The Bowring series of the transverse mercator published in 1989 by Bernard Russel Bowring gave formulas for the Transverse Mercator that are simpler to program but retain millimeter accuracy. Bowring rewrote the fourth order Redfearn series (after discarding small terms) in a more compact notation by replacing the spherical terms, i.e. those independent of ellipticity, by the exact expressions used in the spherical transverse Mercator projection. There was no gain in accuracy since the elliptic terms were still truncated at the 1mm level. Such modifications were of possible use when computing resources were minimal.
Notation
a {\displaystyle a} = radius of the equator of the chosen spheroid (e.g. 6378137 m for GRS80/WGS84)
b {\displaystyle b} = polar semi-axis of the spheroid
k 0 {\displaystyle k_{0}} = scale factor along the central meridian (e.g. 0.9996 for UTM)
ϕ {\displaystyle \scriptstyle \phi } = latitude
ω {\displaystyle \scriptstyle \omega } = difference in longitude from the central meridian, in radians, positive eastward
m {\displaystyle m} = meridian distance, measured on the spheroid from the equator to ϕ {\displaystyle \scriptstyle \phi } (see below) E = distance east of the central meridian, measured on the Transverse Mercator projection N = distance north of the equator, measured on the Transverse Mercator projection
ε = 2 r − 1 ( r − 1 ) 2 = a 2 − b 2 ( b 2 ) {\displaystyle \varepsilon \;=\;{\frac {2r-1}{(r-1)^{2}}}=\;{\frac {a^{2}-b^{2}}{(b^{2})}}\;}
where r is the reciprocal of the flattening for the chosen spheroid (for WGS84, r = 298.257223563 exactly).
Convert Lat-Lon to Transverse Mercator
c = cos ϕ s = sin ϕ {\displaystyle c=\cos \phi \qquad s=\sin \phi }
ν = a 1 + ε 1 + ε c 2 {\displaystyle \nu \;=\;a{\sqrt {\frac {1+\varepsilon }{1+\varepsilon c^{2}}}}} (prime vertical radius of curvature)
z = ε ω 3 c 5 6 {\displaystyle z={\frac {\varepsilon \omega ^{3}c^{5}}{6}}}
tan θ 2 = 2 s c sin 2 ( ω / 2 ) s 2 + c 2 cos ω {\displaystyle \tan \theta _{2}\;=\;{\frac {2sc\sin ^{2}(\omega /2)}{s^{2}+c^{2}\cos \omega }}}
E = k 0 ν [ tanh − 1 ( c sin ω ) + z ( 1 + ω 2 10 ( 36 c 2 − 29 ) ) ] {\displaystyle {\text{E}}\;=\;k_{0}\nu \left[{\tanh }^{-1}(c\sin \omega )+z\left(1+{\frac {\omega ^{2}}{10}}(36c^{2}-29)\right)\right]}
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