Tubes about submanifolds play an important role in Riemannian geometry and their properties have been studied intensively and applied on many occasions. For example, we refer to [\textit{A. Gray}, Tubes, Addison- Wesley, Redwood (1990; Zbl 0692.53001)] and [the reviewer, Rend. Semin. Fac. Sci., Univ. Cagliari, supplemento al vol. 58, 73-176 (1988)] where many results are given together with a large but not exhaustive list of references. In this study the tubes always have constant radius. In this paper the author's purpose is to develop a theory for tubes about submanifolds when the radius is not constant on the submanifold. The main part of the paper is devoted to the derivation of formulae for the shape operator of such a tube when the ambient manifold is Euclidean space. One may expect that this will lead to the construction of examples of special Riemannian hypersurfaces. The author gives as example a construction of hypersurfaces having the property that the number of distinct principal curvatures is constant on the hypersurface or on a dense subset of it.