﻿90 Dr. A. D. Fokker : A Summary of 



11. As long as we deal only with the linear orthogonal 

 transformations of the principle of relativity, the properties 

 of tensors are relatively simple. But our aim is to consider, 

 and we did already consider quite general transformations 

 ,%'i —f { (,!'/, xj, %3, #4')* &o that in the transformation formulae 

 for the differentials 



dd\ = 2 p tK dxf K ', 



K 



the coefficients p tK are not such as to make the transformation 

 a linear orthogonal one. Therefore the coefficients ir lK in. 

 the reciprocal equations 



dx K ' — X ir l< dx 



are not the same functions of the coordinates as p lK , and we 

 must now distinguish between different kinds of tensors of 

 the second order, namely, covariant tensors which transform 

 themselves by the formula 



T' M y = SlV^Ta/3, 



contravariant tensors, for which 



©Vv *■ 2 TTay. 7T0V ©«/3> 

 aj9 



and mixed tensors, which follow the rule 



T y, v = 2 p afl irp v T aj3 . 



For instance, the quantities g^ v which define our invariable 

 line element 



ds 2 = 2 g^dx^ dx v 



fj.v 



form a covariant tensor : 



#11 912 On 9u 



a 2\ #22 #23 f?2i 



9n 9z2 9zz g& (9^ = 9^)- 



9& 9i2 9& 9u 



On the contrary, if <y^ v is the minor of g^ v in the determinant, 

 divided by the determinant itself, then the tensor of the y M „, 



Yn Y12 713 7i4 



721 722 723 724 

 731 732 733 734 

 741 742 743 744, 



is a contravariant tensor. 



