(31)
It has been seen that the orbit is in this case an ellipse; also that if we put μ = K/m the velocity at any point P is v = √μ. OD, where OD is the semi-diameter conjugate to OP. Hence (31) is consistent with the known property of the ellipse that OP2 + OD2 is constant.
The forms assumed by the dynamical equations when the axes of reference are themselves in motion will be considered in § 21. At present we take only the case where the rectangular axes Ox, Oy rotate in their own plane, with angular velocity ω about Oz, which is fixed. In the interval δt the projections of the line joining the origin to any point (x, y, z) on the directions of the co-ordinate axes at time t are changed from x, y, z to (x + δx) cos ω δt − (y + δy) sin ωδt, (x + δx) sin ω δt + (y + δy) cos ω δt, z respectively. Hence the component velocities parallel to the instantaneous positions of the co-ordinate axes at time t are
u = ẋ − ωy, v = ẏ + ωz, ω = ż.
(32)
In the same way we find that the component accelerations are
u̇ − ωv, v̇ + ωu, ω̇
(33)
Hence if ω be constant the equations of motion take the forms
m (ẍ − 2ωẏ − ω2ẋ) = X, m (ÿ + 2ωẋ − ω2y) = Y, mz̈ = Z.