Suppose a particle moves in an ellipse whose semiaxes are a and b in such a manner that it obeys the law of areas with respect to the center of the ellipse as origin ; it is required to find the components of acceleration along and perpendicular to the radius vector. The equation of the ellipse may be written in the parametric form
 (1.30) 
for, if φ is eliminated, the ordinary equation
is found. It follows from (1.30) that
 (1.31) 
On substituting (1.30) and (1.31) in the expression for the law of areas,
it is found that
The integral of this, equation is
and if φ = 0 when t = 0, then φ = c_{1}t.
On substituting the final expression for φ in (1.30), it is found that

If these values of the derivatives are substituted in (1.20) the components of acceleration along and perpendicular to the radius vector are found to be

1. A particle moves with uniform speed along a helix traced on a circular cylinder whose radius is R ; find the components of velocity and acceleration parallel to the x, y and z axes. Answer :

2. A particle moves in the ellipse whose parameter and eccentricity are p and e with uniform angular speed with respect to one of the foci as origin ; find the components of velocity and acceleration along and perpendicular to the radius vector and parallel to the x and y axes in terms of the radius vector and the time. Answer :

3. A particle moves in an ellipse in such a manner that it obeys the law of areas with respect to one of the foci as an origin ; it is required to find the components of velocity and acceleratio i along and perpendicular to the radius vector and parallel to the axes in terms of the coordinates. Answer :

4. The accelerations along the x and y axes are the derivatives of the velocities along these axes ; why are not the accelerations along and perpendicular to the radius vector given by the derivatives of the velocities in these respective directions ? Find the accelerations along axes rotating with the angular velocity unity in terms of the accelerations with respect to fixed axes.