IN INFINITE SERIES. 



]25 



roots of equation (41) in infinite series, the method of article (22) 

 requires the determination of the ns roots of equation (41) to find 

 the n roots of the four-term equation 



This method is therefore to be avoided in practice when possible. 



The following table exhibits the conditions sufficient for the 

 absolute convergence of the infinite series which give the roots of 

 the four-term equation obtained from the four groups of equations. 

 The series obtained from each group of equations determine all the 

 roots of the four-term equation. The convergency conditions must 

 be taken from this table as in article 20, and the limiting convergency 

 conditions must be taken into account. 



A less inclusive set of conditions sufficient for the absolute con- 

 vergence of the series which give the roots of the four-term equa- 

 tion derived from the groups of equations of table (43) is obtained 

 by taking the second member of each inequality from the bottom 

 of the column in which the sign of inequality stands. 



(43) 



I a}'"-\-d}'''x-{-cyji-\-d=^0 

 \ ay^xArby^-\-cy^x-\-d^^Q ] k 



TTT S 'y"+'^j*-^-ho''-l-^-^=o i'^ — ^ 



\ ay''x+dy^x+cy^+d=0 \ I 

 I" ay^A;-by^-\-c)^x-\-dx^^O « — k 

 k—l 

 I 



IV \ ay''x-\-by^-{-cy^-\-dx^O 

 [ ay^x-\-by''x-\-cy'--\-d=0 



It is only when the convergency conditions of the groups I, II, 

 III, IV, together with the corresponding limiting convergency con- 

 ditions fail simultaneously that the use of equation (41) becomes 

 necessary. 



IV. The Five-term Equation. 



24. In the five-term equation 



