Physics. — "On the coniie.vion betiveen yeometnj and mechanics 

 in static i)voblems\ By Prof. J. A. Schoutkn and D. J. Struik. 

 (Communicated by Prof. H. A. Lokentz). 



(Communicated in the meeting of November 30, 1918). 



Connexions between the (/eodetic di[f'eventiations belonginii 

 to diferent fundamental tensors. 



When in a space two difFei-ent fnndamenlai tensors ■! and 'j are 

 given'), we have the following genei-al theoiem: 



The ijeodetic differential quotient of a i/iven afjinor with respect 

 to 'j is equal to the sum of the geodetic diferential quotient with 

 respe'ct to "1 and the product of the afjinor by a simultaneous cora- 

 riant of *j and 'I. The same holds for the geodetic dijferentials. 



Let ns first give tlie proof for a vector. When 



H'z=z a"= a\'= . . . 'j'= z"= z\'= . ')) 



then we have : 



(1) 



a' := M' U z' = *i'i z 



a = -1 U' z =: 'j 1 z' 



(2) 



aa' =: zz' = IJe^e^' *) ..,..-. (3) 



When 7 and (/ are the syml)ols of the geodetic differentiation 

 belonging to '1 and 7 and 'd those belonging to *j, then we have 

 the following relations for an arbitrary scalar/) and for an arbitrary 

 vector V resp. v' : 



'Vp == Vp ......... (4) 



'Vv = 7(z'^ v)z=7(z'i v)z = 7v4 v^ a'7(ai z')2=7v_vi z'7 (z^ a')a 

 'Vv'='V(zl v')z'=V(zi v')z'=Vv'4-v'i a7(a'i z)z'=7v'-v'^ z7 (z'^ a)a'^) 

 or when we introduce the notation ^i : 



1) For the notations used in this paper we refer to: J. A. Schouten, Die Ana- 

 lysis zur neueren Relativitatstheorie, Verh. der Kon. Akad. v. Wet. Dl. XII. N". 6. 

 cited further on as A. R. 



') As we shall have to do with different fundamental tensors, we must distinguish 

 between covariant and contravariant quantities while else this is not necessary. 

 Between two quantities p and p' no relation exists, unless this has been stated 

 especially. 



3) Gomp. A.R. p. 44. 



^) Comp. A.R. p. 89, formula (101). 



