980 
0 m 0 am 
0=Y—gk, + = (alm) aoa WIZ) Egg! — Erie ae (19) 
where 
m m m 1 
Zp = = (Ty + By) =~ = Wo" Gps As) - » (20) 
These might be called the “equations of motion” for the gravita- 
tional field. Comparing this with our former result, we are induced 
to consider the tensor Zy as the dynamical tensor of the stresses, 
momenta and energy in the gravitational field. We see that it is just 
equal and opposite to those of the matter and of the electro-magnetic 
field taken together. 
16. By formula (16) we can prove, that the covariant divergency of 
Zj must be identically zero. The variation of ft ‘—q H de, de, de, dx, 
may also be calculated by means of the formulae of $ 8. If we 
choose the 07? and their first and second derivatives equal to zero 
at the boundary, then according to (16) the variation must vanish. 
From 4c and d together with (17) and (41) we find 
1 
a fwo Hdz,dz,dz,dz2,= Sion) = V-9 (Gar—t9anG)datdx,dx,da,dz,— 
1 0 
=ftededejde Sam 5 | V-g ger (Gark gad G) 4) — 
“im 
— Ora 
d P OV ion 
dam (W-g gen (Ga — $9ab G) mk W-gok! an, 9" (Go — 4916 a) 
é Ta 
This can only be equal to zero if the coefficient of Òre, i.e. 
W—g times the covariant divergency of Z/' is zero, so that 
Le 
= (bllm) SIW —g ge (Gar—t gar CN — 
1 Ok be. 
Sto V-9 a et (Gis — 490 G)—O0.. (21) 
va 
17. This identity, which implies four connexions between the 
components of (Gs } gas), is important because it shows that 
the ten differential equations 
Gap = Fane G =O 
which determine the gravitational field at those places of our extension 
where there is neither matter nor an electro-magnetic field, are not 
independent of each other. In such extensions void of matter the 
gravitation potentials may therefore be subjected arbitrarily to four 
