igoS.J IN INFINITE SERIES. 133 



V. Conclusion. 

 32. In the algebraic equation of / terms 



f(y)=o 



the two terms from which the factor x is to be omitted can be 

 selected in 



/(/-I) 



ways. Each one of the resulting equations defines _v as an algebraic 

 function of x, and each algebraic function of x can be expanded 

 into a power series in .r by the methods used to obtain the corre- 

 sponding expansions for the three-, four- and five-term equations. 

 When .r is made unity in these power series the resulting series 

 become the roots of the /-term equation and a table of convergency 

 conditions for these series analogous to tables (20), (38), (62) can 

 be set up. In fact, this table may be written mechanically by fol- 

 lowing the directions of article 19. 



33. If in the /-term equation the substitution 



is made, a table of convergency conditions analogous to tables (39), 

 (63) can be set up, and the value of s can be determined so that 

 this table of conditions shows that it is possible to obtain all the 

 roots of the transformed equation from the series derived either 

 from the equation in which x is omitted from the first and last 

 terms, or from the two equations in which x is omitted from the 

 first and second, and from the second and last terms respectively. 

 The roots of the given equation are then found from the roots of 

 the transformed equation by substituting in 



34. Finally, tables of convergency conditions analogous to tables 

 (43), (64) can be set up for the f-term equation, and it is necessary 

 to use the transformed equation only when the convergency condi- 



