## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### Numerical Calculation of the First \(O(\frac{1}{c})\) Correction in the Propagation Time
Moreover, if the lengths of the curves \(\gamma\) and \(\alpha\) are equal, then \(\gamma\) and \(\alpha\) coincide as non-parametrizable curves._  
In fact, what does it mean that \"the length of the geodesics from point \(p\) to point \(q\) does not exceed the length of any other curve\" (but only locally, i.e., in the vicinity of the given neighborhood)? This means that this geodesic line is minimal. So, this confirms the theorem in another, third well-known book on differential geometry by Mishtenko and Fomenko [24]:  
**Theorem 3**. : _The geodesic line \(\gamma:[a,b]\to M^{n}\) is minimal if it is not longer than any smooth path, joining the endpoints \(\gamma(a)\) and \(\gamma(b)\)._  
So, if we have, for example, the functional \(E(\gamma(s))=\int\mid\dot{\gamma}^{2}(s)\mid ds=\int g_{ij}(x)\dot{x}^{i}(s) \dot{x}^{j}(s)ds\) for the null cone equation, and we are searching for the extremals, these extremals will be derived from the geodesics--the multitude of the solutions for the extremals from the geodesics turns out to be larger in comparison with the one for the other local curves in the vicinity, in particular, for the functional \(E(\gamma(s))\) for the null cone equation. On the basis of combining the results in the previously mentioned monographs by Fock [19; 26], one easily establishes the validity of the following theorem in [24]:  
**Theorem 4**. : _The extremals of the length functional \(L(\gamma(s))=\int\sqrt{g_{ij}(x)\dot{x}^{i}(s)\dot{x}^{j}(s)}ds\) are smooth functions, derived from the geodesics (by means of smooth change of the parameter along the curve). Then, each extremal of the functional \(E(\gamma(s))=\int\mid\dot{\gamma}^{2}(s)\mid ds=\int g_{ij}(x)\dot{x}^{i}(s) \dot{x}^{j}(s)ds\) (called also action functional of the trajectory) is also an extremal of the functional \(L(\gamma(s))=\int\sqrt{g_{ij}(x)\dot{x}^{i}(s)\dot{x}^{j}(s)}ds\). However, each extremal of the functional \(L(\gamma(s))\) is not necessarily an extremal of the functional \(E(\gamma(s))\)._  
There is a simple inequality \(L^{2}(\gamma(s))\leq E(\gamma(s))\)[24] in confirmation of the above theorem, which can be immediately proved by means of the Schwartz inequality  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]6  
for the following functions [24]:  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]7  
It should be noted that the parameter \(s\) in the book [19] is defined simply as a \"space parameter along a curved line\". In the above formulaes, \(s\) is also a space parameter. But as noted in [24], when \(g(s)=const\), i.e., \(\gamma(s)=const\) and the parameter \(s\) is proportional to the length of the arc of the curved line, an equality can be derived \(L^{2}(\gamma(s))=E(\gamma(s))\). Thus, every solution of the null cone equation \(g_{ij}(x)\dot{x}^{i}(s)\dot{x}^{j}(s)=0\) (i.e., a zero-extremal of \(E(\gamma(s))\)) will also be a zero extremal of \(L^{2}(\gamma(s))\). If this zero extremal corresponds to the propagation of the signal along the geodesics of the minimal length, the propagation time will satisfy Fermat's principle regarding the least time of propagation of light between two space points. Consequently, even if there are other extremals for the length functional, of interest are only those, related to the minimal geodesics, obtained from the action functional \(E(\gamma(s))=\int g_{ij}(x)\dot{x}^{i}(s)\dot{x}^{j}(s)ds\), which ensures the fulfillment of Fermat's principle. New Approach for Analytical Calculation of Elliptic Integrals--Analytic Relation Between Elliptic Integrals in the Weierstrass and in the Legendre Form  
In this section, some new analytical algorithms for the calculation of elliptic integrals will be presented, which have been published in the paper [33].