Recalling that for an elliptical orbit ( and hence also a circular orbit ) the velocity and radius vectors are perpendicular at apoapsis and periapsis, conservation of angular momentum requires h = r _ pv _ p = r _ av _ a = \ text { constant }, thus v _ p = \ frac { r _ a } { r _ p } v _ a:
22.
These irregularities or " residuals ", both in the planet's ecliptic longitude and in its distance from the Sun, or radius vector, might be explained by a number of hypotheses : the effect of the Sun's gravity, at such a great distance might differ from Newton's description; or the discrepancies might simply be observational error; or perhaps Uranus was being pulled, or perturbed, by an as-yet undiscovered planet.
23.
"' Problem 3 "'again explores the ellipse, but now treats the further case where the center of attraction is at one of its foci . " A body orbits in an ellipse : there is required the law of centripetal force tending to a focus of the ellipse . " Here Newton finds the centripetal force to produce motion in this configuration would be inversely proportional to the square of the radius vector . ( Translation :'Therefore the centripetal force is reciprocally as L X SP? that is, ( reciprocally ) in the doubled ratio [ i . e . square ] of the distance . . . .') This becomes Proposition 11 in the " Principia ".
