278 
PAUL BIEFELD 
by the observer within, is at rest and a gravitational field pres- 
ent. This gravitational field may then be legitimately replaced, 
within the limits of the box, by the accelerated coordinate sys- 
tem K'. This principle of equivalence is evidently intimately 
related to the equality between the inertia-mass and gravitational- 
mass, and leads thus to the extension of the principle of relativity 
to frames of reference that are in non-uniform motion with refer- 
ence to each other. Again this conception leads to the unity of 
the nature of inertia and gravitation. 
It is to be noted, however, that a gravitational field can be re- 
placed by a frame of reference having a uniform acceleration only 
within a very limited portion of space ; about in the same way as 
we can think of a plane sheet of paper being in contact with a 
curved surface over only a very limited part of the surface. 
Nevertheless the principle of equivalence gives us the correct 
mode of attack to conquer the difficulties of the general theory of 
relativity. For as Einstein says in his Princeton lectures of 
1921 : “The possibility of explaining the numerical equality of 
inertia and gravitation gives to the general theory of gravitation, 
according to my conviction such a superiority over the conception 
of the classical mechanics, that all the difficulties encountered in 
the development must be considered as small in comparison.’' 
In the general theory then we must do away with preferred 
systems of reference of any kind, the term ‘gravitational field’ 
involving all arbitrarily moving frames of reference of any kind. 
Let us now consider one of the simplest of such fields present 
in case of a rotating system. Let K' be a frame of reference 
whose x'3-axis coincides with the Xg-axis of the system K at rest. 
We put the question : Are the configurations of rigid bodies at 
rest relative to K' (that is, moving with K') in accord with 
Eculidean geometry? 
As K' is not an inertial system we do not know directly the 
configurations of rigid bodies with reference to K'. 
Let us take a circle in the x\ — x '2 plane and a diameter of this 
circle. Further let us place a sufficient number of very small 
rigid rods of the same length along the circle and the diameter ; 
then the ratio of the number in the circle to the number in the 
diameter come out equal to tt. Let now the system K' be rotated 
with uniform angular velocity about x'^-axis . We will now find 
that more rods will have to be placed along the circle while none 
need be added to the diameter. This is easily explained ; for the 
