150 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the matrix of the adjugate has in its elements terms of the 0 th , 1 st , 2 nd , 
3 rd degrees, and thus is partitionable into 
' 1 . 
• 
r 
a 
P 
Z ' 
1 
. 
. 
— a 
• 
y 
€ 
• 
1 
• 
-p 
~y 
• 
8 
\ 
. 
1 
> 
— e 
-8 
• > 
fy 2 + 8 2 + e 2 
'll 
1 
1 
ay - 8^ 
ae + yd8 
8$ 
-e<I> 
y<P 
1 -py-< 
/3 2 + 8 2 + £ 2 
— afi — 8e 
- a£ + y8 
- 8<I> 
• 
/3$ 
ay — 8£ 
— a/? — 8e 
a 2 + £ 2 + e 2 
l 
1 
e$ 
• 
a<l> 
ae -f /3S 
— a£ + y8 
w 
1 
i 
«- + P°- + r J 
, “7^ 
— a'T* 
* . 
where d? stands for ad - /3e + , i.e. for 
i /> i 
y e 
In regard to these we may remark in passing (1) that they are alternately 
axisymmetric and skew, and (2) that the values of their determinants are 
in order 
(11) It is the one, P say, with quadric elements and the value d> 4 that 
specially concerns us. On multiplying it by itself a curious property comes 
to light, the result being 
(y 2 + 8 2 + e 2 )M — <3? 2 — (/3y +• e£)M ( a y ~ 8£)M (ae + /3&) M 
-(fi y + e£) M (/3 2 + 8 2 + £ 2 )M-3> 2 — (a/3 + 8e)M (-a£ + yS)M 
(ay— 8£)M — (a/3 + 8e)M (a 2 + e 2 + £ 2 )M — $ 2 — (/3£ + ye) M 
(ae + /}’8)M (— a^ + yS)^! — (/3£ + ye)M (a 2 + /5 2 + y 2 )M — < J >2 
where M stands for a 2 + /3 2 + y 2 + d 2 + e 2 + f 2 . If the (r, s) th element of P be 
denoted by p r8 , what we thus have reached is 
<£2 i 
Pu~ 
M 
Pl2 
Pis 
Pl4 
Pli 
P12 
Vl3 
Pl4 
2 
<l> 2 
P21 
2 i\ ,r 
P'23 
P24 
P 21 
V'i 2 
P'23 
P 2 4 
= M 4 . 
M 
T O 
P 31 
P 32 
P33 
Ps4 
Psi 
P32 
P33 ~ 
$- 
vr 
P34 
Pil 
P 42 
Pi3 
P 44 
IVl 
\ 
P41 
P 42 
P43 
P 44 — 
consequently when d> is taken 
equal 
to 0 
P 11 
P 12 
P 18 
Pl4 
2 
2 Pn 
2 Pl2 
2 fts 
2?14 
P 21 
P22 
P'2'3 
P21 
f M V 
^P'21 
~‘P k 22 
2^23 
-‘Ph 
P 31 
P32 
P33 
^34 
\2 / 
2 P31 
^P'32 
2^33 
2 P 3 4 
P 41 
P 42 
Pl3 
P44 
2p 4 i 
2 P 4 2 
2?>43 
2i>44 
• 
