 It is in applying Sturm's theorem (28) to (29), through the use of the Euclidean algorithm above that the Routh matrix is formed.We get
and identifying the coefficients of this remainder by formula_141, formula_142, formula_143, formula_144, and so forth, makes our formed remainder 
where
Continuing with the Euclidean algorithm on these new coefficients gives us
where we again denote the coefficients of the remainder formula_148 by formula_149, formula_150, formula_151, formula_152,
making our formed remainder 
and giving us
The rows of the Routh array are determined exactly by this algorithm when applied to the coefficients of (20).An observation worthy of note is that in the regular case the polynomials formula_155 and formula_156 have as the highest common factor formula_157 and thus there will be formula_32 polynomials in the chain formula_119.