Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.¶¶The formulas to derive the projected Easting and Northing coordinates are in the form of a series as follows:¶¶Easting, E =  FE + k0*nu[A + (1 - T + C)A^3/6 + (5 - 18T + T^2 + 72C - 58e'sq)A^5/120]	¶¶Northing, N =  FN + k0{M - M0 + nu*tan(lat)[A^2/2 + (5 - T + 9C + 4C^2)A^4/24 + (61 - 58T + T^2 + 600C - 330e'sq)A^6/720]} ¶where T = tan^2(lat)                nu = a /(1 - esq*sin^2(lat))^0.5¶C = esq*cos^2(lat)/(1 - esq)¶A = (lon - lon0)cos(lat), with lon and lon0 in radians.¶M = a[(1 - esq/4 - 3e^4/64 - 5e^6/256 -....)lat - (3esq/8 + 3e^4/32 + 45e^6/1024+....)sin(2*lat) + (15e^4/256 + 45e^6/1024 +.....)sin(4*lat) - (35e^6/3072 + ....)sin(6*lat) + .....]¶with lat in radians and M0 for lat0, the latitude of the origin, derived in the same way.¶¶The reverse formulas to convert Easting and Northing projected coordinates to latitude and longitude are:¶¶lat = lat1 - (nu1*tan(lat1)/rho1)[D^2/2 - (5 + 3*T1 + 10*C1 - 4*C1^2 - 9*e'^2)D^4/24 + (61 + 90*T1 + 298*C1 + 45*T1^2 - 252*e'^2 - 3*C1^2)D^6/720]¶lon = lon0 + [D - (1 + 2*T1 + C1)D^3/6 + (5 - 2*C1 + 28*T1 - 3*C1^2 + 8*e'^2 + 24*T1^2)D^5/120] / cos(lat1)¶where lat1 may be found as for the Cassini projection from:¶¶lat1 = mu1 + ((3*e1)/2 - 27*e1^3/32 +.....)sin(2*mu1) + (21*e1^2/16 -55*e1^4/32 + ....)sin(4*mu1)+ (151*e1^3/96 +.....)sin(6*mu1) + (1097*e1^4/512 - ....)sin(8*mu1) + ......¶and where¶nu1 = a /(1 - esq*sin^2(lat1))^0.5¶rho1 = a(1 - esq)/(1 - esq*sin^2(lat1))^1.5¶e1 = [1- (1 - esq)^0.5]/[1 + (1 - esq)^0.5]¶mu1 = M1/[a(1 - esq/4 - 3e^4/64 - 5e^6/256 - ....)]¶M1 = M0 + (N - FN)/k0¶T1 = tan^2(lat1)¶C1 = e'^2*cos^2(lat1)¶D = (E - FE)/(nu1*k0), with nu1 = nu for lat1¶¶For areas south of the equator the value of latitude lat will be negative and the formulas above, to compute the E and N,  will automatically result in the correct values. Note that the false northings of the origin, if the equator, will need to be large to avoid negative northings and for the UTM projection is in fact 10,000,000m. Alternatively, as in the case of Argentina's Transverse Mercator (Gauss-Kruger) zones, the origin is at the south pole with a northings of zero. However each zone central meridian takes a false easting of 500000m prefixed by an identifying zone number. This ensures that instead of points in ¶different zones having the same eastings, every point in the country, irrespective of its projection zone, will have a unique set of projected system coordinates. Strict application of the above formulas, with south latitudes negative, will result in the derivation of the correct Eastings and Northings. ¶¶Similarly, in applying the reverse formulas to determine a latitude south of the equator, a negative sign for lat results from a negative lat1 which in turn results from a negative M1.