AN EXAMPLE r.V POTENTIAL THEORY. 529 



quotients of F in T -\- 7", '/'', being the boundary set of T. Then, 

 for any n, U n lies between F\ B LU B)t (P) and F(B) + LU B , t (I'), 

 since its values on the boundary of t n do, and these functions are 

 harmonic. Hence U lies between these functions, for every positive e, 

 and hence approache i / ' B). 



If F is not given as a polynomial, we first extend its definition so 

 that it shall be continuous in the whole plane' 5 We may then approxi- 

 mate to F by a sequence of polynomials over a closed region contain- 

 ing 7" in its interior, the approach being uniform. The correspond- 

 ing harmonic functions of 7, U\, I , I '■, . . . will then approach uni- 

 formly in 7' a b.armonic -function with the required boundary valu< 

 If 7" extends to infinity, we have merely to replace, in the above rea- 

 soning, the polynomials In- rational functions with a single pole not 

 in V. 



Conversely, a barrier set exists for each region T\, one, indeed, 

 independent of e. For, to each boundary point, B, corresponds a 

 harmonic function of T\, Ub(P), determined by the boundary values 



F(P) = HI'. To see that Ub(P) — l' ( P) is uever n< e, we have 



only to recall Gauss' mean value theorem for harmonic functions, and 

 to notice that the above difference, a1 any point, P, of 7, exceeds its 

 arithmetic mean on any circle in 7'i with center at I'. Such a Func- 

 tion can have no minimum in 7'i, and as the present one vanishes 

 on 7V and is continuous in 7\ + 7'/, it can, accordingly, never be 

 negative in this region. 



4. A Logarithmic Spread on S. 



It remains to establish th( i tence of a harrier set for T*. There 

 is do difficulty connected with the barrier functions for the points B 

 of K, since a harrier for the open circle evidently is a harrier for 7*. 

 As a preliminary step in the establishment of barriers for the points 

 of S, we consider the logarithmic potential of a certain spread of 

 attracting matter on S. It will simplify notation to shift the origin 



of coordinates for the present to the point [ — — , ()]. The potential 



r 



in question is then the Stieltjes integral L(P) — J logr d/x(x), where 



I I or the possibility of this, see Lebesgue, I. c. Rendiconti <li Palermo, or, 

 Carath^orody, Vorlesungen iiber reele Funktionen, Leipzig, L918, pp. 617-620. 



