66
ALGEBRAIC INVARIANTS
5. The quotient of the Hessian of a x ” = ft n by n 2 {n — l) 2 equals
I »— 2 o
\a x ar
ft n-2 ftft
n —2
Oil OC\Ot2
n — 2o 2
ft n “ 2 ft 2
n —2
«a; ocioii
ft n_2 ftft
ft" “"ft,
one-half of the sum of which equals \ ai n-2 ft n “ 2 (a/3) 2 .
n— 2 <
«j a 2 ‘
6. ai ft 7i
a 2 ft 72
«x ft 7x
(«(3) 7i + (/3 7)ax + (7«) ft = 0.
41. Evident Covariants. We obtain a covariant K of
f=a x n = № = . . .
by taking a product of co factors of type a x and X factors of
type (a/3), such that a occurs in exactly n factors, /3 in exactly
n factors, etc. On the one hand, the product can be inter
preted as a polynomial in ao, • • • , a n , x\, X2. On the other
hand, the product is a covariant of index X of /, since, by
(1), §40,
(AB) r (AC) s (BCy . . . A a x B\C c x . . .
= (^) x (a(3) r (a7) s (jS7) i . . . a x a (3 x b y x c . . . ,
if \=r+s+t + - ■ • and
A x = A1X1 -\-A2X2, A1 =a$, A2=a v , (AB) =AiB2—A2Bi,
etc. The total degree of the right member in the a’s, /3’s, . . .
is 2X + co = nd, if d is the number of distinct pairs of symbols
ai, a.2', /3i, /S2; • • - in the product. Evidently d is the degree
of K in a 0 , ai, . . ., and co is its order in x\, X2.
Any linear combination of such products with the same
co and X, and hence same d, is a covariant of order 00, index
X and degree d of /.
EXERCISES
1. (a/3)(«7)az 3 ft 4 7x 4 and (a/3) 2 (ay)a x 2 /3 x 3 y 4 x arecovariantsofa x 6 = ft 5 = 7x 5 .
2. (afi) r a x n ~ r ft m -r is a covariant of a x n , ft w .
3. If m=n, ft w = a x n and r is odd, the last covariant is identically zero.
4. a 0 xi 2 -\-2aiX 1 x 2 +a 2 x 2 2 and boX 1 2 +2biXiX2+b 2 x 2 2 have the invariant
(a/3) 2 — dob 2 — 2cL\b\ -)-fl 2 feo.