270 
MATHEMATICS: R. L. MOORE 
established by Kossel between the frequencies of the K and L lines 
hold true for the tungsten lines. 
A more detailed account of these and other experiments on the tung- 
sten spectrum, including the distribution of energy in the continuous 
spectrum, will be published shortly in the Physical Review. 
iMosely, Phil. Mag., 26, 210 and 1024 (1913); 27, 710 (1914). 
2 1. Maimer, Phil. Mag., 28, 787 (1914). 
3 W. H. Bragg, Phil. Mag., 29, 407 (1915). 
4 Duane & Hunt, Physic. Rev., 6, 166 (1915). 
5 Barnes, Phil. Mag., 30, 368 (1915). 
6 Rutherford, Barnes and Richardson, Phil. Mag., 30, 339 (1915). ' 
7 Hull, Physic. Rev., 7, 156 (1916). 
8 Gorton, Physic. Rev., 7, 203 (1916). 
3 Kossel, Ber. D. Physik. Ges. 16, 953 (1914). 
" Webster, these Proceedings, 2, 90 (1916). 
ON THE FOUNDATIONS OF PLANE ANALYSIS SITUS 
By Robert L. Moore 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF PENNSYLVANIA 
Recdved by the Academy, March 30, 1916 
The notions point, line, plane, order, and congruence are fundamental 
in EucHdean geometry. Point, Hne and order (on a Hne) are funda- 
mental in descriptive geometry. Point, Hmit-point and regions (of 
certain types) are fundamental in analysis situs. It seems desirable 
that each of these doctrines should be founded on (developed from) 
a set of postulates (axioms) concerning notions that are fundamen- 
tal for that particular doctrine. EucKdean geometry and descrip- 
tive geometry have been so developed.^ The present paper contains 
two systems of axioms, 22 and 2;,, each of which is sufficient for a consid- 
erable body of theorems in the domain of plane analysis situs. The 
axioms of each system are stated in terms of a class, S, of elements called 
points and a class of sub-classes of S called regions. 
On the basis of 22, the existence of simple continuous arcs"^ is proved as 
a theorem. 
The system 22 contains an axiom (Axiom 1) which postulates the exist- 
ence of a countable sequence of regions containing a set of subsequences 
that close down in a specified way on the points of space. Among other 
things this axiom implies that the set of all points is separable.^ 
The system 23 is obtained from 22 by replacing Axioms 1,2, and 4 by 
three other axioms, Axioms 1', 2\ and 4' respectively. Here Axiom 1' pos- 
tulates the existence, for each point P, of a countable sequence of regions 
