## 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 invariants \(\widehat{g}_{2},\widehat{g}_{3}\) do not contain any useful information--they appeared after the variable change (233)  
\[\overline{s}_{3}=\frac{1}{6}a_{0}a_{2}a_{4}+\frac{1}{48}a_{1}a_{2}a_{3}-\frac {a_{2}^{3}}{6^{3}}-\frac{1}{16}(a_{0}a_{3}+a_{1}^{2}a_{4}). \tag{224}\]8when the equality of the two integrals (234)  
\[\overline{s}_{3}=\frac{1}{6}a_{0}a_{2}a_{4}+\frac{1}{48}a_{1}a_{2}a_{3}-\frac {a_{2}^{3}}{6^{3}}-\frac{1}{16}(a_{0}a_{3}+a_{1}^{2}a_{4}). \tag{224}\]9  
was obtained. Although we found the relation between the roots of the polynomials in both sides of (234), it is more important to find the dependance of the invariants \(g_{2},g_{3}\) on the invariants \(\overline{g}_{2},\overline{g}_{3}\), expressed through polynomials of the modulus \(q\) and calculated in the process of the four-dimensional uniformization, presented in the previous sections. That is why, now, we shall combine the transformations (245) and (233). We obtain  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]0  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]1  
Taking into account (146) for \(a=\frac{g_{3}}{\delta 2}K(q)\) with (148) for \(K(q)\equiv\frac{q^{4}-q^{2}+1}{2q^{4}-5q^{2}+2}\) and substituting in the above equalities, it follows that  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]2  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]3  
Dividing the above two equalities, it can be obtained that  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]4  
After some transformations, the ratio \(\frac{g_{3}}{g_{2}}\) can be expressed as a function of powers of the ratio \(\frac{\overline{\widehat{g}}_{3}}{\widehat{g}_{2}}\) (which is a rational function of polynomials of the modulus parameter \(q\)-see (143) and (213)) and powers of \((9K(q))\), which is also a rational function of higher-order polynomials in the nominator and in the denominator according to (148):  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]5  
This is an important formulae, enabling to express \(g_{3}\) as a linear function of \(g_{2}\) and a complicated function \(N(q)\):  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]6  
In such a way, substituting the above expression in formulae (235) for the (real-valued) first root \(\alpha_{1}=A_{1}+B_{1}\) of the cubic polynomial \(4\sigma^{3}-g_{2}\sigma-g_{3}=0\), one can obtain  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]7  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]8where \(F(q)\) and \(\widetilde{\Delta}\) (obtained from the discriminant formulae (236) \(\Delta=\frac{\widetilde{s}_{3}^{2}}{4}-\frac{1}{27}\widetilde{A}\widetilde{s}_{2 }^{3}>0\)) denote the following functions:  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]9  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]0  
In (259), the following new notations were introduced:  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]1  
Since the \"modified\" discriminant \(\widetilde{\Delta}\) (261) is linearly dependant on the Weierstrass invariant \(g_{2}\), it is evident that \(g_{2}\) can be found if the ratio \(\frac{\widetilde{A}_{1}}{\widetilde{B}_{1}}\) is known. This will be used in the next sections. A Simple Proof for the Positivity of the Weierstrass Invariants \(g_{2}\) for the One Pair of Weierstrass Invariants \(\widetilde{g}_{2}\) (226) and \(\widetilde{g}_{3}\) (227)  
From (261), the condition for a non-zero discriminant can be written as  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]2  
Now, we shall prove that this expression is different from zero and also is positive. The nominator of the expression for (148) \(K(q)\equiv\frac{q^{4}-q^{2}+1}{2q^{4}-\overline{g}q^{2}+2}\) and also the expression for \(\widetilde{g}_{3}\) (227),  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]3  
are different from zero.