• The anomaly is then the angle BFP which the radius vector makes with the major axis.
• Let P, P' be two consecutive positions of the radius vector.
• From the law of angular motion of the latter its radius vector will run ahead of PQ near A, PQ will overtake and pass it at apocentre, and the two will again coincide at pericentre when the revolution is completed.
• The problem of finding a radius vector satisfying this condition is one which can be solved only by successive approximations, or tentatively.
• This is the simplest case of generation of a plane figure by a moving ordinate; the corresponding figure for generation by rotation of a radius vector is a circle.
• Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant angular velocity about the pole, in the direction of diminishing radius vector.
• Hence equal areas are swept over by the radius vector in equal times.
• If p be the radius-vector 0J of the momental ellipsoid Ax+By+Czf=Me4 (I)
• The motion of the body relative to 0 is therefore completely represented if we imagine the momental ellipsoid at 0 to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact.
• Hence the relation between the radius vector and the perpendicular on the tangent of the rolling curve must be identical with the relation between the normal PN and the ordinate PR of the traced curve.
• In the case of the satellites it is the period relative to the radius vector from the sun.
• Johann Kepler had proved by an elaborate series of measurements that each planet revolves in an elliptical orbit round the sun, whose centre occupies one of the foci of the orbit, that the radius vector of each planet drawn from the sun describes equal areas in equal times, and that the squares of the periodic times of the planets are in the same proportion as the cubes of their mean distances from the sun.
• The co-ordinates of P will then be the following three quantities: - (I) The length of the line OP, or the distance of the body from the origin, which distance is called the radius vector of the body.
• (2) The angle XOQ which the projection of the radius vector upon the fundamental plane makes with the initial line OX.
• (3) The angle QOP, which the radius vector makes with the fundamental plane.