974 
8. When gy changes by dg”, then ie and get change by 
Odgt 0? dg 
02, ag 02,0xg 
and their derivatives, the variation of ’—y H becomes 
0H O /V-gòH 
d(V-gH) & (abed)| dg ae ee 
West =de do rf oe 
SE ee OE ee 
dada dae? (ta OE: )+ 
Q (Ge 4) = nae oe 0 (GE) - (4) 
Ox dg? Ox, | Oxg òge? 
If H=G/2z, it can be proved’) that 
0H 1 0 (V—q0H 1 0° (VY—g0H 1 
ae) De =—— Gq (4d) 
dg? Y—g dae dg? Y Ògaf 2x 
Summarizing and choosing the variations dg? arbitrarily with 
only this condition that both they and their first derivatives vanish 
at the boundaries of the extension, HamiltTon’s principle requires that 
If we consider ’—-g H as a function of the g” 
ri LV -99aH+Vy-g9 
gedra 
m 
0 = | ae, de, dx, dv, 2 (abm) |} 1 V—9 dam (Tf, +i.) + 
1 
a ge Mo Gane ae “| ag |. … (ah 
Hence we find the well-known equations for the gravitational field 
Gab — 4 9a G-= — x (Tas + Ea). . - - . ~ (6) 
The origin of the second term of the left hand side is apparent; 
it appears by the variation of 1 —g in the principal function. 
Virtual displacement of the matter. 
9. The third variation we shall consider will be caused by giving 
to the molecules of our gas virtual displacements. We do not choose 
these displacements different for each individual moiecule, but to all 
molecules which at a certain moment are present in a definite element 
of volume we give the same virtual displacement, characterized by 
the infinitesimal vector dr? (comp. § 1), which may be an arbitrary 
function of the coordinates. The variation gives directly 
0 mm m 
far de, de, dae, & (almp) an Ws (—d, R — Tz; ) arl + 
Lm 
pin Jam 
& 
0 3 0 yn 
+ arn Voo + 5 (V—9 Tp) —4V—99" Ti | Pen KS 
Lm 5 P 
1) Comp. Lorentz, i.c. XXV, p. 472. 
