Page:LorentzGravitation1916.djvu/49

 of the second derivatives of those quantities. This latter involves that, if we replace (91) by

$R=Q_{1}+Q_{2}-\sum(abfe)\left(\frac{\partial Q}{\partial g_{ab,fe}}g_{ab,fe}\right)-\sum(abfe)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,fe}}\right)g_{ab,f}$|undefined

the second and the third term annul each other. Thus

If now we define a complex $$\mathfrak{v}$$ by the equation

we have

If finally we put

$\mathfrak{t'=t+u+v}$

we infer from (90) and (94)

and from (88), (89), (93) and (92)

and for $$e\ne h$$

Formula (95) shows that the quantities $$\mathfrak{t}_{h}^{'e}$$ can be taken just as well as the expressions (88) for the stress-energy-components and we see from (96) and (97) that these new expressions contain only the first derivatives of the coefficients $$g_{ab}$$; they are homogeneous quadratic functions of these differential coefficients.

This becomes clear when we remember that $$Q_{1}$$ is a function of this kind and that only $$Q_{1}$$ contributes something to the second term of (96) and the first of (97); further that the derivatives of $$Q$$ occurring in the following terms contain only the quantities $$g_{ab}$$ and not their derivatives.

§ 55. 's stress-energy-components have a form widely different from that of the above mentioned ones. They are