MATHEMATICS: A. A. BENNETT 
595 
the range of the product is the matrix sum of the signatures of the 
ranges of the factors. 
The notation II z \\ will be used for the norm of z. We shall require 
only that (1)|| z \\, = \\—z ||, isa uniquely defined real non-negative 
number such that 3 = 0 implies || z \\=0, but not necessarily converse- 
ly; (2) II zi +22 II ^ II zi 11 -\- II Z2 II, II zi Z2 II ^ II zi II 11^2 II, and in 
particular, if II Zi \\ = 0, \\ Zi-\-Z2 || = ll Z2 \\, the sum and the product 
being defined as above; (3) the signature of z being given and a positive 
number e, being given, there exists a z, such that 0< || s || < e; and (4) 
if Zi, 22, . . . , . . . be a denumerable sequence of vectors on the 
same range, and such that II Si || + II 22 || + . . . + II 2„ || + . . . 
converges to a finite limit, then there is at least one z on the same range 
such that given any positive number e, there exists a number m such 
that n >w implies that || z-l^Ui \\ < € and lim„=^ || S?2, || = || (z) \\ . 
Such a z will be called a limit of 21 + 22 + • • . + 2;„ + . . . , and we notice 
that for any two such limits z and z\ \\ z—z^ \\ = 0. We shall define 
two vectors z and s' for which || 2 — 2' || = 0 as equivalent. It is by 
virtue of this last-named property that the vectors of a given range form 
in some sense a closed set. 
A vector b with the signature ( — 1, 1) will be said to be non-singular j 
if and only if there exists a unique vector b~^, with the signature (1,-1) 
such that the equation a = bx where a is of signature (0, 1), and x is 
required to be of signature (1, 0), always has one and only one solution 
given hy X = b~^a. 
We shall extend the usual definition of continuity to the case where 
g{x) and X are of any signature as follows : — The function g{x) is con- 
tinuous in X Sit x' \i for every assigned positive constant e, there exists 
a positive constant b, such that || x — || < 5 implies that || g{x)~ 
g{x') II < 
Description of Newton's method. — In describing Newton^s Method, we 
presuppose that we are given initially a function fix), where x is of sig- 
nature (1,0) and f{x) of signature (0, 1), and such that we may expand 
f(x+h) as follows: — 
fix + h) =/(x) +nx)h+i rm, w 
where h and ^ are of signature (1, 0), of signature (2, 0),/' of signature 
( — 1, 1),/" of signature ( — 2,1), the expansion being valid in a known 
domain. The ^ is supposed to be dependent on the x and h, but such 
that simultaneously || ^ — x\\ ^ \\ h \\ and || (x-\-h)—^ \\ ^ \\ h \\. The 
functions /, f, and /" are all supposed to be continuous in the domain 
considered. Equation (1) may also be written, for convenience, in the 
form f(x-\-h) =a-\-bh-{-ch'^. 
