977 
18. In order to find what has to be put for Ogap and dy, 
if they are to represent a virtual displacement of the fields in 
agreement with the geometrical character of the potentials Ja, and 
Pm just discussed, we shall proceed in the following way. First 
we sball describe the world by means of somewhat altered coordi- 
nates. We introduce the transformation 
Cn Em Or" (nm = 1, 2, 3, 4), 
where dr” represent the infinitesimal components of the displace- 
ment, the squares of which will be neglected, so that in differentiating 
a quantity which contains this dr" or is to be multiplied by it, we 
need make no difference between partial differentiations with respect 
to 2, and to 2'n. 
After the transformation of the coordinates we shall deform the 
net of coordinates together with the field in such a way that the 
surfaces @' =a, come at the place where originally were found 
the surfaces «,, =a, . This is evidently reached by a virtual displa- 
cement of the field characterized everywhere by dr”. In order to 
find what we have after the displacement we have only to omit the accents. 
For the indicated transformation we have 
0 dpm 
da' ed 
OEP) p 
Vp 
The geometrical character of the gas implies that the form 
= (ab) Jas da'g da'y = & (ab) gas dag res = 
a 0 fr” 
== (GD) ger | dra — = oe de) dx', — X(p) at One 
Oc! P p 
is invariant. 
Hence we deduce easily that 
i OdrP 
gab = gar — EP) 96 > — = (P) Jap dn 
Here gs is the same function of den which gas was of 
En. Therefore 
ddr? dr” 
° fr? + phx Yap => 
Or’, 
Up wa 
J'ab = Jab (em) — & ( » 15 
If, omitting the accents, we now express that the new net of 
Coordinates can be made to coincide with the original one, when 
the field is displaced at the same time, we find for the variation at 
a definite point-instant: 
Òdrp ddr? | 
dgar = — = (p) ee drP + gpb =— 7 + Jap =— an vat Lae 
Lp 
