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

DETERMINANTS IN GENERAL (UNTERHUBER, 1870) 31

UNTERHUBER, A. (1870): HOUEL, J. (1871): MAURER, F. (1872):
UNTERHUBER, A. (1872).
[Einleitung in die Theorie der Determinanten. Leoben.]
[Notions elementaires sur les determinants. Paris. (Lithographed.)]
[Grundztige der Determinantenlehre. Budweis.]
[Ueber Determinanten. 14 pp. Leoben.]
The second of these accurately describes itself, and the others
are school-programs of the simplest type.
CAYLEY, A. (1871).
[Question 3329. Educ. Times, xxiii. p. 261, xxiv. p. 17; or lMath.
from Educ. Times, xv. pp. 66-67.]
The requirement here is to show that every permutation of 12345
can be obtained by means of the cyclical substitution (12345), and
the interchange (12); and it is met by giving the equivalents of.
other interchanges in terms of the said two operations, K and a
say. For example, the interchange (15) can be effected by performing in succession the six operations K, a, K, K, K, K, or, as
this is written,
(15) = K4aK.
Similarly (13) = aKaK4a- aK2=KaK:I...,
the variant forms being due to the facts
K5= 1, a2 = 1, (Ka)4 = 1.
BRILL, A. (1871).
[Ueber diejenigen Curven eines Biischels,.... Math. Annalen,
iii. pp. 459-468.]
Brill has incidentally to consider an identity which is the case
of Schlafli's theorem of 1851, in which n, r = 3, 2, that is to say,
the case which gives a six-line determinant, A,,2 say, as the equiva