364 MATHEMATICS: R. L. MOORE 
Theorems V and VI hold word for word when the ternary continued frac- 
tion has a finite number of non-periodic partial quotients. 
Some progress has been made in the problem of finding a periodic ternary 
continued fraction which shall be the development of a given cubic irration- 
ality, but the results are not yet in final form. 
1 Lagrange, Mem. Berlin, 24, 1770 Oev. II, 74. 
2 Jacobi, Ges. Werke, VI, 385-426. 
3 Bachmann, Crelle, 75, 1873, (25-34). 
4 Berwick, Proc. London Math. Soc, (Ser. 2), 12, 1913. 
A CHARACTERIZATION OF JORDAN REGIONS BY PROPERTIES 
HAVING NO REFERENCE TO THEIR BOUNDARIES 
By Robert L. Moore 
Department of Mathematics, University of Pennsylvania 
Communicated by E. H. Moore, September 18, 1918 
SchoenfTies 1 has formulated a set of conditions under which the common 
boundary of two domains will be a simple closed curve. A different set has 
been given by J. R. Kline. 2 Caratheodory 3 has obtained conditions under 
which the boundary of a single domain will be such a curve. In each of these 
treatments, however, conditions are imposed 1) on the boundary itself, 2) re- 
garding the relation of the boundary to the domain or domains in question. 
In the present paper I propose to establish the following theorem in which 
all the conditions imposed are on the domain R alone. 
•Theorem. In order that a simply 4 " connected, limited, two-dimensional do- 
main R should have a simple closed curve as its boundary it is necessary and suf- 
ficient that R should be uniformly connected im kleinen. 5 
Proof. Suppose the simply connected domain R is connected im kleinen. 
Let M denote the boundary of R, that is to say the set of all those limit points 
of R that do not belong to R. 
I will first show that M can not contain two arcs that have in common only 
one point, that point being an endpoint of only one of them. Suppose it 
does contain two such arcs EFG and FK with no point in common except F. 
Let a and /3 denote circles with common center at F, and with radii n and r 2 , 
respectively, such that n>r 2 and such that E, K and G lie without a. By 
hypothesis there exists a positive number 5 r such that if X and Y are two points 
of R at a distance apart less than d r then X and F lie together in a connected 
subset of R that lies wholly within some circle of radius r 2 . Let y denote a 
circle with center at F and with a radius less than one half of d r and also less 
than fi — r 2 . It is clear that if two points of R are both within y then they 
lie together in a connected subset of R that lies wholly within a. There 
