## 2 A Brief Review of the Shapiro Delay Formulae and the Aims of This Paper
### Standard Shapiro Delay Formulae, Derived from the Null Cone Equation
#### 2.2.7 Fourth Aim of the Paper: Finding New Analytical Methods for Expressing the  
Two Weierstrass Invariants in the Elliptic Integrals of Zero Order and in the Legendre Form  
Essentially, in view of the application of elliptic integrals in GPS inter-satellite communications and also in several other areas of physics (cosmology, nonlinear equations, etc. ), in the paper [33], general types of elliptic integrals were considered:  
\[J_{n}^{(4)}(y)=\int\frac{y^{n}dy}{\sqrt{a_{0}y^{4}+4a_{1}y^{3}+6a_{2}y^{2}+4a_ {3}y+a_{4}}}\text{ and }\]  
\[J_{n}^{(3)}(x)=\int\frac{x^{n}dx}{\sqrt{ax^{3}+bx^{2}+cx+d}}. \tag{17}\]  
Some more general types of transformations,  
\[x=\frac{a}{y^{2}}+b, \tag{18}\]  
were applied, so that \(a\) and \(b\) represent complicated functions of the modulus of the elliptic integrals. The transformation (18) has been applied only with respect to a special type of elliptic integrals--the s.c. elliptic integrals of zero order (i.e., \(n=0\))--and in the Legendre form. This is the left integral in (19). The advantage of such an approach is that there is no restriction for the modulus of the elliptic integral to be small. Also, the theory of elliptic functions uses the standard parametrization of an elliptic curve of the type \(y^{2}=4x^{3}-g_{2}x-g_{3}\) in terms of the Weierstrass function and its derivative, meaning that the function \(x=\rho(z)\) and \(y=\rho^{\prime}(z)\) satisfy the above equation. In this paper, it will be proved that the propagation time \(T\) will be expressed by means of elliptic integrals of the types (with some coefficients in front)  
\[\int\frac{dy}{\sqrt{(1-y^{2})(1-q^{2}y^{2})}}\text{ and }\int\frac{y^{2}dy}{ \sqrt{(1-y^{2})(1-q^{2}y^{2})}}. \tag{19}\]  
Since it is known that integrals in the Weierstrass form \(\int\frac{dx}{\sqrt{4x^{3}-g_{2}x^{2}-g_{3}}}\) can be solved analytically, it is evident that the transformation (18) can be used to find analytical expressions for the first integral in (19), which is called an elliptic integral of the first kind and of zero order in the Legendre form. The second integral in (19) can also be solved analytically due to the recurrent chain of elliptic integrals of different orders, which usually is written as [34]  
\[y^{n}Z=a_{0}(n+2)J_{n+3}+2a_{1}(2n+3)J_{n+2}\] \[+6a_{2}(n+1)+2a_{3}(2n+1)J_{n}+na_{4}J_{n-1}, \tag{20}\]  
where \(Z\) denotes the square of the fourth degree polynomial in the integral \(\int_{n}^{(4)}(y)\) in (17). \[Z:\equiv\sqrt{a_{0}y^{4}+4a_{1}y^{3}+6a_{2}y^{2}+4a_{3}y+a_{4}}. \tag{21}\]  
Further, it will be shown that the invariant \(g_{2}\) and the Weierstrass function \(\rho(z,g_{2},g_{3})\) can be found from a complicated system of two nonlinear equations. Then, if the function \(\rho(z,g_{2},g_{3})\) is known, one can apply the formulas (217)-(220), given in the monograph by Whittaker and Watson [35] with the purpose of finding a parametrization of the polynomials (21) and (201) \(Z^{2}=f(y)=a_{0}y^{4}+a_{1}y^{3}+a_{2}y^{2}+a_{3}y+a_{4}=Y\) in terms of expressions, depending on the Weierstrass function and its derivative. Propagation Time for the Case of a Signal-Emitting Satellite, Moving Along a Plane Elliptical Orbit  
New Calculation of the Propagation Time in Terms of the Eccentric Anomaly Angle \(E\) Without Any Approximations  
As mentioned in the introduction of the paper, the null cone metric is taken in the standard form (6) (see also [36; 37]):  
\[ds^{2}=-c^{2}\bigg{(}1+\frac{2V}{c^{2}}\bigg{)}(dT)^{2}+\bigg{(}1-\frac{2V}{c^ {2}}\bigg{)}\bigg{(}(dx)^{2}+(dy)^{2}+(dz)^{2}\bigg{)}=0,\]  
where \(V=\frac{G_{0}M_{0}}{r}\) is the standard gravitational potential of the Earth. No account is taken of any harmonics due to the spherical form of the Earth since the GPS orbits are situated at a distance more than 20,000 km (considered from the centre of the Earth).