1918-19.] 
Factors of Circulants. 
41 
V. — Factors of Circulants. By Professor W. H. Metzler. 
(MS. received December 18, 1918. Read May 5, 1919.) 
1. Among the well-known theorems for the breaking up of a circulant into 
factors are the following : — 
Theorem I.* * * § . — The circulant 
C(aq, &2 , . . ., CL„) = Il(tq + (X>2@ + + . . . + CL n O n 1 ), 
where 0 is one of the nt h roots of unity. 
Theorem Il.f — 
Q/(CL i , &2 j • • • j ^n) ~ iP l ”b ^2^ d" +••••+ Cl n O n 1 )(A^ + A 2 $ w 1 + Ag0b 2 4* ... + A n $), 
where A k is the signed complementary minor of a k in the first row of C. 
In the case where 0 = 1, the cofactor of s = a 1 + a 2 + . . . +a n in C is 
A x + A 2 + . . . +A n , and has been expressed in persymmetric form by 
Catalan, J and when the order of C is odd, in symmetric form by Muir.§ 
A similar theorem is also true for circulants of even order. That is, the co- 
f actor of s . s', (s' = a x — a 2 + a z — . . . ), in the circulant C(a 1 , a 2 , . . . , a n ), 
of order 2 n + 2, can be expressed as a symmetric determinant of order n. 
A proof quite similar to that used by Muir for odd orders may be used to 
show this. 
Theorem III. 1 1— Every circulant of order 2m can be expressed as the 
product of a circulant and a skew circulant, each of order m. 
Thus, 
C(cq , ? • • • 5 m) — G(flq + , (%2 d~ ®"m+2 j • • • j ^ m ~b ^2m) ^ 
G (fq — Q'm + 1 j ^2 ~ ^m+2 j • • • j — ^ 2 m)- 
Theorem IV.1I — A circulant of order n = r .m can be expressed as a 
product of m circulants of order r, each involving an mth root of unity. 
* Catalan, “ Recherches sur les determinants,” Bull, de VAcad. R. des Sci., etc., de 
Belgique , 1 ser., xiii. 
t Stern, “Einige Bemerkungen liber eine Determinante,” Crellds Journal, lxxiii, 
pp. 374-380. 
f Catalan, loc. cit. 
§ Muir, “ On Circulants of Odd Order,” Quarterly Jour. Math., xviii, pp. 261-265. 
j| Scott, “Note on a Determinant Theorem of Mr Glaisher’s,” Quarterly Jour. Math., 
xvii, pp. 129 -132. 
IT Torelli, “ Sui determinant! circolanti,” Rendic. Accad. delle Sci. Fis. e Mat. (Napoli), 
1882, pp. 3-11. 
