  Considered is quantum tunnelling in anisotropic spin systems in a magnetic
field perpendicular to the anisotropy axis. In the domain of small field the
problem of calculating tunnelling splitting of energy levels is reduced to
constructing the perturbatio n series with degeneracy, the order of degeneracy
being proportional to a spin value. Partial summation of this series taking
into account ''dangerous terms'' with small denominators is performed and the
value of tunnelling splitting is calculated with allowance for the first
correction with respect to a magnetic field.
