The theory of determinants in the historical order of development, by Sir Thomas Muir.

CIRCULANTS (ROBERTS, 1861)

373

ROBERTS, M. (1861).
[Question 581. Nouv. Annales de Math., xx. p. 139. Solution by
E. Beltrami in (2) iii. (1864, February), pp. 64-66.]
Roberts' theorem concerns the circulant C whose elements are
the terms of the expansion of
(t + 1)n - 1
t
and is to the effect (1) that there are no odd powers of t in the
development of the circulant, and (2) that if in the said development
S be put for t2, the equation in e
Ct2= = 0
has for its roots the squares of the differences of the roots of the
equation
xn — = 0.
If W1, w2,... be the ntl roots of 1 we have identically
(t-Ol)(t-. 2)... (t-.n) = tn-1,
and therefore also
{t-(WI-Wr)} {t-(W2-0)} * * {t- (n -()} = (t +)) nwhere the rt' factor on the left is simply t itself. Hence the
expression
(t + l)n-l (t + o2)n-l (t o)n-1
t t t
consists of n(n —1) factors, which, if suitably combined in pairs, are
replaceable by,n(n-1) factors of the form t2-(wo —w)2. But the
said expression being equal to C by Spottiswoode's theorem (which,
however, Beltrami does not assume*) the desired result at once
follows.
ZEHFUSS, G. (1862).
[Anwendungen einer besonderen Determinante. Zeitschriftf. Math.
u. Phys., vii. pp. 439-445.]
Zehfuss proves Spottiswoode's result by multiplying the rows in
order by OC, 0~,-1.., 0r respectively, and the columns in order by
1, 0,..., -1, a procedure which amounts to multiplying the
* For his mode of proof see under Baltzer (1864).