975 
If the dre are zero at the boundary, then Hamitton’s principle 
demands that the integral always vanishes, so that 
Ò Ògan 
V—g hy + = (aml) | (V—9 Tp) —4V—g EET | =0 . (8) 
Oxm , dep 
These are the equations of motion of the matter in an invariant 
form. & is a covariant vector and the form between brackets is 
W—g times the covariant divergency of the mixed tensor 77". 
10. Consider now the virtual variation of the electrie current. 
If each electric particle undergoes a displacement dop, then the 
variation of the intensity of the current at a definite point-instant, is 
0 
d(V—g W™) = = (a) Seg. aa as orn — —g W™ dr) ,") 
so that the integral is varied by: 
0 m 
few dx, dx, dx, = (map) E 2 | VY —9(W™ pa—de S) aval IE 
&m 
: 04 m dq ) 
V—ght++VY-g We (5) - Ph )| « (9) 
da, O2n 
+ dr 
If dr? vanishes at the boundary of our extension, we must have 
therefore 
Opie uwsl 
V—gky + 2imV—9 We (ee ze) Pe 7E RESO) 
Lp Ön 
This may be called the “equation of motion’ for the electric 
current. The second term may be said to represent the force exerted 
by the electric field on the carrier of the charge. 
Virtual displacements of the fields. 
11. Before calculating the variation which is obtained by a 
virtual displacement of the electro-magnetic field or of the gravitational 
field, we have to state what will be meant by this. 
Doubtlessly we can say: to give a virtual displacement to the 
electro-magnetic field means to assume that the four potentials which 
originally occur at the point-instant x, (p = 1, 2, 3,4) will be found 
after the displacement at the point-instant «, + dr (p= 1, 2, 3, 4). 
From. this follows that there will be at one and the same point- 
instant a variation dp, 
Pm 
Opm = — = (p) —— dr. 
xp 
1) Comp. Lorentz, l.c. XXIII, p. 1077. 
62* 
