According to Equations (11.138) and (11.150),
According to Equations (11.139) and (11.151),
It follows from Equations (11.76), (11.122)-(11.124), (11.170), (11.171), (11.182), and (11.183), as well as the previous expressions for , , , and , that the net perturbation to the lunar orbit due to terms in the solution of the lunar equations of motion that depend linearly on is
The first term on the right-hand side of Equation (11.259) is known as the parallactic inequality. The parallactic inequality attains its maximum amplitude when the Moon in half illuminated (i.e., when or ). Conversely, the amplitude of the parallactic inequality is zero when the Moon is either fully illuminated or not illuminated at all (i.e., when or ). According to Equation (11.259), the parallactic inequality generates a perturbation in the lunar ecliptic longitude that oscillates with a period of a synodic month, and has an amplitude (calculated using , , and ) of arc seconds (Yoder 1995). As before, the oscillation period is in good agreement with observations, whereas the amplitude is somewhat inaccurate [it should be arc seconds (Chapront-Touzé and Chapront 1988)] because of the omission of higher-order (in ) contributions.