{l\frac {\partial f_0}{\partial
J_R}+ m\frac {\partial f_0}{\partial J_\phi} \over
l\Omega_R+m\Omega_\phi },\end{aligned}$$ where $$\Omega_R(\textbf{\textit{J}})=\frac {\partial {\cal H}_0}{\partial J_R},~~
\Omega_\phi(\textbf{\textit{J}})=\frac {\partial {\cal H}_0}{\partial J_\phi}.$$ Equation (\[eq:trial-functions-one\]) suggests to choose $$\Phi^{ml}_j(\textbf{\textit{J}})=\varrho^{ml}_{0}(\textbf{\textit{J}})
\Psi^{ml}_j(\textbf{\textit{J}}),\label{eq:trial-functions-two}$$ as the trial functions (in the space of actions) for an unsteady $f_1(\Theta,\textbf{\textit{J}},t)$. These functions have integrable singularities for resonant orbits with $l\Omega_R+m\Omega_{\phi}=0$. For unidirectional disks with only prograde orbits, they also include a term with the Dirac delta function $\delta(J_\phi)$ (see JH). One should therefore avoid the partial derivatives of $\Phi^{ml}_j$ with respect to the actions by evaluating the weighted residual form of $[f_1,{\cal H}_1]$ through integration by parts (Appendix A).

For deriving the relation between $a^m_j$ and $d^{ml}_{j}$ in (\[eq:a-versus-d\]), the fundamental equation (\[eq:fundamental-equation\]) was multiplied by the complex conjugates of the basis functions used in the expansion of $V_1(R,\phi,t)$. One may follow a similar approach for obtaining the weighted residual form of the CBE and set $$W^{ml}_j(\Theta,\textbf{\textit{J}})=\Psi^{ml}_j(\textbf{\textit{J}})
e^{-{\rm i}(m\theta_\phi+l\theta_R)},\label{eq:weight-functions}$$ which are the complex conjugates of the basis functions used in the expansion of $V_1(\Theta,\textbf{\textit{J}},t)$ in the action-angle space. The trial and weighting functions introduced as above, are not orthogonal but they result in a simple form for the linear part of the reduced CBE as I explain in 