382 
Proceedings of Royal Society of Edinburgh . [sess. 
and therefore into 
ftl ftg 
a 2 
-a 4 
l 
CO 
e 
«5 
a 1 
a 3 
a 
to 
1 
a 4 
a 3 a 5 
a 2 -a 4 
a i 
-a 5 
a 2 
a 4 
a 2 
-a 4 
a l ~ 
a b 
a 2 - a 4 
a 3 — a 5 
a 2 
-a 4 
oq - 
a 3 
i 
~ a 5 
a 2 - 
a 4 
ftj “““ ft^ 
and are thus identical, as was to he proved. 
By comparison with the result of § 12, we obtain the identity 
a 4 - a 3 a 2 — a 4 a 3 - a 5 
&2 ft 4 ft^ ^5 ft 4 
0^2 ft 4 ft^ ft^ 
ftg ft^ ft 4 ft2 ft^ 
ft. 
<x 3 
ftj ft 4 ft2 ft^ ftg 
ft.q ftj ft^ ft2 ftp^ ft^ 
ft ^ ft 4 ft 2 
ft^ ft-^ 
The determinant form has the advantage of showing that the 
factor is further resolvable, for being centro - symmetric it is 
equal to 
a i ~ a 3 + a s ~ a 5 a 2 ~ a \ 
9 a 4 "I - a^9 <x 4 a 4 
a 
K - % - % + a 5 ) 
{(ctj - a b ) 2 - 2 (a 2 - af 2 }{a l - 2« 3 + a 5 ) : 
i.e., 
so that 
C (a 1 a 2 a 3 a 4 a 5 a 4 a 3 a 2 ) 
= (a 4 + 2 a 2 + 2 a 3 + 2 a 4 + a 5 ) • (a 4 - 2 a 2 + 2 a 3 - 2a 4 + a 5 ) 
• (a 4 - 2a 3 + a 5 ) 2 • {(oq - a 5 ) 2 - 2 (a 2 - a 4 ) 2 } 2 . 
15. Finally, it must be observed that, although it has been 
shown that 
C(tq , , «2n-l) ^ ( a i d 1* ®2n-l) 
and C(oq , • • • • , a 2n ) (oq + • • • + a 2n )(oq — a 2 + • • • — a 2n ) 
are expressible as persymmetric determinants, which become zero- 
axial skew determinants when a 2 , • • • • , a 2n _ 1 and a 2 , • • • , a 2n 
are the same on being read backwards as on being read forwards, 
it has not been formally proved that all the factors of these per- 
symmetric determinants can also be expressed as persymmetric 
determinants having the same property. This, however, would 
seem to be the case. 
