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lower bulb. With regard to (7), it is indeed most necessary to employ a factor, different for each temperature, to convert observed readings to temperature; and this factor may be criticized as inconvenient to use, inaccurate in value and laborious of computation. But the use of a similar factor is likewise necessary, albeit frequently neglected, in the case of the Beckmann thermometer, whose degree, if true at 0°, is, for example, about 3% in error at 80°.5 The accuracy of the conversion factor for the newer type is dependent in part on the accuracy with which the vapor pressure of the filling liquid is known. For such liquids as would be employed, and within the ranges of temperature that come here in question, this quantity, the vapor pressure, can now be measured to better than one part in one thousand. The table that will be published for the filling liquid water, in a more comprehensive article elsewhere, was derived by differentiation from the highly satisfactory, if cumbrous, equation connecting temperature and vapor pressure of water given in the 1918 edition of the Smithsonian Meteorological Tables, and has an accuracy of just this order. The process of computing factors for, perhaps, each tenth part of a degree over a considerable range of temperature may indeed be laborious; but, once published, the factor table may be used by everyone. This inconvenience, therefore, is shifted from the shoulders of the user of the thermometer to those of him who first computed the table. It may be added that the Beckmann type is considerably more cumbrous as well as very much more fragile than the type here described, which one constructs from stout-walled Pyrex tubing.

In certain respects, therefore, it would appear that this type of differential thermometer has advantages over the Walferdin metastatic type as elaborated by Beckmann; and the question arises as to whether such other factors as are peculiar to a given application are favorable to its use. In studying its application in ebullioscopy, for example, as outlined in the article following, one finds that the important disturbing factor, peculiar to ebullioscopy in its incidence, of barometric fluctuation does not measurably affect the readings of the newer type, while such pressure fluctuations are one of the chief outstanding sources of error when the metastatic type is used. Another application in a different field may be described in the near future.

1 Menzies, A. W. C., Easton, Pa., J. Amer. Chem. Soc., 42, 1920 (1951-1956).

2 Cavendish, London, Phil. Trans. Roy. Soc., 50, 1757 (300); Walferdin, M., Bull.

Soc. geol. de France, 13, 1841-2 (113).

3 Beckmann, E., Leipzig, Zeit. physik. Chem., 2, 1888 (644); 51, 1905 (329).

Cf. Staehler, A., Leipzig, Arbeitsmethoden in der Anorg. Chemie, 3, i, 1913 (106). 'Staehler, A., Ibid., p. 108.

Cf. Smith, A., and Menzies, A. W. C., Easton, Pa., J. Amer. Chem. Soc., 32, 1910 (1412-1434).




Communicated by E. H. Moore, January 18, 1921

The notion of norm or numerical value of a complex quantity, c = a + b√-1, namely, c❘ = √a2 + b2, as it arises in algebra, has a more or less immediate generalization to more extensive matric systems. The three important properties: (1) c1 + C2| ≤ │c1| + |c2|; (2) |C1.C2| = |C1|.|C2|; (3) c is the positive square root of a positive definite quadratic form, are carried over at the expense only of replacing (2) by (2') C1.02 ≤ |61|-|02| and allowing in place of (3), (3') c is the positive square root of a positive definite form, Hermitian or quadratic. By C1.C2 in these geometrical examples is meant the inner product1 or a generalization of it. Two other generalizations of norm have been of great importance. The first of these is that of the theory of algebraic numbers, where (1) is dropped, (2) is retained, and in place of (3) one has, Norm of c is a certain function of the nth degree, n being the order of the algebraic field. The second is that of a general theory of sets as treated for example by Fréchet, where (1) is retained, (2) and (3) are dropped. The theory of integral equations as usually developed is geometrical in an infinity of dimensions and retains (1), (2), (3′). It is noted that instances in which (1) and (2) are retained usually keep (3') also. Now the importance of (3′) is chiefly that it implies (1) and (2′) with the conventions as to linearity and so forth usually assumed. The converse that (1) and (2′) imply (3') is false. It is of interest to show that most of the familiar properties of the norm may be retained, in particular (1) and (2′), when the norm is positive definite but otherwise largely arbitrary.

Three discussions bearing on this topic may be referred to. First, a geometrical study involving points but not their duals, by Minkowski, in his Geometrie der Zahlen. The great generality of the idea of norm is there beautifully developed although it is not carried so far as it is here; but since the concept of the point dual is not brought in by Minkowski, most of the ideas here discussed are not found there. A second discussion, involving inner products, but treating only a very special case of the non-quadratic norm for an infinite number of variables is given by F. Riesz in examining the convergence of bilinear forms. The third discussion involving only a scalar system, and hence without inner products, between elements of different systems is given by Kürschàk. It is perhaps the most suggestive system of scalars in the literature in which In may be less than n, for n a natural number.

The following treatment relates under one head the notions of convex region, the triangle property,' the linearly homogeneous property of distance or norm, conjugate norms," the inner product,1 convergence of

a bilinear form,5 Minkowski's gauge form and Schwarz's inequality9 (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 admitting 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, S, 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 S is replaced by T. We may employ 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, and 1 are defined. The “Addition Proposition,"11 Add. (R): (i) R is a system comprising elements, r, and a rule of binary combination, +. (ii) For any 71, 72, of R, r1+ r2 is a uniquely defined element of R. (iii) For any 71, 72, of R, r1 + 12 = r2+ r1. (iv) For any 71, 72, 73, of R, r1 + (r2 + r3) (r1 + r2) + r3. (v) There is an element OR of R (also denoted by merely O), such that for every r of R, 1 + OR = r. (vi) For a given r1 of R, there is not more than one element r of R, such that r1 + r = r1. (vii) For a given r1 of R, there is one and only element, r, of R such that r1 + T = OR-this element, r, is denoted by -71.





The "Multiplication Proposition," Mult. (R, S, T. P): (i) There are systems R, S, 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. (ii) For any r of R, s of S, t of T, r.s and s.t are uniquely defined, and s.t is an element of P. (iii) For any r1, r2 of R, and S1, S2 of S, it is true that 71. (S1 + S2) 71.51 + 71.52 and (r1 + r2).S1 = r1.S1 + 72.S1. (iv) For any (iv) For any r of R, s of S, t of T, it is true that r.(s.t) = (r.s).t. (v) For any r of R, and s of S, it is true that OR.S Or, and r.Os Or The "Commutative Proposition," Com. (C. M. L.): (i) C is a subsystem of M. (ii) For each c of C and l of L, c.l=l.c. (iii) There exists an element 1c, of C (also denoted merely by 1) such that for every l, 1c.ll. The "Regular Proposition," Reg. (L): (i) The elements of L fall into two mutually exclusive subsets Lo and L∞o. (ii) The elements of Lo fall into two mutually exclusive subsets, L. and L. (iii) OL is an element of L。. (iv) 1 if it has been defined and exists, is an element of L*. (v) For every 1 of L, -l is in the same set as l. (vi) There exists at least one l other than 1.

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The "Regular Multiplication Proposition," R. Mult. (R, S, T, P): (i) Reg. (R), Reg. (S), Reg. (T), Reg. (P). (ii) Mult. (R, S, T, P). (iii) Mult. (Ro*, So*, To*, Po*). (iv) Mult. (R。, S。, To, P.).

The "Regular Commutative Proposition," R. Com. (C, M, L): (i) Com. (C, M, L). (ii) For every c of C and I of L, co-lo, Col and c.l。 are of (iii) lc is of L*.



Lo; c*.l* is of L*; cool*, c*.l∞, and cool are of L∞.
(iv) For every l1 and ↳1⁄2 of L, if neither ↳, nor ↳ is in Loo, l+
L∞, and if both 1 and 2 are in Lo, l + l is in L。.

is not in

For each

The "Regular Conjugate Proposition," R. Conj. (C, R, S): r of R, there is a c of C and an s of S, such that (i) r.s = s.r = c.c; (ii) if r is of Ro, c is of Co, s is of S。; if r is of R*, c is of C✶, s is of S*; if r is of R c is of C s is of S

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The "Normal Proposition," Nrm. (L): (i) Each 1 of L has associated with it a unique value, . (ii) Reg. (L). (iii) For each l。, = 0; for each l*, ||1|| is a positive real finite number; for each l∞, ∞ is positively infinite. (iv) For each 1 of L, ||-|| = ||||. (v) If l exists, = 1. (vi) There exists at least one l for which ||l|| is different from unity.

The "Normal Multiplication Proposition," N. Mult. (R, S, T, P): (i) Nrm. (R), Nrm. (S), Nrm. (T), Nrm. (P). (ii) Mult. (R, S, T, P). (iii) ||s.t|| ≤ ||s||-||t|| .

The "Normal Commutative Proposition," N. Com. (C, M, L). (ii) For every c of C and I of L, ||c.l|| = ||c||.||4||. (iii) ||1c|| (iii) ||1c|| = 1. (iv) For every ↳ ≤ ||4|| + ||22||.

Com. (C, M, L): (i) for which ||c||-|||| exists, and l1⁄2 of L, ||l1 + la||

The "Normal Conjugate Proposition," N. Conj. (C, R, S): For each r of R, there is a c of C and an s of S, such that (i) r.s = s.r = c.c, (ii) ||r|| = ||c|| = ||s||.

The "Linear Proposition," Lin. (C, M, L): (i) Mult. (M, M, L, L). (ii) Mult. (M, L, M, L). (iii) Com. (C, M, L).

In the above, L is called a linear system, with M as system of multipliers, and C as commutative subsystem of M.

The "Hypernumber Proposition," Hyp. (C, H): (i) Lin. (C, C, C). (ii) Lin. (C, H, H).

In the above, H is called a system of hypernumbers with C as commutative subsystem.

The "Vector Proposition," Vect. (C, H, V): (i) Hyp. (C, H). (ii) Lin. (C, H, V).

In the above, V is called a system of vectors, with H as associated system of hypernumbers.

The "Regular Linear Proposition," R. Lin. (C, M, L): (i) R. Mult. (M, M, L, L). (ii) R. Mult. (M, L, M, L). (iii) R. Com. (C, M, L). The "Regular Hypernumber Proposition," R. Hyp. (C, H): (i) R. Lin. (C, C, C). (ii) R. Lin. (C, H, H).

The "Regular Vector Proposition," R. Vect. (C, H, V): (i) R. Hyp. (C, H). (ii) R. Lin. (C, H, V).

The "Normal Linear Proposition," N. Lin. (C, M, L): (i) N. Mult. (M, M, L, L). (ii) N. Mult. (M, L, M, L). (iii) N. Com. (C, M, L). The "Normal Hypernumber Proposition," N. Hyp. (C, H): (i) N. Lin. (C, C, C). (ii) N. Lin. (C, H, H).

The "Normal Vector Proposition," N. Vect. (C, H, V): (i) N. Hyp. (C, H). (ii) N. Lin. (C, H, V).

The "Geometric Proposition," Geom. (C, H, X, U): (i) Vect. (C, H, X). (ii) Vect. (C, H, U). (iii) Mult. (H, 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). (iii) R. Mult. (H, X, U, H). (iv) R. Mult. (H, U, X, H). (v) R. Conj. (C, X, U). (vi) R. Conj. (C, U, X).

The "Normal Geometric Proposition," N. Geom. (C, H, X, U): (i) N. Vect. (C, H, X). (ii) N. Vect. (C, H, U). (iii) 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. (C, H, R, S): (i) For each of RG there is an s of SG, such that 1c, while for this s and any other element r' of RG, ||r'.s||| ||s.r'|| ≤ 1. (ii) For each r✶ of R* r* there is an r of Rc, and a c of C✶, such that r* = c.rr.c. The "Gauge Geometric Proposition," Gge. R. Geom. (C, H, X, U). (ii) N. Hyp. (C, H). XG of X, and a subset UG of U*. (iv) Gge. (C, H, U, X).

Geom. (C, H, X, U): (i)

(iii) There exists a subset (C, H, X, U). (v) Gge.

The "Division Proposition," Div. (C, S, T): For each s of S., there is at least one t of T*, such that (i) s.t = t.s. = 1c, (ii) ||s||.||t|| = 1.

The "Norm Product Proposition," N. Prod. (R): For every r1, 72 of R, for which ||rl|-|r2|| is defined, ||r1.72||


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A system (C, 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. (C, H, X, U) also is valid. To normalize a geometric system is to define ||x|| and ||u|| in such a manner that N. Geom. (C, 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. (C, C, C), Div. (C, H, H), N. Prod. (C), N. Prod. (H), and for the normalized system obtained the following also are provable: Div. (C, X, U), Div. (C, U, X). For the general normal geometric system none of these six propositions may be true.

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