The method of obtaining the estimates of the maximal t-interval (ω_-, ω₊) on which the solution of the N-body problem exists and which is such that some fixed mutual distance (e. g. Δ₁₂) exceeds some fixed non-negative lower bound, for all t contained in (ω_-, ω₊), is considered. For given masses and initial data, the increasing sequences of the numbers γ_k, each of which provides the estimate ω₊ >γ_k, are constructed. It appears that if ω₊ = +∞, then {Mathematical expression}.