  We present in a full analytic form the partial widths for the lepton flavour
violating decays $\mu^\pm \to e^\pm e^+ e^-$ and $\tau^\pm \to \ell^\pm
\ell'^{+} \ell'^{-}$, with $\ell,\ell'=\mu,e$, mediated by neutrino
oscillations in the one-loop diagrams. Compared to the first result by Petcov
in [1], obtained in the zero momentum limit $\mathcal{P}\ll m_{\nu} \ll M_W$,
we retain full dependence on $\mathcal{P}$, the momenta and masses of external
particles, and we determine the branching ratios in the physical limit $m_\nu
\ll \mathcal{P} \ll M_W$. We show that the claim presented in [2] that the
$\tau \to \ell \ell' \ell'$ branching ratios could be as large as $10^{-14}$,
as a consequence of keeping the $\mathcal{P}$ dependence, is flawed. We find
rates of order $10^{-55}$, even smaller than those obtained in the zero
momentum limit, as the latter prediction contains an unphysical logarithmic
enhancement.
