## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### Another Integral in the Weierstrass Form After Applying the Four-Dimensional Uniformization
If \(e_{1},e_{2},e_{3}\) are the roots of the above polynomial, i.e.,_  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]1  
_then the periods \((\omega_{1},\omega_{2})\) on the complex lattice \(\Lambda\) (177) can be found from the following integrals:_  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]2  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]3  
_In the first integral, the unique branch \(\Gamma_{1}\) of the square root is chosen with cuts from \(e_{1}\) to \(e_{2}\) and from \(e_{3}\) to \(\infty\), and in the second integral, the branch \(\Gamma_{2}\) is with cuts from \(e_{2}\) to \(e_{3}\) and from \(e_{1}\) to \(\infty\)._  
Moreover, for every lattice \(\Lambda\) on the complex plane, defined according to (177), a biholomorphic mapping is determined as \(\varphi:\mathbb{C}/\Lambda\mapsto E(\mathbb{C})\), where \(E(\mathbb{C})\) denotes the elliptic curve \(y^{2}=4x^{3}-g_{2}(\Lambda)x-g_{3}(\Lambda)\). The mapping \(\varphi\) is defined by means of the Weierstrass elliptic function \(\rho(z)\) and its derivative \(\rho^{\prime}(z)\) and the inverse mapping--by the corresponding elliptic integral. Since \(\rho(z)\) and \(\rho^{\prime}(z)\) are uniformization functions, they will satisfy the cubic polynomial [34]  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]4  
from where the following relations are easily derived [34], following from the algebraic properties of the roots of a cubic equation  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]5  
This property is very important also for the theory of elliptic curves (see also the monograph [60]) since if the property \(g_{2}^{3}-27g_{3}^{2}\
eq 0\) is fulfilled, then a lattice \(\Lambda\) on the complex plane exists, so that the Weierstrass invariants \(g_{2}\) and \(g_{3}\) can be defined as (184)  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]6  
At this point, it becomes clear why the correspondence \((g_{2}(\Lambda),g_{3}(\Lambda))\Leftrightarrow\Lambda\) in the cited monograph [59] is in both directions--previously, we proved that if the lattice \(\Lambda\) is given, then the Weierstrass invariants \(g_{2}\) and \(g_{3}\) can be defined, i.e., \(\Lambda\Rightarrow(g_{2}(\Lambda),g_{3}(\Lambda))\). 8.1 The Consistency Problem for a Non-Zero Discriminant for the Case, When the Invariants in the Weierstrass Integrals are Expressed Through the Modulus Parameter \(q\) of the Integral in the Legendre Form  
Now, the statement is proved in the opposite directions--if the Weierstrass invariants are given, then a complex lattice can be defined \((g_{2}(\Lambda),g_{3}(\Lambda))\Rightarrow\Lambda\). Note that this statement is not trivial and it can be reformulated in the following way: If \(g_{2}\) and \(g_{3}\) are given (real or complex) numbers or even functions of some parameter (such examples will be shown further), then a two-dimensional lattice \(\Lambda\) on the complex plane with periods \(\omega_{1}\) and \(\omega_{2}\) can be defined, so that the sums in the defining equalities for \(g_{2}\) and \(g_{3}\) in (184) will be convergent. This is also a problem, which needs further investigation. Further, it shall be understood why this is important in view of the fact that an elliptic integral in the Legendre form (with a modulus parameter \(q\)) shall be transformed to an elliptic integral in the Weierstrass form \(\int\frac{dx}{\sqrt{4x^{3}-g_{2}x^{2}-g_{3}}}\). The Weierstrass integral is characterized by invariants \(g_{2}\) and \(g_{3}\), which shall be expressed by complicated polynomials, depending on the modulus parameter \(q\), i.e., \(g_{2}=g_{2}(q)\) and \(g_{3}=g_{3}(q)\).