930 Proceedings of Eoyal Society of Edinburgh. [sess. 
of the numbers 1,2, . . . , n , and the positive or negative 
sign is to be taken according to the well-known dichotomous 
law.” 
An obvious extension of the notation is also indicated, whereby 
what he calls “ compound ” determinants may be appropriately 
represented. Since, in accordance with the foregoing, 
i: ;} 
is used to denote 
he considers that 
aa . b/3 — a/3 . ba 
{ 
ab cd ) 
a/3 yS ( 
“will naturally denote 
that is 
ab cd ab cd 
a (3 y8 y8 a/3 
(aa x b/3 | ( cy x dS)j 
(a/3 x ba)\ x 1 - (cS x dy)\ 
And in general the compound determinant 
) J 
( (ay X bij) > 1 
r (ca x d/3) a 
f - 1 
( - (aS x by)\ x 1 
I - (c/3 x da) f 
a 1 b 1 . . l Y 
bo 
. L 
a l fil * ’ ^1 $2 * ' ’ ^2 * * 
will denote 
a r (3 r 
... A, 1 
i a i \ • 
"h \ 
x , a 2 J . 
1 
-X 1 
a r b r . 
„ l > 
• “«! ft J ' 
’ a e. 2 • 
•• v 
1 
a 6 r Po r ' 
where, as before, we have the disjunctive equation 
0 15 0 2 , ... ,0 r = 1,2, ... , r .” 
As an example of the power of this notation he gives the theorem 
a Y 
a 2 . . . 
a r 
a r + 1 
a x a. 2 . 
. . a^ Ojy | o 
a 1 a 2 . 
• • u r a r + s 
«i 
a 2 . . . 
a r 
a r + 1 
a l a 2 • 
• . a r a r _^_2 
.... a, a 2 . 
■ • a r a r+s 
= 
i a Y a 2 
. a r i 
t s_1 
> x 
(a x a 2 ... 
a r a r+ 1 a r+ 2 . . 
. a r+s [ 
1 a i a 2 
. a r 
1 
1 a i a 2 . • • 
a r a r a r _^2 • • 
• a r+s ( 
adding, in his enthusiasm, that without the aid of his “ system of 
umbral or biliteral notation, this important theorem could not be 
