A converse of the second Noether them can also be established. Specifically, suppose that the Euler-Lagrange expressions E σ σ of the system are subject to q differential relations 0 = D a [ E ] = ∑ ∑ | I | = 0 s F a σ σ , I d I E σ σ . Letting λ λ = ( λ λ 1 , … … , λ λ q ) be an arbitrary q -tuple of functions, the formal adjoint of the operator D a acts on these functions through the formula E σ σ ( D + ) σ σ [ λ λ ] − − λ λ a D a [ E ] = d i B λ λ i , which defines the adjoint operator ( D + ) σ σ uniquely. The coefficients of the adjoint operator are obtained through integration by parts as before, specifically ( D + ) σ σ [ λ λ ] = ∑ ∑ | I | = 0 s R a σ σ , I λ λ I a , where R a σ σ , I = ∑ ∑ | J | = 0 s − − | I | ( − − 1 ) | I | + | J | ( | I | + | J | | I | ) d J F a σ σ , I J . Then the definition of the adjoint operator together with the relations 0 = D a [ E ] state that for each q -tuple of functions λ λ , the value of the adjoint on the functions when contracted with the Euler-Lagrange expressions is a total divergence, viz. E σ σ ( D + ) σ σ [ λ λ ] = d i B λ λ i , therefore if we define the variations δ δ λ λ u σ σ := ( D + ) σ σ [ λ λ ] = ∑ ∑ | I | = 0 s R a σ σ , I λ λ I a , the variation δ δ λ λ L = E σ σ δ δ λ λ u σ σ + d i W λ λ i = d i ( B λ λ i + W λ λ i ) of the Lagrangian is a total divergence, hence the variations δ δ λ λ u σ σ are quasi-symmetries for every value of the functions λ λ a .