  The paper proposes 4-dimensional equations for the proper characteristics of
a rigid reference frame \[ {{{W}'}^{\gamma }}=\Lambda
^{0}_{\;\;i}\frac{d\Lambda ^{\gamma i}}{d{t}}\,\,,\] \[ {{{\Omega }'}^{\gamma
}}=-\frac{1}{2}\,{{e}^{\alpha \mu \gamma }}\Lambda _{\;\;i}^{\mu
}\frac{d{{\Lambda }^{\alpha i}}}{d{t}}\,\,.\]
  From these conditions follow the law of motion of the proper tetrad and the
equations of the inverse problem of kinematics, i.e., differential equations
that solve the problem of restoring the motion parameters of a rigid reference
frame from known proper acceleration and angular velocity. In particular, it is
shown that when boosted, a moving reference frame that has proper Thomas
precession relative to the new laboratory frame will have a combination of two
rotations: the new Thomas proper precession and the Wigner rotation, which
together give the original frequency of Thomas precession $$
\frac{1-\sqrt{1-v^{2}}}{v^{2}\sqrt{1-v^{2}}}\,\,e^{\beta\mu\nu}v^{\mu}\dot{v}^{\nu}=\omega'^{\beta}_{W}
+
b^{\beta\alpha}_{W}\,\frac{1-\sqrt{1-v^{*2}}}{v^{*2}\sqrt{1-v^{*2}}}\,\,e^{\alpha\mu\nu}v^{*\mu}\dot{v}^{*\nu}\,.$$
