(95) 
ISTOTE ON UNIMODULAE AND OTHEE PEESYMMETEIC 
DETEEMINANTS. 
By Sir Thomas Muie, LL.D. 
(1) On pp. 201-204 of vol. xxxiv of the ' Edinburgh Mathematical 
Society's Proceedings there is given, in regard to persymmetric deter- 
minants, an interesting set of three theorems, which on more than one 
account are deserving of comment. In the first place, unfortunately they 
are all inaccurately stated in a matter of sign, and the second of them is 
presented in a form that obscures its real character. 
(2) For conviction on this latter point we have only to recall the fact 
that the determinant concerned, namely, 
«1 
«4 
«5 
H 
— ai 
% 
— (Xo 
— 
«1 
— % 
— % 
— tto 
— Ciji 
-^3 
— Cto - 
is, like all other even-ordered determinants, unaltered in substance by 
changing the signs of the main diagonal and every alternate parallel 
diagonal, this change giving 
«2 
~<^3 
«'4 
-H 
— ttj 
— ai 
0 
-^3 
— a^ 
— «3 
— 
— a-x 
do 
— til 
and showing that the negative signs belong simply to the variables a^, a^, a^, 
and do not contribute in any way to the specialisation of the determinant. 
Putting +a^, -fao, -f«5for —a^, —a.^, —a^, and writing the determinant in 
the w^ay adopted for persymmetrics — that is to say, with its columns in the 
reverse order — we obtain the equality. 
* Datta, H., " On Symmetrical Determinants and Pfaffians," ' Proc. Edin. Math. 
Soc./ xxxiv, pp. 197-204. 
