90 
MATHEMATICS: G. C. EVANS 
Proc. N. A. S. 
b 2 u b 2 u 
+ d y 2 = (4) 
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 neces- 
sarily absolutely continuous as in (3) or even continuous), or any sort 
of additive function of curves in the plane /(s), 6 of limited variation. 
More precisely, we consider the equation 
V n uds = F{s) (3 r ) 
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 
u(Mi) = 1 f log I df(e) (5) 
Air J s 7 
defined in terms of a Lebesgue-Stieltjes integral. In this equation Mi 
is the point of coordinates %\, yi, M the point of coordinates 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 MiM. 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 
V B «(M0 jj^d}{e). (6) 
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 2' is a region enclosed in 2, with boundary S' (a curve of class T), we 
may speak of the function 
«'CMi) = ~ f log - dF(s), (5') 
which is identical with the function given by (5) for all points interior to 
2', provided that the equality 
F( s ) — f( e ) ( e — se t of points inside s), (5 ;/ ) 
holds for any 5 of T on which F(s) is continuous as a function of curves. 
Methods of making correspondences of type (5'0 will be discussed below. 
2. Various Forms of Green's Theorem. — Let F(s) and G(s) be two func- 
tions of curves with discontinuities of the first kind, and U(M) and V(M) 
two solutions of the equations 
