## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### A New Theorem, Concerning the Equivalence Between Some of the Equations in the Recurrent System
#### Combining the Two Recurrent Systems of Equations  
From the two recurrent system of equations in terms of the \(y\)-variable and the \(x\)-variable, the following equation can be obtained:  
\[\widetilde{X}^{(\widetilde{k})}=N_{1}(\widetilde{a}_{2},\widetilde{b}_{2} \widetilde{k})J_{2}^{(\widetilde{k})}(y)+N_{2}(\widetilde{a}_{2},\widetilde{ b}_{2}\widetilde{k})J_{0}^{(\widetilde{k})}(y)+N_{3}(\widetilde{a}_{2}, \widetilde{b}_{2}\widetilde{k})= \tag{169}\]  
\[=N_{1}^{(\widetilde{k})}J_{2}^{(\widetilde{k})}(y)+N_{2}^{(\widetilde{k})} \biggl{[}-\frac{1}{2\widetilde{a}_{2}}J_{0}^{(q)}(y)\biggr{]}+N_{3}^{( \widetilde{k})}, \tag{170}\]  
where  
\[N_{1}(\widetilde{a}_{2},\widetilde{b}_{2}\widetilde{k})\equiv N_{1}^{( \widetilde{k})}\equiv 8\widetilde{a}_{2}^{3}\widetilde{k}^{2}(1+\widetilde{k}^{2 })+24\widetilde{a}_{2}^{2}\widetilde{b}_{2}\widetilde{k}^{2}, \tag{171}\]  
\[N_{2}(\widetilde{a}_{2},\widetilde{b}_{2}\widetilde{k},)\equiv N_{2}^{( \widetilde{k})}\equiv 12\widetilde{a}_{2}\widetilde{b}_{2}^{2}-\frac{g_{2} \widetilde{a}_{2}}{(\sqrt{a})^{(q)}}-4\widetilde{a}_{2}^{3}\widetilde{k}^{2}, \tag{172}\]  
\[N_{3}(\widetilde{a}_{2},\widetilde{b}_{2}\widetilde{k})\equiv N_{3}^{( \widetilde{k})}\equiv 4\widetilde{a}_{2}^{3}\widetilde{k}^{2}y^{2}y^{2} \widetilde{y}^{(\widetilde{k})}. \tag{173}\]  
In (169), the expression for \(\widetilde{X}^{(\widetilde{k})}\) is defined as \(\widetilde{X}^{(\widetilde{k})}\equiv 4\widetilde{x}^{3}-g_{2}^{( \widetilde{k})}\widetilde{x}-g_{3}^{(\widetilde{k})}\). The notations \(g_{2}^{(\widetilde{k})}\) and \(g_{3}^{(\widetilde{k})}\) mean that these Weierstrass invariants are expressed as complicated functions of the parameter \(\widetilde{k}\).  
Now, it can be noted that if it is possible to make the replacement \(\widetilde{k}\iff q\), then another equation will be obtained:  
\[\widetilde{X}^{(q)}=N_{1}^{(q)}J_{2}^{(q)}(y)+N_{2}^{(q)}J_{0}^{(q)}(y)+N_{3}^ {(q)}\quad, \tag{174}\]  
and in this way, the Equations (170) and (174) will give the following solution for the zero-order elliptic integral in the Legendre form:  
\[J_{0}^{(4)}(y)\equiv I_{0}^{(q)}(y)=\frac{2\widetilde{X}^{(q)}}{\sqrt[3]{2}N _{2}^{(q)}}-\frac{N_{3}^{(q)}}{N_{2}^{(q)}}\]  
\[-\frac{\sqrt[3]{2}\widetilde{X}^{(\widetilde{k})}\left(\sqrt[3]{2}-2N_{2}^{(q )}\right)}{N_{2}^{(q)}\left(\sqrt[3]{2}-2\right)}-\frac{N_{1}^{(q)}\left(2 \widetilde{X}^{(q)}-\sqrt[3]{2}N_{3}^{(q)}\right)}{N_{2}^{(q)}\sqrt[3]{2}}. \tag{175}\]As a consequence of the replacement \(\widetilde{k}\Longleftrightarrow q\), one also has the following equality, which follows from the defining relations (168):  
\[\frac{\widetilde{k}^{4}-\widetilde{k}^{2}+1}{3g_{2}(\widetilde{k})}=\frac{q^{4} -q^{2}+1}{3g_{2}(q)}. \tag{176}\]  
If this is a quite a general condition (not imposing any restrictions), then the replacement \(\widetilde{k}\Longleftrightarrow q\) will be justified. Evidently, this will depend on the determination of \(g_{2}^{(q)}\equiv g_{2}(q)\), which will be given in the next sections.