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

PERSYMMETRIC DETERMINANTS (SYLVESTER, 1851) 337
and that ae -4bd + 3c2 and the determinant here appearing are
the two invariants * of the quartic under investigation.
The reduction of the octavic (a0, a,..., a.8 x, y)8 to the form
18 + U28 + Z%8 + '% 8 + 7Oe'612'22'324 2,
where U%.=prx+qry, is shown in similar fashion to depend on
the solution of the quintic equation
cao a1 a2 a3 a4-V
CO1 CO2 CL3 1 4a + 4V C5
a1 a2 a a a5
a2 a3 a4 - - 5 a6 = 0,
C3 t4 + 4 5 a6 a7
Ct4 - V C5 Ca6 Ca7 ct
where
= 72ep12p22p32p42I and I = s4- {ss + - s22,
I being the quadratic invariant of
X4 + sxSy + S2xy2 + s3xy3 + s4y4 or (x + X1y)(x + X2y)(x + X\y)(x + Xy).
The fact that the coefficients of v3, v2. Vl, vO are invariants of the
octavic is insisted on, and generalisations are effected for functions
of the degree 4m and the degree 4rn+ 2.
Further, it is pointed out that when the said even-degreed
functions after transformation are without the last (or unique)
term,-that is to say, are in Sylvester's phraseology "meiocatalectic,"-the last of the series of invariants must vanish:
for example, the condition that (a0, ac,..., a6 x, y)6 may be
expressible as the sum of three sixth powers is
C0 a1 c 2 a3
Cto 1 a2 as3
t Ct2 a3 a4
a2 ct3 a4 a~5
a3 Ct4 (5 a6
This, of course, may be proved independently, but is seen to be
a conclusion from putting e= 0 in the foregoing.
* The term " invariant" is first used in this paper.
M.D. II. Y