| ε = | s² | sin α cos a, η = | s² | sin² α tan φ1, |
| 2ρn | 2ρn |
| φ′ − φ = | s | cos (α − 2⁄3ε) − η, |
| rho0 |
| ω = | s sin (alpha − 1⁄3ε) | , |
| n cos (φ′ + 1⁄3η) |
α* − α = ω sin (φ′ + 2⁄3η) − ε + 180°.
Here n is the normal or radius of curvature perpendicular to the meridian; both n and ρ correspond to latitude φ1, and ρ0 to latitude ½(φ + φ′). For calculations of latitude and longitude, tables of the logarithmic values of ρ sin 1″, n sin 1″, and 2 n ρ sin 1″ are necessary. The following table contains these logarithms for every ten minutes of latitude from 52° to 53° computed with the elements a = 20926060 and a : b = 295 : 294 :—
| Lat. | Log. 1/ρ sin 1″. | Log. 1/n sin 1″. | Log. 1/2ρn sin 1″. |
| ° ′ | |||
| 52 0 | 7.9939434 | 7.9928231 | 0.37131 |
| 10 | 9309 | 8190 | 29 |
| 20 | 9185 | 8148 | 28 |
| 30 | 9060 | 8107 | 26 |
| 40 | 8936 | 8065 | 24 |
| 50 | 8812 | 8024 | 23 |
| 53 0 | 8688 | 7982 | 22 |
The logarithm in the last column is that required also for the calculation of spherical excesses, the spherical excess of a triangle being expressed by a b sin C/(2ρn) sin 1″.
It is frequently necessary to obtain the co-ordinates of one point with reference to another point; that is, let a perpendicular arc be drawn from B to the meridian of A meeting it in P, then, α being the azimuth of B at A, the co-ordinates of B with reference to A are
AP = s cos (α − 2⁄3ε), BP = s sin (α − 1⁄3ε),
where ε is the spherical excess of APB, viz. s² sin α cos α multiplied by the quantity whose logarithm is in the fourth column of the above table.