MATHEMATICS: A. A. BENNETT 
597 
The proof of the above theorem is immediate. We shall use the pre- 
vious notation. From (3),a;,-+i ^(C/B^) a], and from (2), || hi \\ ^ 
Hence || So"' k II ^ \\ h II ^ (V^) [«o + (C/^^)ce'o + {C/B'Y + • • • 
+ (C/52)'^'-^a2^'+ . . . ] ^ (^/C)^[(C/5^)ao]. Thus if li So II = 
II (xo + S^^j) — ^0 II < D, we have by hypothesis x' = Xq-\-11,q hi as a 
vector in the domain, for which also lim,=oo on = 0. 
Theorem 2. // is a root of f{x), and \\ f'{x')~^ \\ exists and is equal 
to \/^' and if \\ || ^ ^'/D' in the region defined by \\ x-x' || ^ D' 
and a f(x) is expansible in the form (1) in this region^ then there is no 
other nonequivalent root of f(x) in this region. 
Suppose if possible x'^ is another nonequivalent root in the region. 
Then/(x'0=/(xO + y(^0 (x'-xO+i f'iO [x" -x'f and since x' and x" 
are roots of /(x), the equation/' (x') (x"-xO =f{x") -f{x') -^f {^') 
{x'^-x'Y reduces to || x"-x' \\ = \\f (x')-^ \\ . || [-if (^0] (x'-x^ \\ 
or il x"-x' II ^ (1/^0 (^7^0 II x'-x' II 2, so that since || x'-x' \\ 4= 0, 
II x'^ — x' II ^ Z>', contrary to hypothesis. 
The following theorem is a corollary of Theorems 1 and 2 where 
D^ = 2D. The proof is trivial and will be omitted. 
Theorem 3. Let Xo be given and f{x) be expansible in the form (1) in 
the domain \\ x—x \\ < B/(2C), these being defined as in Theorem 1. If 
II f{xQ) II ^ B'^/{3C), then Newton's Method yields a root x' in the domain 
and any root in the domain is equivalent to x' . 
Conclusion. — It may be noted that in these theorems no explicit use 
is made of ^o, and similar terms, derivable but not initially given. 
This feature and the statement of the general Theorems 1, and 2, as 
against Theorem 3, distinguish the form of these results from those of 
Dean H. B, Fine, already referred to. The steps in the proofs are in 
essential taken from the article by Dean Fine, where however the 
proofs hold only for a finite range, and are here extended to a general 
range, with a consequent notational simplification. The method here 
used of obtaining a general result by a mere reinterpretation of the case 
of one variable, offers several features of novelty and is suggested aSy 
perhaps, of even more interest than the results obtained by its particu- 
lar application to the present problem. 
The present theory applies to nonlinear functionelles and integral 
equations with quadratic terms. A complete expansion in integral 
power series is not presupposed. In addition to interpretations of 
II z II as \/S z^n) and its generalizations, || z || may be interpreted as 
max„ I I or as S I I or in various other ways for the range (1, 0), 
with obvious extensions to the other cases. We may even with Riesz 
