## 13 Conclusions
Due to this, the propagation time will not be the (presumed to be known at each moment of time) Euclidean distance between the satellites, divided by the velocity of light. The nontrivial moment in such a problem is that both satellites are moving (at the moments of emission and receiving of the signal and also during the time of propagation of the signal) so that the moments of emission and reception of the signal (they are denoted by \(T_{1}\) and \(T_{2}\)) should incorporate these important peculiarities. In other words, the curvature of the signal trajectory will have to be such that at a certain space point of the second satellite, the signal will be intercepted, provided also that the signal is emitted at the proper position by the first (signal emitting) satellite. The last three sections of this paper (Sections 10.1, 11.1 and 12) are dedicated to the problem of finding new solutions of elliptic integrals in the Legendre form by means of transforming them to elliptic integrals in the Weierstrass form, which, after inversion, have as a solution the Weierstrass elliptic function. The necessity of studying such integrals (at first, in the pure mathematical aspect, but subsequently, applications in problems related to physics might be searched out) is dictated not only by their important application in GPS inter-satellite communications with account also of the general relativity effects, but also by the wide application of elliptic integrals in various problems of mechanics, cosmology, black hole physics, integrable systems, etc. A short review of these different applications has been given in the paper [33]. In this paper (as well in [33]), the representation of the integral is exposed in two different ways; one of them is based on the transformation \(x=\frac{q}{y^{2}}+b\) (\(a,b\) are proved to be functions of the modulus parameter \(q\) of the integral in the Legendre form), and the second method is related to the s.c. \"four-dimensional uniformization\", exposed in the monograph by Whittaker and Watson [35]. After comparison of the Weierstrass invariants in the two representations, the main result is the representation (257) for the ratio of the two invariants  
\[\frac{\delta_{3}}{\delta_{2}}=\left(\frac{\overline{\delta}_{3}}{\overline{ \delta}_{2}}\right)^{\frac{81}{86}}(9K(q))^{\frac{25}{86}},\]  
where \(K(q)\) is the function (148), \(K(q)\equiv\frac{q^{4}-q^{2}+1}{2q^{4}-Sq^{4}+2}\), \(\overline{\delta}_{2}\) and \(\overline{\delta}_{3}\) are the Weierstrass invariants, obtained after the four-dimensional uniformization [35]. This result is not trivial and previously not investigated in the monographs on elliptic functions. Further, it was proved also that the Weierstrass invariant \(g_{2}\) and the Weierstrass function \(\rho(z)\) can be found from a complicated system of two nonlinear equations. Another interesting result was obtained in this paper, suggested by a theorem in the monograph by Koblitz [60]. If \(g_{2}\) is a real number (or a real and positive function) and there is one real root and two imaginary ones for the cubic polynomial \(4x^{3}-g_{2}x-g_{3}\) (the case investigated in this paper), then the other invariant \(g_{3}\) should be imaginary and the discriminant \(\Delta=g_{2}^{3}-2\mathcal{F}g_{3}^{2}\) should be negative (a case not investigated in this paper). It should be kept in mind also that the case \"negative discriminant and imaginary \(g_{3}\)\" does not follow directly from the formulation of the theorem, but rather than from the \"non-fulfillment\" of the theorem, which is an unusual moment. No doubt, more strict mathematical treatment is necessary. The case of a negative discriminant would correspond to three real roots of the cubic polynomial (see [61; 68]) and this case needs a separate investigation.