Physics. — “On the application of Einstein's theory of gravitation 
to a stationary field of gravitation.” By H. A. Kramers. (Com- 
municated by Prof. H. A. Lorentz). 
(Communicated at the meeting of September 25, 1920). 
1. Definition and invariant properties of a stationarr 
prop y 
field of gravitation. 
We will call a field of gravitation stationary when the expression 
for the line element can be put into such a form ds’ = g,,dx,dz,') 
(a, time-coordinate, 2,,2,,7, space-coordinates) that the gravitation 
potentials g,, do not depend on the time z,. A special case of the 
stationary field of gravitation, defined in this way, forms the so- 
called “static” field of gravitation, which appears when it is possible 
by a suitable transformation to make the quantities go1, gor and go3 
equal to zero. It is simply seen, that when the line element of a 
stationary field’ of gravitation is brought in the above mentioned 
form, the most general transformation of coordinates, for which the 
g””s remain independent of the time, and for which a point at rest 
remains at rest, is given by the formulae 
Br PUG eco ols (== 4,3) (1) 
Ly = ast, i wp (e'‚, V's &',). 
Here pz; and w are arbitrary functions of w',, 2',, z'‚, while a is 
a positive constant. The quantities g,, and their derivatives show, 
with regard to the transformation group expressed by (1), certain 
invariant and covariant properties, which we will now investigate. 
The line element may be written in the following form, 
1 
ds? =9,,dx,da,=-& Gy dapdx (+ — (goede, Horde, Hodes +954)’, 
Joo 
. (2) 
gok Jol 
Gri = — gu Hs 
1) Just as EINSTEIN we have omitted the signs of summation for summations 
which have to be extended over indices, which occur twice in a product. 
