 This difference, formula_33, of negative and positive jumping incongruities encountered while traversing formula_38 from formula_87 to formula_88 is called the Cauchy Index of the tangent of the phase angle, the phase angle being formula_42 or formula_90, depending as formula_91 is an integer multiple of formula_41 or not.To derive Routh's criterion, first we'll use a different notation to differentiate between the even and odd terms of formula_8