## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### Numerical Calculation of the First \(O(\frac{1}{c})\) Correction in the Propagation Time
The main advantage of the theoretical method is that (1) it will be applicable for various values of the modulus \(q\) of the elliptic integral, no matter whether \(q\) is a small or a big number, and (2) the integrals, encountered in the calculation of the propagation time in the preceding sections, were in the Legendre form, which contains the specific fourth-order polynomial \((1-y^{2})(1-q^{2}y^{2})\) under the square root in the denominator. However, in the previous Section 4, the definitions for elliptic integrals with arbitrary polynomials of the third and of the fourth degree were given. Such integrals are very often encountered in cosmology, nonlinear evolution equations, black hole physics, etc. For example, in the definition for the luminosity in cosmology, the more general polynomial of the fourth degree under the square root in the denominator was used. Also, there is an interesting interplay between the methods of the standard three-dimensional uniformization and of the four-dimensional uniformization, described in the monograph by Whittaker and Watson [35]. This method will be exposed in detail further and will allow not only the analytical computation of the Weierstrass invariants \(g_{2}\) and \(g_{3}\) for four-dimensional elliptic integrals of the type \(J_{n}^{(4)}(y)\) but also the comparison of elliptic integrals in the Weierstrass form \(J_{n}^{(3)}(x)\), both of them defined by formulaes (48)-(51) in Section 4.1, concerning the definitions of higher-order elliptic integrals. Some physical applications of elliptic integrals were presented in the paper [33]. Transforming an Elliptic Integral in the Legendre Form into an Elliptic Integral in the Weierstrass Form  
The purpose of this section will be to propose a transformation, which will transform the elliptic integral  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]8  
in the Legendre form into an elliptic integral in the Weierstrass form  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]9  
For the purpose, the transformation  
\[T_{1}^{(E_{6})}=\frac{a}{c}\int\limits_{0}^{E_{(6)}}\sqrt{1-e^{2}\cos^{2}E}dE=-0. 0281342485273829\;\text{[s]},\]0shall be applied. It is possible to calculate the parameters \(a\) and \(b\), so that the integral (144) is satisfied. These parameters represent complicated functions of the modulus parameter \(q\) of the integral  
\[T_{1}^{(E_{6})}=\frac{a}{c}\int\limits_{0}^{E_{(6)}}\sqrt{1-e^{2}\cos^{2}E}dE=-0. 0281342485273829\;\text{[s]},\]1  
\[T_{1}^{(E_{6})}=\frac{a}{c}\int\limits_{0}^{E_{(6)}}\sqrt{1-e^{2}\cos^{2}E}dE=-0. 0281342485273829\;\text{[s]},\]2  
In (147), \(K(q)\) is a rational function, depending on the modulus parameter \(q\) of the elliptic integral in the Legendre form  
\[T_{1}^{(E_{6})}=\frac{a}{c}\int\limits_{0}^{E_{(6)}}\sqrt{1-e^{2}\cos^{2}E}dE=-0. 0281342485273829\;\text{[s]},\]3  
However, \(\mathcal{S}_{2}\) and \(\mathcal{S}_{3}\) remain undetermined. This will turn out to be an advantage of the applied formalism, since, further, another representation of the integral in the Weierstrass form shall be found with different Weierstrass invariants \(\overline{\mathcal{S}}_{2}\) and \(\overline{\mathcal{S}}_{3}\), which will depend explicitly on the modulus parameter \(q\). Then, by comparing both representations in the Weierstrass form and proving a new theorem, \(\mathcal{S}_{2}\) and \(\mathcal{S}_{3}\) will be expressed through \(\overline{\mathcal{S}}_{2}\) and \(\overline{\mathcal{S}}_{3}\) in a complicated way. Further, the following important property will be very important. Corollary 1: _The parameter \(b\) satisfies the cubic equation, i.e.,_  
\[T_{1}^{(E_{6})}=\frac{a}{c}\int\limits_{0}^{E_{(6)}}\sqrt{1-e^{2}\cos^{2}E}dE=-0. 0281342485273829\;\text{[s]},\]4  
The proof is straightforward, if the expressions for \(\mathcal{S}_{2}\) and \(\mathcal{S}_{3}\) are expressed from (146) and (147) and are substituted into the above cubic equation.