32 Proceedings of the Royal Society of Edinburgh. [Sess. 
become 
0 0 0 0 c-b+f-e d-c+a-f 
0000 d-c+a-f e-d+b-a 
and so by a final step we reach the full resolution corresponding to the 
resolution 
x 6 — 1 = (afi - a? + x - \){x 2 + x + 1), 
and with both determinant factors in persymmetric form. 
(18) A little additional interest attaches to this mode of resolution 
when the two factors are of the same degree ; for example, the resolution 
P(<z + 2h + 2c + d , . . . ) . P(a - 2b + 2c - d , ... ) 
corresponding to 
x 6 —l = (x 3 — 2x 2 + 2x - 1)(£ 3 + 2x 2 + 2x + 1) . 
In such a case any element of the one persymmetric determinant and 
the corresponding element of the other are such that it is possible to view 
them as the sum and difference respectively of one and the same pair of 
quantities; for example, the elements in the (ll) th places are 
(a + 2c) + (b + 2d) , ( a + 2c) - (b + 2d) . 
This peculiarity at once recalls Zehfuss’ theorem regarding centro- symmetric 
determinants, from which we at once obtain 
a + 2c 
b + 2d 
c + 2e 
f+m_ 
e + 2 c 
d + 2b 
b + 2d 
c +2e 
d + 2f 
a + 2e 
f+ 2d 
dh 
+ 
<55 
c + 2e 
d + 2f 
e + 2 a 
b + 2f 
a + 2e 
f+2d 
f+2d 
a + 2e 
b + 2f 
e + 2 a 
d + 2f 
c + 2e 
e + 2c 
f+ 2 d 
a + 2e 
d + 2f 
c + 2e 
b + 2d 
d + 2b 
e + 2 c 
f + 2d 
c + 2e 
b + 2d 
a + 2c 
as an equivalent of the product of the two P’s, and therefore an equivalent of 
C (a ,b,c,d, e ,f) . 
(19) Lastly, it has to be noted that each one of the persymmetric 
determinant factors has alternative forms which are obtainable by cyclical 
substitution ; for example, corresponding to x 3 — 1 we have 
a - d 
b - e 
£-f 
b ~e 
e-f 
d - a 
b - e 
c~f 
d - a 
= - 
d — a 
e-b 
c -f 
d -a 
e-b 
d — a 
e-b 
f~ c 
In the case of an mline circulant with an m-line persymmetric factor 
the number of such alternatives is n if 2m — 1 is prime to n ; otherwise 
it is less. 
Kondebosch, S.A., 
llth September 1919. 
{Issued separately February 3, 1920.) 
