  The fact that repulsive Rutherford scattering casts a paraboloidal shadow is
rarely exploited in introductory mechanics textbooks. Another rarely used
construction in such textbooks is the Hamilton vector, a cousin of the more
famous Laplace-Runge-Lenz vector. We will show how the latter (Hamilton's
vector) can be used to explain and clarify the former (paraboloidal shadow).
