532 KELLOGG. 



It is also negative at all points within a circle C with S as diameter, 

 for the integrand is never positive in this circle. On and outside any 



larger concentric circle, of radius R, L{P) > log ( R — — J. 



5. The Barriers for T*. 



We now shift our origin of coordinates back to its position of sym- 

 metry in the set S, and denote by Ui(x, y) the function which L{P) + wi 

 thus becomes. We then consider the subset, s, of S, namely the points 

 common to S and one of the intervals, i, of S„; let (a, 0) be the center 

 of this interval. Then Uz(x,y) = l\ {a + 3".r, 3"//) is harmonic in 

 the whole plane except at the points of s, it is never negative, and 

 within the circle c, which has i as diameter, c7 2 (.r, ?/) ^ m. On and 

 without any circle with (a, 0) as center, and radius R/3 n , f T 2 (.r, y) > 



log ( R — - ). Or, with S n R in place of R, we may state that on and 



outside a circle of radius R about (a, 0), fV.r, //) > ralog3 + log 



( ^ — o~lw )• ^ ne ^ east va l ue possible of R on K is - + t . Hence, 



on /v, f : 2 (x, ?/) > «log3. 



Now let 5 be any point of S. We identify the above subset s with 



one containing #. The maximum of BP on K is 5/2. Hence tjCr, y) 



5U 2 (x,y) i dd »- • i • + 



= ; — -—exceeds i>i J on 7v, is harmonic except on s, is never nega- 



//21og3 



tivc, but within c is less than 5?»/?;21og3. Finally, U B , n (.r, y) = 



U3 (%> y) + ^T exceeds BP on s and 7v, and hence in T*. As P ap- 

 o 



proaches B, this function approaches a value which vanishes with l/?i, 

 and thus has all the properties of the barrier functions, so that the ex- 

 istence of the barrier set, and with it the fact that T* belongs to [7"i] 

 have been established. 



6. Remarks. 



The preceding considerations give an example of the possible use- 

 fulness of the notion of barrier. As the nature of the region T which 

 makes it a 1\ depends on the boundary "im Kleinen," a generalization 

 of the idea of barrier may be indicated which may be useful, namely a 



