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

296 HISTORY OF THE THEORY OF DETERMINANTS

WEIHRAUCH, K. (1876): FROBENIUS, G. (1878).
[Zur Construction einer unimodularen Determinante. Zeitschrift
f. Math. u. Phys., xxi. pp. 134-137.]
[Theorie der linearen Formen mit ganzen Coefficienten. Crelle's
Journ., lxxxvi. pp. 146-208.]
What is here given by Weihrauch is an alternative solution of
the problem which Hermite set himself in 1849 (Hist., ii. p. 306).
Frobenius devotes a section (~3, pp. 150-151) of his extensive
memoir to the same subject, referring his solution back to Gauss
(Disq. Arith., ~ 279), and drawing attention to four solutions given
in a paper of Jacobi's published in 1868, but written probably early
in 1850 (see Crelle's Journ., lxix. pp. 1-28).
BARDELLI, G. (1876).
[Alcune proprieta dei coefficienti di uno sostituzione ortogonale.
Rendic.... Istituto Lombardo... (2) ix. pp. 167-174.]
The basis of the work here is any orthogonant A which has all
its elements functions of one and the same variable; and the
investigation is mainly concerned with the determinant A, each
element of which a',. is the derivate of the corresponding element
as of A.
From the fact that
a2 l+a2+. +a2 = 1)
a,.l asl + a, as2.-. +a,.,,a,, = of
there is obtained by differentiation
alar, [ -+.... - +a-aQ, =a
(a,., aL +.... + arnsn) +(asl a.l + +asna n) = OJ,
which equalities, in view of the fact that the expressions on their
left are exactly such as occur when A, is multiplied rowwise by A,
may be conveniently shortened into
Pr, = 0, pR' +Psr = 0.
We thus have AA, i.e. A, equal to the determinant of the p's,
P say: and, this latter being zero-axial and skew, it follows that