1904-5.] Vanishing Aggregates of Determinant Minors. 859 
If 
123456 
123456 
is axisymmetric, the aggregates 
2 
345 
126 ’ 
2 
245 
126 ’ 
123 
124 
all vanish, and therefore the aggregate on the left vanishes. It 
will be observed that this is entirely independent of the elements 
in the seventh and eighth rows and columns, which may be 
any whatever. This is the example given in article 23 of Muir’s 
paper. 
We may obtain various other interesting results as special 
cases of this same theorem. For instance, let us form a 
determinant of order 2 p + q according to the following diagram : 
p + r 
( 0 ) 
p + q - r 
(B) 
(X) 
(B) 
(X) 
(0) 
(0) 
(B') 
j>p + A: 
r — k 
h 
> p + q - r - h 
p + k r- k h p+q- r-h 
where in the upper left-hand corner there is a square of p + r rows 
and column of zeros. The complementary minor of this square 
has h rows and h columns of zeros, and for convenience they are 
represented as the first h rows and columns of the minor, but may 
be any h whatever. The remaining elements of the minor are 
