JJ82 



, 'd d\' 



we can deduce the iriütioii just as well as from (J 8). In tins con- 

 nexion we may therefore say that the perilieliuni motion of Mercnrius 

 is "due" to the fact that the gravitation is only in first approximation 

 proportional to the square of the distance an,d that a correction has 

 to be added which is a function of the velocity and of two affinors 

 'j and A " , detinitely given in each point of the euclidian space. 

 This gives another "explanation" of the phenomenon, from the point 

 of view of classic euclidian mechanics with a collection force. 

 When another arbitrary fundamental tensor is introduced instead 

 of "j, another force has to be added. This gives again another "ex- 

 planation", this time from the point of view of classic mechanics in 

 a space with an arbitrary line element with a correcting force. It 

 need hardly to be remarked that from the point of view of the 

 theory of relativity all these "explanations", accurate to quantities 

 of the order e, are [perfectly equivalent. 



The remark (irst made by Riemann and accentuated so strongly 

 later on among (Uhers by Poincake, viz. that the question of the 

 validity of a definite geometry is iiisepai-able from the question of the 

 \alidity of certain laws of nature, is e\ident for mechanics here. 

 Equations (35) and (37) enable us to pass to new mechaincs belonging 

 to any new definition of measures. 



Geodetic curvature of fhf path. 



The equations of motion for '1 and ^j can be written in the 



following form -. 



1 dH d\' /diy d dx' 



-"I'l K = + 1 . • • • . (38) 



m dt'' dl \dtj dl dl ' 



i (/';• dx' fdJX 'd dx' 



in dt' dj V''V d? dj 



where <// and (// represent the line-elements measuied with '1 resp. 



*j. Therefore the curvature vector (viz. the vector perpendicular to 



the path which has a length equal to the geodetic curvature and 



a direction towards the side of the curvature ^), is, measured in "1, 



d dx' 

 equal to——- and to the «'omponent of '1'^ K [)erpendicular (measured 

 dl dl 



^) When ij determines a congruency. the curvature vector is i; ivi;. 



Gomp. G. Ricci and T. Levi Civita, Calcul difïérentiel absolu, Math. Ann. 54- 

 (01) p. 154 or J. E Wrioht, Invariants of quadratic differentia] forms, Cambridge 

 (08), p. 78. 



