Physics. — "In what ivay does it become manifest in the funda- 

 mental laws of phifsics that space has three dimensions?" 

 By Prof. Dr. P. Ehrenfest. (Communicated bj Prof. Dr. 



H. A. LORENTZ). 



(Communicated in the meeting of May 26, 1917). 



Introduction. 



"Why has our space just three dimensions?" or in other words: 

 "Bj which singular characteristics do geometries and physics in 

 E^ distinguish themselves from those in the other /?„'s?" When put 

 in this way the questions have perhaps no sense. Surely they are 

 exposed to justified criticism. For does space "exist"? Is it three- 

 dimensional? And then the question "why"! What is meant by 

 "physics" of R, or i^^? 



I will not try to find a better form for these questions. Perhaps 

 others will succeed in indicating some more singular properties of /?, 

 and then it will become clear to what are the "justified" questions 

 to which our considerations are fit answers. 



§ 1. Gravitation and planetary motion. 



As to the planetary motion, roe shall see, that there is a difference 

 between R^ and R^ cis loell as between R^ and the higher Rn's ivith 

 respect to the stability of the circuiar trajectories, hi R^ a.small disturb- 

 ance leaves the trajectory finite if the energy is not too great; in R, 

 on the contrary this is the case for all values of the energy. In R„ 

 for ?^>3 the planet falls on the attracting centre or flies away 

 infinitely. In Rn for n ^ ^ there do not exist motions comparable 

 iviih the elliptic motion in R^, — all trajectories have the character of 

 spirals. 



For the attraction under the influence of which a planet circulates in 



the space /?„, we put x -^^ ; to this corresponds tor n ^ 2 a potential 



energy : 



Mm 

 V(r) = -^ (1) 



{n — }i)r" ^ 



We deduce this law of attraction from the differential equation 

 of Laplace — Poisson. The means : we assume the force to be 



