Generalized series

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The above series, to eighth order in eccentricity or fourth order in third flattening, are adequate for most practical applications. Each can be written quite generally. For example, Kazushige Kawase (2009) derived following general formula.:[1]

 

where

 

The first result is for Bessel's series. This is derived by regarding the integrand of   as a generating function of Gegenbauer polynomials   and expanding it to series of trigonometric functions in accordance with well known formula: (See for instance NIST library[1])

 

That is, the integrand becomes

 

Integrating term by term and rearranging the indices of the summation and product give the final result of the first general formula.

The second result is for Helmert's series. This is derived by firstly taking account of the well known relation between a special case of an incomplete elliptic integral of the third kind and that of the second kind with the same arguments as follows:

 

Substituting  , the right side of above equation becomes

 

Since the integrand of the first term of above corresponds to a generating function of  , the first term is identical with the first result with substituting   to   analogically. The contribution from the second term remains only trigonometric terms with the coefficient which is multiplied   as a result of the second derivative of  . Consolidating the first and second terms gives the final result of the second general formula.

Truncating the summation of the second result at   gives Helmert's approximation.

  1. ^ Kawase, K. (2009): A General Formula for Meridional Distance from the Equator to Given Latitude, Journal of the Geographical Survey Institute, 119, 45–55 (ISSN 0430-9081, in Japanese)