13 



(&l=^'"-^'"'= 



in the differentiation on the left hand side the coordinates of the 

 material points are kept constant. To show that Tab and tj" satisfy 

 equation {QQ>) we must now show that 



and for b =1= c 





If here the value (72) is substituted for L and if (70) is taken 

 into account, these equations say that for all values of b and c we 

 must have 



Now this relation immediately follows from a condition, to which 

 L must be subjected at any rate, viz. that Lc/aS is a scalar quantity. 

 This involves that in a definite case we must find for H always 

 the same value whatever be the choice of coordinates. 



§ 45. Let us suppose that instead of only one coordinate Xc a 

 new one Xc' has been introduced, which differs infinitely little from 

 Xc, with the restriction that if 



the term %c depends on the coordinate Xh only and is zero at the 

 point in question of the field- figure. The quantities g°^ then take 

 other values and in the new system of coordinates the world-lines 

 of the material points will have a slightly changed course. 



By each of these circumstances separately H would change, but 

 all together must leave it unaltered. As to the first change we 

 remark that, according to the transformation formula for (/«'', the 

 variation ög^^^ vanishes when the two indices are different from c, while 



d9«"c = 2^^^ - — 

 and for a ==:|=:c 



(f^«C ^^ (fgCa -:^ gi 



ah 



The change of B due to these variations is 

 2-^^(a)^«M -— . 



