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 $\mathbb{R}{E}^3$ we consider a surface $\varPhi %\colon \vect{x} = \vect{x}(u^1,u^2) $withpositionvectorfield$\vect{x}$, which is relatively normalized by a relative normalization $\text{vect}y$. A surface ${\Phi }^{\ast }$ with position vector field $\text{vect}{x}^{\ast }=\text{vect}x+\mu \text{vect}y$, where $\mu$ is a real constant, is called a relatively parallel surface to $\Phi$. Then $\text{vect}y$ is also a relative normalization of ${\Phi }^{\ast }$. 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}).