The one-speed, time-dependent, source-free Boltzmann integro-differential neutron-transport equation is used to study the time dependence of monoenergetic neutrons in a spherical homogeneous medium. By applying the Marshak boundary condition at the outer face instead of the usual vanishing of the scalar flux at some extrapolated boundary, two coupled characteristic equations are derived which relate the time constants and space eigenvalues of the sphere in terms of its geometric radius and the nuclear parameters of the medium. Tables and graphs of the fundamental space eigenvalue and time constant are given for 0.82- and 1.24-MeV neutrons in lead. Numerical values of the time constant as a function of the size of the system are compared for several PN approximations ranging from P1 to P15. The results of fitting experimental data with the characteristic equation of the P7 approximation are given; they compare favorably with published values obtained by others. A method is given for determining the angular moments of a Legendre polynomial expansion of the scattering kernel from pulsed-neutron data.