## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### An Earlier Example for an Elliptic Integral, Expressed Also Analytically  
However, from the formulaes in the preceding section, it cannot be concluded that only abelian integrals can be treated analytically (in the sense of being expressed by elementary functions) and the elliptic integrals cannot be expressed by analytical functions. A typical example was given earlier, related to the calculation of the integral (68)  
\[\widetilde{T}_{1}=\frac{na}{c\sqrt{1-e^{2}}}\int\sqrt{1+e^{2}+2e\cos fdf},\]  
which was proved also to be the elliptic integral (83) \(\widetilde{T}_{1}=\frac{2 na(1+e)}{cq\sqrt{1-e^{2}}}\int\frac{\tilde{y}^{2}d \widetilde{y}}{\sqrt{(1-\tilde{y}^{2})\left(\frac{\tilde{y}^{2}}{q^{2}}-1 \right)}}\) of second order in the Legendre form in terms of the variable (81) \(\widetilde{y}=\frac{\sqrt{1+2e\cos f+e^{2}}}{1+e}\). Also, in terms of the variables (74)  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]and (75)  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]  
the integral \(\widetilde{T}_{1}\) was expressed as an elliptic integral of the second order and in the Legendre form (76)  
\[\widetilde{T}_{1}=-2i\frac{na}{c}q^{\frac{3}{2}}\int\frac{y^{2}dy}{\sqrt{(1-y^ {2})(1-q^{2}y^{2})}}=-2i\frac{na}{c}q^{\frac{3}{2}}\widetilde{J}_{2}^{(4)}(y; q).\]  
Earlier, by means of the transformations (69) and (70), it was proved that the same integral can be expressed analytically by means of the integrals (71)  
\[\widetilde{T}_{1}=-\frac{nai}{c\sqrt{(1-e^{2})q}}\left[\int\frac{d\overline{m }}{\left(\overline{m}-\frac{q^{2}}{2}\right)\sqrt{\overline{m}-\frac{q^{2}}{2 }}}+\int\frac{d\overline{m}}{\left(\overline{m}+\frac{q^{2}}{2}\right)\sqrt{ \overline{m}-\frac{q^{2}}{2}}}\right],\]  
which can be integrated. Thus, it can be concluded that when expressed in two different variables \(\widetilde{y}\) and \(\overline{m}\), one and the same integral (68) can be considered to be an elliptic one, but also, it can be written in another form and even calculated exactly. From the transformations (69) and (70), the relation between the variables \(f\) and \(\overline{m}\) can be found to be  
\[\cos f=1+\frac{1}{2e}\frac{(2\overline{m}+q^{2})B_{1}}{(2\overline{m}-q^{2})}. \tag{93}\]  
If account is taken also of (81) for \(\widetilde{y}\), the relation between the variables \(\widetilde{y}\) and \(\overline{m}\) is calculated to be a quadratic algebraic curve with respect to the variable \(\widetilde{y}\):  
\[\widetilde{y}^{2}=1+\frac{1}{(1+e)^{2}}.\frac{(2\overline{m}+q^{2})B_{1}}{(2 \overline{m}-q^{2})}. \tag{94}\]  
Then, if the integral (68) in terms of the variable \(\overline{m}\) can be integrated, it can be asked if there is any motivation for the assertion in the monographs [38; 41; 42] that elliptic integrals cannot be integrated in elementary functions. Further, it shall be proved that analytical formulaes can be found for the determination of the s.c. Weierstrass invariants \(g_{2}\) and \(g_{3}\), depending on complicated polynomials of the modulus \(q\) of the elliptic integral in the Legendre form. But, then, the s.c. \"four-dimensional parametrization\" of an elliptic curve will allow the parametrization of an fourth-order polynomial in terms of expressions, depending ot the Weierstrass function \(\rho(z)\) and its derivative \(\frac{d\rho(z)}{dz}\), which, of course, are not elementary functions. This will be proved further, but evidently, depending on the variables used in the calculations, a certain elliptic integral can be expressed analytically in elementary functions, but also can be expressed in an analytical form, containing non-elementary functions.