231 • 



this Qq is a constant. Locally however it may be variable: the 

 world-matter can be condensed to bodies of greater density, or it 

 can have a smaller density than the average, or be absent altogether. 

 According to Einstein's view we mnst assume that all ordinary 

 matter (sun, stars, nebulae etc.) consist of condensed world-uiatter, 

 and pei'haps also that all world-matter is thus condensed. 



3. To begin with we will neglect gravitation aud consider only 

 the inertial field. The three-dimensional line-element is in the two 

 systems A and B: 



do'' = dr' -f i^" sm* — [<Zi|r 4- sin' ^^ dtP]. 

 R 



If W is positive and finite, this is the line-element of a three- 

 dimensional space with a constant positive curvature. There are 

 two forms of this, viz: the space of Riemann ^), or sphetHcal space, 

 and the elliptical space, which has been investigated by Newcomb "). 

 In the spherical space all "straight" li.e. geodetic) lines which start 

 from one point, intersect again in another point: the "antipodal 

 point", whose distance from the first point, measured along any 

 of these lines, is nR. In the elliptical space any two straight lines 

 have only one point in common. In both spaces the straight line is 

 closed; in the spherical space its total length is 2jtR, in the ellip- 

 tical space it is nR. In the spherical space the largest possible 

 distance between two points is nR, in the elliptical space ^nR. 

 Both spaces are finite, though unlimited. The volume of the whole 

 of spherical space is 2a'R^, of elliptical space ji^R^. For values of 

 r which are small compared with R, the two spaces differ only inap- 

 preciably form the euclidean space. 



The existence of the antipodal point, where all rays of light 

 starting from a point again intersect, and where also, as 

 will be shown below, the gravitational action of a material point 

 (however small its mass may be) becomes infinite, certainly is a 

 drawback of the spherical space, and it will be preferable to assume 

 the true physical space to be elliptical. 



The elliptical space can be projected on euclidean space by the 

 transformation 



r ^= Rta7iyi . . . . . . . . (5) 



The line-element in the systems A and B then becomes 



1) Ueber die Hypothesen welche der Geometrie zu Grunde liegen (1854), 

 ■^) Elementary theorems relating to geometry of three dimensions and of 

 uniform jiositive curvature, Grelle's Journal Bd. 8.S, p. 293 (1877). 



