496 ON A PROPERTY OF COMMUTANTS. [369 
The property in question is a generalization of a property of determinants, viz. we 
have 
2AA' , \/jl' + X'fjL, \v' + AT,. . 
A/T + , ¡.lv' 4- fXV, . . 
\v + AT, jjuv' 4- ¡XV , 2vv' , 
= 0 
whenever the order of the determinant is greater than 2. 
To enunciate the corresponding property of commutants, let 
i \l, Xb.. 'j 
1^ \ 
l • ) 
or, in a notation analogous to that of a commutant, 
" + A + " 
1 l 
2 2 
P P - 
denote a function formed precisely in the manner of a determinant (or commutant of 
two columns), except that the several terms (instead of being taken with a sign 
4- or — as above) are taken with the sign 4-: thus 
each denote 
f A u A 12 1 
- + A+ " 
1 ( or 
1 1 
(A-21 A22J 
_ 2 2 _ 
An A22 T A 12 A21. 
This being so, the theorem is that the commutant 
111.. w 
222 
_ p p p 
where 
A 
v st . . (9) 
whenever p > 6, is = 0. 
' A lr , 
A ls . 
.(0)> 
- t\ +- 
Aar, 
A 2s 
V 1 
s 2 
• 
t 3 
^ A er> 
A es 
• ) 
. p J 
To prove this, consider the general term of the commutant, viz. this is