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ON THE GENERALIZED THEORY OF GRAVITATION 355
a way as in the case of the symmetrical field. It has been disturbing
to find that it can be achieved in two different ways.
These variational principles furnished two systems of equations
-let us denote them by E, and E,,-which were different from
each other (although only slightly so). each of them exhibiting
specific imperfections. Consequently even the condition of compatibility
was insufficient to determine the system of equations
uniquely.
It was. in fact. the formal defects of the systems E, and Eo that
indicated a possible way out. There exists a third system of
equations. Ea. which is free of the formal defects of the systems
E, and Eo and represents a combination of them in the sense
that every solution of E, is a solution of E, as well as of Eo. This
suggests that E, may be the system we have been looking for.
Why not postulate E,. then. as the system of equations? Such
a procedure is not justified without further analysis. since the
compatibility of E, and that of E2 do not imply compatibility of
the stronger system E,. where the number of equations exceeds
the number of field components by four.
An independent consideration shows that irrespective of the
question of compatibility the stronger system. Ea. is the only
really natural generalization of the equations of gravitation.
But E, is not a compatible system in the same sense as are
the systems E, and E2• whose compatibility is assured by a sufficient
number of iqentities. which means that every field that
satisfies the equations for a definite value of the time has a continuous
extension representing a solution in four-dimensional
space. The system E,. however. is not extensible in the same
way. Using the language of classical mechanics. we might say:
in tlle case of the system E, the "initial condition" cannot be
freely chosen. What really matters is the answer to the question:
is the manifold of solutions for the system E, as extensive as
must be required for a physical theory? This purely mathematical
problem is as yet unsolved.
The skeptic will say: "It may well be true that this system of
equations is reasonable from a logical standpoint. But this does
not prove that it corresponds to nature." You are right. dear
skeptic. Experience alone can decide on truth. Yet we have