Journal of Geodesy

, Volume 92, Issue 5, pp 573–573 | Cite as

Correction to: Spherical gravitational curvature boundary-value problem


1 Correction to: J Geod (2016) 90:727–739

The right-hand side of Eq. (35) in the original article equals one. Thus, Eq. (35) is incorrect, and we suggest the following correct form instead:
$$\begin{aligned} V_{{n,m}}^{k} \left( r \right)= & {} \frac{1}{{4\pi ~a_{k} }} \int _{{{{\Omega ^{\prime }}}}} {\mathbf {V}}^{{\left( k \right) }} \left( {r,{{\Omega ^{\prime }}}} \right) {\vdots }\,{\mathbf {Z}}_{{n,m}}^{k} \left( {{{\Omega ^{\prime }}}} \right) ~{\text {d}\Omega ^{\prime }},\\&\quad k = 0,~1,~2,~3. \end{aligned}$$
The error in Eq. (35) affects also Eq. (40). The correct form of Eq. (40) is:
$$\begin{aligned}&V\left( {r,{{\Omega }}} \right) \\&\quad = \frac{{ GM }}{{4\pi R~a_{k} }}\int _{{{{\Omega ^{\prime }}}}} \left[ {\mathbf {V}}^{{\left( k \right) }} \left( {R,{{\Omega ^{\prime }}}} \right) {\vdots }\,\mathop \sum \limits _{{n = k}}^{\infty } \mathop \sum \limits _{{m = - n}}^{{ + n}} \frac{1}{{v_{n}^{k} \left( R \right) }}\left( {\frac{R}{r}} \right) ^{{n + 1}}\right. \\&\qquad \times \left. {\mathbf {Z}}_{{n,m}}^{k} \left( {{{\Omega ^{\prime }}}} \right) \right] \bar{Y}_{{n,m}} \left( {{\Omega }} \right) ~{\text {d}\Omega ^{\prime }},\quad k=0,1,2,3. \end{aligned}$$
The degree-dependent coefficients \(v_n^k \left( R \right) \) are:
$$\begin{aligned} v_n^k \left( R \right) =\frac{V_{n,m}^k \left( R \right) }{\bar{C}_{n,m}}, \end{aligned}$$
and their explicit forms are obvious from Eqs. (8)–(11).

The other equations in the original article are unaffected by the errors in Eqs. (35) and (40). The authors are thankful to Reiner Rummel for pointing out this issue.

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.NTIS - New Technologies for the Information Society, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic
  2. 2.School of Engineering, Faculty of Engineering and Built EnvironmentUniversity of NewcastleCallaghanAustralia

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