530 
MATHEMATICS: J. R. KLINE 
Proc. N. a. S. 
that its end-points are in AB and CD, (2) each arc of Hes wholly within 
A BCD A except that its end-points are on BC and DA, (3) each point of 
ABCDA, except A, B, C, and D, is an end-point of just one arc of ai or of 
just one arc of a2, (4) through each point within ABCDA there is just 
one arc of ai and just one arc of a2, (5) each arc of ai has just one point 
in common with each arc of a2. It is the purpose of the present note to 
show how this theorem may be used to establish the following theorem 
due to Schoenfiies^ — Suppose the closed curves Ji and J 2 are in continuous 
one-to-one correspondence under a transformation ir. Let Ri {i = i, 2) de- 
note the point set Ji + Jj, the interior of /,-. Then there exists a continuous 
one-to-one correspondence ir' between the points of Ri and R2, such that points 
of J I and f2 correspond as under the transformation x. 
Proof. — Let Ai, Bi, Ci and Di be four points on the closed curve /i 
such that Ai and Ci separate B]_ and Di on Ji. Let 7r(Ai) = A2, ir{Bi) = ^2, 
7r(Ci) = C2 and 7r(Di) = D2. It may easily be proved, with the use of the 
definition of a simple closed curve and the properties of continuous one-to- 
one correspondences, that A2 and C2 separate B2 and D2 on J 2- Cover 
Ri {i = 1, 2) with sets of arcs (q;,,i) and (aj,2) having the properties 
described in Professor Moore's theorem. Consider the square whose 
vertices are A (o, o), B (o, 1), C (1, 1) and D (1, o). There exists a con- 
tinuous one-to-one correspondence S between the points of that arc of 
J I from Ai to Bi which does not contain Ci and the points of the interval 
AB of the Y axis, such that = A and = B. Likewise there 
exists a continuous one-to-one correspondence between the points of 
that arc of Ji from Ai to Di, which does not contain Bi, and the interval 
AD of the X-axis such that S'(Ai) = A and 2'(Di) = D. Let us as- 
sign to every point Pi of Ri coordinates (^'pj , >''pi) in the following 
manner: Suppose Pi is a point of Ri lying neither on the interval AiBi 
nor on the interval CiDi of /i. Let Xi denote the end-point on AiDi of 
that arc of (0:1,2) which contains Pi. Then x' shall be the abscissa 
of S'(Xi). If Hi is a point of Ri on the interval AiBi of /i, then x'hi 
is zero, while, if Ki is a point of Ri on the interval CiDi of /i, x' is one. 
Suppose Qi is a point of Ri lying neither on the interval AiDi nor on the 
interval BiCi of /i. Let Yi denote the end-point on AiBi of that arc of 
(0:1,1) which contains Qi. Then ^'q^ shall be the ordinate of ^{Yi). If 
Ml is a point of Ri lying on the interval AiDi of fi, then y^Mi is zero; while, 
if Ml is on the interval BiCi of Ji, then y^Mi is one. We shall say that the 
point Pi of Ri corresponds to the point P of R, if and only if, :r'pi = Xp 
and y'pi = yp. In this manner, a continuous one-to-one correspondence 
r is defined transforming the points of R into the points of Ri. 
If H2 is a point of the interval ^2^2 of J2, then we shall say that H2 
corresponds to the same point of AB as t~'^{H2) corresponds to under 
while M2, a point of A2D2, corresponds to the same point of AD as 
7r~^(M2) corresponds to under (S')~^ In this manner, we define con- 
