igos.] IN INFINITE SERIES. 131 



30. The substitution 



(64) y = s', 



where s is a positive integer, transforms the five-term equation 



ay'' + by^ + O'' + ^y"^ + / ^ o 

 into another five-term equation 



(65 ) ay"-' + by''' -\~ cy^' -{- dy"'' -f / = o. 



An examination of the table of convergency conditions for equa- 

 tion (65) corresponding to table (6^), shows that it is always pos- 

 sible so to determine ^ that the convergency conditions for the series 

 obtained from the equation 



(66) ay"' + by^'x + cy^'x -f dy"''x + / = o 

 or from the pair of equations 



(67) ay"' + by^' -\- cy^'x + dy"''x + Ix = o 



(68) ay"'x + by^' + cy^'x + dy"''x -\- l = o 



are satisfied. Hence it is always possible to determine all the roots 

 of a five-term equation by means of series. 



31. The method of article 30 requires the determination of the ns 

 roots of equation (65) in order to find the « roots of the five-term 

 equation 



ayn -|- by^ + O'' 4~ dy"' -j- / = o. 



The use of this method becomes necessary only when the conver- 

 gency conditions of the seven groups of equations of the following 

 table, together with the corresponding limiting convergency condi- 

 tions fail simultaneously. 



The convergency conditions must be taken from this table as in 

 article 20. 



A less inclusive set of congruency conditions may be taken from 

 this table as in article 23. 



