507 



of motions which we may call ''geodesic displacements" of the 

 infinitesimal rigid. Thej are the substitutes for parallel displacements') 

 in Euclidean extensions. We may assign the name "compass-rigid" 

 to a small rigid body that cannot move but in the geodesic manner 

 defined. It must be understood that a compass-rigid which, after 

 a displacement, returns to its starting-point by the same way, will 

 on arrival be in its starting-position too. If, however, it returns by 

 a circuit, it generally will not be in its starting-position again on 

 arrival. 



3. Geodesic differential. If we want to compare two vectors in 

 neighbouring points P and Q, we can proceed as follows. We put 

 a compass-rigid with its centre in P and by marking one of its 

 points we delineate the vector in it. Now displacing the compass- 

 rigid to Q it is reasonable to say that the marked point defines the • 

 vector displaced geodesically from P to Q. By comparing this 

 vector with the one present in Q we immediately see the meaning 

 of the geodesic di[jereutial of a vector. If this is known, it is clear 

 what Christoffel's covariant differentiation means. 



In the same way we can displace our vector-units from F to Q. 

 In geneial these will differ from the vector-units in Q. A set of 

 geodesically displaced vectoi'-units is what Prof. Schouten defined 

 as a system of axes moving geodesically. 



4. (reodesic line. We can easily iniagine what we have to do 

 in order to prolong a given line-elenwni geodesically. We put the 

 centre of the compass-rigid in the starting-point and mark the end 

 of the line-element by an arrow in the compass-rigid. After the 

 centre has been displaced along the line-element, the arrow will 

 point in another definite direction. This is the geodesic prolongation 

 of the element. Continuing to move the compass-rigid in the direction 

 of the arrow, the centre will gradually describe a geodesic line. 



In this case, during displacements along a geodesic line, vectors 

 moving geodesically will continue to include constant angles with 

 the geodesic (cf. IjEVi-Ci vita's article), these angles being fixed angles 

 in the compass-i-igid. 



5. RiemAiNn's measure of cnrvaiure. Let us now sup|)Ose that we 



M Taking another starling-point. T. Levi-Civita arrives at a definition of 

 parallelism which comes to ttie same thing: "Nozlone di parallelismo in una 

 varieta qualumjue., e conseguente speciflcazioiie geometrica delta curvatura 

 Riemanniana". Rend. Circ. Mat. Palermo, XLII p. i, 1917. His geometrical 

 explanation of the measure of curvature, however, is totally different from the one 

 we shall give in section 5. 



