Hıe purpose of this paper is, first, to preseni the definition of the a-pedal hypersurface of a given smooth, connecled and orienlable surface M in E as a hypersurface with consıanl support funclion, then ,n+l (chapter I) and if M is laken we will give Ihe sotne new characlerislic properties of the a-pedal hypersurface of M relaled to support function, the area element, the higher order Gaussian curvatures, the characteristic polynom, the first, second and third fundamental forms and their coefficients of M (chapter 11).