756 
arbitrary variations that need not vanish at the limits of S, becomes 
0 (Ua 
JL + S(a)Kadtq = — (ab) ee é u) nee nn 
TN 
§ 6. We can derive from this the equations for the momenta 
and the energy. 
Let us suppose that only one of the four variations dw, differs 
from O and let this one, say Ov,, have a constant value. Then (14) 
shows that for each value of a that is not equal to c 
fac = —— Wa ho,, Yea. Walaa 
while all #’s without an index c vanish. 
Putting first 4c and then a—c, and replacing at the same 
time in the latter case 4 by a, we find tor the right hand side 
of (20) 
Uata s, UcWa 5 7 
z@ en pre = (a) ~(" P san, ) 
But, er to a 5) and (16), 
UqWa 
BN oe 
so that (20) becomes 
OL 0 (Uta 
dL + Koda, = — — du, — Za) —— Ide. . « (22) 
Oa, Ora P 
It remains to find the value of dL. 
The material system together with its state of motion has been 
shifted in the direction of the coordinate x, over a distance du. If 
the gravitation field had participated in this shift, dl; would have 
OL so 
been equal to RTE dr. As, however, the gravitation field has been 
we 
left unchanged, 
in this last expression must be diminished by 
Ve 
dL 
(Ga) , the index w# meaning that we must keep constant the 
Ue) u 
quantities w‚ and consider only the variability of the coefficients gus. 
Hence 
dL = == | - = +- Ge) (on ° 
Ox, 02, w 
Substituting this in (22) and putting 
1) The circumstance that (21) does not hold for a=c might lead us to exclude 
this value from the two sums. We need not, however, introduce this restriction, 
as the two terms that are now written down too much, annul each other. 
ee 
