This paper presents an analytic methodology for determining the temperature and radiosity distributions in a rectangular, gray fin array through a primitive formulation that explicitly couples the two distributions. The primitive formulation, used in conjunction with the Green's function method, produces a set of coupled nonlinear Fredholm integral equations for temperature and radiosity that must be solved simultaneously. Accurate numerical results are produced rapidly and efficiently using the trapezoidal method and a standard iterative scheme. Initially, an integrodifferential equation is derived from the one-dimensional steady-state energy balance. This mixed mode equation displays the coupling between the temperature and radiosity. In deriving the energy balance, a temperature-dependent thermal conductivity is included in terms of a Taylor series expansion. The Green's function method is used to convert the integro-differential equation describing the conservation of energy into an equivalent integral equation. The second integral equation is developed from the balance of radiant energy for a diffuse-gray surface. The coupled integral equations are solved simultaneously by a simple numerical integration scheme. A parametric study considering the effects of the conduction-radiation number, emissivity, spacings, and thermal conductivity is presented.