THE first-order perturbation theory of satellite orbits can be formulated simply and elegantly in terms of constants of the unperturbed motion using vector methods. For let the equation of motion be: where μ = GM (M = mass of the earth), and F is a general perturbing force (per unit mass). The energy E and angular momentum h (both per unit mass) defined as: are constants of the unperturbed motion. Hamilton's integral, which may be written in the form1: provides a further constant of the unperturbed motion under an inverse square law force. Taking scalar products with h and r, k is seen to be a vector of length μe lying along the major axis in the direction of apogee. The orbit is over-determined by E, h and k which give seven scalar equations.