The polarisation potential acting on an electron (or positron) in the field of an atom or ion is calculated. Neglecting the exchange between the core and valence electrons, the wave function Φ of the core is written as Φ = Φ0+χ, where Φ0 is the unperturbed core wave function, and χ corresponds to the perturbation of the core by the valence electron, and is required to be orthogonal to Φ0. Then eq. (7) gives the expression for the polarisation potential. The perturbation χ of the core is calculated from a stationary perturbation theory of first order [eq. (11)] for hydrogen-like core orbits of principle quantum number 1 and 2; the corresponding polarisation potentials are calculated and are shown in Fig. 1, 2 and 3 (sec. 5). An approximation for many electron cores is given in sec. 6 and applied to helium- and neon-like core configurations (Fig. 4); simple analytical approximations of the polarisation potential are given for these cases. The results for Si4+ are discussed and compared with the results of other authors (sec. 7). The last section of the paper contains some critical remarks on the stationary approximation. It is shown that the influence of the “kinetic terms” enlarges the polarisation effects, contrary to the usual supposition. — An appendix contains a derivation of an expression for the polarisation potential acting on two valence electrons for large distances from the core.