For linear dynamical systems with constant coefficients, Routh gave a set of stability criteria in terms of the coefficients of (i) the characteristic equation and (ii) the equation with roots which are the sums of the characteristic roots taken in pairs. This set of criteria involves ½n(n+1) inequalities, wherenis the system order. The present paper investigates whether some of these inequalities ore redundant. Forn< 0 it is found that ½n(n−1) inequalities are indeed redundant, i.e. can be omitted, leaving a set ofnconditions which fire both necessary and sufficient for stability. Also, forngeneral, certain of the inequalities are shown to be non-redundant, i.e. they cannot be omitted without destroying the sufficiency of the criteria.