A solution of the ideal resonance problem has already been exhibited (Jupp, 1972) explicitly in terms of the ‘mean’ elements; to second order in the small parameter in the case of libration, and to first order in the case of deep circulation. Both representations possess a singularity when the ‘mean’ modulus of the Jacobi elliptic functions is unity; this corresponds to motions on or close to the separatrix of the phase plane of the dynamical system.