1906-7.] Dr Muir on Axisymmetric Determinants. 145 
[n]-* 
[12] .... 
[i»] 
[21] 
[22] - x .... 
[2»] 
= 0 
[«i] 
[»2] • • • • 
\nn\ - x 
are the p th powers of the roots of the equation 
(ll)-z 
(12) .... 
{In) 
(21) 
(22) - x .... 
<2») 
= 0. 
(» 1) 
(ni) .... 
(nn) - x 
The “ demonstration 
. ” leaves a 
1 good deal to be desired. 
In effect it amounts 
to saying that if 
£i > £2 > • • 
. , be the p th 
roots of unity, and D, 
rn the 
determinant got from |(11), 
(22) , . . . , (nn) \ by subtracting £ m x 
from 
each of the diagonal elements, then 
Di D 2 D 3 . . 
.D = 
[11] -x» 
[12] . . 
. . [In] 
[21] 
1 1 
LO 
to 
1 
[2»] 
01] 
02] • • 
. . \nn\ - 
Now it is well known that the multiplication of D x , D 2 , . . . , enables 
us to arrive at the equation whose roots are the p th powers in question, 
but this and Sylvester’s statement are not by any means identical. The 
separate points to be established are (1) that the element in the r th row 
and s th column of the determinant which is the product of D 1 , D 2 , . . . , D p 
consists of [rs] and a tail of other terms, (2) that this tail vanishes in 
the case of every non-diagonal element, (3) that in the case of the diagonal 
elements it reduces to — x and Sylvester’s only justificatory statements 
are that the product of the p determinants is independent of the order in 
which they are taken, and that all the terms containing f in any other 
power than the p th will vanish. 
Another true proposition made on insufficient foundation is that the p th 
power of an axisymmetric determinant is itself axisymmetric. The 
foundation here is the incorrect proposition of § 6. 
Cayley (1852, Dec.). 
[On the rationalisation of certain algebraical equations. Cambridge 
and Dub . Math. Journ., viii. pp. 97-101 : or Collected Math . 
Papers , ii. pp. 40—44.] 
The equations first concerned with are of the type 
cu 1 + b 2 + & + . • . . = 0 , 
VOL. XXVII. 
10 
