1915-16.] The Theory of Circulants from 1880 to 1900. 171 
at once in cyclical procession. For example, if we desire to have the 
factor of C(a 1 , a 2 , . . . , a 10 ) corresponding to the factor & 4 -}-cc 3 +x 2 + £ + l of 
x 10 — 1, and know from the table that the first element of it is a 7 — a 8 + a 2 — a 3 , 
we have the whole of it immediately, namely, 
| + C&2 — d 3 dg — CIq + dg — d 4 dg — d-^Q "1“ d 4 dg d^Q dj -I - dg dg 
dg - dg + dg - d 4 dg — djg + d 4 - dg djQ - ^ + dg - dg dj - dg + dg - ^ 
dg “ djg + d 4 - dg djg ~ dj + dg - dg dj - dg + dg ~ dy d 2 - dg + d^ - dg 
dio - dj + dg - dg dj - d 2 + dg - d 7 d 2 - dg + d 7 - dg dg - d 4 + dg - dg . 
The process is seen to be general, and to consist in deducing from the 
equation of higher degree, by means of the other equation, an equation 
of lower degree than the latter, and in then using cyclical substitution 
and elimination. 
The next part of the paper, however, makes it clear that the same 
results can be obtained, when known, by operating on the circulants 
themselves in accordance with the laws of determinants. Thus, from the 
eight-line circulant it is shown how to remove the second linear factor 
after the first, and how then to resolve the remaining six-line determinant 
into the two previously found persymmetric determinants of the second 
and fourth orders respectively. 
Finally, the results are applied to the special case of circulants whose 
elements, when the first is left aside, read forward the same as backward : 
and it is shown that, when from such circulants the rational linear factor 
or factors are removed, the remaining factor is a complete square. This is 
possible from the fact that the persymmetric cofactors above obtained are 
then zero-axial and skew. Thus it is found that 
C(«1 5 ^2 > a 3 > a i > 5 > a z) 
= (ctj + 2 a 2 + 2a 3 + 2a 4 ) 1 a 3 - a 4 a 2 - d 3 a x - d 2 
a 2 -a x 
a 3 
d 2 
a 3 “ ^4 a 2 — % 
ci x — d 2 
a 2 - 
a l 
a Z ~ a 4 
a i~ 
d 2 
a 2 — 
d 3 
a 3 ~ 
a 4 
and that 
C(«i j &2 > a 3 ’ a 4 > a 5 J ^4 > 
= (d x + 2 a 2 + 2 a 3 + 2a 4 + a 5 )(a 1 - 2a 2 + 2a 3 - 2a 4 + a 5 ) 
d 3 -di a 4 —d 2 
a 5~ 
d 3 
a 3~d 1 
a 4 - 
d 2 
d 5 -d 3 
a 3 ~ 
a i 
d 4 -d 2 
«5 - a 3 
a 3 ~ a \ 
a 4 -d 2 
a 3 ~ a 1 
