202 



Remarks : 



I'^f . In this connexion we may remind of the following theorem of 

 Bertrand ^) : The trajectories of a material point described under the 

 influence of a force which is directed towards a tixed centre and 

 a function of the distance to that centre are only then closed when 

 the force is proportional to that distance or inversely proportional 

 to its square. 



2"^. It is remarkable that also in a non-euclidic three-dimensional 

 space the pla"netary trajectories corresponding to the elliptic ones 

 prove to be closed, if the changes in the gravitation law and in the 

 equations of mechanics corresponding to the curvature of the space 

 are introduced. (Comp. Liebmann) ^). 



3«^ We may put the question: what does of Bohr's deduction of 

 the series in the spectra in R„ become, if ?i =|= 3. Let us change in this 

 deduction the law of electric attraction in the same way as that 

 of gravitation, and like Bohr quaniisize the moment of momentum. 

 From the preceding it it clear that for ;i> 3 only circular trajectories 

 can occur. For n ^ 4 we find infinite series and for 7i = 4 a 

 singular case which is particularly remarkable with respect to the 

 theory of quanta. (See appendix II). » 



^ 2. Translation — rotation, force — pair of forces, electric 

 field — magnetic field. 



In Rt there is dualism between rotation and translation, in so 

 far as both are defined by three characterizing numbers. This is 

 closely connected to the fact that the number of planes through the 

 pairs of axes of coordinates equals the number of axes themselves. 



In every other Rn these two numbers are not equal. The number 



of axes of coordinates is ?z. Taking two of these at a time we 



w(« — 1) n{n — 1) 



can draw through them planes. Evidently — - — > n for 



n{n— 1) 

 n > 3, while n > for n <^ö e.g. : 



for n = 2 we have only one rotation and two translations, 

 for n =: 4 we have 6 rotations and 4 translations. 

 This corresponds to the dualism which exists only for ?i =: 3 

 between the three components of the force and the three components 

 of a pair of forces which together can replace an arbitrary system 

 of forces. 



1) J. Bertrand, Gomptes Rendus. T. 77, 1873, p. 849. 



>) H. Liebmann, Nichl-euklidische Geometrie. 2e Aufl. 1912, p. 207. 



