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

AXISYMMETPJC DETERMINANTS (SYLVESTER, 1853) 127

SYLVESTER, J. J. (1853, March).
[on the relation between the volume of a tetrahedron and the
product of the sixteen algebraical values of its superficies.
Cambridge and Dub. Math. Journm, viii. pp. 171-178; or
Nom~v. Annales de Math., xiii. pp. 203-209; or Collected
Math. Papers, i. pp. 404-410.1
Denoting the vertices of a tetrahedron by a, b, c, d, its volume
in terms of the edges by V, and the areas of its faces by b/F
b/G, b/H, b/,we know that
144 V2 - (bc)2 (da)2 {(ca)2 + (db)2 + (ab)2 ~ (cd)2 — (bC)2 - (da)2}
+ (ca)2(db)2 {(ab) + (cd) + (be) + (da) (ca)' - (db)2}
+ (ab)2 (cd)2 { (bc)2 + (da)2 + (ca)2 + (db)2 - (ab)2 - (cd)2}
-( bc)' (ca)2 (ab)2 - (be)2 (db)2 (cd)2 - (ca)2 (cd)2 (da)2 _(ab)2(da)2 (db)2
= W say,
and
F =-(bc)4 - (cd)4 - (db)4 + 2(cd)2 (db)2 + 2 (db)2 (bc)2 + 2 (bc)2 (cd)2
G =- (ac)4 - (cd )4 - (da)4 + 2(cd)2 (da)2 + 2 (da)2 (aC)2 ~ 2 (ac)2 (Cd)2
11= - (ab)4 _ (bd)4 - (da)4 ~ 2 (bd)2 (da)2 ~ 2(da)2 (ab)2 ~ 2(ab )2 (bd)2
K =- (ab)4 - (bc)4 - (ca)4 + 2 (bc)2 (ca)2 + 2 (ca)2 (ab )2 + 2 (ab)2 (b C)2.
With this notation Sylvester points out that the condition f or
the vanishing of the surface of the tetrahedron is,/-F+ / + ff /K= 0,
and that this when freed of root-signs is
EF- 4EF3G + 6EF 2G 2 + 4EF2GH - 40FGHK = 0,
or say
N = 0,
where N consequently is of the eighth degree in the squared
edges. His reasoning then is that as the vanishing of the
surface and the vanishing of the volume are necessarily coincident, it follows that W, having no rational factors, must itself
be a factor of N; and that, W being of the third degree in the
squared edges, the quotient N/W must be of the fifth degree.