end of the century, Pierre-Simon de Laplace rediscovered the conservation of A, deriving it analytically, rather than geometrically. In the middle of the nineteenth century, William Rowan Hamilton derived the equivalent eccentricity vector defined below, using it to show that the momentum vector p moves on a circle for motion under an inverse-square central force (Figure 3). At the beginning of the twentieth century, Josiah Willard Gibbs derived the same vector by vector analysis. Gibbs' derivation was used as an example by Carl Runge in a popular German textbook on vectors, which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom. In 1926, Wolfgang Pauli used the LRL vector to derive the energy levels of the hydrogen atom using the matrix mechanics formulation of quantum mechanics, after which it became known mainly as the Runge–Lenz vector. Mathematical definition An inverse-square central force acting on a single particle is described by the equation The corresponding potential energy is given by . The constant parameter describes the strength of the central force; it is equal to for gravitational and for electrostatic forces. The force is attractive if > 0 and repulsive if < 0. The LRL vector A is defined mathematically by the formula where is the mass of the point particle moving under the central force, p is its momentum vector, L = r × p is its angular momentum vector, r is the position vector of the particle (Figure 1), is the corresponding unit vector, i.e., , and is the magnitude of r, the distance of the mass from the center of force. The SI units of the LRL vector are joule-kilogram-meter (J⋅kg⋅m). This follows because the units of p and L are kg⋅m/s and J⋅s, respectively. This agrees with the units of (kg) and of (N⋅m2). This definition of the LRL vector A pertains to a single point particle of mass moving under the action of a fixed force. However, the same definition may be extended to two-body problems such as the Kepler problem, by taking as the reduced mass of the two bodies and r as the vector between the two bodies. Since the assumed force is conservative, the total energy is a constant of motion, The assumed force is also a central force. Hence, the angular momentum vector L is also conserved and defines the plane in which