## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### Second Formulation of the Theorem for Four-Dimensional Uniformization
The denominator can be decomposed as  
\[\overline{\overline{g}}_{3}=-\frac{1}{6}q^{2}(1+q^{2})+\frac{(1+q^{2})^{3}}{6^{3}}. \tag{227}\]3Now, if one assumes that this expression is positive, then this is possible only when the following inequalities are simultaneously fulfilled:  
\[\overline{\overline{g}}_{3}=-\frac{1}{6}q^{2}(1+q^{2})+\frac{(1+q^{2})^{3}}{6^{3}}. \tag{227}\]4  
The other two inequalities with the positive sign should be discarded because \(0<q<1\). But if the inequalities (271) are fulfilled and \(K(q)\), \(N(q)\) and \(g_{3}\) are positive, then from the real-valuedness of \(g_{2}\) and \(g_{2}\) and also the positivity and real-valuedness of the discriminant \(\Delta=g_{2}^{3}-27g_{3}^{2}>0\) from the theorem in [60], it should follow that the roots \(e_{1},e_{2},e_{3}\) of the polynomial \(4x^{3}-g_{2}x-g_{3}=0\) should all be real. But this will be in contradiction to the case of one real root and two imaginary roots, investigated in this paper. Consequently, an important conclusion can be made that there can be two cases:  
1 st case: positive invariant \(g_{2}\), imaginary \(g_{3}\) (or both imaginary invariants \(g_{2}\) and \(g_{3}\)) and positive discriminant \(\Delta=g_{2}^{3}-27g_{3}^{2}>0\). Then, not all the roots should be real ones (the presently investigated case). 2 nd case: real roots \(\alpha_{1},\beta_{1},\gamma_{1}\) (the case not investigated in this paper). Then, the invariants \(g_{2}\) and \(g_{3}\) should be real and the discriminant \(\Delta\) should be negative. From the point of view of practical applications for the inter-satellite communications, it is evident that the case of a negative discriminant and modulus parameter \(q<\frac{1}{\sqrt{2}}\approx 0.707106\) (remember that for a GPS orbit \(e\approx 0.01\) and \(q=\frac{1-e}{1+e}\) ), such a case seems to be realistic and should be investigated. Note that in the strict mathematical sense, the theorem in the monograph by Koblitz [60] should be reformulated and proved. The case of a negative discriminant for a general cubic equation \(x^{3}+px+\widetilde{q}=0\) is investigated in details in the monograph by Obreshkoff [61] and the roots have been expressed by the concise formulae  
\[\overline{\overline{g}}_{3}=-\frac{1}{6}q^{2}(1+q^{2})+\frac{(1+q^{2})^{3}}{6^{3}}. \tag{227}\]5  
and the angular variable \(\varphi\) is given by the formulae  
\[\overline{\overline{g}}_{3}=-\frac{1}{6}q^{2}(1+q^{2})+\frac{(1+q^{2})^{3}}{6^{3}}. \tag{227}\]6  
We have not investigated the case of the other pair of invariants (212) \(\widetilde{g}_{2}^{(1,2)}\) (identical with the invariant (226)) and (214) \(\widetilde{g}_{3}^{(1,2)}\); when the second invariant is given by a polynomial of the sixth degree and depending on its roots, it can be either positive or negative. Finding the Weierstrass Function \(\rho(z)\) and the Weierstrass Invariant \(g_{2}\) From the Known Roots of the Cubic Equation and the Group-Theoretical Law for Summing up of Points on the Cubic Curve  
The main goal of every investigation, related to the cubic algebraic equation in its parametriizable form, is to find the expression for the Weierstrass function \(\rho(z)\) by known Weierstrass invariants \(g_{2}\) and \(g_{3}\). Since we started from a Weierstrass integral, derived from an elliptic integral in the Legendre form, previously, we found how the invariant \(g_{3}\) can be expressed through the other invariant \(g_{2}\) (258)  
\[\overline{\overline{g}}_{3}=-\frac{1}{6}q^{2}(1+q^{2})+\frac{(1+q^{2})^{3}}{6^{3}}. \tag{227}\]7Now, it remains to find the other Weierstrass invariant \(g_{2}\) and the Weierstrass function itself.