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

PERSYMMETRIC DETERMINANTS (SCOTT, 1879)

325

SCOTT, R. F. (1879).
[On some symmetrical forms of determinants. Messenger of Math.,
viii. pp. 131-138, 145-150.]
Beginning with the evaluation of the persymmetric continuant
K(b bb.. ) (see chap. xvii. at end), Scott passes on by two easy
stages to the result
C 06 Ct.... (t
e a a.... ac,
b c a a
)b c a.~ *a(c-b)'-b(c-a)n
b b c... a = a —
b b b.... C
adding that the determinant got from this by bordering it axisymmetrically with 0, 1, 1, 1,..., 1 is equal to
(c - a)n-1 - -(c-b)n-1
a-b
and that the determinant, no longer persymmetric, got by changing
the c's into c1,,..., cn, is equal to
a (b)- b —(), where 4 (x) = (c1-x)(c2-x)... (c,-x).
He next devotes considerable space to the determinant which is
identical with the above as regards its main diagonal and the two
contiguous minor diagonals, but differs in having a c in all its other
places. This is followed by one still more fanciful, and then he
returns to forms like Sardi's of 1868.
On account of the connection of persymmetric determinants with
Sturm's Functions, the following writings dealing with the latter
and reproducing previous results in more or less fresh forms are
worth noting:
1862. HATTENDORF, K. Ueber die Sturm'schen Functionen.
Dissert. 54 pp. G6ttingen. 2te Aufl., Hannover, 1874.