## 2 A Brief Review of the Shapiro Delay Formulae and the Aims of This Paper
### Can Elliptic Integrals Be Considered to be Abelian Ones? The elliptic integrals (50) and (51) cannot be considered to be a generalization of the s.c. abelian integral,  
because after setting up  
\[y=\frac{\sqrt{1-q^{2}x^{2}}}{\sqrt{1-x^{2}}}, \tag{57}\]  
\[y=(1+nx^{2})\sqrt{(1-x^{2})(1-q^{2}x^{2})}, \tag{58}\]  
it can easily be seen that these algebraic curves are different from the curves (31) and (32). Consequently, the elliptic integrals (54)-(56) are not abelian integrals. Similarly, it can be proved that the found integral (29)  
\[T=\frac{a}{c}\int\sqrt{\frac{a_{1}y^{3}+a_{2}y^{2}+a_{3}y}{b_{1}y^{3}+b_{2}y^{ 2}+b_{3}y+b_{4}}}dy\]  
is not also an abelian integral. Again, it should reminded that the above integral is found as a consequence of the approximation \(\beta=\frac{2V}{c^{2}}\ll 1\), which physically means that the gravitational potential is small. Mathematical Structure of the Expression for the Propagation Time of the Signal, Emitted by a Satellite on a Plane Elliptical Orbit:  
The Propagation Time as a Sum of Zero-Order Elliptic Integrals of the First, Second and Third Rank  
The first term in expression (41), after performing the simple transformation \(\frac{\pi}{2}-E=\overline{E}\), can be brought to the following form:  
\[T_{1}=\int\limits_{0}^{E}\sqrt{1-e^{2}\cos^{2}E}\,dE=\int\limits_ {0}^{E}\sqrt{1-e^{2}\sin^{2}(\frac{\pi}{2}-E)}dE\] \[=-\int\limits_{\frac{\pi}{2}}^{\frac{\pi}{2}-\overline{E}}\sqrt{ 1-e^{2}\sin^{2}\overline{E}}d\overline{E}. \tag{59}\]  
This represents an elliptic integral of the second kind. Note that the s.c.modulus of the elliptic integral is equal to the eccentricity \(e\) of the orbit. The second term in Equation (41) after performing the substitution  
\[\sqrt{\frac{1+e\cos E}{1-e\cos E}}=\overline{y} \tag{60}\]  
and introducing the notations  
\[\widetilde{k}^{2}=\frac{1-e}{1+e}=q\,,\ \frac{\overline{y}}{\widetilde{k}}=y\ \ \,\ \ \left(\frac{ \overline{y}}{\widetilde{k}}\right)^{2}=\widetilde{y}_{1}\,\ \widetilde{y}_{1}=- \widetilde{\widetilde{y}} \tag{61}\]  
can be written as a sum of two integrals  
\[T_{2}=-\frac{2G_{\oplus}M_{\oplus}}{c^{3}}\int\sqrt{\frac{1+e\cos E}{1-e\cos E }}dE=I_{2}^{A})+I_{2}^{(B)}. \tag{62}\]  
It should be noted that \(\widetilde{k}\) in Equation (61) is simply a notation, representing the real expression \(\widetilde{k}=\sqrt{\frac{1-e}{1+e}}=\sqrt{q}\), since \(0<\frac{1-e}{1+e}<1\) (the eccentricity of the orbit \(e\) is always less than one, \(0<e<1\)). We shall call this second part (62) of the expression for the propagation time the \(O(c^{-3})\) correction. Mathematical Proof of the Real-Valuedness of the Separate Expressions for the Propagation Time for the Case of Plane Elliptic Orbits  
1.1 The First Real-Valued Elliptic Integral \(I_{2}^{(A)}\) of the First Kind in the Weierstrass Form in the \(O(c^{-3})\) Correction of the Propagation Time  
The first term \(I_{2}^{(A)}\) in Equation (62) can be represented as  
\[I_{2}^{(A)}=\frac{4 GM}{c^{3}}\frac{1}{\widetilde{k}\sqrt{e^{2}-1}}\int\frac{d \widetilde{y}_{1}}{\sqrt{\widetilde{y}_{1}}\left(\widetilde{y}_{1}-\frac{1}{ \widetilde{k}^{4}}\right)(\widetilde{y}_{1}-1)}. \tag{63}\]  
In terms of the variable  
\[\widetilde{y}_{1}=\bigg{(}\frac{1+e}{1-e}\bigg{)}\cdot\bigg{(}\frac{1+e\cos E }{1-e\cos E}\bigg{)}, \tag{64}\]  
the integral (63) represents an elliptic integral of zero order and of the first kind, written in the Weierstrass form (i.e., a polynomial of third order under the square root in the denominator). Due to the presence of the \(\sqrt{e^{2}-1}\) term in the denominator (again, \(0<e\leq 1\)), it might seem that expression (63) is imaginary. However, this is not true, because it can be rewritten as  
\[y=(1+nx^{2})\sqrt{(1-x^{2})(1-q^{2}x^{2})}, \tag{58}\]0  
thus representing an elliptic integral of zero order and of the first kind. The classification of elliptic integrals, depending on the degree of the polynomial in the denominator, is given in the monographs [34; 38; 45].