  We present an analytical derivation of the on-axis performance of Adaptive
Optics systems using a given number of guide stars of arbitrary altitude,
distributed at arbitrary angular positions in the sky. The expressions of the
residual error are given for cases of both continuous and discrete turbulent
atmospheric profiles. Assuming Shack-Hartmann wavefront sensing with circular
apertures, we demonstrate that the error is formally described by integrals of
products of three Bessel functions. We compare the performance of Adaptive
Optics correction when using natural, Sodium or Rayleigh laser guide stars. For
small diameter class telescopes (~5m), we show that a few number of Rayleigh
beacons can provide similar performance to that of a single Sodium laser, for a
lower overall cost of the instrument. For bigger apertures, using Rayleigh
stars may not be such a suitable alternative because of the too severe cone
effect that drastically degrades the quality of the correction.
