92 
ALGEBRAIC INVARIANTS 
To the various powers, whose product is any one term of K, 
(pi — P2H (pi — pa) 3 , (p2 — pa) k , • . 
(«1 —Pi« 2 ) fc , (xi-p2X 2 ) h , . . ., 
we make correspond the symbolic factors 
Wb (ac)b (bc) k , . . . , dx\ b x h , . . . 
of the corresponding covariant of /: 
C = {ab) t (dc) J (bc) t . . . d x l bj*cj* . . ., 
of degree p (since there are p symbols a, b, c, . . ., cor 
responding to pi, . . . , pp) and having the same order 
/1+Z2+/3 + . . .as K. Conversely, C determines K. 
EXAMPLES 
Let p = 2. To K = ao 2s (p\ — pz) 2s corresponds the invariant C— (ab) 2s 
of degree 2 of f=a x 2s = b x 2s . Again, to the covariant Kof $ corresponds 
the covariant (ab) 2s ajbj of the form a x 2s+t =b x 2s+t . 
Concomitants of Ternary Forms in Symbolic Notation, 
§§ 65-67 
66. Ternary Form in Symbolic Notation. The general 
ternary form is 
/= S-TTTl drstXl r X2 S X^, 
where the summation extends over all sets of integers r, s, t, 
each = 0, for which r-\-s-{-t=n. 
We represent / symbolically by 
f =(X x n = Px n • • . , a x = aiXi-\-a2X2-\-asX3, • • • • 
Only polynomials in ai, (*2, «3 of total degree n have an inter 
pretation and 
a \ r <X2 S OLZ t = a rs t- 
Just as aifa —cx2f3i was denoted by (a/3) in §39, we now 
write 
0:2 
182 
72 
(a/37) = 
Oil 
71 
as 
/33 
73