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

302 HISTORY OF THE THEORY OF DETERMINANTS
and then making the four familiar deductions (Hist., i. pp. 437-438),
Igel takes as his second datum the reverse substitution
e = XV,+ miy + VlZ
'7 = X2X+AI.2y1+Y2Z
~ = X3x + Y + 3yv'z
and deduces the rest. (But see Hesse's Vorlesungen of 1861, p. 192.)
Similarly, he starts with the two substitutions,

y;= /k%+M2?1+b(3`l
z = V1l+ "'271+ l'3V

Si1 = fJ.lx+ L2y+X3-z
='] v ~41X+MO2y+1430
S~ = VlX + V2Y + 3y J

and deduces as far as he can analogous results, thereafter making
an application to Aronhold's theory of ternary quantics. He uses
throughout n variables.
GRAVELAAR, N. L. W. A. (1878).

[Eene bijzondere vergelijking. Nieuw
113-124.1
The equation in question is

A rchief v. Wisk-., iv. pp.

a,,-x a1.
all a22- X

ain bit
a.L b2,

b12
b22,.... bim
bm

aL, a2,.... a,, —X b,1

bn2... bnm

= 0,

blm
b12
bbm

b2m

bn2

bnm

or, say, Ln,m.(X) = 0,
being thus a generalization of Hesse's of 1862 and Geiser's of 1876.
The object of course is to prove that all the n-m roots are real
and the procedure followed (pp. 118-124) is similar to that of 1875
in the case of rn = 0.