To the $RH$ (Ramification Hypothesis = Proposition 1 in Kurepa [1935:2,3 p.~130] we join here proposition $P_0'(s.~3:2)$, $P_{18},P_{19},\ldots,P_{45}$, each equivalent to $RH$; we stress in particular $P_{18} := P_s:$ For every branching tree $T$ the width $p_sT^2$ of the cardinal square of $T$ equals $p_sT$. (s.~1:0) and is attained (s.~No.~3).