  In the Kuramoto model, a uniform distribution of the natural frequencies
leads to a first-order (i.e., discontinuous) phase transition from incoherence
to synchronization, at the critical coupling parameter $K_c$. We obtain the
asymptotic dependence of the order parameter above criticality: $r-r_c \propto
(K-K_c)^{2/3}$. For a finite population, we demonstrate that the population
size $N$ may be included into a self-consistency equation relating $r$ and $K$
in the synchronized state. We analyze the convergence to the thermodynamic
limit of two alternative schemes to set the natural frequencies. Other
frequency distributions different from the uniform one are also considered.
