igoS.] IN INFINITE SERIES. 129 



^« 71" ^" ^^ ^» 



(59) -^^7^k<^^j^_^j-k, -j-^^,<n\ -^^i^^^,<n\ 



2y. When the conditions (59) are satisfied the first group of 

 terms of (54) gives the roots of the four-term equation 



ayn ^Jyyl^ J^cy^ + / = 



expressed in the series obtained from the equation 



ay"' -J- hy^x -\- cyKv -\- l^o 



and the successive groups of (54) are the series of corrections which 

 must be applied to the roots of this four-term equation to obtain the 

 roots of the five-term equation 



ay" + hy^^ -f cy^ + dy'" -\-l = o. 



28. If in the first row of (54) either of the terms 



c d 



is placed first and the consequent changes in (54) are made, the 

 convergency conditions of the two new series are found to be 



(60) -^T^;^<n\ -^^^ < JiQ^Zri^! ' ^z'V^'^^^"' 



c" ^/" «" 

 -^ 1," .-^ 



In the limiting convergency conditions the signs of inequality in 

 the first inequality of (59), in the second inequality of (60) and in 

 the third inequality of (61) must be replaced by the equality sign. 



The conditions sufficient for the absolute convergence of (54) 

 may be written from equation (44) by the method stated in article 19. 



In like manner the conditions sufficient for the absolute con- 

 vergence of the series obtained from equations (45) to (53) may 

 be written. 



The convergency conditions for all these series may be taken 

 from the following table. The convergency conditions are taken 

 from the table by the method stated in article 20, except that the 

 right-hand members of two inequalities must be determined from 



