552 
MATHEMATICS: H. B. FINE 
(P 2 
dx bx 
Similarly 
Therefore 
f{z, + hj)^f(zj)+f(zj)h,^^-i±^v\hj\\ -i<e„e2<i, (23) 
Hence if hj be so taken as to satisfy the equation f (zj) + (z^) hj — 0 
and if, as in the proof of Theorem 1 , we set \hj\ = pj, \ f (zj) | = 4>j, we 
shall have 
But these are the same as the inequalities (5) and (7) in the proof of The- 
orem 1 for the case n = 1 except that v is here replaced by \/2v. It 
therefore at once follows from that proof^ and in particular from the in- 
equalities (15) and (18), that if <po<XVV2j' and therefore p = po< X/'l^^ 2 
the equation f(z)=0 has one and but one root is 5; also that the equations 
(b)' will yield a sequence of approximations to the root if instead of Zo 
any other point in 5 be taken as the point of departure.^ 
^ This paper was read before the American Mathematical Society, April 29, 1916. 
^ The distinctive feature of the method used in this paper is the determination of a 
number C such that if <po <C, then lim^- «> fj = 0. In this respect it diBfers from other 
discussions of Newton's method which I have been able to find. Of such discussions the 
following should be mentioned. Cauchy (Oeuvres [Ser. 2, Vol. 4], p. 573) obtained p </\2p 
as the condition for the existence in 5 of a real root of a real equation obtainable by Newton's 
method, and p<C\/4v as the corresponding condition for a complex root. Quite recently 
O. Faber (/. Math., Berlin, 138) has proved, for an analytic equation f(z)=0, that if 
\f(z)f"iz)/[f'(z)]^\ <a<l in the circle whose center is zo and radius p/(l— a), a root of 
f{z) = 0, obtainable by Newton's method, exists in this circle. I have been unable to dis- 
cover any previous proofs of the existence of a solution for the case »> 1. But Runge 
(Encyc. Math. Wiss., 1, p. 446) and E. Blutel (Paris, Acad. Sci., C. R. 151, 1109) have given 
proofs that if a solution exists at a certain point C, there must also exist a region R about C 
from any point of which a steady approach to C will be made by Newton's method Blutel's 
method of proof (an extension of one employed by Cauchy) being that which I have used 
in the proof of the corresponding part of Theorem 1. 
