Vol.. 7, 1921 
MATHEMATICS: G. C. EVANS 
89 
Consult A. D. Pitcher and E. W. Chittenden, "On the Foundations of the Calcul 
Fonctionelle of Frechet," Trans. Amer. Math. Soc, 19, 1918 (66-78). 
4 Minkowski, H., Geometrie der Zahlen, p. 9 (Ed. 1910), Leipzig 
5 Riesz, F., Les syste~mes d' equations lineaires, Paris (1913). 
6 Kiirschak, 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). 
PROBLEMS OF POTENTIAL THEORY 
By Professor G. C. Evans 
Department of Mathematics, Rice Institute 
Communicated by E. H. Moore, January 18, 1921 
1. The Equations of Laplace and Poisson. — As is well known, Boeher 
considered the integral form of Laplace's equation: 1 
and showed that it was entirely equivalent to the differential form 
for functions u continuous with their first partial derivatives over any 
"Weierstrassian" region (as Borel would call it 2 ); 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 func- 
tion for its gradient vector Vw," 3 the components of the latter being sum- 
mable superficially in the Lebesgue sense, and if the equation 
V n uds = 0 (1) 
is satisfied for every curve of a certain class, 4 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. 
Let us pass on to the equation 
is* -J* (3) 
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 
