594 
MATHEMATICS: A. A. BENNETT 
as to satisfy certain useful inequalities. We shall define in the three 
cases respectively; (l) || \\ = max,^,^ | h {si) h {s^ |, || F' [x{s), r] \\ = 
max, f,' I F [x{s), r] \ dr, \\ \ F" {x (s), n, rj || = max, f^' ^ \ h F" 
[X {s),n, fj I dn dr,- (2) II W^f,' fo'h^in) h^{r,) dr, dr,, \\ F'[x{s), r] ||^ 
Vfo' fo' F' [x {s), rY ds dr, || 4 F^' [x {s), r„ r,] \\ ^ V fo' fo' fo'{i F'^ 
[x{s), ^2]}^ ds dr, dr,- (3) || \\ ^ f^' \ h{r,) h{r,) \ dr, dr,, 
II F' [x{s), r] II ^ f,' {max, | F [x {s), r]\} ds, \i F" [x {s), r„ r,] \\ ^ 
{max.j;.^ I \ F" \x [s), r,, r,] \ } ds. With these definitions jj z, + z, \\ 
= II ^1 II + II ^2 II and II Zi z, \\ ^ || z, \\ \\ z, \\ , where z, z, means the 
symbolic product. Furthermore we may define || z \\ for z of range 
(0, 0) as identical with | 2 | . 
The case of Newton's method for one variable and n variables has 
been discussed by Dean Fine in the paper referred to. These cases 
and the case of integration in which the ranges are continuous may be 
treated by the methods discussed in the present paper. But the condi- 
tions here used are of an abstract sort and may be used in much more 
extensive cases as will be clear to those familiar with the recent work of 
E. H. Moore, M. Frechet, F. Riesz, V. Volterra, etc. The scalars, or 
vectors of signature (0, 0) , which in all of the classical instances are ordi- 
nary real numbers, may be taken as Hensel j5>-adic numbers or elements 
in any perjekte bewertete Korper of Kurschak,^ but more generally do not 
need to constitute a field as division is not essential. To avoid repeti- 
tion the further discussion of this case will not be treated separately 
but may be regarded as included in the following sections. 
Preliminary concepts. — Starting with an arbitrarily chosen range, 
which we shall refer to by the signature (1, 0), we shall suppose that we 
may construct ranges of the signatures (0, 0), (0, 1), (1, 1), ( — 1, 1), 
( — 2, 1), (2, 0), respectively, where the range of signature (0, 0) contains 
but one element. An expHcit definition of signature will not be re- 
quired. 
By the term vector or function on a range, will be meant a correspond- 
ence from the elements of a range to a set of scalars where each element 
determines one and only one scalar. A vector of signature (0, 0) is by 
definition a scalar, and conversely. The sum and the difference of two 
scalars will be required to exist uniquely, addition being associative. 
The sum and the difference of two vectors will be the usual vector or 
matrix sum and difference, respectively, and will be required to exist 
uniquely in the cases considered. The product of two vectors will be 
required to exist uniquely in the cases considered, and to be (1) asso- 
ciative, (2) completely distributive with respect to addition — so that 
addition is proved to be commutative, (3) continuous in each factor, 
continuity being defined as below, and (4) such that the signature of 
