  In this paper, we elaborate on correctly predicting \'Echelle spectrograms by
employing the fully three-dimensional representation of Snell's law to model
the effects of prisms as cross-dispersers in \'Echelle spectrographs. We find
that it is not sufficient to simply apply the frequently used trigonometric
prism dispersion equation to describe recorded spectra. This vector equation
approach is not limited to a single dispersive element when modelling
multi-prism cross-disperser configurations. Our results help to understand the
main levers in an \'Echelle spectrograph as well as contribute to
auto-calibration algorithms for minimizing calibration efforts in daily
operation.
