1344 
coordinates will be of secondary importance; with a single exception 
($ 13) it only serves for short calculations which we have to inter- 
calate (for the proof of certain geometrie propositions) and for 
establishing the final equations, which have to be used for the 
solution of special problems. In the discussion of the general prin- 
ciples coordinates play no part; and it is thus seen that the formu- 
lation of these principles can take place in the same way whatever 
be our choice of coordinates. So we are sure beforehand of the 
general covariancy of the equations that was postulated by EiNsrrrn. 
§ 4. Einstein ascribes to a line-element PQ in the field-figure a 
length ds defined by the equation 
== 5 (ab) Jap dba day oe (1) 
(Jab = ba) 
Here dv,...du, are the changes of the coordinates when we pass 
from P to (}, while the coefficients gq, depend in one way or another 
on the coordinates. The gravitation field is known when these 10 
quantities are given as functions of .c, ...2,. Here it must be remarked 
that in all real cases the coordinates can be chosen in such a way 
that for one point arbitrarily chosen (1) becomes 
ds* = — dz,* — dx,? — da,’ + de. 
This requires that the determinant g of the coefficients of (1) be 
always negative. The minor of this determinant corresponding to 
the coefficient g,, will be denoted by Gs. 
Around each point P of the field-figure as a centre we may now 
construct an infinitesimal surface’), which, when P is chosen as 
origin of coordinates, is determined by the equation 
oD heb) Gas that SES cd len (2) 
where « is an infinitely small positive constant which we shall fix once 
for all. This surface, which we shall call the @dicatríz, is a hyper- 
boloid with one real axis and three imaginary ones. We shall also 
introduce the surface determined by the equation 
| te te NEN 
which differs from (2) only by the sign of «?. We shall call this 
the conjugate indicatrix. It is to be understood that the indicatrices 
and conjugate indicatrices. take part in the changes to which the 
field-figure may be subjected. As these surfaces are infinitely small, 
= (ab) gu, vaa, = -— EE 
1) A “surface” determined by one equation between the coordinates is a three- 
dimensional extension. It will cause no confusion if sometimes we apply the name 
of “plane” to certain two-dimensional extensions, if we speak e.g. of the “plane” 
determined by two line-elements. 
