In this paper, we study continuous index Gaussian reciprocal processes (Grp's) (or two-point boundary valued processes) with Dirichlet boundary conditions. Our main contributions are 1) deriving first order white noise driven representations from the second order correlated noise driven representations of Grp's given by Krener, Frezza, and Levy; 2) deriving Kalman-Bucy like recursive filtering equations for Grp's with continuous indices; and 3) deriving recursive smoothing equations for Grp's with continuous indices.