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

294 HISTORY OF THE THEORY OF DETERMINANTS

Finally, we draw attention to the fact that the orthogonants
involved are all here taken as positive, and that if they be not so
taken Siacci's enunciations will require modification.
BELTRAMI, E. (1873).
[Sulle funzioni bilineari. Giornale di Mat., xi. pp. 98-106.]
In the treatment of the problem * of the transformation of
~C,SxyS into the canonic form y7,,^enam, it is inferred that if
rs. nl
a f e.... a' f' e....
f b d.... f' b' d'.
e d c.... e' d' c....
be two forms of the square of any determinant, the one obtained by rowmultiplication and the other by column-multiplication, then
a-x f e.... a'-x f' e'.....
f b-x d.... f' b'-x d'..
e d c-x..... e d' c'-x............, i..........,
viewed as functions of x, are identical. The ground for the inference
is that while the functions are of the ntl degree in x, they are found
to be equal for n+1 values of x.
A closely allied theorem by Hamburger will be found dealt with
under the same year in chap. i.
GRAVELAAR, N. L. W. A. (1875).
[Neuer Beweis fur die Realitat der Wurzeln einer wichtigen
Gleichung. Archiv d. Math. u. Phys., lviii. pp. 301-318.]
Gravelaar's proof is not really new: what his paper provides is a
criticism of a previous proof and a mode of removing the defects.
In discussing the character of the roots of the equation
al -X a1 a 3a....
a12 a22-x a23 * - - - = 0 = L,(x), say,
a13 a23 a33-x............... n
* Jacobi's of 1831 (see Hist., i. p. 436): but Beltrami begins by showing generally
how the determinant of a bilinear form is altered by the performance of the two
linear substitutions (see Cayley's memoir referred to in Hist., ii. p. 313).