§38] 
DIFFERENTIAL OPERATORS 
61 
Theorem. If l and f are any covariants of a system of 
binary forms, we obtain a covariant {or invariant) of the system 
of forms by operating on f with the operator obtained from l by 
replacing x by 9 / 9 y and yby—d/ d%, i-e-, x r y s byf — l) s 9 r+s / dy r dx s . 
EXERCISES 
1. Taking l=f=ax 2 +2bxy+cy 2 , obtain the invariant 4{ac—b 2 ) of /. 
2. If /=/ is the binary quartic, the invariant is 2 -4! / of § 31. 
3. Using the binary quartic and its Hessian, obtain the invariant J. 
4. Taking l=doX P -\-, . ,,f=b 0 x p +. . ., obtain their simultaneous 
invariant 
If also l =f, we have an invariant of /, which vanishes if p is odd. For 
p — 2 and p — 4:, deduce the results in Exs. 1, 2. 
5. A fundamental system of covariants of a quadratic and cubic 
Q = z Ax 2 -\-2Bxy-\-Cy 2 , f=ax 3 -\-3bx 2 y+3c*y 2 -\-dy 3 
is composed of 15 forms. We may take Q and its discriminant AC—B 2 -, 
f, its discriminant and Hessian h, given by (5) and (2) of § 8, the Jacobian 
/ of / and H; 
J— {a 2 d—3abc+2b 3 )x 3 +3(abd+b 2 c—2ac 2 )x 2 y 
+3(2 b 2 d—acd — be 2 ) xy 2 + (3 bed—ad 2 —2c 3 )y 3 ] 
the Jacobian of/and Q: 
{Ab—Ba)x 3 + {2Ac—Bb—Cd)x 2 y + {Ad+Bc—2Cb)xy 2 + {Bd—Cc)y 3 ] 
the Jacobian of Q and h: 
{As—Br)x 2 -\-{At—Cr)xy+{Bt—Cs)y 2 ] 
the result of operating on / with the operator obtained as in the theorem 
from l=Q: 
L x = {aC-\-cA — 2bB)x + {bC +dA — 2 cB)y\ 
the result of operating on Q with the operator obtained from Lx: 
L t = \aBC-b{2B 2 +AC) +3 cAB-dA 2 ]x 
+ \aC 2 -3bBC+c{AC+2B 2 )-dAB\y]