Integrating,
ρ = k(tan 1⁄2z)h,
(i.)
where k is a constant.
Now h is at our disposal and we may give it such a value that two selected parallels are of the correct lengths. Let z1, z2 be the co-latitudes of these parallels, then it is easy to show that
| h = | log sin z1 − log sin z2 | . |
| log tan 1⁄2z1 − log tan 1⁄2z2 |
(ii.)
This projection, given by equations (i.) and (ii.), is Lambert’s orthomorphic projection—commonly called Gauss’s projection; its descriptive name is the orthomorphic conical projection with two standard parallels.
The constant k in (i.) defines the scale and may be used to render the scale errors along the selected parallels not nil but the same; and some other parallel, e.g. the central parallel may then be made errorless.
The value h = 1⁄3, as suggested by Sir John Herschel, is admirably suited for a map of the world. The representation is fan-shaped, with remarkably little distortion (fig. 24).