Vol. 6, 1920 
MATHEMATICS: J. R. KLINE 
531 
tinuous one-to-one correspondences, T and T', between the points of 
A2B2 and AB and between the points of A1D2 and AD, respectively. We 
shall assign to every point P2 of coordinates {%" y" follow- 
ing manner. Suppose P2 is a point of R2, lying neither on the interval 
A2B2 nor on the interval C2D2 of J 2- Let X2 be the end-point on A2D2 
of that arc of (0:2,2) which contains P2. Then x" p.^ shall be the abscissa 
of (rO~^^2- If H2 is a point of Ri on the interval A2B2 of J2, then ^t;"^^ 
is zero, while, if H2 is a point of on the interval C2-D2 of 72, then x'^ 
is one. Suppose O2 is a point of R2, lying neither on the interval A2D2 
nor on the interval ^2^2 of /2. Let Y2 denote the end-point on A2B2 
of that arc of (q!2,i) which contains Q2. Then y"Q^ shall be the ordinate 
of (7) "^72. If M2 is a point of R2 on the interval of J 2, then :v''ji/2 is 
zero; while, if M2 is a point of R2 on the interval 52C2 of h, then y" Mi is 
one. We shall say that a point of P2 of i?2 corresponds to a point P oi R 
if and only if, :r"p, = ^t:p and y'^p.^ = yp. In this manner a continuous 
one-to-one correspondence V' is defined transforming the points of i^into 
the points of R2. It should be noted that the correspondence FT"^ is a 
continuous one-to-one correspondence transforming the points of Ri 
into the point of R2 in such a manner that the points of BiAiDi and 
B2A2D2 correspond as under ir. This is, however, not necessarily true of 
the points of BiCiDi and B2C2D2. 
Let P2 be any point of ^2^2, while 7r~i(P2) = Pi. Let Qi denote the 
end-point on CiDi of that arc of (an) which contains Pi. Let 7r(0i) = O2. 
Let (rO~HQ2) = Q. Let PQ denote the interval from P to Q of the 
straight line joining P and 0, where P = (rO"H^2). Let r'(PQ) be the 
arc P2Q2. Do this for every point of ^2^2. Then let P'2 denote any 
point of ^2-^2 and let -k"^ (P'2) = P'l- Let 0\ denote the end-point on 
jBiCi of that arc of (0:1,2) which contains P'l. Let Tr{Q'i) = Q'2, while 
(rO-H^y = P' and (rO'KQy = Q'- Let P'Q' denote the interval 
between P' and Q' of the straight line joining these two points. Let 
V'{P'0') be the arc P'2(3'2. Do this for every point P'2 of A2D2. In this 
manner two new sets of arcs (o:'2i) and (0:^22) are obtained having all the 
properties of the original sets of arcs (a2i) and (0:22) and having the addi- 
tional property (6) that if t is any arc of either set with end-points H2 
and K2 and if x~'H^2) = Hi and 7r~^(i^2) = Ki, then one of the arcs of 
the set (oiii) -f (0:12) joins Hi and Ki. 
Using these new sets of arcs, (q:'2,i) and {a' 22), we may define in a manner 
similar to that used in defining F', a new continuous one-to-one corre- 
spondence F" transforming the points of R into the points of R2. It is 
clear that F'^F)"^ is a continuous one-to-one correspondence transform- 
ing the points of Ri into the points of R2 in such a manner that the points 
of J I and J2 correspond as fixed by tt. 
1 Cf. Moore, R. L., Trans. Amer. Math. Soc, 20, 1919 (176). 
2 Cf. Schoenflies, A., Math. Ann. Leipzig, 62, 1906 (324). 
