893 
The other quantities with indices will always be vector components 
and the letters will have the same meaning that is generally ascribed 
to them in the theory of gravitation. 
With an expression as qaw” in which some indices occur twice 
we denote the sum of a number of such terms, which are obtained 
by giving to those indices all values from 1 to 4. , 
In a four-dimensional extension S we consider an electro-magnetic 
field and moving electrons. Let the state be characterized by the values 
of the potential g, and of the intensity of the current w' and let a 
force A; act on the charges per unit of volume. Then the principle 
may be formulated as follows: The equations which exist between 
these quantities are such that a virtual displacement of the charges 
and a virtual change of the potential cause a variation of certain 
two integrals over the volume S, which is equal to the negatiye 
work of the external forces during that change. The gravitation 
potentials are kept constant. 
In a formula this may be expressed as follows 
aft Vg dS + aft, Vg dS | ROAN ASD Oa 
Here HirBeRT’s invariant occurs 
dl Ogm 
Li = E (Gn—Qnm) (quo) OO UE where Jinn = an: 
Un 
and a new one 
men Tl 
Let the potentials be changed by the amounts dg; and suppose 
the intensity of the current to be varied by a displacement of the 
charges over a distance dw. Then Lorentz gives for the changes 
of the current 
Oad 
dwt =" where Yap = w! Owa—wtdan 
vy . 
This is easily found by considering the changes in density and 
velocity which are caused by the displacements de 
We shall write 
. d OL V q 0 Lg 
JL ae EAD ER ne ‘fig; Le ey | 
WJ zl Ogik ) one Ov}: ( Ogik at ) C) 
; 0 
OLY g = w' 09; — fib qib + an: BG) reld) 
At the limits the potentials are not changed and the charges are 
