36 
ALGEBRAIC INVARIANTS 
identically in the a’s, where S2 is the differential 
Œ —(-2ui~-—h3ii2~——K . . T-pa 
dai 002 0«3 
operator 
0 
p-l~_ ’ 
Qdp 
In other words, I must satisfy the partial differential 
equation Œ7 = 0. In Sylvester’s phraseology, I must be anni 
hilated by the operator Î2. 
From this section and the preceding we have the important 
Theorem. A rational integral function I of the coefficients 
of the binary form f is an invariant of f if and only if I is iso- 
baric, is unaltered by the replacement (l), and is annihilated 
by Ü. 
EXAMPLE 
An invariant of degree d of the binary quartic (§6) is of weight 2d 
(end of § 16). For d= 1, the only possible term is ka 2 ; since 0=fi(^a 2 ) 
= 2£ai, we have ¿ = 0. For d=2, we have 
I=ra 0 ai+sa,ia 3 +/o 2 2 , 
ill = (s+4r)a 0 a 3 + (41 +3s) d\d 2 = 0, 
s = — 4r, /=3r, /=r(a 0 (u-iaidi+3o 2 J ). 
EXERCISES 
1. Every invariant of degree 3 of the binary quartic is the product of a 
constant by 
J ~ dod 2 di-\~2dld 2 d 3 — d 0 d 3 2 — d\ 2 di— d 3 3 . 
2. The invariant of lowest degree of the binary cubic 
d 0 X 3 +3diX 2 y+3diXy 2 +diy 3 
is its discriminant {d 0 a 3 —aio 2 ) 2 —i(a 0 d 2 —di 2 ){d 3 d 3 —o 2 2 ). 
3. An invariant of two or more binary forms 
a 0 * Pl +. . . , b 0 x P2 +. , ., ctX P3 +. . . 
is annihilated by the operator 
Si2 = Oo h* 
Soi da 2 
• Abo~r-\-2bi—— +. 
dbi a&2 
• ~rCo 
dci 
4. Every invariant of 
daX 2 +2dixy-\-diy 2 , b 0 x 2 -\-2b!xy+b2y 2