## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### Another Integral in the Weierstrass Form After Applying the Four-Dimensional Uniformization
In the next sections on the basis of the methods for four-dimensional uniformization, a primary goal will be to find the Weierstrass invariants \(g_{2}\) and \(g_{3}\) not only for integrals in the Weierstrass form, but also for integrals in the Legendre form and also for more general integrals (50) (for \(n=0\))  
\[J_{0}^{(4)}(y)=\int\frac{dy}{\sqrt{a_{0}y^{4}+4a_{1}y^{3}+6a_{2}y^{2}+4a_{3}y +a_{4}}},\]  
with an arbitrary polynomial of the fourth-degree under the square in the denominator. Here, it shall be clarified that finding the Weierstrass invariants will be possible after transforming the integral initially given in the Legendre form into an integral in the Weierstrass form, and in such a way, the Weierstrass invariants \(g_{2}\) and \(g_{3}\) will depend in a complicated manner on the modulus \(q\) of the integral in the Legendre form. Thus, as it will be demonstrated by a theorem in the monograph by [59], for known \(g_{2}\) and \(g_{3}\), it will be possible to determine the solution (and the roots) of the cubic equation \(y^{2}=4x^{3}-g_{2}x-g_{3}\). Then, the periods \(\omega_{1}\) and \(\omega_{2}\) of the two-dimensional lattice of the fundamental parallelogram (177) can be calculated. Let us first note that finding a solution \(z-z_{0}\) of the integral \(\int\frac{dx}{\sqrt{4x^{3}-g_{2}x-g_{3}}}\) in the Weierstrass form is equivalent to knowing the roots of the cubic polynomial \(y^{2}=4x^{3}-g_{2}x-g_{3}\). This \"equivalency\" can be written as [57]  
\[z-z_{0}=\int\limits_{x_{0}}^{x}\frac{dx}{\sqrt{4x^{3}-g_{2}x-g_{3}}}\Leftrightarrow \ y^{2}=4x^{3}-g_{2}x-g_{3}. \tag{185}\]In fact, the equality \(x=\rho(z)\) is a solution also of the elliptic integral in (185) and this is the known problem of \"inversion\" for elliptic integrals. In (183), \(\rho(z)\) is the Weierstrass function [57]:  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]0  
From the second representation (186) for the Weierstrass function \(\rho(z)\), it becomes clear why, in the literature, it is denoted by \(\rho(z;\delta_{2},\delta_{3})\). The following theorem, proved in Ch. 6 of the monograph by Knapp [59], allows us to understand how the result of integration for the elliptic integral (143) \(\int_{0}^{(d)}(y)=\int\frac{dy}{\sqrt{(1-y^{2})(1-d^{2}y^{2})}}\) can be represented in an analytical form by means of the Weierstrass integral, after calculating the Weierstrass invariants \(g_{2}\) and \(g_{3}\) by means of the method of four-dimensional uniformization. This method will be presented in the next sections. A Theorem about the Unique Correspondence Between the Periods \(\omega_{1}\) and \(\omega_{2}\) on the Two-Dimensional Complex Lattice and the Weierstrass Invariants \((g_{2},g_{3})\) for Elliptic Integrals in the Weierstrass Form  
**Theorem 7** ([59]). : _There exists an unique correspondence \((g_{2}(\Lambda),g_{3}(\Lambda))\Leftrightarrow\Lambda\) between the lattices \(\Lambda\) on the complex plane \(\mathbb{C}\) and the pair \((g_{2},g_{3})\) of the complex numbers, the Weierstrass invariants, such that the discriminant \(\Delta\equiv g_{2}^{3}-27g_{3}^{2}\) of the cubic polynomial \(4x^{3}-g_{2}x-g_{3}=0\) is different from zero.