851 



of a covariant vector, and thus llie unambigiioiis chaiacler of this 

 qiiolient would vanisli. But when tlie second expression is used the 

 transvection v' rvy of two vectors ;;■' and /(;, is no more an in\ ariant 

 with a pseudo-parallel displacement, so that the differential quotient 

 of the first formula occupies a well defineil preferied position. 



We will not consider the most general case, but the seini-sym- 

 metricdl case in which the alternating part of the parameters has 

 the form 



V, (r'l, - r';>) = v,(6', a; - 6V A]) ; Ai=\l'''"~^^ 



\0,v ^ X, 



in which -S'; is a general covariant vector')- It will he shown that 

 already with this simplified supposition the above mentioned difiiculty 

 can be made to disappear. 



About the specinl fonu of the world function -p, nothing will be 

 supposed, so that the resulting expressions are quite general. 



2. l)i;ductioH of the fiehl equations. The /)^. of a semi-symmetrical 

 displacement can always be divided into a symmetrical and an 

 antisymmetrical part: 



(1) r'L = . ilu + S[; ^,1] : y/I,„ = ./;> '). 



Be R'm/ the curvature quantity belonging to r'y^-. 



... a Ö '„ d '.J '„ ', '., 'y 



Raiij.' the curvature quantity formed in the same way with the 



parameters yl;^, R',,^ the quantity obtained from R,.,uV' by con- 

 tracting, o)= 1': 



o ^'^ o 'at 'a( '/ 'a 'x 



d.v/' dx" 



and R*a) the quantity obtained in the same way from Rtai', then 

 we can easily deduce the relation 



1) That the differences r."_^ — PJ^ always are the components of a quantity of 

 the third rank may be supposed as known. Gf. the author's paper in Math. 

 Zeitschrifl 13 (1922), p. 56—81, Nachtrag 15 (1922) p. 168. 



•) In this paper the symbol 'V- w,^ means 'jlt'^M-^ — r ^ tü.). 



