1342 
developing it and in applying it to special problems. It will also be 
desirable to present. the fundamental ideas in a form as simple as 
possible. ‘ 
In this communication it will be shown that a four-dimensional 
geometric representation may be of much use for this latter purpose ; 
by means of it we shall be able to indicate for a system containing 
a number of material points and an electromagnetic field (or even- 
tually only one of these) the quantity H, which oeeurs in the variation 
theorem, and which we may call the principal function. This quantity 
consists of three parts, of which the first relates to the material 
points, the second to the electromagnetic field and the third to the 
gravitation field itself. 
As to the material points, it will be assumed that the only con- 
nexion between them is that which results from their mutual gravi- 
tational attraction. 
§ 2. We shall be concerned with a four-dimensional extension R,, 
in which “space” and “time’’ are combined, so that each point P 
in it indicates a definite place A and at the same time a definite 
moment of time ¢. If we say that P refers to a material point we 
mean that at the time ¢ this point is found at the place A. In the course 
of time the material point is represented every moment by a new 
point P; all these points lie on the “world-line’, which represents 
the state of motion (or eventually the state of rest) of the material 
point’). In the same sense we may speak of the world-line of a 
propagated light-vibration. An intersection of two world-lines means 
that the two objects to which they belong meet at a certain moment, 
that a “coineidence” takes place’). Now Einstein has made the 
striking remark*) that the only thing we can learn from our 
observations and with which our theories are essentially concerned, 
is the existence of these coincidences. Let us suppose e.g. that we 
have observed an occultation of a star by the moon or rather the 
reappearance of a star at the moon’s border. Then the world-line of 
a certain light-vibration starting from a point on the world-line of 
the star has in its further course intersected the world-line of a 
‘) It will be known that in the theory of relativity Minkowskr was the first who 
used this geometric representation in an extension of four dimensions. The name 
‘“‘world-line’” has been borrowed from him. 
*) For the sake of simplicity we shall imagine the two motions not to be 
disturbed by this coincidence, so that e.g. two material points penetrate each other 
or pass each other at an extremely small distance without any mutual influence. 
5) In a correspondence I had with him. 
