## 13 Conclusions  
In this paper, the propagation time of a signal was analytically calculated by making use of the null cone equation \(ds^{2}=0\) from the metric element, defined for the near-Earth space. Two different cases were considered, corresponding to two different parametrizations of the space coordinates \((x,y,z)\) of the satellite orbit: The first parametrization is in terms of the eccentric anomaly angle \(E\), which is the dynamical parameter for the case of a plane elliptic orbit, and the second parametrization is in terms of all the six Kepler parameters \((f,a,e,\Omega,I,\omega)\) for the case of space-oriented orbit with only the true anomaly angle \(f\) considered to be the dynamical parameter for the motion of the satellite. The propagation time for the first case is proved to contain the three known elliptic integrals of the first, second and the third kind; all of them are zero-order elliptic integrals. The definitions of the different kinds of elliptic integrals are given in Section 4.3. The propagation time for the second case is investigated in detail in Section 6. It is expressed in more complicated elliptic integrals of the second and of the fourth order. These are higher-order elliptic integrals, the mathematical definitions of which are given in Section 4.1. It should be mentioned these integrals have been known in the mathematical literature for a long time, but the known physical applications are not so many compared to the elliptic integrals of the first, second and third kind. For both cases, the sum of the \(O(c^{-1})\) and \(O(c^{-3})\) propagation time corrections are real-valued expressions. For the case of plane elliptical orbits, this was proved in Section 5.1, and for the second case of space-oriented orbits, this was proved in Section 7. This proves the correctness of the theoretical approach of parametrizing the space coordinates \(x,y\) or \(x,y,z\) in the null cone equation with the orbital coordinates, known from celestial mechanics. Numerical calculations for the propagation time for the first case of plane elliptical orbit are performed in Section 8 and they confirm that during the propagation time, the signal travels a much greater distance than the distance travelled by the satellite. It turned out also that theorems from general relativity theory [19] and from differential geometry [24; 26] confirm the conclusion that various parametrizations of the space coordinates, related to the orbital motion of the satellite, can be used. This proof comes in three stages: The real-valuedness of all the expressions for the propagation time for all the two cases, the theorems from general relativity theory and differential geometry and, lastly, the numerical calculations confirm the substantial fact that no matter whether a parametrization of the space coordinates (related to the satellite orbits) is used, the propagation time for a signal, expressed from the null cone equation, gives the typical values for signal propagation and not for the moving satellite. It is important to clarify that the calculated propagation time gives the correspondence between the initial and final values of the dynamical parameter (the eccentric anomaly angle \(E\) or the true anomaly angle \(f\)) and the initial \(T_{init}\) and final \(T_{fin}\) moments of propagation of the signal. However, in such a theoretical setup, \(T_{init}\) and \(T_{fin}\) are not related to any real processes of transmission of signals between satellites on one and the same orbit or on different space-oriented orbits. In such a modified setting, the initial position of the signal-emitting satellite and the final position of the signal-receiving satellite have to be given and they will correspond to the initial \(T_{init}\) and final \(T_{fin}\) moments of propagation of the signal. They have to be calculated in view of the fact that the trajectory of the signal is curved due to the action of the gravitational field.