377 
1896 - 97 .] Dr Muir on Resolution of Circulants. 
Similarly, by adding to each element of the 1st row the correspond- 
ing element of all the odd-numbered rows, and thereafter subtract- 
ing the corresponding element of all the even-numbered rows, the 
second factor can be removed, with the result that the co-factor 
left is 
0 - 
-1 
1 - 
1 
1 
-1 
1 
- 1 
1 
a L 
«2 
a 3 
a 4 
a 5 
a 6 
1 
a 8 
a 1 
a 2 
a. 3 
a 4 
a 5 
«6 
1 
a* 
i 
Cl 8 
a 1 
(i 2 
«8 
a 4 
1 
a a 
a 7 
a s 
a i 
«2 
u 3 
a 4 
1 
% 
a 6 
a 7 
a 8 
a l 
« 2 
a :] 
1 
a 4 
a 5 
% 
a 7 
«8 
a 1 
a 2 
1 
«3 
a 4 
a o 
a 6 
<h 
a 8 
a x 
cting 
from 
each 
element 
of the 
2nd 
row 
corresponding element of the 3rd, and from each element of the 
3rd the corresponding element of the 4tli, and so on, we obtain 
-1 
1 
— 
1 . . 
. 
- 1 
<h 
— a s 
a 2 
- a Y 
do 
o 
- « 2 • • 
. a 7 - 
-«6 
a s 
- a 7 
a Y 
-a 8 
a 2 
- a Y . 
• • - 
a 7 
- a <0 
- a 7 
a l 
- a 8 . 
• • « 5 “ 
~ a 4 
a 6 
-«5 
a 7 
— a Q 
a s - 
- a 7 
• . « 4 - 
~a s 
a b 
-a 4 
a 6 
a 7 - 
~ a G • • 
• • « 8 - 
- a 2 
a 4 
— « 3 
a 5 
-a 4 
a Q ' 
’ « 5 • • 
• • Cl 2 - 
- a Y 
and, similarly, by adding to each element of the 1st column the 
corresponding element of the 2nd column, to each element of the 
2nd column the corresponding element of the 3rd column, and so 
on, there 
results 
$2 $g 
dg “ CL-^ 
a 4 
~a 2 
«5 
“«4 
a 7 
a x - a 7 
a 2 ~~ l ^8 
«3 
« 4 
-d 2 
% 
— « 3 
a 6 
-a 4 
'O 
e 
l 
00 
e 
CL - ^ CL*j 
a 2 
— a 8 
«8 
~ a i 
— « 2 
a 5 
-a 3 
a 7 - a 5 
a 8 ~~ a 6 
a x 
- a 7 
«2 
«3 
d 4 
-a 2 
€Lq ci 4 
a 7 -a 5 
«s 
-«6 
- 
«2 
— a 8 
«3 
~ a i 
a ro -a z 
ftg ct 4 
a 7 
— a b 
« 8 
-*e 
«1 
- a 7 
«2 
— a 8 
or, on reversing the order of the rows and changing the signs of 
all the elements, 
VOL. XXL 
2 B 
29 / 3/97 
