viz. we 
utant of 
a sign 
ON A PROPERTY OP COMMUTANTS. 
the general term of A rst _ is \ ar X 6g \ ct . . , where a, b, c, . . represent some permutation 
of the numbers 1, 2, 3.. 6. Substituting the like values for each of the factors 
A 1 &c., the general term of the commutant is 
= Pg it • • • ^re'i Xj,'s' Vi' • • X a " 2 X5"s' ~^c"t" • • ~^a p p ~^b p s p ^c p t p • • • 
Taking the sum of this term with respect to the quantities s', s",.. s p , which denote 
any possible permutation of the numbers 1, 2 ...p; again, with respect to the quantities 
t', t",..tP, which denote any possible permutation of the numbers 1, 2, ...p; and the 
like for each of the (6 — 1) series of quantities, the sum in question is 
which is 
i X^" a • • • ^a p p —"¿s Xy s' • • Xb p s p S ¿£ Xy p X c " ... 'KqP^p . . , 
= X a . i X № " 2 • • • X a 3q, 
rx+ “ 
"X+ " 
b’ 1 
& 1 
b" 2 
c" 2 
- b p p - 
c v p 
but p being greater than 6, since the numbers b!, b", ... b p are all of them taken out 
of the series 1, 2 ... 6, some of these numbers must necessarily be equal to each other, 
and we have therefore 
X* 
6' i 
b" 2 
*- bvp-* 
= 0; 
whence finally the commutant is =0. 
In the case where p = 6 = 2, we have for a determinant of the order 2 the theorem 
2XX' , 
\fjb + X'fi = — 
\fJL + X'/U, 
2gg ! | 
A, fi 
and it is probable that there exists a corresponding theorem for the commutant 
where 
L '/' St .. (p) 
r A f 
1 ii..(p) 
2 2 2 
L p p P -J 
^ X lr , X ls ... (p) 
x 2r , X 2S 
v \p r , X^ 
but I have not ascertained what this theorem is. 
Cambridge, October 26, 1865. 
C. V. 
^ _ 
• + x + - 
r 1 
- 
s2 
t 3 
) 
L • P J 
63