58 
ALGEBRAIC INVARIANTS 
Thus each term of the sum involves every root exactly d times. 
The signs agree since 
dp = œ-\-2(W — œ), 
as follows by counting the total number of a’s. 
Any covariant of degree d, order co and weight W of 
ao{x—aiy) . . . (x—a v y) 
equals the product of ao d by a sum of products of constants and 
co differences like x—a r y and W — co differences like a r —a s , such 
that every root occurs in exactly d factors of each product; more 
over, the sum equals a symmetric function of the roots. Conversely, 
the product of ao d by any such sum equals a rational integral 
covariant. 
EXERCISES 
1. f=aoX 3 -\-3aiX 2 y+3a2xy 2 +a 3 y 3 has the covariant 
K = a 0 2 2(x—aiy) 2 (ct2 — as) 2 . 
3 
Show that the coefficient of x 2 in K equals — 18(a 0 «2 —Ui 2 ). Why may we 
conclude that K= —18H, where H is the Hessian of /? 
2. The same binary cubic has the covariant 
ao 2 2(x—oiif) {x—aif) («2—0:3) («3—«1) = 9U. 
3 
3. Every rational integral covariant of the binary quadratic / is a prod 
uct of powers of / and its discriminant by a constant. 
37. Covariant with a Given Leader S. If the seminvariant 
S has the factor ao, and S = aoQ, and if Q is the leader of a 
covariant K of f, then, since ao is the leader of f, S is the leader 
of the covariant fK. Hence it remains to consider only a 
seminvariant 5 not divisible by ao. If S is of degree d and 
weight w, 
5 = ao d SCi(product of w factors like a T — <x s ), 
where each product is of degree at most d in each root, and 
of degree exactly d in at least one root (§ 34). If each product 
is of degree d in every root, S is an invariant (§ 35) and hence 
is the required covariant. In the contrary case, let «2, for 
example, enter to a degree less than d; we supply enough 
factors x—oi2y to bring the degree in «2 up to d. Then ao d