233. 



although the system of reference was determined by tlie condition 

 that it should be constant in the radial direction. 



In this system of reference also all //„, are zero at infinity in the 

 system B, and in A all g,,., excepting (/,„ which remains 1. 



To /i=roo corresponds r = ^ ji U. The whole of elliptical space 

 is therefore by the transformation (7) projected on the whole of 

 hyperbolical space. B'or values of r exceeding {:tR, h becomes 

 negative. Now a point (— A, ij% »'^) is the same as [h, jt — 1|', rr-h/)-). 

 The projection of the spherical space therefore fills the hyperbolical 

 space twice. The same thing is true of the projection, by (5), of 

 the elliptical and spherical spaces on the euclidian space. 



5. Let the sun be placed in the origin of coordinates, and let 

 the distance from the sun to the earth be a. We still neglect all 

 gravitation. 



In the system A the rays of light are straight lines, when de- 

 scribed in -the coordinates >-, if% ^, 1- e. in the elliptical or spherical 

 space. 



In the system B the same is true for the coordinates h, i|', ^ 

 (hyperbolical space). 



In the system A, consequently to triangles formed by rays of 

 light, the ordinary formulas of spherical trigonometry are applicable. 

 The parallax p of a star whose distance from the sun is r, is thus 

 given bv the formula 



a r 



tan p = sm — cot — , 



The square of ajR being negligible, we can write this 



ara 

 f = R""R=^ <«^> 



In the system B we have similarly, in the coordinates h,xp,^: 



a h 



tan p ■=. sink — coth — , 

 ^ R R 



or 



ah a a \ y/ r* 



^ R R Rsini X V ^ R^ 



(95) 



In the system A we have consequently p = for r = ^ tiR, i.e. 

 for the largest distance which is possible in the elliptical space. If 

 we admitted still larger distances, which are only possible in the 

 spherical space, then p would become negative, and for r = nR 

 we should find p = — 90°. 



16 



Proceedings Royal Acad. Amsterdam. Vol. XX. 



