60 Proceedings of tlie Poyal Society of Edinburgh. [Sess. 
V - *■ 
fl^l2 2 
n ^ 
0-13 
a H a i3 
a \2 a i3 
a H a i2 
a ii a i3 
a i2^13 
a 21 
a 23 2 
$2J $99 
$ 21 $9 3 
^22^23 
a 2\ a 22 
^21^*23 
a 22 a 23 
a z\ 
^32^ 
a 33 2 ~ ^ 
$31$ 32 
a 3\ a 33 
a 32 a 3'3 
a 3\ a 32 
a 31 tt 33 
a 32 a 33 
a ii a 21 
$j9$99 
a i3 a 23 
$H$99 A 
a ii a 23 
$^9$2 3 
$12$91 
a i3 a 21 
^13^22 
a n a 3\ 
$^9$ 3 9 
a i3 a 33 
$H$g9 
a i\ a 33 ~~ ^ 
«12«33 
a i2 a 31 
a i3 a 31 
a \3 a 32 
°2\ a 3\ 
a 23 a 33 
Cl 2\ a 32 
a 2i a 33 
a 22 a 33 ~ ^ 
a 22 a 31 
a 23 a 31 
a 23 a 32 
a 2i a il 
a 22 a i2 
a 23 a i3 
a 2i a \2 
Cl2\ a i3 
«23^i3 
ctc)C)Ct 1 1 A 
a 23 a il 
a 23 a i2 
a 3i a il 
^32^12 
a 33 a \3 
^31^12 
a 3\ a \3 
a 3 2^13 
$ 39 $ if 
^38^11 — ^ 
a 33 a \2 
a 3i a 21 
$g9$99 
a 33 a 23 
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a 3i a 23 
a 32 a 23 
^32^21 
a 33 a 2 1 
a 33 a 22 
which by performing the following operations 
then 
Col. 4 + col. 7 ; col. 5 + col. 8 ; col. 6 + col. 9 ; 
Row 7 - row 4 ; row 8 — row 5 ; row 9 — row 6 ; 
gives the result 
Djj A j . | A a A | — 0, 
and since the roots of \Au — \\ = 0 are g 1 g 2 , g ± g 3 , g 2 g 3 , it follows that the 
roots of | D ti -A | = 0 are g 2 , g 2 , g 3 2 , g 3 g 2 , g 1 g 3> g z g 3 . 
If we denote the square of A (row by column) by A 2 , and if | A*, 2 — A | = 0 
denotes the determinantal equation formed by subtracting A from the 
elements along the principal diagonal of A 2 , then, since the roots of 
Aii 2 — A | = 0 are g j 2 , g 2 2 , g 3 , it follows that 
D t i — A | = | Ajj 2 — A | . | An — A 
Syracuse University, 
April 1917. 
(. Issued separately April 16, 1918.) 
