1904 - 5 .] Vanishing Aggregates of Determinant Minors. 855 
As this has the effect of increasing the subscripts of every 
element in each minor of the identity by a + /3, the truth of the 
theorem follows. 
Theorem IY. Given any identical relation between the minors of 
order m of a persymmetric determinant of order n, where k 
of the row numbers are invariant ( the same for each minor), 
we may obtain another identity by increasing by a each of 
the other m — k row numbers in each minor , leaving the column 
numbers the same. 
The effect in this case is to increase by a the subscript of each 
element in each row of every minor except in those rows whose 
row numbers are invariant throughout the identity. If we expand 
by Laplace’s theorem each minor of the identity in terms of minors 
of order k, with their complementaries, formed from the k in- 
variant rows, then since these complementaries are minors of a 
persymmetric determinant, it follows that the total coefficient of 
any minor of order k formed from the k invariant rows vanishes,* 
and the theorem is established. 
Theorem Y. Given any identical relation between the minors of 
order m of a persymmetric determinant of order n, where k 
of the row numbers and h of the column numbers are in- 
variant (k + h<n), we may obtain another identity by 
adding a to each of the other m - k roio numbers and /3 to 
each of the other m - h column numbers. 
The original identity in this case is the extensional of another 
having either no column numbers or no row numbers invariant 
according as h>k, and therefore the theorem follows as in IY. 
It is obvious that instead of increasing the numbers as has been 
supposed in these theorems, we might have diminished them. It 
is also obvious that certain restrictions are placed upon the 
magnitude of a and /3. For instance, in theorem I. the largest 
row number plus a^f>n, and similarly for the others. 
4. If now we apply these theorems to the vanishing aggregates 
for axisymmetric determinants, we get results true for per- 
symmetric determinants, but in general no longer true for axi- 
symmetric determinants. For instance, if we take the Kronecker 
aggregate, 
Metzler, l.c., paper I., theorem (1). 
