208 
STECKER. 
which should contain only simple and easily overlooked geo¬ 
metric concepts , and which should not assume the differentia¬ 
bility of the function defining displacements .” The method of 
proof is different from Lie’s, making use in particular of 
Cantor’s theory of point-assemblages and of C. Jordan’s 
theory that a continuous closed curve, free from double 
points divides the plane into an inner and outer region. 
Hilbert had earlier * given the first nine pages of this 
paper. We can only give the important results of this paper, 
referring the reader to the memoir for proofs. 
DEFINITIONS. 
A number-plane means the ordinary plane with a rectangu¬ 
lar coordinate system x, y. 
By a Jordan curve is understood a curve free from double 
points which is continuous inclusive of its end points. If 
such a curve is closed, its interior is termed a Jordan region. 
A plane is a system of points representable in a one-valued, 
reversible way upon the points of a finite portion of the num¬ 
ber-plane, or upon a certain partial system of the same. 
We will term a point of the plane and the corresponding 
point of the number-plane images each of the other. 
To every point A of our plane corresponds a Jordan re¬ 
gion, within which the image of A lies, and which we terni 
the domain of the point A. 
For two points A and B of our plane always exists a com¬ 
mon domain. 
A displacement is a one-valued, reversible, continuous 
transformation of the images of the points of the number- 
plane into themselves, such that the direction in which a 
closed Jordan curve is traversed is unaltered. 
A displacement by which a point M remains unchanged 
is termed a rotation about M. 
* Gottingen Nachrichten for November 8, 1901, pp. 234-241. 
