Let \(S^{n-1}=\{g\in\mathbb{R}^ n: \| g\|=1\}\) be the Euclidean sphere in \(\mathbb{R}^ n\). Let \(H:=(L^ 2(S^{n- 1},\lambda),\langle\cdot,\cdot\rangle)\) be the Hilbert space of square- integrable functions on \(S^{n-1}\) with respect to the Lebesgue measure \(\lambda\) in which the inner product is canonically given. This space admits a complete orthonormal system \(\{\psi_ i;\;i\in I\}\) formed by a countable number of spherical harmonic polynomials [cf. \textit{K. Müller}, ``Spherical polynomials'', Springer Lectures Notes in Mathematics 17 (1966)]. Consider now a nonsmooth function \(f\) defined on an open set \(U\subset\mathbb{R}^ n\) and directionally differentiable at \(x_ 0\in U\). If the directional derivative \(g\in S^{n-1}\mapsto d_{x_ 0} f(g)\) belongs to \(H\), then the coefficients \(a_ i\) in the expansion \(d_{x_ 0} f=\sum_{i\in I} a_ i\psi_ i\) play an important role in the understanding of the behaviour of the function \(f\) around \(x_ 0\). This is, roughly speaking, the main idea behind this interesting and original paper.