  Recently, we showed that eigenphase shifts and eigentime delays near a
resonance for a system of one discrete state and two continua are functionals
of the Beutler-Fano formula using appropriate dimensionless energy units and
line profile indices and identified parameters responsible for the avoided
crossing of eigenphase shifts and eigentime delays and also identified
parameters responsible for the eigentime delays due to a change in frame
transformation.
  In this paper, the geometrical realization of the Beutler-Fano formulas is
considered in the three-dimensional Liouville space spanned by the Pauli
matrices, where dynamic operators are vectors. Vectors corresponding to the
background scattering matrix, the S matrix, and the time delay matrix Q form a
spherical triangle whose vertex and edge angles are parameters pertaining to
the frame transformations among eigenchannels of those matrices and eigenphase
shifts of the scattering matrices and the phase shift due to a resonance
scattering. The cotangent laws of the spherical triangle yield Beutler-Fano
resonance formulas appearing in eigenphase shifts and time delays. Duality
holding for the spherical triangle explains the symmetry observed in the
relations among parameters and provides a systematic way of defining conjugate
dynamic parameters. The spherical triangle also shows the rule of combining the
channel-channel couplings in the background scattering with the resonant
interaction to give the avoided crossing interactions in the curves of
eigenphase shifts as functions of eneryg.
  The theory developed in the previous and present papers is applied to the
vibrational predissociation of triatomic van der Waals molecules.
