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

PERSYMMETRIC DETERMINANTS (HANKEL, 1861) 313

he points out, are fundamentally related. By dividing each row
of a persymmetric determinant by the first element of the row a
determinant of the form (P') is evolved, and conversely, by multiplying each row of (P') by the first element of the corresponding
column, persymmetry makes its appearance, the connecting equation
being
a1 2-2... a1I P'(a1, a2,..., a2-2)
= P(1, cl, a1a2, ala2a3,.... a.. 2 (2-2)
By performing on the persymmetric determinant of
a0, a,,,..., a2.nthe operations
col-col_1, ol_-co_, 1-...., col,-col1,
then on the resulting determinant the operations
Coln-Col,_-, coln_1-coli,_2,..., Co13-col2,
and so on, there being one operation fewer each time, there is
obtained the determinant
l ^o X1 A().,,.. (.n-1)
0 A AO
al (1) A(2). A(-1)
ail,-1 a 1 —1 A71-1
Next this is treated in an exactly similar way, save that it is now
the rows that are operated on, the outcome being
a(0) a() A(2) a("-1)
A(1) A() A() A('
A A* A*. * * * v vA 1
(-1) () A(+1) ('2. -2)
Hankel has thus his fundamental theorem, namely, that the
persymmetric determinant of ao, a1, a2,..., a2,_2 is equal to the
persymmetric determinant of the first members of the 2n-2 differenceseries of the a's. As illustrations he uses the well-known case