The analysis of the mathematical criteria defining the conditions of the changeover of the hydrodynamic regimes in a three-phase reactor was performed. The object of the analysis were criteria derived from the volume-averaged macroscopic equations of continuity and momentum balance for both phases [11-13]. Analysing the way of derivation of these criteria and investigating their characteristic features, main similarities and discrepancies were discussed. It was established, that the condition of the stability change, C-1 = 0, can be treated as a necessary and sufficient criterion of the transition from the trickling flow to the pulsing one in the reactor.