1181 



we introduce tlie fniidauieiital tensors : 



'j =: z* —- z,^ =...=: e,- e,- + ?•■- 5m' St-^ t-^ ] r* e^ 69 . 



1 1 (29) 



'j' = z" = z\-^ = ... = e',.eV+ -T-;-^eVeV4--e'9e'9 



r sur a r' 



with the i-eiations : 



Z= Y. z' , Z'=:ri". Z , ^j^ *j' = M' M' . . . (30) 



Evidently the t'tiiidanientiil tensors 'j and '^j' give a euclidian 

 line-element. 



F'or the indicated values ^1 and 'j we (hen have: 



3.., 



=— ^-f 1 — - We,eV— »■( 1 — ; Je^e^e'rf re^Cee',- — 



—rsiv'^ei 1 — ^^ je^e-^eV+r «m'''6'e..e.^eV ; (31) 



— sinO cos dQ.^^Q^fj-^.-iinS cosó^Q-^Q-^t'e = 

 = — è -^1 ], je, e, eV + i(Q(,ege',-{-usin^OQ'^e.^Q\- 



Equations of motion for the euclidian fundamental tensor 'j. 



-Applying the relations (9) to the equations of motion: 



d c/x' 

 at at 



we obtain : 



'd d\' dx' dx' a-. 



'ri\7i^ = 2 A ..... (33) 



dt dt dt dt 



In classic mechanics the equations for ^j would be: 



• r 'ddx' 



■''••^^' = ..^,' ....... (34) 



and, as V ^' = V 6', the problem can therefore also be regarded as 

 a problem of euclidian classic mechanics with an additional force: 



I dx' dx' ^ -) 



K/=.^ii j(M'-T)^7rf --2A j .... (35) 



|)er unit of mass. 

 As: 



c.c'' 1 ftC^ 



'^'=Y '',-='- 2?'' '^"^ 



and eV has the same direction hs e, , K/ has the direction of e,. 

 From tlie eqiuitions : 



