972 
Y—gS=2Z(m)V—g W" Qn. 
Vg Wm (m = 1, 2, 3,4) denotes what is usually indicated by 
QV: WV, ev: and go. Here, as in other places, the factor V—g 
occurs because we take the different quantities per units of time 
and volume, expressed in the coordinates and not in natural units. 
It is to be noted that at a change of the gas, V —g W™ remains unchanged. 
This corresponds to the fact that for a single charged particle the 
term > (m)ep,dx, is independent of the gravitation potentials. 
For the electro-magnetic field the scalar may be constructed in 
the following way. From the potentials the covariant field-intensities 
are derived : 
From these we form the contravariant intensities of the field: 
Fab — > (mn) tg de gen rae 
Finally we form the scalar: 
M=— + = (abmn) gen gin Fab Fn ’ 
= —12 (ab) FX fas 
Further it may be remarked here, that 
OM aM ef Se: 
ze} fob, and —— == — 4 pe (bn) 905 fas Som 
Ofab 
Ògem 
ScHwARzscHuL.D*) has already used the integrand W—g S in the 
variation law. Recently Tresiinc”) has communicated to the Academy 
of Sciences how this term may be used in Hamitton’s principle. 
Except as to the sign, the term W—gM corresponds to the term 
used by Lorentz, who writes Waz for what has been called here 
V—gF® and yay for fas. 
Variations of the field quantities. 
6. In the first place we shall consider the variation which is 
obtained by varying the electric jield in such a way, that everywhere 
the potentials y,, are changed to an amount dy. 
The Òp„ (m= 1, 2, 3,4) will be infinitesimal continuous functions 
of the coordinates. : 
1) K. ScHWARZSCHILD, Zur Elektrodynamik. I. K. Ges. Wiss. Göttingen, 
Math. phys. 1903. 
2) J. Trestine, The equations of the theory of electrons in a gravitation 
field of Einstein deduced from a variation principle. The principal function 
of the motion of the electrons. These Proceedings. XIX p. 892. 
