Letdbe a positive integer, andFbe a field of characteristic zero. Suppose that for each positive integern,In, is aGLn,(F)- invariant of forms of degreedin x1, …,xn, over F. We call {In} anadditive family of invariantsifIp+q(f⊥g) =Ip(f).Iq(g) wheneverf;gare forms of degreedoverFinxl, …,xp; …,xqrespectively, and where (f⊥g)(xl, …,xp+q) =f(x1, …,xp,) +g(xp+1, …,xp+q). It is well-known that the family of discriminants of the quadratic forms is additive. We prove that in odd degree d each invariant in an additive family must be a constant. We also give an example in each even degree d of a nontrivial family of invariants of the forms of degree d. The proofs depend on thesymbolic methodfor representing invariants of a form, which we review.