Equations in terms of the Curvature of the World. 175 



origin, for which cubes of the coordinates are negligible, 

 might be represented in five dimensions ; but this is not 

 generally true, and unfortunately the simplification is not 

 admissible in the cases of greatest practical importance, e. g. 

 the gravitational field of the sun. The theorem itself is 

 correct ; but it requires a more general proof. 



Take a point on the four-dimensional surface as origin. 

 Let [x u #2? x z, x i) be rectangular coordinates in the tangent 

 plane at the origin. The normal is a six-dimensional con- 

 tinuum in which we take rectangular axes z±, z 2 ....z G . Six 

 equations will be necessary to define the fonr-dimensional 

 surface,, and at a regular point these may be taken to be 



2z r = a 7>v x ljL Xy-j- higher powers (r = 1, 2 ... 6). 



There are no linear terms in x^ since deviations perpen- 

 dicular to the tangent plane are of the second order compared 

 with displacements parallel to the tangent plane. According 

 to the usual summation convention (applying only to Greek 

 suffixes) a rixl/ x lx x v is summed for values of /ul and v from 1 to 4, 

 and represents a general quadratic function of the a?'s. By 

 differentiation 



dzr = CLr^x^lXy ; dz? = a rfXV x lJL dx v . ar^x^dxr. 



Since the geometry of the ten-dimensional space is 

 Euclidean, 



- ds 2 = dx? + dx 2 2 + dx£ + dx? 4- dz, 2 + . . . + dz 2 



— dxi 2 + dx 2 2 + dx B 2 -f dx^ + £ {fi rixv a r(J7 x iX x a \dx v dx T + terms 



r 



containing cubes of the coordinates. 



Equating this to —g^dx^dx^ we have at the origin 



911 = 022 = $33= 9u= ~ h #12 = > etc - 



All first derivatives of the g^ v vanish. 

 The second derivatives are given by 



-n -n == ^i K^r^v^rar "i~ a rfJLT a rav ) . 

 X^ Xcr r 



When the first derivatives vanish, the Riemann-Christoffel 

 tensor simplifies to 



B va = 1 ( ^ 9(Tp | ^ 2 £W - ^Vi d 2 ,? yp \ 



LV<TP 2\dx^dx v 'dXrdXp 'dXy'dXp 'dXfdxJ 



^ y^ryLv^ra-p O'rfjujO'i vp) • 

 r 



It will be seen that the six z r 's contribute terms to B MI/<J . 

 which are simply additive. * Our subsequent formulae will 



