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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 eventually only one of these) the quantity $H$, which occurs 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 connexion between them is that which results from their mutual gravitational attraction.

§ 2. We shall be concerned with a four-dimensional extension $R_{4}$, 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 $t$. If we say that $P$ refers to a material point we mean that at the time $t$ 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^[1]. 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 "coincidence" takes place^[2]. Now Einstein has made the striking remark^[3] 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

1. It will be known that in the theory of relativity Minkowski was the first who used this geometric representation in an extension of four dimensions. The name "world-line" has been borrowed from him.
2. 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.
3. In a correspondence I had with him.