In this study, the idea employed in the article of  to treat the problem of the arc-length of meridian is applied to the arc-length determined by the third flattening (n) instead of the eccentricity (e) of an Earth-reference ellipsoid . With WGS 84 ellipsoid datum used in modern navigation, the meridian arc-length is a crucial factor for map projections and geodetic and distance calculations such as Mercator charts, the universal transverse Mercator, great circle (ellipse), rhumb line, and geodesic. Because navigation software lacks officially standardized calculation methods, these “black box solutions” used in navigational systems are unknown. Therefore members of the general public must be provided with logical and simpler formulas. Because the previous general formulas of meridian arc-length are unnecessarily complicated and difficult to understand, this paper deduces a new general formula of meridian arc-length by using the binomial theorem and general terms of sinusoidal even power. The general formula presented herein is suitable for computer algorithm programming and other common navigational uses.
Tseng, Wei-Kuo; Tsai, King-Cheng; Liou, Chian; Guo, Jiunn-Liang; and Sun, Yi-Jia
"Algorithms for the Generalized Inverse Solution and Direct Solution: Using an Algebra Computer-Based System to Obtain Meridian Arc Length,"
Journal of Marine Science and Technology: Vol. 31:
2, Article 9.
Available at: https://jmstt.ntou.edu.tw/journal/vol31/iss2/9