5()H 



direction onlv one displacenient can be effected in which tins approxi- 

 uiation to the exact ecpialii.v is reah«ed?Thi.s, however, cannot be expect- 

 ed, since in the special case of I^]uclideaii space not only pui-e transla- 

 tions but screw-disphiceinents loo are allowed by leaving this margin. 

 Therefore a second propeiMy of pure translations is required, fit to 

 exclude those screvv-displacenients. 



This property is found in the fact that the sliifts are not only 

 equal, l»ut also parallel to one another. This amounts to a certain 

 reciprocity between translations in different directions. Consider two 

 translations. i)y which a point /-^ is ti-ansferred to neighbouring points 

 Q and H respectively. The first translation will carry point R to 

 the same place where the second translation will carry point Q. 

 This property indeed excludes screw-displacemei.ls. 



In the following pages we shall first give a summary of the 

 results arrived at in this paper, and afterwards (§ 6) give the ana- 

 lytical formulae. For exam|)les we will maiidy take those of three- 

 dimensional space. The results, however, will hold good, independent 

 of whatever nund^er {n) of dimensions we choose to ascribe to 

 our space. 



2. (Teodesic displaceuieiU, Let us define an infinitesimal r/(;z</ as a7i 

 aggregate of particles, which keep their mutual distances unchanged éwv'm^ 

 their motions. (Jne of these points we may choose as a central, and 

 imagine the other |)oints defined by the ends of infinitesiuial vectors 

 from this central |>oint, these vectors having constant lengths (of 

 the order s) and including constant angles. The number of points 

 must be no less than (2?i--2), hence the number of vectors (2?? — 3), 

 no n of them being situated together in a space of (/i — J) dimensions. 



We imagine this rigid to execute motions so as to shift the 

 central particle from a starting point /•* to neighbouring points over 

 distances of the order A. 



It appears possible {^ 7) to indicate a certain variety of motions 

 m which, firstly the shifts of all the other points of the rigid, up to & 

 margin of the order Ag", equal the shift of the central point, and, 

 secondly, there exists a certain reciprocity which becomes apparent 

 when we observe tiüo arbitrarily chosen motions belonging to the 

 variety, which shift the central particle, let us say, from P to Q 

 and from P to R, and when we notice the displacements of the 

 particles havi)ig their starting points in R and Q respectively. The 

 particle from R in the motion [PQ) niill reach the same point attained 

 by the particle from Q in the other motion [PR). 



The two conditions specified determine without ambiguity a variety 



