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This paper deals with reciprocal static line-elements, previously defined by the author, the condition that their Ricci tensors vanish being no longer imposed. If the indices i, k run from 1 to n with the exception of the fixed index a (the line-elements being static with respect to xa) a certain quantity  appears with the remarkable property that in a reciprocal transformation  is invariant, whilst  merely changes sign.  is closely related to the Hamiltonian derivative of the Gaussian curvature, so that the general results obtained may be applied to the field equations of General Relativity Theory, with n=a=4.  is then the total energy density; and formally every static distribution of matter has a " reciprocal distribution" associated with it. In particular, the equation of state of a distribution of fluid reciprocal to a distribution of fluid possessing a given equation of state may be obtained directly from the latter, i.e. without the solution of the field equations being known.

Reciprocal Static Solutions of the Equations of the Gravitational Field