Meridian Arc (with gifs)

The below is my work from 1976-1977 which was finally approved for submission to the Defense Mapping Agency. My own work starts with the binomial expansion and includes the integration of the mapping integral and that of the even powers of the sine function. The equations show the derivation of the series expansion for the value of the meridian arc from the equator to latitude phi (which was required to compute the Universal Transverse Mercator [UTM] coordinates used on military maps.) Enjoy!
[Equations 1 thru 3]