514 



element. It is easily seen that tiiis entails for the geodesic the 

 equation 



1771 \ 



= d\v'^ + 2:{lm) 



dx^dw'" . 



This (covariant) equation coincides with what we get from the 

 familiar definition of a geodesic as the sliortest line between two 

 points. 



11. We shall now displace geodesically a pai-ticle P\ which, 

 with relation to P, is defined by a vector u, to a point S' near T, 

 by shifting the compass rigid in two steps fioni P to T, along PQ 

 and Q7\ Then, a second time, we displace the particle to S" neai' 

 T, taking the steps along PK and KP, the «lüadrilatei-al PQTk 

 being a (quasi-) parallelogram with sides dx {PQ and A' 7') and ox 

 {PK and QT). 



After the displacement along PQ the cooi'dinates of the particle 

 considered relative to Q have become 



f a \ 



At the second step we must be careful to take the values of 

 Ckistoffei/s symbols at point Q, so that after the displacements 

 along PQ and QT the coordinates relative lo T are 



u"-2: 



I in 

 a 



d.vlu'" - 2J 



Im j 

 a 



(fw' 



iP'h 



dxi'ui 



d Ua 

 dxi') c 



dxrö.v'ii'" . 



If the displacements had- been performed along PK and KT, the 

 coordinates relative to 7' would have been 



u"— 2: 



hn 

 a 



(fxhi'" — 2 



hn 

 a 



dx' 



\P9\ 



(fxi'nu 



bx}'\ a 



ö.ri'dx'u'". 



Taking the difference we find 



g« = i:{lmp) 



' d 



dxi' 



1 771 



a 



Si 



11 tii i pin 



or 



S^' — i 2:{lmp) 



{dx'óxi' — dxi'öx') u'" 

 pn) I / in I 



In 



\P^ 



Öj7' \ a \ dx' \ a \ 



X{dxlöxi'—dxJ'<ixl)u"' ........ (6) 



The first factor is seen to be a Riemann's four-index-symbol, 

 of the second kind. Availing ourselves of his notation we can put 



^(f z= ^ 2:{lmp)\maJpUdx'(Kvr - dxröx')H"' . ... (6) 



The 'C" {a r= 1, 2, . . . ji) are the components of the displacement- 



