Many nonlinear field equations can be written in Hamiltonian form. Thus the equation ∂tu=K(u) can be written ∂tu =[u, H], where H is an appropriate functional and [ , ] is a Poisson bracket. Frequently one is interested in the solution of the equation linearized about a given solution, i.e., the equation ∂t τ=K′(τ), where K′(τ)=(d/dε) K(u+ετ)‖ε=0. It is known that if a functional I is a constant of motion then τ=[u, I] is a solution. Recently, more general solutions of this form have been found. To prove these results, it is very useful to have the answer to the following question: If Ii, Ij are two functionals, and Ki =[u, Ii], what is K′i(Kj)? The answer is Ki(Kj) =[[u, Ii], Ij]. Previously, this was proved assuming that canonical coordinates can be introduced. Here a proof is given without any such assumption.