In the framework of classical dielectric theory, the hemispherical geo
metry is studied. Calculations on surface modes are carried out for is
olated Drude-like hemispheres. The convergence of the results with res
pect to the number of coupled terms in the expressions of the potentia
l is discussed. The electron-energy-loss probability is studied for Al
and Ag particles involving this geometry. The surface modes and hence
the energy-loss probability are given by coupled expressions, the phy
sical meaning of which is the coupling among multipolar terms, because
of the particular geometry. The results obtained here present a good
quantitative agreement with experiments in the case of clear surfaces
(Ag) and provide a qualitative understanding for the experiments in Al
, in terms of the position and impact parameter of the beam. This allo
ws us to set the validity of the dielectric theory for cases that seem
ed to question it.