124 LAMBERT— SOLUTION OF ALGEBRAIC EQUATIONS [April 25, 



If the inequality 



c 



7Vn-.<«" 



of table (39) is not satisfied, it is always possible to take s suffi- 

 ciently large so that the corresponding inequality 



of table (42) will be satisfied. 



In like manner, if the inequalities 



of table (39) are not satisfied simultaneously, it is always possible 

 to take .y sufficiently large so that the corresponding inequalities of 

 table (42) 



b'' if ^"-' I (^ J 



will be satisfied simultaneously. 



To the convergency conditions of table (42) must be added the 

 limiting convergency conditions obtained by replacing in the first 

 column of inequality signs of table (42) each inequality sign by the 

 equality sign. 



It follows that it is always possible to determine ^ so that all the 

 roots of the four-term equation 



(41 ) a>s:"« + hz^' 4- cz^' -\-d = o 



may be derived from the roots of the three-term equation 



(42 ) av?"s + bs^^ -[-d==o. 

 The roots of the four-term equation 



ay^ -j- by^ -\- cy^ -\- d ^= o 

 are found from the roots of equation (41) by substituting in 

 (40) ' y = z'. 



23. While table (42) shows the possibility of expressing all the 



