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

PERSYMMETRIC DETERMINANTS (WHITWORTH, 1865) 319
which lead to the determination of the /'s and the finding of the
actual scale as enunciated. Now, by multiplying the series by the
scale of relation, it naturally is found that the product, unlike
the multiplicand, is limited in the number of its terms, and we have
then little more to do than to take the quotient of this product by
the scale in order to reach the desired expression for the sum.
If the idea of a 'formula of recurrence' be used instead of a
'scale of relation,' the first of these results takes the form: If
cG, c1, Cg,... be a recurring series of the r"t order, the recurrenceformula is

Co. C1
cl c2
Or-1 Cel
Cn -r Cn -r+1

C2
C3
cr+1
Cn -r +2.. c1. Cr+1
= 0.
C2r —1.. Cl

Zeipel's memoir we deal with under its own heading (Chap. XX.);
its single persymmetric determinant
(mn)34.,. (m)p+2-l ('-2.* '* * (m )
(m%)+-l.- (m),._. (n)).-_3.... (n))-l
o(? ),+r-2 ( +, _3 (o ),,+,o -4 I.. ( )-2
(mt), (n), l_ (?0),)-2.... (M)cannot well be considered apart from other forms.
FONTEBASSO, D. (1872).
[I PRIMI ELEMENTI DELLA TEORIA DEI DETERMINANTI,.
viii+134 pp Treviso.]
Fontebasso proves that the n-line determinant whose elements
in the secondary diagonal of the complementary minor of the
(n, n)th element are zeros and all the others units is _ 1. He does
not observe that the set of operations
row-, rown_, row,-ro, ro....
gives at once the result
(_1)?(,-i.