Four films (A, B, C and D) of discontinuous platinum films (D(Pt)Fs) whose mass thicknesses (dm) are 10, 20, 30 and 40 A, respectively were deposited onto Corning 7,059 glass substrates at ambient temperature via the thermal evaporation technique. Each film was γ-irradiated by different doses, namely, 100, 200, 300, 500 and 700 Gy; this was done by using 137Cs (0.662 MeV) radiation source of dose rate 0.5 Gy/min. For each dose, the d. c. and total resistance of the Pt films were measured; in that way the a.c. conductance Gac of the films could be determined. It was found that: (1) Gac increases as the dose, dm and the angular frequency ω of the voltage imposed on the film increases (2) the γ-irradiation has modified the shape of islands such that they are elongated parallel to the substrate surface and thus the inter-island spacings have decreased. This elongation has been confirmed via micrographs taken by the atomic force microscope. To account qualitatively for the results of Gac it was assumed that, the a.c. conductance of D(Pt)Fs is due to the hopping of electrons through the sites which exist on the substrate surface between two adjacent islands.