Vol. 7, 1921 
MATHEMATICS: A. A, BENNETT 
87 
The "Regular Vector Proposition," R. Vect. (C, H, V): (i) R. Hyp. 
(C, H). (ii) R. Lin. (G, H, V). 
The "Normal Linear Proposition," N. Lin. (C, M, L) : (i) N. Mult. 
(My M, L, L). (ii) N. Mult. (M, L, M, L). (Hi) N. Com. (C, M, L). 
The "Normal Hypernumber Proposition," N. Hyp. (C, H) : (i) N. Lin. 
(C, C, G). {ii) N. Lin. (C, H, H). 
The "Normal Vector Proposition," N. Vect. (C, H, V): (i) N. Hyp. 
(C, H). (ii) N. Lin. (G, if, F). 
The "Geometric Proposition," G^m. (C, if, X, U): (i) Vect. (C, H, 
X). (ii) Vect. (C, H, U). (Hi) Mult, (if, X, U, H). (iv) Mult. (H, U, 
X, H). 
The "Regular Geometric Proposition," R. Geom. (C, H, X, U): (i) R. 
Vect. (C, H, X). (ii) R. Vect. (C, H, U). (Hi) R. Mult. (H, X, U, H). 
(iv) R. Mult. (H, V, X, H). (v) R. Conj. (C, X, U). (vi) R. Conj. 
(C, U, X). 
The "Normal Geometric Proposition," AT. Geom. (C, H, X, U): (i) N, 
Vect. (C, H, X). (ii) N. Vect. (C, H, U). (Hi) N. Mult. (H, X, U, H). 
(iv) N. Mult. (H, U, X, H). (v) N. Conj. (C, X, U). (vi) N. Conj. 
(C, U, X). 
The "Gauge Proposition," Gge. (G, H, R, S): (i) For each r of R G 
there is an 5 of S^, such that r.s = s.r — lc, while for this 5 and any 
other element r' of R G , \\r'.s\\ = \\s.r'\\ < 1. («) For each r* of i^* 
there is an r of i? G , and ac of C*, such that r* = c.r = r.c. 
The "Gauge Geometric Proposition," Gge. Geom. (C, H, X, U): (i) 
R. Geom. (G, H, X, U). (ii) N. Hyp. (C, H). (Hi) There exists a subset 
X G of X*, and a subset c/ G of U*. (iv) Gge. (C, if, X, U). (v) Gge. 
(C, H, U, X). 
The "Division Proposition," Div. (G, 5, T): For each 5 of 5*, there is 
at least one t of T*, such that (i) s.t = t.s. = l c , (ii) \\s\\.\\t\\ = 1. 
The "Norm Product Proposition," N. Prod. (R): For every n, r 2 
of R, for which Ihll.yi is defined, 11^.^11 = ||fi||.||r 2 ||. 
A system (G, H, X, U) is said to be a geometric system if Geom. (C, H, 
X, U) is valid with respect to it. It is further said to admit a gauge set, 
if Gge. Geom. (G, H, X, U) also is valid. To normalize sl geometric system 
is to define \\x\\ and \\u\\ in such a manner that N. Geom. (G, H, X, U) 
is valid, while any geometric system for which N. Geom. (C, H, X, U) 
holds is said to be a normal geometric system. 
A geometric system admitting a gauge set may be normalized but 
leads to a special type of normal geometric system, since for any geometric 
system admitting a guage set, the following additional propositions may 
be proved: Div. (G, G, C), Div. (G, H t H), N. Prod. (G), N. Prod, (if), 
and for the normalized system obtained the following also are provable: 
Div. (G, X, U), Div. (C, U, X). For the general normal geometric system 
none of these six propositions may be true. 
