An analysis is made of the motion of a spherical Brownian particle whose surface can diffusely reflect the molecules of an equilibrium host gas. The analysis is based on Newton's second law and a limiting form of Markov's method. It is shown, both for specular and diffuse reflections, that equipartition of energy is a consequence of the dynamics and randomness of the motion. In addition, it is demonstrated that the diffusion coefficient can depend on the temperature of the particle. The entire analysis is restricted to the case for which the Knudsen number of the particle is large compared to unity.