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To normalize a geometric system admitting a guage set, define for every XG and G, ||xc|| = 1, ||uc|| 1, and for every c.xG and, ||C.XG|| ||c.xall = llell, llc.ucllcl. The theorem then follows without difficulty.



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For the general normal geometric system, the relative inclination, 0, of two elements, x of X*, and u of U*, may be defined by, ||x.u|| = ||x||.||u|| cos 0. Thus x of X, and u of U are mutually orthogonal if and only if ||x.u = 0. Two elements, x of X and u of U*, are mutually conjugate if and only if x.uu.x c.c while also ||x|| = ||u|| = ||cll. Two elements, x of X*, and u of U*, are mutually reciprocal if and only if x.u = u.x = lc while also |x|||||| 1. The sets (x) and (u) for which ||x|| 1, and ||u||| 1, respectively, are Convex. The relation ||r1 + r2|| ≤ ||r1|| + ||r2|| is called the Triangle Inequality of the norm. The relation ||c.l|| c. is called the Linearly Homogeneous Property of the norm. The expressions ||x|| and ||ul| are said to be Conjugate Norms and when continuous, each is a “Gauge Form." The expressions x.u and u.x are Inner Products. Each of the bilinear forms x.u and u.x




"converges" if |x|| and ||u|| are finite, since ||x.u|| ≤ ||x||.||u||, and ||u.x|| ≤ ||u||.||x||, which are statements of the general form of Schwarz's Inequality.

The following possibilities for a normal geometrical system may be emphasized: (i) The product among hypernumbers need not be commutative. (ii) The product involving two x's or two u's need not have a meaning. (iii) The system C may be an integral system in which division is not in general possible. (iv) ||nc|| may be less than n, where by nc is meant lc + 1o + ... + lc to n terms, for n > 1. (v) The conjugate of a given element need not be unique. (vi) The set of elements (x) for which ||x||| 1, may be dense but not continuous, as for example the rational points on a circle. In particular, the above theory is applicable to the Geometry of Numbers of Minkowski, to a system where C is a Kürschak valuated field, and to a system where H is a quaternion field.12



While in every normal geometric system each x of X✶ has a conjugate, u of U*, which has again the given x as its conjugate, the conjugate relation is not in general a simple one. The geometric systems for which (3') is satisfied are identified by the semi-linear property that if x1 and u1 be conjugates, and x2, u2 be conjugates, then C1.X1+ C2x2 and c'1.u1 + c' will be conjugates where C1, C2, c'1, c'2 are of C, while c and c' are themselves conjugates in a more elementary sense.

1 Due to Grassmann, Geometrische Analyse, 1847, paragraph 7, p. 16; cf. Gibbs, Vector Analysis (E. B. Wilson).

2 Suggested by Gauss, Dirichlet, Werke, I, p. 539. Extended use by Kronecker, "De unitatibus complexis," Werke I, p. 14.

Fréchet, M., Sur quelques points..., Rend. Circ. M. Palermo, 22, 1906 (1–74).

Consult A. D. Pitcher and E. W. Chittenden, "On the Foundations of the Calcul
Fonctionelle of Fréchet,” Trans. Amer. Math. Soc., 19, 1918 (66–78).
'Minkowski, H., Geometrie der Zahlen, p. 9 (Ed. 1910), Leipzig.
Riesz, F., Les systèmes d'equations linéaires, Paris (1913).

• Kürschak, J., Ueber Limesbildung..., Crelle, 142, 1913 (211).

7 Proved in elementary geometry for triangles, namely (1) above.

8 Identified since C. Wessel (1799), Gauss, and Hamilton.

9 Moore, E. H., Fifth Int. Cong. Math. 1912, I (253).

10 Moore, E. H., Bull. Amer. Math. Soc., (Ser. 2), 18, 1912 (334-362), and later papers. 11 Cf. Moebius, Der barycentrische Calcul, (1897), Werke, I.

12 Cf. Hurwitz, A., Zahlentheorie der Quaternionen, Berlin (1919).




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

1. The Equations of Laplace and Poisson.-As is well known, Bôcher considered the integral form of Laplace's equation:1

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and showed that it was entirely equivalent to the differential form

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for functions u continuous with their first partial derivatives over any "Weierstrassian" region (as Borel would call it); in fact he showed that any such solution of (1) possessed continuous derivatives of all orders and satisfied (2) at every point. One can go still further, and consider solutions of (1) which have merely summable derivatives, and of the first order, with practically the same result.

THEOREM I.—If the function u is what we shall call a "potential function for its gradient vector Vu,' ," the components of the latter being summable superficially in the Lebesgue sense, and if the equation

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is satisfied for every curve of a certain class, then the function u has merely unnecessary discontinuities, and when these are removed by changing the value of u in the points at most of a point set of superficial measure zero, the resulting function has continuous derivatives of all orders and satisfies (2) at every point.

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in which a curvilinear integral on the left is equal to a superficial integral on the right, and this equality is a generalization of Poisson's equation

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But we need not limit ourselves in the right hand member to quantities which are the integrals of functions p; instead we may take the right hand member to be any sort of additive function of point sets f(e)5 (not necessarily absolutely continuous as in (3) or even continuous), or any sort of additive function of curves in the plane f(s), of limited variation. More precisely, we consider the equation

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in which F(s) is an additive function of curves in the plane, of limited variation, with discontinuities "of the first kind."

The difference of any two solutions of (3') is a solution of (1'), and, therefore, part of the study of (3′) is the study of a particular solution. Such a particular solution is found to be the function

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defined in terms of a Lebesgue-Stieltjes integral. In this equation Mi is the point of coördinates x1, y1, M the point of coördinates x, y which is the argument of the integration, and r the distance between them; in measuring angles the sense of r is taken as M,M. This function may be shown to be the potential function of a gradient vector of which the component in the direction a is given by

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The vector given by (6) may be shown to be a solution of (3′) for every curve of the class T (specified below).

In two dimensions, a Stieltjes integral may be defined equally well in terms of both sorts of functional, and a function of curves used as well as a function of point sets for the function of limited variation. And thus if Σ' is a region enclosed in 2, with boundary S' (a curve of class г), we may speak of the function

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Slog dF(s),

2π Σ


which is identical with the function given by (5) for all points interior to Σ', provided that the equality

F(s) = f(e) (e
f(e) (e = set of points inside s),


holds for any s of I on which F(s) is continuous as a function of curves. Methods of making correspondences of type (5") will be discussed below. 2. Various Forms of Green's Theorem.-Let F(s) and G(s) be two functions of curves with discontinuities of the first kind, and U(M) and V(M) two solutions of the equations

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which may be written in the form given by (5) or (5′) plus harmonic functions (i. e., solutions of (1) after the unnecessary discontinuities have been removed). With respect to F(s) and G(s) let us introduce the symbols Tr(s) and Tc(s) to stand for their total variation functions. Then we have the following rather general result.

THEOREM II.-If the discontinuities of F(s) and G(s) are of the first kind, a sufficient condition that the equation

S. (v,Uv,V+v,UV,V)de=SUdG(s)-SUV.Vds

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shall hold for all regions σ and all boundaries s of T, internal to ', is the existence of the quantity

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A vector which satisfies the equation corresponding to (3')

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for every curve of I may be called a polarization vector for the distribution F(s); it is not necessary that such vectors shall, as in the examples already given, have potential functions. Those that we have already given satisfy also certain restrictions of integrability which we denote by condition N, as follows:

Condition N. Any component of is summable superficially, and the normal component is summable along any curve of T; in particular, the integral of the absolute value of ❤n, along any curve of I or finite number of mutually exclusive curves remains finite, less than some fixed number N, provided that the total length of the curves remains finite, less than some fixed number so.

We have then the following theorems:

THEOREM III.-Let G(s) be an arbitrary function of curves, additive and of limited variation,' and let (M) be a polarization vector for it which satisfies condition N; further let u(M) be a function continuous over Σ' with its first partial derivatives. Then for every s of class r the following equation is valid:

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THEOREM IV.-If (M) satisfies condition N and u(M) is continuous and a potential function for its vector gradient in 2, the equation

SudG(s) - Syds + S.,1,
(√xux + xu¥)do


remains valid provided that one of the following hypotheses (a), (B) or

(7) be imposed:

(a) Vu(M) is bounded.

(B) (M) is bounded.

(7) The quantities {Vu(M)}2 and {4(M)}2 are summable superficially. If in this theorem we put u(M)


log 1/r, we obtain:

THEOREM V.-If (M) satisfies condition N the equation

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is valid, given the region σ, for all points M1 except possibly those which form a point set of superficial measure zero.

One or two more special theorems are perhaps interesting. If in Theorem II we put V = log 1/r, we obtain the equation

2πU(M1) = S,{

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-log V


S{U + Slog dF(s) (11)

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which holds whenever U(M1) is the derivative of its own superficial integral, and, therefore, except for the points of a set of superficial measure

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where X is a scalar point function, so that the vector satisfies the equation

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then equation (9') reduces to the following:

S_^{Vue + V ̧ue,}do = −Sλup,dø +S_udG(5)


These theorems provide for the Stieltjes integral with respect to a function of curves a representation in terms of a Lebesgue integral and a curvilinear integral. The curvilinear integral is essentially a curvilinear integral, depending only on the contour, and differs thus from the integral around general boundaries defined by P. J. Daniell, which is a "frontier integral" not uniquely determined by the contour itself, but by some superficial point set.

The theorems of this section remain true if for σ is substituted a set of points measurable superficially in the sense of Borel, for s its frontier, and for F(s), G(s) the corresponding additive functions of point sets f(e), g(e), the frontier integrals being defined according to the method of Daniell. We shall come later to another possible method of determining frontier integrals.

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