SOLUTION OF ALGEBRAIC EQUATIONS IN INFINITE 



SERIES. 



By PRESTON A. LAMBERT. 



(Read April 2^, 1908.) 



I. Introduction. 



1. The object of this investigation is to develop a method for 

 determining all the roots, real and imaginary, of an algebraic equa- 

 tion by means of infinite series. 



2. Suppose the given equation to be represented by f(y) =0. 

 The method consists in introducing a factor .v into all the terms but 

 two of the given equation ; expanding y, which now is an algebraic 

 function of x, into a power series in x ; placing x equal to unity in 

 this power series. The resulting value of 3?^ if convergent, is a root 

 of the given equation expressed in terms of the coefficients and expo- 

 nents of the equation. 



3. The method presupposes the solution of the two-term equation 



ay'^ -\-b = o. 

 In fact the roots of this equation when written in the form 



y = = r(cos 6 -\- i sin 6) 



a ^ ' 



are found to any required degree of approximation from the formula 



i / 2S'Tr -\- 2S7r + 0\ 



y = r" I cos h I sin I , 



\ ;/ n J 



where 



^ = 0,1,2,3,4, ■■■,n—i. 



4. The method proceeds step by step from the two-term equation 

 to the three-term equation, from the three-term equation to the four- 

 term equation, and so on. 



PROC. AMER. PHIL. SOC. XLVII. l88 H, PRINTED JULY l8, I908. 



