13] 
ON THE THEORY OF LINEAR TRANSFORMATIONS. 
81 
I have already employed the notation 
a , /3 , 
7 > 
8 ,... 
/3', 
7 > 
3', 
... 
// 
7 . 
8", 
(1) 
(where the number of horizontal rows is less than that of vertical ones) to denote 
the series of determinants, 
•(2), 
a , /3 , 7 , 
/3', 7', 
a", /3", y", 
which can be formed out of the above quantities by selecting any system of vertical 
rows; these different determinants not being connected together by the sign +, or in 
any other manner, but being looked upon as perfectly separate. 
The fundamental theorem for the multiplication of determinants gives, applied to 
these, the formula 
(3), 
A , 
B, 
G, 
D ,... 
= E 
a , /3 , 
7 > 
fi,... 
A\ 
B\ 
O', 
D', 
a', /3', 
/ 
7 > 
S', 
A", 
B\ 
G", 
D", 
a", /3", 
// 
7 > 
B", 
where 
A =\ol + X'cl +\"a" + ... ^ 
B =A/3 + A73' + A"/3" + ... 
A — ¡Ji OL fX OL fjt, OL -|-... 
B'=m,3+ / *'£' + /08" + ... 
&c. 
E= /j,, ... 
V, //, 
(4), 
•(5), 
and the meaning of the equation is, that the terms on the first side are equal, each 
to each, to the terms on the second side. 
This preliminary theorem being explained, consider a set of arbitrary coefficients, 
represented by the general formula 
rst (6), 
in which the number of symbolical letters r, s,... is n, and where each of these is 
supposed to assume all integer values, from 1 to m inclusively. 
C. 
11