B. F. Jones (Ph.D. Thesis, Rice University, 1961) proved the existence and uniqueness of a solution of a one space variable diffusion equation ut a(t) uxx , where a(t) is an unknown function of time. This article considers the analogous problem for a cylindrical region with symmetry with respect to 8 . In particular, we consider the system (separately for r>1 and r<1) We take the five theorems in Jones' paper as Properties a through e ; and, by taking the appropriate bounds on the function M, we show that L defined by (5) satisfies the five properties. Thus, (1) has a unique solution.