64 
ALGEBRAIC INVARIANTS 
«0^2 and ai 2 ) is now excluded from consideration; likewise 
for q;i 2 q:2 4 and the second relation (4). 
In brief, the general binary cubic (1) may be represented 
in the symbolic form (2) since the products (3') of the symbols 
ai, cc2 are in effect independent quantities, in so far as we 
permit the use only of linear combinations of these products. 
But we shall of course have need of other than linear 
functions of ao, ... , ds. To be able to express them sym 
bolically, we represent / not merely by (2), but also in the 
symbolic forms 
(5) {P1X1+P2X2) 3 , (ti*i+72#2) 3 , . . 
so that 
(6) /3i 3 = ao, /3i 2 /32 = 0i, /3i/32 2 = 02, /S2 3 = «3; 7i 3 = uo, .... 
Thus d Q d 2 is represented by either ai 3 piP 2 2 or /3i 3 ai a 2 2 , while 
neither of them is identical with the representation ai 2 a 2 ^\ 2 ^ 2 
of d\ 2 . Hence 
dQd2 di 2 — 2(ai 3 /3i/32 2 + Pi 3 aiOC2 2 ~‘2‘Oi\ 2 Oi2&l 2 &2) 
= \oLi&\{a.i$2 — OC2P1) 2 . 
We shall verify that this expression is a seminvariant of 
/. If 
Xi -Xi~\-tX2, X2=X2, 
then /becomes ^ = yloXi 3 -)-. . . , where 
Aq = do, Ai=di-\-tdo, A2 = d2-\-2tdi-\-t 2 do, 
A3=d3-\-Std2+3t 2 di-\-t 3 do. 
Hence, by (3), 
F = {(X\Xl-\-OC 2X2Y, CL 2 = Oi2~\~t<X\. 
Similarly, the transform of (5i) is 
(filXi -)- ft'2X2) 3 , (i r 2 =P2+tf3i. 
Hence we obtain the desired result 
AoA2—Ai 2 = %ai(3i{ail3'2 — a'2Pi) 2 
— h a l&l (<X102 —OL2/3i) 2 = dod2 — d\ 2 .