The solutions of the Dirichlet problem for the Laplace equation in cylindrical coordinates are
obtained for an axisymmetric quadrupole with boundary conditions specified on the surface of a circular cylinder. The
solution is reduced to calculating the harmonic potential, for the determination of which the TFKP methods are used. A
simple analytical formula is found that accurately describes the electrostatic potential of the field of a quadrupole with
cylindrical electrodes. Using the formulas found for the potential and its derivatives, it is possible to accurately calculate
the stability regions of quadrupole ion traps when additional radio-frequency fields are turned on.