Voi,. 7, 1921 
MATHEMATICS: A. A. BENNETT 
85 
a bilinear form, 5 Minkowski's gauge form 4 and Schwarz's inequality 9 
(so-called), as these occur in geometry, hypercomplex number theory, 
integral equations, and more generally, general linear analysis. 10 
The following theorem may be proved: Every geometric system ad- 
mitting a gauge set may be normalized by means of that set. In order that 
there may be no ambiguity an extended sequence of definitions will be 
given to cover all terms used. 
If P is a proposition concerning a system, 5, it may be that another 
system, T, is such that the proposition P has a meaning for the system T. 
The content of the proposition may be seriously altered while its form is 
not affected except in the sense that 5 is replaced by T. We may em- 
ploy the functional notations P(S) and P(T) to suggest that the form of 
the proposition is carried over unaltered from one system to the other. 
To avoid repetition certain propositional functions will be here listed for 
reference. These serve as definitions, whenever all of the terms appearing 
in a proposition have been themselves defined, for example, "Reg (L)," 
below, acquires a meaning only in a system in which 0 L and 1 L are defined. 
The "Addition Proposition," 11 Add. (R): (i) R is a system comprising 
elements, r, and a rule of binary combination, +. (H) For any n, r 2 , 
of R, Y\ + r 2 is a uniquely defined element of R. (Hi) For any r h r 2 , 
of R, Y\ + r 2 = T2 + n. (iv) For any n, r 2 , n, of R, n + (r 2 + r 3 ) = 
(r\ + r 2 ) + r 3 . (v) There is an element 0 R of R (also denoted by merely 
0), such that for every r of R, r + 0 R = r. (vi) For a given n of R, 
there is not more than one element r of R, such that n + r = n. (vii) 
For a given n of R, there is one and only element, r, of R such that r\ + 
r = 0 R — this element, r, is denoted by — n. 
The "Multiplication Proposition," Mult. (R, 5, T. P): (i) There are 
systems R, 5, T, P, comprising elements, respectively, r, s, t, p, and two 
rules of binary combination + and . , and such that with respect to +, 
the addition proposition is valid for each of the systems, (it) For any 
r of R f s of S, t of T, r.s and s.t are uniquely defined, and s.t is an element 
of P. (Hi) For any n, r 2 of R } and Si, s 2 of 5, it is true that ri.($i + s 2 ) = 
fi-Si + and (n + r 2 ).5i = fi.-si + r 2 .Si. (iv) For any r oi R, s of 5, 
* of 7, it is true that r.(s.t) = (r.s).t. (v) For any r of i?, and s of 5, 
it is true that 0 R .s = 0 T , and r.0 s = 0 T . 
The "Commutative Proposition," Com. (C.M.L.): (i) C is a subsystem 
of M. (H) For each c of C and / of L, c./ = /.c. (m) There exists an ele- 
ment l c , of C (also denoted merely by 1) such that for every /, l c -l = l. 
The "Regular Proposition," Reg. (L): (i) The elements of L fall into 
two mutually exclusive subsets L Q * and Loo. (H) The elements of L Q * fall 
into two mutually exclusive subsets, L Q and L*. (m) 0^ is an element 
of L Q . (w) 1 L if it has been defined and exists, is an element of L*. 
(v) For every I of L, —/is in the same set as /. (vi) There exists at least 
one Z* other than 1^ . 
