AN EXAMPLE IN POTENTIAL THEORY. 533 



barrier for the region bounded by a large circle and an arbitrarily 

 small piece of the boundary set. 



As to the degree of generality of the regions T\, the following may 

 be said. An isolated point cannot form part of the boundary (see, 

 for instance, Osgood, Funktionentheorie, p. 565). Hence the bound- 

 ary must be dense on itself, and as it is closed, it must be perfect. The 

 above developments suggest that a criterion as to which perfect sets 

 may enter the boundary, may be found in the possibility of logarith- 

 mic spreads on the boundary set, of only positive masses, which have 

 potentials that are continuous on the boundary. 



