  In this Article, we give a simple criterion for the regularity of a
tri-linear mapping. We provide if $f:X\times Y\times Z\longrightarrow W $ is a
bounded tri-linear mapping and $h:W\longrightarrow S$ is a bounded linear
mapping, then $f$ is regular if and only if $hof$ is regular. We also shall
give some necessary and sufficient conditions such that the fourth adjoint
$D^{****}$ of a tri-derivation $D$ is again tri-derivation.
