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We present in a full analytic form the partial widths for the lepton flavourviolating decays $\mu^\pm \to e^\pm e^+ e^-$ and $\tau^\pm \to \ell^\pm\ell'^{+} \ell'^{-}$, with $\ell,\ell'=\mu,e$, mediated by neutrinooscillations in the one-loop diagrams. Compared to the first result by Petcovin [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 externalparticles, 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 findrates of order $10^{-55}$, even smaller than those obtained in the zeromomentum limit, as the latter prediction contains an unphysical logarithmicenhancement.

$\tau \to \mu\mu\mu$ at a rate of one out of $10^{14}$ tau decays?