 2 , say, 
