515 



vectors which would become manifest after a geodesical displacement 

 of the compass-rigid about the circuit TKPQT. The displacement 

 cannot be anything else but a rotation, the lengths and angles 

 remaining the same. We see how Rirmann's symbols delermine the 

 rotation in terms of the components of the area of the circuit. This 

 rotation is characteristic of the curvature of space. 



12. It now remains to prove the statement of section 5 as to 

 the interpretation of Riemann's measure of curvature. 



The measure of curvature with respect to the plane of dx and 

 on is defined to be ^) 



i:{alpmq)g„g \ma, lp\ {dx^ (fwl' — dxP(fx^){da'" (fx9 —dxld.i^"') 

 ^ ^ l^lp mq)igt,n9i,g--9^gim){d^'^-^i' - d,i-P(U^{dx^'^ 6x'i - dx^öx^'^]' 

 The denominator is four times the square of the area of the 

 parallelogi-am formed by dx and dx. For by changing indices with- 

 out changing the sum we get four times 



2:{lp mq) ! ^'"' ^'"^ d.vJdx^^^ óxl'iSx'i. 



Writing d for the length of dx, and rf for the length of rfx, we find 



for the denominator 



rf* d(S cos (do) 



d(f cos {do) (P 



this being four times the square of the area of the parallelogram. 



We shall discuss the numerator for the case of three-dimensional 

 space and show that it represents four times the volume of the 

 parallelepiped formed by the axis of rotation and the parallelogram. 



Proceeding with some caution, the analogon in more-dimensional 

 cases is readily found in the same way. We will put for the numerator 



2 2:{amq)g„glC{d''r!"'dxQ—dx9(fx"^) (7) 



denoting by R"m the coefficients of the rotation of curvature (6): 



C,n = 2(m) R^ M"» . 



or, 



How are the numbers Rj related to the components of the axis 

 of rotation? If we suppose the components of the hitter equal to /•, 

 then the rotation is represented, as will be presently shown, by 



linJ (8) 



4 



5c = 2- 2:{ij) 



9ni yti 



9oj gbj 



') Cf for example Bianchi, Lect. on Diff. Geometry, section 819. 

 Proceedings Royal Acad. Amsterdam, Vol. XXI. 



