Mathematics. — "-On the nuuiher of dei^iee.^ of freedom of the 

 i/eodetica/ly iiiovlng sjjsteiti and the enclodny eucJidlan space 

 loith the least possible nuinher of dhnenslons' . By I^-of. J. A. 

 ScHOUTKN. ((Jomiiiunicated by Prof. J. Cardinaal). 



(Communicated in the meeting of May 25, 1918). 



Suppose k to be a non-special curve in a Unite part A'„ of a 

 general space of n dimensions, containing no singnlar points and 

 where onl>' one geodetic line exists between two arbitrary points. 

 Assuming in a point on k a system of n mutnallj independent 

 directions, we can move this system geodetically along k. 



This geodetic moving can be geometrically defined in the following 



1/ 1 i 1 J • . ■ ,. n{ii-\- 1) 

 way. An can always be placed m a euclidian space of 



dimensions, without changing its linear element. There exists in this 

 space a space Yn developable on a euclidian space of n dimensions, 

 tangent to Xn in k. The geodetically moving directions will now 

 coincide at any moment with the directions moving parallel to 

 themselves in the euclidian space Yn. It appears analytically that 

 the known covariant differential of a directed quantity e.q. a vector 

 is a common differential judged from a geodetically moving system 

 of directions. Hence if v is a vector stationary with respect to this 

 system, v satisfies the differential-equation : 



or in co-ordinates : 



dv' 4- \v^ dx" = 0, 



I >^ I 



ami this equation then gives the analytical definition of the notion 



geodetic moving. ') A geodetic line is characterised by the property 



that its lijiear element forms at every point the same angles with 



a system moving geodetically along the line. 



') The covariant notations in this paper are the customary ones, but the contra- 

 variant characteristic numbers of tlie linear element rfx are written contravariant 

 agreeing to G. Hesslnberg, but contrary to G. Ricci and T. Levi Givita. For the 

 invariant notations, the here used direct analysis, cl'. 'Ueber die direkte Analysis 

 der neueren Relativitatstheorie", a paper presented to the "Koninkl. Akademie v. 

 W." together with this note. (Verb. Vol. 12 N'. 6). 



