234 

 III the system B /> has a minimum value 



a 



Po — 



R' 



which it reaches for // = cc, i.e. r={:nR. For values of r exceeding 

 this distance /> increases again, and for r=^:rR we siionld find 

 p = -{- 90°. 



Already in J 900 Schwahzschii.d ') gave a discussion of the possible 

 curvature of space, starting from the formulae (9 A) and (9/i). For 

 the system B we can from the observed paiallaxes') derive a lower 

 limit for R. Schwarzschild finds /?>4.10" astronomical units. In 

 the system A the measured parallaxes cannot give a limit for R. 



In both systems we can, of course, derive such a limit from 

 distances which fiave been determined, or estimated, otherwise than 

 from the measured- parallaxes. These distances must, in the elliptical 

 space, be smaller than ^ rr/s*. This undoubtedly leads to a much 

 higher limit, of the order of 10'" or more. 



6. The straight line being closed, we should, at the point of 

 the heavens 180° from the sun, see an image of the back side of 

 the sun. This not being the case, practically all the light must be 

 absorbed on the long "voyage round tlie universe". Schwarzschild 

 estimates that an absorption of 40 magnitudes would be sufficient *). 

 If we adopt the result found by Shaplky'), viz. that the absorption 

 in intergalactic space is smaller than 0"\01 in a distance of 1000 

 parsecs, then for an absorption of 40 mags we need a distance of 

 7.10^' astronomical units. In the elliptical space we have thus 

 «>|.10-. 



In the system A we can suppose that this absorption is produced 



1) Ueber das zulassige Krümniungsmauss des Raumes, Vierteljahrsschrifl der 

 Astron. Gesellschaft, Bd. 35. p. 337. 



2) The meaning is of course actually measured parallaxes, not parallaxes derived 

 by the formula p — air from a distance which is determined from other sources 

 (comparison of rad al and transversal velocity, absolute magnitude, etc.). Schwarz- 

 schild assumes that there are certainly stars having a parallax of 0" 05. All 

 parallaxes measured since then are relative parallaxes, and consequently we must 

 at the present time still use the same limit. 



^) It might be argued that we should not see the back of the actual sun but 

 of the sun as it was when the light left it. We could thus do without absorption, 

 if the time taken by light to traverse the distance ttR exceeded the age of the 

 sun. With any reasonable estimate of this age, we should thus be led to still 

 larger values of R. 



*) Contributions from the Mount Wilson Solar Observatory Nrs. 115 — 117. 



