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). If any parallel of co-latitude z is true to scale hk(tan ½z1)^h = sin z, if this parallel is the equator, so that z1 = 90°, kh = 1, then equation (i.) becomes [rho] = (tan ½z)^h/h, and the radius of the equator = 1/h. The distance r of any parallel from the equator is 1/h - (tan ½z)^h/h = (1/h){1 - (tan ½z)^h}. If, instead of taking the radius of the earth as unity we call it a, r = (a/h){1 - (tan ½z)^h}. When h is very small, the angles between the meridian lines in the representation are very small; and proceeding to the limit, when h is zero the meridians are parallel--that is, the vertex of the cone has removed to infinity. And at the limit when h is zero we have r = a log_e cot ½z, which is the characteristic equation of Mercator's projection. [Illustration: FIG. 25.--Elliptical equal-area Projection, showing the whole surface of the globe.]