D O C U M E N T 3 6 5 O N U N I F I E D F I E L D T H E O R Y 3 4 5 . (7) We make use of the identity (5), applying it to the tensor density Bα kl , whose lower indices we suppose to be raised. We thus find, as the only nontrivial identity,[17] , [18] which, taking (3b) into account, can be brought into the form: . (8) § 3. The Field Equations After I had discovered the identity (3b), it became clear to me that for a naturally limited characterization of a manifold of the kind that we had in mind, the tensor density Vα kl would have to play a significant role. Since its divergence Vα kl/α van- ishes identically, it occurred immediately to establish the requirement (field equa- tions) that the other divergence Vα kl/l should also vanish. In this manner, one indeed arrives at equations that yield to first order the well-known vacuum field law for gravitation, as found from the present general theory of relativity. However, no vector condition for the  α was obtained, in such a way that all the  α with vanishing divergence would be compatible with those field equations. This is due to the fact that to first order (owing to the commutation of ordinary differen- tiation), the identity is obeyed, but the quantity on the right-hand side vanishes identically owing to (3b). This leads to the loss of 4 equations from the system . Nevertheless, I recognized that this defect could readily be eliminated, if instead of the vanishing of , one postulates the condition , in which is a tensor that differs from Bα kl/l by an arbitrarily[19] small amount:1 . (9) Then one indeed obtains the Maxwell equations (still to first order) by calculating the divergence of the field equations (with respect to the index α). Furthermore, by going to the limit ε = 0, one can derive the equations that yield the correct law of gravitation, Bα kl/l = 0 as before, likewise to first order.  1 This is indeed the method that is always applied to lift degeneracy when it occurs in singular cases. T i k T k i T   ik  –  – V kl l     V kl//l  V kl   l    / –  – V kl l   V – k       / 0  [p. 5] V kl l     V kl//l   Va kl/l 0 = Va kl/a V kl l   0 = Vkl/a a V kl  V kl  h  l  k   k  l  –  – =