979 
extension, Hamitton’s principle requires that 
0 ) m 
0=Y-—gkh +242 (m) Vg Wn it jen Ohm + 
OL» day 
Ògam 
Oz, 
These may be called the equations of motion for the field. We 
see that the acting external force and the force which the carrier 
of the charges exerts on the field’) must be opposite to the co- 
variant divergeney of a tensor multiplied by #/—g. The equations 
correspond exactly to those which we found for the matter. For that 
0 m m 
+ 2 (Ima) 5, V9, )- 3 41 Y— 9 gt! — E; OE 
‚p 
reason we are justified in considering the tensor E; as the dynamical 
tensor of the stresses, momenta and energy in the electro-magnetic 
field. 
15. For the virtual displacement of the gravitational field it is 
easy to find the variation of the part of the integral containing 
V—g H. The integral being a scalar, we have 
[v-9 H dx,dz,dz,dzx, = (vdo, de, de, de, 
for the transformation of $ 13. H being a scalar, we also have 
H' = H (@'p— dre). 
0 seed es OdrP 
Vn Up Prat 
7 (@,'..0,) 
So that after the displacement we find (by omitting ie accents) 
0 
6fv—a Hdx,dz,dzx,dz, = fae, dx, dx, de, = (p) PO ee ad (16) 
Ep 
In what follows we shall use the results of § 8. With 
dg = — = (mn) gg" dgn, + « « - « (17) 
we apply the formulae (4, 46, 11) and find after a short trans- 
formation for the total variation 
yn m 
fan da, dx, dee,  (alnp)| 5° —9 (- da H + Te + Eq) dr} + 
yn m 0, am yn m 
+ On" = ae +Y-gLE, ) 4 W-99! 5 (T) 4. Er 4 (18) 
el el 
As in the preceding cases HamiLton’s principle now teaches us 
that, whenever the displacement vanishes at the boundary of the 
extension, we must have 
1) Per unit of volume. 
