Vol.. 7, 1921 
MATHEMATICS: G. C. EVANS 
97 
A different sort of application of these ideas is to the Dirichlet problem : 
to determine a harmonic function throughout a Weierstrassian region by 
assigned frontier values. For this purpose consider an arbitrary simply 
connected T region as defined by Osgood 12 whose boundary consists of 
more than one point ; for simplicity we may as well restrict it to the finite 
domain. 
Let M 0 be an interior point of T and g(M 0 ,M) its Green's function. 
Let h(M 0 ,M) be the conjugate function of g(M 0 ,M). These functions are 
both harmonic at every point of T, the point M 0 excepted ; and the follow- 
ing facts can be stated with reference to the frontier of T : 
To an accessible point on the frontier corresponds a value of h , h = 
C (and perhaps more than one value), but to two different accessible 
frontier points cannot correspond the same value C 13 We see then 
that if a value of h corresponds to an accessible frontier point, the point 
is uniquely determined. But it can be shown that all values of h except 
possibly some which constitute a set of measure zero actually do corre- 
spond to accessible points on the frontier. 
For a given point M 0 of T we thus set up a correspondence between 
the curves h (M 0 ,M) = C and the accessible points on the frontier of T. 
Consider functions u{M) which are limited and continuous in T, and 
whose first partial derivatives with their squares are summable over T. 
Form the integral 
along a curve joining a point, P h on h = C\ to a point, P 2 , on h = C 2 and 
composed of points M given by the relation g(M 0f M) = ^/{h) t where 
\p(h) is an arbitrary continuous function of values of h between C\ and C 2 . 
As \p{k) is let approach zero remaining always not-negative and less than 
some finite N, it is seen by comparison with additive functions of point 
sets that the integral (24) has a limiting value; and the Stieltjes integral 
in the right hand member defines in the limit an absolutely continuous 
function U(h) of h 2 , which is therefore the integral of its derivative with 
respect to h, this derivative being a summable function u(h). It may be 
shown that as M travels along a curve, h = C, we have lim u(M) = u(C) 
except for values of C which constitute a set of linear measure zero. In 
other words, the frontier values of u(M), taken in this sense, are summable 
with respect to h, and the frontier integral is properly a Stieltjes integral 
of frontier values. 
Vice versa, given u limited and summable with respect to h on the 
frontier of T there is one and only one bounded function u(M), con- 
tinuous and harmonic in T, with first partial derivatives and their squares 
summable over T, and such that it takes on the frontier values u(h) 
"almost everywhere" in the above-described manner. 
(24) 
