The conservation of energy is satisfied in de~Sitter's world; but from the practical standpoint it is abrogated in large scale problems such as that of the system of the spirals, since these are able to withdraw kinetic energy from a source not generally taken into account. Equation~\Eq{(39.44)} \[ \frac{1}{\gamma} \left(\frac{h}{r^{2}}\, \frac{dr}{d\phi}\right)^{2} + \frac{h^{2}}{r^{2}} - \frac{c^{2}}{\gamma} = -1 \] becomes on substituting for~$\gamma$ \[ \left(\frac{h}{r^{2}}\, \frac{dr}{d\phi}\right)^{2} + \frac{h^{2}}{r^{2}} = c^{2} - 1 - \tfrac{1}{3}\lambda h^{2} + \tfrac{1}{3} \lambda r^{2}, \] or writing $u = 1/r$ \[ \left(\frac{du}{d\phi}\right)^{2} + u^{2} = \frac{c^{2} - 1}{h^{2}} - \tfrac{1}{3}\lambda + \frac{\tfrac{1}{3} \lambda}{h^{2} u^{2}}. \] Whence, differentiating \[ \frac{d^{2}u}{d\phi^{2}} + u = -\frac{\tfrac{1}{3} \lambda}{h^{2}} u^{-3}. \Tag{(70.3)} \] The orbit is the same as that of a particle under a repulsive force varying directly as the distance. (This applies only to the form of the orbit, not to the velocity in the orbit.) For the motion of light the constant of areas~$h$ is infinite, and the tracks of light-rays are the solutions of \[ \frac{d^{2}u}{d\phi^{2}} + u = 0, \] i.e.\ straight lines. Determination of distance by parallax-measurements rests on the assumption that light is propagated in straight lines, and hence the method is exact in this system of coordinates. In so far as the distances of celestial objects are determined by parallaxes or parallactic motions, the coordinate~$r$ will agree with their accepted distances. This result may be contrasted with the solution for the field of a particle in \SecRef{38} where the coordinate~$r$ has no immediate observational significance. Radial distances determined by direct operations with measuring-rods correspond to~$R\chi$, not~$r$.