Physics. — "On the equivalent of parallel translation in non- 

 Euclidean space and on Riemann's measure of curvature." 

 By Dr. A. D. F'okkek. (Comiminicated by Prof. H. A. I^okentz.) 



(Communicated in the meeting of April 26, 1918). 



I. Introduction. In tlie following pages I shall try to give a 

 nnental picture of some ideas recently developed by prof. J. A. 

 Schouten before the Mathematical Society at Amsterdam which will 

 help to illustrate the meaning of a "system of axes moving geode- 

 sically", and the "geodesic differential", together with a few applica- 

 tions. ^) The great point will be to realise in a new way wat kind 

 of displacement in non-Euclidean space must be considered to corre- 

 spond to a parallel translation, tliis being an opei-ation indispensable 

 in vector-analysis to conipare vectors in different points. 



One of the characteristic properties of pure translations is this, 

 that all points of a rigid body are thereby transferred over an 

 equally long distance. This property might be used to define a 

 parallel translation, provided the rigid consists of a number of points 

 exceeding a certain minimum. If, for example, in Ihree-dimensional 

 space, we give a pi-escribed displacement lo one of the points of a 

 rigid system consisting of two or three points, it is not enough to 

 demand an equal displacement for the other jioint or points to define 

 a translated position without ambiguity. But in a Euclidean space 

 of n dimensions other motions than pui-e trauslatious are excluded, 

 if for a rigid body of no less than {In — 2) points we want all points 

 to run through equal distances. 



This will be our starting-point. We know, however, that in general 

 no body of tinite dimensions can move in curved space without 

 changing the mutual distances of its |)oints. In order to retain thie 

 idea of a rigid body we shall have to confine ourselves to bodies 

 with dimensions of the order of an infinitesimal e. 



Another and more serious difficulty arises from the fact, that we 

 cannot get all points to shift over exactly the same intinitesimal 

 distance L. We cannot but leave a mai'gin of the order of Lh^ for 

 the separate distances. Here the question arises whether in a certain 



^) Of. a treatise offered by Prof. Schouten lo be published in the transactions 

 of the Kon. Akademie: 'Die directe Analysis zur neueren Relativiteitstheorie". 



