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

392 HISTORY OF THE THEORY OF DETERMINANTS

Brioschi states that similarly, if the determinants of
2 2 +
alx, + a2X2 + a33 3+ 2bx1x2 + 2b2x1x3 ~ 2clx~x2,
alx, + a2X2 ~ 2b1X1X2,
CagjC + a3x3 + 2b2x1x3,
all vanish, the ternary quadric is equal to
I (aCx, + b1x2 + b2X3)2;
a1
and if the determinants of
ax ~ aX2 + aX2 +4 + 2b XX2 + 2b~xx3+ 2b~xx4
1 2 2 3 3 4 4 1X1X2+ 2b2XCIX3 +2b
+ 2CIX2X3+ 2C2XC2X4+ - 2c11X3X4
2 2 2
alxl + a2x2 + a ce 3+ 2b1X1X2 + 2b2xIx3 + 2cIx~x2,
a a~ C62X2 + a4x4 + 2b1x1x2 + 2b3xIx4 + 2c2x2x4,
atXl + a2x~ + 2bxlx2,
a,x, + a3X3 + 2b2X1X3,
aix, + a4X4 + 2b3X1X4,
all vanish, the quaternary quadric is equal to
1 )
(aCx1 + b1x2 ~ b2x3 + bX4)2;
Ca,
and so on generally. An alternative set of conditions is referred
to, and is exemplified by the ease of the ternary quadric, where
the vanishing of a,c1 - b1b2 is substituted for the vanishing of
ctCC2a1 + 2b~b~c1 - a1c a - 2 - aCb,
the latter being equal to
{(ala - b )2c(aa - b2) - (aic, - bb)2 } a
SYLVESTER, J. J. (1853).
[On the conditions necessary and sufficient to be satisfied in order
that a function of any number of variables may be linearly
equivalent to a function of any less number of variables.
Philos. Magazine, v. pp. 119-126; or Collected Math. Papers,
i. pp. 587 —594.]
The title at once suggests a connection with Hesse's converse