120 LAMBERT— SOLUTION OF ALGEBRAIC EQUATIONS [ApriUs, 



i6. The infinite series composed of the terms of the left-hand 

 cohimn of the value of y is convergent when 



and if condition (28) is satisfied this infinite series furnishes the 

 solution of the three-term equation 



(29) 03;" -j- by^ -\- d = o. 



It is found that each one of the infinite series composed of the 

 terms of the respective columns of (2y) is convergent when (28) is 

 satisfied. It follows that (27) may be written 



3 



(29) f = X^ -f --y^X^ + ^2 jV'-^2 + -3 .3 Jo''^3 + 



where Xq, X-^, Xn, X^, •••, stand for the sums of convergent series. 

 If now X is the largest of the numbers X^, X-^, Xo, X^, ■■•, 



(30) yzx{.+^j,: + -^,j.-+-f^,y-+..), 



and this last value of 3; is convergent when 



Affecting both sides of this inequality by the exponent n, this con- 

 vergency condition may be written 



(32) ^^■<"'- 



17. Conditions (28) and (32) are sufficient for the absolute con- 

 vergence of (27). Condition (28) shows that the series which 

 determines the roots of the three-term equation 



(29) aj'" -\-by^ -\- d = o 



is found from 



( 33 ) ay'^ -\- hy^x -{- d = o. 



The columns of (27) after the first are the corrections which 

 must be applied to the roots of the three-term equation (29) to 

 obtain the roots of the four-term equation 



ay" + by^ -]- cy^ -]- d^o 



