380 Proceedings of Royal Society of Edinburgh. [sess. 
• 
u 3 
-«1 
« 4 
-a 2 
« 5 
-«3 
a 3 
- flq 
a Y 
— a 3 
• 
% 
- oq 
“«2 
«5 
-«3 
• 
a 2 
~ a i 
flq 
-«8 
• 
«8 
- flq 
« 4 
“«2 
a 5 
— a 3 
a 3 
«2 
-« 4 
~ a 3 
• 
«8 
- oq 
« 4 
-«2 
• 
«8 
«2 
-« 4 
flq 
-«8 
• 
«3 
— oq 
°5 
“«8 
• 
a 3 
— « 5 
«2 
- a 4 
flq 
-« 3 
• 
which, 
again, by Cayley ? 
s theorem is 
equal to 
| 
« 8 - 
flq 
« 4 ~ 
«2 
% 
« 3 
. 
« 8 “ 
^5 
2 
« 3 “ 
«1 
« 4 ~ 
a 2 
a 5 ~ 
«8 
• 
« 8 “ 
flq 
a 4 - 
« 5 - 
a 3 
« 3 - 
flq 
« 4 ~ 
a 2 
flq 
. 
(13) Another mode of attaining these results is worthy of atten- 
tion, not merely because of the interest attaching to a different 
method, hut because it gives the square in a totally different form. 
Taking, as before, the circulant of the 7 th order, and observing 
that it is centro-symmetric, we resolve it at once into two deter- 
minants, viz., 
oq 4- oq 
flq -4- Oq 
oq 4- oq 
« 4 
a, - 
oq 
a 2 
CO 
e 
1 
flo 
- oq 
a 2 4- a s 
flq 4- flq 
oq 4- oq 
a 3 
A 
o 
T 
oq - 
a <y 
a , 
- oq 
oq 
-« 4 
oq 4- oq 
flq 4- oq 
oq 4- oq 
«2 
A 
o 
1 
A 
2 oq 
2oq 
2% 
oq 
> 
a 3 ~ 
a i 
oq 
-« 4 
oq 
— flq 
The former of these, however, is equal to 
(oq 4- 2 ci 9 4- 2 a 3 4~ 2 u 4 ) 
oq 4- oq 
flq 4” Oq 
oq4-oq 
1 
oq 4- oq 
oq4-oq 
oq4-oq 
1 
oq 4- oq 
oq4-oq 
flq 4- oq 
1 
2 oq 
2a 3 
'2a 2 
1 
and, by subtracting each element of the 2nd, 3rd, and 4th rows 
from the corresponding element in the row preceding it, this is 
readily transformed into 
(tiq 4" 2a 2 4- 2 a 3 4” 2 ct^) 
&2 
a 2 + a S ~ a i ~ a 4 a S ~ a 2 
a 1 - a 2 a 2 + a 4 - a 1 - a 3 
a 2 + a±- 2a 3 oq 4- a 3 - 2a 2 
Increasing each element of the last column by the corresponding 
