512 



to the same displacement-law, we only have to interchange the 

 vectors dx and u 



Now we see, the determinant changing its sign, that this position 

 will nevei- be the same as that reached by the former particle from 

 R, nnless to = 0. 



So the application of the condition of reciprocity exclndes screw- 

 movements^). 



9. Now in the following way we can see that the condition of 

 the body's rigidity and the eqnality of the covered distances together 

 with the condition of reciprocity are snfficient to define the variety 

 of geodesic displacements without ambiguity. 



From eq. (1) and (2) we learn that the required "corrections" 

 du" must be proportional both to the components of the displacement 

 dx and of the vector u. Therefore let us put 



dn" — 2 hi d.V at . . ...... (4') 



Now, according to the condition of leciprocity we must apparently 

 have 



'hi = fits . 

 Substitute (4') in (3), and we get 



= ^ (ahyi) -'-"'" dxl u" v'" -[- ^ {a m s t) (/„,„ jA."| u<' dw v' -j h',^i »•"' dx^ uA. 

 Ox' 



Taking other indices and putting 

 we get 



K,lm = ^{b) Pal, hn , (^ajni ^= ha,ml )i 



0x1 ) 



In this equation we may regard the forms in brackets as unknown vari- 

 ables. Becauseof thesymmetry in the indices a and m theie are i ^i' (n — J) 

 of them. As the equation is to hold for an arbitrary cond^ination 



') Dr. Droste remarked to me that a screw-motion might be excluded in a 

 ditlerent maimer. Let PQ be part of a geodesic. In P and in Q take two infini- 

 tesimal planes perpendicular to the geodesic. Draw the geodesies perpendicular to 

 the first plane and intersecting if in a line-element PR. These together form a 

 "geodesic strip", which will intersect the second plane in an flement QR', PR 

 and QR' can be called "parallel" and in the same way each pair of elements in 

 the samp geodesic strip including equal angles with the geodesic PQ. 



In our chain of thought, however, geodesic lines are defined by making use of 

 the idea of geodesic displacements (see section 10), and so we cannot avail 

 ourselves of the above suggestion. 



