Consider a strong Markov process $X^0$ that has continuous sample paths in the closed cone $ar{G}$ in $R^d(d geq 3)$ such that the process behaves like a ordinary Brownian motion in the interior of the cone, reflects instantaneously from the boundary of the cone and is absorbed at the vertex of the cone. It is shown that $X^0(t)$ has a representation $R(t) ominus (t)$ where $R(t) in [0, infty)$ and $ominus(t) in S^{d-1}$, the surface of the unit ball.