Mathematics. — "On the arising of a precesnon-motion oioing to 

 the non-euclidian linear element of the space in the vicinity 

 of the sun". By Prof. J. A. Schouten. (Communicated bj' 



Prof'. LORENTZ). 



(Communicated in the meeting of June 29, 191 cS). 



If k be ail curve in an n dimensional space A„ of arbitrary form, 



,n{n -\- 1) 

 tiiere will be in the euclidian space of dmiensioiis , into 



which A„ can always be placed without chawging its linear element, 

 a euclidian space Vt,, i. e. a space V,i develojiable on a plane 

 space tangent lo X„ along k. If in the euclidian space Yn a system 

 of n mutual _|_ directions be moved with its origin along k parallel 

 to itself, we find that these directions in X„ define a "geodesically 

 moving system" ^). If two arbitrary spaces are tangent to each other 

 in a curve k, it follows from this definition, that a system geodesi- 

 cally moving along k for one space, will geodesically move for the 

 other space too. A volume-element covered with mass can move in 

 A^„ as a solid body, but for some infinitesimals of a higher order. 

 If a suchlike element always remains at rest with regard to a 

 geodesically co-moving system of directions we will call \t compass- 

 body. Hence the com passbody mechanically realizes the geodesically 

 moving system. 



If k be a closed curve, the initial position will as a rule not 

 coincide with the final position, if A» is non-euclidian. Thus (he 

 position of the compassbody is changed with every rotation. Now 

 according to the investigations of K. vSchwarzschild ^) the space in 

 the vicinity of the sun is not euclidian, l)ut very slightly curved. 

 The linear element is of the form 



ds- = dW -\~R' d6^ \ R' sin^ O d(f' . . . . . (1) 



') Gf. for a more detailed exposition of the geodesically moving system : ''Die 

 direkte Analysis der neueren Relativitatstheorie". Verli. of llie Kon. Akad. v. Wet. 

 Vol. 12. No. 6 and "On the numher of degrees of freedom of the geodesically 

 moving system and the enclosing euclidian space with the least possible number 

 of dimensions". Proc. of the Kon. Akad. v. Wet. May 25, f918. 



2) Ueber das Giavitationsfeld eines Massenpunktes nach der Einstein'schen 

 Theorie, Berl. Sitzungsber. 1916, p. 189—196. 



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