A Theorem of Frobenius’. 
343 
1921-22.] 
term on the left, and that the purpose served, as it were, by all the other 
terms on the left is to nullify the undesired remaining part. 
(3) In necessary illustration of the theorem and of what has just been 
said, let us consider the case where n = S, r = S, s = 4, the determinant from 
whose minors all the second factors D^' are taken being 
I 1 ’ 
and the minor array of it that provides all the first factors being 
«2 ^3 ^4 
Seeking first the right-hand member of the equality, we see it to be 
a, 
00 
h 
h 
. 
^7 
«8 
rfl 
d=i 
cl. 
d, 
dg 
K 
*2 
h. 
K 
h 
Jiq 
h, 
hg 
Then on the left our first term of the equivalent is the product 
1 . 1 \ , 
and this is followed by twelve products like 
eighteen products like 
+ i I • i I ’ 
and finally four products like 
- 1 H ^-2 P 3 I • I ^465/6^7^8 1 • 
A mere glance at these suffices to make clear that thirty-four out of the 
thirty-five products contribute no part of the equivalent of the gnomon-like 
determinant G, and that their net effect is merely to cancel that part of 
the first product which is not a part of G. 
In passing it may be noted that had we taken n equal to 7 or 6, the 
number of products on the left would not have been altered, and further 
that in the latter of these two cases G would be 0, and the equality would 
revert to a very old type in the history of our subject, namely, that of a 
vanishing aggregate of products of pairs of determinants. 
(4) With less symbolism, therefore, Frobenius’ theorem may be enunci- 
ated as follows : — If an aggregate of ijroducts he formed in which each 
first factor is a minor of a fixed rectangular array of | a^, | , and the 
corresponding second factor is the co factor of the first in | aj^ | , every minor 
