Physics. — ''On Einstein's Theory of gravitation'" . III. By Prof. 



H. A. LORENTZ. 



(Communicated in the meeting of April 1916.)") 



§ 32. In the two preceding papers') we have tried so far as 

 possible to present the fundamental principles of the new gravitation 

 theory in a simple form. 



We shall now show how Einstein's differential equations for the 

 gravitation field can be derived from Hamilton's principle. In this 

 connexion we shall also have to consider the energy, the stresses, 

 momenta and energy-currents in that field. 



We shall again introduce the quantities gab formerly used and we 

 shall also use the "inverse" system of quantities for which we shall 

 now write g"^. It is found useful to introduce besides these the 

 quantities 



^ab :^3: \/ — g gab^ 



Differential coefficients of all these variables with respect to the 

 coordinates will be represented by the indices belonging to these 

 latter, e.g. 



_ ^gab ^^9ab 



9nh,i} — '^ 1 Pab,pg 



We shall use Christoffel's symbols 

 a h 



d.vtjdwfj 



= ^ {gac, b + gbc, a — gab, c) 



and Riemann's symbol 



{ik, Im) = i {gim,M + gkl,i,i 



Further we put 



il' 



km' 

 b 



Girn = :^{kl)g^^{ik,lm) (40) 



G = :S (im) gi^- Gi,n (41) 



This latter quantity is a measure for the curvature of the field- 

 figure. The principal function of the gravitation field is 



^) Published September 1916, a revision having been found desirable. 

 2) See Proceedings Vol. XIX, p. 1341 and 1354. 



