## 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  
This theorem is also exposed in the monograph by E.T. Whittaker and G. N. Watson [35]. In another monograph by the same author Whittaker [66], many useful examples are given about the solutions in terms of elliptic functions of known equations from theoretical mechanics and analytical dynamics. We shall review the corresponding theorem by making a slight modification to the fourth-degree polynomial (201)--the numbers \(4,6\) and \(4\) in the coefficients are removed. **Theorem 10**. : _If \(f(s)\) is a fourth-order polynomial_  
\[f(y)=a_{0}y^{4}+a_{1}y^{3}+a_{2}y^{2}+a_{3}y+a_{4} \tag{221}\]  
_and the variable transformation \(z=\int\limits_{y_{0}}^{y}\frac{d_{8}}{\sqrt{f(s)}}\) is performed in the \(4D\) elliptic integral_  
\[I_{0}^{(4)}(y)=\int\frac{dy}{\sqrt{a_{0}y^{4}+a_{1}y^{3}+a_{2}y^{2}+a_{3}y+a_{ 4}}}, \tag{222}\]  
_then the Weierstrass invariants \(\overline{s}_{2}\) and \(\overline{s}_{3}\) in the integral (203) \(I_{0}^{(3)}(s)=\int_{s}^{\infty}\frac{d\sigma}{\sqrt{4\sigma^{3}-\overline{s} _{2}\sigma^{-\overline{s}_{3}}}}\) are uniquely calculated as_  
\[\overline{s}_{2}=a_{0}a_{4}-\frac{1}{4}a_{1}a_{3}+\frac{1}{12}a_{2}^{2}, \tag{223}\]  
\[\overline{s}_{3}=\frac{1}{6}a_{0}a_{2}a_{4}+\frac{1}{48}a_{1}a_{2}a_{3}-\frac {a_{2}^{3}}{6^{3}}-\frac{1}{16}(a_{0}a_{3}+a_{1}^{2}a_{4}). \tag{224}\]  
_One may note that in this formulation of the theorem, the Weierstrass invariants \(\overline{s}_{2}\) and \(\overline{s}_{3}\) do not depend on any root \(y_{0}\) of the polynomial (221)._  
Four-Dimensional Uniformization of the Elliptic Integral in the Legendre Form and Its Representation in the Weierstrass Form--Another Calculation by Using the Second Formulation of the Theorem  
Let us take, as an example, which will be used further, the polynomial  
\[f(y)\equiv(1-y^{2})(1-q^{2}y^{2})=1-(1+q^{2})y^{2}+y^{4}. \tag{225}\]  
The two Weierstrass invariants are calculated to be  
\[\overline{s}_{2}=q^{2}+\frac{1}{12}(1+q^{2})^{2}\ >0, \tag{226}\]\[\overline{\overline{g}}_{3}=-\frac{1}{6}q^{2}(1+q^{2})+\frac{(1+q^{2})^{3}}{6^{3}}. \tag{227}\]  
Clearly, the result for \(\overline{\overline{g}}_{2}\) in inequality (226) coincides with the result for \(\overline{\overline{g}}_{2}^{(1,2)}\equiv q^{2}+\frac{1}{12}(1+q^{2})^{2}>0\) in (212), but formulae (227) for \(\overline{\overline{g}}_{3}\) is quite different from formulaes (213) and (214) for \(\overline{\overline{g}}_{3}^{(1,2)}\). The same refers also to the formulae for \(\overline{\overline{g}}_{3}^{(3,4)}\), not presented here. The important consequences are as follows:  
1. For this case, there is only one pair of Weierstrass invariants \(\overline{\overline{g}}_{2}\) and \(\overline{\overline{g}}_{3}\) instead of the two pairs \(\overline{\overline{g}}_{2}^{(1,2)},\overline{\overline{g}}_{3}^{(1,2)}\) and \(\overline{\overline{g}}_{2}^{(3,4)},\overline{\overline{g}}_{3}^{(3,4)}\) for the previous case, considered in the preceding Section 11. 2. The expression for \(\overline{\overline{g}}_{2}\) (226) coincides with the expression for \(\overline{\overline{g}}_{2}^{(1,2)}\) in (212), but expression \(\overline{\overline{g}}_{3}\) in (227) is different from expression (213) for \(\overline{\overline{g}}_{2}^{(1,2)}\). It is not clear whether this will be so for the case of an elliptic integral with an arbitrary polynomial in the denominator. 3. The two invariants \(\overline{\overline{g}}_{2}\) and \(\overline{\overline{g}}_{3}\) are symmetric with respect to the interchanges \(a_{0}\rightleftarrows a_{4}\) and also \(a_{1}\rightleftarrows a_{3}\). The polynomial \(f(y)\) is also invariant with respect to these changes. 12 Comparison of Elliptic Integrals in Different Variables and with Different Weierstrass Invariants  
In this section, the method for calculating the Weierstrass invariants \(g_{2}\) and \(g_{3}\) will be presented, which is based on the combination of two approaches.