1180 



Reduction of the fow-diniensional problem to a three-dhiiensional 07ie. 



Comparing (17) and (21) we see, that the motion of a material 

 point can be convsidered as a motion according to classic mechanics 

 ' in a space with a line-element dl and a force-function: 



^' = |o^ ...... . ^)(24) 



Using the line-element calculated by Schwarzschild: 



ds^ — c^\ 1—'^ \dt' — f//' 



di' = (l + " Jt/r' +r- sin^ Odif' ^- r'dS' \ 



(25; 



for the gravitation field of a single centre, we find 



L= =r X* , . . . . . . (20) 



2/- /■ 



where x represents the gravitation constant and .1/, the mass of the 

 centre, while the reduction holds for velocities of the order of 



« , «* 



magnitude ot c, when quantities of the order - are neglected. 

 r r' 



Therefore e(piatlons (18) can give us the motion, and thus also e.g. 

 the perihelium motiou of" Mercurius just as well as equations (15). 

 In this connexion we may therefore say, that the perihelium motion 

 is "line'" to the being iion euclidian of the space in the neighbour- 

 hood of ill e sun and this gixes an "explanation" of the phenomenon 

 from the point of view of classic though non-euclidian mechanics. 



Passage to another Ihie-e/ement. 



Besides the fundamental tensors : 



e,- Cr f r- sm' <9 e-j; Qv + f^ e^ e^ 1 



1 1 



e',e';- \ e'vc'v i--— e'e e'e 



(27) 



M'l a , =11 M' = 'S e, e'. , . . (28) 



1) in his paper Statica Einsteiniana, Rend, del Lincei 21 (17) 449— 470, T. Levi 

 CiviTA has shown that without neglection of quantities of the order c' the four- 

 dimensional problem can be reduced to a three dimensional one. A new auxiliary 

 variable l* however is used then as "time". A. Palatini has applied this to the 

 line-element of Schwarzschild and the perihelium motion : Lo spostamento del 

 pericliu ili niercurio etc, Nuovo (jimenlo 1 -i (17) 1"2 — 54. 



