40 
ALGEBRAIC INVARIANTS 
By operating on K by (2), we must have 
{OS—5i)*"+(05i — 25 2 )^" ~+. . H-(05 - _i-»S , J*y-- 1 
identically in x, y. Hence K becomes +05*3* — 0, 
(3) ic =5*"+05*"~ 1 y+^0 2 5*" _2 y 2 + . . .+ 0"5y", 
co! 
whik , by 05* = 0, 
(4) O" +1 5 = 0. 
Hence a covariant is uniquely determined by its leader S. 
(Cf. § 25). 
Similarly, K is annihilated by (1) if and only if 
(5) 125 = 0, 05i = co5, aS 2 =(«-l)5i, i25*=5*_i. 
The function 5 of ao, . . ., a p must be homogeneous and 
isobaric (§§ 21, 22). If such a function S is annihilated by 
i2, it is called a seminvariant. If we have 5*, we may find 
5*-i by (5), then 5*-2, . . . , and finally Si. But if K is 
a covariant, we can derive 5 W from S. For, by § 20, the 
transformation x = — rj, y = £ replaces / by a form in which 
A i ={ — \) i a v - t \ by the covariance of K, 
5U)T+. • .=5(T)y“ + . . .=50)*"+. . .+5*0)3+ 
so that 5*0) =5(^4). Hence 5* is derived from 5 by the 
replacement (l) in § 20. 
When the seminvariant leader 5 is given, and hence also o> 
(see Ex. 1), the function (3) is actually a covariant of/; likewise 
the function whose coefficients are given by (5). Proof will be 
made in § 25. In the following exercises, indirect verification 
of the covariance is indicated. 
EXERCISES 
1. The weight of the leader 5 of a covariant of order co of a binary form 
/ is W-u=\ and hence (§ 21) is %{pd—u). Thus S and / determine co, 
2. The binary cubic has the seminvariant S = a 0 a 2 —Ui 2 . A covariant 
with 5 as leader of is order co = 2 and is 
(a 0 a 2 —di 2 )x 2 + {a 0 a 3 —a^xy+(aia 3 —c 2 2 ) y 2 . 
Since this is the Hessian of the cubic, it is a covariant.