  In this paper we investigate the gravothermal instability of spherical
stellar systems endowed with a radially anisotropic velocity distribution. We
focus our attention on the effects of anisotropy on the conditions for the
onset of the instability and in particular we study the dependence of the
spatial structure of critical models on the amount of anisotropy present in a
system. The investigation has been carried out by the method of linear series
which has already been used in the past to study the gravothermal instability
of isotropic systems.
  We consider models described by King, Wilson and Woolley-Dickens distribution
functions. In the case of King and Woolley-Dickens models, our results show
that, for quite a wide range of amount of anisotropy in the system, the
critical value of the concentration of the system (defined as the ratio of the
tidal to the King core radius of the system) is approximately constant and
equal to the corresponding value for isotropic systems. Only for very
anisotropic systems the critical value of the concentration starts to change
and it decreases significantly as the anisotropy increases and penetrates the
inner parts of the system. For Wilson models the decrease of the concentration
of critical models is preceded by an intermediate regime in which critical
concentration increases, it reaches a maximum and then it starts to decrease.
The critical value of the central potential always decreases as the anisotropy
increases.
