The expression for the coefficient \(A_m^{(rr)}\) given in the third row of the first column of Table 5 should be corrected to include the factor \(\bar{C}_{nm}\) as follows:
$$\begin{aligned} A_m^{(rr)} = \sum _{n=m}^N \left( \frac{a}{r}\right) ^n (n+1)(n+2)\bar{C}_{nm}\bar{P}_n^m. \end{aligned}$$
Similarly, the expression for the coefficient \(A_m^{(rrr)}\) given in the fourth row of the first column of Table 6 should be corrected to include the factor \(\bar{C}_{nm}\) as follows:
$$\begin{aligned} A_m^{(rrr)} = \sum _{n=m}^N \left( \frac{a}{r}\right) ^n (n+1)(n+2)(n+3)\bar{C}_{nm}\bar{P}_n^m. \end{aligned}$$
The right-hand sides of the formulas appearing in Table 10 should be corrected to include the factor \(\left( 1/r \right) \) as follows:
$$\begin{aligned} a_1= & {} \displaystyle \frac{1}{r} \sum _{m=1}^N m \left( A_m^{(1)} c_{m-1} + B_m^{(1)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_2= & {} \displaystyle \frac{1}{r} \sum _{m=1}^N m \left( B_m^{(1)} c_{m-1} - A_m^{(1)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_3= & {} \displaystyle \frac{1}{r} \sum _{m=0}^N \left( A_m^{(2)} c_{m} + B_m^{(2)} s_{m}\right) \cos ^{m}\phi \\ a_4= & {} - \displaystyle \frac{1}{r} \sum _{m=0}^N \left( A_m^{(3)} c_{m} + B_m^{(3)} s_{m}\right) \cos ^{m}\phi . \end{aligned}$$
The right-hand sides of the formulas appearing in Table 11 should be corrected to include the factor \(\left( 1/r \right) ^2\). The expression of the coefficient \(a_{11}\) contains further misprints. Hence, the formulas of Table 11 should read:
$$\begin{aligned} a_{11}= & {} \displaystyle \frac{1}{r^2} \sum _{m=2}^N m(m-1) \left( A_m^{(1)} c_{m-2} + B_m^{(1)} s_{m-2}\right) \cos ^{m-2}\phi \\ a_{12}= & {} \displaystyle \frac{1}{r^2} \sum _{m=2}^N m(m-1) \left( B_m^{(1)} c_{m-2} - A_m^{(1)} s_{m-2}\right) \cos ^{m-2}\phi \\ a_{13}= & {} \displaystyle \frac{1}{r^2} \sum _{m=1}^N m \left( A_m^{(2)} c_{m-1} + B_m^{(2)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{14}= & {} -\displaystyle \frac{1}{r^2} \sum _{m=1}^N m \left( A_m^{(3)} c_{m-1} + B_m^{(3)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{22}= & {} -a_{11}\\ a_{23}= & {} \displaystyle \frac{1}{r^2} \sum _{m=1}^N m \left( B_m^{(2)} c_{m-1} - A_m^{(2)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{24}= & {} -\displaystyle \frac{1}{r^2} \sum _{m=1}^N m \left( B_m^{(3)} c_{m-1} - A_m^{(3)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{33}= & {} \displaystyle \frac{1}{r^2} \sum _{m=0}^N \left( A_m^{(4)} c_{m} + B_m^{(4)} s_{m}\right) \cos ^{m}\phi \\ a_{34}= & {} -\displaystyle \frac{1}{r^2} \sum _{m=0}^N \left( A_m^{(5)} c_{m} + B_m^{(5)} s_{m}\right) \cos ^{m}\phi \\ a_{44}= & {} \displaystyle \frac{1}{r^2} \sum _{m=0}^N \left( A_m^{(6)} c_{m} + B_m^{(6)} s_{m}\right) \cos ^{m}\phi . \end{aligned}$$
The right-hand sides of the formulas appearing in Table 12 should be corrected by including the factor \((1/r)^3\). The expressions of the coefficients \(a_{111}\), \(a_{143}\), \(a_{222}\) and \(a_{444}\) contain further misprints. Hence, the formulas of Table 12 should read:
$$\begin{aligned} a_{111}= & {} \displaystyle \frac{1}{r^3} \sum _{m=3}^N m(m-1)(m-2)\\&\times \left( A_m^{(1)} c_{m-3} + B_m^{(1)} s_{m-3}\right) \cos ^{m-3}\phi \\ a_{112}= & {} -a_{222}\\ a_{113}= & {} \displaystyle \frac{1}{r^3} \sum _{m=2}^N m(m-1) \left( A_m^{(2)} c_{m-2} + B_m^{(2)} s_{m-2}\right) \cos ^{m-2}\phi \\ a_{114}= & {} -\displaystyle \frac{1}{r^3} \sum _{m=2}^N m(m-1) \left( A_m^{(3)} c_{m-2} + B_m^{(3)} s_{m-2}\right) \cos ^{m-2}\phi \\ a_{123}= & {} \displaystyle \frac{1}{r^3} \sum _{m=2}^N m(m-1) \left( B_m^{(2)} c_{m-2} - A_m^{(2)} s_{m-2}\right) \cos ^{m-2}\phi \\ a_{124}= & {} -\displaystyle \frac{1}{r^3} \sum _{m=2}^N m(m-1) \left( B_m^{(3)} c_{m-2} - A_m^{(3)} s_{m-2}\right) \cos ^{m-2}\phi \\ a_{143}= & {} -\displaystyle \frac{1}{r^3} \sum _{m=1}^N m \left( A_m^{(5)} c_{m-1} + B_m^{(5)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{221}= & {} -a_{111}\\ a_{222}= & {} -\displaystyle \frac{1}{r^3} \sum _{m=3}^N m(m-1)(m-2)\\&\times \left( B_m^{(1)} c_{m-3} - A_m^{(1)} s_{m-3}\right) \cos ^{m-3}\phi \\ a_{223}= & {} -a_{113}\\ a_{224}= & {} -a_{114}\\ a_{243}= & {} -\displaystyle \frac{1}{r^3} \sum _{m=1}^N m \left( B_m^{(5)} c_{m-1} - A_m^{(5)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{331}= & {} \displaystyle \frac{1}{r^3} \sum _{m=1}^N m \left( A_m^{(4)} c_{m-1} + B_m^{(4)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{332}= & {} \displaystyle \frac{1}{r^3} \sum _{m=1}^N m \left( B_m^{(4)} c_{m-1} - A_m^{(4)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{333}= & {} \displaystyle \frac{1}{r^3} \sum _{m=0}^N \left( A_m^{(7)} c_{m} + B_m^{(7)} s_{m}\right) \cos ^{m}\phi \\ a_{334}= & {} -\displaystyle \frac{1}{r^3} \sum _{m=0}^N \left( A_m^{(8)} c_{m} + B_m^{(8)} s_{m}\right) \cos ^{m}\phi \\ a_{441}= & {} \displaystyle \frac{1}{r^3} \sum _{m=1}^N m \left( A_m^{(6)} c_{m-1} + B_m^{(6)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{442}= & {} \displaystyle \frac{1}{r^3} \sum _{m=1}^N m \left( B_m^{(6)} c_{m-1} - A_m^{(6)} s_{m-1}\right) \cos ^{m-1}\phi \\ a_{443}= & {} \displaystyle \frac{1}{r^3} \sum _{m=0}^N \left( A_m^{(9)} c_{m} + B_m^{(9)} s_{m}\right) \cos ^{m}\phi \\ a_{444}= & {} -\displaystyle \frac{1}{r^3} \sum _{m=0}^N \left( A_m^{(10)} c_{m} + B_m^{(10)} s_{m}\right) \cos ^{m}\phi . \end{aligned}$$
The expression for the coefficient \(A_m^{(2)}\) in Table 13 should read:
$$\begin{aligned} A_{m}^{(2)} = \displaystyle \sum _{n=m}^N \rho _n \bar{C}_{nm} \mathrm{d} \bar{H}_n^m / \mathrm{d}u . \end{aligned}$$
There are missing terms in the expression for the function \(\bar{W}_n^m\) in Table 14. The formula should read:
$$\begin{aligned} \bar{W}_n^m= & {} \displaystyle (n+m+1)(n+m+2)(n+m+3) \bar{H}_n^m \\&+\, 3 u (n+m+2)(n+m+3) \mathrm{d} \bar{H}_n^m / \mathrm{d}u \\&+\, 3 u^2\mathrm{d}^2 \bar{H}_n^m / \mathrm{d}u^2 \\&+\, u^3 \mathrm{d}^3 \bar{H}_n^m / \mathrm{d}u^3 . \end{aligned}$$
The letter l in Eqs. (76) and (77) should be replaced by the letter r, so that the two equations read:
$$\begin{aligned} V_{nm}= & {} \displaystyle \frac{(-1)^n}{(n-m)!} \left( \frac{\partial }{\partial x} + \mathrm{i} \frac{\partial }{\partial y}\right) ^m \left( \frac{\partial }{\partial z}\right) ^{(n-m)} \left( \frac{1}{r}\right) ,\\ V_{nn}= & {} \displaystyle (-1)^n \left( \frac{\partial }{\partial x} + \mathrm{i} \frac{\partial }{\partial y}\right) ^n \left( \frac{1}{r}\right) , \end{aligned}$$
respectively.
A \(+\) sign should be replaced by a \(-\) sign in Eq. (94), in the following way:
$$\begin{aligned} \bar{S}_{nm}^{(\alpha \beta \gamma )}= & {} \displaystyle \frac{(-1)^{\beta /2}}{(-2)^{\alpha +\beta }} \sum _{p=0}^{\alpha } \sum _{q=0}^{\beta } \left[ \epsilon ^+ A^+ \bar{S}_{n-\eta ,m+\sigma } \right. \\&- \left. (1-\delta _{0m}) \epsilon ^- A^- \bar{S}_{n-\eta ,-m+\sigma } \right] . \end{aligned}$$
Finally, the term \(3\eta \) appearing in the upper limits of the summations in Eqs. (101), (102) and (103) and related text should be replaced by \(\eta \):
$$\begin{aligned}&\displaystyle \frac{\partial ^{\eta } V}{\partial x^{\alpha } \partial y^{\beta } \partial z^{\gamma }} \!=\! \frac{GM}{a^{1+\eta }} \sum _{m=0}^{N+\eta } \left( A_m^{(\alpha \beta \gamma )} \cos m \lambda + B_m^{(\alpha \beta \gamma )} \sin m\lambda \right) , \\&\displaystyle A_m^{(\alpha \beta \gamma )} = \sum _{n=m}^{N+\eta } \bar{C}_{nm}^{(\alpha \beta \gamma )} \bar{E}_{nm},\\&\displaystyle B_m^{(\alpha \beta \gamma )} = \sum _{n=m}^{N+\eta } \bar{S}_{nm}^{(\alpha \beta \gamma )} \bar{E}_{nm}.\\ \end{aligned}$$
