  A systematic study is performed of fully integrated particle multiplicities
in central Au-Au and Pb-Pb collisions at beam momenta of 1.7A GeV, 11.6A GeV
(Au-Au) and 158$A$ GeV (Pb-Pb) by using a statistical-thermal model. The close
similarity of the colliding systems makes it possible to study heavy ion
collisions under definite initial conditions over a range of centre-of-mass
energies covering more than one order of magnitude. In order to further study
the behaviour of strangeness production, an updated study of Si-Au collisions
at 14.6A GeV is also presented. The data analysis has been performed with two
completely independent numerical algorithms giving closely consistent results.
We conclude that a thermal model description of particle multiplicities, with
additional strangeness suppression, is possible for each energy. The degree of
chemical equilibrium of strange particles and the relative production of
strange quarks with respect to u and d quarks are higher than in e+e-, pp and
ppbar collisions at comparable and even at lower energies. The behaviour of
strangeness production as a function of centre-of-mass energy and colliding
system is presented and discussed. The average energy per hadron in the
comoving frame is close to 1 GeV per hadron despite the fact that the energy
increases more than 10-fold.
