Baecklund transformation for the Liouville equation and gauge conditions in the relativistic string theory

It is shown that the gauge conditions in the relativistic string theory which make it possible to use here the d'Alembert equation instead of the nonlinear Liouville equation are the direct consequence of the Backlund transformation relating solutions of these equations. A purely geometrical derivation of the Backlund transformation for the Liouville equation is given also. By making use of the co-moving frame method and the exterior differential forms in the surface theory the classical theory of the relativistic string in the gauge t=tau is constructed. The moving frame on the string trajectory is chosen in a special form. As a result, the theory of the free relativistic string in the four-dimensional space-time is reduced to the d'Alembert equation for one scalar function