1888.] Dr T. Muir on Aggregates of Determinants. 103 
«3 
«4 a^ ttg 
^4 ^5 ^6 
h h h 
-t 
^1 ^2 ^3 
+ 
^4 ^^5 ^6 
C4 C5 Cq 
O4 Cg Cg 
^2 ^3 
and the second set 
O5 ttg 
Ct^ ^2 ^6 
a^ «3 
h h h 
&4 &2 ^6 
+ 
&4 h^ Z>3 
Cl Cg Cq 
C4 Cg Cq 
O4 Cq Cg 
Expressing each determinant of the first set in terms of the 
elements of a row and their complementary minors, viz., the first 
determinant in terms of the elements of the first row, the second 
determinant in terms of the elements of the second row, and so on, 
we obtain the nxn terms 
- «2l^4«6l + “ 3 IM 5 I. 
- h,}a^c^\, 
+ c^\aj}^\ - + CjlVsI. 
But the sum of all the first terms of the expansions is expressible as 
a determinant of the third order, so also is the sum of all the 
second terms, and so on ; the result being 
I I + I I \ 5 
as was to be proved. 
8. This new identity, it will be seen, depends for its existence on 
the possibility of a double application of a certain expansion- 
theorem ; and as this theorem is but the simplest case of Laplace’s 
expansion-theorem, we are prepared to find that the dependent 
identity likewise is widely generalisable, so as, in fact, to be co- 
extensive with the theorem of Laplace. The generalisation is as 
follows : — 
If any two determinants A and Bo/ the n^’^ order he taken, and 
from them two sets of determinants he formed, viz., first, a set of 
n(n-l) . . . (n-r-f 1)/1.2 . . . . r determinants, each of which 
is in r roivs identical with A, and in the remaining rows with B, and 
secondly, a set of the same number of determinants, each of which is 
in r columns identical with A, and in the remaining columns ivith B, 
then the sum of the first set of determinants is equal to the sum of 
the second set. 
