240 



For r = ^ jt R, i.e. for the largest distance which is possible in 

 the elliptical space, we have thus y = 0. For still larger distances, 

 which are only possible in the spherical space, y becomes positive, 

 and finally for r ^=z n R we should have g^^ = oo, however small 

 the mass of the sun may be, as has already been remarked above 

 (art. 3). 



If now from (J4) and (15) we endeavour to determine n and /i, 

 we are met by difficulties. It appears that the equations (13), (14), (15) 

 are contradictory to each other. If we make the combination 



(13) + (14) -2. (15) -i^ «an x-y^ 



dr 



we find 



y'tani = i) (17) 



which is absurd. If the equations were exact, they should, in con- 

 sequence of the invariance, be dependent on each other. They are 

 however not exact, since on the right-hand-sides terms of the order 

 of f . >t ^ have been neglected, f being of the order of «, /?, y. In 

 the world-matter we have^) x (> r= x (>o = 2-?., and these corrections 

 can only be neglected if X is also of (he order 8. This has not been 

 assumed in the equations (13), (14), (15). If we wish to assume it, 

 then we must also develop in powers of /. We can then use the 

 coordinates r, lJ^ x>. We put thus 



a=\-\-a , h^ X\\ -\- ii) , f= 1 + y. 

 The equations, in which now the accents denote differentiations 

 with respect to r, then become, to the first order 



2 



r" A — y' ^ ^9i^ 



r 



^" + - ^' («• + y') rz. — X(), - 2;., 



r r 



/? - « + r (I?' + y') = - Xx\ 



which are easily verified to be dependent on each other. 

 We can thus add an arbitrary condition. If we take e.g. 



a = 2^, 

 then we find, to the first order, outside the sun 



a=-2Xx' ^~ , ii? = — Ar^+i- , y = --, 

 r r r 



1) Of course, if beside the world-matter there is also "ordinary matter", i. e. 

 if the density of the world matter is not constant, this relation is also only 

 approximatively true, and requires a correction of the order A . f. (See also art. 11). 



