=] 
~j 
Oo 
The variation becomes): 
0 
af de, de, de, de, =1 du, dv, de, dv, © (mq) E (V—g Fa bpm) + 
it 
0 
=~ Sfn | W—g Wm — aes W—g ra) | 
&y 
If at the boundaries of the four-dimensional extension the dg» 
are chosen equal to 0, while within this extension they have arbitrary 
values, then HAmitton’s principle demands that 
0 
V—g Wn = & (9) Tan W/—g Fg), (nm 1 2, Dy B) ed 
vg 
These are the four equations of the field in an invariant form. 
7. The second variation to be considered is a variation of the 
gravitational field. At each point-instant of the extension it may be 
determined by the changes dg” of the quantities g@. 
If we have to do with an ideal gas, we may deduce directly that 
now the variation of V—g R is: 
Mg A) = E (abm) kW —a gina Ty dg?) « . . (do) 
Taking into consideration that, 
SV —9 = — = (ab) 5 V— 9 gar Fy” , 
dM = — & (abdn) } gt" fad fon gt = — 4 Z(abedmn) gam 9" gt" fea fon Og”, 
dM = — & (abmn) } gam F™ fon Sg™ , 
we easily find for the variation of V—g M 
Ad (Vv —g M) = & (abm) 4 V —9 Ina Es OGEE Nr DE 
where we have put 
Ey = — AZ (n) Fer fin —ids M. 
d is a mixed tensor, the components of which are 1 or 0 
according as m—b6 or m=b. We shall also introduce the notation 
Hi =(m) Jam ES. 
We shall see further on that V—g E; are the stresses ete. in 
the electro-magnetic field in the same way as W—g1'; are those in 
the matter. 
For the above mentioned reason the variation of ¥—gS will 
be zero. 
Òdp, IdPn 
1) It should be kept in mind that dfng = —— and that Fg — Fom 
fn Uy 
Comp. for the deduction TRESLING, l.c. 
2) Comp. Lorentz, le. XXV, p. 476, form. (63). 
62 
Proceedings Royal Acad. Amsterdam. Vol. XIX. 
