535 



naturally measured distance relative to the centre of the sun, while 

 « = 2.945.10^' cm. The circle described by a definite point P: 



(5) 



z z= 2\/a{R~a) 



consequently represents a circle in the diametral surface, having 

 the same centre as tiie sun. if the sun be looked upon as a globe, 

 filled with an incompressible liquid, a diametral surface within the 

 sun will have the same linear element as the globe-surface, which 

 touches the described rotation-surface in a parallel-ciicle with a radius 

 R„. This radius Ra too may approximately be equalized with the 

 astronomically measured radius of the sun. If the described rotation- 

 surface (4) is rotated in the four-dimensional xyzn space around 

 the yz plane, there arises a curved three-dimensional space with the 

 linear element (I) when 6 is the angle of rotation, measured from, 

 the yzu space. 



We shall now investigate the motion of a compassbody moving 

 in the circle (5) around the sun. For this purpose it is sufficient to 

 find a space, tangent to (1) in (5), and in which the geodesic motion 

 can conveniently be indicated. We now make the tangent line PQ 

 rotate with the parabola. That tangent line describes a cone with 

 the linear element: 



dR^ 



ds^= h ^'^(/ ' ....... (6) 



cos* / 



in which equation / only depends on the definitely selected point 

 P, and therefore is a constant. With the second i-otation there 

 arises from this cone a space with a linear elemeiit : 



dR' 



ds^ = h R'd&' + R' sin' 6 d<f' ..... (7) 



cos* X 



in which -/ is once more a constant. The linear element of a eucli- 



dian space may be expressed (in polar-coordinates) R', (p', 6' -. 



ds" = dR!' + R!* d6'^ + E- sin' 6' d^f' .... (8) 



and by the substitution : 



R = R cos •/ \ 



cosy^ (9) 



cosx 

 (7) passes into : 



ds* =dR* + R'* dS'* 4 R* sin' d<p" . . . (10) 



cos X 



