Notwithstanding the facility of construction, the stereographic projection is not much used in map-making. It is sometimes used for maps of the hemispheres in atlases, and for star charts. _External Perspective Projection._--We now come to the general case in which the point of vision has any position outside the sphere. Let abcd (fig. 10) be the great circle section of the sphere by a plane passing through c, the central point of the portion of surface to be represented, and V the point of vision. Let pj perpendicular to Vc be the plane of representation, join mV cutting pj in f, then f is the projection of any point m in the circle abc, and ef is the representation of cm. Let the angle com = u, Ve = k, Vo = h, ef = [rho]; then, since ef: eV = mg:gV, we have [rho] = k sin u/(h + cos u), which gives the law connecting a spherical distance u with its rectilinear representation [rho]. The relative scale at any point in this system of projection is given by [sigma] = d[rho]/du, [sigma]´ = [rho]/sin u, [sigma] = k(1 + h cos u)/(h + cos u)²; [sigma]´ = k/(h + cos u), the former applying to measurements made in a direction which passes through the centre of the map, the latter to the transverse direction. The product [sigma][sigma]´ gives the exaggeration of areas. With respect to the alteration of angles we have [Sigma] = (h + cos u)/(l + kcos u), and the greatest alteration of angle is /h - 1 u \ = sin^(-1) ( ----- tan² --- ). \h + 1 2 / This vanishes when h = 1, that is if the projection be stereographic; or for u = 0, that is at the centre of the map. At a distance of 90° from the centre, the greatest alteration is 90° - 2 cot^(-1) [root]h. (See _Phil. Mag._ 1862.) _Clarke's Projection._--The constants h and k can be determined, so that the total misrepresentation, viz.: _ / [beta] M = | {([sigma] - 1)² + ([sigma]´ - 1)²} sin u du, _/ 0 shall be a minimum, [beta] being the greatest value of u, or the spherical radius of the map. On substituting the expressions for [sigma] and [sigma]´ the integration is effected without difficulty. Put [lambda] = (1 - cos [beta])/(h + cos [beta]); [nu] = (h - 1)[lambda], H = [nu] - (h + 1) log_e ([lambda] + 1), H´ = [lambda](2 - [nu] + (1/3)[nu]²)/(h + 1). Then the value of M is M = 4 sin² ½[beta] + 2kH + k²H´. When this is a minimum, dM/dh = 0; dM/dk = 0 :. kH´ + H = 0; 2 dH/dh + k dh H´/dh = 0.