  In a recent paper new upper and lower limits were given, in the context of
the Schr\"{o}dinger or Klein-Gordon equations, for the number $N_{0}$ of S-wave
bound states possessed by a monotonically nondecreasing central potential
vanishing at infinity. In this paper these results are extended to the number
$N_{\ell}$ of bound states for the $\ell$-th partial wave, and results are also
obtained for potentials that are not monotonic and even somewhere positive. New
results are also obtained for the case treated previously, including the
remarkably neat \textit{lower} limit $N_{\ell}\geq \{\{[\sigma
/(2\ell+1)+1]/2\}\}$ with $% \sigma =(2/\pi) \underset{0\leq r<\infty}{\max}[r|
V(r)| ^{1/2}]$ (valid in the Schr\"{o}dinger case, for a class of potentials
that includes the monotonically nondecreasing ones), entailing the following
\textit{lower} limit for the total number $N$ of bound states possessed by a
monotonically nondecreasing central potential vanishing at infinity: $N\geq
\{\{(\sigma+1)/2\}\} {(\sigma+3)/2\} \}/2$ (here the double braces denote of
course the integer part).
