Therefore M = 4 sin² ½[beta] - H²/H^1, and h must be determined so as to make H²:H´ a maximum. In any particular case this maximum can only be ascertained by trial, that is to say, log H² - log H´ must be calculated for certain equidistant values of h, and then the particular value of h which corresponds to the required maximum can be obtained by interpolation. Thus we find that if it be required to make the best possible perspective representation of a hemisphere, the values of h and k are h = 1.47 and k = 2.034; so that in this case 2.034 sin u [rho] = ------------. 1.47 + cos u For a map of Africa or South America, the limiting radius [beta] we may take as 40°; then in this case 2.543 sin u [rho] = -------------. 1.625 + cos u For Asia, [beta] = 54, and the distance h of the point of sight in this case is 1.61. Fig. 11 is a map of Asia having the meridians and parallels laid down on this system. [Illustration: FIG. 11.] Fig. 12 is a perspective representation of more than a hemisphere, the radius [beta] being 108°, and the distance h of the point of vision, 1.40. [Illustration: FIG. 12.--Twilight Projection. Clarke's Perspective Projection for a Spherical Radius of 108°.] The co-ordinates xy of any point in this perspective may be expressed in terms of latitude and longitude of the corresponding point on the sphere in the following manner. The co-ordinates originating at the centre take the central meridian for the axis of y and a line perpendicular to it for the axis of x. Let the latitude of the point G, which is to occupy the centre of the map, be [gamma]; if [phi], [omega] be the latitude and longitude of any point P (the longitude being reckoned from the meridian of G), u the distance PG, and [mu] the azimuth of P at G, then the spherical triangle whose sides are 90° - [gamma], 90° - [phi], and u gives these relations-- sin u sin [mu] = cos [phi] sin [omega], sin u cos [mu] = cos [gamma] sin [phi] - sin [gamma] cos [phi] cos [omega], cos u = sin [gamma] sin [phi] + cos [gamma] cos [phi] cos [omega]. Now x = [rho] sin [mu], y = [rho] cos [mu], that is, x cos [phi] sin [omega] --- = -------------------------------------------------------------, k h + sin [gamma] sin [phi] + cos [gamma] cos [phi] cos [omega] y cos [gamma] sin [phi] - sin [gamma] cos [phi] cos [omega] --- = -------------------------------------------------------------, k h + sin [gamma] sin [phi] + cos [gamma] cos [phi] cos [omega]