854 
Proceedings of Royal Society of Edinburgh. 
2 
1234 
5678 
= - 2 
1236 
4578 
= 2 
6781 
4523 
( 4 ) 
(b) For persymmetric determinants : 
1 k : 
: 1 l 
1 h 
h l \ 
\h k + 
\ k l j 
1 k+ 1 i 
: 1 /+1 
; 1 
h l 
! h k 
+ ! k 
( 5 ) 
( 6 ) 
Other aggregates may be obtained by applying the law of 
complementaries and the law of extensible minors to these.§ 
3. In what follows the umbral notation will be used, and the 
usual law for the sign factor of a term observed. 
Theorem I. Given any identical relation between the minors of 
order m of a persymmetric determinant of order n, we may 
obtain another identity by adding a to the column numbers 
of each minor , leaving the row numbers the same. 
The effect of increasing the column numbers is evidently to 
increase by a the subscript of each element entering the minors 
of the given identity, and hence the theorem follows at once. 
Theorem II. Given any identical relation between the minors of 
order m of a persymmetric determinant of order n, we may 
obtain another identity by adding a to each row number of 
each minor , leaving the column numbers the same. 
This has the same effect as increasing the column numbers, and 
hence the theorem. 
Theorem III. Given any identical relation between the minors of 
order m of a persymmetric determinant of order n, we may 
obtain another identity by increasing every row number by a 
and every column number by ft. 
* Nansen, “Minors of Axisymmetric Determinants,” Amer. Jour, of 
Math., January 1905. — Metzler, “Variant Forms of Vanishing Aggregates 
of Minors of Axisymmetric Determinants,” Proc. Roy. Soc. Edin., xxv., 
1905, p. 717. 
t Muir, “The Automorphic Linear Transformation of a Quadric,” Trans. 
Roy. Soc. Edin., xxxix. L.c., paper I., theorem (A). 
X Cazzaniga Tito, “Relazioni fra i minori di un determinante di Hankel,” 
Rendiconti del R. 1st. Lorrib. di sc., e lett., Serie ii., vol. xxxi., 1898. 
§ Muir, l.c., paper I. “The Law of Extensible Minors and Certain 
Determinants, ” Proc. Edin. Math. Soc., vol. xx., 1901-1902. 
