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.