MATHEMATICS: H. B. FINE 551 
do<2p. And the condition 5o<2p is satisfied if {a^^\ af\ . . . , af^) 
be taken anywhere in S. 
Again since ^2 = ^< 1 we have, as in (\2),bjj^i^ (^2 = ^j^^j- 
Hence each approximation in the sequence {a^\ a[^\ . . . , a^n^), (; = 
0, 1, 2, . . closer than the one which precedes it. 
Finally, by (21), ^^V^2- Let j' be any value of j such that 
r^''/k2 < 0.1 and f-'' < 0.1 and set 7 =7' + / (/ =0, 1, 2, . . . ). We then 
have 
Hence | ^ will coincide withj ^ cl | and therefore i?/"^^ with 
Ck {k =^ \,2, , . . , /^), to at least the 2^th decimal figure. 
Theorem 2. Let f{z) he any function of the complex variable z which 
is analytic at z = Zq, and h the number determined by the equation 
f(zo) -Vf'{z,)h 0. 
Let S be the circular region whose center is Zo-\-h and whose radius is 
p = \h\, and suppose that, in S, f(z) is everywhere analytic, X(>0) is the 
lower bound of the values of \f{z)\ and ^(< 00) is the upper bound of the 
values of \f'{z)\. Then, if 
\f{z,)\<^, {a") 
the equatiofi f (s) = 0 has one and but one root in S, and this root will be 
approximated to uninterruptedly by successive determinations of hj and 
3^+1 by the formulas 
f (%) +r fe) hj = 0, zj^, = zj + hj. U = 0, 1, 2, . . . ), (^0 
(where h^ = h), the root being limy.= a, %• 
For let 
Zj = x + iy, f(Zj)=(p{x,y)+if(x,y), hj = ^-{-ir]. 
Then if Zj and Zj^ 1 are in S, we shall have 
where x\ x lie between x and x + y' , y" between }• and y + But 
