14 



Further, in the new system of coordinates the figure formed by 

 the world-lines differs from that figure in the old system by the 

 variation dxc = êc which is a function of Xb only. Therefore accord- 

 ing to (73) the second variation of i:^ is 



'• dwb ' 



By putting equal to zero the sum of this expression and the 

 preceding one we obtain (76). 



§ 46. We have thus deduced for some cases the equations of 

 the gravitation field from the variation theorem. Probably this can 

 also be done for thermodynamic systems, if the Lagrangian function 

 is properly chosen in connexion with the thermodynamic functions, 

 entropy and free energy. But as soon as we are concerned with 

 irreversible phenomena, when e.g. the energy-current consists in a 

 conduction of heat, the variation principle cannot be applied. We 

 shall then be obliged to take Einstein's field-equations as our point 

 of departure, unless, considering the motions of the individual atoms 

 or molecules, we succeed in treating these by means of the gene- 

 ralized principle of Hamilton. 



§47. Finally we shall consider the stresses, the energy etc. which 

 belong to the gravitation field itself. The results will be the same 

 for all the systems treated above, but we shall confine ourselves to 

 the case of § § 44 and 45. We suppose certain external forces /C to 

 act on the material points, though we shall see that strictly speaking 

 this is not allowed. 



For any displacements öXa of the matter and variations of the 

 gravitation field we first have the equation which summarizes what 

 we found above 



+ 2{ab) ^ (J/=7 Va (^.^«) - ^{<^b) —- (1/=^ F«) dXa + 



+ 2iab) (-—^ dg-^ + ^ <f,Q + ^ d,Q -{- 2 (a) Kaéxa- 



In virtue of the equations of motion of the matter, the terms 

 with (fxa cancel each other on the right hand side and similarly, 

 on account of the equations of the gravitation field, the terms with 

 ög^^ and r)\ Q. Thus we can write ^) 



1) To make the notation agree with that of § 38 & has been replaced by e. 



