122 



LAMBERT— SOLUTION OF ALGEBRAIC EQUATIONS lApriUs, 



giving this difference an exponent equal to itself. It will be found 

 that when the sign of inequality is > in convergency conditions 

 corresponding to conditions (32) and (35) the right-hand member 

 is the reciprocal of what it is when the sign of inequality is <. 



20. In like manner two sets of conditions sufficient for the abso- 

 lute convergence of the infinite series giving the roots of the four- 

 terni equation obtained from each one of the equations (21), (22), 

 (23), (24), (25), (26) may be written. 



The convergency conditions for all these infinite series may be 

 taken from the following table, in which the signs of equality of 

 the limiting conditions of convergence have been omitted. 



In this table the signs of the two inequalities which constitute 

 the convergency co.nditions of the series obtained from the equa- 

 tions (21) to (26) are placed to the right of the respective equations. 

 The left-hand member of each inequality is at the top of the column 

 in which the sign of inequality stands. The right-hand member of 

 one inequality must be taken at the bottom of the column in which 

 the sign of inequality stands ; the right-hand member of the second 

 inequality is the expression at the right of the row in wlfich the 

 sign of inequality stands when the sign of inequality is <, when the 

 sign of inequality is > the right-hand member of the inequality is 

 the reciprocal of this expression. 



21. The following table exhibits one set of convergency condi- 

 tions of the infinite series which give the roots of the three-term 

 equation 



av" -f- hy^ -(- rf = o 



together with the equations from which these series are derived and 



