  A relationship between partial derivatives of the associated Legendre
function of the first kind with respect to its degree, $[\partial
P_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$, and to its order, $[\partial
P_{n}^{\mu}(z)/\partial\mu]_{\mu=m}$, is established for $m,n\in\mathbb{N}$.
This relationship is used to deduce four new closed-form representations of
$[\partial P_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$ from those found recently for
$[\partial P_{n}^{\mu}(z)/\partial\mu]_{\mu=m}$ by the present author [R.
Szmytkowski, J. Math. Chem. 46 (2009) 231]. Several new expressions for the
associated Legendre function of the second kind of integer degree and order,
$Q_{n}^{m}(z)$, suitable for numerical purposes, are also derived.
