## 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  
Theorem 6: From (153), the corresponding equation for \(m=-3\)  
\[\frac{\widetilde{y}}{y^{3}}=-q^{2}J_{0}(y)+2(1+q^{2})H_{2}^{(4)}(y)-3H_{4}^{(4 )}(y) \tag{158}\]  
can be transformed into the equation for \(m=1\)  
\[y\widetilde{y}^{(q)}=J_{0}(y)-2(1+q^{2})I_{2}(y)+3q^{2}J_{4}(y) \tag{159}\]  
by means of the transformation \(y\Rightarrow\frac{1}{qy}\) and the simple relations (for the concrete case for \(m=2\))  
\[H_{m}(y)=-q^{m}J_{m}(\frac{1}{qy}),\ \ H_{m}(\frac{1}{qy})=-q^{m}J_{m}(y). \tag{160}\]  
In such a way, only one independent equation can be obtained from the \(4D\) recurrent system of equations\[2(1+q^{2})I_{2}^{(q)}(y)=3q^{2}\tilde{I}_{4}^{(q)}(y)+J_{0}^{(q)}(y)-y\widetilde{y} ^{(q)}. \tag{161}\]  
_10.4. Relations Between the Recurrent System of Equations in Terms of the \(y\) and \(x\) Variables_  
10.4.1. The Recurrent System of Equations in Terms of the \(x\) Variable  
Now, we shall write the recurrent system of equations for the elliptic integrals  
\[I_{m}^{(3)}(x)\equiv\int\frac{x^{m}dx}{\sqrt{4x^{3}-g_{2}x-g_{3}}},\]  
\[G_{m}^{(3)}(x)\equiv\int\frac{dx}{x^{m}\sqrt{4x^{3}-g_{2}x-g_{3}}},\,x=\frac{ \widetilde{a}_{2}}{y^{2}}+\widetilde{b}_{2}, \tag{162}\]  
as this has been performed in the monographs [38; 45]. Generally, the polynomial of a fourth-order under the square root is denoted as  
\[\gamma^{2}=a_{0}y^{4}+4a_{1}y^{3}+6a_{2}y^{2}+4a_{3}y+a_{4}, \tag{163}\]  
but for the concrete case of the integrals (162), the coefficients will be given by  
\[a_{0}=0\,,\,\,a_{1}=1\,\,,\,\,\,\,a_{2}=0\,\,,\,\,\,a_{3}=-\frac{g_{2}}{4}\, \,,\,\,\,\,a_{4}=-g_{3}. \tag{164}\]  
For this choice of the coefficients, due to \(a_{0}=0\), the polynomial becomes a cubic one, so we change the variables in (163) from \((Y,y)\) to \((X,x)\), i.e., \(X^{2}=4x^{3}-g_{2}x-g_{3}\), where the Weierstrass invariants (coefficients) \(g_{2}\) and \(g_{3}\) will be defined in the next sections. The recurrent system of equations is easily found to be (omitting the upper indice \"\((3)\)\" in \(I_{m}^{(3)}\))  
\[x^{n}X=2(2n+3)I_{n+2}-\frac{g_{2}}{2}(2n+1)I_{n}-ng_{3}I_{n-1}. \tag{165}\]  
It should be noted also that the variable transformation \(x=\frac{\widetilde{a}_{2}}{y^{2}}+\widetilde{b}_{2}\) in (162) contains other parameters \(\widetilde{a}_{2}\) and \(\widetilde{b}_{2}\), which are different from the parameters \(a_{2}\) and \(b_{2}\) in the transformation (145). Now, we shall transform the three-degree polynomials under the square root in (162) into the known four-degree polynomials for the elliptic integral in the Legendre form. The transformation \(\,\,x=\frac{\widetilde{a}_{2}}{y^{2}}+\widetilde{b}_{2}\) (as well as the previously applied transformation (145) \(x=\frac{a}{y^{2}}+b\)) are generalizations of the transformation \(x=e_{3}+\frac{(e_{1}-e_{3})}{y^{2}}\) in the paper [54], where \(e_{1},e_{2},e_{3}\) are constant roots of a cubic polynomial. The transformation transforms the integral \(\int\frac{dx}{\sqrt{4(x-e_{1})(x-e_{2})(x-e_{3})}}\) into the integral \(-\frac{1}{\sqrt{e_{1}-e_{3}}}\int\frac{dy}{\sqrt{(1-y^{2})(1-q^{2}y^{2})}}\), where \(q^{2}=\frac{(e_{2}-e_{3})}{(e_{1}-e_{3})}\). Under another transformation, \(y\Rightarrow\frac{1}{ky}\), the \(x\)-variable will transform as  
\[\widetilde{x}\equiv\widetilde{a}_{2}\widetilde{k}^{2}y^{2}+\widetilde{b}_{2}, \tag{166}\]  
where a new variable \(\widetilde{x}\) has been introduced. The two recurrent systems of equations, (153) (written in terms of the \(y\)-variable) and the recurrent system (165) for the \(x\)-variable, are not independent because  
\[y\widetilde{y}^{(q)}=J_{0}(y)-2(1+q^{2})I_{2}(y)+3q^{2}J_{4}(y) \tag{159}\]0  
Two central problems remain to be solved:  
1. How to combine the derived equations in terms of the \(x\) variable with the found relation (161);
2. How to estimate the parameter \(g_{2}\) in Equation (168).