## 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
So, a natural question arises: Will the integral in the Weierstrass form with the calculated values for the Weierstrass invariants \(g_{2}\) and \(g_{3}\) be correctly defined? The answer will be affirmative, because from the above correspondence, it will follow that the lattice \(\Lambda\) can be defined. However, if \(g_{2}\) and \(g_{3}\) are defined as \(g_{2}=g_{2}(q)\) and \(g_{3}=g_{3}(q)\), then the condition \(\Delta\equiv g_{2}^{3}-27g_{3}^{2}\
eq 0\) for the discriminant should also be fulfilled, which, in fact, will mean that the higher-order \(g_{2}^{3}-27g_{3}^{2}\) polynomial of \(q\) should have no zeroes! Further, in the section for four-dimensional uniformization, we shall obtain the parametrization for \(g_{2}\) and \(g_{3}\) (212) in the form of the following higher-order polynomials of the modulus parameter \(q\) of the elliptic integral (143) in the Legendre form  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]7  
and (213) (the notation in both equalities will be \(\overline{g}_{2}\) and \(\overline{g}_{3}\))  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]8Then, will the higher-order polynomial \(\overline{S}_{2}^{3\,(1,2)}-2\overline{7}\overline{S}_{3}^{2\,(1,2)}\) have the property to be without any roots? The same question appears with respect to the found second pair of Weierstrass invariants (226) (the same as (212)),  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]9  
and also (227)  
\[f(z+n\omega_{1}+m\omega_{2})=f(z) \tag{179}\]0  
In view of the theorems, which have been proved, it follows that in both cases, the higher-order polynomial \(\overline{g}_{2}^{3\,(1,2)}-2\overline{g}_{3}^{2\,(1,2)}\) should have no roots! Since both the polynomials \(\overline{g}_{2}^{3\,(1,2)}-2\overline{7}\overline{g}_{3}^{2\,(1,2)}\) depend on higher degrees of \(q\), which is defined for \(0\leq q<1\), it is known from higher algebra that there are many theorems [61] investigating the case when an arbitrary polynomial of the \(n\)-th order will have (or will not have) roots in the circle \(q<1\). One such theorem, formulated in the monograph by Obreshkoff [61], is the theorem by Schur [62], published in 1918. The concrete application of this theorem requires cumbersome analytic calculations, because it should be applied to a chain of algebraic equations of diminishing degrees. For example, for the case of the polynomials (226) and (227), the polynomial \(\overline{g}_{2}^{3\,(1,2)}-2\overline{7}\overline{g}_{3}^{2\,(1,2)}\) will be of the 12 th degree! For polynomials of the fourth degree, the Schur theorem has been applied several times in the papers [20,40] and the results have shown excellent consistency with the physical meaning, related to these polynomials. Moreover, in [20], it has been demonstrated that the Schur theorem can be used both for the case of proving that a certain polynomial has roots in the circle \(q<1\) and for the case of proving that it does not have such roots. In the next sections, we shall give only the formulation of this theorem. On the other hand, if in one of the cases for the polynomial \(\overline{g}_{2}^{3\,(1,2)}-2\overline{7}\overline{g}_{3}^{2\,(1,2)}\), it is proved that there are roots, then this will mean that either the corresponding parametrization should be rejected, or that the elliptic integral in the Legendre form is not defined for the values of \(q\), satisfying the above higher-order polynomial \(\overline{g}_{2}^{3\,(1,2)}-2\overline{7}\overline{g}_{3}^{2\,(1,2)}=0\). In standard monographs on elliptic functions and integrals, this interesting problem is not investigated.