Generalization of two Bonnet's Theorems to the relative Differential Geometry of the 3-dimensional Euclidean space

This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space $\R{E} ^3 $ we consider a surface $\varPhi %\colon \vect{x} = \vect{x}(u^1,u^2) $ with position vector field $\vect{x}$, which is relatively normalized by a relative normalization $\vect{y}% (u^1,u^2) $. A surface $\varPhi^*% \colon \vect{x}^* = \vect{x}^*(u^1,u^2) $ with position vector field $\vect{x}^* = \vect{x} + \mu \, \vect{y}$, where $\mu$ is a real constant, is called a relatively parallel surface to $\varPhi$. Then $\vect{y}$ is also a relative normalization of $\varPhi^*$. The aim of this paper is to formulate and prove the relative analogues of two well known theorems of O.~Bonnet which concern the parallel surfaces (see~\cite{oB1853}).