The hypothesis adopted in this work is that any permissible metric field whatsoever must satisfy the field equations deduced from an action principle in which the Lagrangian is quadratic in the components of the Riemann curvature tensor. The adoption of such a hypothesis is motivated by the precariousness of the general relativistic interpretation of Mach's principle, which is often used to justify a phenomenological approach to the theory. The quadratic action principle is chosen to provide the fundamental equations of the gravitational field because it is logically and aesthetically appealing, and causes us to lose nothing of the standard relativity theory based on Einstein's vacuum equations. The set of relationships, Rρσ - ½9ρσR = -k Tρσ (Equation) retained as a definition of the matter tensor Tρσ. Attention is concentrated on the solutions of the (generally fourth order) fundamental field equations in the static, spherically symmetric case. Sets of exact, series and numerical solutions are obtained corresponding to certain boundary conditions, or with certain properties in common. Study of the geometrical, topological and physical properties of several of the universes obtained as a result of our hypothesis leads us to believe that our theory is not implausible. We conclude by considering the further possibilities of the theory.