xgoS.] IN INFINITE SERIES. 121 



1 8. If the two terms in the second row of (27) are interchanged 

 and the consequent changes are made throughout {2y), the left- 

 hand column in the resulting value of y is convergent if 



(34) 



c 



and the entire expression for y is convergent if in addition 



(35) «V"-^- ^ ^^^' 



Conditions (34) and (35) are sufficient for the absolute conver- 

 gence of the new series for y. 



Condition (34) shows that the series which determines the solu- 

 tions of the three-term equation 



(36) ay"^ ^ cy^ -\- d^^o 

 is found from 



( 37 ) aj'" + ^3' '-^ + ^ = O- 



This series is the left-hand column of the value of y. 



Condition (35) shows that the series of corrections which must 

 be applied to the roots of the three-term equation (36) to obtain 

 the solution of the four-term equation 



av" -\- hy^ -\- cy^ -\- d = o ' 

 is convergent. 



19. From equation (21) by omitting in succession each of the 

 terms containing x are obtained the equations 



(33) af'^hyKv^d^o 



( 37 ) ay^^ -\- cy^x -\- d = o 



The convergency conditions (28) and (34) may be written from 

 equations (33) and (37) respectively by following the directions 

 (a), {h), (c) of article 13. The left-hand members of the condi- 

 tions (32) and (35), together with the character of the signs of 

 inequality, may be written from equations (37) and (33) respec- 

 tively by following the same directions. The right-hand member 

 of conditions (32) and (35) is formed by writing the difference of 

 the exponents of the two terms of (21) which do not contain x and 



