igoS.] 



IN INFINITE SERIES. 



123 



the number of the roots given by each series, and also the conditions 

 sufficient for the absokite convergence of the series of corrections 

 which must be appHed to the roots of this three-term equation to 

 obtain the roots of the four-term equation 



22. The substitution 



(40) y = z\ 



where ^ is a positive integer, transforms the four-term equation 



ayn _j_ JyyJc _|_ q. J _|_ J =: q 



into the four-term equation 



(41 ) ac"' + b::^' + f^'« + cf = o. 



The table of convergency conditions for equation (41) corre- 

 sponding to table (39) is 



(42) 



I a2"'-{-b2'''x+d=o 

 ^as"'-{-b2'''+dx=o 

 \az'^x+bz^'+d=o 



{sn—sl-y"- 



{k'%n-k) 

 The three-term equations 



ay'^ -\- by^ -{- d = o 

 a::"" -f bc''^ -{-d^o 

 for all values of .y have the same convergencv conditions 



