## 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
For the nominator of \(K(q)\), the positivity is easily established, because it can be represented in the form  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]4  
which, evidently, is greater than zero. Consequently, whether \(g_{2}\) will be positive depends on the ratio \(\frac{\overline{g}_{3}}{\widetilde{g}_{2}}\). The invariant \(\overline{g}_{2}\) (226)  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]5  
is evidently positive. Let us check the sign of expression (227) for \(\overline{g}_{3}\), which can be rewritten as  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]6  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]7  
This expression will be positive when the inequalities  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]8and also  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]9  
are fulfilled. The first case (267) should be disregarded, because it means that \(q>6.804\), which is impossible because \(q\) should be in the range \(0<q<1\). The second expression (268) is positive when \(q>2.168\), which is also impossible. Consequently, the expression in the square brackets in (266) cannot be positive when \(q>6.804\) and \(q>2.168\). However, when \(q<6.804\) and \(q<2.168\) (always fulfilled since \(0<q<1\)), the expression  
\[\overline{\overline{g}}_{3}=-\frac{1}{6}q^{2}(1+q^{2})+\frac{(1+q^{2})^{3}}{6^{3}}. \tag{227}\]0  
is positive, because both expressions in the round brackets are negative and (269) is with a plus sign. Therefore, we proved that for all values of \(q\), the invariant \(g_{2}\) (263) is positive and from the assumption for the positivity of the discriminant (236)  
\[\overline{\overline{g}}_{3}=-\frac{1}{6}q^{2}(1+q^{2})+\frac{(1+q^{2})^{3}}{6^{3}}. \tag{227}\]1  
it follows also that \(\mid g_{3}\mid>\frac{1}{3\sqrt{3}}\mid g_{2}^{\frac{3}{2}}\mid\). The Possibility for a Positive and Real Invariant \(g_{2}\), Imaginary Invariant \(g_{3}\) and a Negative Discriminant--A Theorem in [60]  
Formally, it may seem that if \(g_{3}\) is negative but the condition for the moduli is fulfilled, then the discriminant \(\Delta\) will be positive. In fact, here, we may have the case when \(g_{2}\) is positive but \(g_{3}\) can be imaginary. This problem is not investigated in the mathematical literature, but it is important that the possibility for \(g_{3}\) to be imaginary does not contradict a theorem in problem 12, paragraph 6 of chapter 1 in the monograph by Koblitz [60]:  
**Theorem 12**. : _The roots \(e_{1},e_{2},e_{3}\) of the polynomial \(4x^{3}-g_{2}x-g_{3}=0\) are all real ones if and only if the invariants \(g_{2}\) and \(g_{3}\) are all real ones and the discriminant \(\Delta=g_{2}^{3}-27g_{3}^{2}>0\) is positive._  
In this paper, we are not investigating the case when all the roots of the polynomial \(4x^{3}-g_{2}x-g_{3}=0\) are real; we deal with the case when only the first root \(\alpha_{1}\) is a real one and the other two roots \(\beta_{1}\) and \(\gamma_{1}\) are imaginary. Consequently, it is not obligatory that the invariants \(g_{2}\) and \(g_{3}\) should be real ones. We obtained that the invariant \(g_{2}\) is positive, but let us remember that for the invariant \(g_{3}\), we found the formulae (258)  
\[\overline{\overline{g}}_{3}=-\frac{1}{6}q^{2}(1+q^{2})+\frac{(1+q^{2})^{3}}{6^{3}}. \tag{227}\]2  
We already proved that the ratio \(\frac{\widetilde{g}_{3}}{\widetilde{g}_{2}}\) is positive and real; also, we proved that the nominator of the function (148) \(K(q)\equiv\frac{q^{4}-q^{2}+1}{2q^{4}-5q^{2}+2}\) is positive.