141]
A SECOND MEMOIR UPON QUANTICS.
267
43. For a cubic, the number of asyzygetic invariants of the degree 6 is
coefficient x 0 in —— ,
1 - a; 4
i.e. there is a single irreducible invariant of the degree 4.
44. For a quartic, the number of asyzygetic invariants of the degree 6 is
coefficient x e in
(1 — x 2 ) (1 — x 3 ) *
i.e. there are two irreducible invariants of the degrees 2 and 3 respectively.
45. For a quin tic, the number of asyzygetic invariants of the degree 0 is
The numerator is the irreducible factor of 1 — x 36 , i.e. it is equal to (1 - x 36 ) (1 - x 6 )
-h (1 — # 18 ) (1 — x 12 ) ; and substituting this value, the number becomes
1 — OLf 6
coefficient x e in
(1 — ¿r 4 ) ( 1 — xi 8 ) (1 — x 1 ' 2 ) (1 — x 18 ) ’
i.e. there are in all four irreducible invariants, which are of the degrees 4, 8, 12 and
18 respectively; but these are connected by an equation of the degree 36, i.e. the
square of the invariant of the degree 18 is a rational and integral function of the
other three invariants; a result, the discovery of which is due to M. Hermite.
46. For a sextic, the number of asyzygetic invariants of the degree 6 is
coefficient x 0 in
the second factor of the numerator is the irreducible factor 1 — af°, i. e. it is equal
to (1 — x 30 ) (1 — x 5 ) (1 — x 3 ) (1 — x 2 ) -r- (1 — a? 15 ) (1 — x 10 ) (1 — x 6 ) (1 — ¿u); and substituting this
value, the number becomes
1 -x 80
coefficient x 30 in
(1 - x 2 ) (1 - O (1 - x 6 ) (1 - x 10 ) (1 - Æ 15 ) ’
i.e. there are in all five irreducible invariants, which are of the degrees 2, 4, 6, 10
and 15 respectively; but these are connected by an equation of the degree 30, i.e.
the square of the invariant of the degree 15 is a rational and integral function of
the other four invariants.
47. For a septimic, the number of asyzygetic invariants of the degree 6 is
coefficient x 0 in
1 — x G 4- 2x s — x 10 + 5x 12 + 2x u + Qx 16 + 2a? 18 + 5x 20 — x 22 + 2ar 4 — x‘ M + x 3i
(1 — ¿c 4 ) (1 — x 6 ) (1 - a?) (1 - x 10 ) (1 - a? 12 )
34—2
