The general spherically symmetric, static solution of +"T""=0 in the exterior Schwarzschild metric is expressed in terms of two integration constants and two arbitrary functions, one of which is the trace of T",. One constant is the magnitude of T,"at infinity, and the other is determined if the physically normalized components of T", are finite on the future horizon. The trace of the stress tensor of a conformally invariant quantum field theory may be nonzero (anomalous), but must be proportional (here) to the Wey) scalar, 48M'r; we fix the coefficient for the scalar field by indirect arguments to be (2880m') '. In the twodimensional analog, the magnitude of the Hawking blackbody effect at infinity is directly proportional to the magnitude of the anomalous trace (a multiple of the curvature scalar); a knowledge of either number completely determines the stress tensor outside a body in the final state of collapse. In four dimensions, one obtains instead a relation constraining the remaining undetermined function, which we choose as T f)-T /4. This, plus additional physical and mathematical considerations, leads us to a fairly definite, physically convincing qualitative picture of (T",). Groundwork is laid for explicit calculations of (T",).

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