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

CIRCULANTS (GLAISHER, 1872)

379

it is found that
AUo P.2 4. '-1
- _-1 f o rV. n-2
-4'n-2 -..-. * *. p.-3 = 1.
- 1 - k - -V 3.. 0
It is properly pointed out that the case of the q's where n = 3
had been obtained in 1827 by Louis Olivier (see Crelle's Journ.,
ii. pp. 243-251).
GUNTHER, S. (1875).
[LEHRBUCH DER DETERMINANTEN-THEORIE,... viii+236 pp.
Erlangen.]
Giinther devotes two pages (~ 11, pp. 93-95) to the subject, but
they contain nothing fresh save the proposal to call the determinant
'doppeltorthosymmetrisch,' a quite unsuitable name which accurately describes a very different special form.*
BALTZER, R. (1875).
[THEORIE UND ANWENDUNG DER DETERMINANTEN. 4te verbesserte Aufl..... viii-247 pp. Leipzig.]
The only fresh matter in this edition is the statement that in the
case of the evanescent determinant
C(a0- (, a,, a2,. * * a_),
where
r = ao + al + a202 +.. + anllL-1,
the signed complementary minors of the elements of any row form
* A determinant which is doubly 'orthosymmetric' can have only two different
elements, and must have all its odd-numbered columns identical, and all its evennumbered columns identical. This is readily seen on starting with the first two
elements and then carrying out the requirements of double 'orthosymmetry.'
For example,
a b a b
b a b a
a b a b
b a b a.
b aba.