508 



make a compass-rigid describe a small circuit, e.g. along a vanishing 

 (quasi-)|)arallelogram. We already pointed out that in general it will 

 noi return to its Parting- positio?i. The difference between the 

 two positions is such as might have been produced by an 

 infinitesimal rotation ai'ound (he starting-jjoint. The amount of' this 

 rotation is proportional to the area of' the circuit described, the 

 orientation of' the "axis" of' rotation (which in higher extensions is 

 of' {u. —2) diujensions) being determined bv the orientation of the 

 plane of' the circuit. The rotation is intimately connected with the 

 curvature of space. When this rotation of curvature, as it may be 

 called, vanishes in all points for every arbitrary circuit, then the 

 space is Euclidean. ') ^ 



The components of the o[)eraioi' by which from the data of the 

 area included by the circuit the rotation of curvature for the 

 compass-rigid can be derived, are Rikmann's four-index-symbols, of 

 the second kind. 



Further — to confine ourselves to three-dimensional space — if 

 we take the length of the axis of i-olation equal to the amount of 

 the angle of rotation, and construct a paiallelepiped with this axis 

 and the parallelogram of the circuit, we can consider the ratio of 

 its volume to the scpiare of the parellelogram as a measure for the 

 curvature of space. Indeed, in the limit, for a vanishing circuit, 

 this ratio is jusi (he number indicated bv Kikmann as tke measure 

 of curvature of the space milk respect to the plane of the circuit 

 considered. 



6. Now we shall proceed (o give (he i-eijuired formulae. We take 

 the length of a line-element as defined by 



(/*■"== ^{ab) g„i4x" dw'' , 

 dx" cAr* representing increments of the coordinates along the line- 

 element dx, (j/^h {= cp,„) being regulai- functions of the coordinates of 

 the starting-point, each index in the sum going through all the 

 values from 1 to )i., where n is the number of dimensions of space. 

 The algebraical complements of (/„b will be denoted by ^rf^'>, so that 



9 'he fundamental idea of a recent article by H. Wevl (Gravitation \md 

 EleJdrlzitdt, Berl. Sitz. Ber. May. 1918) may be considered the hypothesis that a 

 small rigid (— "Vektorraum"^ after turning about an infinitesimal circuit of "trans- 

 lations" (of a somewhat more general kind) not only will have got in a changed 

 position, but in gener 3i\\yi\\\[a.v e changed its dimensions as weW. In four-dimensional 

 space-time the linear dilatation of the (4 dimensional) rigid would be half the scalar 

 product of the alternating electromagnetic tensor and the area included by the 

 circuit. (Note added during the revisal of the proofs). 



