order. From this we can deduce that ( rfg"^ j is also such a 



tensor. 

 Writing for it 6«^ we lind according to (46) and (47) that 



:S {ab) Mab e«^ 



is a scalar for every choice of (e"*). 



This involves that (Mab) is a covariant tensor of the second order 

 and as the same is true for (Gab) we must prove the equation 



^ab ^^ Gab 



only for one special choice of coordinates. 



^ 37. Now this choice can be made in such a way that at the 

 point P of the field-tigure //,, = g^^ =g^^z= ~~^, g^^ = -f- 1, ^„^ = 

 for a=\=b and that moreover all first derivatives gaf>,e vanish. If 

 then the values gaf> at a point Q near P are developed in series 

 of ascending powers of the differences of coordinates .Va{Q) — •i'a{P) 

 the terms directly following the constant ones will be of the second 

 order. It is with these terms that we are concerned in the calcula- 

 tion both of 3fab and of Gab for the point P. As in the results 

 the coefficients of these terms occur to the first power only, it is 

 sufficient to show that each of the above mentioned terms separately 

 contributes the same value to M„h and to Gab- 



From these considerations we may conclude that 



ö,Qz= :E{ab)G,b(fr^ (48) 



Expressions containing instead of dg»* either the variations dg'^^ 

 or (fgah might be derived from this by using the relations between 

 the different variations. Of these we shall only mention the formula 



1 q(^^ 

 (fgab — ___ (f^ab L 2 ^cd) (fed dv]«^ . . . (49) 



^—9 2 V—g 



§ 38. In connexion with what precedes we here insert a con- 

 sideration the purpose of which will be evident later on. Let the 

 infinitely small quantity (i be an arbitrarily chosen continuous func- 

 tion of the coordinates and let the variations fkjab be defined by 

 the condition that at some point P the quantities g^i have after the 

 change the values which existed before the change at the point Q, 

 to which P is shifted when xh is diminished by §, while the three 

 other coordinates are left constant. Then we have 



dgab = — gab,h § 

 and similar formulae for the variations rfg"^. 



