A Precise Measurement of the $\tau$ Polarisation at LEP-1

The $\tau$ polarisation has been studied with the ${\rm e^+e^-}\to \tau^+\tau^-$ data collected by the DELPHI detector at LEP in 1993, 1994 and 1995 around the Z resonance firstly through the exclusive decay channels ${\rm e}\nu\bar{\nu}$ , $\mu\nu\bar{\nu}$ , $\pi\nu$ , $\rho\nu$ and ${\rm a}_1\nu$ and secondly with an inclusive hadronic analysis which benefits from a higher efficiency and a better systematic precision. The results have been combined with those previously published on 1990 to 1992 DELPHI data, to produce results which reflect the full LEP-1 statistics. The fit of the $\tau$ polarisation dependence on the production angle yielded the polarisation parameters ${\cal A}_{_{\sc…