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

CIRCULANTS (CATALAN, 1846)

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on a A'= (n - 2) 2-A". Enfin, d'apres le n~ 13, et en observant que
les quantites A - A2, A2 - As,... ont ici change de signe
A" = n.
On deduit, de ces diverses formules
A = (n - 2)(- 2)This result, which at a later date would have been written
C(-I 1, 1,....,) = (n-2)(-2)n-l,,
and which, we may point out in passing, could also be reached by
the operations
rowl + row2 +... + rown,
removal of factor n - 2,
rown - row,,_, rowt1 rOWn - 2...
is then attempted to be generalised (~18) by withdrawing the
restriction as to the number of negative units in a row. The
reasoning, however, seems to have been incautiously conducted,
the extension arrived at being
C(-1, -1,.., 1, 1,_ = (-2p)(-2)1-1,
where the number of consecutive negative units in the first row
is p, and the number of positive units n-p.
Catalan then passes (~ 19) to the consideration of the similar
circulant whose first row consists of p consecutive positive units
followed by in-p zeros, separating the investigation into two
parts, (1) the case where p and n have a common factor other
than unity, (2) where they are mutually prime. In the former
case he shows that the equations which have the circulant in
question for determinant are "indeterminees ou incompatibles";
in the latter case he shows that the equations are determinate.
He thereupon goes on to supplement the information in the
second case by proving that the circulant is equal to p: he omits,
however, any similar proof that in the first case the circulant is
zero.*
*Catalan proposed a question on this subject in Nouv. Annales de Math. xv.
(1856), p. 257, and returned again to it in Nouv. Corresp. Math. iv. (1868), p. 78.