Special cases of the Routh array

If there is a zero in the first eolumn, then further ealeulation eannot normally proeeed sinee it will involve dividing by zero. The problem is solved by replaeing the zero with a small number e whieh ean be assumed to be either positive or negative. When the array is eomplete, the signs of the elements in the first eolumn are evaluated by allowing e to approaeh zero. 

Irrespeetive of whether e is a small positive or negative number in array (5.38), there will be two sign ehanges in the first eolumn. 

If all the elements of a partieular row are zero, then they are replaeed by the derivatives of an auxiliary polynomial, formed from the elements of the previous row. 

The eoeffieients of equation (5.40) beeome the elements of the. v row, allowing the array to be eompleted. 

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